Author name code: weiss ADS astronomy entries on 2022-09-14 author:"Weiss, Nigel O." ------------------------------------------------------------------------ Title: On long-term modulation of the Sun's magnetic cycle Authors: Beer, J.; Tobias, S. M.; Weiss, N. O. Bibcode: 2018MNRAS.473.1596B Altcode: We utilize reconstructions based on cosmogenic radionuclides as well as direct observations of solar magnetic activity, to argue that the solar dynamo has operated similarly to the present day for at least the past 10 000 yr. The persistence of the 87-yr Gleissberg cycle throughout supermodulation events suggests that the Hale and Schwabe cycles continue independently of the modulational mechanism for activity. We further analyse behaviour of solar activity during the Spörer and Maunder Minima. Such grand minima recur with the characteristic de Vries period of approximately 208 yr but their incidence is modulated by the Hallstatt cycle with a characteristic period of around 2300 yr. We ascribe the latter to supermodulation, caused by breaking the symmetry of the dynamo pattern. Finally, we emphasize the need for further calculations in order to determine the effects of changes in solar field morphology and symmetry on the solar wind and on cosmic ray deflection. Title: Supermodulation of the Sun's magnetic activity: the effects of symmetry changes Authors: Weiss, N. O.; Tobias, S. M. Bibcode: 2016MNRAS.456.2654W Altcode: In this paper, we argue that the solar activity record, as revealed by telescopic observations and proxy data from the abundances of cosmogenic isotopes, is consistent with the action of a deterministic non-linear chaotic dynamo. In particular, we claim that the long time-scale `supermodulation' apparent in the isotopic record can be ascribed to switching of the dynamo between two different modulational patterns. The first (which is currently in operation) involves deep grand minima and occasional changes in symmetry triggered by these minima. The second, which exhibits only weak modulation and no grand minima, arises as a consequence of symmetry breaking. These processes are demonstrated for highly idealized simple models of the non-linear dynamo equations. Title: Solar Activity in the Past and the Chaotic Behaviour of the Dynamo Authors: Arlt, Rainer; Weiss, Nigel Bibcode: 2015sac..book..525A Altcode: No abstract at ADS Title: Solar Activity in the Past and the Chaotic Behaviour of the Dynamo Authors: Arlt, Rainer; Weiss, Nigel Bibcode: 2014SSRv..186..525A Altcode: 2014SSRv..tmp...27A; 2014arXiv1406.7628A The record of solar activity is reviewed here with emphasis on peculiarities. Since sunspot positions tell us a lot more about the solar dynamo than the various global sunspot numbers, we first focus on the records of telescopic observations of sunspots leading to positional information. Then we turn to the proxy record from cosmogenic isotope abundances, which shows recurrent grand minima over the last 9500 years. The apparent distinction between episodes of strong modulation, and intervening episodes with milder modulation and weaker overall activity, hints at the solar dynamo following a variety of solutions, with different symmetries, over the course of millennia. Title: Sunspots and Starspots Authors: Thomas, John H.; Weiss, Nigel O. Bibcode: 2012sust.book.....T Altcode: Preface; 1. The sun among the stars; 2. Sunspots and starspots: a historical introduction; 3. Overall structure of a sunspot; 4. Fine structure of the umbra; 5. Fine structure of the penumbra; 6. Oscillations in sunspots; 7. Sunspots and active regions; 8. Magnetic activity in stars; 9. Starspots; 10. Solar and stellar activity cycles; 11. Solar and stellar dynamos; 12. Solar activity, space weather, and climate change; 13. The way ahead; Appendices; References; Index. Title: Convectively driven dynamo action in the quiet Sun Authors: Bushby, P. J.; Favier, B.; Proctor, M. R. E.; Weiss, N. O. Bibcode: 2012GApFD.106..508B Altcode: Observations of the quiet solar surface indicate that localised concentrations of vertical magnetic flux tend to accumulate in the convective downflows. Furthermore, there is some evidence to suggest that mesogranular boundaries are preferred locations for the formation of these flux concentrations. This implies that these magnetic fields are organised on scales that are larger than the granular scale. One possible explanation for the existence of quiet Sun magnetic features is that they are continuously regenerated by the near-surface convective motions. Motivated by this, we consider dynamo action in a local Cartesian model of convection in a compressible electrically conducting fluid. The horizontal scale of this domain is large enough to model mesoscale behaviour. Dynamo action occurs provided that the magnetic Reynolds number exceeds some critical value. In the kinematic regime the presence of mesogranules seems to be beneficial for dynamo action: compared to similar smaller aspect ratio calculations, we find higher kinematic growth rates for the magnetic energy, as well as a lower value for the critical magnetic Reynolds number. In the nonlinear regime the peak magnetic field strengths compare very favourably to observations, greatly exceeding the equipartition value at the surface, as observed in the quiet Sun. However, there is no evidence to suggest that the presence of mesogranules significantly increases the saturation level of the dynamo in this nonlinear regime, which (in the highest magnetic Reynolds number case) is comparable to that found in similar calculations in smaller domains. Title: Reflections on magnetoconvection Authors: Weiss, N. O. Bibcode: 2012GApFD.106..353W Altcode: The interaction between magnetic fields and convection can be observed directly in stars like the Sun. This selective review uses idealized numerical experiments to illustrate and extend results obtained from bifurcation theory. It covers a wide variety of steady and time-dependent behaviour, in incompressible and compressible fluids, in two and three dimensions. These results illuminate behaviour that appears in recent "realistic" simulations of solar magnetoconvection. Title: Magnetic buoyancy instabilities in the presence of magnetic flux pumping at the base of the solar convection zone Authors: Barker, Adrian J.; Silvers, Lara J.; Proctor, Michael R. E.; Weiss, Nigel O. Bibcode: 2012MNRAS.424..115B Altcode: 2012arXiv1204.5432B We perform idealized numerical simulations of magnetic buoyancy instabilities in three dimensions, solving the equations of compressible magnetohydrodynamics in a model of the solar tachocline. In particular, we study the effects of including a highly simplified model of magnetic flux pumping in an upper layer ('the convection zone') on magnetic buoyancy instabilities in a lower layer ('the upper parts of the radiative interior - including the tachocline'), to study these competing flux transport mechanisms at the base of the convection zone. The results of the inclusion of this effect in numerical simulations of the buoyancy instability of both a preconceived magnetic slab and a shear-generated magnetic layer are presented. In the former, we find that if we are in the regime that the downward pumping velocity is comparable with the Alfvén speed of the magnetic layer, magnetic flux pumping is able to hold back the bulk of the magnetic field, with only small pockets of strong field able to rise into the upper layer.

In simulations in which the magnetic layer is generated by shear, we find that the shear velocity is not necessarily required to exceed that of the pumping (therefore the kinetic energy of the shear is not required to exceed that of the overlying convection) for strong localized pockets of magnetic field to be produced which can rise into the upper layer. This is because magnetic flux pumping acts to store the field below the interface, allowing it to be amplified both by the shear and by vortical fluid motions, until pockets of field can achieve sufficient strength to rise into the upper layer. In addition, we find that the interface between the two layers is a natural location for the production of strong vertical gradients in the magnetic field. If these gradients are sufficiently strong to allow the development of magnetic buoyancy instabilities, strong shear is not necessarily required to drive them (cf. previous work by Vasil & Brummell). We find that the addition of magnetic flux pumping appears to be able to assist shear-driven magnetic buoyancy in producing strong flux concentrations that can rise up into the convection zone from the radiative interior. Title: The Solar Tachocline Authors: Hughes, D. W.; Rosner, R.; Weiss, N. O. Bibcode: 2012sota.book.....H Altcode: Preface; Part I. Setting the Scene: 1. An introduction to the solar tachocline D. O. Gough; 2. Reflections on the solar tachocline E. A. Spiegel; Part II. Observations: 3. Observational results and issues concerning the tachocline J. Christensen-Dalsgaard and M. J. Thompson; Part III. Hydrodynamic Models: 4. Hydrodynamic models of the tachocline J.-P. Zahn; 5. Turbulence in the tachocline M. S. Miesch; 6. Mean field modelling of differential rotation G. Rudiger and L. L. Kitchatinov; Part IV. Hydromagnetic Properties: 7. Magnetic confinement of the solar tachocline P. Garaud; 8. Magnetic confinement and the sharp tachopause M. E. McIntyre; 9. ß-Plane MHD turbulence and dissipation in the solar tachocline P. H. Diamond, K. Itoh, S.-I. Itoh and L. J. Silvers; Part V. Instabilities: 10. Global MHD instabilities of the tachocline P. A. Gilman and P. S. Cally; 11. Magnetic buoyancy instabilities in the tachocline D. W. Hughes; 12. Instabilities, angular momentum transport and magnetohydrodynamic turbulence G. I. Ogilvie; Part VI. Dynamo Action: 13. The solar dynamo and the tachocline S. M. Tobias and N. O. Weiss; Part VII. Overview: 14. On studying the rotating solar interior R. Rosner; Index. Title: Global model of differential rotation in the Sun Authors: Balbus, Steven A.; Latter, Henrik; Weiss, Nigel Bibcode: 2012MNRAS.420.2457B Altcode: 2011arXiv1111.3809B The isorotation contours of the solar convective zone (SCZ) show three distinct morphologies, corresponding to two boundary layers (inner and outer), and the bulk of the interior. Previous work has shown that the thermal wind equation (TWE) together with informal arguments on the nature of convection in a rotating fluid could be used to deduce the shape of the isorotation surfaces in the bulk of the SCZ with great fidelity, and that the tachocline contours could also be described by relatively simple phenomenology. In this paper, we show that the form of these surfaces can be understood more broadly as a mathematical consequence of the TWE and a narrow convective shell. The analysis does not yield the angular velocity function directly; an additional surface boundary condition is required. However, much can already be deduced without constructing the entire rotation profile. The mathematics may be combined with dynamical arguments put forth in previous works to the mutual benefit of each. An important element of our approach is to regard the constant angular velocity surfaces as an independent coordinate variable for what is termed the 'residual entropy', a quantity that plays a key role in the equation of thermal wind balance. The difference between the dynamics of the bulk of the SCZ and the tachocline is due to a different functional form of the residual entropy in each region. We develop a unified theory for the rotational behaviour of both the SCZ and the tachocline, using the solutions for the characteristics of the TWE. These characteristics are identical to the isorotation contours in the bulk of the SCZ, but the two deviate in the tachocline. The outer layer may be treated, at least descriptively, by similar mathematical techniques, but this region probably does not obey thermal wind balance. Title: Solar and Stellar Dynamos Authors: Weiss, Nigel O. Bibcode: 2011IAUS..271..247W Altcode: Records of the solar magnetic field extend back for millennia, and its surface properties have been observed for centuries, while helioseismology has recently revealed the Sun's internal rotation and the presence of a tachocline. Dynamo theory has developed to explain these observations, first with idealized models based on mean-field electrodynamics and, more recently, by direct numerical simulation, notably with the ASH code at Boulder. These results, which suggest that cyclic activity relies on the presence of the tachocline, and that its modulation is chaotic (rather than stochastic), will be critically reviewed. Similar theoretical approaches have been followed in order to explain the magnetic properties of other main-sequence stars, whose fields can be mapped by Zeeman-Doppler imaging. Of particular interest is the behaviour of fully convective, low-mass stars, which lack any tachocline but are nevertheless extremely active. Title: The influence of stratification upon small-scale convectively-driven dynamos Authors: Bushby, Paul J.; Proctor, Michael R. E.; Weiss, Nigel O. Bibcode: 2011IAUS..271..197B Altcode: In the quiet Sun, convective motions form a characteristic granular pattern, with broad upflows enclosed by a network of narrow downflows. Magnetic fields tend to accumulate in the intergranular lanes, forming localised flux concentrations. One of the most plausible explanations for the appearance of these quiet Sun magnetic features is that they are generated and maintained by dynamo action resulting from the local convective motions at the surface of the Sun. Motivated by this idea, we describe high resolution numerical simulations of nonlinear dynamo action in a (fully) compressible, non-rotating layer of electrically-conducting fluid. The dynamo properties depend crucially upon various aspects of the fluid. For example, the magnetic Reynolds number (Rm) determines the initial growth rate of the magnetic energy, as well as the final saturation level of the dynamo in the nonlinear regime. We focus particularly upon the ways in which the Rm-dependence of the dynamo is influenced by the level of stratification within the domain. Our results can be related, in a qualitative sense, to solar observations. Title: Chaotic behaviour in low-order models of planetary and stellar dynamos Authors: Weiss, N. O. Bibcode: 2011GApFD.105..256W Altcode: The behaviour of the geodynamo and the solar cycle can be modelled by low-order systems of coupled nonlinear differential equations. The Earth's magnetic field reverses aperiodically, and similar behaviour is exhibited by disc dynamos that are described by the Lorenz equations. Chaotic behaviour is also a characteristic feature of coupled disc dynamos. In stars like the Sun, magnetic activity varies cyclically, with regular reversals of magnetic fields, but the cyclic activity is modulated on longer timescales. This behaviour can be described by normal form equations that account for symmetry-breaking as well as for variations in amplitude. The Von Kármán Sodium (VKS) experiment has successfully demonstrated magnetic reversals in the laboratory, and these results can be represented by evolution equations also. Title: Effects of boundary conditions on the onset of convection with tilted magnetic field and rotation vectors Authors: Proctor, M. R. E.; Weiss, N. O.; Thompson, S. D.; Roxburgh, N. T. Bibcode: 2011GApFD.105...82P Altcode: The problem of the onset of thermal convection is considered, firstly when a uniform tilted magnetic field is present, and secondly in a frame rotating about an oblique axis. If up-down symmetry is broken we expect to find only bifurcations that lead to travelling waves. Numerical studies show, however, that in a Boussinesq fluid the spectrum of eigenvalues can be symmetrical about the real axis, even when the boundary conditions are asymmetrical. Here we show analytically that this symmetry property indeed holds for a wide range of boundary conditions and hence that both steady solutions and standing waves are allowed. Title: Small-scale Dynamo Action in Compressible Convection Authors: Bushby, P. J.; Proctor, M. R. E.; Weiss, N. O. Bibcode: 2010ASPC..429..181B Altcode: Motivated by observations of magnetic fields at the surface of the Sun, we consider direct numerical simulations of dynamo action in highly-stratified, three-dimensional compressible convection. Whether or not a convective flow can drive a dynamo depends crucially upon the magnetic Reynolds number. If this parameter is large enough that the inductive effects of the fluid motions outweigh the dissipative effects of magnetic diffusion, then dynamo action can occur. Simulating convection with a Reynolds number of approximately 150, we find that it is possible to excite a dynamo with computationally accessible values of the magnetic Reynolds number. In the kinematic regime, the growth rate of the dynamo has a logarithmic dependence upon the magnetic Reynolds number. Following these dynamos into the nonlinear regime, we find that intense, partially-evacuated concentrations of vertical magnetic flux form near the upper surface of the computational domain. This partial evacuation has a profound influence upon the efficiency of the (explicit) numerical scheme that is used in these calculations. These results can be related (in a qualitative sense) to recent observations of magnetic fields in the quiet Sun. Title: Modulation of the sunspot cycle Authors: Weiss, Nigel Bibcode: 2010A&G....51c...9W Altcode: Solar activity has been abnormally high for the last half-century, but the extremely feeble start of the latest 11-year cycle suggests that this episode is probably coming to an end. This prospect raises some fascinating and important issues, involving an extremely wide range of timescales, writes Nigel Weiss. Title: Differential rotation in fully convective stars Authors: Balbus, Steven A.; Weiss, Nigel O. Bibcode: 2010MNRAS.404.1263B Altcode: 2010MNRAS.tmp..279B; 2010arXiv1001.3542B Under the assumption of thermal wind balance and effective entropy mixing in constant rotation surfaces, the isorotational contours of the solar convective zone may be reproduced with great fidelity. Even at this early stage of development, this helioseismology fit may be used to put a lower bound on the mid-latitude radial solar entropy