Author name code: gilman ADS astronomy entries on 2022-09-14 author:"Gilman, Peter" ------------------------------------------------------------------------ Title: Simulating Solar Near-surface Rossby Waves by Inverse Cascade from Supergranule Energy Authors: Dikpati, Mausumi; Gilman, Peter A.; Guerrero, Gustavo A.; Kosovichev, Alexander G.; McIntosh, Scott W.; Sreenivasan, Katepalli. R.; Warnecke, Jörn; Zaqarashvili, Teimuraz V. Bibcode: 2022ApJ...931..117D Altcode: Rossby waves are found at several levels in the Sun, most recently in its supergranule layer. We show that Rossby waves in the supergranule layer can be excited by an inverse cascade of kinetic energy from the nearly horizontal motions in supergranules. We illustrate how this excitation occurs using a hydrodynamic shallow-water model for a 3D thin rotating spherical shell. We find that initial kinetic energy at small spatial scales inverse cascades quickly to global scales, exciting Rossby waves whose phase velocities are similar to linear Rossby waves on the sphere originally derived by Haurwitz. Modest departures from the Haurwitz formula originate from nonlinear finite amplitude effects and/or the presence of differential rotation. Like supergranules, the initial small-scale motions in our model contain very little vorticity compared to their horizontal divergence, but the resulting Rossby waves are almost all vortical motions. Supergranule kinetic energy could have mainly gone into gravity waves, but we find that most energy inverse cascades to global Rossby waves. Since kinetic energy in supergranules is three or four orders of magnitude larger than that of the observed Rossby waves in the supergranule layer, there is plenty of energy available to drive the inverse-cascade mechanism. Tachocline Rossby waves have previously been shown to play crucial roles in causing seasons of space weather through their nonlinear interactions with global flows and magnetic fields. We briefly discuss how various Rossby waves in the tachocline, convection zone, supergranule layer, and corona can be reconciled in a unified framework. Title: Origin of Rossby waves observed near the solar surface Authors: Gilman, Peter; Dikpati, Mausumi; Guerrero, Gustavo; Kosovichev, Alexander; McIntosh, Scott; Sreenivasan, Katepalli; Warnecke, Joern; Zaqarashvili, Teimuraz Bibcode: 2021AGUFMSH53C..04G Altcode: Differential rotation and toroidal magnetic bands in the tachocline are unstable to MHD Rossby waves and may be responsible for patterns of solar activity seen in the photosphere. Helioseismic and surface velocity measurements reveal energetically neutral Rossby waves in the supergranulation layer. To explore plausible sources of energy for these Rossby waves, we study nonlinear dynamics of horizontal flows in the supergranular layer in thepresence of rotation and differential rotation. With a shallow-water model we show that kinetic energy, put into smallest resolved spatial scales, very quickly 'reverse cascades' to largest scales, exciting energetically neutral Rossby-Haurwitz type waves, as well as energetically active ones with low longitudinal spectral modes, depending on differential rotation. Horizontal velocities in supergranules are known to be much larger than their vertical motions; our shallow-water system includes a similar ratio. If supergranules are responsible for Rossby waves seen in photosphere, it paradoxically follows that (i) stable stratification of a thin rotating spherical shell may be a sufficient but not a necessary condition for Rossby waves, and (ii) small-scale convection producing global Rossby waves in a thin differentially rotating fluid may be the first ever example found in a celestial body. Title: Observational evidence of spot-producing magnetic ring's split during MHD evolution Authors: Norton, Aimee; Dikpati, Mausumi; McIntosh, Scott; Gilman, Peter Bibcode: 2021AGUFMSH55D1876N Altcode: Spot-producing toroidal rings of 6-degree latitudinal width, with peak field of 15 kG, have been found to undergo dynamical splitting due to nonlinear MHD. Split-time depends on the latitude-location of the ring. Ring-splitting occurs fastest, within a few weeks, at latitudes 20-25 degrees. Rossby waves work as perturbations to drive instability of spot-producing toroidal rings. The ring-split is caused by the `mixed stress' or cross correlations of perturbation velocities and magnetic fields, which arise due to the interaction of Rossby Waves. Mixed stress carries magnetic energy and flux from the ring-peak to its shoulders, eventually leading to the ring-split. The two split-rings migrate away from each other, the high latitude counterpart slipping poleward faster, due to migrating mixed stress and magnetic curvature stress. Broader toroidal bands do not split. Much stronger rings of 35 kG, despite being narrow, don't split, due to rigidity from stronger magnetic fields within the ring. The analysis of magnetograms from MDI during solar cycle 23 indicates emergence of active regions sometimes at the same longitudes but separated in latitude by 20-degrees or more, which could be evidence of active regions emerging from split-rings, which consistently contribute to occasional high latitude excursions of observed butterfly wings during ascending, peak and descending phases of a solar cycle. In the future, observational studies using much longer term magnetograms including GONG and SDO/HMI can determine how often new spots are found at higher latitudes than their lower latitude counterparts, and how the combinations influence solar eruptions and space weather events. Title: Dynamical Splitting of Spot-producing Magnetic Rings in a Nonlinear Shallow-water Model Authors: Dikpati, Mausumi; Norton, Aimee A.; McIntosh, Scott W.; Gilman, Peter A. Bibcode: 2021ApJ...922...46D Altcode: We explore the fundamental physics of narrow toroidal rings during their nonlinear magnetohydrodynamic evolution at tachocline depths. Using a shallow-water model, we simulate the nonlinear evolution of spot-producing toroidal rings of 6° latitudinal width and a peak field of 15 kG. We find that the rings split; the split time depends on the latitude of each ring. Ring splitting occurs fastest, within a few weeks, at latitudes 20°-25°. Rossby waves work as perturbations to drive the instability of spot-producing toroidal rings; the ring split is caused by the "mixed stress" or cross-correlations of perturbation velocities and magnetic fields, which carry magnetic energy and flux from the ring peak to its shoulders, leading to the ring split. The two split rings migrate away from each other, the high-latitude counterpart slipping poleward faster due to migrating mixed stress and magnetic curvature stress. Broader toroidal bands do not split. Much stronger rings, despite being narrow, do not split due to rigidity from stronger magnetic fields within the ring. Magnetogram analysis indicates the emergence of active regions sometimes at the same longitudes but separated in latitude by 20° or more, which could be evidence of active regions emerging from split rings, which consistently contribute to observed high-latitude excursions of butterfly wings during the ascending, peak, and descending phases of a solar cycle. Observational studies in the future can determine how often new spots are found at higher latitudes than their lower-latitude counterparts and how the combinations influence solar eruptions and space weather events. Title: Reconciling various solar Rossby wave observations using global (M)HD models Authors: Dikpati, M.; Gilman, P. A.; McIntosh, S. W.; Zaqarashvili, T. V. Bibcode: 2021AAS...23811319D Altcode: Various observational analyses indicate that the Sun has Rossby waves, which include hydrodynamic Rossby waves like that on the Earth's atmosphere as well as magnetohydrodynamic ones, which do not have their counterparts on the Earth. Many of us are showing, from observations and model-calculations, that solar hydrodynamic Rossby waves have retrograde speed and they follow Rossby-Haurwitz type dispersion relation. However, it has also recently been shown MHD Rossby waves can have both retrograde as well as prograde speed; if MHD Rossby waves are retrograde they are more retrograde than their HD counterparts, whereas if they are prograde, they are relatively slow. Helioseismically determined Rossby waves detected so far are HD waves that are energetically neutral. Coronal bright points as well as long-lived coronal holes' longitudinal drift-patterns with time show evidence of HD as well as MHD Rossby waves. We know from nonlinear simulations of global MHD waves and instabilities that Rossby waves can be energetically active and hence, can nonlinearly interact with the solar differential rotation and spot-producing toroidal magnetic fields, very much like nonlinear Orr mechanism in fluid dynamics. Is it possible to reconcile the various solar Rossby waves observations? Certainly different observational techniques applied to different elevations on the Sun may be measuring different Rossby waves. Furthermore, the different measurements may be detecting waves that originate at different depths inside the Sun. Solar atmosphere magnetic features may have roots deep in the convection zone and reflect MHD Rossby waves at those depths. We will present model-simulations to show what physical conditions are responsible for producing sectoral modes and what are responsible for generating energetically active Rossby waves, which have important implications in causing short-term variability in solar activity, and in turn, in space weather. This work is supported by the NCAR, sponsored by the NSF under cooperative agreement 1852977. MD acknowledges support from several NASA grants, namely the LWS award 80NSSC20K0355 to NCAR, subaward to NCAR from NASA's DRIVE Center award 80NSSC20K0602 (originally awarded to Stanford) and NASA-HSR subaward 80NSSC18K1206 (originally awarded to NSO). Title: Deciphering the Deep Origin of Active Regions via Analysis of Magnetograms Authors: Dikpati, Mausumi; McIntosh, Scott W.; Chatterjee, Subhamoy; Norton, Aimee A.; Ambroz, Pavel; Gilman, Peter A.; Jain, Kiran; Munoz-Jaramillo, Andres Bibcode: 2021ApJ...910...91D Altcode: In this work, we derive magnetic toroids from surface magnetograms by employing a novel optimization method, based on the trust region reflective algorithm. The toroids obtained in this way are combinations of Fourier modes (amplitudes and phases) with low longitudinal wavenumbers. The optimization also estimates the latitudinal width of the toroids. We validate the method using synthetic data, generated as random numbers along a specified toroid. We compute the shapes and latitudinal widths of the toroids via magnetograms, generally requiring several m's to minimize residuals. A threshold field strength is chosen to include all active regions in the magnetograms for toroid derivation, while avoiding non-contributing weaker fields. Higher thresholds yield narrower toroids, with an m = 1 dominant pattern. We determine the spatiotemporal evolution of toroids by optimally weighting the amplitudes and phases of each Fourier mode for a sequence of five Carrington Rotations (CRs) to achieve the best amplitude and phases for the middle CR in the sequence. Taking more than five causes "smearing" or degradation of the toroid structure. While this method applies no matter the depth at which the toroids actually reside inside the Sun, by comparing their global shape and width with analogous patterns derived from magnetohydrodynamic (MHD) tachocline shallow water model simulations, we infer that their origin is at/near the convection zone base. By analyzing the "Halloween" storms as an example, we describe features of toroids that may have caused the series of space weather events in 2003 October-November. Calculations of toroids for several sunspot cycles will enable us to find similarities/differences in toroids for different major space weather events. Title: Derivation of Toroid Patterns from Analysis of Magnetograms And Inferring Their Deep-origin Authors: Chatterjee, S.; Dikpati, M.; McIntosh, S. W.; Norton, A. A.; Ambroz, P.; Gilman, P.; Jain, K.; Munoz-Jaramillo, A. Bibcode: 2020AGUFMSH0020013C Altcode: We employ a novel optimization method based on Trust Region Reflective algorithm to derive magnetic toroids from surface magnetograms. Toroids obtained are combinations of Fourier modes (amplitudes and phases) with low longitudinal wavenumbers. After validating the method using synthetic data generated as random numbers along a specified toroid, we compute shapes and latitudinal-widths of toroids from magnetograms, usually requiring several m 's to minimize residuals. By comparing properties of these toroids with patterns produced in the bottom toroidal band undergoing MHD evolution in a 3D thin-shell shallow-water type model, we infer their deep origin at/near convention-zone's base or tachocline. A threshold field-strength is chosen to include all active regions in magnetograms for toroid derivation, while avoiding non-contributing weaker fields. Higher thresholds yield narrower toroids, with m = 1 dominant, implying that stronger active regions are erupting from the core of the toroids at bottom. We determine the spatio-temporal evolution of toroids by optimally weighting amplitudes and phases of each Fourier mode for a sequence of 5 Carrington Rotations (CRs) to get the best amplitude and phases for the middle CR in the sequence. Taking more than 5 causes 'smearing' or degradation of toroid structure. As an example case, we analyze 'Halloween' storms toroids, and describe the features that might have caused the series of space weather events in October-November of 2003. We compare features of these toroids with analogous patterns derived from model-output. To find similarities/differences in toroids for different major space weather events, we will analyze long-term magnetograms for several solar cycles. Title: Tachocline Instabilities and Solar Activity: a Fifty-Six-Year Personal Perspective Authors: Gilman, P. Bibcode: 2020AGUFMSH006..01G Altcode: There are several hydrodynamic and magnetohydrodynamic instabilities likely to be at work in the solar tachocline. Theoretical models of these instabilities have been developed over the past 25 years, ever since the tachocline was discovered. My primary focus will be on instabilities that arise from rotation and differential rotation, and/or toroidal magnetic fields, and have global scale at least in longitude. I will give a personal perspective on these instabilities, their physical significance, and how they may help determine evidence of magnetic activity we observe at the solar surface. I will also discuss their possible helioseismic signatures, and what they might contribute to the workings of the solar dynamo. A few possible paradoxes will be outlined, as well as how the theories could be made physically more realistic. Title: Physics of Magnetohydrodynamic Rossby Waves in the Sun Authors: Dikpati, Mausumi; Gilman, Peter A.; Chatterjee, Subhamoy; McIntosh, Scott W.; Zaqarashvili, Teimuraz V. Bibcode: 2020ApJ...896..141D Altcode: Evidence of the existence of hydrodynamic and MHD Rossby waves in the Sun is accumulating rapidly. We employ an MHD Rossby wave model for the Sun in simplified Cartesian geometry, with a uniform toroidal field and no differential rotation, to analyze the role of each force that contributes to Rossby wave dynamics, and compute fluid particle trajectories followed in these waves. This analysis goes well beyond the traditional formulation of Rossby waves in terms of conservation of vorticity. Hydrodynamic Rossby waves propagate retrograde relative to the rotation of the reference frame, while MHD Rossby waves can be both prograde and retrograde. Fluid particle trajectories are either clockwise or counterclockwise spirals, depending on where in the wave pattern they are initiated, that track generally in the direction of wave propagation. Retrograde propagating MHD Rossby waves move faster than their hydrodynamic counterparts of the same wavelength, becoming Alfvén waves at very high field strengths. Prograde MHD Rossby waves, which have no hydrodynamic counterpart, move more slowly eastward than retrograde MHD Rossby waves for the same toroidal field, but with a speed that increases with toroidal field, in the high field limit again becoming Alfvén waves. The longitude and latitude structures of all these waves, as seen in their velocity streamlines and perturbation field lines as well as fluid particle trajectories, are remarkably similar for different toroidal fields, rotation, longitudinal wavelength, and direction of propagation. Title: The solar benchmark: rotational modulation of the Sun reconstructed from archival sunspot records Authors: Morris, Brett M.; Davenport, James R. A.; Giles, Helen A. C.; Hebb, Leslie; Hawley, Suzanne L.; Angus, Ruth; Gilman, Peter A.; Agol, Eric Bibcode: 2019MNRAS.484.3244M Altcode: 2019arXiv190104557M; 2019MNRAS.tmp..205M We use archival daily spot coverage measurements from Howard et al. to study the rotational modulation of the Sun as though it were a distant star. A quasi-periodic Gaussian process measures the solar rotation period Prot = 26.3 ± 0.1 d, and activity cycle period Pcyc = 10.7 ± 0.3 yr. We attempt to search for evidence of differential rotation in variations of the apparent rotation period throughout the activity cycle and do not detect a clear signal of differential rotation, consistent with the null results of the hare-and-hounds exercise of Aigrain et al. The full reconstructed solar light curve is available online. Title: Magnetic Buoyancy and Magnetorotational Instabilities in Stellar Tachoclines for Solar- and Antisolar-type Differential Rotation Authors: Gilman, Peter A. Bibcode: 2018ApJ...867...45G Altcode: We present results from an analytical model for magnetic buoyancy and rotational instabilities in full spherical shell stellar tachoclines that include rotation, differential rotation of either solar or antisolar type, and toroidal field. We find that in all cases, for latitudes where the tachocline vertical rotation gradient is positive, toroidal fields can be stored against magnetic buoyancy up to a limit that is proportional to the square root of the local vertical rotation gradient. For solar magnitude differential rotation, this limit is about 9 kG. For fixed percentage differential rotation, storage capacity varies linearly with the rotation rate. Faster rotators with the same percentage differential rotation can store larger fields, and slower rotators can store smaller fields. At latitudes where the vertical rotation gradient is negative, vigorous magnetorotational instability for even weak (≪1 kG) toroidal fields prevents such storage. We infer from these results that for stars with solar-type latitudinal differential rotation (fast equator, slow poles), any starspots present should be found in low latitudes, similar to the Sun. For antisolar differential rotation, any spots present should be found in mid- and high latitudes, perhaps with a peak of occurrence near 55°. These results hopefully provide some guidance for making and interpreting observations of stellar activity and differential rotation on stars with convection zones and tachoclines. Title: Phase Speed of Magnetized Rossby Waves that Cause Solar Seasons Authors: Dikpati, Mausumi; Belucz, Bernadett; Gilman, Peter A.; McIntosh, Scott W. Bibcode: 2018ApJ...862..159D Altcode: Motivated by recent analysis of solar observations that show evidence of propagating Rossby waves in coronal holes and bright points, we compute the longitudinal phase velocities of unstable MHD Rossby waves found in an MHD shallow-water model of the solar tachocline (both overshoot and radiative parts). We demonstrate that phase propagation is a typical characteristic of tachocline nonlinear oscillations that are created by unstable MHD Rossby waves, responsible for producing solar seasons. For toroidal field bands placed at latitudes between 5° and 75°, we find that phase velocities occur in a range similar to the observations, with more retrograde speeds (relative to the solar core rotation rate) for bands placed at higher latitudes, just as coronal holes have at high latitudes compared to low ones. The phase speeds of these waves are relatively insensitive to the toroidal field peak amplitude. Rossby waves for single bands at 25° are slightly prograde. However, at latitudes lower than 25° they are very retrograde, but much less so if a second band is included at a much higher latitude. This double-band configuration is suggested by evidence of an extended solar cycle, containing a high-latitude band in its early stages that does not yet produce spots, while the spot-producing low-latitude band is active. Collectively, our results indicate a strong connection between longitudinally propagating MHD Rossby waves in the tachocline and surface manifestations in the form of similarly propagating coronal holes and patterns of bright points. Title: Role of Interaction between Magnetic Rossby Waves and Tachocline Differential Rotation in Producing Solar Seasons Authors: Dikpati, Mausumi; McIntosh, Scott W.; Bothun, Gregory; Cally, Paul S.; Ghosh, Siddhartha S.; Gilman, Peter A.; Umurhan, Orkan M. Bibcode: 2018ApJ...853..144D Altcode: We present a nonlinear magnetohydrodynamic shallow-water model for the solar tachocline (MHD-SWT) that generates quasi-periodic tachocline nonlinear oscillations (TNOs) that can be identified with the recently discovered solar “seasons.” We discuss the properties of the hydrodynamic and magnetohydrodynamic Rossby waves that interact with the differential rotation and toroidal fields to sustain these oscillations, which occur due to back-and-forth energy exchanges among potential, kinetic, and magnetic energies. We perform model simulations for a few years, for selected example cases, in both hydrodynamic and magnetohydrodynamic regimes and show that the TNOs are robust features of the MHD-SWT model, occurring with periods of 2-20 months. We find that in certain cases multiple unstable shallow-water modes govern the dynamics, and TNO periods vary with time. In hydrodynamically governed TNOs, the energy exchange mechanism is simple, occurring between the Rossby waves and differential rotation. But in MHD cases, energy exchange becomes much more complex, involving energy flow among six energy reservoirs by means of eight different energy conversion processes. For toroidal magnetic bands of 5 and 35 kG peak amplitudes, both placed at 45° latitude and oppositely directed in north and south hemispheres, we show that the energy transfers responsible for TNO, as well as westward phase propagation, are evident in synoptic maps of the flow, magnetic field, and tachocline top-surface deformations. Nonlinear mode-mode interaction is particularly dramatic in the strong-field case. We also find that the TNO period increases with a decrease in rotation rate, implying that the younger Sun had more frequent seasons. Title: Magnetic Buoyancy and Rotational Instabilities in the Tachocline Authors: Gilman, Peter A. Bibcode: 2018ApJ...853...65G Altcode: We present results from an analytical model for magnetic buoyancy and rotational instabilities in a full spherical shell tachocline that includes rotation, differential rotation close to that observed helioseismically, and toroidal field. Perturbation solutions are found for the limit of large latitudinal wave number, a limit commonly used to maximize instability due to magnetic buoyancy. We find that at all middle and high latitudes vigorous rotational instability is induced by weak toroidal fields, particularly for high longitudinal wave number, even when the vertical rotation gradient is marginally stable without toroidal field. We infer that this instability will prevent much storage of toroidal fields in the tachocline at these latitudes, but could be responsible for the appearance of ephemeral active regions there. By contrast, the low-latitude vertical rotation gradient, opposite in sign to that at high latitudes, is not only stable itself but also prevents magnetic buoyancy instability until the peak toroidal field is raised above a threshold of about 9 kG at the equator, declining to zero where the vertical rotation gradient changes sign, at 32\buildrel{\circ}\over{.} 3 in our model. Thus this rotation gradient provides a previously unnoticed mechanism for storage of toroidal fields until they amplify by dynamo action to order 10 kG, whereupon they can overcome the rotation gradient to emerge as sunspots. These results provide a new explanation for why sunspots are seen only at low latitudes. The purely rotational instability at latitudes above 50°, even without toroidal fields, also suggests that the high-latitude tachocline should be much thicker, due to HD turbulence, than has been inferred for lower latitudes from helioseismic measurements. Title: Baroclinic Instability in the Solar Tachocline for Continuous Vertical Profiles of Rotation, Effective Gravity, and Toroidal Field Authors: Gilman, Peter A. Bibcode: 2017ApJ...842..130G Altcode: We present results from an MHD model for baroclinic instability in the solar tachocline that includes rotation, effective gravity, and toroidal field that vary continuously with height. We solve the perturbation equations using a shooting method. Without toroidal fields but with an effective gravity declining linearly from a maximum at the bottom to much smaller values at the top, we find instability at all latitudes except at the poles, at the equator, and where the vertical rotation gradient vanishes (32.°3) for longitude wavenumbers m from 1 to >10. High latitudes are much more unstable than low latitudes, but both have e-folding times that are much shorter than a sunspot cycle. The higher the m and the steeper the decline in effective gravity, the closer the unstable mode peak to the top boundary, where the energy available to drive instability is greatest. The effect of the toroidal field is always stabilizing, shrinking the latitude ranges of instability as the toroidal field is increased. The larger the toroidal field, the smaller the longitudinal wavenumber of the most unstable disturbance. All latitudes become stable for a toroidal field exceeding about 4 kG. The results imply that baroclinic instability should occur in the tachocline at latitudes where the toroidal field is weak or is changing sign, but not where the field is strong. Title: Baroclinic Instability in the Solar Tachocline. II. The Eady Problem Authors: Gilman, Peter A. Bibcode: 2016ApJ...818..170G Altcode: We solve the nongeostrophic baroclinic instability problem for the tachocline for a continuous model with a constant vertical rotation gradient (the Eady problem), using power series generated by the Frobenius method. The results confirm and greatly extend those from a previous two-layer model. For effective gravity G independent of height, growth rates and ranges of unstable longitudinal wavenumbers m and latitudes increase with decreasing G. As with the two-layer model, the overshoot tachocline is much more unstable than the radiative tachocline. The e-folding growth times range from as short as 10 days to as long as several years, depending on latitude, G, and wavenumber. For a more realistic effective gravity that decreases linearly from the radiative interior to near zero at the top of the tachocline, we find that only m = 1, 2 modes are unstable, with growth rates somewhat larger than for constant G, with the same value as at the bottom of the tachocline. All results are the same whether we assume that the vertical velocity or the perturbation pressure is zero at the top of the layer; this is a direct consquence of not employing the geostrophic assumption for perturbations. We explain most of the properties of the instability in terms of the Rossby deformation radius. We discuss further improvements in the realism of the model, particularly adding toroidal fields that vary in height, and including latitudinal gradients of both rotation and toroidal fields. Title: Effect of Toroidal Fields On Baroclinic Instability in the Solar Tachocline Authors: Gilman, Peter A. Bibcode: 2015ApJ...801...22G Altcode: Using an MHD generalization of a two-layer hydrostatic but non-geostrophic model, we show that a toroidal field tends to stabilize baroclinically unstable modes in the solar tachocline. In the hydrodynamic (HD) case, baroclinic instability occurs at almost all latitudes in both the radiative and overshoot tachoclines. The toroidal field creates stable bands of latitude near where the vertical rotation gradient changes sign, as well as near the equator and pole, which widen with increasing field until, by ∼2.25 kG, all latitudes are stable. The stable bands center on where the local latitudinal entropy gradient is smallest. This result is independent of how subadiabatic the local stratification is, provided it is not so subadiabatic that baroclinic instability is absent in the HD case. Growth rates and most unstable longitudinal wavenumbers remain close to their HD values until the toroidal field gets within ∼ 20% of the value that totally suppresses the instability. The results are similar to those found in the 1960s from an MHD geostrophic model, but apply to a much wider range of latitudes and subadiabatic stratifications. Where tachocline toroidal fields are weak enough to allow baroclinic instability, magnetic patterns in longitude should be produced that could be transmitted through the convection zone to be seen in the photosphere. The results also show it should be possible to construct a baroclinic wave dynamo for the solar tachocline. Title: Baroclinic Instability in the Solar Tachocline Authors: Gilman, Peter; Dikpati, Mausumi Bibcode: 2014ApJ...787...60G Altcode: The solar tachocline is likely to be close to a geostrophic "thermal wind," for which the Coriolis force associated with differential rotation is closely balanced by a latitudinal pressure gradient, leading to a tight relation between the vertical gradient of rotation and the latitudinal entropy gradient. Using a hydrostatic but nongeostrophic spherical shell model, we examine baroclinic instability of the tachocline thermal wind. We find that both the overshoot and radiative parts of the tachocline should be baroclinicly unstable at most latitudes. Growth rates are roughly five times higher in middle and high latitudes compared to low latitudes, and much higher in the overshoot than in the radiative tachocline. They range in e-folding amplification from 10 days in the high latitude overshoot tachocline, down to 20 yr for the low latitude radiative tachocline. In the radiative tachocline only, longitudinal wavenumbers m = 1, 2 are unstable, while in the overshoot tachocline a much broader range of m are unstable. At all latitudes and with all stratifications, the longitudinal scale of the most unstable mode is comparable to the Rossby deformation radius, while the growth rate is set by the local latitudinal entropy gradient. Baroclinic instability in the tachocline competing with instability of the latitude rotation gradient established in earlier studies should be important for the workings of the solar dynamo and should be expected to be found in most stars that contain an interface between radiative and convective domains. Title: Theory of Solar Meridional Circulation at High Latitudes Authors: Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2012ApJ...746...65D Altcode: 2011arXiv1112.1107D We build a hydrodynamic model for computing and understanding the Sun's large-scale high-latitude flows, including Coriolis forces, turbulent diffusion of momentum, and gyroscopic pumping. Side boundaries of the spherical "polar cap," our computational domain, are located at latitudes >= 60°. Implementing observed low-latitude flows as side boundary conditions, we solve the flow equations for a Cartesian analog of the polar cap. The key parameter that determines whether there are nodes in the high-latitude meridional flow is epsilon = 2ΩnπH 2/ν, where Ω is the interior rotation rate, n is the radial wavenumber of the meridional flow, H is the depth of the convection zone, and ν is the turbulent viscosity. The smaller the epsilon (larger turbulent viscosity), the fewer the number of nodes in high latitudes. For all latitudes within the polar cap, we find three nodes for ν = 1012 cm2 s-1, two for 1013, and one or none for 1015 or higher. For ν near 1014 our model exhibits "node merging": as the meridional flow speed is increased, two nodes cancel each other, leaving no nodes. On the other hand, for fixed flow speed at the boundary, as ν is increased the poleward-most node migrates to the pole and disappears, ultimately for high enough ν leaving no nodes. These results suggest that primary poleward surface meridional flow can extend from 60° to the pole either by node merging or by node migration and disappearance. Title: A Family of Simple Solar Dynamo Models for Testing Data Assimilation Methods Authors: Gilman, P.; Dikpati, M. Bibcode: 2011AGUFMSH51B2013G Altcode: We present a family of simplified flux-transport type dynamo models that can be used for testing various data assimilation schemes. This family of models contain one, two or three layers in the vertical, and are in cartesian geometry. More layers allow more detailed representation of important dynamo processes, including transport by meridional flow, production of poloidal fields by the alpha-effect, and magnetic diffusivity that varies with depth. The models are capable of simulating resonance effects, which make them sensitive to variations in inputs by assimilation of data. The one-layer version of this family is closely related to that used by Kitiashvili & Kosovichev (2008) (ApJ 688,L49) to test Kalman filter methods on solar dynamo equations. Title: Resonance in Forced Flux-transport Dynamos Authors: Gilman, Peter A.; Dikpati, Mausumi Bibcode: 2011ApJ...738..108G Altcode: 2011arXiv1107.2436G We show that simple two- and three-layer flux-transport dynamos, when forced at the top by a poloidal source term, can produce a widely varying amplitude of toroidal field at the bottom, depending on how close the meridional flow speed of the bottom layer is to the propagation speed of the forcing applied above the top layer, and how close the amplitude of the α-effect is to two values that give rise to a resonant response. This effect should be present in this class of dynamo model no matter how many layers are included. This result could have implications for the prediction of future solar cycles from the surface magnetic fields of prior cycles. It could be looked for in flux-transport dynamos that are more realistic for the Sun, done in spherical geometry with differential rotation, meridional flow, and α-effect that vary with latitude and time as well as radius. Because of these variations, if resonance occurs, it should be more localized in time, latitude, and radius. Title: Physical Origin of Differences Among Various Measures of Solar Meridional Circulation Authors: Dikpati, Mausumi; Gilman, Peter A.; Ulrich, Roger K. Bibcode: 2010ApJ...722..774D Altcode: 2010arXiv1008.2772D We show that systematic differences between surface Doppler and magnetic element tracking measures of solar meridional flow can be explained by the effects of surface turbulent magnetic diffusion. Feature-tracking speeds are lower than plasma speeds in low and mid latitudes, because magnetic diffusion opposes poleward plasma flow in low latitudes whereas it adds to plasma flow at high latitudes. Flux-transport dynamo models must input plasma flow; the model outputs yield estimates of the surface magnetic feature tracking speed. We demonstrate that the differences between plasma speed and magnetic pattern speed in a flux-transport dynamo are consistent with the observed difference between these speeds. Title: Impact of changes in the Sun's conveyor-belt on recent solar cycles Authors: Dikpati, Mausumi; Gilman, Peter A.; de Toma, Giuliana; Ulrich, Roger K. Bibcode: 2010GeoRL..3714107D Altcode: Plasma flowing poleward at the solar surface and returning equatorward near the base of the convection zone, called the meridional circulation, constitutes the Sun's conveyor-belt. Just as the Earth's great oceanic conveyor-belt carries thermal signatures that determine El Nino events, the Sun's conveyor-belt determines timing, amplitude and shape of a solar cycle in flux-transport type dynamos. In cycle 23, the Sun's surface poleward meridional flow extended all the way to the pole, while in cycle 22 it switched to equatorward near 60°. Simulations from a flux-transport dynamo model including these observed differences in meridional circulation show that the transport of dynamo-generated magnetic flux via the longer conveyor-belt, with slower return-flow in cycle 23 compared to that in cycle 22, may have caused the longer duration of cycle 23. Title: Length of a minimum as predictor of next solar cycle's strength Authors: Dikpati, Mausumi; Gilman, Peter A.; Kane, Rajaram P. Bibcode: 2010GeoRL..37.6104D Altcode: 2010GeoRL..3706104D Motivated by a prevailing view that a long minimum leads to a weak sunspot cycle, we estimate the correlation coefficients between the length of a cycle minimum and (i) the following cycle's peak, (ii) the preceding cycle's peak, (iii) following peak minus preceding peak and (iv) depth of minimum. Using both sunspot number and spot area data, we find that a long minimum is both followed and preceded by weak cycles. Similarly short minima are followed and preceded by strong cycles. Consistent with these results, we find no correlation between the length of a cycle minimum and the difference in peaks of the following and preceding cycles. From sunspot number data, for longer-than-average minima, five following cycle peaks were lower than that of the preceding cycles' peaks, while four were higher. Following shorter-than-average minima, seven cycle peaks were higher than the preceding peaks and seven were lower. Therefore one cannot predict from the length of a minimum whether the next cycle will be stronger or weaker than the preceding cycle. Thus we cannot predict whether cycle 24 will be stronger or weaker than 23. We also find that there is a strong anticorrelation between the length of a solar cycle minimum and the depth of that minimum. We define the depth as the least spot number or spot area (13-rotation averaged) within the span of a cycle minimum. We speculate that this anticorrelation is due to the longer time available for annihilation of late cycle toroidal flux across the equator in the case of a longer minimum. Title: Axisymmetric MHD Instabilities in Solar/Stellar Tachoclines Authors: Dikpati, M.; Cally, P. S.; Gilman, P. A.; Miesch, M. S. Bibcode: 2009ASPC..416..525D Altcode: We show that banded toroidal fields in the tachoclines of the Sun and other stars should be unstable to 3-D axisymmetric overturning modes if the peak toroidal field is ∼100 kG or more. This instability should fragment and limit the amplitude of toroidal fields in tachoclines. Title: Obituary: John W. Firor (1927-2007) Authors: Gilman, Peter A. Bibcode: 2009BAAS...41.1204G Altcode: John W. Firor, a former Director of the High Altitude Observatory and the National Center for Atmospheric Research, and a founder of the Solar Physics Division of the American Astronomical Society, died of Alzheimer's disease in Pullman, Washington on November 5, 2007, he was 80. He was born in Athens Georgia on October 18, 1927, where his father was a professor of agricultural economics.