gradient, which is in good accord with standard mixing length theory. In this paper, we generalize this solar calculation to fully convective stars (and potentially planets), retaining the assumptions of thermal wind balance and effective entropy mixing in isorotational surfaces. It is found that each isorotation contour is of the form R2 = A + BΦ(r), where R is the radius from the rotation axis, Φ(r) is the (assumed spherical) gravitational potential, and A and B are constants along the contour. This result is applied to simple models of fully convective stars. Both solar-like surface rotation profiles (angular velocity decreasing toward the poles) as well as `antisolar' profiles (angular velocity increasing toward the poles) are modelled; the latter bear some suggestive resemblance to numerical simulations. We also perform exploratory studies of zonal surface flows similar to those seen in Jupiter and Saturn. In addition to providing a practical framework for understanding the results of large-scale numerical simulations, our findings may also prove useful in dynamical calculations for which a simple but viable model for the background rotation profile in a convecting fluid is needed. Finally, our work bears directly on an important goal of the CoRoT programme: to elucidate the internal structure of rotating, convecting stars. Title: Summary and Perspective Authors: Weiss, N. O. Bibcode: 2010ASSP...19..346W Altcode: 2010mcia.conf..346W This meeting celebrated a double anniversary. A 100 years ago, George Ellery Hale had a hunch: he had noticed the vortical structure in Hα above sunspots, and he interpreted this as evidence for a vortex flow, which (he supposed) carried electric charges round a spot, and so provided an azimuthal electric current that would generate a magnetic field like that in a solenoid (Hale 1908a). His hunch was actually wrong since (as we now know) the solar plasma is electrically neutral - but when he looked for spectroscopic evidence of a magnetic field, using the recently discovered Zeeman effect, he found that there were kilogauss magnetic fields in sunspots (Hale 1908b). This was the first demonstration that magnetic fields were present outside the earth. Title: Differential rotation and convection in the Sun Authors: Balbus, Steven A.; Bonart, Julius; Latter, Henrik N.; Weiss, Nigel O. Bibcode: 2009MNRAS.400..176B Altcode: 2009arXiv0907.5075B We show that the differential rotation profile of the solar convection zone, apart from inner and outer boundary layers, can be reproduced with great accuracy if the isorotation contours correspond to characteristics of the thermal wind equation. This requires that there be a formal quantitative relationship involving the entropy and the angular velocity. Earlier work has suggested that this could arise from magnetohydrodynamic stability constraints; here, we argue that purely hydrodynamical processes could also lead to such a result. Of special importance to the hydrodynamical solution is the fact that the thermal wind equation is insensitive to radial entropy gradients. This allows a much more general class of solutions to fit the solar isorotation contours, beyond just those in which the entropy itself must be a function of the angular velocity. In particular, for this expanded class, the thermal wind solution of the solar rotation profile remains valid even when large radial entropy gradients are present. A clear and explicit example of this class of solution appears to be present in published numerical simulations of the solar convective zone. Though hydrodynamical in character, the theory is not sensitive to the presence of weak magnetic fields. Thus, the identification of solar isorotation contours with the characteristics of the thermal wind equation appears to be robust, accommodating, but by no means requiring, magnetic field dynamics. Title: The Solar Dynamo Authors: Weiss, N. O.; Thompson, M. J. Bibcode: 2009SSRv..144...53W Altcode: 2008SSRv..tmp..155W It is generally accepted that the strong toroidal magnetic fields that emerge through the solar surface in sunspots and active regions are formed by the action of differential rotation on a poloidal field, and then stored in or near the tachocline at the base of the Sun’s convection zone. The problem is how to explain the generation of a reversed poloidal field from this toroidal flux—a process that can be parametrised in terms of an α-effect related to some form of turbulent helicity. Here we first outline the principal patterns that have to be explained: the 11-year activity cycle, the 22-year magnetic cycle and the longer term modulation of cyclic activity, associated with grand maxima and minima. Then we summarise what has been learnt from helioseismology about the Sun’s internal structure and rotation that may be relevant to our subject. The ingredients of mean-field dynamo models are differential rotation, meridional circulation, turbulent diffusion, flux pumping and the α-effect: in various combinations they can reproduce the principal features that are observed. To proceed further, it is necessary to rely on large-scale computation and we summarise the current state of play. Title: The Solar Dynamo Authors: Weiss, N. O.; Thompson, M. J. Bibcode: 2009odsm.book...53W Altcode: It is generally accepted that the strong toroidal magnetic fields that emerge through the solar surface in sunspots and active regions are formed by the action of differential rotation on a poloidal field, and then stored in or near the tachocline at the base of the Sun's convection zone. The problem is how to explain the generation of a reversed poloidal field from this toroidal flux—a process that can be parametrised in terms of an α-effect related to some form of turbulent helicity. Here we first outline the principal patterns that have to be explained: the 11-year activity cycle, the 22-year magnetic cycle and the longer term modulation of cyclic activity, associated with grand maxima and minima. Then we summarise what has been learnt from helioseismology about the Sun's internal structure and rotation that may be relevant to our subject. The ingredients of mean-field dynamo models are differential rotation, meridional circulation, turbulent diffusion, flux pumping and the α-effect: in various combinations they can reproduce the principal features that are observed. To proceed further, it is necessary to rely on large-scale computation and we summarise the current state of play. Title: Oded Regev (2006): Chaos and Complexity in Astrophysics. Cambridge University Press, 468 pp., $89. ISBN-13 978-0-521-85534-1 (ISBN-10 0-521-85534-9) Authors: Weiss, Nigel Bibcode: 2008ThCFD..22..485W Altcode: 2008ThCFD.tmp...16W No abstract at ADS Title: Sunspots and Starspots Authors: Thomas, John H.; Weiss, Nigel O. Bibcode: 2008sust.book.....T Altcode: Preface; 1. The sun among the stars; 2. Sunspots and starspots: a historical introduction; 3. Overall structure of a sunspot; 4. Fine structure of the umbra; 5. Fine structure of the penumbra; 6. Oscillations in sunspots; 7. Sunspots and active regions; 8. Magnetic activity in stars; 9. Starspots; 10. Solar and stellar activity cycles; 11. Solar and stellar dynamos; 12. Solar activity, space weather, and climate change; 13. The way ahead; Appendices; References; Index. Title: For how long will the current grand maximum of solar activity persist? Authors: Abreu, J. A.; Beer, J.; Steinhilber, F.; Tobias, S. M.; Weiss, N. O. Bibcode: 2008GeoRL..3520109A Altcode: Understanding the Sun's magnetic activity is important because of its impact on the Earth's environment. The sunspot record since 1610 shows irregular 11-year cycles of activity; they are modulated on longer timescales and were interrupted by the Maunder minimum in the 17th century. Future behavior cannot easily be predicted - even in the short-term. Recent activity has been abnormally high for at least 8 cycles: is this grand maximum likely to terminate soon or even to be followed by another (Maunder-like) grand minimum? To answer these questions we use, as a measure of the Sun's open magnetic field, a composite record of the solar modulation function Φ, reconstructed principally from the proxy record of cosmogenic 10Be abundances in the GRIP icecore from Greenland. This Φ record extends back for almost 10,000 years, showing many grand maxima and grand minima (defined as intervals when Φ is within the top or bottom 20% of a Gaussian distribution). We carry out a statistical analysis of this record and calculate the life expectancy of the current grand maximum. We find that it is only expected to last for a further 15-36 years, with the more reliable methods yielding shorter expectancies, and we therefore predict a decline in solar activity within the next two or three cycles. We are not able, however, to predict the level of the ensuing minimum. Title: Flux Pumping and Magnetic Fields in the Outer Penumbra of a Sunspot Authors: Brummell, Nicholas H.; Tobias, Steven M.; Thomas, John H.; Weiss, Nigel O. Bibcode: 2008ApJ...686.1454B Altcode: The filamentary structure of a sunspot penumbra is believed to be magnetoconvective in origin. In the outer penumbra there is a difference in inclination of up to 30°-40° between the magnetic fields associated with bright and dark filaments, and the latter fields plunge downward below the surface toward the edge of the spot. We have proposed that these fields are dragged downward by magnetic pumping caused by the external granular convection. In this paper we model this process in a more elaborate idealized configuration that includes the curvature force exerted by an arched magnetic field in addition to magnetic buoyancy, and demonstrate that magnetic pumping remains an efficient mechanism for holding flux submerged. We discuss the implications of these results for the magnetic structure of the outer penumbra. Title: Convective intensification of magnetic fields in the quiet Sun Authors: Bushby, P. J.; Houghton, S. M.; Proctor, M. R. E.; Weiss, N. O. Bibcode: 2008MNRAS.387..698B Altcode: 2008arXiv0804.1238B Kilogauss-strength magnetic fields are often observed in intergranular lanes at the photosphere in the quiet Sun. Such fields are stronger than the equipartition field Be, corresponding to a magnetic energy density that matches the kinetic energy density of photospheric convection, and comparable with the field Bp that exerts a magnetic pressure equal to the ambient gas pressure. We present an idealized numerical model of three-dimensional compressible magnetoconvection at the photosphere, for a range of values of the magnetic Reynolds number. In the absence of a magnetic field, the convection is highly supercritical and characterized by a pattern of vigorous, time-dependent, `granular' motions. When a weak magnetic field is imposed upon the convection, magnetic flux is swept into the convective downflows where it forms localized concentrations. Unless this process is significantly inhibited by magnetic diffusion, the resulting fields are often much greater than Be and the high magnetic pressure in these flux elements leads to their being partially evacuated. Some of these flux elements contains ultraintense magnetic fields that are significantly greater than Bp. Such fields are contained by a combination of the thermal pressure of the gas and the dynamic pressure of the convective motion, and they are constantly evolving. These ultraintense fields develop owing to non-linear interactions between magnetic fields and convection; they cannot be explained in terms of `convective collapse' within a thin flux tube that remains in overall pressure equilibrium with its surroundings. Title: Convection-driven Emergence of Small-Scale Magnetic Fields and their Role in Coronal Heating and Solar Wind Acceleration Authors: Isobe, H.; Proctor, M. R. E.; Weiss, N. O. Bibcode: 2008ApJ...679L..57I Altcode: Recent observations by the Solar Optical Telescope (SOT) on board Hinode have revealed that the surface of the Sun is, on average, covered with small-scale horizontal magnetic fields. Frequent emergence of horizontal magnetic flux on a granular scale is found in the quiet Sun and in plage regions. In this Letter we present the results of magnetohydrodynamic simulations that cover the upper convection zone and the corona. It is found that, even when the initial magnetic field is uniform and vertical, a disordered magnetic field is produced in the convection zone. The photospheric magnetic field is then characterized by strong vertical fields concentrated in the intergranular lanes and relatively weak, horizontal fields both in the granules and in the intergranular lanes. Occasionally, fragments with large magnetic fluxes are driven above the photosphere by the upward convective flows. These characteristics are consistent with the SOT observations. Moreover, the simulated flux emerging on a granular scale undergoes magnetic reconnection with the expanding vertical magnetic concentrations in the chromosphere. These reconnection events heat the local plasma and emit high-frequency waves that propagate into the corona. Such an interplay between the small-scale horizontal fields and the vertical flux may play an important role in coronal heating and wind acceleration in the Sun and stars. Title: The Solar Tachocline Authors: Hughes, D. W.; Rosner, R.; Weiss, N. O. Bibcode: 2007sota.book.....H Altcode: Preface; Part I. Setting the Scene: 1. An introduction to the solar tachocline D. O. Gough; 2. Reflections on the solar tachocline E. A. Spiegel; Part II. Observations: 3. Observational results and issues concerning the tachocline J. Christensen-Dalsgaard and M. J. Thompson; Part III. Hydrodynamic Models: 4. Hydrodynamic models of the tachocline J.-P. Zahn; 5. Turbulence in the tachocline M. S. Miesch; 6. Mean field modelling of differential rotation G. Rudiger and L. L. Kitchatinov; Part IV. Hydromagnetic Properties: 7. Magnetic confinement of the solar tachocline P. Garaud; 8. Magnetic confinement and the sharp tachopause M. E. McIntyre; 9. ß-Plane MHD turbulence and dissipation in the solar tachocline P. H. Diamond, K. Itoh, S.-I. Itoh and L. J. Silvers; Part V. Instabilities: 10. Global MHD instabilities of the tachocline P. A. Gilman and P. S. Cally; 11. Magnetic buoyancy instabilities in the tachocline D. W. Hughes; 12. Instabilities, angular momentum transport and magnetohydrodynamic turbulence G. I. Ogilvie; Part VI. Dynamo Action: 13. The solar dynamo and the tachocline S. M. Tobias and N. O. Weiss; Part VII. Overview: 14. On studying the rotating solar interior R. Rosner; Index. Title: Sunspot Structure and Dynamics Authors: Weiss, N. O. Bibcode: 2007sdeh.book...13W Altcode: No abstract at ADS Title: The Solar Tachocline Authors: Hughes, David W.; Rosner, Robert; Weiss, Nigel O. Bibcode: 2007sota.conf.....H Altcode: No abstract at ADS Title: The solar dynamo and the tachocline Authors: Tobias, Steven; Weiss, Nigel Bibcode: 2007sota.conf..319T Altcode: No abstract at ADS Title: The Puzzling Structure of a Sunspot Authors: Weiss, N. O. Bibcode: 2006ASPC..354..213W Altcode: Sunspots are characterized by the presence of a filamentary penumbra but it is only within the last few years that the fine structure of penumbral magnetic fields, and of the associated Evershed outflow, has been definitively established. High resolution observations show that bright filaments in the inner penumbra possess slender dark cores with fields and flows that are nearly horizontal, while the ambient fields are inclined at 40° to the vertical. In the outer penumbra the fields in bright and dark filaments differ in inclination by about 30° and recent observations confirm that the Evershed flow is along the almost horizontal fields in dark filaments. Moreover, these two families of field lines remain distinct. This intricate magnetic geometry poses major theoretical problems. How can such a structure be maintained and how does it originate? How do penumbral fields relate to the photospheric granulation outside the spot? What drives the Evershed flow within dark filaments? What form does convection take in the umbra, in bright filaments and in dark filaments? What causes the fine structure within bright filaments? What is the subsurface structure of a sunspot and how does it relate to outflows and inflows in the moat cell that surrounds it? Although a general theoretical picture is beginning to emerge, these questions can only be properly answered through detailed computational investigations, guided by further observations both from the ground and from space. Title: Unpredictable Sun leaves researchers in the dark Authors: Tobias, Steven; Hughes, David; Weiss, Nigel Bibcode: 2006Natur.442...26T Altcode: No abstract at ADS Title: On the fine structure of magnetic fields in sunspot penumbrae Authors: Thomas, J. H.; Weiss, N. O.; Tobias, S. M.; Brummell, N. H. Bibcode: 2006A&A...452.1089T Altcode: Recent observations have revealed the interlocking-comb structure of the magnetic field in the outer penumbra of a sunspot. We have argued that this structure owes its origin in part to downward pumping of magnetic flux by vigorous granular convection in the region surrounding the spot. Here we stress the difference between the inner and outer penumbra, and correct some misleading assertions in a recent paper by Spruit & Scharmer. Title: Sunspot Structure and Dynamics Authors: Weiss, N. O. Bibcode: 2006SSRv..124...13W Altcode: 2006SSRv..tmp...87W Sunspots are the most prominent magnetic features on the Sun but it is only within the last few years that the intricate structure of their magnetic fields has been resolved. In the penumbra the fields in bright and dark filaments differ in inclination by 30°. The field in the bright filaments is less inclined to the vertical, while the field in dark filaments becomes almost horizontal at the edge of the spot. Recent models suggest that this interlocking-comb structure is maintained through downward pumping of magnetic flux by small-scale granular convection, and that filamentation originates as a convective instability. Within the bright filaments convection patterns travel radially owing to the inclination of the field. A proper understanding of these processes requires new observations, from space and from the ground, coupled with large-scale numerical modelling. Title: Hughes, Rosner, Weiss: Stellar MHD: Magnetohydrodynamics of stellar interiors Authors: Hughes, David; Rosner, Robert; Weiss, Nigel Bibcode: 2005A&G....46d..35H Altcode: David Hughes, Robert Rosner and Nigel Weiss describe what was achieved during a programme on stellar magnetic fields at the Isaac Newton Institute in Cambridge. Over a four-month period more than 90 participants visited the Institute for a mixture of structured workshops and informal collaboration. Title: Linear and nonlinear dynamos Authors: Weiss, N. O. Bibcode: 2005AN....326..157W Altcode: More than eighty years have passed since the idea of a solar dynamo was first put forward, and almost fifty years since Parker's pioneering paper. Yet dynamo theory remains a fertile subject and continues to raise new challenges, both physical and mathematical. So far, nearly all treatments have relied on mean field electrodynamics. The earliest models were linear but recent treatments have been nonlinear and increasingly sophisticated. During his career, Michael Stix has made major contributions to all branches of this subject. In this survey I shall review the historical development of dynamo theory, as applied to stars like the Sun, and then focus on some issues of current interest, ranging from axisymmetric calculations in spherical geometry to low-order models. Finally, I shall look ahead to what is needed in order to develop fully consistent, three-dimensional numerical models of the solar cycle. Title: Book review of The magnetic universe, geophysical and astrophysical dynamo theory, by G. Rüdiger and R. Hollerbach, Wiley-VCH, Weinheim, 2004, XI+332 pp., £90.00, $161.00, &euro;135.00, hardback (ISBN: 3-527-40409-0). Authors: Weiss, Nigel Bibcode: 2005GApFD..99..347W Altcode: No abstract at ADS Title: Fluid Dynamics and Dynamos in Astrophysics and Geophysics: reviews emerging from the Durham Symposium on Astrophysical Fluid Mechanics held July 29 to August 8, 2002 Authors: Soward, Andrew M.; Jones, Christopher A.; Hughes, David W.; Weiss, Nigel O. Bibcode: 2005fdda.conf.....S Altcode: No abstract at ADS Title: Solar and stellar physics - II Authors: Weiss, N. O. Bibcode: 2004Obs...124..348W Altcode: No abstract at ADS Title: Fine Structure in Sunspots Authors: Thomas, John H.; Weiss, Nigel O. Bibcode: 2004ARA&A..42..517T Altcode: Important physical processes on the Sun, and especially in sunspots, occur on spatial scales at or below the limiting resolution of current solar telescopes. Over the past decade, using a number of new techniques, high-resolution observations have begun to reveal the complex thermal and magnetic structure of a sunspot, along with associated flows and oscillations. During this time remarkable advances in computing power have allowed significant progress in our theoretical understanding of the dynamical processes, such as magnetoconvection, taking place within a sunspot. In this review we summarize the latest observational results and theoretical interpretations of the fine structure in sunspots. A number of projects underway to build new solar telescopes or upgrade existing ones, along with several promising new theoretical ideas, ensure that there will be significant advances in sunspot research over the coming decade. Title: Sunspots: The puzzling structure of a sunspot Authors: Tobias, Steven; Weiss, Nigel Bibcode: 2004A&G....45d..28T Altcode: High-resolution observations of fine structure in sunspots have at last made it possible to answer a very old question: what causes the filamentary structure of the penumbra, with magnetic fields forming an interlocking comb-like configuration, that surrounds the dark central umbra of a sunspot? This unexpected geometry can only be explained by an understanding of the interaction between magnetic fields and small-scale turbulent convection outside the spot. This interaction leads to downwards transport of magnetic flux, submerging magnetic flux tubes below the solar surface and resulting in a strongly fluted structure. An idealized version of this process can be demonstrated in numerical experiments. Title: On the origin of filamentary structure in sunspot penumbrae: non-linear results Authors: Tildesley, M. J.; Weiss, N. O. Bibcode: 2004MNRAS.350..657T Altcode: As the magnetic flux in a pore increases, it grows in radius and the magnetic field at its outer edge becomes increasingly inclined to the vertical. It is believed that this increased field inclination causes filamentary convection to set in and that this new pattern of convection eventually develops into the highly fluted azimuthal structure of sunspots. The linear instability was investigated in a highly idealized Boussinesq model by Tildesley. A non-linear extension of this work is carried out here and the saturation of the instability is explored. These solutions are contrasted with those obtained when the upper-boundary condition is modified by matching the magnetic field to a potential field above the convecting layer. In the non-linear regime an alternating pattern of hot and cool stripes is produced at the upper surface. The results from these model calculations are related to the transition from pores to protospots and to the development of penumbral filaments in sunspots. Title: Global Magnetorotational Instability with Inflow. I. Linear Theory and the Role of Boundary Conditions Authors: Kersalé, Evy; Hughes, David W.; Ogilvie, Gordon I.; Tobias, Steven M.; Weiss, Nigel O. Bibcode: 2004ApJ...602..892K Altcode: We formulate a model system suitable for the systematic numerical investigation of global aspects of the magnetorotational instability and nonlinear dynamo action in accretion disks. The model consists of a cylindrical annulus occupied by an incompressible fluid with explicit viscosity and resistivity. Boundary conditions are imposed that permit an accretion flow appropriate to the stresses acting within the fluid to develop freely through the annulus. A steady basic state is identified in which a slow, steady accretion flow is driven by the explicit viscosity. We investigate the linear theory of this state subject to different choices of boundary conditions. The choice of boundary conditions is a crucial factor in determining the nature and growth rate of the instabilities. It is found that very rapidly growing wall modes occur generically, drawing energy artificially from outside the computational domain. However, by carefully selecting boundary conditions for which the total pressure is constrained at the radial boundaries, the wall modes are found to have growth rates bounded by the local properties of the magnetorotational instability. The resulting model provides the basis for a systematic exploration of nonlinear behavior in our future work. Title: The Origin of Penumbral Structure in Sunspots: Downward Pumping of Magnetic Flux Authors: Weiss, Nigel O.; Thomas, John H.; Brummell, Nicholas H.; Tobias, Steven M. Bibcode: 2004ApJ...600.1073W Altcode: This paper offers the first coherent picture of the interactions between convection and magnetic fields that lead to the formation of the complicated filamentary structure of a sunspot penumbra. Recent observations have revealed the intricate interlocking-comb structure of the penumbral magnetic field. Some field lines, with associated Evershed outflows, plunge below the solar surface near the edge of the spot. We claim that these field lines are pumped downward by small-scale granular convection outside the sunspot. This mechanism is demonstrated in numerical experiments. Magnetic pumping is a key new ingredient that links several theoretical ideas about penumbral structure and dynamics; it explains not only the abrupt appearance of a penumbra as a pore increases in size but also the behavior of moving magnetic features outside a spot. Title: The Sun, An Introduction, by M. Stix, Astronomy and Astrophysics Library, Springer Berlin, Heidelberg, New York, Barcelona, Hong Kong, London, Milan, Paris, Tokyo, 2nd ed. 2002, XVI+490 pp., EUR 79,95, hardback (ISBN: 3-540-42886-0) Authors: Weiss, Nigel Bibcode: 2003GApFD..97..421W Altcode: Available from http://taylorandfrancis.metapress.com/openurl.asp?genre=article&issn=0309-1929&volume=97&issue=5&spage=421 Title: On the Interaction between Convection and Magnetic Fields Authors: Cattaneo, Fausto; Emonet, Thierry; Weiss, Nigel Bibcode: 2003ApJ...588.1183C Altcode: Turbulent convection in the solar photosphere can act as a small-scale dynamo, maintaining a disordered magnetic field that is locally intense. On the other hand, convection is inhibited in the presence of a strong, externally imposed magnetic field, as for instance, in a sunspot. Large-scale, three-dimensional, numerical experiments on highly nonlinear magnetoconvection in a Boussinesq fluid show that there is a continuous transition from a dynamo regime through a convective regime to an oscillatory regime as the strength of the imposed magnetic field is progressively increased. The patterns found in these different regimes are described and analyzed. Title: Modelling solar and stellar magnetoconvection Authors: Weiss, Nigel Bibcode: 2003safd.book..329W Altcode: Numerical experiments on three-dimensional convection in the presence of an externally imposed magnetic field reveal a range of behaviour that can be compared with that observed at the surface of the Sun (and therefore expected to be present in other similar stars). In a strongly stratified compressible layer small-scale convection gives way to a regime with flux separation as the field strength is reduced; with a weak mean field magnetic flux is concentrated into narrow lanes enclosing vigorously convecting plumes. Small-scale dynamos, generating disordered magnetic fields, have been found in Boussinesq calculations with very high magnetic Reynolds numbers; there is a gradual transition from dynamo action to magnetoconvection as the strength of the imposed field is increased. Title: Modelling Stellar Magnetoconvection Authors: Weiss, N. O. Bibcode: 2003IAUS..210..127W Altcode: No abstract at ADS Title: Magnetic flux separation in photospheric convection Authors: Weiss, N. O.; Proctor, M. R. E.; Brownjohn, D. P. Bibcode: 2002MNRAS.337..293W Altcode: Three-dimensional non-linear magnetoconvection in a strongly stratified compressible layer exhibits different patterns as the strength of the imposed magnetic field is reduced. There is a transition from a magnetically dominated regime, with small-scale convection in slender hexagonal cells, to a convectively dominated regime, with clusters of broad rising plumes that confine the magnetic flux to narrow lanes where fields are locally intense. Both patterns can coexist for intermediate field strengths, giving rise to flux separation: clumps of vigorously convecting plumes, from which magnetic flux has been excluded, are segregated from regions with strong fields and small-scale convection. A systematic numerical investigation of these different states shows that flux separation can occur over a significant parameter range and that there is also hysteresis. The results are related to the fine structure of magnetic fields in sunspots and in the quiet Sun. Title: Downward pumping of magnetic flux as the cause of filamentary structures in sunspot penumbrae Authors: Thomas, John H.; Weiss, Nigel O.; Tobias, Steven M.; Brummell, Nicholas H. Bibcode: 2002Natur.420..390T Altcode: The structure of a sunspot is determined by the local interaction between magnetic fields and convection near the Sun's surface. The dark central umbra is surrounded by a filamentary penumbra, whose complicated fine structure has only recently been revealed by high-resolution observations. The penumbral magnetic field has an intricate and unexpected interlocking-comb structure and some field lines, with associated outflows of gas, dive back down below the solar surface at the outer edge of the spot. These field lines might be expected to float quickly back to the surface because of magnetic buoyancy, but they remain submerged. Here we show that the field lines are kept submerged outside the spot by turbulent, compressible convection, which is dominated by strong, coherent, descending plumes. Moreover, this downward pumping of magnetic flux explains the origin of the interlocking-comb structure of the penumbral magnetic field, and the behaviour of other magnetic features near the sunspot. Title: Umbral and penumbral magnetoconvection Authors: Weiss, N. O. Bibcode: 2002AN....323..371W Altcode: Compressible convection in a strong vertical magnetic field can take two forms: either there is a regular pattern of small plumes or else the field separates from the motion giving regions of vigorous convection and regions that are magnetically dominated. In an inclined field, patterns are bound to travel. When the magnetic field is almost horizontal, convection takes the form of rolls aligned with the field. In the umbra of a sunspot convection will take the form of spatially modulated oscillations, giving rise to umbral dots. The penumbral field shows an interlocking-comb structure. In the bright filaments the magnetic field is inclined to the horizontal and modulated travelling waves, moving inwards or outwards, can give rise to the migration of bright grains. In the dark filaments the field is nearly horizontal, allowing only inefficient overturning convection. Title: Magnetic flux pumping and the structure of a sunspot penumbra Authors: Thomas, J. H.; Weiss, N. O.; Tobias, S. M.; Brummell, N. H. Bibcode: 2002AN....323..383T Altcode: We propose an overall scenario for the development and maintenance of a sunspot penumbra, in which turbulent magnetic flux pumping plays a key role. Recent high-resolution observations have revealed arched, "returning" magnetic flux tubes that emerge in the inner or middle penumbra, dive back down below the solar surface near the outer edge of the penumbra, and carry much of the Evershed flow. Some mechanism is required to keep the outer parts of the returning flux tubes submerged in spite of their magnetic buoyancy. We have proposed that the relevant mechanism is downward turbulent pumping of magnetic flux by granular convection in the moat outside the sunspot. This mechanism is demonstrated by means of an appropriate three-dimensional numerical simulation of turbulent compressible convection in the strongly superadiabatic granulation layer. We suggest that a penumbra first forms through a convectively driven instability at the outer edge of a growing pore. The nonlinear development of this instability pUSA)roduces the filamentary penumbra with its interlocking-comb magnetic field geometry. Downward flux pumping of some of the nearly horizontal magnetic flux in the dark filaments produces the returning flux tubes, with their associated Evershed flow, and also establishes the subcritical nature of the bifurcation that produces the filamentary penumbra, which explains why there are pores larger than the smallest sunspots. Title: Presidential Address: Dynamos in planets, stars and galaxies Authors: Weiss, Nigel Bibcode: 2002A&G....43c...9W Altcode: Nigel Weiss discusses dynamos in settings as diverse as galaxies and planets in this his 2000 Presidential Address to the Ordinary (A&G) Meeting of the Royal Astronomical Society. <heading id="h10" level="1"</heading>

Global magnetic fields in the Earth and other major planets, in the Sun and other active stars, and also in spiral galaxies like the Milky Way, are apparently maintained by hydromagnetic dynamos. This lecture will contrast the various dynamo models that have been put forward. Powerful supercomputers have now made it possible to simulate the geodynamo in considerable detail. In the case of the Sun, we have yet to explain its internal rotation and models still have to rely on mean field dynamo theory. There are many features that can be explained but extrapolation to other stars remains uncertain. For galaxies, even the need for a dynamo is still controversial. Title: On the Origin of the Solar Mesogranulation Authors: Cattaneo, Fausto; Lenz, Dawn; Weiss, Nigel Bibcode: 2001ApJ...563L..91C Altcode: The observed properties of mesogranules are related to structures found in idealized numerical experiments on turbulent convection. We describe results obtained for three-dimensional Boussinesq convection in a layer with a very large aspect ratio. There are two distinct cellular patterns at the surface. Energy-transporting convection cells (corresponding to granules in the solar photosphere) have diameters comparable to the layer depth, while macrocells (corresponding to mesogranules) are several times larger. The motion acts as a small-scale turbulent dynamo, generating a disordered magnetic field that is concentrated at macrocellular corners and, to a lesser extent, in the lanes that join them. These results imply that mesogranules owe their origin to collective interactions between the granules. Title: Sustaining the Sun's Magnetic Network with Emerging Bipoles Authors: Simon, G. W.; Title, A. M.; Weiss, N. O. Bibcode: 2001ApJ...561..427S Altcode: The Michelson Doppler Imager experiment on SOHO has revealed a ``magnetic carpet'' dominated by the emergence of bipolar magnetic flux in ephemeral active regions, which subsequently split into small flux elements that drift into the magnetic network. The effects of granular and supergranular convection on these flux elements are represented here by kinematic modeling: Elementary flux tubes are transported passively by the supergranular flow, while experiencing small random displacements produced by granulation. They end up in the magnetic network that surrounds the supergranules, where they eventually meet oppositely directed fields and are annihilated. The model calculations show that the total unsigned magnetic flux will decay within a few days unless it is continually replenished. A statistically steady state with a total unsigned flux of 2-3×1023 Mx over the whole solar surface can be maintained if bipolar flux emerges at a rate of 7×1022 Mx day-1, as indicated by published measurements of the rate at which ephemeral active regions appear. Title: Presidential Address: Turbulent magnetic fields in the Sun Authors: Weiss, Nigel Bibcode: 2001A&G....42c..10W Altcode: Nigel Weiss recounts his Presidential Address 2001, given to the RAS A&G Ordinary Meeting on 9 February 2001.