John had an unusually diverse scientific career. His interest in physics and astrophysics began while serving in the army, during which time he was assigned to the Los Alamos National Laboratory, where he guarded highly radioactive materials (many have heard him describe how informal the protections were compared to later times). After his service he returned to college and graduated in physics from Georgia Tech in 1949. He received his Ph.D. from the University of Chicago in 1954, writing his thesis on cosmic rays under John Simpson. John Firor would later remark that: "If you needed cosmic rays to actually do anything, you are sunk." That thought, partly in jest, may help explain his motivation for moving to so many new scientific and management pursuits.

John moved from cosmic ray physics to radio astronomy (particularly of the Sun) when he began work at the Carnegie Institution of Washington's Department of Terrestrial Magnetism, where he remained until 1961. During this time, he met Walter Orr Roberts, then the Director of the High Altitude Observatory (HAO) in Boulder, Colorado. HAO was then affiliated with the University of Colorado. In 1959, a movement began to upgrade the atmospheric sciences in the United States by establishing a National Center, where the largest, most important atmospheric research problems could be addressed. Roberts became the first Director of NCAR, as well as the first president of the University Corporation for Atmospheric Research (UCAR), the consortium of universities that was commissioned to manage and staff the new Center. HAO became a scientific division of NCAR (Roberts made it a requirement if he was to be NCAR Director). He chose John to be the new HAO Director. John was 33 then; his leadership potential was recognized very early.

While Director of HAO, John presided over a very active scientific program that focused on non-equilibrium thermodynamics, radiative transfer, plasma physics, coronal spectroscopy, geomagnetism, physics of the Earth's upper atmosphere, and solar activity. Prominent scientists such as Jack Evans, Sydney Chapman, Gene Parker, Leo Goldberg and Donald Menzel visited to share their insights and enthusiasm with the HAO staff. John was also an active participant in HAO's well-known solar eclipse expeditions, traveling to New Guinea and to Lake Chad in Africa. Late in his tenure as HAO Director, he took the lead in helping to improve the involvement of the AAS in solar physics, and solar physicists in the AAS. Having led the AAS committee that planned its birth, he was the founding chair of the new Solar Physics Division. Much less importantly, he gave me a summer position in 1966 when I was joining the faculty at CU--that is when I first met John.

In 1968, John became the Director of NCAR when Walter Roberts decided to split the directorship from the Presidency of UCAR. As NCAR Director, John had responsibility for a vastly broadened program compared to his HAO days. NCAR, the largest NSF supported research center, conducted research covering all of atmospheric sciences, as well as oceanography and solar-terrestrial physics. It also provided major observational and computational facilities to the atmospheric sciences community. John showed very quickly he understood and could guide all of it. I came to know John best during this period, because in 1971 he chose me to head the Advanced Study Program, which supported (and still supports) postdoctoral and graduate education.

In that time, budgets were good, and NCAR was a relatively collegial, informal place. But in December 1973, that all changed suddenly. NCAR management was reviewed by an NSF-appointed Joint Evaluation Committee (JEC), whose report raised many questions. Walter Roberts resigned as UCAR President, and John had the task of working with the UCAR Board to make major changes. During this time, John asked me to put aside my ASP duties to assist him. Over the next 18 months, NCAR was restructured, several Division Directors replaced, a new, more rigorous scientific appointment system was developed, and a major budget cut was absorbed. John did all of this; in my view, it was heroic. Then the UCAR Board appointed a new UCAR President and NCAR Director, Francis Bretherton, and John became the NCAR Executive Director. John used to say that many people asked him how he felt about being 'demoted'--always gracious, he said he never saw it that way. In fact, he was mentoring Bretherton, who was very junior to John in experience, never having held a major management position. The Bretherton-Firor 'team' managed UCAR until 1980.

Then a new UCAR and NCAR management was chosen, and John had to decide what to do next. He took a sabbatical and became more deeply involved in environmental issues. But this interest had been kindled much before. As early as 1970, he was writing and speaking about environmental questions while NCAR Director. When he returned to NCAR, he became the Director of the Advanced Study Program (ASP), what I had been a decade before. He served NCAR in that role until 1996. When he stepped down, he remarked that he had been a member of the NCAR Directors Committee for 34 years; a record that he judged should never be broken.

While John was attentive to the needs of the post-docs and graduate students his program supported, ASP was a platform for him to pursue his environmental interests. He spoke and wrote clearly and influentially on environmental issues. He served on the Boards of several important environmental organizations and wrote two well received environmental books: "The Changing Atmosphere: A Global Challenge" (1990), and, with his wife Judith Jacobsen "The Crowded Greenhouse: Population, Climate Change and Creating a Sustainable World" (2002). After ASP, he continued his focus on environmental issues as a member of the Environmental and Societal Impacts group at NCAR. John retired from NCAR in 2005.

John had many active pursuits beyond his professional work. He was an accomplished pilot, with licenses for flying single and multiengine aircraft, sailplanes, and balloons. He piloted a sailplane in at least one meteorological field program. He also was an avid river rafter.

John faced the disease that took his life as he did all events in his life, with grace and dignity. He endured the loss of two spouses to cancer, Merle Jenkins Firor in 1979, and Judith Jacobsen in 2004.

John is survived by his four children with his first wife, Daniel Firor of Seattle, Washington; Kay Firor of Cove, Oregon; James Firor of Hotchkiss, Colorado, and Susan Firor of Moscow, Idaho; a sister; a brother; and three grandchildren. His children and his many friends in Boulder and elsewhere gave him loving support during his days battling Alzheimer's.

John used to define a 'southern gentleman' as a man dressed in white linen suit on a hot dusty summer day in a small Georgia town who could cross the street without breaking a sweat. John and his intellect and his management ability were like that; he could deal gracefully and successfully with almost anything that came his way. A man of great accomplishment, he rarely showed an ego to match. In the darkest days following the JEC Report, he almost single-handedly invented a new NCAR scientific appointment system. He chose the first cadre of 'senior scientists' to populate the top rank. There were about eighteen members in this group, but there was one name conspicuously absent - his own. This 'error' was quietly corrected by the UCAR Board. Title: Sequential Data-assimilation in a Flux-transport Dynamo Model Authors: Dikpati, Mausumi; de Toma, G.; Gilman, P. A.; Anderson, J. L.; Ulrich, R. K.; Boyden, J. E. Bibcode: 2009SPD....40.1114D Altcode: Applying a very simplified data-nudging technique in a flux-transport dynamo, Dikpati, de Toma and Gilman predicted solar cycle amplitude and onset-timing of cycle 24 seperately. In order to simultaneously predict cycle amplitude and timing we have developed a sequential data-assimilation technique, in a similar way used in atmospheric and oceanic prediction models. However two major difficulties in applying this technique in solar dynamo models are, (i) equatorward return meridional circulation is unknown, (ii) time-varying surface flow measurements have not been available for years prior to 1996. With recent progress of Mount Wilson Observatory's flow-data analysis by Ulrich and colleagues, we can now go back to 1985. We build sequential data-assimilation into a flux-transport dynamo model by (i) solving mean and perturbation equations by incorporating time-varying meridional flow since 1985; (ii) investigating transport of assimilated poloidal magnetic fields from surface to tachocline, where they are sheared by differential rotation to create spot-producing fields; (iii) updating model after a finite time-interval, by comparing model-output with observations; (iv) forecasting simultaneously cycle-amplitude, duration and shape. We form an ensemble of model-runs whose outputs calibrate best with surface magnetic observations. The ensemble-average gives the simultaneous prediction of solar cycle timing, amplitude and shape.