Recent high-resolution observations, from the ground and from space, have revealed the fine structure of magnetic features at the surface of the Sun. At the same time, advances in computing power have at last made it possible to develop models of turbulent magnetoconvection that can be related to these observations. The key features of flux emergence and annihilation, as observed by the MDI experiment on SOHO, are reproduced in kinematic calculations, while three-dimensional numerical experiments reveal the dynamical processes that are involved. The pattern of convection depends on the strength of the magnetic field: as the mean field decreases, slender rising plumes give way to a regime where magnetic flux is separated from the motion and then to one where locally intense magnetic fields nestle between broad and vigorously convecting plumes. Moreover, turbulent convection is itself able to act as a small-scale dynamo, generating disordered fields near the solar surface. Title: How the Sun Maintains its Magnetic Network Authors: Simon, G. W.; Title, A. M.; Weiss, N. O. Bibcode: 2001AAS...198.8601S Altcode: 2001BAAS...33..913S The MDI experiment on SOHO has revealed a `magnetic carpet' dominated by the emergence of bipolar magnetic flux in ephemeral active regions, which subsequently split into small flux elements that drift into the magnetic network. The effects of granular and supergranular convection on these flux elements are represented here by kinematic modeling: Elementary flux tubes are transported passively by the supergranular flow, while experiencing small random displacements produced by granulation. They end up in the magnetic network that surrounds the supergranules, where they eventually meet oppositely directed fields and are annihilated. The model calculations show that the total unsigned magnetic flux will decay within a few days unless it is continually replenished. A statistically steady state with a total unsigned flux of 2-3 x 1023 Mx over the whole solar surface can be maintained if bipolar flux emerges at a rate of 7 x 1022 Mx d-1, as indicated by published measurements of the rate at which ephemeral active regions appear. Title: Flux Separation in Photospheric Magnetoconvection Authors: Weiss, N. O.; Proctor, M. R. E. Bibcode: 2001IAUS..203..219W Altcode: Numerical experiments on three-dimensional magnetoconvection in a stratified compressible layer yield results that can now be compared with the high resolution observations of granulation and intergranular magnetic fields obtained at La Palma, and related to the slender loops revealed by TRACE. As the imposed magnetic field strength is decreased there is a transition from small-scale plumes, in the magnetically dominated regime, to large-scale vigorous plumes when the field is dominated by the motion. In the intermediate regime magnetic flux separates from the motion, so that there are almost field-free regions, with clusters of vigorous plumes, surrounded by regions where the Lorentz force is strong enough to control the dynamics. There is a range of field strengths where either small-scale plumes or flux-separated solutions can persist, depending on initial conditions for the computation. The patterns revealed in these calculations can be related to convection in sunspot umbrae (where there is a strong vertical field, and the appearance of umbral dots, to the formation of plage regions and to the behaviour of intergranular fields in the quiet Sun. In the weak field limit, turbulent convection is able to act as a dynamo and to maintain a disordered field. Title: Generation of Coronal Currents by the Solar Convection Zone Authors: Galloway, D. J.; Uchida, Y.; Weiss, N. O. Bibcode: 2001PASA...18..329G Altcode: : Solar flares are thought to be caused by reconnection of magnetic fields and their associated electric currents in the solar corona. The currents have to be there to provide available energy over and above the current-free minimum energy state, but what generates them has been little discussed. This paper investigates the idea that twisting motions in the turbulent convection zone below may provide a natural source for the currents and explain some of their properties. The twists generate upward-propagating Alfvén waves with a Poynting flux of the right order of magnitude to power a flare. Depending on the depth it takes place, the twisting event that initiates a particular flare may occur hours, days or even months before the flare itself. Title: Physical Causes of Solar Activity Authors: Weiss, N. O.; Tobias, S. M. Bibcode: 2000SSRv...94...99W Altcode: The magnetic fields that dominate the structure of the Sun's atmosphere are controlled by processes in the solar interior, which cannot be directly observed. Magnetic activity is found in all stars with deep convective envelopes: young and rapidly rotating stars are very active but cyclic activity only appears in slow rotators. The Sun's 11-year activity cycle corresponds to a 22-year magnetic cycle, since the sunspot fields (which are antisymmetric about the equator) reverse at each minimum. The record of magnetic activity is aperiodic and is interrupted by episodes of reduced activity, such as the Maunder Minimum in the seventeenth century, when sunspots almost completely disappeared. The proxy record from cosmogenic isotopes shows that similar grand minima recur at intervals of around 200 yr. The Sun's large-scale field is generated by dynamo action rather than by an oscillator. Systematic magnetic cycles are apparently produced by a dynamo located in a region of weak convective overshoot at the base of the convection zone, where there are strong radial gradients in the angular velocity Ω. The crucial parameter (the dynamo number) increases with increasing Ω and kinematic (linear) theory shows that dynamo action can set in at an oscillatory (Hopf) bifurcation that is probably subcritical. Although it has been demonstrated that the whole process works in a self-consistent model, most calculations have relied on mean-field dynamo theory. This approach is physically plausible but can only be justified under conditions that do not apply in the Sun. Still, mean-field dynamos do reproduce the butterfly diagram and other key features of the solar cycle. An alternative approach is to study generic behaviour in low-order models, which exhibit two forms of modulation, associated with symmetry-breaking and with reduced activity. Comparison with observed behaviour suggests that modulation of the solar cycle is indeed chaotic, i.e. deterministically rather than stochastically driven. Title: Sunspots Authors: Weiss, N. Bibcode: 2000eaa..bookE2050W Altcode: Sunspots are the sites of strong magnetic fields at the surface of the Sun. They are visible as features that appear dark because they are cooler than the surrounding SOLAR PHOTOSPHERE, owing to partial suppression of convective energy transport by the magnetic field. Sunspots are distinguished from SUNSPOT PORES by having a filamentary SUNSPOT PENUMBRA surrounding a dark SUNSPOT UMBRA. Their dia... Title: Resonance in a coupled solar-climate model Authors: Tobias, S. M.; Weiss, N. O. Bibcode: 2000SSRv...94..153T Altcode: The 11-year solar activity cycle is magnetic in origin and is responsible for small changes in solar luminosity and the modulation of the solar wind. The terrestrial climate exhibits much internal variability supporting oscillations with many frequencies. The direct effect of changing solar irradiance in driving climatic change is believed to be small, and amplification mechanisms are needed to enhance the role of solar variability. In this paper we demonstrate that resonance may play a crucial role in the dynamics of the climate system, by using the output from a nonlinear solar dynamo model as a weak input to a simplified climate model. The climate is modelled as oscillating about two fixed points (corresponding to a warm and cold state) with the weak chaotically modulated solar forcing on average pushing the solution towards the warm state. When a typical frequency of the input is similar to that of the chaotic climate system then a dramatic increase in the role of the solar forcing is apparent and complicated intermittent behaviour is observed. The nonlinear effects are subtle however, and forcing that on average pushes the solution towards the warm state may lead to increased intervals of oscillation about either state. Owing to the intermittent nature of the timeseries, analysis of the relevant timeseries is shown to be non-trivial. Title: Turbulent Convection, Rotation, and the Solar Dynamo Authors: Tobias, Steve; Toomre, Juri; Weiss, Nigel Bibcode: 2000astu.progE..30T Altcode: No abstract at ADS Title: The Influence of Internal Heating on Nonlinear Compressible Convection Authors: Hurlburt, N. E.; Weiss, N. O. Bibcode: 2000SPD....31.0506H Altcode: 2000BAAS...32..837H In the bulk of the solar convection zone we expect convection to be efficient and therefore maintain an adiabatic temperature gradient. In most numerical simulations of solar convection the total energy flux within this region is due to the conduction down this gradient (which is small) and the various contributions due to the convective motions. What has often been neglected is the contribution that is transported by radiation. The contribution of this flux decreases across the layer and thereby deposits a significant amount of thermal energy in the midst of the convection zone. This is in contrast to most simulations of the convection where the input of energy is supplied exclusively by conduction from the boundaries. Mixing length models predict that approximately half of the total energy input to the solar convection zone is deposited, more-or-less uniformly over the convection zone, with the remaining half being conducted from the lower boundary. Thus the study of the behavior of internally-heated compressible convection is warranted. Previous studies of internally heated compressible convection have been inconclusive due to the shearing instabilities that arise in simple, periodic domains. Here we suppress these instabilities by considering flows in axisymmetric geometries. We conduct surveys of the structure and dynamics of the resulting flows and present possible applications to observed solar and stellar phenomena. Title: Resonance in a Coupled Solar-Climate Model Authors: Tobias, S. M.; Weiss, N. O. Bibcode: 2000svc..book..153T Altcode: No abstract at ADS Title: Physical Causes of Solar Activity Authors: Weiss, N. O.; Tobias, S. M. Bibcode: 2000svc..book...99W Altcode: No abstract at ADS Title: Symmetry Considerations in Stellar Dynamos Authors: Tobias, S. M.; Knobloch, E.; Weiss, N. O. Bibcode: 1999ASPC..178..185T Altcode: 1999sdnc.conf..185T No abstract at ADS Title: Modelling photospheric magnetoconvection Authors: Blanchflower, S. M.; Rucklidge, A. M.; Weiss, N. O. Bibcode: 1998MNRAS.301..593B Altcode: The increasing power of computers makes it possible to model the non-linear interaction between magnetic fields and convection at the surfaces of solar-type stars in ever greater detail. We present the results of idealized numerical experiments on two-dimensional magnetoconvection in a fully compressible perfect gas. We first vary the aspect ratio lambda of the computational box and show that the system runs through a sequence of convective patterns, and that it is only for a sufficiently wide box (lambda>=6) that the flow becomes insensitive to further increases in lambda. Next, setting lambda=6, we decrease the field strength from a value strong enough to halt convection and find transitions to small-scale steady convection, next to spatially modulated oscillations (first periodic, then chaotic) and then to a new regime of flux separation, with regions of strong field (where convection is almost completely suppressed) separated by broad convective plumes. We also explore the effects of altering the boundary conditions and show that this sequence of transitions is robust. Finally, we relate these model calculations to recent high-resolution observations of solar magnetoconvection, in plage regions as well as in light bridges and the umbrae of sunspots. Title: Flux expulsion by inhomogeneous turbulence Authors: Tao, L.; Proctor, M. R. E.; Weiss, N. O. Bibcode: 1998MNRAS.300..907T Altcode: Flux expulsion is an important consequence of the interaction of magnetic fields with fluid convection and has been well studied for particular cases of steady, single-cell flows. Here we examine a related phenomenon in inhomogeneous turbulence using direct numerical simulations. To understand our numerical results, we analyse average properties of our model, and obtain mean transport coefficients which can be used to describe the approach of the system to its final state. For the kinematic problem these transport coefficients give an excellent prediction of the expulsion process; however, the enhanced transport is suppressed by dynamical back-reaction of the Lorentz force. Finally, we discuss the astrophysical implications for magnetic fields in stellar convection zones. Segregation of magnetic fields from turbulent motion not only allows strong toroidal fields to accumulate in regions of convective overshoot but also permits significant poloidal fields to be maintained by dynamo action in stars like the Sun. Title: Modulation and symmetry changes in stellar dynamos Authors: Knobloch, E.; Tobias, S. M.; Weiss, N. O. Bibcode: 1998MNRAS.297.1123K Altcode: Stellar dynamos are governed by non-linear partial differential equations (PDEs) which admit solutions with dipole, quadrupole or mixed symmetry (i.e. with different parities). These PDEs possess periodic solutions that describe magnetic cycles, and numerical studies reveal two different types of modulation. For modulations of Type 1 there are parity changes without significant changes of amplitude, while for Type 2 there are amplitude changes without significant changes in parity. In stars like the Sun, cyclic magnetic activity is interrupted by grand minima that correspond to Type 2 modulation. Although the Sun's magnetic field has maintained dipole symmetry for almost 300 yr, there was a significant parity change at the end of the Maunder Minimum. We infer that the solar field may have flipped from dipole to quadrupole polarity (and back) after deep minima in the past and may do so again in the future. Other stars, with different masses or rotation rates, may exhibit cyclic activity with dipole, quadrupole or mixed parity. The origins of such behaviour can be understood by relating the PDE results to solutions of appropriate low-order systems of ordinary differential equations (ODEs). Type 1 modulation is reproduced in a fourth-order system while Type 2 modulation occurs in a third-order system. Here we construct a new sixth-order system that describes both types of modulation and clarifies the interactions between symmetry-breaking and modulation of activity. Solutions of these non-linear ODEs reproduce the qualitative behaviour found for the PDEs, including flipping of polarity after a prolonged grand minimum. Thus we can be confident that these patterns of behaviour are robust, and will apply to stars that are similar to the Sun. Title: An Active Sun Throughout the Maunder Minimum Authors: Beer, Jürg; Tobias, Steven; Weiss, Nigel Bibcode: 1998SoPh..181..237B Altcode: Measurements of 10Be concentration in the Dye 3 ice core show that magnetic cycles persisted throughout the Maunder Minimum, although the Sun's overall activity was drastically reduced and sunspots virtually disappeared. Thus the dates of maxima and minima can now be reliably estimated. Similar behaviour is shown by a nonlinear dynamo model, which predicts that, after a grand minimum, the Sun's toroidal field may switch from being antisymmetric to being symmetric about the equator. The presence of cyclic activity during the Maunder Minimum limits estimates of the solar contribution to climatic change. Title: Flux Separation in Stellar Magnetoconvection Authors: Tao, L.; Weiss, N. O.; Brownjohn, D. P.; Proctor, M. R. E. Bibcode: 1998ApJ...496L..39T Altcode: The effect of a strong magnetic field on photospheric convection in a cool star like the Sun can be established by relating high-resolution solar observations to results from nonlinear models that rely on computation. The patterns of motion in numerical experiments on three-dimensional, compressible magnetoconvection depend not only on the strength of the imposed vertical magnetic field but also on the aspect ratio λ of the computational box. In a wide box (λ=8) with a moderately strong field, the flow organizes itself so that magnetic flux is separated from the motion. There are regions with strong fields and small-scale oscillatory convection next to almost field-free regions with clusters of broad and vigorous convective plumes. In the solar photosphere, this corresponds to the difference between the patterns of granulation in plage regions (with fields greater than 100 G) and in the adjacent quiet Sun. Title: Kinematic Modeling of Vortices in the Solar Photosphere Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1997ApJ...489..960S Altcode: The application of local correlation tracking to the proper motions of granules yields the horizontal velocity field at the solar surface. The divergence of the velocity reveals a pattern of sources and sinks. The vorticity is concentrated at sinks to form local swirls (with either sense of motion). A simple kinematic model of such a vortex, in which the radial inflow is balanced by an eddy viscosity, predicts that the vorticity should have a Gaussian profile. This prediction is confirmed by comparison with three sets of high-resolution observations, obtained from Spacelab 2 and from the Swedish Vacuum Solar Telescope on La Palma. Finally, we develop a more precise version of the model and provide an estimate of the eddy viscosity due to small-scale granulation. Title: Physics of the solar dynamo Authors: Weiss, N. O. Bibcode: 1997ppvs.conf..325W Altcode: Introduction Stellar magnetic cycles Solar magnetic activity Stellar activity Origins of the Sun's magnetic field Dynamo theory The kinematic dynamo problem Nonlinear equilibration Mean-field dynamo theory Modulation of activity cycles Toy models Global models Solar activity and climatic change Observational correlations Mechanisms and speculations Title: Modulation of Solar and Stellar Activity Cycles Authors: Weiss, N. O.; Tobias, S. M. Bibcode: 1997LNP...489...25W Altcode: 1997shpp.conf...25W Cyclic magnetic activity in the Sun and other similar stars is interrupted by episodes of reduced activity such as the Maunder Minimum. This pattern is reproduced in mean field (αθ) dynamo models, where growth of the field is limited by the nonlinear action of the Lorentz force on differential rotation. The observed aperiodicity can be ascribed to deterministic rather than stochastic processes, and chaotic modulation is demonstrated for a low-order system, for two-dimensional dynamo waves and for a simple global model. Amplitude modulation that leads to grand minima is distinct from modulation associated with changes in the symmetry of the field. Title: Magnetoconvection Authors: Weiss, N. O. Bibcode: 1997ASPC..118...21W Altcode: 1997fasp.conf...21W Two recent computational treatments of the nonlinear interaction between convection and magnetic fields are described and contrasted. The first is a sequence of numerical experiments on an idealized model of three-dimensional compressible magnetoconvection, with an emphasis on changes in the magnetic field structure and the velocity pattern as the strength of the imposed field is reduced. The second is a two-dimensional simulation of the formation and evolution of a highly dynamical magnetic flux sheet, in a realistic model of the solar photosphere. The results are related to high-resolution observations of fine structure in sunspots and to the behaviour of intergranular magnetic fields. Title: Photospheric Convection in Strong Magnetic Fields Authors: Weiss, N. O.; Brownjohn, D. P.; Matthews, P. C.; Proctor, M. R. E. Bibcode: 1996MNRAS.283.1153W Altcode: The effect of magnetic fields on convection at the surfaces of cool stars can be explored by comparing the results of detailed numerical experiments with high-resolution solar observations. We have investigated non-linear three-dimensional magnetoconvection in a fully compressible perfect gas. In this paper we study the effect of an imposed magnetic field on the pattern of convection in a deep stratified layer. When the field is strong enough to dominate the motion we find steady convection with rising plumes on a deformed hexagonal lattice, and a magnetic network at the upper boundary. This gives way to spatially modulated oscillations for weaker fields. As the field strength is further reduced the oscillations become more violent and irregular, and their horizontal scale increases. Magnetic flux moves