This work is partially supported by NASA grant NNX08AQ34G. Title: Flux-Transport Solar Dynamos Authors: Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2009SSRv..144...67D Altcode: Large-scale solar dynamo models were first built by Parker (1955). Over the past half a century these models have evolved significantly. We discuss here the development of a class of large-scale dynamo models which include, along with the α-effect and Ω-effect, an important third process, flux transport by meridional circulation. We present the properties of this ‘flux-transport’ dynamo, including the crucial role meridional circulation plays in giving this dynamo predictive power. Title: Axisymmetric MHD Instabilities in Solar/Stellar Tachoclines Authors: Dikpati, Mausumi; Gilman, Peter A.; Cally, Paul S.; Miesch, Mark S. Bibcode: 2009ApJ...692.1421D Altcode: Extensive studies over the past decade showed that HD and MHD nonaxisymmetric instabilities exist in the solar tachocline for a wide range of toroidal field profiles, amplitudes, and latitude locations. Axisymmetric instabilities (m = 0) do not exist in two dimensions, and are excited in quasi-three-dimensional shallow-water systems only for very high field strengths (2 mG). We investigate here MHD axisymmetric instabilities in a three-dimensional thin-shell model of the solar/stellar tachocline, employing a hydrostatic, non-Boussinesq system of equations. We deduce a number of general properties of the instability by use of an integral theorem, as well as finding detailed numerical solutions for unstable modes. Toroidal bands become unstable to axisymmetric perturbations for solar-like field strengths (100 kG). The e-folding time can be months down to a few hours if the field strength is 1 mG or higher, which might occur in the solar core, white dwarfs, or neutron stars. These instabilities exist without rotation, with rotation, and with differential rotation, although both rotation and differential rotation have stabilizing effects. Broad toroidal fields are stable. The instability for modes with m = 0 is driven from the poleward shoulder of banded profiles by a perturbation magnetic curvature stress that overcomes the stabilizing Coriolis force. The nonaxisymmetric instability tips or deforms a band; with axisymmetric instability, the fluid can roll in latitude and radius, and can convert bands into tubes stacked in radius. The velocity produced by this instability in the case of low-latitude bands crosses the equator, and hence can provide a mechanism for interhemispheric coupling. Title: Flux-Transport Solar Dynamos Authors: Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2009odsm.book...67D Altcode: Large-scale solar dynamo models were first built by Parker (1955). Over the past half a century these models have evolved significantly. We discuss here the development of a class of large-scale dynamo models which include, along with the α-effect and Ω-effect, an important third process, flux transport by meridional circulation. We present the properties of this `flux-transport' dynamo, including the crucial role meridional circulation plays in giving this dynamo predictive power. Title: Solar Dynamo and Magnetic Self-Organization Authors: Kosovichev, A. G.; Arlt, R.; Bonanno, A.; Brandenburg, A.; Brun, A. S.; Busse, F.; Dikpati, M.; Hill, F.; Gilman, P. A.; Nordlund, A.; Ruediger, G.; Stein, R. F.; Sekii, T.; Stenflo, J. O.; Ulrich, R. K.; Zhao, J. Bibcode: 2009astro2010S.160K Altcode: No abstract at ADS Title: Three-dimensional magneto-shear instabilities in the solar tachocline - II. Axisymmetric case Authors: Cally, Paul S.; Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2008MNRAS.391..891C Altcode: 2008MNRAS.tmp.1248C A Boussinesq model of the development of non-axisymmetric (in particular m = 1) three-dimensional magneto-shear instabilities in the solar tachocline was presented in Paper I. However, there it was erroneously concluded that the axisymmetric (m = 0) modes are stable, and they were not discussed further. Here it is shown that, although m = 0 modes are indeed stable for broad magnetic profiles, they are strongly unstable to radial shredding (high radial wavenumber) instabilities on the poleward shoulders of toroidal magnetic bands at high field strengths (roughly 40-100kG depending on bandwidth and latitude). These instabilities have growth rates comparable to or greater than those for tipping instabilities (m = 1) in many cases, but both are strongly stabilized by gravitational stratification characteristic of the upper radiative core. Weaker fields are m = 0 stable (though weakly m = 1 unstable), even in neutral gravitational stratification (convection zone). Title: A solar mean field dynamo benchmark Authors: Jouve, L.; Brun, A. S.; Arlt, R.; Brandenburg, A.; Dikpati, M.; Bonanno, A.; Käpylä, P. J.; Moss, D.; Rempel, M.; Gilman, P.; Korpi, M. J.; Kosovichev, A. G. Bibcode: 2008A&A...483..949J Altcode: Context: The solar magnetic activity and cycle are linked to an internal dynamo. Numerical simulations are an efficient and accurate tool to investigate such intricate dynamical processes.
Aims: We present the results of an international numerical benchmark study based on two-dimensional axisymmetric mean field solar dynamo models in spherical geometry. The purpose of this work is to provide reference cases that can be analyzed in detail and that can help in further development and validation of numerical codes that solve such kinematic problems.
Methods: The results of eight numerical codes solving the induction equation in the framework of mean field theory are compared for three increasingly computationally intensive models of the solar dynamo: an αΩ dynamo with constant magnetic diffusivity, an αΩ dynamo with magnetic diffusivity sharply varying with depth and an example of a flux-transport Babcock-Leighton dynamo which includes a non-local source term and one large single cell of meridional circulation per hemisphere. All cases include a realistic profile of differential rotation and thus a sharp tachocline.
Results: The most important finding of this study is that all codes agree quantitatively to within less than a percent for the αΩ dynamo cases and within a few percent for the flux-transport case. Both the critical dynamo numbers for the onset of dynamo action and the corresponding cycle periods are reasonably well recovered by all codes. Detailed comparisons of butterfly diagrams and specific cuts of both toroidal and poloidal fields at given latitude and radius confirm the good quantitative agreement.
Conclusions: We believe that such a benchmark study will be a very useful tool since it provides detailed standard cases for comparison and reference. Title: Global solar dynamo models: Simulations and predictions Authors: Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2008JApA...29...29D Altcode: No abstract at ADS Title: Polar Flux, Cross-equatorial Flux, and Dynamo-generated Tachocline Toroidal Flux as Predictors of Solar Cycles Authors: Dikpati, Mausumi; de Toma, Giuliana; Gilman, Peter A. Bibcode: 2008ApJ...675..920D Altcode: We evaluate the skill of three solar cycle predictors, namely, polar magnetic flux, flux crossing the equator, and tachocline toroidal flux, using both observations and a calibrated dynamo model. Polar flux measurements are available only for the past three sunspot cycles, implying poor statistics. However, the correlation between observed north and south polar flux peaks, and peaks of the next sunspot cycle is r = 0.785. We find that the correlation between the observed cross-equatorial flux and the observed peak of the next solar cycle is also high, close to that of Cameron & Schüssler, and the statistics are more reliable. Thus, the cross-equatorial flux is a better predictor of the next cycle than is the polar flux. From the dynamo model, the correlations with observed cycle peaks for polar flux, cross-equatorial flux, and toroidal flux are 0.48, 0.76, and 0.96, respectively. All these correlations decline when the northern and southern hemispheres are simulated separately, as well as with shortening of the averaging length in input data. A very high correlation between the model polar flux at the end of a cycle and the observed peak of that cycle implies that, within a calibrated flux transport dynamo, the polar flux follows the sunspot cycle, rather than being a precursor to it. With short-term averaging of input data, the polar and cross-equatorial fluxes retain much more short-term variability than does the toroidal flux. This is because the long traversal time of the input poloidal fields to the bottom of the convection zone in a mean-field model smooths out the short-term variability in the toroidal flux. The observed slowdown in meridional flow during 1996-2004 leads to a weaker polar flux, but a stronger cross-equatorial flux compared to the case with steady meridional flow. We infer that it is unlikely that both the cross-equatorial and the polar fluxes can be good predictors of solar cycle peaks. Title: The Waldmeier Effect: An Artifact of the Definition of Wolf Sunspot Number? Authors: Dikpati, Mausumi; Gilman, Peter A.; de Toma, Giuliana Bibcode: 2008ApJ...673L..99D Altcode: We show that the well-known "Waldmeier effect," or anticorrelation between the peak in sunspot number of a cycle and the time from minimum to reach that peak, is not present in sunspot area data averaged in the same way. With a 13-rotation Gaussian running average applied to both sunspot number and sunspot area, we find the correlation between cycle rise time and cycle peak is r = - 0.71 for sunspot number and r = 10-4 for sunspot area. Hathaway et al. previously showed that the Waldmeier effect is much weaker (r = - 0.34) in sunspot group number. Thus the Waldmeier effect may be specific to only Wolf sunspot number. Title: Simulating Solar Cycles in Northern and Southern Hemispheres by Assimilating Magnetic Data into a Calibrated Flux-Transport Dynamo Authors: Dikpati, Mausumi; Gilman, Peter A.; de Toma, Giuliana; Ghosh, Siddhartha S. Bibcode: 2007SoPh..245....1D Altcode: We use the flux-transport dynamo prediction scheme introduced by Dikpati, de Toma, and Gilman (Geophys. Res. Lett.33, L05102, 2006) to make separate simulations and predictions of sunspot cycle peaks for northern and southern hemispheres. Despite the division of the data, the skill level achieved is only slightly lower than that achieved for the sum of both hemispheres. The model shows skill at simulating and predicting the difference in peaks between North and South, provided that difference is more than a few percent. The simulation and prediction skill is achieved without adjustment to any parameters of the model that were used when peaks for the sum of North and South sunspot areas was simulated. The results are also very insensitive to the averaging length applied to the input data, provided the simulations and predictions are for peaks defined by averaging the observations over at least 13 rotations. However, in its present form, the model is not capable of skillfully simulating or predicting short-time-scale features of individual solar cycles. Title: Global MHD Instabilities in a Three-dimensional Thin-Shell Model of Solar Tachocline Authors: Gilman, Peter A.; Dikpati, Mausumi; Miesch, Mark S. Bibcode: 2007ApJS..170..203G Altcode: We generalize the linear analysis of the global instability of coexisting differential rotation and toroidal magnetic fields in the solar tachocline to include continuous radial stratification, thermodynamics, and finite tachocline thickness, as perturbed by three-dimensional disturbances of longitudinal wavenumbers m=1, 2. For radiative tachocline stratification, the instability for both banded and broad toroidal field profiles is similar to the two-dimensional and shallow water cases studied previously, even though the unstable modes have substantial vertical structure. For overshoot tachocline stratification, instability for banded toroidal fields with peaks <~20 kG is similar to the corresponding shallow water case, but for substantially higher (perhaps unrealistic) toroidal fields there is no low-subadiabaticity cutoff, and modes appear with much higher growth rate and increasingly negative phase velocities in longitude, analogous to those found earlier by Cally. All of these results are only modestly sensitive to the tachocline thickness chosen. For broad toroidal field profiles, the instability results are similar to the two-dimensional and shallow water cases for toroidal field peaks up to at least 80 kG, unless the shell has a thickness that is a substantial fraction of a pressure scale height. For thinner shells, only above about 94 kG do the high growth rate, low phase velocity modes appear, with structure similar to the ``polar kink'' instability Cally found. But even in this case, we find that the slower growing ``clam-shell'' instability eventually replaces the faster growing polar kink modes. We conclude that for conditions most likely to occur in the solar tachocline, such as peak toroidal fields limited by the dynamo to 20 kG or less, the two-dimensional and shallow water type unstable modes are likely to predominate even when the tachocline has finite thickness and the modes can have radial structure. Title: Magneto-Shear Instabilities in the Solar Tachocline Authors: Miesch, Mark S.; Gilman, P. A.; Dikpati, M. Bibcode: 2007AAS...210.4607M Altcode: 2007BAAS...39..161M Previously, linear theory has demonstrated that toroidal magnetic fields in the solar tachocline are destabilized by the presence of latitudinal differential rotation. For strong fields, such that the magnetic energy is comparable to the kinetic energy of the differential rotation, the most unstable modes are those with longitudinal wavenumber m=1 and these have been referred to as clam-shell or tipping instabilities for broad and banded toroidal field profiles respectively. Here we present nonlinear, three-dimensional simulations of these instabilities under both freely-evolving and forced conditions. Although the instabilities are allowed to have an arbitrary vertical structure, the dynamics remain quasi-2D, proceeding on horizontal surfaces which are nearly decoupled. The clam-shell instability saturates by opening up completely until horizontal loops of field become perpendicular to the equatorial plane. By contrast, the tipping instability saturates at relatively moderate tipping angles of 6-12 degrees by forming a jet of fluid along the axis of the tipped band. When the rotational shear and the magnetic field are maintained indefinitely by external forcing, clam-shell instabilities can operate continually, exhibiting a quasi-periodic behavior as mean toroidal fields alternately build up and destabilize. Title: Steps for Building a Calibrated Flux-Transport Dynamo for the Sun Authors: Dikpati, M.; Gilman, P. A. Bibcode: 2007SoPh..241....1D Altcode: Given the complexity involved in a flux-transport-type dynamo driven by both Babcock - Leighton and tachocline α effects, we present here a step-by-step procedure for building a flux-transport dynamo model calibrated to the Sun as a guide for anyone who wishes to build this kind of model. We show that a plausible sequence of steps to reach a converged solution in such a dynamo consists of (i) numerical integration of a classical α - ω dynamo driven by a tachocline α effect, (ii) continued integration with inclusion of meridional circulation to convert the model into a flux-transport dynamo driven by only a tachocline α effect, (iii) final integration with inclusion of a Babcock - Leighton surface α effect, resulting in a flux-transport dynamo that can be calibrated to obtain a close fit of model output with solar observations. Title: Nonlinear Evolution of Global Magnetoshear Instabilities in a Three-dimensional Thin-Shell Model of the Solar Tachocline Authors: Miesch, Mark S.; Gilman, Peter A.; Dikpati, Mausumi Bibcode: 2007ApJS..168..337M Altcode: We investigate global instabilities of toroidal fields and differential rotation in the solar tachocline using a three-dimensional thin-shell model. We initiate our nonlinear numerical simulations by superposing random, high-wavenumber perturbations on an equilibrium state and we then allow the system to evolve freely as the instabilities develop, grow exponentially, and saturate. For broad toroidal field profiles the dominant mode is the clamshell instability previously identified in two-dimensional and shallow-water investigations in which loops of field in the northern and southern hemispheres tilt and reconnect, eventually becoming perpendicular to the equatorial plane. If the initial toroidal field is instead confined to thin bands of alternating polarity in the northern and southern hemispheres, the evolution is more complex. At early times, a previously unidentified instability occurs near the edges of each band that is characterized by a high longitudinal wavenumber m. These edge instabilities are later superseded by an m=1 tipping instability whereby field lines tilt in latitude as in the broad-field case. The tipping instability saturates by forming a jet of fluid within each band, which provides gyroscopic stabilization. Both the clamshell and the tipping instabilities are quasi-two-dimensional and proceed most efficiently near the impenetrable boundaries where the vertical velocity vanishes. Strong stable stratification enhances this tendency by decoupling horizontal layers. These simulations demonstrate the robustness of global m=1 instabilities in differentially rotating spherical shells threaded by toroidal magnetic fields and as such have important implications for tachocline dynamics, including dynamo processes and tachocline confinement. Title: Global MHD instabilities of the tachocline Authors: Gilman, Peter A.; Cally, Paul S. Bibcode: 2007sota.conf..243G Altcode: No abstract at ADS Title: Dynamo-based simulations of solar activity differences between north and south hemispheres and forecasts for cycle 24 Authors: Dikpati, M.; Gilman, P. A.; de Toma, G. Bibcode: 2006AGUFMSH22A..06D Altcode: Recently we (GRL, 2006, vol.33, L05102, doi:10.1029/2005GL025221) built a predictive tool based on a Babcock-Leighton type flux-transport dynamo model of solar cycle. By assimilating the observed surface magnetic source data since cycle 12, we ran the model and showed that the model can correctly simulate the relative peaks of cycles 16 through 23. Extending the simulation into the future we predicted that cycle 24 will be 30-50 percent stronger than current cycle 23, because a strong 'seed' for cycle 24 is being formed, from the combination of latitudinal fields from past three cycles, 21, 22 and 23, in the dynamo layer in our model. We are currently exploring the data assimilation in our model for the N and S hemispheres separately, in order to simulate hemispheric asymmetries in the solar cycle features. Preliminary results show that the model has skill in simulating the N and S hemispheres separately, as well as the difference between the two hemispheres. We will discuss which hemisphere of the Sun should become more active in cycle 24 and which hemisphere should reach cycle peak first. This work is partially supported by NASA's LWS grant NNH05AB521, SR&T grant NNH06AD51I and the NCAR Director's Opportunity Fund. National Center for Atmospheric Research is sponsored by National Science Foundation. Title: Simulating and Predicting Solar Cycles Using a Flux-Transport Dynamo Authors: Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2006ApJ...649..498D Altcode: We construct a predictive tool based on a Babcock-Leighton-type flux-transport dynamo model of a solar cycle, run the model by updating the surface magnetic source using old cycles' data since cycle 12, and show that the model can correctly simulate the relative peaks of cycles 16-23. The simulations use the first four cycles to load the meridional circulation conveyor belt to create the Sun's memory about its past magnetic fields. Extending the simulation into the future, we predict that cycle 24 will be 30%-50% stronger than the current cycle 23. These simulations and predictions are robust for a wide range of convection zone magnetic diffusivity values between 3×1010 and 2×1011 cm2 s-1. Our model predictions are the same for three different treatments of the unknown surface magnetic source for the cycles to be predicted, namely (1) assuming some cyclic pattern, (2) incorporating ``zero'' surface source, or (3) including a surface source derived from the self-excited version of the dynamo model. Technique 3, for treating the surface source for cycles to be predicted, also shows significant skill in predicting two cycles ahead. Analyzing the evolution of magnetic field patterns over a full magnetic cycle, we show that the key to success of our prediction model lies in the formation of a ``seed'' for producing cycle n from the combination of latitudinal fields at high latitudes from three past cycles, n-1, n-2, and n-3, instead of the previous cycle's polar fields. These results have many implications for both solar and stellar dynamo modeling. Title: Simulating And Predicting Solar Cycles Using A Flux-transport Dynamo Authors: Dikpati, Mausumi; Gilman, P. A. Bibcode: 2006AAS...208.6507D Altcode: 2006BAAS...38Q.145D We construct a predictive tool based on a Babcock-Leighton type flux-transport dynamo model of solar cycle, run the model by updating the surface magnetic source using old cycles' data since cycle 12, and show that the model can correctly simulate the relative peaks of cycles 16 through 23. The simulations use the first 4 cycles to load the meridional circulation conveyor belt to create the Sun's memory about its past magnetic fields. Extending the simulation into the future we predict that cycle 24 will be 30-50% stronger than current cycle 23. These simulations and predictions are robust for a wide range of convection zone diffusivity values. Analyzing the evolution of magnetic field patterns over a full magnetic cycle, we show that the key to success of our prediction model lies in the formation of a 'seed' for producing cycle <i>n</i> from the combination of latitudinal poloidal fields at high latitudes from past three cycles, <i>n</i>-1, <i>n</i>-2 and <i>n</i>-3, instead of previous cycle's polar fields. These results have many implications for both solar and stellar dynamo modeling. The quality of our simulations and predictions implies that the flux-transport models must contain most of the important physical processes acting in the solar dynamo. Title: A 42 Year Quest to Understand the Solar Dynamo and Predict Solar Cycles Authors: Gilman, Peter A. Bibcode: 2006SPD....37.3101G Altcode: 2006BAAS...38..257G <table border="1" cellpadding="1" class="DisplayTable" id="{C4167B88-99D7-4F70-A4E7-C3D16AAB71FF}"><caption></caption><tr><td rowspan="1" colspan="1">For me this quest began in 1964, when I was a graduate student in meteorology at MIT. Most recently I have participated in a collaboration led by Mausumi Dikpati that has successfully simulated and ''predicted'' the relative peaks of the past 8 solar cycles, using a ''flux-transport'' solar dynamo. We are also ''hot on the trail'' of a theory for active longitudes on the sun, that involves global MHD instability of the solar tachocline. Over the 42 years, I have seen the solar dynamo problem declared ''solved'' by mean field dynamo theory in the 1970''s, followed by near total rejection of this conclusion in the 1980''s due to helioseismic measurements of solar rotation at depth, followed by a spectacular comeback in the 1990''s to the present, in the form of flux transport models in which meridional circulation is an essential component. I have seen 3D global MHD models for solar convection and differential rotation work well as dynamos, but fail as solar dynamos, yielding backward ''butterfly diagrams'', or no butterfly diagrams at all. In the near future, we will search for the limits of skill of axisymmetric flux transport models to predict details of individual cycles, and we will generalize the theory to include longitude dependence, by which we hope to produce a unified theory of the solar cycle and active longitudes. Since flux-transport dynamos work so well for the sun, they ought to work very well for many other stars.</td></tr></table> Title: A 42 Year Quest To Understand The Solar Dynamo And Predict Solar Cycles Authors: Gilman, Peter A. Bibcode: 2006AAS...208.3101G Altcode: 2006BAAS...38..108G For me this quest began in 1964, when I was a graduate student in meteorology at MIT. Most recently I have participated in a collaboration led by Mausumi Dikpati that has successfully simulated and 'predicted' the relative peaks of the past 8 solar cycles, using a 'flux-transport' solar dynamo. We are also 'hot on the trail' of a theory for active longitudes on the sun, that involves global MHD instability of the solar tachocline. Over the 42 years, I have seen the solar dynamo problem declared 'solved' by mean field dynamo theory in the 1970's, followed by near total rejection of this conclusion in the 1980's due to helioseismic measurements of solar rotation at depth, followed by a spectacular comeback in the 1990's to the present, in the form of flux transport models in which meridional circulation is an essential component. I have seen 3D global MHD models for solar convection and differential rotation work well as dynamos, but fail as solar dynamos, yielding backward 'butterfly diagrams', or no butterfly diagrams at all. In the near future, we will search for the limits of skill of axisymmetric flux transport models to predict details of individual cycles, and we will generalize the theory to include longitude dependence, by which we hope to produce a unified theory of the solar cycle and active longitudes. Since flux-transport dynamos work so well for the sun, they ought to work very well for many other stars. Title: Predicting the strength of solar cycle 24 using a flux-transport dynamo-based tool Authors: Dikpati, Mausumi; de Toma, Giuliana; Gilman, Peter A. Bibcode: 2006GeoRL..33.5102D Altcode: We construct a solar cycle strength prediction tool by modifying a calibrated flux-transport dynamo model, and make predictions of the amplitude of upcoming solar cycle 24. We predict that cycle 24 will have a 30-50% higher peak than cycle 23, in contrast to recent predictions by Svalgaard et al. and Schatten, who used a precursor method to forecast that cycle 24 will be considerably smaller than 23. The skill of our approach is supported by the flux transport dynamo model's ability to correctly 'forecast' the relative peaks of cycles 16-23 using sunspot area data from previous cycles. Title: Penetration of Dynamo-generated Magnetic Fields into the Sun's Radiative Interior Authors: Dikpati, Mausumi; Gilman, Peter A.; MacGregor, Keith B. Bibcode: 2006ApJ...638..564D Altcode: Any large-scale magnetic fields present in solar/stellar radiative interiors have so far been thought to be primordial or residuals from extinct dynamos. We show that a regular cyclic dynamo can also be the origin of strong magnetic fields in the solar radiative tachocline and interior below. By exploiting a kinematic, mean-field flux-transport dynamo, we show that for a wide range of core-diffusivity values, from 109 cm2 s-1 down to a molecular diffusivity of 103 cm2 s-1, oscillatory dynamo fields penetrate below the tachocline. Amplitudes of these fields are in the range of ~1 kG to 3×103 kG, depending on core diffusivity value, when the dynamo produces ~100 kG peak toroidal fields in the overshoot tachocline. For a low enough core diffusivity (<~107 cm2 s-1), there is also a steady (nonreversing) dynamo in the radiative tachocline and below, which generates strong toroidal field of amplitude ~1 kG to 3×103 kG or more there. The key elements in this dynamo are the low diffusivity, the differential rotation near the bottom of the tachocline, and an assumed tachocline α-effect. The Lorentz force feedback may limit oscillatory dynamo fields to ~30 kG, for which the mean nonreversing toroidal fields is still ~300 kG, for the lowest core diffusivity value. The presence of strong oscillatory and steady toroidal fields in the radiative tachocline implies that there cannot be a slow tachocline; the dynamics should always be fast there, dominated by MHD. These results are obtained using solar parameters, but they should also apply generally to stars with convecting shells and perhaps also with convective cores. Title: Jets in the Solar Tachocline as Diagnostics of Global MHD Processes Authors: Christensen-Dalsgaard, J.; Corbard, T.; Dikpati, M.; Gilman, P. A.; Thompson, M. J. Bibcode: 2005ASPC..346..115C Altcode: Multiple theories predict the existence of prograde fluid jets in the solar tachocline. We find helioseismic evidence of a prograde jet near 60° latitude in N and S hemispheres that persists through almost all of the current solar cycle. This evidence favors a hydrodynamic origin for the jet, from global instability of the differential rotation of the tachocline. We see no evidence for jets that migrate toward the equator with the advancing solar cycle, which tends to rule out jets associated with toroidal field bands in the tachocline. Title: A Shallow-Water Theory for the Sun's Active Longitudes Authors: Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2005ApJ...635L.193D Altcode: We show that the global MHD shallow-water instability of differential rotation and toroidal field bands in the solar tachocline provides a possible mechanism for the formation and evolution of active longitudes seen in synoptic maps of photospheric magnetic fields. The mechanism involves the production of upward bulges at selected longitudes in the overshoot tachocline that contain significant toroidal fields. Title: Constraints on the Applicability of an Interface Dynamo to the Sun Authors: Dikpati, Mausumi; Gilman, Peter A.; MacGregor, Keith B. Bibcode: 2005ApJ...631..647D Altcode: Taking into account the helioseismically inferred interior structure, we show that a pure interface-type dynamo does not work for the Sun if the skin effect for poloidal fields does not allow them to penetrate the tachocline. Using a simple mean-field kinematic α-Ω dynamo model, we demonstrate that, in the absence of tachocline radial shear participating in the dynamo process, a latitudinal differential rotation can provide the necessary Ω-effect to drive an oscillation in an interface dynamo, but it alone cannot produce the latitudinal migration. We show that to make an interface dynamo work with the constraints of interior structure and skin depth, a meridional circulation is essential. We conclude that a flux-transport dynamo driven by both the Babcock-Leighton and interface/bottom α-effects is a robust large-scale dynamo for the Sun. Title: Recovering Solar Toroidal Field Dynamics from Sunspot Location Patterns Authors: Norton, Aimee A.; Gilman, Peter A. Bibcode: 2005ApJ...630.1194N Altcode: 2005astro.ph..6025N We analyze both Kitt Peak magnetogram data and MDI continuum intensity sunspot data to search for the following solar toroidal band properties: width in latitude and the existence of a tipping instability (longitudinal m=1 mode) for any time during the solar cycle. In order to determine the extent to which we can recover the toroidal field dynamics, we forward-model artificially generated sunspot distributions from subsurface toroidal fields that we have assigned certain properties. Sine-curve fitting of Kitt Peak magnetogram data provided an upper limit of 15° to the tipping amplitude but could not adequately separate the tip from the width of the toroidal band. We then analyzed two sunspot distribution parameters using MDI and model data: the average latitudinal separation of sunspot pairs as a function of longitudinal separation and the number of sunspot pairs creating a given angle with respect to the east-west direction. A toroidal band of 10° width with a constant tipping of 5° best fits MDI data early in the solar cycle, when the sunspot band is at high latitudes (>18.5d). A toroidal band of 20° width with a tipping amplitude decreasing in time from 5deg to 0deg best fits MDI data late in the solar cycle when the sunspot band is at low latitudes (<18.5d). Model data generated by untipped toroidal bands cannot fit MDI high-latitude data using χ2 goodness-of-fit criteria and can fit only one sunspot distribution parameter at low latitudes. Tipped toroidal bands satisfy χ2 criteria at both high and low latitudes for both sunspot distribution parameters. We conclude that this is evidence to reject the null hypothesis-that toroidal bands in the solar tachocline do not experience a tipping instability-in favor of the hypothesis that the toroidal band experiences an m=1 tipping instability for a significant portion of the solar cycle. Our finding that the band widens from 10° early in the solar cycle to 20° late in the solar cycle may be explained in theory by magnetic drag spreading the toroidal band due to altered flow along the tipped field lines. Higher m modes, most notably m=2 and 6, are apparent in MDI data, but further analysis is needed to determine this property in detail. Title: Concentration of Toroidal Magnetic Field in the Solar Tachocline by η-Quenching Authors: Gilman, Peter A.; Rempel, Matthias Bibcode: 2005ApJ...630..615G Altcode: 2005astro.ph..4003G We show that if the turbulent magnetic diffusivity used in solar dynamos is assumed to be ``quenched'' by increasing toroidal fields, much larger amplitude and more concentrated toroidal fields can be induced by differential rotation from an assumed poloidal field than if there is no quenching. This amplification and concentration mechanism is weakened and bounded by jXB feedbacks on the differential rotation. Nevertheless, it is strong enough to contribute to the creation of ~100 kG toroidal fields near the base of the convection zone, perhaps in conjunction with the ``exploding flux tube'' process. Such high fields are necessary for sunspots to occur in low solar latitudes. Title: Comments on "Full-sphere simulations of circulation-dominated solar dynamo: Exploring the parity issue" Authors: Dikpati, M.; Rempel, M.; Gilman, P. A.; MacGregor, K. B. Bibcode: 2005A&A...437..699D Altcode: Using two distinct simulation codes that respectively apply semi-implicit and fully explicit schemes, we perform calculations of a 2D kinematic Babcock-Leighton type flux-transport dynamo with Chatterjee et al.'s parameter settings. We show that their solutions are diffusion-dominated, rather than circulation-dominated as their title implies. We also have been unable to reproduce several properties of their dynamo solutions, namely we obtain a much faster cycle with ~ 4 times shorter period than theirs, with highly overlapping cycles; a polar field value of ∼ 2 kG if one has to produce a ~ 100 kG toroidal field at convection zone base; and quadrupolar parity as opposed to Chatterjee et al.'s dipolar parity solutions. Title: The tachocline and the solar dynamo Authors: Gilman, P. A. Bibcode: 2005AN....326..208G Altcode: The solar tachocline contains a rich variety of physics, and contributes in many ways to the workings of the solar dynamo. It includes complex quasi-equilibrium states supported by the near balances of gas and magnetic pressure gradients, gravity, magnetic curvature stresses, Coriolis and other forces. The dynamics include overshooting convection, waves of several types, hydrodynamic and MHD instabilities on several spatial and temporal scales, several types of boundary layers and dynamo action. The dynamo processes present include the in situ generation of poloidal fields by global MHD and/or magnetic buoyancy instabilities; the advection of these poloidal fields by meridional circulation; the generation of toroidal fields from shearing by the differential rotation; and storage of these fields and their subsequent eruption into the convection zone. The tachocline probably also plays a significant role in creating magnetic patterns that are seen in the photosphere. This talk will of necessity focus on a subset of these topics, including particularly global MHD instabilities and whether we can see evidence of them in surface magnetic data, meridional circulation and the boundary layers that limit it, possible jets in the tachocline, and certain solar dynamo questions. Title: The solar photograph archive of the Mount Wilson Observatory. A resource for a century of digital data Authors: Lefebvre, S.; Ulrich, R. K.; Webster, L. S.; Varadi, F.; Javaraiah, J.; Bertello, L.; Werden, L.; Boyden, J. E.; Gilman, P. Bibcode: 2005MmSAI..76..862L Altcode: The solar telescopes and spectroheliographs of the Mount Wilson Observatory were among the earliest modern facilities for the study of the solar surface. The photographic collection of the solar program at Mt. Wilson begins in 1894 and continues to the present day. A program to digitize and distribute the images in this collection was begun at UCLA in 2003 and is now making available the first of the catalogued and catagorized images from the CaK sequence. Most of the instrumentation with which the images were obtained is still available although in a disassembled form. Original log books have been digitized and associated with the images so that a maximum of scientific return can be obtained from the data base. The present range of images available from www.astro.ucla.edu/~ulrich extends from late 1915 to 1952. Each image has been digitized with 12-bit precision and represented in a 16-bit format. These images are each 13 Mbytes in size and larger than will be the final product images since not all image defects have been mitigated at this time. The radii and centers of the solar images have been determined and are included in the available data files. Optical vignetting by the system introduces an intensity gradient of known magnitude that can be used to help characterize the photograph plates. The roll angle of the images has yet to be determined. Title: The Solar Photograph Archive of the Mount Wilson Observatory - A Resource for a Century of Digital Data Authors: Ulrich, R. K.; Webster, L. S.; Varadi, F.; Javaraiah, J.; Lefebvre, S.; Gilman, P. Bibcode: 2004AGUFMSH52A..03U Altcode: The solar telescopes and spectroheliographs of the Mount Wilson Observatory were among the earliest modern facilities for the study of the solar surface. The photographic collection of the solar program at Mt. Wilson begins in 1894 and continues to the present day. A program to digitize and distribute some of the images in this collection was begun at UCLA in 2003 and is now making available the first of the catalogued and catagorized images from the Ca K sequence. Most of the instrumentation with which the images were obtained is still available although in a disassembled form. Original logbooks have been digitized and associated with the images so that a maximum of scientific return can be obtained from the data base. The present range of images available from: http://www.astro.ucla.edu/&~slash;ulrich/MW&_slash;SPADP/CaK&_slash;fits/ extends from late 1915 to mid-1925. Each image has been digitized with 12-bit precision and represented in a 16-bit format. These images are each 13 Mbytes in size and larger than will be the final product images since not all image defects have been mitigated at this time. The radii and centers of the solar images have been determined and are included in the available data files. Optical vignetting by the system introduces an intensity gradient of known magnitude that can be used to help characterize the photograph plates. The roll angle of the images has yet to be determined. Title: Detection of Jets and Associated Toroidal Fields in the Solar Tachocline Authors: Christensen-Dalsgaard, J.; Corbard, T.; Dikpati, M.; Gilman, P. A.; Thompson, M. J. Bibcode: 2004ESASP.559..376C Altcode: 2004soho...14..376C No abstract at ADS Title: Global MHD Instabilities in a Thin-Shell Model of the Solar Tachocline Authors: Gilman, P. A.; Dikpati, M.; Miesch, M. S. Bibcode: 2004ESASP.559..440G Altcode: 2004soho...14..440G No abstract at ADS Title: Limits to Penetration of Meridional Circulation below the Solar Convection Zone Authors: Gilman, Peter A.; Miesch, Mark S. Bibcode: 2004ApJ...611..568G Altcode: We show that meridional circulation, such as that observed at the top of and in the solar convection zone (CZ) by direct doppler and helioseismic techniques, cannot penetrate significantly below the bottom (~0.7 Rsolar) of the overshoot layer at the bottom of the CZ. Therefore, solar dynamo models that rely on penetration as deep as to 0.6 Rsolar are ruled out. The analysis we carried out to reach this conclusion elucidates two boundary layers, one of which we have not seen applied to astrophysical problems before. This analysis should be relevant to understanding interfaces between convective and radiative zones in stellar interiors generally. Title: Linear Analysis and Nonlinear Evolution of Two-Dimensional Global Magnetohydrodynamic Instabilities in a Diffusive Tachocline Authors: Dikpati, Mausumi; Cally, Paul S.; Gilman, Peter A. Bibcode: 2004ApJ...610..597D Altcode: We develop a more realistic two-dimensional model for global MHD instabilities in the solar tachocline, by including diffusion in the form of kinetic and magnetic drag (following Newton's cooling law formulation). This instability has previously been studied by us and others for an idealized tachocline with no kinematic viscosity and magnetic diffusivity. Since radial diffusion is more important than latitudinal diffusion in the thin solar tachocline, diffusive decay of flow and magnetic fields can be considered as proportional to those variables. We find that, for solar-like toroidal magnetic fields of ~100 kG, instability exists for a wide range of kinetic and magnetic drag parameters, providing a mechanism for enhanced angular momentum transport in latitudes, which could explain how thin the solar tachocline is. From a detailed parameter space survey, we set upper limits of 5×1011 and 3×1010 cm2 s-1 for kinematic viscosity and magnetic diffusivity, respectively, such that this instability occurs in the solar tachocline on a timescale shorter than a sunspot cycle. We find that magnetic drag has much more influence than kinetic drag in damping this instability. This happens because the sink due to magnetic drag dissipates perturbation magnetic energy faster than the vorticity sink from kinetic drag dissipates perturbation kinetic energy. Consequently, in the presence of a large enough magnetic drag, the nonsolar-like clamshell pattern, found by Cally to be an inevitable final state of a broad profile undergoing an ideal MHD tachocline instability, is suppressed, whereas a banded profile still tips with no reduction in tip angle. We discuss how tipping may affect various surface manifestations of magnetic features, such as the latitudes and orientations of bipolar active regions. Title: Deciphering Toroidal Field Dynamics from Sunspot Statistics Authors: Norton, A. A.; Gilman, P. A. Bibcode: 2004AAS...204.5304N Altcode: 2004BAAS...36..756N We are interested in what solar surface magnetism can tell us about the interior toroidal magnetic fields. Because some fraction of solar surface magnetism must be a direct result of the dynamics of the interior toroidal field, we feel it is worthwhile to study the patterns of flux which emerge and attempt to recover the basic properties of the toroidal bands and their time dependent behavior.