rapidly along the network that encloses the ephemeral plumes; when the imposed field is relatively weak, intense fields appear at junctions in the network, where the magnetic pressure is comparable to the gas pressure and an order of magnitude greater than the dynamic pressure. This behaviour is related to convection in sunspots and plages and to the structure of intergranular magnetic fields on the Sun. Title: Dynamos in Different Types of Stars Authors: Weiss, N. O. Bibcode: 1996mpsa.conf..387W Altcode: 1996IAUCo.153..387W No abstract at ADS Title: Double-diffusive convection with two stabilizing gradients: strange consequences of magnetic buoyancy. Authors: Hughes, D. W.; Weiss, N. O. Bibcode: 1995JFM...301..383H Altcode: Instabilities due to a vertically stratified horizontal magnetic field (magnetic buoyancy instabilities) are believed to play a key role in the escape of the Sun's internal magnetic field and the formation of active regions and sunspots. In a star the magnetic diffusivity is much smaller than the thermal diffusivity and magnetic buoyancy instabilities are double-diffusive in character. The authors have studied the nonlinear development of these instabilities, in an idealized two-dimensional model, by exploiting a non-trivial transformation between the governing equations of magnetic buoyancy and those of classical thermosolutal convection. The main result is extremely surprising. They have demonstrated the existence of finite-amplitude steady convection when both the influential gradients (magnetic and convective) are stabilizing. This strange behaviour is caused by the appearance of narrow magnetic boundary layers, which distort the mean pressure gradient so as to produce a convectively unstable stratification. Title: Chaotically modulated stellar dynamos Authors: Tobias, S. M.; Weiss, N. O.; Kirk, V. Bibcode: 1995MNRAS.273.1150T Altcode: Slowly rotating, late-type stars show intermittent cyclic magnetic activity, interrupted by grand minima. The record of the solar cycle is apparently chaotic. We construct a simple third-order model of a stellar dynamo following two different procedures. The first uses simple physical arguments based on the processes that occur in a dynamo. The second relies on bifurcation theory, with no explicit reference to the dynamo equations, and displays the essential mathematical structure of the model. Modulation of the basic cycle and chaos are found to be a natural consequence of the transition from a non-magnetic state to one with periodically reversing fields. The model is related to more general normal forms and it is shown that the behaviour described is generic and therefore robust. Title: Kinematic Models of Supergranular Diffusion on the Sun Authors: Simon, G. W.; Title, A. M.; Weiss, N. O. Bibcode: 1995ApJ...442..886S Altcode: We develop kinematic models of diffusion generated by supergranulation at the solar surface. These models use current observations for the size, horizontal velocity, and lifetime of supergranules. Because there is no observational description of the appearance and disappearance of supergranules, we investigate models using several plausible evolution processes, including the effect of different lifetime distribution functions for the cells. The results are quite insensitive to the methods chosen to replace old supergranules, the distribution of cell lifetimes, and even the cell lifetime itself, for mean lifetimes between 15 to 30 hr. Calculated diffusion coefficients range between 500 and 700 sq km/s, in agreement with the best fit diffusion coefficients used by Sheeley and his collaborators to model the large-scale distribution of magnetic fields over the solar surface. However, our models do not explain the field distribution in plage, and they predict that virtually all the strong field in quiet Sun exists in relatively isolated clumps. We suggest possible mechanisms for the creation of plage and the bright network seen in quiet Sun. Title: The Structure of Sunspots Authors: Weiss, N. O. Bibcode: 1995SPD....26..301W Altcode: 1995BAAS...27..952W No abstract at ADS Title: The abrupt development of penumbrae in sunspots Authors: Rucklidge, A. M.; Schmidt, H. U.; Weiss, N. O. Bibcode: 1995MNRAS.273..491R Altcode: A sunspot is distinguished from a pore by having a filamentary penumbra, corresponding to convective motions that carry energy into the spot from the surrounding field-free plasma. A simplified model of energy transport in sunspots is developed in order to model the transition from pores to spots as the magnetic flux is varied. The observed overlap between the radii of large pores and small spots implies that the filamentary convective mode sets in suddenly and rapidly, as in the idealized case where pore solutions lose stability at a bifurcation. Title: Kinematic modelling of magnetoconvection Authors: Simon, G. W.; Title, A. M.; Weiss, N. O.; Ginet, G. P. Bibcode: 1994smf..conf..276S Altcode: No abstract at ADS Title: Solar and Stellar Dynamos Authors: Weiss, N. O. Bibcode: 1994lspd.conf...59W Altcode: No abstract at ADS Title: Modeling Solar Magnetoconvection Authors: Weiss, N. O. Bibcode: 1994ASPC...68...45W Altcode: 1994sare.conf...45W No abstract at ADS Title: Magnetoconvective patterns Authors: Weiss, N. O. Bibcode: 1994ASIC..433..287W Altcode: No abstract at ADS Title: Kinematic Modeling of Magnetic Field Diffusion at the Solar Surface Authors: Title, Alan M.; Simon, George W.; Weiss, Nigel O. Bibcode: 1994ASPC...68...87T Altcode: 1994sare.conf...87T No abstract at ADS Title: Nonlinear three-dimensional magnetoconvection in a compressible atmosphere Authors: Matthews, P. C.; Proctor, M. R. E.; Rucklidge, A. M.; Weiss, N. O. Bibcode: 1994smf..conf..279M Altcode: No abstract at ADS Title: Nonlinear stellar dynamos. Authors: Tobias, S. M.; Weiss, N. O. Bibcode: 1994cmcp.conf...46T Altcode: Slowly rotating late-type stars show intermittent cyclic activity, interrupted by grand minima. A simple third-order dynamo model describes the transition from a non-magnetic state to one with periodically reversing fields. As the rate of rotation is increased, subsequent bifurcations lead to periodically modulated cycles and then to chaotically modulated oscillations. Title: Diffusion of "Corks" Over the Solar Surface Authors: Title, A. M.; Simon, G. W.; Weiss, N. O. Bibcode: 1993BAAS...25Q1183T Altcode: No abstract at ADS Title: Book-Review - Sunspots - Theory and Observations Authors: Thomas, J. H.; Weiss, N. O.; Parkinson, J. H. Bibcode: 1993Obs...113..145T Altcode: No abstract at ADS Title: Diffusion of ``Corks'' Over the Solar Surface Authors: Title, A. M.; Simon, G. W.; Weiss, N. O. Bibcode: 1993AAS...182.4804T Altcode: 1993BAAS...25Q.880T Test particles in flow fields generated by correlation tracking of movies of the solar surface and kinematic models of the solar surface quickly collect in stagnation points of the flow fields and remain there. Test particles do not form a quasi-stable network pattern. The diffusion coefficients generated from a net of kinematic models with a range of cell sizes and lifetimes are not proportional to the cell size squared divided by the cell lifetime as commonally assumed. Reasonable estimates of cell sizes and lifetimes yield diffusiion coefficients that are lower than the 600 km(2) /s used by Sheeley and his collaborators in their surface diffusion models. We conclude that: 1) The appearance of plages and enhanced network can not be explained by adjustment of the cell sizes or surface velocities; and 2) diffusion is not sufficient to explain the appearance of plages and enhanced network. Title: Magnetic geometry of sunspots Authors: Weiss, Nigel Bibcode: 1993Natur.362..208W Altcode: No abstract at ADS Title: Solar and Stellar Dynamos Authors: Weiss, N. O. Bibcode: 1993ASSL..183..541W Altcode: 1993pssc.symp..541W No abstract at ADS Title: Oscillations and secondary bifurcations in nonlinear magnetoconvection Authors: Rucklidge, A. M.; Weiss, N. O.; Brownjohn, D. P.; Proctor, M. R. E. Bibcode: 1993GApFD..68..133R Altcode: Complicated bifurcation structures that appear in nonlinear systems governed by partial differential equations (PDEs) can be explained by studying appropriate low-order amplitude equations. We demonstrate the power of this approach by considering compressible magnetoconvection. Numerical experiments reveal a transition from a regime with a subcritical Hopf bifurcation from the static solution, to one where finite-amplitude oscillations persist although there is no Hopf bifurcation from the static solution. This transition is associated with a codimension-two bifurcation with a pair of zero eigenvalues. We show that the bifurcation pattern found for the PDEs is indeed predicted by the second-order normal form equation (with cubic nonlinearities) for a Takens - Bogdanov bifurcation with Z2 symmetry. We then extend this equation by adding quintic nonlinearities and analyse the resulting system. Its predictions provide a qualitatively accurate description of solutions of the full PDEs over a wider range of parameter values. Replacing the reflecting (Z2) lateral boundary conditions with periodic [O(2)] boundaries allows stable travelling wave and modulated wave solutions to appear; they could be described by a third-order system. Title: Bifurcations and Symmetry-Breaking in Simple Models of Nonlinear Dynamos Authors: Weiss, N. O. Bibcode: 1993IAUS..157..219W Altcode: No abstract at ADS Title: Symmetries of time-dependent magnetoconvection Authors: Proctor, M. R. E.; Weiss, N. O. Bibcode: 1993GApFD..70..137P Altcode: In the presence of a magnetic field, convection may set in at a stationary or an oscillatory bifurcation, giving rise to branches of steady, standing wave and travelling wave solutions. Numerical experiments provide examples of nonlinear solutions with a variety of different spatiotemporal symmetries, which can be classified by establishing an appropriate group structure. For the idealized problem of two-dimensional convection in a stratified layer the system has left-right spatial symmetry and a continuous symmetry with respect to translations in time. For solutions of period P the latter can be reduced to Z2 symmetry by sampling solutions at intervals of ½P. Then the fundamental steady solution has the spatiotemporal symmetry D2 = Z2 Z2 and symmetry-breaking yields solutions with Z2 symmetry corresponding to travelling waves, standing waves and pulsating waves. A further loss of symmetry leads to modulated waves. Interactions between the fundamental and its first harmonic are described by the group D2h = D2 ⊗ Z2 and its invariant subgroups, which describe solutions that are either steady or periodic in a uniformly moving frame. For a Boussinesq fluid in a layer with identical top and bottom boundary conditions there is also an up-down symmetry. With fixed lateral boundaries the spatiotemporal symmetries, again described by D2h and its invariant subgroups, can be related to results obtained in numerical experiments and analysed by Nagata et al. (1990). With periodic boundary conditions, the full symmetry group, D2h⊗Z2, is of order 16. Its invariant subgroups describe pure and mixed-mode solutions, which may be steady states, standing waves, travelling waves, pulsating waves or modulated waves. Title: X-Ray Observations of Sunspot Penumbral Structure Authors: Sams, B. J., III; Golub, L.; Weiss, N. O. Bibcode: 1992ApJ...399..313S Altcode: High-resolution coronal observations with the Normal Incidence X-ray Telescope (NIXT) reveal previously unobserved structure in the magnetic fields above a sunspot. The X-ray images are precisely aligned with a continuum photospheric image taken at the same time. The X-ray brightness traces magnetic field lines and shows: (1) that none of the bright loops originate in the spot umbra and (2) that some field lines emerging from the inner penumbra connect to regions far away from the spot. Such large-scale structures must remain distinct from the shallowly inclined fields in the outer penumbra. In particular, they cannot be involved in any interchange between the bright and dark filaments. This imposes constraints on models of penumbral convection. Title: Kinematic Modeling of Active Region Decay Authors: Simon, G. W.; Title, A. M.; Weiss, N. O. Bibcode: 1992AAS...180.1101S Altcode: 1992BAAS...24..746S No abstract at ADS Title: Convective motion on the Sun Authors: Weiss, Nigel Bibcode: 1992Natur.356..287W Altcode: No abstract at ADS Title: The Theory of Sunspots Authors: Thomas, John H.; Weiss, Nigel O. Bibcode: 1992ASIC..375....3T Altcode: 1992sto..work....3T This review covers the present state of our theoretical understanding of the physics of sunspots, along with the principal observational results that need to be explained. The topics covered range from the detailed structure of an individual sunspot to the broad connection between sunspots and the global solar magnetic field and the solar cycle. Our aim is to give a critical discussion of the theoretical ideas and models without presenting mathematical details. After outlining the historical development of the basic concepts associated with the magnetohydrodynamic theory of sunspots, we discuss recent treatments of their properties and structure, placing special emphasis on developments that have occurred within the last ten years. There have been remarkable improvements in the theoretical modelling of sunspots, led by new ideas and by more elaborate and realistic numerical simulations. At the same time, new observations have raised new theoretical questions or caused old ones to be reconsidered. In particular, measurements of oscillations in and around sunspots have opened up the new field of sunspot seismology, while recent high-resolution observations have forced us to rethink the structure of a sunspot penumbra. Title: Sunspots - Theory & Observations: NATO Cambridge, 1992 Authors: Thomas, John H.; Weiss, Nigel O. Bibcode: 1992ASIC..375.....T Altcode: 1992sto..work.....T The papers contained in this volume focus on theoretical problems associated with sunspots and present results of recent high-resolution observations of sunspots. In particular, attention is given to the evolution of sunspots, overall structure and fine structure of sunspots, waves and oscillations in sunspots, and the relation of sunspots to the global solar magnetic field. Specific topics discussed include continuum observations and empirical models of the thermal structure of sunspots, fine structure of umbrae and penumbrae, magnetohydrodynamic waves in structural magnetic fields, and the motion of magnetic flux tubes in the convection zone and the surface origin of active regions. Title: The Origin of the Solar Cycle Authors: Rosner, R.; Weiss, N. O. Bibcode: 1992ASPC...27..511R Altcode: 1992socy.work..511R No abstract at ADS Title: What is a stellar dynamo? Authors: Cattaneo, F.; Hughes, D. W.; Weiss, N. O. Bibcode: 1991MNRAS.253..479C Altcode: Numerical simulations of turbulent stellar dynamos are now feasible. The characteristic time-scale for kinematic behavior is related to the turnover time of the turbulent eddies. Results from idealized 2D models show that the Lorentz force alters the velocity field, allowing transient magnetic activity to persist for intervals much longer than the expected turbulent decay time. For 3D flows a characteristic time Te is defined for turbulent diffusion to act, based on the rate at which magnetic energy is dissipated, and it is asserted that there is a dynamo only if the field survives for times much longer than Te. This criterion is then applied to cyclic magnetic activity in late-type stars. Title: Convective structures in the sun Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1991MNRAS.252P...1S Altcode: Observed patterns of motion at the solar surface reflect the structure of subphotospheric convection, which controls the distribution of angular momentum and magnetic fields. Mesogranules are interpreted as secondary features associated with supergranular circulation, and that coupling between mesogranules and granulation triggers the spasmodic formation of exploding granules. Supergranules are expected to generate isolated sinking plumes. It is argued that these plumes can penetrate to the base of the convective zone, and that there is no organized structure on larger scales. Title: Symmetry breaking in stellar dynamos Authors: Jennings, R. L.; Weiss, N. O. Bibcode: 1991MNRAS.252..249J Altcode: The generation of magnetic fields in stars like the sun can be described by an azimuthally averaged dynamo model. Solutions of the linear (kinematic) problem have pure dipole or quadrupole symmetry, i.e., toroidal fields that are antisymmetric or symmetric about the equator. These symmetries can be broken only at bifurcations in the nonlinear regime, which lead to the appearance of spatially asymmetric mixed-mode solutions. The symmetries of dipole, quadrupole and mixed-mode solutions, whether steady or periodic, form the same group for any axisymmetric dynamo. To establish the bifurcation structure it is necessary to follow unstable as well as stable solutions. This is feasible only for simple systems and a minimal nonlinear alpha(omega) dynamo is studied in detail in order to illustrate the formation of mixed-mode periodic solutions and to distinguish between their symmetries. The results are applied to the sun (where there are persistent deviations from dipole symmetry) and to other late-type stars. Title: Modeling Mesogranules and Exploders on the Solar Surface Authors: Simon, G. W.; Title, A. M.; Weiss, N. O. Bibcode: 1991ApJ...375..775S Altcode: Radial outflows in exploders and mesogranules can be modeled by superposing Gaussian source functions. This model is used to explore the relationship between mesogranules and exploders. Although it is demonstarted that there is a mathematical equivalence between mesogranules and exploders distributed normally about the mesogranule centers, the results indicate that the observed mesogranular velocity pattern is not consistent with a flow pattern generated by exploders dropped randomly on the solar surface. Detailed comparisons with observations suggest that the averaged mesogranular velocity is produced by a combination of a persistent outflow from a source together with exploders distributed randomly about its center. Similar analysis also shows supergranules are not the result of random occurrences of mesogranules. Title: Magnetoconvection Authors: Weiss, N. O. Bibcode: 1991GApFD..62..229W Altcode: Cowling investigated the effect of an imposed magnetic field on convection in order to explain the origin of sunspots. After summarizing the classical linear theory of Boussinesq magnetoconvection, this review proceeds to more recent nonlinear results. Weakly nonlinear theory is used to establish the relevant bifurcation structure, which involves steady, oscillatory and chaotic solutions. Behaviour found in numerical experiments can then be related to these analytical results. Thereafter, attention is focused on the astrophysically relevant problem of fully compressible magnetoconvection. Steady two-dimensional nonlinear solutions show two important effects: stratification introduces an asymmetry between rising and falling fluid, while compressibility leads to evacuated magnetic flux sheets. Time-dependent behaviour includes transitions between standing waves and travelling waves, as well as changes in horizontal scale, leading to the development of more complicated spatial structures. Work on three-dimensional models, which is now in progress, will lead to a better understanding of the structure of a sunspot. Title: Simulating exploding granules and mesogranular flows Authors: Simon, G. W.; Title, A. M.; Weiss, N. O. Bibcode: 1991AdSpR..11e.259S Altcode: 1991AdSpR..11..259S Cellular convective motion at the solar surface can be simulated by a suitable distribution of axisymmetric sources /1/. With this model we represent randomly distributed exploding granules or mesogranules. The effect of a given velocity field on the magnetic field is modelled by inserting test particles (``corks'') and following their resultant motions. An important question raised by the observations is whether mesogranular flows are generated entirely by exploding granules which recur in approximately the same location or whether there is a persistent underlying circulation.

Our model calculations suggest that a combination of systematic cellular motion on a mesogranular scale and recurrent exploding granules located near the mesogranular centers is compatible with observed magnetic structures. We also generate randomly-distributed exploders not tethered to mesogranular sites, and the resulting cork patterns do not resemble those observed on the solar surface. Finally we introduce a large-scale persistent supergranular flow which transports the mesogranules and exploders towards the supergranular network and obtain patterns not unlike those seen on the Sun.