New theory predicts a global instability resulting in a tipping of the toroidal bands with respect to the equatorial plane. We search for evidence of a tipped toroidal field for some phases of the solar cycle by examining the dominant latitude of emerging flux as a function of longitude. In order to determine the extent to which we can recover the toroidal field dynamics from observations, we use a model to artificially generate sunspot distributions from subsurface toroidal fields that we have assigned certain properties such as latitudinal width, peak field strength and degree of tipping with respect to the equatorial plane. Observational studies set an upper limit of 15 degrees to the tipping angle. Model results which best fit the observed data are those having a toroidal band with a tipping angle of 10 degrees at high latitudes (early in the sunspot cycle) gradually decreasing to 0 degrees as the sunspot band migrates towards the equator. Title: Global MHD Instabilities In a 3D Thin-shell Model Of Solar Tachocline Authors: Gilman, P. A.; Dikpati, M.; Miesch, M. S. Bibcode: 2004AAS...204.5303G Altcode: 2004BAAS...36..755G Previous work has shown that in model 2D and 'shallow water' tachoclines containing latitudinally varying differential rotation and toroidal fields, a global MHD instability is present for tachocline type differential rotations and a wide range of toroidal field profiles (both broad and narrow in latitude), in which longitudinal wave number m=1 usually predominates, leading to a tipping of the toroidal field away from a purely E-W orientation. We show here that the same instability occurs in model tachoclines with similar differential rotation and toroidal fields, but for which the stratification is continuous in the vertical. The unstable disturbances here have periodic vertical structures up to wave numbers of at least 10, depending on the degree of 'subadiabaticity' of the stratification. For overshoot layer conditions, the modes of shallow water type dominate and may be the only ones present, requiring vertical displacement of the top boundary of the tachocline, while in the radiative tachocline, modes with a wide range of vertical wavenumbers grow, with the same growth rate in the asymptotic limit of strongly subadiabatic temperature gradient. The latitude and longitude variations of the unstable modes are identical to the corresponding structures of 2D or shallow water modes. They also have similar kinetic helicity profiles in latitude, but with vertical structure determined by the vertical wavenumber. These modes should also contribute to driving the solar dynamo. This work is partially supported by NASA grants W-10107 and W-10175. National Center for Atmospheric Research is sponsored by National Science Foundation. Title: The Role of Time-varying Meridional Flow Pattern During Past 20 Years In Influencing Upcoming Solar Cycle Features Authors: Dikpati, M.; de Toma, G.; Gilman, P. A.; Corbard, T.; Rhodes, E. J.; Haber, D. A.; Bogart, R. S.; Rose, P. J. Bibcode: 2004AAS...204.5305D Altcode: 2004BAAS...36..756D Given the success of a recently built flux-transport dynamo-based scheme (ApJ, 2004, 601, 1136) in reproducing observed polar field features in cycle 23 including a) why polar reversal as well as polar field build-up after reversal were unusually slow, and b) why S-pole reversed a year after N-pole did, we apply this scheme to predict some features of solar cycle 24. It has been demonstrated (ApJ, 2000, 543, 1027) that the duration of the Sun's memory of its own magnetic field is governed primarily by the meridional flow speed in flux-transport dynamos, and is no less than two solar cycles. Therefore, observations of the Sun's magnetic field patterns over at least the past two cycles, and dynamical changes in the Sun's large-scale mass-flow in which the solar magnetic fields are partially frozen, should play important roles in determining certain features in the upcoming solar cycle. We first demonstrate theoretically how a N-S asymmetry in meridional flow pattern can produce asymmetry in sunspot maxima in N & S hemispheres, hence causing double peaks, as observed in cycles 22 and 23. We also show how deceleration in meridional flow, during the rising phase of cycle 23 produced a slower rise in this cycle compared to cycles 21 and 22. We then discuss the team-effort for extracting observed changes in meridional flow over the past 20 years, using helioseismic archive of MWO. By incorporating this long-term dynamical variation in flow-pattern in our prediction model, if we can tune the model to successfully reproduce various "anomalies" in solar cycle 23, we can comment further that cycle 23 is going to be a longer cycle if meridional flow does not accelerate during its declining phase, hence causing onset of cycle 24 around 2007. This work is supported by NASA grants W-10107 and W-10175. National Center for Atmospheric Research is sponsored by National Science Foundation. Title: Thin-Shell Magnetohydrodynamic Equations for the Solar Tachocline Authors: Miesch, Mark S.; Gilman, Peter A. Bibcode: 2004SoPh..220..287M Altcode: We present a new system of equations designed to study global-scale dynamics in the stably-stratified portion of the solar tachocline. This system is derived from the 3D equations of magnetohydrodynamics in a rotating spherical shell under the assumption that the shell is thin and stably-stratified (subadiabatic). The resulting thin-shell model can be regarded as a magnetic generalization of the hydrostatic primitive equations often used in meteorology. It is simpler in form than the more general anelastic or Boussinesq equations, making it more amenable to analysis and interpretation and more computationally efficient. However, the thin-shell system is still three-dimensional and as such represents an important extension to previous 2D and shallow-water approaches. In this paper we derive the governing equations for our thin-shell model and discuss its underlying assumptions, its context relative to other models, and its application to the solar tachocline. We also demonstrate that the dissipationless thin-shell system conserves energy, angular momentum and magnetic helicity. Title: Magnetic Field-Minimum Intensity Correlation in Sunspots: A Tool for Solar Dynamo Diagnostics Authors: Norton, Aimee A.; Gilman, Peter A. Bibcode: 2004ApJ...603..348N Altcode: Within a sunspot umbra, the continuum intensity is known to be inversely proportional to the magnetic field strength. Studied less is the relationship between the minimum continuum intensity and the maximum field strength of different sunspots. We conduct a test of this global relationship, using brightness ratios and magnetic field data from the Advanced Stokes Polarimeter and the Michelson Doppler Imager (MDI) for 10 sunspot umbrae of various sizes observed 1998 May-2003 June. We determine that the peak field strengths of sunspots can be ascertained from a fit to their corresponding brightness ratios with an accuracy of ~100 G, nearly twice the accuracy that a fit to the MDI magnetogram values can provide. We then analyze continuum intensity data from the MDI to characterize the distribution of sunspots as a function of latitude. We hand-select 331 and 321 umbrae, respectively, in the northern and southern hemispheres during Carrington rotations 1910-2003. Although the average location of sunspot eruption moves equatorward throughout the solar cycle, the northern hemisphere shows darker umbrae located systematically closer to the equator, while brighter umbrae are found at higher latitudes. These findings confirm the results of simulations that show strong flux emerging radially while weak flux emerges nonradially, causing weak flux to emerge poleward of its original toroidal field position. The average umbral intensity decreased in the north through the solar cycle, reaching a minimum intensity around sunspot maximum, possible evidence of the toroidal field strength peaking at solar maximum. This finding is in opposition to previous observations suggesting an increase late in the cycle. The southern hemisphere umbral distribution appears more disorganized and periodic in nature. Title: Diagnostics of Polar Field Reversal in Solar Cycle 23 Using a Flux Transport Dynamo Model Authors: Dikpati, Mausumi; de Toma, Giuliana; Gilman, Peter A.; Arge, Charles N.; White, Oran R. Bibcode: 2004ApJ...601.1136D Altcode: Motivated by observed anomalous features in cycle 23, as inferred from records of photospheric magnetic flux, we develop a flux transport dynamo-based scheme in order to investigate the physical cause of such anomalies. In this first study we focus on understanding anomalies occurring in the polar field evolutionary pattern in cycle 23, namely, why the polar reversal in cycle 23 was slow, why after reversal the buildup of the polar field was slow, and why the south pole reversed approximately a year after the north pole did. We construct a calibrated flux transport dynamo model that operates with dynamo ingredients such as differential rotation, meridional circulation, and large-scale poloidal field source derived from observations. A few other dynamo ingredients, such as diffusivity and quenching pattern, for which direct observations are not possible, are fixed by using theoretical guidance. By showing that this calibrated model can reproduce major longitude-averaged solar cycle features, we initialize the model at the beginning of cycle 22 and operate by incorporating the observed variations in meridional circulation and large-scale surface magnetic field sources to simulate the polar field evolution in cycle 23. We show that a 10%-20% weakening in photospheric magnetic flux in cycle 23 with respect to that in cycle 22 is the primary reason for a ~1 yr slowdown in polar reversal in cycle 23. Weakening in this flux is also the reason for slow buildup of polar field after reversal, whereas the observed north-south asymmetry in meridional circulation in the form of a larger decrease in flow speed in the northern hemisphere than that in the southern hemisphere during 1996-2002 and the appearance of a reverse, high-latitude flow cell in the northern hemisphere during 1998-2001 caused the north polar field to reverse before the south polar field. Title: The Solar Tachocline: Limiting Magneto-Tipping Instabilities Authors: Cally, P. S.; Dikpati, M.; Gilman, P. A. Bibcode: 2004IAUS..219..541C Altcode: 2003IAUS..219E.172C Two dimensional magneto-shear instabilities in the solar tachocline have been extensively explored in recent years. One of their most notable traits over a wide range of shear and magnetic profiles is a propensity for the magnetic field to tip substantially from its initial axisymmetric configuration with possible implications for patterns of flux emergence. However it is found that modifications of the standard models to include either kinetic and magnetic drag or prograde toroidal velocity jets associated with magnetic bands can suppress the instabilities or considerably reduce their nonlinear development. In the case of tip reduction by jets for a toroidal field of ~100kG in the tachocline (required for sunspots to emerge in sunspot latitudes) simulations indicate that jets capable of reducing tipping below the limits of detection from sunspot patterns at the surface are potentially detectable by helioseismic methods and should be looked for. Establishing an upper limit to the jet may result in a lower limit for the amount of tipping to be expected. Title: Penetration of Meridional Circulation into a Stably-Stratified Region Authors: Miesch, M. S.; Gilman, P. A. Bibcode: 2003AGUFMSH21B0128M Altcode: Several recent solar dynamo models have been proposed which require the axisymmetric component of the solar meridional circulation (MC) to penetrate substantially below the convective envelope, well into the stably-stratified interior. In this poster we investigate this possibility using a thin-shell approximation which has recently been developed to study the stably-stratified portion of the solar tachocline. Analytic and numerical results are presented for the extent of MC penetration as a function of amplitude and background stratification (represented in the nondimensional thin-shell system as the Rossby and Froude numbers). The results indicate that substantial penetration of the meridional circulation below the convective envelope is unlikely. Furthermore, the analysis is closely related to the classical Ekman layer analysis and may have additional geophysical applications. Title: Stability Analysis of Tachocline Latitudinal Differential Rotation and Coexisting Toroidal Band Using a Shallow-Water Model Authors: Dikpati, Mausumi; Gilman, Peter A.; Rempel, Matthias Bibcode: 2003ApJ...596..680D Altcode: Recently global, quasi-two-dimensional instabilities of tachocline latitudinal differential rotation have been studied using a so-called shallow-water model. While purely hydrodynamic shallow-water type disturbances were found to destabilize only the overshoot tachocline, the MHD analysis showed that in the presence of a broad toroidal field, both the radiative and overshoot parts of the tachocline can be unstable. We explore here instability in the shallow-water solar tachocline with concentrated toroidal bands placed at a wide range of latitudes, emulating different phases of the solar cycle. In equilibrium, the poleward magnetic curvature stress of the band is balanced either by an equatorward hydrostatic pressure gradient or by the Coriolis force from a prograde jet inside the band. We find that toroidal bands placed almost at all latitudes make the system unstable to shallow-water disturbances. For bands without prograde jets, the instability persists well above 100 kG peak field, while a jet stabilizes the band at a field of ~40 kG. The jet imparts gyroscopic inertia to the toroidal band inhibiting it from unstably ``tipping'' its axis away from rotation axis. Like previously studied HD and MHD shallow-water instabilities in the tachocline, unstable shallow-water modes found here produce kinetic helicity and hence a tachocline α-effect these narrow kinetic helicity profiles should generate narrowly confined poloidal fields, which will help formation of the narrow toroidal field. Toroidal bands poleward of 15° latitude suppress midlatitude hydrodynamic α-effects. However, even strong toroidal bands equatorward of 15° allow this hydrodynamic α-effect. Such bands should occur during the late declining phase of a solar cycle and, thus, could help the onset of a new cycle by switching on the mid latitude α-effect. Title: The Search for a Tipped Toroidal Field Authors: Norton, A. A.; Gilman, P. A.; Henney, C. J.; Cally, P. S. Bibcode: 2003SPD....34.1903N Altcode: 2003BAAS...35..842N A magnetic tipping instability of the tachocline toroidal field has been predicted (Cally et al., 2003) that could produce a toroidal field tipped with respect to the equatorial axis. One result of a tipped toroidal band is the eruption of magnetic flux over a range of latitudes from the same toroidal ring. The longitudinal dependence of this flux emergence would contribute to non-axisymmetry of the whole flux emergence pattern by giving it a longitudinal wavenumber m=1 dependence. We search for evidence (or lack thereof) of a tipped toroidal field for some phases of the solar cycle by examining the dominant latitude of emerging flux as a function of longitude. We use the existing observational data from Kitt Peak synoptic Carrington Rotation magnetograms to identify the location of strong flux, independently in each hemisphere, and test whether the location of the flux reveals a pattern compatible with the tipping instability m=1. Title: Meridional motion and the slope of isorotation contours Authors: Gilman, P. A.; Howe, R. Bibcode: 2003ESASP.517..283G Altcode: 2003soho...12..283G As helioseismic inversions have continued to improve in precision, it has become apparent that in the bulk of the solar convection zone the contours of rotation, while certainly not parallel to the rotation axis as predicted in earlier global convection simulations, are also not radial, as has often been stated based on earlier inversions. Instead, between 15 and about 55 degrees latitude (measured at r = 0.8R) the rotation contours make an angle with the rotation axis of about 25 degrees, which does not appear to vary systematically with latitude in this latitude range. What then determines this angle? By use of an extremely simple dynamical equilibrium model, we show that Coriolis forces from the observed meridional circulation could be responsible for this angle if, without this effect, the contours of rotation would be radial. Title: Clamshell and Tipping Instabilities in a Two-dimensional Magnetohydrodynamic Tachocline Authors: Cally, Paul S.; Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2003ApJ...582.1190C Altcode: Building on Cally's nonlinear model of two-dimensional MHD tachocline instability, we further explore the evolution of a wide variety of toroidal field profiles due to this instability. Cally showed in a recent study that an initially broad toroidal field opens up into a ``clamshell'' pattern because of nonlinear evolution of MHD tachocline instability. Various other toroidal field profiles-single toroidal bands, double bands, and mixed profiles with a band in addition to broad profiles-may also occur in the Sun during various phases of the solar cycle. Detailed study of the evolution of banded profiles shows no occurrence of clamshell instability, but the bands commonly tip relative to the axis of rotation. The higher the latitude location of the band, the more it tips. Extreme tipping results when the band is at 60° latitude or higher-the magnetic ring hangs from the pole on one side of the Sun. For bands of 10° latitude width at sunspot latitudes (<=40°), the band tip is within +/-10° about the mean latitude of the band. This tipping could either enhance or reduce the observed tilt in bipolar active regions. Double bands, or profiles consisting of a band and a broad profile, may exist at certain phases of the solar cycle. We find that double-band systems with two oppositely directed bands separated widely (>15°) in latitude, as well as two close bands of same polarities, do not interact in the same hemisphere-the higher latitude band tips, while the lower latitude band hardly responds. A significant interaction between two individual bands in one hemisphere takes place only when the band separation is <=15° and the bands are oppositely directed, which is a nonsolar case. In this case, the band system either tips or forms the clamshell pattern depending on the dominant mode symmetry. We also show that a mixed profile with oppositely directed narrow fields close to the equator in addition to the broad fields evolve in such a way as to oppose the reconnection of the broad fields across the equator, and thus inhibiting the clamshell formation, at least at certain phases of the solar cycle. Finally, we note that the tipping and clamshell instabilities strongly inhibit differential rotation. Title: Solar Multiscale Convection and Rotation Gradients Studied in Shallow Spherical Shells Authors: De Rosa, Marc L.; Gilman, Peter A.; Toomre, Juri Bibcode: 2002ApJ...581.1356D Altcode: 2002astro.ph..9054D The differential rotation of the Sun, as deduced from helioseismology, exhibits a prominent radial shear layer near the top of the convection zone wherein negative radial gradients of angular velocity are evident in the low- and midlatitude regions spanning the outer 5% of the solar radius. Supergranulation and related scales of turbulent convection are likely to play a significant role in the maintenance of such radial gradients and may influence dynamics on a global scale in ways that are not yet understood. To investigate such dynamics, we have constructed a series of three-dimensional numerical simulations of turbulent compressible convection within spherical shells, dealing with shallow domains to make such modeling computationally tractable. In all but one case, the lower boundary is forced to rotate differentially in order to approximate the influence that the differential rotation established within the bulk of the convection zone might have upon a near-surface shearing layer. These simulations are the first models of solar convection in a spherical geometry that can explicitly resolve both the largest dynamical scales of the system (of order the solar radius) as well as smaller scale convective overturning motions comparable in size to solar supergranulation (20-40 Mm). We find that convection within these simulations spans a large range of horizontal scales, especially near the top of each domain, where convection on supergranular scales is apparent. The smaller cells are advected laterally by the larger scales of convection within the simulations, which take the form of a connected network of narrow downflow lanes that horizontally divide the domain into regions measuring approximately 100-200 Mm across. We also find that the radial angular velocity gradient in these models is typically negative, especially in the low- and midlatitude regions. Analyses of the angular momentum transport indicate that such gradients are maintained by Reynolds stresses associated with the convection, transporting angular momentum inward to balance the outward transport achieved by viscous diffusion and large-scale flows in the meridional plane, a mechanism first proposed by Foukal & Jokipii and tested by Gilman & Foukal. We suggest that similar mechanisms associated with smaller scale convection in the Sun may contribute to the maintenance of the observed radial shear layer located immediately below the solar photosphere. Title: Angular momentum transport in the solar tachocline Authors: Miesch, Mark S.; Gilman, Peter A. Bibcode: 2002ESASP.505..509M Altcode: 2002IAUCo.188..509M; 2002solm.conf..509M Angular momentum transport by non-axisymmetric motions in the solar tachocline likely play a key role in maintaining the global rotation profile in the solar interior. As such, it has important implications with regard to the solar dynamo and to the amplification and emergence of magnetic flux. We report results from several types of numerical experiments designed to study angular momentum transport due to stratified turbulence and shear instabilities in the tachocline and we discuss what implications this may have for its latitudinal and vertical structure. Title: Analysis of Instability of Latitudinal Differential Rotation and Toroidal Field in the Solar Tachocline Using a Magnetohydrodynamic Shallow-Water Model. I. Instability for Broad Toroidal Field Profiles Authors: Gilman, Peter A.; Dikpati, Mausumi Bibcode: 2002ApJ...576.1031G Altcode: We examine the global MHD instability of solar tachocline latitudinal differential rotation and the coexisting broad toroidal magnetic field, using a ``shallow-water'' model that captures the simplest effects of subadiabatic stratification. We assume a single fluid shell that has a fixed bottom but variable thickness. This model is the MHD generalization of a hydrodynamic model that we have previously applied to the tachocline, although the solution method is somewhat different. Stratification in the model is characterized by an ``effective gravity'' G (G=0 for adiabatic stratification). The radiative (lower) part of the tachocline thus has high G (~102) and the overshoot part, low G (less than 1). We obtain growth rates, phase velocities, and spatial structures of unstable modes for a wide range of toroidal field strengths and effective gravities, as well as differential rotations that are consistent with helioseismic observations. We recover known two-dimensional MHD stability results in the limit of large G and hydrodynamic instability results in the limit of vanishing toroidal field. For strong magnetic fields, only longitudinal wavenumber m=1 is unstable, but for weak fields m=2 is also. For peak toroidal fields of 20 kG and above, the growth rates and disturbance structures are essentially independent of the effective gravity, until it becomes so small that the fluid shell shrinks to zero in low latitudes, whereupon the instability is cut off. In contrast, the instability evolves radically at low G when toroidal field is increased from zero. In both overshoot and radiative parts of the tachocline, unstable modes grow fastest for toroidal fields of the order of 102 kG. The structure of the unstable disturbances is always governed by the latitude location of singular or critical points at which the Doppler-shifted phase velocity of the disturbance equals the local (angular) Alfvén speed. All unstable disturbances possess kinetic helicity, narrowly concentrated in the neighborhood of the same critical points. Just as shown by Dikpati & Gilman for the hydrodynamic case, such disturbances could provide an ``α-effect'' for the solar dynamo. But unlike the hydrodynamic case, this α-effect would be a function of the toroidal field itself. Title: Flux Transport Solar Dynamos with Near-Surface Radial Shear Authors: Dikpati, Mausumi; Corbard, Thierry; Thompson, Michael J.; Gilman, Peter A. Bibcode: 2002ApJ...575L..41D Altcode: Corbard & Thompson analyzed quantitatively the strong radial differential rotation that exists in a thin layer near the solar surface. We investigate the role of this radial shear in driving a flux transport dynamo operating with such a rotation profile. We show that despite being strong, near-surface radial shear effectively contributes only ~1 kG (~30% of the total) to the toroidal fields produced there unless an abnormally high, surface α-effect is included. While 3 kG spot formation from ~1-2 kG toroidal fields by convective collapse cannot be ruled out, the evolutionary pattern of these model fields indicates that the polarities of spots formed from the near-surface toroidal field would violate the observed polarity relationship with polar fields. This supports previous results that large-scale solar dynamos generate intense toroidal fields in the tachocline, from which buoyant magnetic loops rise to the photosphere to produce spots. Polar fields generated in flux transport models are commonly much higher than observed. We show here that by adding enhanced diffusion in the supergranulation layer (originally proposed by Leighton), near-surface toroidal fields undergo large diffusive decay preventing spot formation from them, as well as reducing polar fields closer to the observed values. However, the weaker polar fields lead to the regeneration of a toroidal field of less than ~10 kG at the convection zone base, too weak to produce spots that emerge in low latitudes, unless an additional poloidal field is produced at the tachocline. This is achieved by a tachocline α-effect, previously shown to be necessary for coupling the north and south hemispheres to ensure toroidal and poloidal fields that are antisymmetric about the equator. Title: Observational constraints on the solar dynamo Authors: Gilman, Peter A. Bibcode: 2002ESASP.508...25G Altcode: 2002soho...11...25G There are basically two types of observations that should constrain dynamo models for the sun. One is observations of the sun's magnetic field in all its manifestations, which dynamo models should eventually predict accurately; the other is observations of solar velocities and thermodynamic structure and variations, which constrain the inductive processes available for the dynamo. With respect to magnetic properties, the level of detail in current observations of fields and inferred field patterns far outstrips current dynamo capabilities to reproduce, but models have moved beyond comparison to just the "classical" solar cycle characteristics such as Hale's polarity laws and the butterfly diagram. With respect to velocity constraints, much more is known now about differential rotation and meridional circulation in the convection zone and tachocline, than when early results from helioseismology overturned the prevailing α-ω dynamos of the 1970's that required angular velocity increasing inwards. Meridional circulation has become a particularly important component, since recent models show it can determine the solar cycle period correctly for observed flow speeds. By contrast, so-called giant cells, which have been particularly difficult to detect, may not be as relevant to the dynamo problem as they were once thought to be. For the future, improved inference from helioseismic analysis of structure, velocities and magnetic fields in the solar tachocline would be particularly valuable. Title: Effect of subsurface radial differential rotation on flux-transport solar dynamo Authors: Corbard, T.; Dikpati, M.; Gilman, P. A.; Thompson, M. J. Bibcode: 2002ESASP.508...75C Altcode: 2002soho...11...75C A near-surface radial gradient of rotation was recently inferred and quantified from MDI f-modes observations by Corbard & Thompson (2001). We show, from our preliminary simulation of the large-scale solar magnetic field, by using a flux-transport type dynamo, that despite being strong, this gradient plays only a small role compared to the tachocline in shearing the poloidal fields to produce toroidal fields. This happens primarily because the turbulent diffusivity near the surface wins in the competition of generation versus decay of the magnetic fields unless an abnormally high α-effect is considered there. This supports the results of previous flux-transport as well as interface and overshoot layer dynamo models that the major toroidal fields of the Sun are generated in the tachocline. Title: Equilibrium And Instability Of Toroidal Field Bands And Rotational Jets In The Solar Tachocline Authors: Gilman, P. A.; Rempel, M.; Dikpati, M. Bibcode: 2002AAS...200.0416G Altcode: 2002BAAS...34..645G Recently Dikpati & Gilman (2001, ApJ, 552, 348) have shown, using a shallow-water model of the solar tachocline that allows the top surface to deform, that a tachocline with the observed broad differential rotation and a strong toroidal field is prolate. A strong toroidal field ring requires extra mass on its poleward side to provide a hydrostatic latitudinal pressure gradient to balance the poleward curvature stress. In a parallel study using a different approach, Rempel et al (2000, A&A, 363, 789) have shown that a weakly subadiabatic stratification leads to a complementary equilibrium state of the overshoot tachocline in which the magnetic curvature stress is balanced by a prograde rotational jet inside the toroidal ring. We show that the shallow water model yields a similar equilibrium state if we suppress the shell deformation and allow the differential rotation to be modified. We are analyzing the stability of such an equilibrium tachocline by using the MHD shallow-water model of Gilman & Dikpati (2002, ApJ, submitted). We expect to show that the combination of toroidal band and rotational jet is virtually always unstable to disturbances with longitudinal wave number m>0, except perhaps when the band is extremely narrow. This instability could wipe out the jet, and lead to some poleward migration of the toroidal field, as well as the excitation of longitudinally periodic magnetic patterns that might provide sites for magnetic bouyancy to produce spots as well as other photospheric magnetic features. This work is supported by NASA grants W-19752 and S-10145-X. The National Center for Atmospheric Research is sponsored by the National Science Foundation. Title: Solar rotation: observations, theory, modeling, and implications for solar magnetism Authors: Gilman, Peter Bibcode: 2002ocnd.confE..12G Altcode: No abstract at ADS Title: Progress and Problems in Solar Dyanmo Theory Authors: Gilman, P. A. Bibcode: 2001AGUFMGP22B..02G Altcode: In the 1970's it was widely believed that an application of mean-field dynamo theory to the bulk of the solar convection zone had 'solved' the solar dynamo problem, but the advent of helioseismology provided measures of solar differential rotation within the convection zone that contradicted the required assumption of an angular velocity increasing inward. This and requirements for storage of magnetic flux while it amplified led to new theories focussed on the solar tachocline, a sharp shear layer seen at the base of the solar convection zone. Most recently, new bulk convection zone dynamo models, the so-called flux transport models, have shown considerable success in reproducing many features of the solar cycle, but such models still contain assumptions concerning processes that are not subject to observational test. A crucial improvement in the these models has been the inclusion of (measured) meridional circulation, which is now seen as determining the solar cycle period. Despite these successes, many uncertainties remain. Little is known about the global dynamics and MHD of the solar tachocline, where toroidal magnetic flux is likely stored while it amplifies enough to generate sunspots. Global convection theory still has not yielded a fully convincing theory of the maintenance of the differential rotation in the convection zone, though all such models rely on organized transport of angular momentum toward the equator to make it spin faster in the face of turbulent diffusion. The history of the solar dynamo problem makes it clear that observational constraints are crucial to developing a plausible, durable theory; presumably the same is also true for the geodynamo. Title: Flux-Transport Dynamos with α-Effect from Global Instability of Tachocline Differential Rotation: A Solution for Magnetic Parity Selection in the Sun Authors: Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2001ApJ...559..428D Altcode: We propose an αΩ flux-transport dynamo for the Sun that is driven by a tachocline α-effect. This α-effect comes from the global hydrodynamic instability of latitudinal differential rotation in the tachocline, as calculated using a shallow-water model. Growing, unstable shallow-water modes propagating longitudinally in the tachocline create vortices that correlate with radial motion in the layer to produce a longitude-averaged net kinetic helicity and, hence, an α-effect. We show that such a dynamo is equally successful as a Babcock-Leighton-type flux-transport dynamo in reproducing many large-scale solar cycle features. The success of both dynamo types depends on the inclusion of meridional circulation of a sign and magnitude similar to that seen on the Sun. Both α-effects (the Babcock-Leighton-type and tachocline α-effect) are likely to exist in the Sun, but it is hard to estimate their relative magnitudes. By extending the simulation to a full spherical shell, we show that the flux-transport dynamo driven by the tachocline α-effect selects a toroidal field that is antisymmetric about the equator, while the Babcock-Leighton flux-transport dynamo selects a symmetric toroidal field. Since our present Sun selects antisymmetric fields, we argue that the tachocline α-effect must be more important than the Babcock-Leighton α-effect. Title: Erratum: Shallow-Water Magnetohydrodynamic Waves in the Solar Tachocline Authors: Schecter, D. A.; Boyd, J. F.; Gilman, P. A. Bibcode: 2001ApJ...556L.127S Altcode: An error occurred in the printing of Figure 2 in the Letter ``Shallow-Water Magnetohydrodynamic Waves in the Solar Tachocline'' by D. A. Schecter, J. F. Boyd, and P. A. Gilman (ApJ, 551, L185 [2001]). The f in the abscissa label of Figure 2b was grossly displaced. The corrected figure is shown here. Title: Latitudinal Transport of Angular Momentum by Cellular Flows Observed with MDI Authors: Hathaway, D. H.; Gilman, P. A.; Beck, J. G. Bibcode: 2001AGUSM..SP21C09H Altcode: We have analyzed Doppler velocity images from the MDI instrument on SOHO to determine the latitudinal transport of angular momentum by the cellular photospheric flows. Doppler velocity images from 60-days in May to July of 1996 were processed to remove the p-mode oscillations, the convective blue shift, the axisymmetric flows, and any instrumental artifacts. The remaining cellular flows were examined for evidence of latitudinal angular momentum transport. Small cells show no evidence of any such transport. Cells the size of supergranules (30,000 km in diameter) show strong evidence for a poleward transport of angular momentum. This would be expected if supergranules are influenced by the Coriolis force, and if the cells are elongated in an east-west direction. We find good evidence for just such an east- west elongation of the supergranules. This elongation may be the result of differential rotation shearing the cellular structures. Data simulations of this effect support the conclusion that elongated supergranules transport angular momentum from the equator toward the poles. Cells somewhat larger than supergranules do not show evidence for this poleward transport. Further analysis of the data is planned to determine if the direction of angular momentum transport reverses for even larger cellular structures. The Sun's rapidly rotating equator must be maintained by such transport somewhere within the convection zone. Title: Prolateness of the Solar Tachocline Inferred from Latitudinal Force Balance in a Magnetohydrodynamic Shallow-Water Model Authors: Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2001ApJ...552..348D Altcode: Motivated by recent helioseismic observations concerning solar tachocline shape and thickness and by the theoretical development of MHD shallow-water equations for the tachocline, we compute the prolateness of the tachocline using an MHD shallow-water model, in which the shape and thickness are determined from the latitudinal force balance equation. We show that a strong toroidal magnetic field stored at or below the overshoot part of the tachocline leads to a pileup of fluid at high latitude, owing to the poleward magnetic curvature stress which has to be balanced by an equatorward latitudinal hydrostatic pressure gradient. For toroidal fields of solar amplitude (~100 kG), results for differentially rotating and uniformly rotating tachoclines are almost the same. In contrast, the unmagnetized differentially rotating tachocline would always be weakly oblate. We propose that a strong toroidal field in the overshoot part of the tachocline should tend to suppress the overshooting, thereby increasing the magnetic storage capacity of the layer since the stratification there should become more subadiabatic. We illustrate the effect of this process on the shape and thickness of the layer by assuming its effective gravity is a function of field strength. If toroidal fields are concentrated in relatively narrow bands which migrate toward the equator with the advance of the sunspot cycle, then they should be accompanied by a ``thickness front'' advancing at the same rate. Applying our model to the prolateness estimate of Charbonneau et al. yields toroidal fields of 60-150 kG in the overshoot layer, consistent with other considerations. Their prolateness in the radiative part of the tachocline would require ~600 kG fields to be present. Title: Flux-transport Dynamos Driven by a Tachocline α -effect; a Solution to Magnetic Parity Selection in the Sun Authors: Dikpati, M.; Gilman, P. A. Bibcode: 2001AGUSM..SP31A19D Altcode: We propose here an α Ω flux-transport dynamo driven by a tachocline α -effect, produced by the global hydrodynamic instability of tachocline differential rotation as calculated using a shallow-water model (Dikpati & Gilman, 2001, ApJ, Mar.20 issue). Growing, unstable shallow-water modes propagating longitudinally in the tachocline create alternate vortices which correlate with upward/downward radial motion of top boundary, associated with convergence/divergence of the disturbance flow to produce a longitude-averaged net kinetic helicity, and hence an α -effect. We show that a flux-transport dynamo driven by a tachocline α -effect is equally successful as a Babcock-Leighton flux-transport dynamo (Dikpati & Charbonneau 1999, ApJ, 518, 508) in reproducing many large-scale solar cycle features, including the most difficult feature of phase relationship between the subsurface toroidal field and surface radial field. In view of the success of flux-transport dynamos, whether the α -effect is at the surface or in the tachocline, we argue that the solar dynamo should be considered to involve three basic processes, rather than two (α -effect and Ω -effect only). The third important process is the advective transport of flux by meridional circulation. In reality, both α -effects (Babcock-Leighton type and tachocline α -effect) are likely to exist, but it is hard to estimate their relative magnitudes. We show, by extending the simulation in a full spherical shell model that a flux-transport dynamo driven by a tachocline α -effect selects toroidal field that is antisymmetric about the equator, while a Babcock-Leighton flux-transport dynamo selects symmetric toroidal field. Since our present Sun selects antisymmetric toroidal fields, we argue that the flux-transport solar dynamo is primarily driven by a tachocline α -effect. Acknowledgements: This work is supported by NASA grants W-19752 and S-10145-X. National Center for Atmospheric Research is sponsored by National Science Foundation. Title: ``Shallow Water'' MHD Waves in the Solar Tachocline Authors: Schecter, D. A.; Gilman, P. A.; Boyd, J. F. Bibcode: 2001AGUSM..SP31A18S Altcode: Peter Gilman recently introduced ``Shallow water'' magnetohydrodynamics (SMHD) as a simple model for understanding predominantly horizontal flows in the solar tachocline. Here, we present analytic solutions to SMHD waves in the tachocline. We show that there are two branches of such waves: Alfvén and magneto-gravity. Finite Alfvén and magneto-gravity waves can both propagate without change of shape in regions of the tachocline where the Coriolis force is finite. At the equator, where the Coriolis force is zero, magneto-gravity waves steepen and develop singularities. We show that the time required for a singularity to develop increases monotonically with the ambient magnetic field strength. Currently, a numerical model is being developed for an extensive study of nonlinear SMHD in the tachocline. This model is based on the β -plane approximation of geophysical fluid dynamics. The β -plane approximation removes some of the complications of spherical geometry, but keeps latitudinal variation in the Coriolis parameter. Early results from this model should be available for this presentation. Title: Analysis of Global MHD Instability in the Tachocline Using a Shallow-water Model Authors: Gilman, P. A.; Dikpati, M. Bibcode: 2001AGUSM..SP22A05G Altcode: Results from studying 2D models indicate that tachocline differential rotation is hydrodynamically stable, but is magnetohydrodynamically unstable with a coexisting toroidal field (Gilman & Fox 1997, ApJ,484, 439). In a recent study, Dikpati & Gilman (2001, ApJ, Mar.20 issue) have shown, by including simplified 3D effects through use of a shallow-water model, that the overshoot part of the tachocline can be hydrodynamically unstable even without a toroidal field. Hydrodynamic shallow-water equations have an MHD analog (Gilman 2000, ApJ, 544, L79). We study here the linear MHD shallow-water instability of tachocline latitudinal differential rotation with a variety of coexisting toroidal field profiles. We show that both the radiative and overshoot parts of the tachocline become unstable in presence of almost all toroidal field profiles, from broad to narrow, with a wide range of field strengths, generalizing the 2D MHD instability, as well as the shallow-water HD instability. Instability occurs for a wide range of subadiabatic tachocline stratifications. Unstable MHD shallow-water modes possess kinetic helicity which, unlike the kinetic helicity of HD shallow-water modes, depends on field strength and latitude-location of the peak toroidal field. Therefore, the α -effect produced by global MHD shallow-water instability contains its own quenching, as well as time-dependence, if we identify the latitude of the peak magnetic field as a proxy for solar-cycle phase. We also demonstrate how the toroidal field band can be twisted in the tachocline due to the first order effect of the instability. Acknowledgements: This work is supported by NASA grants W-19752 and S-10145-X. National Center for Atmospheric Research is sponsored by National Science Foundation. Title: ``Shallow-Water'' Magnetohydrodynamic Waves in the Solar Tachocline Authors: Schecter, D. A.; Boyd, J. F.; Gilman, P. A. Bibcode: 2001ApJ...551L.185S Altcode: This Letter discusses waves in a rotating magnetized fluid layer, governed by ``shallow-water'' magnetohydrodynamics. Such waves likely exist in the solar tachocline, and we focus on this application. A dispersion relation is derived, giving two branches of waves: Alfvén and magnetogravity. In general, finite Alfvén and magnetogravity waves can propagate without change of shape. However, if the Coriolis force is absent, as on the equator of the tachocline, finite magnetogravity waves steepen and develop singularities in a time τs. It is shown that τs increases monotonically with the ambient magnetic field strength. Title: Analysis of Hydrodynamic Stability of Solar Tachocline Latitudinal Differential Rotation using a Shallow-Water Model Authors: Dikpati, Mausumi; Gilman, Peter A. Bibcode: 2001ApJ...551..536D Altcode: We examine the global, hydrodynamic stability of solar latitudinal differential rotation in a ``shallow-water'' model of the tachocline. Charbonneau, Dikpati, & Gilman have recently shown that two-dimensional disturbances are stable in the tachocline (which contains a pole-to-equator differential rotation s<18%). In our model, the upper boundary of the thin shell is allowed to deform in latitude, longitude, and time, thus including simplified three-dimensional effects. We examine the stability of differential rotation as a function of the effective gravity of the stratification in the tachocline. High effective gravity corresponds to the radiative part of the tachocline; for this case, the instability is similar to the strictly two-dimensional case (appearing only for s>=18%), driven primarily by the kinetic energy of differential rotation extracted through the work of the Reynolds stress. For low effective gravity, which corresponds to the overshoot part of the tachocline, a second mode of instability occurs, fed again by the kinetic energy of differential rotation, which is primarily extracted by additional stresses and correlations of perturbations arising in the deformed shell. In this case, instability occurs for differential rotation as low as about 11% between equator and pole. If this mode occurs in the Sun, it should destabilize the latitudinal differential rotation in the overshoot part of the tachocline, even without a toroidal field. For the full range of effective gravity, the vorticity associated with the perturbations, coupled with radial motion due to horizontal divergence/convergence of the fluid, gives rise to a longitude-averaged, net kinetic helicity pattern, and hence a source of α-effect in the tachocline. Thus there could be a dynamo in the tachocline, driven by this α-effect and the latitudinal and radial gradients of rotation. Title: Symmetry Selection in Solar Cycle Dynamo Models Authors: Dikpati, M.; Gilman, P. A. Bibcode: 2001ASPC..248..125D Altcode: 2001mfah.conf..125D No abstract at ADS Title: The solar tachocline and its variation (?) Authors: Corbard, T.; Jiménez-Reyes, S. J.; Tomczyk, S.; Dikpati, M.; Gilman, P. Bibcode: 2001ESASP.464..265C Altcode: 2001soho...10..265C; 2000astro.ph.11367C The solar tachocline, located at the interface between the latitude-dependent rotation of the convection zone and the rigid radiative interior, presents high gradients of angular velocity which are of particular interest for the models of the solar dynamo and angular momentum transport. Furthermore, latitudinal and temporal variations of the tachocline parameters, if any, are also of particular interest in order to constrain models. We present a review of some of the theories of the tachocline and their predictions that may be tested by helioseismology. We describe the methods for inferring the tachocline parameters from obervations and the associated difficulties. A review of results previously obtained is given and an analysis of the new 6 years database of LOWL observations is presented which yields no compelling evidence of variations or general trend of the tachocline parameters during the ascending phase of the current solar cycle (1994-2000). Title: Magnetohydrodynamic ``Shallow Water'' Equations for the Solar Tachocline Authors: Gilman, Peter A. Bibcode: 2000ApJ...544L..79G Altcode: We argue that the classical ``shallow water'' equations of geophysical fluid dynamics should be useful for studying the global dynamics of the solar tachocline and demonstrate the existence of an MHD analog that would allow taking into account the strong toroidal magnetic field likely to be present there. Title: Solar and Stellar Convection: A Perspective for Geophysical Fluid Dynamicists Authors: Gilman, Peter A. Bibcode: 2000gac..conf...37G Altcode: No abstract at ADS Title: Erratum: Joint Instability of Latitudinal Differential Rotation and Toroidal Magnetic Fields below the Solar Convection Zone. II. Instability for Toroidal Fields That Have a Node between the Equator and Pole Authors: Gilman, Peter A.; Fox, Peter A. Bibcode: 2000ApJ...534.1020G Altcode: There is a repeated error in certain equations in the papers by Peter A. Gilman and Peter A. Fox, ``Joint Instability of Latitudinal Differential Rotation and Toroidal Magnetic Fields below the Solar Convection Zone. II. Instability for Toroidal Fields That Have a Node between the Equator and Pole'' (510, 1018 [1999]) (GF) Mausumi Dikpati and Peter A. Gilman, ``Joint Instability of Latitudinal Differential Rotation and Concentrated Toroidal Fields below the Solar Convection Zone'' (512, 417 [1999]) (DG) and Peter A. Gilman and Mausumi Dikpati, ``Joint Instability of Latitudinal Differential Rotation and Concentrated Toroidal Fields below the Solar Convection Zone. II. Instability of Narrow Bands at All Latitudes'' (528, 552 [2000]) (GD). In GF, equation (3), DG, equation (24), and GD, equation (3), all are missing an additive term that multiplies the variable H. The correct total factor multiplying H is {1/(1-μ2) ( 2 - m2/(1-μ2) + 1/S[2c(ωo-c) + μ d/dμ(S/(1-μ2)]} The second term, involving S in the denominator, is missing in these papers. As a consequence, the expression for k2, shown in equation (6) of GF and equation (10) of DG, is also missing this term. The correct expression for k2 in both cases is 1/(1-μ2) ( 2 - m2/(1-μ2) + ((1/2S)(d2S/dμ2)) + 1/S[2c(ωo-c) + μ d/dμ(S/(1-μ2)] No results or conclusions are affected in any of these papers, because the equations containing errors are used only to identify the location of singular points, where the effective wavenumber increases without bound. All of these singular points are the same in the erroneous forms as in the correct forms, being determined principally by the roots of S. But obviously, if the erroneous forms are used for other purposes, incorrect conclusions might be reached. The authors are indebted to Paul Cally for discovering this error. Title: Shear Flow Instabilities Authors: Gilman, Peter Bibcode: 2000astu.confE..15G Altcode: No abstract at ADS Title: Instability of Tachocline Differential Rotation in a Shallow-Water Model, and the Solar Dynamo Authors: Dikpati, M.; Gilman, P. A. Bibcode: 2000SPD....31.0115D Altcode: 2000BAAS...32..803D We examine the global, hydrodynamic stability of solar latitudinal differential rotation in a "shallow-water" model of the tachocline. Charbonneau, Dikpati and Gilman (1999, ApJ 526, 523) have recently shown that 2D disturbances are stable in the tachocline. In our model, the upper boundary of the thin shell is allowed to deform in latitude, longitude and time, thus including simplified 3D effects. We examine the stability of differential rotation as a function of the effective gravity of the stratification in the tachocline. High effective gravity corresponds to the radiative part of the tachocline; for this case, the instability is similar to the strictly 2D case. For low effective gravity, which corresponds to the overshoot part of the tachocline, a second mode of instability occurs, fed primarily by potential energy stored in the deformed shell. In this case, instability occurs for differential rotation down to about 13% between equator and pole. If this mode occurs in the Sun, it should destabilize much of the tachocline, even without a toroidal field. For the full range of effective gravity, the vorticity associated with the perturbations, coupled with radial motion due to horizontal divergence/convergence of the fluid, gives rise to a longitude-averaged, net kinetic helicity pattern, and hence a source of α -effect in the tachocline. Thus there could be a dynamo in the tachocline, driven by this α -effect and the latitudinal and radial gradients of rotation. Preliminary simulations suggest that an α -ω dynamo with the above α -effect working in the tachocline is capable of producing a reasonable butterfly diagram, particularly when a meridional circulation is included in the convective envelope, as is known to exist. Acknowledgement. This work has been partially supported by the NASA grant S-10145-X. The National Center for Atmospheric Research is sponsored by the National Science Foundation. Title: Fluid Dynamics and MHD of the Solar Convection Zone and Tachocline: Current Understanding and Unsolved Problems - (Invited Review) Authors: Gilman, Peter A. Bibcode: 2000SoPh..192...27G Altcode: We review recent progress and define unanswered scientific questions in five related topics: granulation- to supergranulation-scale convection and magnetic structures; global convection and circulation; the rise of magnetic flux tubes to the photosphere, and their injection into the base of the convection zone; tachocline fluid dynamics and MHD; and the solar dynamo. We close with a set of observational `targets' for helioseismologists to aim for. Title: Three-dimensional Spherical Simulations of Solar Convection. I. Differential Rotation and Pattern Evolution Achieved with Laminar and Turbulent States Authors: Miesch, Mark S.; Elliott, Julian R.; Toomre, Juri; Clune, Tom L.; Glatzmaier, Gary A.; Gilman, Peter A. Bibcode: 2000ApJ...532..593M Altcode: Rotationally constrained convection possesses velocity correlations that transport momentum and drive mean flows such as differential rotation. The nature of this transport can be very complex in turbulent flow regimes, where large-scale, coherent vorticity structures and mean flows can be established by smaller scale turbulence through inverse cascades. The dynamics of the highly turbulent solar convection zone therefore may be quite different than in early global-scale numerical models, which were limited by computational resources to nearly laminar flows. Recent progress in high-performance computing technology and ongoing helioseismic investigations of the dynamics of the solar interior have motivated us to develop more sophisticated numerical models of global-scale solar convection. Here we report three-dimensional simulations of compressible, penetrative convection in rotating spherical shells in both laminar and turbulent parameter regimes. The convective structure in the laminar case is dominated by ``banana cells,'' but the turbulent case is much more complex, with an intricate, rapidly evolving downflow network in the upper convection zone and an intermittent, plume-dominated structure in the lower convection zone and overshoot region. Convective patterns generally propagate prograde at low latitudes and retrograde at high latitudes relative to the local rotation. The differential rotation profiles show some similarity with helioseismic determinations of the solar rotation but still exhibit significantly more cylindrical alignment. Strong, intermittent, vortical downflow lanes and plumes play an important dynamical role in turbulent flow regimes and are responsible for significant differences relative to laminar flows with regard to momentum and energy transport and to the structure of the overshoot region at the base of the convection zone. Title: Joint Instability of Latitudinal Differential Rotation and Concentrated Toroidal Fields below the Solar Convection Zone. II. Instability of Narrow Bands at All Latitudes Authors: Gilman, Peter A.; Dikpati, Mausumi Bibcode: 2000ApJ...528..552G Altcode: Assuming that concentrated toroidal field bands occur in the solar tachocline at different latitudes as the solar cycle progresses, we examine the joint instability of latitudinal differential rotation and coexisting narrow bands placed at a wide range of latitudes. Following the basic formalism developed by Gilman & Fox and employing the numerical technique of Dikpati & Gilman, we show that the instability exists for almost all phases of the solar cycle, i.e., for a wide range of latitudinal positions of the bands of field strengths between 500 G and 200 kG. Modes with longitudinal wavenumber m=1 up to m=7, depending on parameter values, are unstable for both kinds of symmetries. Mid-latitude bands are the most unstable; the instability disappears if the band is at very high or low latitudes. High-latitude bands are highly unstable, with e-folding growth times of a few months, even when the field strength is low (about a few hundred gauss) low-latitude bands are unstable with longer growth times (>=1 yr), but for high field strengths (104-2×105 G). We argue that, because of the instability, the high-latitude bands would undergo turbulent mixing in the latitudinal direction on a timescale that is short compared to their build-up time from shearing of poloidal field by the differential rotation, and thus they may not be buoyant enough to appear as active regions at the surface, but the low-latitude bands can build up simultaneously with this instability and can eventually manifest as active regions. Instability for modes with m>=1 could help determine the longitude distribution of active regions; nonlinear changes in the toroidal field due to the instability may contribute to their decay. Title: The Solar Cycle and the Tachocline: Theories and Observations Authors: Corbard, T.; Jimenez-Reyes, S. J.; Tomoczyk, S.; Dikpati, M.; Gilman, P. Bibcode: 2000ESASP.463...21C Altcode: 2000sctc.proc...21C No abstract at ADS Title: Stability of the Solar Latitudinal Differential Rotation Inferred from Helioseismic Data Authors: Charbonneau, Paul; Dikpati, Mausumi; Gilman, Peter A. Bibcode: 1999ApJ...526..523C Altcode: We revisit the hydrodynamical stability problem posed by the observed solar latitudinal differential rotation. Specifically, we carry out stability analyses on a spherical shell for solar-like two-dimensional inviscid shear flow profiles of the form ν=s0-s2μ2-s4μ4, where μ is the sine of latitude. We find that stability is remarkably sensitive to the magnitude of the μ4 term. This allows us to reconcile apparently conflicting results found in the published literature. We then use latitudinal differential rotation profiles extracted from various helioseismic inversions of the solar internal rotation and investigate their stability as a function of depth from the base of the tachocline to the top of the convective envelope. In all cases considered, we find that the latitudinal differential rotation in the tachocline is stable while that in the bulk of the convective envelope is unstable. Under the assumption that the instability is not impeded by finite Reynolds number or three-dimensional effects not accounted for in our analysis, we speculate on possible observable consequences of the occurrence of the instability in the top half of the convective envelope. Title: Joint Instability of Latitudinal Differential Rotation and Toroidal Magnetic Fields below the Solar Convection Zone. III. Unstable Disturbance Phenomenology and the Solar Cycle Authors: Gilman, P. A.; Fox, P. A. Bibcode: 1999ApJ...522.1167G Altcode: We analyze additional solutions for the two-dimensional instability of coexisting differential rotation and toroidal magnetic fields, organized in families with fixed ratios ER of magnetic to kinetic energy in the unperturbed state. Solutions are found for a wide range of differential rotation amplitudes found in the solar tachocline, for toroidal fields that have a node that ranges in latitude from the pole to the equator, as we expect to exist in the Sun through a sunspot cycle. Fixed ER is a proxy for nonlinear saturation of the solar dynamo due to the reaction of electromagnetic body forces. Since the saturation ratio is not known from either theory or observations, we find solutions in the range 0.1<=ER<=30, corresponding to peak toroidal fields in the solar tachocline of between about 8×103 to 1.4×105 G. We focus on properties of the unstable disturbances that could test the hypothesis that such disturbances in the solar tachocline provide a template for surface features. We show that the symmetry of magnetic pattern about the equator could switch at one or more phases of the magnetic cycle, and for high ER a switch could also occur between two antisymmetric patterns of different latitudinal profiles. In the former case, the pattern rotation rate would be unchanged, but there would be a sudden longitudinal phase shift in one or both hemispheres. In the latter case, there would be no phase shift but instead a substantial change in the rotation rate of the observed magnetic pattern. For a given mode symmetry and type, the rotation rate is the same at all latitudes, with the rate being close to that of the local rotation of the plasma at the latitude where the disturbance amplitude peaks. For ER<~1, the disturbance magnetic patterns have significant tilts upstream away from the equator, reminiscent of similar patterns in synoptic magnetograms. Sharp changes with latitude in the differential rotation and toroidal field are associated with ``critical points'' in the system, where the Doppler-shifted disturbance rotation equals the local (angular) Alfvén speed. These migrate toward the equator with the toroidal field node but increasingly lag it. The higher the magnetic energy for a given differential rotation, the closer the equator is approached. If these sharp changes in differential rotation and toroidal field are related to the torsional oscillations and latitudes of sunspots, then these solutions favor large toroidal fields in the tachocline, of peak amplitude at least 6×104 G. Title: Interpretation of Solar Activity Features Using Joint Instability of Differential Rotation and Toroidal Magnetic Fields Authors: Gilman, P. A.; Dikpati, M. Bibcode: 1999AAS...194.5609G Altcode: 1999BAAS...31..913G Gilman & Fox found that latitudinal differential rotation and a broad toroidal field are jointly unstable to 2D horizontal perturbations in the solar tachocline, while each of them is separately stable there. They argued this instability could provide enhanced angular momentum mixing in latitude, as required by Spiegel and Zahn for the solar tachocline to retain its present thickness. Motivated by observations of active regions that imply subsurface toroidal fields of the Sun occur in narrow bands, and assuming that these bands occur in the tachocline at different latitudes as the solar cycle progresses, we examine the joint instability of latitudinal differential rotation and coexisting narrow bands placed at all latitudes. We show that instability occurs for almost all phases of solar cycle, for a wide range of differential rotation amplitudes and toroidal field strengths (between 500 Gauss and 200 kGauss). Mid-latitude bands are most unstable; instability disappears if the band is at very high or low latitudes. We argue that a highly unstable weak (<10(3) Gauss) high-latitude toroidal band would undergo turbulent mixing in a time short compared to its build-up time due to shearing of poloidal field by differential rotation, and thus may never be buoyant enough to appear as active regions at the surface, providing a possible explanation for why active regions are not seen in high latitudes. Low latitude bands, on the other hand, are unstable with a longer growth time and instability persists for high field strengths (up to 2 x 10(5) Gauss). Therefore, low-latitude bands can still amplify while experiencing instability, and can still erupt at the surface as active regions. Instability for modes with longitudinal wavenumber m equal to or greater than 1 could help determine the longitude distribution of active regions, and initial nonlinear changes in the toroidal field due to this instability may contribute to their decay. *The National Center for Atmospheric Research is sponsored by the National Science Foundation. Title: Joint Instability of Latitudinal Differential Rotation and Concentrated Toroidal Fields below the Solar Convection Zone Authors: Dikpati, Mausumi; Gilman, Peter A. Bibcode: 1999ApJ...512..417D Altcode: Motivated by observations of sunspot and active-region latitudes that suggest that the subsurface toroidal field in the Sun occurs in narrow latitude belts, we analyze the joint instability of solar latitudinal differential rotation and the concentrated toroidal field below the base of the convection zone, extending the work of Gilman & Fox (hereafter GF). We represent the profile of the toroidal field by Gaussian functions whose width is a variable parameter and solve the two-dimensional perturbation equations of GF by relaxation methods. We reproduce the results of GF for broad profiles, and we find instability for a wide range of amplitudes of differential rotation and toroidal fields (103-106 G fields at the base of the solar convection zone), as well as a wide range of toroidal-field bandwidths. We show that the combination of concentrated toroidal fields and solar-type latitudinal differential rotation is again unstable, not only to longitudinal wavenumber m=1 as in GF, but also to m>1 for sufficiently narrow toroidal-field profiles. For a fixed peak field strength, the growth rate first increases as the toroidal-field band is narrowed, reaching a peak for bandwidths between 10° and 20° in latitude, depending on the peak field strength, and then decreases to a cut-off in the instability for toroidal field bands of 3°-4°. Irrespective of bandwidth, the differential rotation is the primary energy source for the instability for weak fields, and the toroidal field is the primary source for strong fields. The weaker (stronger) the peak toroidal field is, the narrower (broader) is the bandwidth for which the toroidal field becomes the primary energy source. The Reynolds, Maxwell, and mixed stresses required to extract energy from the differential rotation and toroidal field are most active in the neighborhood of the singular or turning points of the perturbation equations. This first study focuses on toroidal fields that peak near 45° latitude, as in GF; later papers will place the toroidal-field peak at a wide variety of latitudes, as we might expect to occur at different phases of a sunspot cycle. Title: Joint Instability of Latitudinal Differential Rotation and Toroidal Magnetic Fields below the Solar Convection Zone. II. Instability for Toroidal Fields that Have a Node between the Equator and Pole Authors: Gilman, Peter A.; Fox, Peter A. Bibcode: 1999ApJ...510.1018G Altcode: We generalize results of Gilman and Fox to unperturbed toroidal fields that have a node somewhere between the equator and the pole as we speculate the Sun's field to have for most phases of its magnetic cycle. We use the same solution method as in Gilman and Fox, namely Legendre polynomial expansion and matrix inversion to solve for the eigenvalues and eigenfunctions. The solutions are structured around certain singular or critical points of the suitably transformed and combined vorticity and induction equations. There are singular points at the poles, and singularities where ω0-cr=+/-α0, in which ω0 is the local rotation rate, cr is the longitudinal phase speed of an unstable wave, and α0 is an angular measure of the toroidal field. We survey the instability as a function of toroidal field profile and amplitude as well as differential rotation amplitude, thereby examining reference states that could be characteristic of most phases of the solar cycle, and most depths within the rotational shear layer just below the base of the solar convection zone.