Operated by the National Optical Astronomy Observatories for the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation. Partial Support for the National Solar Observatory is provided by the USAF under a Memorandum of Understanding with the NSF. Title: Momentum boundary conditions and Euclidean field theory Authors: Unruh, W. G.; Weiss, N. Bibcode: 1990CQGra...7.2331U Altcode: No abstract at ADS Title: Kinematic Modeling of the Relations Among Exploders, Mesogranules, and Supergranules Authors: Simon, G. W.; Title, A. M.; Weiss, N. O. Bibcode: 1990BAAS...22R1225S Altcode: No abstract at ADS Title: Oscillatory convection in sunspot umbrae Authors: Weiss, N. O.; Brownjohn, D. P.; Hurlburt, N. E.; Proctor, M. R. E. Bibcode: 1990MNRAS.245..434W Altcode: Subphotospheric convection is partially inhibited by the strong vertical magnetic field at the centre of a sunspot. We investigate the effects of stratification on non-linear magnetoconvection in a fully compressible perfect gas by means of two-dimensional numerical experiments. Behaviour depends critically on the ratio ξ of the magnetic to the thermal diffusivity, which increases with depth. If ξ > 1 throughout the layer we find steady overturning convection with an asymmetry between rising and falling plumes. For ξ > 1 we obtain oscillatory convection with reversals of the flow. When ξ > 1 at the top of the layer but ξ > 1 at its base, convection sets in as steady motion but there is a transition (involving several bifurcations) to mixed-mode periodic solutions as the Rayleigh number is increased. The flow at the bottom of the layer does not reverse but adjacent rising plumes alternate between active and quiescent states. In the umbral photosphere t, > 1 but ξ > 1 at depths below 1500 km owing to the increase in opacity. Our results imply that time-dependent convection immediately below the photosphere is coupled to motion at levels where ξ > 1. They also explain the existence of umbral dots with a lifetime corresponding to the Alfven time for the converting layer. Title: Solar and stellar convection zones. Authors: Weiss, N. O. Bibcode: 1990CoPhR..12..233W Altcode: There are several important astrophysical questions that might be answered by numerical modelling. These involve the kinematic effects of motion on magnetic fields, the dynamics of magnetoconvection, the structure and scale of convection and the global dynamo problem. This review focuses on detailed modelling of nonlinear compressible convection in a strong vertical magnetic field. Techniques range from heuristic models, which may be relatively primitive, through idealized experiments to large scale simulations and each approach has its proponents. There have been systematic investigations of fully compressible two- and three-dimensional convection as well as ambitious simulations. Detailed studies of two-dimensional behaviour reveal complicated bifurcation structures, involving changes of scale and transitions from steady to oscillatory solutions and from standing waves to travelling waves. Models of large-scale behaviour throughout the convection zone of a star like the sun show that dynamo action can occur but are still far from being able to reproduce the observed patterns of differential rotation or magnetic activity. Title: Fine structure on the Sun Authors: Weiss, Nigel Bibcode: 1990Natur.344..815W Altcode: No abstract at ADS Title: Periodicity and Aperiodicity in Solar Magnetic Activity Authors: Weiss, N. O. Bibcode: 1990RSPTA.330..617W Altcode: Solar activity varies irregularly with an 11-year period whereas the magnetic cycle has a period of 22 years. Similar cycles of activity are seen in other slowly rotating late-type stars. The only plausible theory for their origin ascribes them to a hydromagnetic dynamo operating at, or just below, the base of the convective zone. Linear (kinematic) dynamo models yield strictly periodic solutions with dynamo waves propagating towards or away from the equator. Nonlinear (magneto-hydrodynamic) dynamo models allow transitions from periodic to quasi-periodic to chaotic behaviour, as well as loss of spatial symmetry followed by the development of complex spatial structure. Results from simple models can be compared with the observed sunspot record over the past 380 years and with proxy records extending over 9000 years, which show aperiodic modulation of the 11-year cycle. Title: Symmetry Breaking in the Solar Dynamo: Nonlinear Solutions Authors: Jennings, R. L.; Weiss, N. O. Bibcode: 1990IAUS..138..355J Altcode: No abstract at ADS Title: Magnetohydrodynamics of Sunspots Authors: Weiss, N. O. Bibcode: 1990IAUS..142..139W Altcode: Recent numerical investigations of fully compressible nonlinear magnetoconvection have clarified the nature of convection in sunspot umbrae. In a shallow layer with a strong vertical magnetic field oscillations give way to traveling waves as the Rayleigh number is increased but in a deep stratified layer oscillatory behavior only appears after secondary bifurcations. This behavior leads to a model that explains the formation of umbral dots. Penumbral structure is more difficult to explain owing to the apparent presence of adjacent horizontal and inclined fields in dark and bright filaments. The inner penumbra lies above a transition zone where volume currents are needed to maintain an overall static equilibrium; instabilities in this region may be responsible for filamentary structure in the penumbra as well as for fine structure at the umbral-penumbral boundary. Title: Transitions to asymmetry in magnetoconvection Authors: Nagata, M.; Proctor, M. R. E.; Weiss, N. O. Bibcode: 1990GApFD..51..211N Altcode: In two-dimensional Boussinesq magnetoconvection with symmetrical boundary conditions upward and downward motion are equivalent. Hence there exist symmetric solutions with equivalent flux sheets on either side of each convection roll. Numerical experiments show that this symmetry can be broken for both steady and oscillatory solutions. The underlying bifurcation structure is established by studying a truncated seventeenth-order model system. Steady solutions of this relatively low-order system can be obtained explicitly and their stability can be investigated. Primary bifurcations from the trivial static solution lead to pure single-roll and two-roll solutions, both steady and oscillatory; secondary bifurcations give mixed-mode steady and oscillatory branches while tertiary bifurcations allow behaviour that is more complicated but less robust. Properly interpreted, this detailed study of a particular system provides a better understanding of the behaviour of nonlinear solutions of the full partial differential equations. Title: Periodicity and aperiodicity in solar magnetic activity. Authors: Weiss, N. O. Bibcode: 1990ecvs.conf..617W Altcode: Solar activity varies irregularly with an 11-year period whereas the magnetic cycle has a period of 22 years. Similar cycles of activity are seen in other slowly rotating late-type stars. The only plausible theory for their origin ascribes them to a hydromagnetic dynamo operating at, or just below, the base of the convective zone. Linear (kinematic) dynamo models yield strictly periodic solutions with dynamo waves propagating towards or away from the equator. Nonlinear (magnetohydrodynamic) dynamo models allow transitions from periodic to quasi-periodic to chaotic behaviour, as well as loss of spatial symmetry followed by the development of complex spatial structure. Results from simple models can be compared with the observed sunspot record over the past 380 years and with proxy records extending over 9000 years, which show aperiodic modulation of the 11-year cycle. Title: Nonlinear compressible magnetoconvection. I - Travelling waves and oscillations Authors: Hurlburt, N. E.; Proctor, M. R. E.; Weiss, N. O.; Brownjohn, D. P. Bibcode: 1989JFM...207..587H Altcode: Two-dimensional compressible convection in a polytropic layer with an imposed vertical magnetic field is studied in a series of numerical experiments which consider a shallow layer, spanning only a fraction of a scale height in density, and increase the ratio (1/beta) of the magnetic to the thermal pressure in a regime where convection sets in at an oscillatory bifurcation. Initially, there are stable periodic oscillations (standing wave solutions). For moderate values of beta the only deviations from Boussinesq behavior are where the field is locally intense but as beta is decreased magnetic pressure fluctuations become increasingly important. When beta is of order unity at the top of the layer standing waves become unstable at higher Rayleigh numbers and traveling waves are preferred. This is an essentially compressible effect in which magnetic pressure plays a crucial role. The associated bifurcation structure is investigated in some detail. Title: Simulation of Large-Scale Flows at the Solar Surface Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1989ApJ...345.1060S Altcode: A simple analytic axisymmetric function is used to represent the radial outflow associated with an isolated convection plume at the solar surface. The vertical velocity can be deduced from the continuity equation. A regular cellular pattern of convection can be created by superposing a number of such sources. The function is applied to the large-scale horizontal motions observed by the Solar Optical Universal Polarimeter (SOUP) instrument on Spacelab 2. The flow pattern visible in three different regions covered by the SOUP observations is simulated. In each case a superposition of the plume functions mimics the observed mesogranular and supergranular motions well. The model flows are used to compute the motion of passive test particles (corks) which accumulate in a network that outlines mesogranular cells. Detailed comparisons suggest that magnetic flux tubes are affected more by outflow from sources at the centers of mesogranules than by flow into sinks within the network. Title: Simulating plumes and sinks observed at the solar surface Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1989hsrs.conf..529S Altcode: No abstract at ADS Title: Modeling the Flow in Solar Vortices Authors: Simon, G. W.; Weiss, N. O.; Scharmer, G. B. Bibcode: 1989BAAS...21Q.829S Altcode: No abstract at ADS Title: Time Dependent Compressible Magnetoconvection Authors: Weiss, N. O. Bibcode: 1989ASIC..263..471W Altcode: 1989ssg..conf..471W No abstract at ADS Title: Compressible Magnetoconvection Authors: Weiss, N. O. Bibcode: 1989gmca.conf....1W Altcode: No abstract at ADS Title: A Simple Model of Mesogranular and Supergranular Flows Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1989ASIC..263..595S Altcode: 1989ssg..conf..595S No abstract at ADS Title: Dynamo Processes in Stars Authors: Weiss, N. O. Bibcode: 1989ASSL..156...11W Altcode: 1989admf.proc...11W Present understanding of stellar dynamo processes is reviewed. The observational constraints are described, and the basic physical processes of helicity and differential rotation that lead to dynamo action are outlined, stressing the importance of recent helioseismological results. Mean field dynamo theory and nonlinear dynamo waves are discussed. Title: Simulation of Large-Scale Flows at the Solar Surface Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1988BAAS...20.1008S Altcode: No abstract at ADS Title: Is the solar cycle an example of deterministic chaos? Authors: Weiss, N. O. Bibcode: 1988ssgv.conf...69W Altcode: No abstract at ADS Title: The relation between convection flows and magnetic structure at the solar surface Authors: Simon, G. W.; November, L. J.; Acton, L. W.; Title, A. M.; Tarbell, T. D.; Topka, K. P.; Shine, R. A.; Ferguson, S. H.; Weiss, N. O.; Zirin, H. Bibcode: 1988AdSpR...8k.133S Altcode: 1988AdSpR...8..133S We describe recent results from the comparison of data from the Solar Optical Universal Polarimeter instrument on Spacelab 2 and magnetograms from Big Bear Solar Observatory. We show that the Sun's surface velocity field governs the structure of the observed magnetic field over the entire solar surface outside sunspots and pores. We attempt to describe the observed flows by a simple axisymmetric plume model. Finally, we suggest that these observations may have important implications for the prediction of solar flares, mass ejections, and coronal heating. Title: Dynamics of Convection Authors: Weiss, N. O. Bibcode: 1987RSPSA.413...71W Altcode: Thermal convection in a fluid layer is an example of a dynamical system governed by partial differential equations. As the relevant control parameter (the Rayleigh number) is increased, successive bifurcations may lead to chaos and the nature of the transition depends on the spatial structure of the flow. Numerical experiments with idealized symmetry and boundary conditions make it possible to explore nonlinear behaviour in some detail and to relate bifurcation structures to those found in appropriate low-order systems. Two examples are used to illustrate transitions to chaos. In two-dimensional thermosolutal convection, where the spatial structure is essentially trivial, chaos is caused by a heteroclinic bifurcation involving a symmetric pair of saddle foci. When convection is driven by internal heating several competing spatial structures are involved and the transition to chaos is more complicated in both two-and three-dimensional configurations. Although the first few bifurcations can be isolated a statistical treatment is needed for behaviour at high Rayleigh numbers. Title: Magnetic fields and non-uniform rotation in stellar radiatives zones. Authors: Mestel, L.; Weiss, N. O. Bibcode: 1987MNRAS.226..123M Altcode: This paper examines the effects of dynamical and resistive instabilities on magnetic redistribution of angular momentum within a star. It is tentatively concluded that if significant differential rotation survives in a stably stratified radiative zone over a stellar evolution time, then the poloidal field, Bp, cannot exceed an upper limit of order 3×10-2G and is probably less than 10-3G. In the radiative core of the Sun Bp is estimated to be at least of order 5×10-2G and probably 100 G or more. Values greater than 0.1 G cannot easily be reconciled with the rotational shear inferred from frequency splitting of solar oscillations. Title: Rotation and magnetic fields in the Sun. Authors: Weiss, N. O. Bibcode: 1987ppcs.work...46W Altcode: Stellar magnetic fields exert torques which alter the distribution of angular momentum in a star. In the radiative interior of the sun, these torques tend to enforce uniform rotation and the existence of a rapidly rotating core would imply a poloidal field of less than 0.01 G. In the convective envelope, magnetic fields generated by dynamo action produce torques which lead to torsional oscillations. A simple nonlinear model allows both multiply periodic and chaotic behavior. This system demonstrates that both aperiodic magnetic cycles and the irregular modulation responsible for grand minima can be regarded as examples of deterministic chaos. This picture is consistent with the C-14 record but implies that the multiply periodic lamination of some Precambrian varves is not associated with the solar cycle. Title: What can the sun tell us about stellar activity? Authors: Weiss, N. O. Bibcode: 1987LNP...292....1W Altcode: The solar-stellar connection relates high-resolution synoptic solar observations to observations of magnetic activity in stars with different rotation rates and internal structures. Our knowledge of magnetic fields in stellar convection zones is based on detailed observations of field structures in the Sun but recent measurements of magnetic activity in other late-type stars have extended our understanding of the solar dynamo. These observations have stimulated detailed modelling of processes associated with magnetic activity. Modulation of activity cycles in slowly rotating stars can be inferred from terrestrial data extending over the last 104 years, while the evolution of the Sun's magnetic field can be inferred from the behaviour of younger stars. Title: What Can the Sun Tell Us About Stellar Activity? Authors: Weiss, N. O. Bibcode: 1987LNP...292....3W Altcode: 1987ssp..conf....3W The solar-stellar connection relates high-resolution synoptic solar observations to observations of magnetic activity in stars with different rotation rates and internal structures. Modulation of activity cycles in slowly rotating stars can be inferred from terrestrial data extending over the last 104years, while the evolution of the Sun's magnetic field can be inferred from the behaviour of younger stars. Title: Dynamical chaos. Proceedings of a Royal Society discussion meeting held in London, UK, 4 - 5 February 1987. Authors: Berry, M. V.; Percival, I. C.; Weiss, N. O.; Zeeman, E. C. Bibcode: 1987dcpr.book.....B Altcode: The individual contributions within the subject scope of Astronomy and Astrophysics Abstracts are included in their corresponding categories - see abstracts 015.003, 021.002, 021.003, 022.002 - 022.005, 062.006 - 062.009, 091.002. Title: Interaction between Magnetic Fields and Convection Authors: Hurlburt, N. E.; Weiss, N. O. Bibcode: 1987rfsm.conf...35H Altcode: The authors discuss nonlinear convection in the presence of an imposed vertical magnetic field and its influence on the fine structure of the resulting field. They contrast recent results of numerical experiments on steady and oscillatory magnetoconvection with those obtained in the Boussinesq approximation. An attempt is also made to relate idealized model calculations to the structure of observed magnetic fields in the solar photosphere. Title: Simulation of Surface Flows in Supergranulation Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1986BAAS...18R.990S Altcode: No abstract at ADS Title: Energy transport in sunspot penumbrae. Authors: Schmidt, H. U.; Spruit, H. C.; Weiss, N. O. Bibcode: 1986A&A...158..351S Altcode: It is proposed that the magnetic field in the outer penumbra of a sunspot is almost horizontal and that the penumbra itself is very shallow, with a sharp lower boundary. A simple model of energy transport in the outer penumbra predicts that there is a Wilson depression of about 100 km, below which the penumbra extends for only 80 km. A two-component model with bright and dark filaments suggests that the associated differences in observed field strength and corrugations of the visible surface of the penumbra will be small. The authors argue that flows along the field are needed to explain the existence of bright and dark filaments while the Evershed flow is driven by pressure differences along flux tubes. Title: Magnetic torques and differential rotation. Authors: Weiss, N. O. Bibcode: 1986ASIC..169..253W Altcode: 1986ssds.proc..253W The rotation profile deduced from the measured splitting of solar oscillations raises several problems. The slight decrease in angular velocity within the convective zone can be explained and the transition from differential to uniform rotation on spherical surfaces must cause a further reduction in the radiative zone. The bump, associated with harmonics of degree 11, seems implausible and it is hard to reconcile a rapidly rotating core with the presence of a significant magnetic field. A possible explanation is that this core has been magnetically decoupled from outer regions since the Sun evolved on to the Hayashi track. Title: Stellar dynamo characteristics Authors: Weiss, N. O. Bibcode: 1986HiA.....7..385W Altcode: Recent discoveries have shown that magnetic activity is typical of cool stars with deep convective zones and magnetic cycles are found in slowly rotating stars like the sun. The current state of hydromagnetic dynamo theory is reviewed, and simplified models are used in an attempt to isolate the dominant nonlinear processes in stellar dynamos. Title: Differential rotation and magnetic torques in the interior of the Sun Authors: Rosner, R.; Weiss, N. O. Bibcode: 1985Natur.317..790R Altcode: The frequencies of solar oscillations can be measured with extreme precision and 5-min oscillations reveal the internal structure of the Sun1-5. In particular, measurements of rotational splitting4 have provided the first reliable indications of the variation of angular velocity with radius6, while recent observations5 have yielded information on the variation with depth of latitudinal differential rotation. These results confirm theoretical predictions that the