As found in Gilman and Fox, instability occurs for a wide range of both toroidal fields and differential rotations. Differential rotation is again the primary energy source for growing modes when the toroidal field is weak, and the toroidal field is the primary source when it is strong. Unlike in Gilman and Fox, here modes of both symmetries about the equator are unstable for low and high toroidal fields, and for high fields a second antisymmetric mode appears. Which mode symmetry is favored for low fields depends in detail on the relative amplitudes of differential rotation and toroidal field. For low toroidal fields (unstable) modes of both symmetries are energetically active (extracting energy from the unperturbed state) only poleward of the node and an adjacent singularity, but are coupled to energetically neutral velocity perturbations equatorward of that singular point. In transition to higher field strengths, those velocity patterns are damped out when two additional singular points appear in the system, but the energetically active high-latitude disturbances remain. By contrast the second antisymmetric mode is energetically active equatorward of the toroidal field node and closely adjacent singular points, but is coupled to an energetically neutral pattern of both velocities and magnetic fields on the poleward side.

As in Gilman and Fox, we find narrow-latitude bands of sharp changes in both differential rotation and toroidal magnetic field that migrate toward the equator with increasing field strength, but are bounded in their migration by the latitude of the toroidal field node. These sharp changes are always at the locations of the singular points of the system and represent narrow domains where both kinetic and magnetic energy are being extracted from the reference state to drive the instability.

We interpret the instability as a form of resonant overreflection between singular points, analogous to what happens in stratified shear flow, as described for example by Lindzen. The instability may contribute to determining the latitudinal and longitudinal distribution of active regions and other large-scale, magnetic features on the Sun, as well as enable a degree of synchronization of the evolution of the solar cycle between low latitudes and high, and between north and south hemispheres. Title: Fluid Dynamics and MHD of the Solar Convection Zone and Tachocline: Current Understanding and Unsolved Problems Authors: Gilman, Peter A. Bibcode: 1999soho....9E...1G Altcode: My assigned topic is extremely broad, so in this abstract I will state only goals and emphases, not all of which I may be able to discuss in the time allotted. The focus of the talk will be primarily theoretical, but guided by observations. Within the fluid dynamical realm, I will say something about most scales of convection, as well as other classes of fluid processes, particularly global instabilities and waves. I will emphasize the differences in dynamics that should characterize the different domains of differential rotation we now know exist on the Sun: strong radial rotation gradients just below the photosphere; the bulk of the convection zone that has little radial gradient; the tachocline at the bottom; and possible polar vortices. Within the MHD realm, I will focus on the multiple dynamos that may be operating, as well as on processes for getting flux from below the convection zone through it to the solar atmosphere, and the possible role played by global MHD instabilities in the tachocline. A key question is how far below the overshoot layer in the tachocline does the solar cycle magnetic field extend. I will also speculate on the role the equator near the base of the convection zone could play in annihilation of solar cycle flux. In closing, I will try to sketch an integrated picture of the dynamics and MHD of the convection zone and tachocline, identify gaps in our understanding, and hopefully stimulate discussion of which gaps we have the best chance of filling, using the tools of helioseismology and theoretical and numerical calculations. Title: Instability of concentrated toroidal fields in the latitudinal differential rotation below the solar convection zone Authors: Dikpati, M.; Gilman, P. A. Bibcode: 1998ASPC..138...89D Altcode: 1998stas.conf...89D No abstract at ADS Title: Joint Instability of Latitudinal Differential Rotation and Toroidal Magnetic Fields below the Solar Convection Zone Authors: Gilman, Peter A.; Fox, Peter A. Bibcode: 1997ApJ...484..439G Altcode: Below the convection zone, where the stratification is radiatively controlled, large-scale motions should be mainly horizontal, i.e., in spherical shells due to the stabilizing effect of negative buoyancy on radial displacements. Watson showed that the observed surface solar differential rotation is at the threshold for instability to horizontal disturbances. Therefore, since helioseismology tells us the latitudinal differential rotation below the convection zone is less than the surface value, the profile should be stable too. We show that in the presence of a broad, nonuniform toroidal field the solar differential rotation is unstable. This is true for a wide range of kinetic and magnetic energies of the unperturbed state, from well below equipartition to well above it. We find instability for essentially all values of differential rotation and toroidal fields for which we are able to find converged solutions. The instability appears to occur only for longitudinal wave number 1. Disturbance symmetries about the equator and profiles in latitude depend on the amplitude of the toridal field. Peak e-folding times are a few months. The primary energy source for the instability is differential rotation for low field strengths and the toroidal field for high field strengths. The mechanism of energy release from the differential rotation is the poleward transport of angular momentum, by the Maxwell stress rather than the Reynolds stress. For the profiles studied, the Reynolds stress is almost always trying to rebuild differential rotation, the exact opposite of the nonmagnetic case. Second-order perturbation theory predicts that the unstable modes produce zonal jets and fine structure in the toroidal field, the latitude of which migrates toward the equator with increasing magnetic field strength. The instability we have found may play a role in the solar dynamo, although being two-dimensional, it cannot produce a dynamo by itself. Mixing of angular momentum caused by the instability could allow achievement of equilibrium of the solar tachocline hypothesized by Spiegel & Zahn. Title: Joint Instability of Latitudinal Differential Rotation and Concentrated Toroidal Fields below the Solar Convection Zone Authors: Dikpati, M.; Gilman, P. A. Bibcode: 1997SPD....28.0213D Altcode: 1997BAAS...29..895D It was found by Gilman and Fox (July 20 1997 issue of ApJ) that the solar differential rotation below the convection zone is unstable to horizontal disturbances in the presence of a broad, nonuniform toroidal magnetic field. This instability occurs for longitudinal wave number 1. The primary energy source for the instability is the differential rotation for low field strength, and the toroidal field for high field strength. Since solar toroidal magnetic fields are inferred from sunspots to occur in narrow latitude belts, we have now studied the joint instability of solar differential rotation and highly concentrated toroidal field. The instability still occurs, and it occurs for both the longitudinal wave number 1 and higher wave numbers. The growth rates of the instability for some wave numbers greater than 1 are higher, if the toroidal field is in a belt equatorward of 60(deg) latitude. The primary source of energy for the instability is usually the magnetic field, but in some cases extraction of energy from the differential rotation becomes more important particularly for higher wave numbers with weaker fields. The instability occurs with high growth rate even for weak toroidal field when the field is at 30(deg) latitude or higher. This enhanced instability at high latitude may inhibit the build up of toroidal field preventing occurrence of sunspots at higher latitudes. For low latitudes, the much lower growth rate of the instability may allow sufficient toroidal field buildup that sunspots can be formed. The occurrence of this instability for wave numbers m=1 to 5 may help determine the observed distribution of active regions with longitude. Title: Joint Instability of Differential Rotation and Toroidal Magnetic Fields below the Solar Convection Zone, II Authors: Gilman, P. A.; Fox, P. Bibcode: 1997SPD....28.0212G Altcode: 1997BAAS...29..895G At the 1996 AAS/SPD meeting in Madison we reported first results for the joint instability of differential rotation and toroidal magnetic fields to 2D disturbances (see also Gilman and Fox, Paper I, July 20 1997 issue of ApJ). This analysis was for the toroidal field profile B=a*sin(LAT)cos(LAT). This paper reports results for the profile B=(a*sin(LAT)+b*(sin(LAT))(3) ))cos(LAT), which, with b<-a<0, allows for a node in the toroidal field at latitude arcsin (-a/b). This generalization is of interest because we should expect such a node to appear and migrate equatorward as the sun proceeds from one sunspot cycle to the next. As with the simpler profile, instability occurs for virtually all differential rotation amplitudes, and all toroidal field amplitudes and shapes, and remains confined to disturbances with longitudinal wave number m=1. For a, b>0, the instability is enhanced for the same a compared to the b=0 case, particularly in high latitudes. For 0>b>-a (so no node is present) the instability is similar to the b=0 case but with diminished growth rates, due to the reduction of toroidal fields at high latitudes. At b=-a, the symmetric mode of instability vanishes, but the antisymmetric mode remains. For b<-a<0, both symmetric and antisymmetric modes are unstable, but with disturbances confined largely to the domain poleward of the node, unless the toroidal field energy greatly exceeds the kinetic energy of differential rotation. Unstable disturbances spread and migrate toward the equator as the field strength is increased and as the node is moved equatorward. Thus, the instability may still contribute to the existence of the solar butterfly diagram, and to other solar dynamo presses. Title: GONG Observations of Solar Surface Flows Authors: Hathaway, D. H.; Gilman, P. A.; Harvey, J. W.; Hill, F.; Howard, R. F.; Jones, H. P.; Kasher, J. C.; Leibacher, J. W.; Pintar, J. A.; Simon, G. W. Bibcode: 1996Sci...272.1306H Altcode: Doppler velocity observations obtained by the Global Oscillation Network Group (GONG) instruments directly measure the nearly steady flows in the solar photosphere. The sun's differential rotation is accurately determined from single observations. The rotation profile with respect to latitude agrees well with previous measures, but it also shows a slight north-south asymmetry. Rotation profiles averaged over 27-day rotations of the sun reveal the torsional oscillation signal-weak, jetlike features, with amplitudes of 5 meters per second, that are associated with the sunspot latitude activity belts. A meridional circulation with a poleward flow of about 20 meters per second is also evident. Several characteristics of the surface flows suggest the presence of large convection cells. Title: The Global Oscillation Network Group (GONG) Project Authors: Harvey, J. W.; Hill, F.; Hubbard, R. P.; Kennedy, J. R.; Leibacher, J. W.; Pintar, J. A.; Gilman, P. A.; Noyes, R. W.; Title, A. M.; Toomre, J.; Ulrich, R. K.; Bhatnagar, A.; Kennewell, J. A.; Marquette, W.; Patron, J.; Saa, O.; Yasukawa, E. Bibcode: 1996Sci...272.1284H Altcode: Helioseismology requires nearly continuous observations of the oscillations of the solar surface for long periods of time in order to obtain precise measurements of the sun's normal modes of oscillation. The GONG project acquires velocity images from a network of six identical instruments distributed around the world. The GONG network began full operation in October 1995. It has achieved a duty cycle of 89 percent and reduced the magnitude of spectral artifacts by a factor of 280 in power, compared with single-site observations. The instrumental noise is less than the observed solar background. Title: Joint Instability of Latitudinal Differential Rotation and Toroidal Magnetic Fields below the Solar Convection Zone Authors: Gilman, Peter A.; Fox, Peter Bibcode: 1996AAS...188.6916G Altcode: 1996BAAS...28..938G Below the convection zone, where the stratification is radiatively controlled, large-scale motions should be mainly horizontal, i.e. in spherical shells, due to the negative buoyancy radial displacements would experience. Watson (G.A.F.D. 16, 285, 1981) showed that the observed surface solar differential rotation is right at the boundary for instability to horizontal disturbances. Therefore, since helioseismology tells us the latitudinal differential rotation below the convection zone is less than the surface value, it should be stable. We show that in the presence of a broad, nonuniform toroidal field this differential rotation is unstable. This is true for a wide range of kinetic and magnetic energies of the unperturbed state, from well below equipartition, to values above it. The instability appears to occur only for longitudinal wave number 1. Its location in latitude depends on details of the magnetic field profile. Generally, the primary energy source for the instability is the differential rotation, but the toroidal field also contributes. The mechanism of energy release is the poleward transport of angular momentum, in a complex interplay between the perturbation Reynolds and Maxwell stresses. This instability may play a role in the solar dynamo, although being two- dimensional, it cannot produce a dynamo by itself. Mixing of angular momentum caused by the instability could allow achievement of equilibrium of the solar tachocline hypothesized by Spiegel and Zahn. Title: GONG Observations of Solar Surface Flows Authors: Hathaway, D. H.; Gilman, P. A.; Jones, H. P.; Kasher, J.; Simon, G. W.; GONG Nearly Steady Flows Team; GONG Magnetic Fields Team Bibcode: 1996AAS...188.5304H Altcode: 1996BAAS...28..903H Doppler velocity observations obtained by the GONG instruments directly measure the nearly steady flows in the solar photosphere. The Sun's differential rotation profile is accurately determined from single observations. This profile is well represented by a fourth order polynomial which includes a rapidly rotating equator and a slight north-south asymmetry. Rotation profiles averaged over 27 day rotations of the Sun are sufficient to reveal the torsional oscillation signal - weak, 5 m/s, jet-like features associated with the sunspot latitude activity belts. A meridional circulation with poleward flow of about 20 m/s is also found from single observations and its spatial structure is well determined. Several of the observed characteristics of the surface flows suggest the presence of large convection cells. The convection spectrum is measured and found to have peak power for cells with wavelengths of about 50,000 km but the spectrum extends to much larger wavelengths. Day-to-day variations in the observed structure of the differential rotation and meridional circulation profiles indicate the presence of large-scale, nonaxisymmetric velocity signals which may be of solar origin. Studies correlating the convective flow patterns on consecutive days also indicate the presence of large cellular patterns that rotate at the Sun's rotation rate. Title: The Dynamo Problem. (Book Reviews: Lectures on Solar and Planetary Dynamos; Solar and Planetary Dynamos) Authors: Gilman, Peter Bibcode: 1995Sci...269..860P Altcode: No abstract at ADS Title: Book Review: Lectures on solar and planetary dynamos [invited papers] / Cambridge U Press, 1995 Authors: Gilman, P. Bibcode: 1995Sci...269Q.860G Altcode: 1995Sci...269Q.860P No abstract at ADS Title: Book-Review - Solar and Planetary Dynamos Authors: Proctor, M. R. E.; Matthews, P. C.; Rucklidge, A. M.; Gilman, P. Bibcode: 1995Sci...269R.860P Altcode: No abstract at ADS Title: What Can We Learn About Solar Cycle Mechanisms from Observed Velocity Fields? Authors: Gilman, Peter A. Bibcode: 1992ASPC...27..241G Altcode: 1992socy.work..241G No abstract at ADS Title: The solar dynamo. Authors: Deluca, E. E.; Gilman, P. A. Bibcode: 1991sia..book..275D Altcode: The authors discuss the present state of our understanding of the origin of the Sun's magnetic field. They begin with an introduction to the theory of magnetic field generation in rotating, conducting fluids. Next they consider a dilemma that has persisted for some 15 yr, namely the inconsistency of the kinematic and dynamic convection zone dynamo models. A resolution of this dilemma has been suggested by the recent helioseismology observations on the rotation rate as a function of latitude and depth. These observations, together with other observational and theoretical constraints, suggest that the solar dynamo operates not in the convection zone, but rather in a thin layer between the convection zone and the radiative interior. A model of such a dynamo is presented and discussed. The authors conclude by discussing the problems posed by the placement of the dynamo below the convection zone. Title: The Physics of an Interface Dynamo Authors: Deluca, E. E.; Gilman, P. A. Bibcode: 1989BAAS...21R.842D Altcode: No abstract at ADS Title: Angular Momentum Transport and Dynamo Action in the Sun: Implications of Recent Oscillation Measurements Authors: Gilman, Peter A.; Morrow, Cherilynn A.; Deluca, Edward E. Bibcode: 1989ApJ...338..528G Altcode: The implications of a newly proposed picture of the sun's internal rotation (Brown et al., 1989; Morrow, 1988) for the distribution and transport of angular momentum and for the solar dynamo are considered. The new results, derived from an analysis of solar acoustic oscillations, affect understanding of how momentum is cycled in the sun and provide clues as to how and where the solar dynamo is driven. The data imply that the only significant radial gradient of angular velocity exists in a transitional region between the bottom of the convection zone, which is rotating like the solar surface, and the top of the deep interior, which is rotating rigidly at a rate intermediate between the equatorial and polar rates at the surface. Thus the radial gradient must change sign at the latitude where the angular velocity of the surface matches that of the interior. These inferences suggest that the cycle of angular momentum that produces the observed latitudinal differential rotation in the convection zone may be coupled to layers of the interior beneath the convection zone. Title: SYSTEMATIC OBSERVATIONS OF THE SUN (In honour of Helen Dodson Prince): Modeling implications Authors: Wilson, P.; Gilman, P. Bibcode: 1989HiA.....8..675W Altcode: No abstract at ADS Title: Angular momentum transport and dynamo action in the Sun: a report on implications of a recent heliospheric estimate of solar rotation. Authors: Morrow, C. A.; Gilman, P. A.; Deluca, E. E. Bibcode: 1988ESASP.286..109M Altcode: 1988ssls.rept..109M Gilman, Morrow and DeLuca have recently introduced some ideas about how an emerging helioseismic picture of the Sun's internal rotation might affect the understanding of angular momentum transport and turbulent dynamic action in the solar interior. The present paper offers a brief report and commentary on these issues. Analysis of the frequency splittings of solar acoustic oscillations measured by Brown and Morrow has suggested that the latitudinal differential rotation observed at the surface persists throughout the convection zone, and that the rotation is more uniform in the outer layers of the radiative interior. Taking this picture at face value, the authors discuss qualitatively how angular momentum must be transported in order to sustain the observed rotation, the exchange of angular momentum between the convection zone and radiative interior, and the action of a turbulent dynamo which might be operating near the base of the convection zone. This new qualitative scenario for the solar dynamo explains both the equatorward migration of the zone where sunspots appear and the observed poleward migration of non-sunspot magnetic field. Title: Distribution of Sunspot Umbral Areas: 1917--1982 Authors: Bogdan, T. J.; Gilman, Peter A.; Lerche, I.; Howard, Robert Bibcode: 1988ApJ...327..451B Altcode: Over 24,000 measurements of individual sunspot umbral areas taken from the Mount Wilson white-light plate collection covering the period 1917-1982 are used to determine the relative size distribution of sunspot umbras. In the range 1.5-141 millionths of a solar hemisphere, the sunspot umbral areas are found to be distributed lognormally. Moreover, the same distribution is obtained for all phases of the solar cycle (maximum, minimum, ascending, descending), as well as for various individual cycles, between 1917 and 1982. Both the mean and the geometric logarithmic standard deviation of this distribution appear to be intrinsically constant over the entire data set; only the number of spots exhibits the familiar solar cycle variations. If the observed lognormal umbral size distribution is not a particular attribute of the sunspot umbras but is instead of a more fundamental property of emerging magnetic flux, then the data would predict a maximum in the size spectrum of photospheric magnetic structures for flux tubes with radii in the range 500-800 km. The absence of solar cycle variations in the relative distribution of umbral areas and especially the lognormal character of this distribution may both argue for the fragmentation of magnetic elements in the solar envelope. Title: Implications of Recent Estimates of Solar Interior Rotation for Angular Momentum Transport and Dynamo Action in the Sun Authors: Gilman, P. A.; Morrow, C. A.; Deluca, E. E. Bibcode: 1988BAAS...20..701G Altcode: No abstract at ADS Title: Dynamo theory for the interface between the convection zone and the radiative interior of a star part Authors: Deluca, Edward E.; Gilman, Peter A. Bibcode: 1988GApFD..43..119D Altcode: We discuss numerical solutions of nonlinear equations that model magnetic field generation in a thin layer beneath the convection zone of a late type star. The model equations were derived previously in Paper I (DeLuca and Gilman, 1986b). Three main results are found: first, the oscillating, dynamo wave solutions discussed in DeLuca and Gilman (1986a) are shown to be a result of the severe truncation used in those calculations; second, the induced velocity feld is shown to have an important role in determining the spatial structure of the magnetic field solutions; time dependent solutions have been found. These are not wave-like solutions, rather the amplitude of different horizontal wave modes vary in time. Further, we show that the exact solutions found in Paper I are generally unstable, with the exception of those that are independent of (latitude in our Cartesian geometry), which are stable if the transient induced velocity field remains small. We conclude that the induced velocity fields are an important ingredient in any model of dynamo action below the solar convection zone. Title: Inertial Oscillations in the Solar Convection Zone. III. A Cylidrical Model for Nonaxisymmetric Oscillations in a Superadiabatic Gradient Authors: Gilman, Peter A. Bibcode: 1987ApJ...318..904G Altcode: The effects of a superadiabatic gradient and differential rotation upon inertial oscillations that are nonaxisymmetric about the axis of rotation are examined as a generalization of earlier calculations published by Gilman and Guenther in 1985 for a cylindrical model of a stellar convection zone. It is shown that the frequencies of these oscillations are sensitive to both superadiabatic gradient and differential rotation. If such oscillations can be detected on the sun, then they could be used to measure the degree of superadiabaticity of the convection zone, and also to distinguish fine differences in the radial gradient of angular velocity, for example, between rotation independent of depth, and one constant on cylinders parallel to the rotation axis, matched to the surface profile of latitudinal differential rotation, such as predicted for the sun by various global convection models. Title: Distribution of Sunspot Umbral Areas: 1917-1982 Authors: Bogdan, T. J.; Gilman, P. A.; Lerche, I.; Howard, R. Bibcode: 1987BAAS...19..924B Altcode: No abstract at ADS Title: Magnetic Field Generation in the Overshoot Region Beneath the Solar Convection Zone Authors: Deluca, E. E.; Gilman, P. A. Bibcode: 1987BAAS...19..945D Altcode: No abstract at ADS Title: The Influence of the Coriolis Force on Flux Tubes Rising through the Solar Convection Zone Authors: Choudhuri, Arnab Rai; Gilman, Peter A. Bibcode: 1987ApJ...316..788C Altcode: In order to study the effect of the Coriolis force due to solar rotation on rising magnetic flux, the authors consider a flux ring, azimuthally symmetric around the rotation axis, starting from rest at the bottom of the convection zone, and then follow the trajectory of the flux ring as it rises. If it is assumed that the flux ring remains azimuthally symmetric during its ascent, then the problem can be described essentially in terms of two parameters: the value of the initial magnetic field in the ring when it starts, and the effective drag experienced by it. For field strengths at the bottom of the convection zone of order 10,000 G or less, it is found that the Coriolis force plays a dominant role and flux rings starting from low latitudes at the bottom are deflected and emerge at latitudes significantly poleward of sunspot zones. Title: A laboratory model of planetary and stellar convection Authors: Hart, J. E.; Toomre, J.; Deane, A. E.; Hurlburt, N. E.; Glatzmaier, G. A.; Fichtl, G. H.; Leslie, F.; Fowlis, W. W.; Gilman, P. A. Bibcode: 1987STIN...8722108H Altcode: Experiments on thermal convection in a rotating, differentially-heated spherical shell with a radial buoyancy force were conducted in an orbiting microgravity laboratory. A variety of convective structures, or planforms, were observed depending on the magnitude of the rotation and the nature of the imposed heating distribution. The results are in agreement with numerical simulations that can be conducted at modest parameter values, and suggest possible regimes of motion in rotating planets and stars. Title: Laboratory Experiments on Planetary and Stellar Convection Performed on Spacelab 3 Authors: Hart, J. E.; Toomre, J.; Deane, A. E.; Hurlburt, N. E.; Glatzmaier, G. A.; Fichtl, G. H.; Leslie, F.; Fowlis, W. W.; Gilman, P. A. Bibcode: 1986Sci...234...61H Altcode: Experiments on thermal convection in a rotating, differentially heated hemispherical shell with a radial buoyancy force were conducted in an orbiting microgravity laboratory. A variety of convective structures, or planforms, were observed, depending on the magnitude of the rotation and the nature of the imposed heating distribution. The results are compared with numerical simulations that can be conducted at the more modest heating rates, and suggest possible regimes of motion in rotating planets and stars. Title: Meridional Motions of Sunspots and Sunspot Groups Authors: Howard, R.; Gilman, P. A. Bibcode: 1986ApJ...307..389H Altcode: Mount Wilson white-light plate data for north-south sunspot motions are studied, taking both sunspot groups and individual spots into consideration. The average results as a function of latitude show a midlatitude northward flow and an amplitude of a few hundredths of a degree per day in each hemisphere. For sunspot groups, a dependence on latitude is seen that tends generally toward more poleward motions at higher latitudes. A previously reported, systematic variation of the latitude dependence of the meridional motion of sunspot groups with phase in the solar cycle is not confirmed. Title: Nonlinear Convection of a Compressible Fluid in a Rotating Spherical Shell Authors: Gilman, P. A.; Miller, J. Bibcode: 1986ApJS...61..585G Altcode: The deeper layers of the solar or stellar convection zone are modeled in view of extensive calculations of nonlinear convection of a compressible fluid in a rotating spherical shell, using the anelastic approximation for the integrated equations. Solar values are used for the luminosity, rotation, and eddy diffusivities; the Rayleigh number for convection is then determined by the luminosity. In the cases of both model depths used, the dominant convection patterns in low latitudes are a broad spectrum of rolls with north-south axes, approximately symmetric about the equator, with more cellular patterns in high latitudes. The results obtained are similar to those of Glatzmaier (1984, 1985), who used substantially different solution techniques, and they agree with the differential rotation observed at the solar surface. Title: Rotation and Expansion within Sunspot Groups Authors: Gilman, P. A.; Howard, R. Bibcode: 1986ApJ...303..480G Altcode: By superposing data for many sunspot groups measured on the Mount Wilson white-light plate collection, the authors demonstrate that differential rotation, about equal to the ambient rate, occurs between sunspots within the group. It is also shown that the relative motions of leader and follower sunspots can be characterized primarily as a simple expansion of the group along its major axis, with very little rotation of the pattern about group center. Title: The Influence of the Coriolis Force on Flux Tubes Rising through Solar Convection Zone Authors: Choudhuri, A. R.; Gilman, P. A. Bibcode: 1986BAAS...18..703C Altcode: No abstract at ADS Title: Dynamo Theory for a Thin Layer Between the Convection Zone and the Radiative Zone of a Star. Formulation and Preliminary Results Authors: Deluca, Edward E.; Gilman, Peter A. Bibcode: 1986LNP...254..173D Altcode: 1986csss....4..173D No abstract at ADS Title: Dynamo Theory for the Sun and Stars Authors: Gilman, Peter A.; Deluca, Edward E. Bibcode: 1986LNP...254..163G Altcode: 1986csss....4..163G No abstract at ADS Title: The solar dynamo: observations and theories of solar convection, global circulation, and magnetic fields. Authors: Gilman, P. A. Bibcode: 1986psun....1...95G Altcode: Contents: Convective zone characteristics (basic structure of depth, oblateness and latitudinal temperature differences, luminosity and radius changes). Observations and theory of solar convection zone motions (granulation, supergranulation and mesogranulation, related theories of convection, global circulation). Phenomenology of the Sun's magnetic field and related features (flux tubes and network; sunspots, active regions, and global patterns; the solar cycle: manifestations in sunspots, magnetic fields, and other properties). Theories of the solar magnetic field (flux tubes and network, active regions and sunspots, solar dynamo theory). Title: Dynamo theory for the interface between the convection zone and the radiative interior of a star: Part I model equations and exact solutions Authors: Deluca, Edward E.; Gilman, Peter A. Bibcode: 1986GApFD..37...85D Altcode: In this paper we derive a set of equations which model magnetic field generation and maintenance in a thin region, ( 104)km thick, below the solar convection zone, and present some simple exact solutions. Energy to drive the dynamo is assumed to come from helical convection that overshoots into this region. Differential rotation and meridional circulation result only from feedbacks by the induced fields. The equations are derived for a homogeneous incompressible fluid in Cartesian geometry. The momentum equation, magnetic induction equation, and the continuity equation are included in the analysis. We assume velocity and magnetic field patterns have an aspect ratio of 1/10 (radial to horizontal scale), that the large scale velocities are smaller than the convection zone velocities, a few meters per sec, and the large scale magnetic fields are of the order of 104 Gauss. Finally we assume that the time scale of interest is the advective time scale. Using these assumptions, we derive governing equations in which the Coriolis force balances the Lorentz force, the pressure gradient force, and the viscosity, and in which the magnetic fields are maintained by the effect but significantly modified by the above velocity fields. The results of this model will be discussed in following papers. Title: Inertial oscillations in the solar convection zone. II - A cylindrical model for equatorial regions Authors: Gilman, P. A.; Guenther, D. B. Bibcode: 1985ApJ...296..685G Altcode: Axisymmetric inertial oscillations found from a simplified, cylindrical model applicable to equatorial latitudes are presented and compared to results obtained previously by numerical methods for a spherical shell. The asymptotic behavior of inertial mode frequencies are analytically derived in the limits of very large and very small axial wavenumbers, and it is shown that inertial oscillations arise only when the angular momentum per unit mass increases outward. The effects of convection zone depth and differential rotation on the different radial orders of inertial modes are demonstrated. In a convection zone 15 billion cm in depth, rotation rate increasing inward leads to decreased oscillation frequency for a given radial order and axial wavenumber, the opposite of the result for a spherical shell. Deeper zones in the cylindrical case give increased frequencies for rotation rate either decreasing or increasing inward. The differences in the results between spherical and cylindrical models are shown to be due to the different geometries. Title: Inertial oscillations in the solar convection zone. I - Spherical shell model Authors: Guenther, D. B.; Gilman, P. A. Bibcode: 1985ApJ...295..195G Altcode: The solar oscillation spectrum has the potential to provide a great deal of information about the sun's interior. However, an interpretation of the oscillation spectrum is an extremely difficult task. Neither direct nor indirect approaches can give a complete picture of the solar interior. It appears, therefore, advisable to employ a wide variety of methods to probe the sun's interior. In the present paper, a description is provided of a family of solar oscillations, which is called inertial oscillations. It is shown that these oscillations are sensitive (independently) to both the depth and the rotation profile of the convection zone. If these modes can be observed, they will provide an independent determination of the convection zone depth and rotation frequency profile, as a function of radius (and latitude). Title: Rotation rates of leader and follower sunspots Authors: Gilman, P. A.; Howard, R. Bibcode: 1985ApJ...295..233G Altcode: The rotation rates of leader and follower sunspots found on Mount Wilson white light plates have been measured for the years 1917-1983. It is found that at all latitudes, leader spots rotate faster than follower spots by about 0.1 deg per day, or 14 m/s. It is also found that, when examined separately, leaders and followers show the same variations in rotation with cycle phase as do all spots taken together, as reported earlier in Gilman and Howard (1984). Leaders and followers show similar variations in rotation rate even on an annual basis. Thus, while leaders and followers in each group diverge in longitude from each other at an average rate of about 0.1 deg per day, each is separately speeding up or slowing down its rotation according to the phase in the cycle, and by a similar amount. Leaders and followers also give about the same covariance of longitude and latitude motions, indicating that whole sunspot groups participate in tracing the apparent angular momentum transport toward the equator, as previously reported for all spots in Gilman and Howard (1984). Title: What would a dynamo theorist like to know about the dynamics of the solar convection zone? Authors: Gilman, P. A. Bibcode: 1984sses.nasa...41G Altcode: 1984sss..conf...41G Observations about the current status of solar dynamo theory are given. The induction equation for magnetic field is solved using assumed velocities and parametric representations of the inductive or diffusive effects of velocities. The equations of motion governing these flow are not solved in parallel. Results from global compressible convection models are discussed. Differential rotation and convection are also investigated. Title: Techniques for detecting giant cells using spatially resolved solar velocity data Authors: Brown, T. M.; Gilman, P. A. Bibcode: 1984ApJ...286..804B Altcode: Whether giant cells exist in the convection zone of the sun, and what their properties might be, are matters of great importance for the understanding of the dynamics of the solar interior. So far, such cells have escaped detection, probably because of the small amplitude of their associated velocity fields and the large amplitudes of shorter-lived flows. Techniques for improving the detectability of giant cells are presented. These methods are based on the spatial extent and symmetry properties of giant cells as seen in self-consistent dynamical models of the solar convection zone. Simulations suggest that these techniques allow detection of giant cells with photospheric rms velocities of 2 m/s (0.4 m/s per longitudinal wavenumber), given observations spanning about one year. Title: Inertial Oscillations in the Solar Convection Zone Part I Authors: Guenther, D. B.; Gilman, P. A. Bibcode: 1984BAAS...16Q1000G Altcode: No abstract at ADS Title: Inertial Oscillations in the Solar Convection Zone Part II: A Cylindrical Model for Equatorial Regions Authors: Gilman, P. A.; Guenther, D. B. Bibcode: 1984BAAS...16R1000G Altcode: No abstract at ADS Title: Variations in solar rotation with the sunspot cycle Authors: Gilman, P. A.; Howard, R. Bibcode: 1984ApJ...283..385G Altcode: The positions of sunspots as photographed in white light at Mt. Wilson from 1921-82 were analyzed to detect any systematic variations, particularly in relation to the solar cycle. The study analyzed 5 deg sq bins of sunspots for both hemispheres. The residual rotation rates were calculated for individual and grouped sunspots. Peaks in the sunspot rotation rate were detected near the solar maximum and minimum and in high solar latitudes 3 yr before the end of the cycle. The highest sunspot rotation peaks were 0.5 deg/day. The ubiquity of the rotation rate changes over the whole solar disk implied periodic angular momentum exchanges between the photosphere and deeper layers of the convective zone. Finally, the interpretations of the differences observed between variations in sunspot and Doppler rotation are discussed. Title: Rotation of the sun measured from Mount Wilson white-light images Authors: Howard, R.; Gilman, P. I.; Gilman, P. A. Bibcode: 1984ApJ...283..373H Altcode: The instrumentation, data and data reduction procedures used in white light observations of sunspot rotation rates are described. The study covered 62 yr of rotation observations. The data were all gathered using the same Mt. Wilson telescope, which has had three different main lenses in the interval 1981-82. Details of the exposure calibration and lens operation procedures are provided. The data were treated in terms of eight evenly space determinations of the solar limb and account was taken of all sunspots within 60 deg of the central meridian. Spot movements were traced in terms of groups of contiguous individual spots. Large spots rotated slower than small spots, a condition attributed to greater viscous drag in the larger flux tubes in the photosphere. The data tend to confirm theories that the photospheric gas revolves at a different rate than the sunspots. Title: On the Correlation of Longitudinal and Latitudinal Motions of Sunspots Authors: Gilman, P. A.; Howard, R. Bibcode: 1984SoPh...93..171G Altcode: Using new measurements of positions of individual sunspots and sunspot groups obtained from 62 years of the Mt. Wilson white-light plate collection, we have recomputed the correlation between longitude and latitude motion. Our results for groups are similar to those of Ward (1965a) computed from the Greenwich record, but for individual spots the covariance is reduced by a factor of about 3 from the Ward values, though still of the same sign and still statistically significant. We conclude that there is a real correlation between longitude and latitude movement of individual spots, implying angular momentum transport toward the equator as inferred by Ward. The two thirds reduction in the covariance for individual spots as opposed to groups is probably due to certain properties of spot groups, as first pointed out in an unpublished manuscript by Leighton. Title: The Rotation of the Sun from Mount Wilson Sunspot Measurements Authors: Howard, R. F.; Gilman, P. A. Bibcode: 1984KodOB...4....1H Altcode: The authors have completed the measurement and reduction of 62 years of white-light solar images taken at the Mount Wilson Observatory. The data have been analyzed for differential rotation and time variations of this quantity. This is a brief review of the work. Title: Dynamically consistent nonlinear dynamos driven by convection in a rotating spherical shell. II - Dynamos with cycles and strong feedbacks Authors: Gilman, P. A. Bibcode: 1983ApJS...53..243G Altcode: Nonlinear dynamos with cycles and strong feedback are investigated theoretically. The convectively driven dynamo model of Gilman and Miller (1981) is applied to cases with viscosity and thermal diffusivity reduced by a factor of 10, so that weaker global convection drives an equatorial acceleration of solar magnitude and profile, and differential rotation (DR) makes up nearly 80 percent of the system kinetic energy. The schematics of the dynamo process are introduced, and the design of the calculations is outlined. The results are discussed in terms of the onset of dynamo action, the phenomenology of cyclic solutions, symmetry preservation and switching, total energy statistics, the maintenance of convection and DR against dissipation and feedbacks, changes in DR profiles, maintenance of toroidal and poloidal fields, kinetic and magnetic energy spectra, and applications to the sun and stars. It is found that the model gives cyclic dynamos with periods one order of magnitude shorter than that of the sun. Drawings and graphs illustrating the findings are provided. Title: Compressible Convection in a Deep Rotating Spherical Shell Authors: Gilman, P. A. Bibcode: 1983BAAS...15..715G Altcode: No abstract at ADS Title: Variations in Sunspot Rotation and the Activity Cycle Authors: Howard, R.; Gilman, P. A. Bibcode: 1983BAAS...15..698H Altcode: No abstract at ADS Title: Dynamos of the sun and stars, and associated convection zone dynamics Authors: Gilman, P. A. Bibcode: 1983IAUS..102..247G Altcode: Results obtained from a fully 3D nonaxisymmetric nonlinear MHD model of convection in a deep spherical shell initially in uniform rotation and heated uniformly from the inside, with central gravity and constant viscosity and thermal conductivity, are summarized and discussed as they apply to solar and stellar dynamos. The constraints imposed by solar observations are examined, the model results are presented graphically and contrasted with the observations, and general implications for the types of stellar dynamo action to be expected are indicated. Cyclic dynamo action is located primarily at low latitudes outside the tangent cylinder to the inner boundary of the convection zone; this prediction leads to line-of-sight-dependent differences in the observability of cyclic-dynamo-induced changes in Ca II emissions. Title: Compressible convection in a rotating spherical shell. V - Induced differential rotation and meridional circulation Authors: Glatzmaier, G. A.; Gilman, P. A. Bibcode: 1982ApJ...256..316G Altcode: Solutions are described for the mean differential rotation and meridional circulation in a compressible, rotating, spherical fluid shell which are induced by linear, anelastic solutions for global convection. The mean solutions strongly depend on the density stratification, the rotation rate, the convective velocity distribution, and the amount of viscous diffusion relative to thermal diffusion. It is noted that at least one of two conditions must be met in order to obtain an equatorial acceleration that is large in amplitude relative to the meridional circulation (together with a small equator-pole temperature difference) when the density stratification is as large as in the solar convection zone. Either the effect of rotation must be large compared with the effects of viscous diffusion and buoyancy or viscous diffusion must be small relative to thermal diffusion. In either case, the angular velocity increases with depth in the upper part of the convection zone but decreases with depth and is nearly constant on cylinders in the lower part when the global convection reaches deep layers. Title: Convective dynamos for rotating stars. Authors: Gilman, P. A. Bibcode: 1982SAOSR.392A.165G Altcode: 1982csss....2..165G Global dynamo theory is applied to the problem of why some stars have field reversing dynamos, and others do not. It is argued that convectively driven dynamos are the most likely source of magnetic fields in stars that have convection zones. Title: Compressible convection in a rotating spherical shell. IV - Effects of viscosity, conductivity, boundary conditions, and zone depth Authors: Glatzmaier, G. A.; Gilman, P. A. Bibcode: 1981ApJS...47..103G Altcode: The manner in which variations in given assumptions affect the stability and structure of solutions is demonstrated for the case of a linear, anelastic model of compressible convection in a rotating spherical fluid shell described by Glatzmaier and Gilman (1981). Velocity is shown to be enhanced in the lower part of the zone opposed to the upper part, when the kinematic viscosity is assumed to be inversely proportional to the density rather than constant. The velocity is more uniformly distributed in radius when the kinematic viscosity increases with radius at a rate lower than the inverse density, or when viscous diffusion is small relative to thermal diffusion. When constant temperature boundaries are replaced by constant diffusive heat flux boundaries, the most unstable modes are more unstable and have larger longitudinal dimensions; this is in agreement with previous studies of plane-parallel, Boussinesq convection. Title: An improved search for large-scale convection cells in the solar atmosphere Authors: Labonte, B. J.; Howard, R.; Gilman, P. A. Bibcode: 1981ApJ...250..796L Altcode: A reanalysis of Mount Wilson solar velocity observations was made to search for giant cellular patterns. The reanalysis avoids several errors made in a previous search. No cells are detected with sensitivity of 3 to 12 m/s depending upon wavenumber. The observed amplitudes do not conflict with recent model predictions. Title: Global circulation and the solar dynamo. Authors: Gilman, P. A. Bibcode: 1981NASSP.450..231G Altcode: 1981suas.nasa..231G The Sun is apparently rather typical of stars in its spectral class, with a convection zone of substantial depth and a modest rotation rate. These two factors are apparently enough to generate a substantial global circulation, seen so far principally as a differental rotation, as well as a nearly cyclic magnetohydrodynamic dynamo, seen principally as the 22 year "solar cycle". It would be expected therefore that many stars would have such dynamical characteristics. Recent observations of very large scale velocity fields of small velocity amplitude were reviewed. Title: Dynamically consistent nonlinear dynamos driven by convection in a rotating spherical shell Authors: Gilman, P. A.; Miller, J. Bibcode: 1981ApJS...46..211G Altcode: Calculations are presented of a convectively driven hydromagnetic dynamo in a rotating spherical shell in which all of the induction effects arise from the same, self-consistent solutions of the convection equations, and the results obtained are compared with observations of the solar dynamo. A hydromagnetic model for nonlinear convection in a rotating spherical shell is used in which it is possible to solve simultaneously for the evolving velocity and magnetic field and include the full feedbacks of the magnetic field on the motion. The solutions are obtained in stages, with the preliminary perturbation of a state of solid rotation to develop a spectrum of convection and differential rotation, followed by the introduction of a small-seed magnetic field of various magnetic Prandtl numbers to induce dynamo behavior. Although the calculated differential rotation is found to be similar to the observed equatorial acceleration of the sun, the calculated dynamo behaves quite differently, with no global magnetic field reversals, equatorial migration of the toroidal magnetic field or preferred symmetry of the induced magnetic fields, which is attributed to an excessively large model helicity. It is concluded that there as yet does not exist a solar dynamo model which both reproduces the observational characteristics of the solar dynamo and is consistent with the laws of fluid dynamics. Title: Differntial Rotation Induced by Compressible Convection Authors: Glatzmaier, G. A.; Gilman, P. A. Bibcode: 1981BAAS...13..527G Altcode: No abstract at ADS Title: More on Dynamos Driven by Global Convection and Differential Rotation Authors: Gilman, P. A. Bibcode: 1981BAAS...13..907G Altcode: No abstract at ADS Title: Compressible Convection in a Rotating Spherical Shell - Part Two - a Linear Anelastic Model Authors: Glatzmaier, G. A.; Gilman, P. A. Bibcode: 1981ApJS...45..351G Altcode: We study the onset of convection for a compressible fluid in a rotating spherical shell via linear anelastic fluid equations for a depth of 40% of the radius, constant kinematic viscosity and thermometric diffusivity, Taylor numbers up to l05, and density stratifications up to seven e-folds across the zone. The perturbations are expanded in spherical harmonics, and the radially dependent equations are solved with a Newton-Raphson relaxation method.

The most unstable modes are single cells extending from the bottom to the top of the convection zone. As the density stratification and rotation rate increase, the horizontal dimension of these cells decreases, while their prograde longitudinal phase velocity and the enhancement of the velocity near the top of the zone increase. Cylindrical cells arranged symmetrically about the equator develop parallel to the rotation axis as the rotation rate increases, but for large stratifications, their axes bend toward the pole at midlatitude instead of intersecting the outer surface, as they do for small stratifications. The buoyancy force does negative work near the top of the zone, while the pressure force does a net positive amount of work for large stratifications. Helicity profiles and convective heat flux profiles for the most unstable modes do not change significantly as the density stratification increases, although distinct differences develop for modes that are not the most unstable. The radial momentum flux for the most unstable modes at high rotation rates is inward for large stratifications instead of outward as it is for small stratifications; this may have a significant effect on differential rotation and magnetic field generation in nonlinear, compressible models. Title: Compressible Convection in a Rotating Spherical Shell - Part Three - Analytic Model for Compressible Vorticity Waves Authors: Glatzmaier, G. A.; Gilman, P. A. Bibcode: 1981ApJS...45..381G Altcode: A simple analog to compressible convection in a rotating spherical shell is described and solved analytically to enable us to understand the basic physics of the prograde phase velocities that resulted from the numerical calculations of Paper II. The analog is for an inviscid, adiabatically stratified, rotating, equatorial annulus of gas for which a form of potential vorticity is conserved. Linear perturbations in this system take the form of vorticity waves which propagate prograde relative to the rotating reference frame as long as the density decreases outward. Their frequencies depend on the longitudinal wavenumber and density stratification in the same way as the convective modes of Paper II. Simple physical arguments explain these dependences. Title: Compressible convection in a rotating spherical shell. I - Anelastic equations. II - A linear anelastic model. III - Analytic model for compressible vorticity waves Authors: Gilman, P. A.; Glatzmaier, G. A. Bibcode: 1981ApJS...45..335G Altcode: In order to develop equations whose solution will clarify the role played by large solar convection zone density variations in differential rotation transports, anelastic equations for convection of a compressible fluid in a deep, rotating spherical shell are derived in the first part of the study. The model equations represent a generalization of a Boussinesq system that has been studied extensively with the solar differential rotation problem in mind, and are expected to apply best in the deep part of a convection zone where departures of the fluid from an adiabatic atmosphere are smallest. The second part of the study focuses on the onset of convection for a compressible fluid in a rotating spherical shell via linear inelastic fluid equations for a depth of 40% of the radius, constant kinematic viscosity and thermometric diffusivity, Taylor numbers up to 100,000, and density stratifications up to seven e-folds across the zone. The perturbations are expanded in spherical harmonics, and the radially dependent equations are solved with a Newton-Raphson relaxation method. Title: Compressible convection in a rotating spherical shell. Authors: Glatzmaier, G. A.; Gilman, P. A. Bibcode: 1981ASIC...68..145G Altcode: 1981spss.conf..145G No abstract at ADS Title: Effects of certain analysis procedures on solar global velocity signals Authors: Gilman, P. A.; Glatzmaier, G. A. Bibcode: 1980ApJ...241..793G Altcode: The effects of data reduction procedures on the signals obtained for global solar velocities from the Doppler magnetograph at Mount Wilson are examined. The data reduction procedure used by Howard (Howard et al., 1980; LaBonte and Howard, 1979; Howard and LaBonte, 1980) is applied to idealized global velocity data, and it is shown that the effects of removing daily rotation, ears and zero offset signals and the construction of synoptic maps can easily reduce an original periodic east-west flow velocity of peak amplitude 100 m/sec to 10 m/sec or less for any wavenumber. Furthermore, it is shown that a velocity spectrum obtained from a nonlinear spherical convection model is attenuated to rms residual velocities close to or within the upper limits of Howard and LaBonte. It is concluded that there could be a substantially larger global velocity signal present in the Mount Wilson magnetograph data than has so far been demonstrated, and data reduction and observational techniques for obtaining these signals are suggested. Title: Axisymmetric Convection Driven by Latitudinal Temperature Gradients in Rotating Spherical Shells. Authors: Hathaway, D. H.; Gilman, P. A.; Miller, J.; Toomre, J. Bibcode: 1980BAAS...12..686H Altcode: No abstract at ADS Title: Erratum - Model Calculations Concerning Rotation at High Solar Latitudes and the Depth of the Solar Convection Zone Authors: Gilman, P. A. Bibcode: 1980ApJ...236..706G Altcode: No abstract at ADS Title: Dynamics of the Solar Interior and the Solar Dynamo Authors: Gilman, P. A. Bibcode: 1980NASCP2098....3G Altcode: 1980sscs.nasa....3G The solar convection zone is the origin of most of the variations in solar output observed or suspected to occur. The Sun's magnetic field is rooted there, and solar activity and the solar cycle are generated and maintained there. Changes in the magnetic fields which reach into the solar atmosphere and beyond to interplanetary space are largely determined by the dynamo action of velocity fields in the convection zone. If changes in solar luminosity occur on time scales of months to millenia, such changes probably have their origin in the changing dynamics of the convection zone, either as cause of or in response to long term changes in the level of solar activity. Fluctuations would occur in the rate at which energy is brought to the surface by convection, and the solar diameter would be slightly modified. To describe and ultimately understand the global workings of the solar dynamo requires simultaneous high quality photospheric observations of solar velocities, magnetic fields, intensity patterns, luminosity and various radiative outputs. The observations must be nearly continuous in time and of long duration-most or all of a solar cycle. Such a measurement program should be a major part of the proposed Solar Cycle and Dynamics Mission. Title: Convective instability when the temperature gradient and rotation vector are oblique to gravity. II. Real fluids with effects of diffusion Authors: Hathaway, D. H.; Toomre, J.; Gilman, P. A. Bibcode: 1980GApFD..15....7H Altcode: The linear stability analysis of Hathaway, Gilman and Toomre (1979) (hereafter referred to as Paper I) is repeated for Boussinesq fluids with viscous and thermal diffusion. As in Paper I the fluid is confined between plane parallel boundaries and the rotation vector is oblique to gravity. This tilted rotation vector introduces a preference for roll-like disturbances whose axes are oriented north-south; the preference is particularly strong in the equatorial region. The presence of a latitudinal temperature gradient produces a thermal wind shear which favors axisymmetric convective rolls if the gradient exceeds some critical value. For vanishingly small diffusivities the value of this transition temperature gradient approaches the inviscid value found in Paper I. For larger diffusivities larger gradients are required particularly in the high latitudes. These results are largely independent of the Prandtl number. Diffusion tends to stabilize the large wavenumber rolls with the result that a unique wavenumber can be found at which the growth rate is maximized. These preferred rolls have widths comparable to the depth of the layer and tend to be broader near the equator. The axisymmetric rolls are similar in many respects to the cloud bands on Jupiter provided they extend to a depth of about 15,000 km. Title: Global Circulation of the Sun: Where Are We and Where Are We Going? Authors: Gilman, P. A. Bibcode: 1980HiA.....5...91G Altcode: No abstract at ADS Title: Differential rotation in stars with convection zones Authors: Gilman, P. A. Bibcode: 1980LNP...114...19G Altcode: 1980sttu.coll...19G; 1980IAUCo..51...19G The paper discusses quantitatively and qualitatively, differential rotation at the surface of stars that rotate and have a convection zone. Convection zone parameters are considered, particularly the ratio of convection turnover time to the stellar rotation time. In addition, the redistribution of angular momentum by axisymmetric meridional circulation and by eddies is discussed. Some models that have been developed to explain differential rotation in the sun are reviewed, particularly models developed for nonaxisymmetric convection of a stratified liquid in a rotating spherical shell; results from calculations indicate that finite amplitude equatorial acceleration is produced only when the influence of rotation upon convection is strong. Qualitative statements about main sequence stars and red giants are then extrapolated from the results, including the estimation of the turnover time for the convection zone from a mixing length model applied to the whole convection zone. Title: Comment (and Response) on Paper "Nonlinear Dynamics of Boussinesq Convection in a Deep Rotating Shell. III. Effects of Velocity Boundary Conditions" by P.A. Gilman Authors: Busse, F. H.; Gilman, P. A. Bibcode: 1980GApFD..14..251B Altcode: No abstract at ADS Title: Convective Instability in Rotating Layers with Thermal Winds and Application to Jupiter Authors: Hathaway, D. H.; Gilman, P. A.; Toomre, J. Bibcode: 1979BAAS...11Q.618H Altcode: No abstract at ADS Title: Model calculations concerning rotation at high solar latitudes and the depth of the solar convection zone. Authors: Gilman, P. A. Bibcode: 1979ApJ...231..284G Altcode: The author has previously carried out extensive nonlinear numerical simulations of convection in a rotating spherical shell, motivated by the problem of understanding the Sun's differential rotation. The polar vortex found in earlier calculations, but not observed on the Sun, disappears when shell depths as large as 40% of the radius are assumed. The equatorial acceleration is also enhanced, producing a closer fit to solar observations than previous calculations for depths of 20%. The reasons for the disappearance of the polar vortex are that in deeper layers the Reynolds stresses, which transport angular momentum toward the equator to form the equatorial acceleration, reach to higher latitudes, and the moment of inertia of the polar cap region is a smaller fraction of the total for the shell. Although the model used is for a stratified liquid (the Boussinesq approximation), we argue that deep layers are less likely to have a polar vortex than shallow ones in the compressible case too. This result favors a solar convection zone substantially deeper than previously inferred from stellar structure calculations applied to the Sun, but is consistent with recent inferences of a deep convection zone made from measurements of solar oscillations. Subject headings: convection - Sun: interior - Sun: rotation Title: Angular velocity gradients in the solar convection zone. Authors: Gilman, P. A.; Foukal, P. V. Bibcode: 1979ApJ...229.1179G Altcode: Numerical calculations of Boussinesq nonaxisymmetric convection in a rotating spherical shell are reported which were performed to study how convection in the supergranule layer redistributes angular momentum. It is found that supergranules are at best weakly influenced by rotation and can be largely responsible for the radial gradient of angular velocity observed in the thin supergranule layer below the photosphere. The results indicate that convection in a thin spherical shell weakly influenced by rotation can produce a substantial outward decrease in rotational velocity that approaches the limit predicted for radially moving particles that conserve their angular momentum. This phenomenon is shown to provide a plausible explanation for the observed difference in angular velocity between sunspots and the photospheric plasma. Title: Convective Instability when the Temperature Gradient and Rotation Vector are Oblique to Gravity. I. Fluids without Diffusion Authors: Hathaway, D. H.; Toomre, J.; Gilman, P. A. Bibcode: 1979GApFD..13..289H Altcode: No abstract at ADS Title: Long-term variations in the solar dynamo Authors: Gilman, P. A. Bibcode: 1979LPICo.390...39G Altcode: No abstract at ADS Title: Response Authors: Gilman, Peter A. Bibcode: 1979GApFD..14..252G Altcode: This Article does not have an abstract. Title: A review of: "Rotating fluids in geophysics" Authors: Gilman, Peter A. Bibcode: 1979GApFD..12..183G Altcode: By P. H. Roberts and A. M. Soward (Editors). Academic Press Inc. (London) Ltd, $36.25 (£17.50). (ISBN 0 12 589650 6.) Title: Polar Deceleration and Convection Zone Depth for the Sun. Authors: Gilman, P. A. Bibcode: 1978BAAS...10Q.400G Altcode: No abstract at ADS Title: Nonlinear dynamics of boussinesq convection in a deep rotating spherical shell III: Effects of velocity boundary conditions Authors: Gilman, Peter A. Bibcode: 1978GApFD..11..181G Altcode: We perform numerical experiments for convection in a rotating spherical shell with either one or two nonslip boundaries, for the same Rayleigh and Taylor numbers (R = 104 to 105, T = 105) and the same temperature boundary conditions as used in Part II. We find that both convection and differential rotation energies are reduced compared with solutions with stress free boundaries, but the differential rotation is reduced by a much larger factor, demonstrating the constraining influence of rigidly rotating boundaries on the angular momentum profile. Differential rotation ceases to be a prominent, global feature of the flow, and may be difficult to observe in laboratory experiments. The convection spectrum with nonslip boundaries is also broader, and shifts much less toward low longitudinal wave numbers with increased R. What differential rotation remains is driven primarily by Coriolis torques from the axisymmetric meridional circulation, rather than by Reynolds stresses. For R5 × 104, the bouyancy driven axisymmetric meridional circulation is substantially larger than for stress free boundaries, due to Ekman boundary layers breaking the rotational constraints which otherwise suppress this inherently nongeostrophic flow. In the same Rayleigh number range, a greater fraction of the total heat flux is carried by convection than with stress free boundaries, also a result of the destabilizing influence of the Ekman layers.
Solutions with stress free top and nonslip bottom behave similarly to the stress free top and bottom case at low Rayleigh number, because the convection occurs mostly outside the cylinder tangent to the inner boundary equator, so that the inner velocity boundary condition is not strongly felt. As R is increased, the convection and differential rotation feel this boundary much more strongly.