angular velocity decreases inwards in the convective zone7,8 but raise problems for dynamo models of the solar cycle. The suggestion that the core rotates with roughly twice the surface angular velocity has important implications both for the rotational history of the Sun and for other late-type stars, whose magnetic activity is closely correlated with rotation. Such a rapidly rotating core is hard to reconcile with the presence of any significant magnetic field pervading the entire radiative interior. We can only explain it by suggesting that the core contains a fossil field, unaffected by turbulence in the pre-main sequence Hayashi phase, that is decoupled from the rest of the star. Title: Oscillatory Convection in Flux Tubes Pores and Sunspots Authors: Hurlburt, N.; Weiss, N. O. Bibcode: 1985tphr.conf..198H Altcode: No abstract at ADS Title: Buoyant magnetic flux tubes. II - Three-dimensional behaviour in granules and supergranules Authors: Schmidt, H. U.; Simon, G. W.; Weiss, N. O. Bibcode: 1985A&A...148..191S Altcode: A simple model is used to study the interaction of isolated magnetic flux tubes with convection in the sun. Convective motion in granules and supergranules is represented by prescribed flows in three-dimensional cells with square cross-sections and thin flux tubes move under the action of magnetic buoyancy, Lorentz curvature forces and aerodynamic drag. Inflow at the base of a cell competes with outflow at its upper surface; small flux tubes tend to be swept to the cell boundaries while larger, more buoyant tubes are dragged to the axis of the cell. These results are compared with recent observations of small-scale granular and intergranular magnetic fields. Title: Chaotic behavior in stellar dynamos Authors: Weiss, N. O. Bibcode: 1985JSP....39..477W Altcode: Slowly rotating main-sequence stars with deep convective zones have activity cycles like the sun's. The solar cycle is aperiodic and modulated to give intervals of reduced activity. A simple sixth-order system, obtained by truncating the dynamo equations, has solutions that mimic this behavior. The transition to chaos is analyzed and the astrophysical significance of these results is discussed. Title: Book-Review - the Solar Granulation Authors: Bray, R. J.; Loughhead, R. E.; Durrant, C. J.; Weiss, N. O. Bibcode: 1985SoPh...96..423B Altcode: No abstract at ADS Title: Book review Authors: Weiss, N. O. Bibcode: 1985SoPh...96..423W Altcode: No abstract at ADS Title: 3-D Behavior of Buoyant Magnetic Flux Tubes in Granules and Supergranules Authors: Simon, G. W.; Schmidt, H. U.; Weiss, N. O. Bibcode: 1985BAAS...17Q.642S Altcode: No abstract at ADS Title: Oscillatory convection in flux tubes, pores and sunspots. Authors: Hurlburt, N. E.; Weiss, N. O. Bibcode: 1985MPARp.212..198H Altcode: High-resolution images of the solar surface provide a means for probing sub-photospheric structures. The authors combine simple conceptual arguments with numerical models to consider the different flow regimes possible within flux tubes, pores and sunspots which should be found in detailed observations. In the presence of a strong magnetic field convection near the photosphere is likely to be oscillatory. The authors have carried out a series of numerical experiments involving simple atmospheres (polytropic in the absence of convection). Title: Nonlinear dynamos: A complex generalization of the Lorenz equations Authors: Jones, C. A.; Weiss, N. O.; Cattaneo, F. Bibcode: 1985PhyD...14..161J Altcode: Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite. Title: Energy transport in sunspot penumbrae. Authors: Schmidt, H. U.; Spruit, H. C.; Weiss, N. O. Bibcode: 1985MPARp.182.....S Altcode: No abstract at ADS Title: Theoretical Interpretation of Small-Scale Solar Features Authors: Weiss, N. O. Bibcode: 1985LNP...233..217W Altcode: 1985hrsp.proc..217W The structure of photospheric magnetic fields is dominated by the interaction between granular convection and isolated flux tubes. The current understanding of compressible convection, and of both kinematic and dynamical aspects of magnetoconvection is summarized. These theories are related to the formation and location of intense magnetic fields within the photospheric network. The overall structure of sunspots is reconsidered and related to umbral and penumbral features. Title: Magnetoconvection. Authors: Weiss, N. O. Bibcode: 1985ssmf.conf..156W Altcode: The author focuses on three topics. These are, first, the formation of small flux tubes; second, the nature of convection in a strong magnetic field; and, third, the structure of the large-scale magnetic field in the sun. Title: The relation between stellar rotation rate and activity cycle periods. Authors: Noyes, R. W.; Weiss, N. O.; Vaughan, A. H. Bibcode: 1984ApJ...287..769N Altcode: The empirical relation between rotation period, spectral type, and cycle activity period in 13 slowly rotating main-sequence stars is investigated, on the basis of available spectrometric data. It is shown that for slowly rotating stars with similar spectral types, the cycle period P(cyc), and rotation period P(rot) were related such that P(cyc) varies as P(rot) to the nth, where n equals 1.25. In a group of stars with individual spectral types from G2 to K7, the cycle periods were consistent with the relation P(cyc) is approximately equal to (P rot/tauc) exp n, where tauc is the convective turnover time near the bottom of the convection zone. On the basis of the above relations, it is suggested that an increase of P(cyc) with increasing P(rot) does not agree with conventional estimates from nonlinear dynamo models, and is limited by two factors: the quenching of the alpha effect; and differential rotation. Title: Amplification and maintenance of thin magnetic flux tubes by compressible convection. Authors: Proctor, M. R. E.; Weiss, N. O. Bibcode: 1984ESASP.220...77P Altcode: 1984ESPM....4...77P The authors present a model that includes both the effects of diffusion and proper treatment of the Lorentz forces, for a thin tube whose depth is of the order of a scale height. The model includes both magnetic pressure (leading to evacuation of the tube) and curvature forces. It is found that while small tubes are limited principally by pressure effects, tubes with fluxes ≡1018mx exert an important retarding force on the convection that causes them. Title: Convection in sunspots and the origin of umbral dots Authors: Knobloch, E.; Weiss, N. O. Bibcode: 1984MNRAS.207..203K Altcode: Recent studies of non-linear magnetoconvection are used to show that the observed properties of umbral dots can be explained if they are produced by oscillatory convection in the umbrae of sunspots. The overall magnetic field is assumed to be coherent within the flux tube underneath the sunspot. Individual convection cells are about 1500 km deep and 300 km in diameter. The oscillations are highly non-linear, with periods of around 6 hr, but the relatively vigorous upward motion that is responsible for umbral dots lasts for only a fraction of this lifetime. This model is compared with various alternative hypotheses. Title: Periodic and aperiodic dynamo waves Authors: Weiss, N. O.; Cattaneo, F.; Jones, C. A. Bibcode: 1984GApFD..30..305W Altcode: In order to show that aperiodic magnetic cycles, with Maunder minima, can occur naturally in nonlinear hydromagnetic dynamos, we have investigated a simple nonlinear model of an oscillatory stellar dynamo. The parametrized mean field equations in plane geometry have a Hopf bifurcation when the dynamo number D=1, leading to Parker's dynamo waves. Including the nonlinear interaction between the magnetic field and the velocity shear results in a system of seven coupled nonlinear differential equations. For D>1 there is an exact nonlinear solution, corresponding to periodic dynamo waves. In the regime described by a fifth order system of equations this solution remains stable for all D and the velocity shear is progressively reduced by the Lorentz force. In a regime described by a sixth order system, the solution becomes unstable and successive transitions lead to chaotic behaviour. Oscillations are aperiodic and modulated to give episodes of reduced activity. Title: Problems of Flux Tube Formation Authors: Weiss, N. O. Bibcode: 1984ssdp.conf..287W Altcode: Recent theoretical studies of magnetoconvection predict that strong fields will be formed between granules but also suggest that much of the magnetic flux may remain near the center of a granule. Title: Solar and stellar magnetic fields Authors: Priest, E. R.; Weiss, N. O. Bibcode: 1983Obs...103..239P Altcode: No abstract at ADS Title: Simple models for magnetic flux tubes. Authors: Simon, G. W.; Weiss, N. O.; Nye, A. H. Bibcode: 1983SoPh...87...65S Altcode: Known potential field solutions can be used to model the structure of magnetic fields in the solar photosphere. Several two-dimensional and axisymmetric solutions are compared. In the most satisfactory model the vertical component of the field is prescribed on a horizontal plane so as to be uniform within a finite disc and zero outside it. The resulting flux distribution provides a good description of small scale intergranular magnetic fields and of the observed field structure in a pore, but is inadequate for sunspots. Title: A review of: "Finite-difference techniques for vectorized fluid dynamics calculations" Authors: Weiss, N. O. Bibcode: 1983GApFD..23..344W Altcode: Edited by D. L. Book. Springer-Verlag, New York, 1981. 226 pp., DM 72, - , $33.60. (ISBN 0 387 104828) Title: Solar magnetism Authors: Weiss, N. O. Bibcode: 1983spm..conf..115W Altcode: Magnetic fields in the Sun and other late-type stars follow cycles of activity, which are more vigorous in rapidly rotating stars. Mean-field dynamo theory provides a good qualitative description of these magnetic cycles, though it cannot be justified in detail. Future progress will require more elaborate models, related to observations and relying heavily on computation. Title: Simple models for magnetic flux tubes. Authors: Simon, G. W.; Weiss, N. O.; Nye, A. H. Bibcode: 1983BAAS...15R.874S Altcode: No abstract at ADS Title: Future research directions - Theoretical approach and perspective Authors: Weiss, N. O. Bibcode: 1983ASSL..102..639W Altcode: 1983IAUCo..71..639W; 1983ards.proc..639W Theoretical aspects of magnetic activity in red dwarfs are surveyed. Some general remarks about the role of theory in astrophysics are made, and a phenomenological description of red dwarf magnetic fields is briefly presented. Numerical simulation of nonlinear processes in magnetic fields is illustrated with examples of stellar dynamos, turbulent magnetic fields, and the interaction of magnetic fields with granular convection. Examples of simplified problems that can be solved by some combination of analytical or numerical techniques as an alternative to large scale simulations are given, including problems involving alpha-omega dynamo models, formation of flux ropes by magnetic buoyancy, and isolated flux tubes. Title: Periodic and aperiodic behaviour in stellar dynamos Authors: Cattaneo, F.; Weiss, N. O.; Jones, C. A. Bibcode: 1983IAUS..102..307C Altcode: A simple parameterized mean field dynamo model has been constructed that includes the dynamical interaction between the magnetic field and differential rotation. This system of seven coupled nonlinear ordinary differential equations has finite amplitude oscillatory solutions (corresponding to Parker's dynamo waves) when the dynamo number (D) is greater than one. Two regimes were studied. In the first, the velocity shear is reduced by the Lorentz force and there are stable periodic solutions for all dynamo numbers greater than one. In the second there is a transition from strictly periodic oscillations to aperiodic (chaotic) behavior as D is increased. This simple example shows that nonlinear hydromagnetic dynamos can produce aperiodic cycles, with Maunder minima, as observed in the sun and other late-type stars. Title: Magnetic buoyancy and the Boussinesq approximation Authors: Spiegel, E. A.; Weiss, N. O. Bibcode: 1982GApFD..22..219S Altcode: The full Boussinesq equations for hydromagnetic convection are derived and shown to include the effects of magnetic buoyancy. Instabilities caused by magnetic buoyancy are analyzed and their roles in double convection are brought out. Title: REVIEW ARTICLE: Magnetoconvection Authors: Proctor, M. R. E.; Weiss, N. O. Bibcode: 1982RPPh...45.1317P Altcode: The interaction between convection and an externally imposed magnetic field in a Boussinesq fluid is discussed. The equations that govern Boussinesq magnetoconvection are derived and boundary conditions and simplified geometries are discussed. The kinematic effects of prescribed velocity fields on magnetic fields, including flux expulsion and the formation of isolated sheets or tubes of flux, are treated. Dynamical effects are introduced by considering the simpler Oberbeck problem and demonstrating the exclusion of motion from the flux sheets. Linear stability theory for the Rayleigh-Benard problem is summarized. Two-dimensional magnetoconvection is discussed in detail; results obtained by perturbation methods are described and extended into the nonlinear regime by adopting a truncated model system, and numerical results for the full problem are presented. Axisymmetric magnetoconvection is described, and the transition from the kinematic regime to one in which the field is dynamically active is discussed. Extensions of the theory to more exotic effects are briefly reviewed, and astrophysical implications are briefly commented on. Title: Magnetic fields in late-type stars Authors: Knobloch, E.; Rosner, R.; Weiss, N. O. Bibcode: 1981MNRAS.197P..45K Altcode: Observations show that magnetic activity in late-type stars is correlated with rotation rates and that there is a discontinuous change in behavior at a critical rotation period. This can be explained as a consequence of a transition from convection in rolls parallel to the rotation axis to normal convection cells as the angular velocity is decreased. Title: The interplay between magnetic fields and convection Authors: Weiss, N. O. Bibcode: 1981JGR....8611689W Altcode: The development of hydrodynamic dynamo theory is considered, taking into account investigations related to the solar cycle. Before constructing any detailed models of the solar cycle it is necessary to understand both the kinematic effect of convection on the magnetic field and the dynamical effect of the distorted field on the motion. The effect of a convective eddy on the magnetic field can be explored by studying an isolated sphere or cylinder rotating as a solid body in an otherwise uniform field. It is shown that reconnection of the lines of force leads to flux expulsion. A summary is provided of the relevant aspects of magnetoconvection, and a description is presented of some new results, obtained from numerical experiments at fairly high Rayleigh numbers. The computations illustrate the complexity of the nonlinear interaction between magnetic fields and convection. Title: Convection and magnetic fields in late-type stars. Authors: Weiss, N. O. Bibcode: 1981Obs...101...37W Altcode: No abstract at ADS Title: Convection and magnetic fields in stars Authors: Galloway, D. J.; Weiss, N. O. Bibcode: 1981ApJ...243..945G Altcode: Recent observations have demonstrated the unity of the study of stellar and solar magnetic fields. Results from numerical experiments on magnetoconvection are presented and used to discuss the concentration of magnetic flux into isolated ropes in the turbulent convective zones of the sun or other late-type stars. Arguments are given for siting the solar dynamo at the base of the convective zone. Magnetic buoyancy leads to the emergence of magnetic flux in active regions, but weaker flux ropes are shredded and dispersed throughout the convective zone. The observed maximum field strengths in late-type stars should be comparable with the field that balances the photospheric pressure. Title: Stellar magnetic structure and activity (theory). Authors: Weiss, N. O. Bibcode: 1981ASIC...68..449W Altcode: 1981spss.conf..449W Both the overall behavior of the solar cycle and the underlying fine structure of magnetic fields in the sun have been studied mathematically in some detail. These theories are summarized and different phenomenological models of the solar cycle are reviewed. In order to provide a description of the magnetic fields in late-type stars it is necessary to extrapolate boldly from what is known about the sun. In this way field strengths and configurations can be estimated. Title: Solar magnetic fields - The generation of emerging flux Authors: Golub, L.; Rosner, R.; Vaiana, G. S.; Weiss, N. O. Bibcode: 1981ApJ...243..309G Altcode: X-ray observations have provided information about magnetic fields on the sun, and the implications of these observations are discussed. The pattern of small-scale flux emergence is quite different from that of active regions. It is inferred that the small-scale fields originate fairly high in the convective zone, while the fields in active regions have a deeper origin. The small-scale turbulent fields are only loosely related to the fields that define the normal solar cycle. The way in which dynamo models must be modified in the light of these results is indicated. Title: Large aspect ratio cells in two-dimensional thermal convection Authors: Hewitt, J. M.; McKenzie, D. P.; Weiss, N. O. Bibcode: 1980E&PSL..51..370H Altcode: Numerical experiments have been carried out on two-dimensional thermal convection, in a Boussinesq fluid with infinite Prandtl number, at high Rayleigh numbers. With stress free boundary conditions and fixed heat flux on upper and lower boundaries, convection cells develop with aspect ratios (width/depth) λ≿ 5, if heat is supplied either entirely from within or entirely from below the fluid layer. The preferred aspect ratio is affected by the lateral boundary conditions. If the temperature, rather than the heat flux, is fixed on the upper boundary the cells haveλ ≈ 1. At Rayleigh numbers of 2.4 × 10 5 and greater, small sinking sheets are superimposed on the large aspect ratio cells, though they do not disrupt the circulation. Similar two-scale flows have been proposed for convection in the earth's mantle. The existence of two scales of flow in two-dimensional numerical experiments when the viscosity is constant will allow a variety of geophysically important effects to be investigated. Title: Magnetic activity and variations in solar luminosity Authors: Spiegel, E. A.; Weiss, N. O. Bibcode: 1980Natur.287..616S Altcode: Attempts to detect changes in the solar luminosity suggest that the solar constant has been misnamed1-4. Although the Nimbus satellite data4 show no significant fractional variations above 5 × 10-3 during the period 1975-78, results from recent balloon5 and rocket6 flights show changes of 4 × 10-3. Intense magnetic fields in sunspots hamper convection locally7 but active regions are believed not to be directly responsible for long term variations in luminosity of the Sun8 or of RS CVn and BY Dra stars9. The cause of luminosity variations over spot cycles should be sought in more deep-seated global features. Here we indicate how strong magnetic fields at the base of the convective zone can alter the local convection. The resulting changes in thermal energy are large enough to produce variations of order 0.1% in the solar luminosity over the 11-yr sunspot cycle. Title: Problems on Interior Structure, the Solar Dynamo and the Role of SCADM in Providing Interior Diagnostics Authors: Weiss, N. O. Bibcode: 1980NASCP2098...55W Altcode: 1980sscs.nasa...55W What is already known about the structure of the Sun, the motion of its convective zone, and the solar cycle is reviewed. Topics discussed include solar variability, solar 'seismology', velocity patterns, magnetic fields, and the dynamo theory. Observations are needed to determine global properties (solar luminosity and radius), oscillations (p and g models), velocities (variation of rotation with time and depth), and magnetic fields. Title: The nature of solar behaviour Authors: Weiss, N. O. Bibcode: 1980asfr.symp....1W Altcode: Current knowledge of the present state of the sun and of its evolution since formation as a main sequence star 4.7 billion years ago is reviewed as an astrophysical background to studies of the fossil record