Despite the many differences, the convection solutions for different boundary conditions have a number of similarities. These include location of peak convection amplitudes (the equator with a secondary peak at high latitudes), north-south roll orientation near the equator, Reynolds stress patterns, and the form of axisymmetric meridional circulation. Of particular interest is that the helicity profile of the convection is similar for all boundary conditions, but is of larger amplitude with nonslip boundaries.

Our results suggest that differential rotation is likely to be much different, and of smaller amplitude relative to the convection which drives it, in a liquid planetary interior as compared to a stellar convection zone. This difference may result in favoring planetary dynamos of the α2 variety, and stellar dynamos of the α-ω type. These speculations need to be tested by model calculations with magnetic field included. Title: Nonlinear Dynamics of Boussinesq Convection in a Deep Rotating Spherical Shell. II. Effects of Temperature Boundary Conditions Authors: Gilman, P. A. Bibcode: 1978GApFD..11..157G Altcode: No abstract at ADS Title: Abstracts to forthcoming papers Authors: Gilman, Peter A. Bibcode: 1978GApFD..11..155G Altcode: This Article does not have an abstract. Title: Anomalous Solar Rotation in the Early 17th Century Authors: Eddy, J. A.; Gilman, P. A.; Trotter, D. E. Bibcode: 1977Sci...198..824E Altcode: The character of solar rotation has been examined for two periods in the early 17th century for which detailed sunspot drawings are available: A.D. 1625 through 1626 and 1642 through 1644. The first period occurred 20 years before the start of the Maunder sunspot minimum, 1645 through 1715; the second occurred just at its commencement. Solar rotation in the earlier period was much like that of today. In the later period, the equatorial velocity of the sun was faster by 3 to 5 percent and the differential rotation was enhanced by a factor of 3. The equatorial acceleration with declining solar activity is in the same sense as that found in recent Doppler data. It seems likely that the change in rotation of the solar surface between 1625 and 1645 was associated with the onset of the Maunder Minimum. Title: A note on estimating the latitudinal angular momentum transport in the solar photosphere from Doppler velocities. Authors: Gilman, P. A. Bibcode: 1977A&A....58..315G Altcode: If it is assumed that global-scale motions on the sun are predominantly horizontal and their patterns persist largely unchanged during passage from the east to the west limb, there is a relationship between the Doppler signal and the latitudinal angular-momentum transport which can in principle be tested from observations. This relationship implies that if there is a significant angular-momentum flux towards the equator from higher latitudes in each hemisphere, as predicted by convection theory for maintaining the equatorial acceleration, the mean-square Doppler-velocity signal must be larger east of the central meridian than west of it at the same longitude, in both the northern and southern hemispheres. The difference should be largest at middle latitudes in each hemisphere. Title: Nonlinear Dynamics of Boussinesq Convection in a Deep Rotating Spherical Shell. I. Authors: Gilman, P. A. Bibcode: 1977GApFD...8...93G Altcode: No abstract at ADS Title: The Theory of Solar Rotation Authors: Gilman, Peter A. Bibcode: 1977lsms.proc...87G Altcode: No abstract at ADS Title: Coronal holes and the sun's interior. Authors: Gilman, P. A. Bibcode: 1977chhs.conf..331G Altcode: No abstract at ADS Title: Abstracts to forthcoming papers Authors: Gilman, Peter A. Bibcode: 1977GApFD...8...91G Altcode: This Article does not have an abstract. Title: Progress in Modeling Convection in Deep Rotating Spherical Shells, with Implications for Solar Differential Rotation Theory Authors: Gilman, P. A. Bibcode: 1976BAAS....8..299G Altcode: No abstract at ADS Title: Solar rotation during the Maunder minimum. Authors: Eddy, J. A.; Gilman, P. A.; Trotter, D. E. Bibcode: 1976SoPh...46....3E Altcode: We have measured solar surface rotation from sunspot drawings made in a.d. 1642-1644 and find probable differences from present-day rates. The 17th century sunspots rotated faster near the equator by 3 or 4%, and the differential rotation between 0 and ±20° latitude was enhanced by about a factor 3. These differences are consistent features in both spots and groups of spots and in both northern and southern hemispheres. We presume that this apparent change in surface rotation was related to the ensuing dearth of solar activity (the Maunder Minimum) which persisted until about 1715. Title: Theory of Convection in a Deep Rotating Spherical Shell, and its Application to the Sun Authors: Gilman, P. A. Bibcode: 1976IAUS...71..207G Altcode: No abstract at ADS Title: Summary of the Final Discussion on August 29 Authors: Durney, B. R.; Gilman, P. A.; Stix, M. Bibcode: 1976IAUS...71..479D Altcode: No abstract at ADS Title: Linear Simulations of Boussinesq Convection in a Deep Rotating Spherical Shell. Authors: Gilman, Peter A. Bibcode: 1975JAtS...32.1331G Altcode: We present extensive linear numerical simulations of Boussinesq convection in a rotating spherical shell of finite depth. The motivation for the study is the problem of general circulation of the solar convection zone. We solve the marching equations on a staggered grid in the meridian plane for the amplitudes of the most unstable Fourier mode of longitudinal wavenumber m between 0 and 24, for Taylor number T between 0 and 106, at a Prandtl number P=1, for a shell of depth 20% of the outer radius. Stress-free, fixed-temperature boundary conditions are used at the inner and outer bounding surfaces. Modes of two symmetries, symmetric and antisymmetric about the equator, are studied. The principal results are as follows:Increasing Taylor number T splits the most unstable solutions for each m into two classes: a broad band of high m solutions which peak at or near the equator, and a small number of low m solutions which peak at or near the poles. The equatorial modes are unstable at lower Rayleigh number R. The polar modes appear to be similar in many respects to plane-parallel convection with rotation parallel to gravity. Modes symmetric about the equator are unstable at lower R than those which are antisymmetric, by a percentage which increases with T in the range studied.Equatorial modes of both symmetries propagate prograde (frequency >0) in longitude at high T and retrograde (<0) at low T, in agreement with earlier work. Polar modes propagate, too, but very slowly.Critical (first unstable) equatorial modes are shown to have or be closely approaching asymptotic dependence RcT, mcT, cT with increasing T, in agreement with analytical analyses of Roberts and Busse.With increasing T, symmetric equatorial modes take on the form of rolls swirling about an axis parallel to the rotation axis and extending across both Northern and Southern Hemispheres in agreement with earlier results. Antisymmetric modes also assume a roll shape, but with swirl oppositely directed in the two hemispheres, together with fluid pumped across the equator parallel to the rotation axis. Polar modes become a ring of vortices more and more tightly arranged around the pole.Outward radial heat flux peaks at the equator for symmetric equatorial modes, and at a low latitude for antisymmetric modes. Both are suppressed near the equator near the outer boundary at high T. Symmetric modes also transport heat toward the equator, while antisymmetric modes transport heat poleward at the lowest latitudes, equatorward at somewhat higher latitudes. Symmetric equatorial modes transport angular momentum radially inward at low T, radially outward at high T. These modes transport angular momentum toward the equator from higher latitudes at all T. Title: Towards Building a Solar General Circulation Model Authors: Gilman, P. A. Bibcode: 1975BAAS....7..364G Altcode: No abstract at ADS Title: Comments on Schatten's Reply to My Comments on `Solar Polar Spindown' Authors: Gilman, Peter A. Bibcode: 1974SoPh...37..491G Altcode: No abstract at ADS Title: Comments on `solar polar spindown', by Kenneth Schatten Authors: Gilman, Peter A. Bibcode: 1974SoPh...36...61G Altcode: No abstract at ADS Title: Solar rotation. Authors: Gilman, P. A. Bibcode: 1974ARA&A..12...47G Altcode: Questions regarding the significance of solar rotation are investigated, giving attention to its apparently small influence on certain solar phenomena. The sun (a G star) is a slow rotator with an equatorial velocity of about 2 km/sec. The rotation of the solar interior is considered along with the rotation of the solar wind and aspects of a differential rotation of the solar surface and atmosphere. Theories for solar rotation are discussed, taking into account the problem of angular momentum history and the problem of surface and atmospheric differential rotation. An outline of interesting problems regarding solar rotation for future studies is also presented. Title: Nonlinear Boussinesq Convective Model for Large Scale Solar Circulations Authors: Gilman, Peter A. Bibcode: 1972SoPh...27....3G Altcode: We present extensive numerical calculations for a model of thermal convection of a Boussinesq fluid in an equatorial annulus of a rotating spherical shell. The convection induces and maintains differential rotation and meridian circulation. The model is solved for an effective Prandtl number P = 1, with effective Taylor number T in the range 102 <T <106, and effective Rayleigh number R between the critical value for onset of convection, and a few times that value. With Ω = 2.6 × 10−6 s−1, d = 1.4 × 1010 cm (roughly the depth of the solar convection zone) the range of Taylor number is equivalent to kinematic viscosities between 1014 and 1012 cm2 s−1, which encompasses eddy viscosities estimated from mixing length theory applied to the Sun. Title: A Method for Constructing Streamlines for the Sun's Large Scale Flow from Doppler Velocities Authors: Gilman, Peter A. Bibcode: 1971SoPh...19...40G Altcode: We show that, if the large scale departures from the mean differential rotation, measured by Howard and Harvey, represent nearly horizontal flow, we may under certain assumptions deduce a pattern of streamlines for these motions from the doppler line of sight velocities. This can be done with data from a single day, without having to construct the total flow from different projections of the (assumed) same velocity vectors seen on different days. Mathematically the method involves integrating a single first order inhomogeneous partial differential equation along a set of characteristic curves which are circles concentric with the center of the solar disk. Title: Instability of Magnetohydrostatic Stellar Interiors from Magnetic Buoyancy. I. Authors: Gilman, Peter A. Bibcode: 1970ApJ...162.1019G Altcode: We show that magnetic buoyancy gives rise to a normal-mode instability in fluids in magnetobydrostatic balance whose diffusivity of heat is large compared with viscosity and magnetic diffusivity, such as in stellar interiors. In an unbounded compressible fluid acted upon by external gravity containing a straight horizontal magnetic field; the instability occurs if the magnetic field decreases with height. The unstable modes are narrow cross-section loops of magnetic field and matter whose finite wavenumber k along the magnetic field lies in the range 0 < k2 < - (g/S2) a/az ln B, where B(z) is the basic magnetic field, g is gravity, and S is the isothermal sound speed. Thus, in a star with a toroidal magnetic field, the analysis for which we are giving in a later paper, this would be a nonaxisymmetric instability. A detailed analysis is made of the special case in which the Alfven and sound speeds are independent of height and which thus elucidates the role of magnetic buoyancy. Sufficiently large rotation perpendicular to gravity is shown to be stabilizing in this case. The instability simultaneously releases potential and magnetic energy, through downward transport of mass and upward transport of magnetic flux, respectively. The instability gives a mechanism by which magnetic flux may be expelled from stellar interiors. Title: Nonaxisymmetric Instabilities in Magnetohydrostatic Stars from Magnetic Buoyancy. I. Plane-Parallel Model Authors: Gilman, Peter A. Bibcode: 1970BAAS....2R.317G Altcode: No abstract at ADS Title: Nonaxisymmetric Instabilities in Magnetohydrostatic Stars from Magnetic Buoyancy. II: Spherical Model Authors: Cadez, Vladimir; Gilman, Peter A. Bibcode: 1970BAAS....2Q.300C Altcode: No abstract at ADS Title: On Large-Scale Solar Convection Authors: Davies-Jones, Robert P.; Gilman, Peter A. Bibcode: 1970SoPh...12....3D Altcode: We examine the effects of rotation about a vertical axis on thermal convection with a simple model in which an inviscid, incompressible fluid of zero thermal conductivity and electrical resistivity is contained in a thin annulus of rectangular cross-section. The initial steady state assumed is one of no motion relative to the rotating frame with constant (unstable) vertical temperature gradient and uniform toroidal magnetic field. Small periodic disturbances are then introduced and the linearized perturbation equations solved. We also determine the second-order mean circulations and magnetic fields that are forced by non-zero Reynolds and thermal stresses and magnetic field transports. Title: Sunspot motion statistics for 1965 67 Authors: Coffey, Helen E.; Gilman, Peter A. Bibcode: 1969SoPh....9..423C Altcode: No abstract at ADS Title: Baroclinic, Alfvén and Rossby Waves in Geostrophic Flow. Authors: Gilman, P. A. Bibcode: 1969JAtS...26.1003G Altcode: We present here further results concerning geostrophic baroclinic flow with a zonal magnetic field, in a -plane channel, as extensions of earlier work by the writer. We find a necessary condition for instability that proves the existence of a short wave cutoff of unstable baroclinic waves for finite magnetic field. We also place bounds on the phase velocities of neutral waves in the system.For zero basic baroclinic and a uniform magnetic field, the neutral oscillations are zonally propagating geostrophic Alfvén waves. The lowest mode is purely horizontal and travels at the ordinary Alfvén speed, but all higher modes, which contain vertical as well as horizontal motion, move more slowly. With 0, we get, in general, combined Alfvén-Rossby waves, which, in the limit of large rotation, yield a class of waves very similar to those found by Hide.Finally, we solve the Eady problem (stability of baroclinic flow with constant shear) in a uniform magnetic field, without and with the effect. We find, for = 0, that the field stabilizes the baroclinic wave. Neutral Alfvén-like waves are found for large field and/or short wavelengths. We also find a pair of `edge waves.'For = 0, the longest baroclinic waves are stabilized for small field strengths, as in the nonmagnetic problem, but very weakly unstable waves occur for larger fields. A set of neutral Rossby waves also is present for long wavelengths and the short Alfvén-like waves are modified, but only slightly. Title: A Rossby-Wave Dynamo for the Sun, II Authors: Gilman, Peter A. Bibcode: 1969SoPh....9....3G Altcode: To make the analysis more tractable, we simplify the equations of Part I to apply to two superposed layers of fluid, with horizontal variations in the motion and magnetic field represented by a small number of Fourier harmonics. The resulting set of eighteen ordinary nonlinear differential equations in time for the Fourier amplitudes is integrated numerically. We analyze in detail the dynamo action from a typical Rossby wave motion and compare it with the solar cycle. Title: A Rossby-Wave Dynamo for the Sun, I Authors: Gilman, Peter A. Bibcode: 1969SoPh....8..316G Altcode: There is increasing interest in the possible existence of large horizontally flowing eddies or `Rossby waves' in the sun's convection zone and photosphere. We present here and in Part II a mathematical model which shows that flows of this type, driven by an assumed latitudinal temperature gradient, can act as hydromagnetic dynamos to induce magnetic fields that periodically reverse. Title: Hydromagnetic Solar Spin Down Authors: Benton, E. R.; Gilman, P. A.; Loper, D. Bibcode: 1969BAAS....1Q.273B Altcode: No abstract at ADS Title: Interaction of Giant Convection Cells with the Differential Rotation Authors: Davies-Jones, Robert; Gilman, Peter Bibcode: 1969BAAS....1..282D Altcode: No abstract at ADS Title: Thermally Driven Rossby-Mode Dynamo for Solar Magnetic-Field Reversals Authors: Gilman, Peter A. Bibcode: 1968Sci...160..760G Altcode: There is increasing interest in the possible existence of large eddies or "Rossby waves" in Sun's convection zone and photosphere. It is shown that many flows of this type, driven by an equator-pole temperature difference, act as hydromagnetic dynamos to produce magnetic fields that periodically reverse. The periods and field amplitudes agree with solar phenomena within an order of magnitude. Title: The General Circulation of the Solar Atmosphere: Large Thermally Driven Rossby Waves. Authors: Gilman, Peter A. Bibcode: 1968AJS....73R..61G Altcode: A set of hydromagnetic equations has been derived to describe very large-scale flow and magnetic fields induced in response to a latitudinal temperature gradient in the surface layer of a rotating gas sphere (such as the convective zone of the sun). The flow is assumed hydrostatic and nearly heliostrophic, and of horizontal scale comparable to the solar bipolar magnetic regions. Maxwell stresses are assumed not larger than Reynolds stresses. A linear study has been made of the wavelike modes (Rossby waves) that transport heat horizontally in response to the temperature gradient, for the highly idealized case of an adiabatic, inviscid, perfectly electrically conducting fluid. From a toroidal magnetic field, these disturbances produce weak, large-scale vertical magnetic fields, which qualitatively look like and amplify at a rate comparable to the solar BMR's. The disturbances also transport these vertical fields meridionally to form a weak axisymmetric poloidal field, in the same manner as postulated by Babcock and observed by Bumba and Howard. The disturbances also transport momentum toward the equator, in the manner deduced by Ward from sunspot statistics. At gas densities of 10-~ g/cm3 (within the convection zone), growth of the disturbances in a toroidal field of 100 gauss requires an equatorpole temperature difference of a few tens of degrees. Currently, a more general, nonlinear, dissipating dynamo model (i.e., including thermal and electrical conductivity and viscosity) is being developed. First results indicate that dynamo cycles with periods roughly comparable to the solar cycle are included among the solutions. Title: The Circulation of the Sun's Atmosphere Authors: Starr, Victor P.; Gilman, Peter A. Bibcode: 1968SciAm.218a.100S Altcode: 1968SciAm.218..100S No abstract at ADS Title: Stability of Baroclinic Flows in a Zonal Magnetic Field: Part III. Authors: Gilman, Peter A. Bibcode: 1967JAtS...24..130G Altcode: The study of the stability of two-layer baroclinic flows in a zonal magnetic field (Part II) is generalized to include a parabolic flow profile in the upper layer. Contrary to the nonmagnetic case, angularities may be present in the equations even when the potential vorticity gradient does not change sign, if the magnetic field is increased beyond a certain strength. Below this strength, power series solutions show that, just as in the nonmagnetic use, horizontal shear renders shorter waves unstable. Reynolds stresses are seen to transport momentum up the gradient, while smaller Maxwell stresses oppose them.Properties of the solutions in this paper and in Part II are compared to solar observations. At gas densities 104 gm cm3, a 35K equator-pole temperature difference would be enough to give baroclinically unstable disturbances in a zonal field of 100 gauss. The perturbation vertical fields produced from this magnitude zonal field will have magnitude 1 gauss, as well as a zonal wave number of 6-8, `e folding' times of a few solar rotations, and a pronounced tilt upstream away from the maximum of zonal flow. All these characteristics are qualitatively consistent with the solar observations.The Reynolds and Maxwell stresses are seen to act in such a way as to be capable of maintaining the solar differential rotation in the manner proposed by Starr and Gilman.It is suggested that the same process could be used to maintain differential rotation in the dynamo theory. The production of perturbation vertical and mean poloidal fields, and the changes in the toroidal field by the disturbances are also seen to correspond to dynamo processes. Suitable generalizations of the equations may allow a complete dynamo cycle to take place. If so, the system would have the important computational advantage of putting all the dynamics on essentially the same time and space scales.Finally, the `magnetic modes' recently suggested by Hide to be responsible for the slow westward drift of the geomagnetic field are very unlikely to be baroclinically unstable. They could, however, be fed energy from baroclinically unstable `inertial modes.'Suggestions are made for further studies of the scaled equations presented. Title: Stability of Baroclinic Flows in a Zonal Magnetic Field: Part II. Authors: Gilman, Peter A. Bibcode: 1967JAtS...24..119G Altcode: The equations of Part I are approximated to represent a two-layer model. All the theorems for the continuous case are shown to hold in the two-layer case as well. In addition, bounds are placed on the phase velocities of neutral waves.The stability of purely baroclinic flow (no horizontal shear) in a uniform zonal magnetic field is then studied. The minimum vertical shear needed for instability no longer depends upon the -effect or the static stability, but rather is determined by the zonal field strength. Short waves are destabilized by the magnetic field, long waves stabilized. Unstable waves convert available potential energy into kinetic energy of the disturbances, part of which in turn is converted into disturbance magnetic energy. Nonmagnetic changes in the initial state are similar to those of Phillips. Perturbation vertical magnetic fields and a single-celled meridional (poloidal) field are produced. Title: Stability of Baroclinic Flows in a Zonal Magnetic Field: Part I. Authors: Gilman, Peter A. Bibcode: 1967JAtS...24..101G Altcode: Because of its possible applications to the solar differential rotation, hydromagnetic dynamo theory, and the geomagnetic secular variation, the dynamics of baroclinic, geostrophic -plane flow in a zonal (toroidal), magnetic field are studied. The equations describing the flow in the absence of a magnetic field are the same as those formulated by Charney and Stern and by Pedlosky. Due to the horizontal magnetic field, potential vorticity is no longer conserved. Vertical magnetic fields can be produced, but the scaling excludes their feedback on the motions and horizontal fields. Thus, the system in its present simplest form can not complete a dynamo cycle, but suitable relaxation of some scaling restrictions may overcome this difficulty.The scaled equations are perturbed about a steady axially symmetric zonal flow and zonal (toroidal) magnetic field. Changes in the initial state are inferred from products of perturbation quantities. These include the growth in the meridional plane of axially symmetric circulations and magnetic fields (i.e., poloidal fields). The energetics of the system are examined. It is shown that the potential vorticity theorem of Charney and Stern no longer holds due to the magnetic field. For short wavelength disturbances, the field should dominate in determining the stability properties. Bounds are placed on the complex phase velocities of normal mode disturbances. For flows with vertical and horizontal shear, these are the same as found by Pedlosky for the nonmagnetic case. The bounds on growth rates of disturbances in barotropic flows are tightened by the magnetic field. Such flows will be stable to all wavelengths if the field is large enough. Title: Energetics of the Solar Rotation. Authors: Starr, Victor P.; Gilman, Peter A. Bibcode: 1965ApJ...141.1119S Altcode: The energetics of the solar differential rotation is considered in the light of recent observational studies of sunspot motions by Ward. It is shown that Ward's calculations of the correlation of longitudinal and latitudinal sunspot displacements imply a systematic conversion of horizontal-eddy kinetic energy into kinetic energy of the mean zonal flow This conversion is therefore directly opposed to the effect of molecular and eddy viscosity. The conversion rate is such that, if the conversion were totally suppressed while other processes were allowed to continue, virtually solid-body rotation would be achieved within a few solar rotations Some of the energy sources available to maintain the kinetic energy of the large-scale horizontal eddies against these conversion losses are briefly considered. Title: On the Structure and Energetics of Large Scale Hydromagnetic Disturbances in the Solar Photosphere Authors: Starr, V. P.; Gilman, P. A. Bibcode: 1965Tell...17..334S Altcode: 1965TellA..17..334S It is suggested that the solar differential rotation is maintained against frictional dissipation by large scale quasi-horizontal hydromagnetic disturbances, which possess tilted structures similar to upper level troughs in the westerlies of the earth's atmosphere. On the sun, these tilts would be directed from the equator toward the east limb in each hemisphere. Horizontal Reynolds stresses in these disturbances thus convert eddy kinetic energy into kinetic energy of the zonal flow, while horizontal Maxwell stresses convert the mean zonal kinetic energy into "eddy" magnetic energy. In order for the disturbances as a whole to feed kinetic energy into the differential rotation, the horizontal eddy magnetic fields must be somewhat less than 7 gauss at the level considered. Measurements of the large scale patterns of the line of sight component of the magnetic field, made by Howard, show clearly the required tilt in the patterns toward the east limb in each hemisphere. These measurements also show that the patterns do have a very large horizontal scale, possessing a longitudinal wave number of about 6, and a latitudinal extent of about 60°. The magnitudes of these fields appear to be compatible with horizontal eddy magnetic fields less than 7 gauss. The braking action by the symmetric steady component of the horizontal Maxwell stress appears from the observations to be small compared with that due to the eddy component.