of solar behavior. Attention is given to the internal structure of the sun as a reflection of the thermonuclear energy generation processes and convective energy dissipation processes occurring there, and to the magnetic fields caused by motions in the convective zone, which are responsible for solar activity. The long-term variability of the periodic changes in solar magnetic fields explained by kinematic dynamo theory is then considered in light of the sunspot record, terrestrial records, and the relationship between magnetic activity, solar luminosity and terrestrial climate. The evident magnetic properties of the sun are compared with those observed in other stars, and it is concluded that solar activity may have been 10 to 100 times more vigorous than at present when the sun reached the main sequence, and even more violent before then. Title: Andrew Marvell and the Maunder Minimum Authors: Weiss, J. E.; Weiss, N. O. Bibcode: 1979QJRAS..20..115W Altcode: No abstract at ADS Title: Buoyant magnetic flux tubes in supergranules. Authors: Meyer, F.; Schmidt, H. U.; Simon, G. W.; Weiss, N. O. Bibcode: 1979A&A....76...35M Altcode: The motion of filamentary flux tubes is described by a simplified model. Relative to the prescribed supergranular flow, the tubes drift at a rate determined by magnetic buoyancy, Lorentz forces and aerodynamic drag. The effect of buoyancy is most pronounced near the surface: small flux tubes are swept towards the network at the edge of a cell but those with larger fluxes float vertically at the center. New flux emerges at the center of the cell. These results are related to the emergence of active regions and ephemeral active regions, and to the slow decay of sunspots. Title: The pattern of convection in the sun. Authors: Weiss, N. O. Bibcode: 1979psa..conf..121W Altcode: No abstract at ADS Title: The nature of solar behavior Authors: Weiss, N. O. Bibcode: 1979LPICo.390..106W Altcode: No abstract at ADS Title: Magnetic flux ropes and convection Authors: Galloway, D. J.; Proctor, M. R. E.; Weiss, N. O. Bibcode: 1978JFM....87..243G Altcode: Three-dimensional cellular convection concentrates magnetic flux into ropes when the magnetic Reynolds number is large. Amplification of the magnetic field is limited by the Lorentz force and the maximum field in a flux rope can be estimated. Boundary-layer analysis yields a completely self-consistent solution for a model of convection driven by imposed horizontal temperature gradients, and the transition from a kinematic to a dynamic regime can be followed in detail. The maximum value of the amplified field is proportional to the square root of the ratio of the viscous to the magnetic diffusivity. Title: Small-scale magnetic fields and convection in the solar photosphere. Authors: Weiss, N. O. Bibcode: 1978MNRAS.183P..63W Altcode: The detection of solar magnetic structures with diameters less than 300 km, as reported by Ramsey et al. (1977), is related to theoretical studies of the nonlinear interaction between magnetic fields and convection. In particular, it is shown that the location of the small-scale magnetic fields at the centers of granules cannot be readily explained on the basis of theoretical investigations of vigorous Boussinesq convection in a magnetic field. Title: On the dynamic interaction between magnetic fields and convection. Authors: Peckover, R. S.; Weiss, N. O. Bibcode: 1978MNRAS.182..189P Altcode: A simple two-dimensional model of convection in an electrically conducting Boussinesq fluid with an externally imposed magnetic field has been investigated in a series of numerical experiments. Convection is driven by horizontal gradients in an imposed temperature field which is unaffected by the motion. The transition from a kinematic to a dynamic regime is studied: magnetic flux is concentrated into ropes and the maximum field strength depends on the ratio of the viscous and magnetic diffusivities. The magnetic energy density in the ropes is unrelated to, and may be much greater than, the kinetic energy density outside. These results are relevant to the interaction between small-scale magnetic fields and photospheric granulation in the sun. Title: The stability of sunspots. Authors: Meyer, F.; Schmidt, H. U.; Weiss, N. O. Bibcode: 1977MNRAS.179..741M Altcode: The energy principle of Bernstein et al. (1958) is used to demonstrate that a magnetic flux tube in equilibrium with an external stratified field-free gas is stable to interchanges, provided the magnitude of the radial field component decreases upward on the tube boundary. Stability conditions on the external pressure in the case where a vacuum field is inside the flux tube are investigated, and it is found that both stable and unstable vacuum-field configurations can exist. A vacuum-field model is employed to show that tubes with fluxes greater than about 10 to the 19th power Mx are stable in the sun, and this result is applied to observations of small-scale solar magnetic fields as well as to sunspot groups. Analysis of a simple sunspot model indicates that a sunspot can be stable in and immediately below the photosphere, that this stability is related to the potential energy associated with the Wilson depression, and that twisted fields need not be invoked. Title: Formation of intense magnetic fields near the surface of the sun Authors: Galloway, D. J.; Proctor, M. R. E.; Weiss, N. O. Bibcode: 1977Natur.266..686G Altcode: Possible mechanisms for the concentration of magnetic flux in the solar photosphere in the form of ropes of average magnetic field strengths of about 1500 gauss are discussed. Some upper limits to amplification of magnetic fields by convection are obtained on the assumption that the field strength cannot exceed the value for which the power needed to contain the flux rope is equal to the rate of turbulent energy dissipation. It is shown that the maximum field strength produced by flux concentration is typically an order of magnitude higher than the equipartition limit. Title: Small scale solar magnetic fields: theory. Authors: Weiss, N. O. Bibcode: 1977IAUS...62..241W Altcode: No abstract at ADS Title: Numerical methods in convection theory Authors: Weiss, N. O. Bibcode: 1977LNP....71..142W Altcode: 1977stco.coll..142W; 1977IAUCo..38..142W; 1977psc..conf..142W Nonlinear astrophysical convection is analyzed numerically, with attention to the relationship between convective heat transport and the super-adiabatic temperature gradient. Cellular patterns revealed by direct solar observations are modeled by two and three dimensional methods. The stability of the models is then evaluated on the basis of axisymmetric flow subjected to nonaxisymmetric perturbations. Title: Magnetic fields and convection Authors: Weiss, N. O. Bibcode: 1977LNP....71..176W Altcode: 1977IAUCo..38..176W; 1977stco.coll..176W; 1977psc..conf..176W In a highly conducting plasma convection is hindered by the imposition of a magnetic field. Convection may set in as direct or overstable modes and behavior near the onset of instability depends on the ratio of the magnetic to the thermal diffusivity. Vigorous convection produces local flux concentrations with magnetic fields that may be much greater than the equipartition value. The interaction between magnetic fields and convection can be observed in detail on the sun and is essential to any stellar dynamo. Title: The calibration of stellar convection theories. Authors: Gough, D. O.; Weiss, N. O. Bibcode: 1976MNRAS.176..589G Altcode: Any formula used to calculate the temperature gradient in a stellar convection zone must be calibrated, for example, by evolving 1-solar-mass stellar models to fit the present age, luminosity, and effective temperature of the sun. When this procedure is followed for various convection theories, including those of Opik (1950) and Boehm-Vitense (1958), the corresponding models become almost indistinguishable. In particular, they predict the same depth, around 150,000 km, for the solar convective zone. Title: Convective Instability in a Compressible Atmosphere. II Authors: Gough, D. O.; Moore, D. R.; Spiegel, E. A.; Weiss, N. O. Bibcode: 1976ApJ...206..536G Altcode: The onset of steady convection in polytropic atmosphere with constant viscosity is studied numencally. Subject headings: convection : atmospheres Title: Magnetohydrodynamics (Book Review) Authors: Mestel, L.; Weiss, N. O. Bibcode: 1976ApL....17..152M Altcode: No abstract at ADS Title: The Solar Magnetic Field - Observation and Theory Authors: Weiss, N. O. Bibcode: 1976IAUTB..16..240W Altcode: 1976IAUT...16B.240W No abstract at ADS Title: Stability of sunspots Authors: Meyer, F.; Schmidt, H. U.; Weiss, N. O. Bibcode: 1976MitAG..40..160M Altcode: No abstract at ADS Title: The Pattern of Convection in the Sun Authors: Weiss, N. O. Bibcode: 1976IAUS...71..229W Altcode: No abstract at ADS Title: Axisymmetric convection in a cylinder Authors: Jones, C. A.; Moore, D. R.; Weiss, N. O. Bibcode: 1976JFM....73..353J Altcode: The geometrical properties of axisymmetric convection in a Boussinesq fluid contained in a cylindrical cell with free boundaries are investigated. The range of unsteady behavior requiring a full three-dimensional solution of the governing equations is not considered. The solution near the critical Reynolds number is obtained from a perturbation expansion. For values of the Nusselt number not greater than 2, solutions are obtained from an expansion in a finite number of vertical modes. For Prandtl numbers less than unity the solution becomes independent of the Prandtl number at large Reynolds numbers. As the Prandtl number approaches 0 the Nusselt number is a function of the Rayleigh number only and there is an effective critical Rayleigh number equal to 1.32 times the critical Rayleigh number. Numerical results obtained for Rayleigh numbers up to 100 times the critical Rayleigh number and Prandtl numbers not in excess of 0.01 are similar to those for two-dimensional rolls. For Prandtl numbers greater than unity there is a viscous regime. At high Rayleigh numbers a large isothermal region develops in which the ratio of vorticity to distance from the axis is almost constant. Title: What drives the solar cycle? Authors: Weiss, Nigel Bibcode: 1975Natur.258..197W Altcode: No abstract at ADS Title: The growth and decay of sunspots. Authors: Meyer, F.; Schmidt, H. U.; Weiss, N. O.; Wilson, P. R. Bibcode: 1974MNRAS.169...35M Altcode: Sunspots are formed between supergranules, at junctions in the chromospheric network. Many spots disappear rapidly. However, some large spots enter a phase of slow decay: an annular cell develops, centered on the spot and with a systematic outward velocity directed from the penumbra toward the nearest faculae. This flow forms a moat around the spot swept clear of magnetic field except for magnetic features which migrate from the penumbra to the surrounding network. This phase may persist for several months before the spot is finally destroyed. Throughout this period the sunspot area, and hence its flux, decrease at a constant rate. The evolution of a sunspot is related to subphotospheric convection, with particular emphasis on a model of the slow decay phase which explains the steady decay of magnetic flux. Title: A Theoretical Model for the Convection of Magnetic Flux in and Near Sunspots Authors: Meyer, F.; Schmidt, H. U.; Weiss, N. O.; Wilson, P. R. Bibcode: 1974IAUS...56..235M Altcode: No abstract at ADS Title: Introduction to Magnetohydrodynamics Authors: Weiss, N. O. Bibcode: 1974magn.conf....2W Altcode: 1974SAAS....4....2W Field in conducting fluids Maxwell's equations and the magnetohydrodynamic approximation Kinematic magnetohydrodynamics and the magnetic Reynolds number Perfectly conducting fluids Flux conservation: Alfvén's theorem Consequences of flux-freezing Hydromagnetic waves Alfvén waves Magneto-sonic waves Equilibria and stability Magnetohydrostatic equilibria Force-free fields The Virial theorem Stability problems Effects of finite resistivity Static diffusion Effects of motion Currents in ionized gases Fully ionized gas: two-fluid model Slightly ionized gas: three-fluid model Other dissipative processes References Title: Convection in the earth's mantle: towards a numerical simulation Authors: McKenzie, D. P.; Roberts, J. M.; Weiss, N. O. Bibcode: 1974JFM....62..465M Altcode: No abstract at ADS Title: Dynamo Maintenance of Magnetic Fields in Stars Authors: Weiss, N. O. Bibcode: 1974magn.conf..185W Altcode: 1974SAAS....4..185W Astrophysical magnetic fields Introduction Observed magnetic fields Hydromagnetic dynamos The solar cycle Observations Phenomenological model Kinematic dynamo theory Cowling's theorem Simple dynamo models Computational techniques Finite differences Expansion in toroidal and poloidal spherical harmonics Macroscopic dynamos Mean field electrodynamics The dynamo equations Mean field electrodynamics Dynamo models Constant α α2 dynamos αω dynamos Oblique rotators Magnetohydrodynamic dynamos Mean field dynamos Macroscopic motions Convective models Dynamo action in the sun and stars Astrophysical dynamos Convection and magnetic fields in the sun Problems of the solar dynamo Stellar dynamos References Title: Magnetohydrodynamics Authors: Mestel, L.; Weiss, N. O. Bibcode: 1974magn.conf.....M Altcode: 1974mssa.book.....M; 1974QB353.M47......; 1974SAAS....4.....M No abstract at ADS Title: Two-dimensional Rayleigh-Benard convection Authors: Moore, D. R.; Weiss, N. O. Bibcode: 1973JFM....58..289M Altcode: No abstract at ADS Title: The Dynamo Problem Authors: Weiss, N. O. Bibcode: 1971QJRAS..12..432W Altcode: No abstract at ADS Title: Theories of Large Scale Fields and the Magnetic Active Cycle Authors: Weiss, N. O. Bibcode: 1971IAUS...43..757W Altcode: No abstract at ADS Title: On the Magnetic Field in Pores Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1970SoPh...13...85S Altcode: The magnetic field in an axisymmetric pore is current free and can be represented by a flux tube with a magnetic potential of the formAJ0(kr)e-kz. For a given magnetic flux the field in this pore model is uniquely defined if the magnetic pressure balances the gas pressure at two levels. For models with fluxes of 0.5-3.0 × 1020 mx the surface radius varies from 1100-2700 km (diameters of 3-8 arc-sec) and the Wilson depression is estimated at 200 km. As the flux increases, the field becomes nearly horizontal at the edge of the pore and eventually a penumbra is formed. The distinction between pores and sunspots is investigated; the critical flux is about 1020 Mx, corresponding to a radius of 1500 km. Title: On the Magnetic Field in Pores Authors: Weiss, N. O.; Simon, G. W. Bibcode: 1969BAAS....1S.295W Altcode: No abstract at ADS Title: Sunspots and photospheric convection Authors: Weiss, N. O. Bibcode: 1969pia..conf..153W Altcode: No abstract at ADS Title: Supergranules and the Hydrogen Convection Zone Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1968ZA.....69..435S Altcode: No abstract at ADS Title: Supergranules and the Hydrogen Convection Zone. Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1968AJS....73S..77S Altcode: The strong magnetic fields observed between supergranules indicate that there must be subphotospheric convection in cells with a preferred diameter of about 30 000 km. Orthodox mixing length theory assumes that the dimensions of cells are limited by the density scale height. This is adequate fot explaining granules but cannot account for supergranulation. A model is therefore proposed in which cellular motions extend over several scale heights. In addition to granules ~nd supergranules, this model predicts a third characteristic scale of motion, with giant cells around 300 000 km in diameter, These cells may produce a pattern of magnetic fields like that suggested by Bumba and Howard for complexes of activity. Title: Concentration of Magnetic Fields in the Deep Convection Zone Authors: Simon, G. W.; Weiss, N. O. Bibcode: 1968IAUS...35..108S Altcode: No abstract at ADS Title: Convection in the Earth's mantle Authors: Allan, D. W.; Thompson, W. B.; Weiss, N. O. Bibcode: 1967metp.conf..507A Altcode: No abstract at ADS Title: Concentration of magnetic fields by convection in the Sun Authors: Weiss, N. O. Bibcode: 1967maco.conf..262W Altcode: No abstract at ADS Title: The Expulsion of Magnetic Flux by Eddies Authors: Weiss, N. O. Bibcode: 1966RSPSA.293..310W Altcode: 1966RSLPS.293..310W A convective eddy imposed on an initially uniform magnetic field in a highly conducting fluid distorts the lines of force and amplifies the field. Flux is concentrated outside the eddy; within it, the field grows and its scale of variation decreases until resistive effects become important. Closed lines of force are then formed by reconnexion. The central field decays and a steady state is reached. Within a period, small compared with the characteristic time for resistive decay, magnetic flux is almost entirely expelled from regions of rapid motion and concentrated at the edges of convection cells. This process is demonstrated from numerical experiments. The results are applied to the sun, where the concentrated fields are strong enough to inhibit convection locally. Title: Convection and the differential rotation of the Sun Authors: Weiss, N. O. Bibcode: 1965Obs....85...37W Altcode: No abstract at ADS Title: Magnetic flux tubes and convection in the sun Authors: Weiss, N. O. Bibcode: 1965IAUS...22..330W Altcode: No abstract at ADS Title: Convection in the Presence of Restraints Authors: Weiss, N. O. Bibcode: 1964RSPTA.256...99W Altcode: In the presence of rotation or a magnetic field, the linearized convection problem reduces to a cubic characteristic equation. In part I, general methods are given for determining the onset of convection; in particular, the transition from oscillatory to steady modes is considered. The importance of this transition arises from evidence that oscillatory modes are inefficient at transporting heat. These methods are then applied to a rotating system where the critical Rayleigh number can be expressed in terms of a Taylor number. It is found that overstable modes develop into steady unstable modes before the exchange of stabilities for Prandtl numbers less than one-third. The nature of the motions is discussed and a similar treatment is provided for convection in a magnetic field. In part II, criteria for the onset of instability are derived from physical arguments. Convection can be treated by balancing the work done by buoyancy forces against the energy dissipated. In a rotating system, the effect of the Coriolis forces is to restrict the cell width and thus to enhance dissipation and promote stability. A magnetic field similarly attenuates the cells and prevents steady convection until the liberated kinetic energy exceeds the energy in the field. In part III, a cellular model is proposed for turbulent convection in a fluid of negligible viscosity, where the motion is limited by the non-linear transfer of energy to smaller-scale motions. If the Rayleigh number R3 = gα β d44kappa 2>> 1 the convective transport varies as R31/2, while it varies as R32 when R3<< 1. The discussion is extended to convection in the presence of rotation or a magnetic field; it is shown that overstable perturbations cannot develop into steady turbulent convection unless the system is already unstable to non-oscillatory modes. The transition from overstable to steady modes should therefore correspond to a sharp increase in convective transport. Title: Magnetic flux tubes and convection in the Sun Authors: Weiss, N. O. Bibcode: 1964MNRAS.128..225W Altcode: Convection in the Sun is apparently cellular and the convective velocities can be estimated. The interaction of cellular convection with a weak magnetic field leads to concentrated ropes of flux in the convective zone with fields of about 5000 gauss and fluxes of 1021 maxwells in agreement with observation. These flux tubes tend to float outwards from the centre of the Sun. The magnetic pressure associated with the fields required to halt photospheric convection is comparable with the gas pressure and so fields will generally be diffuse as in bipolar magnetic regions. Occasionally the flux is concentrated to form a sunspot but the cooling is insignificant at more than 1000 km below the surface of the Sun. Title: Nouvelles de la Science. - Varietes. La prochaine comete. Authors: Weiss Bibcode: 1886LAstr...5..109W Altcode: No abstract at ADS