Author name code: durney ADS astronomy entries on 2022-09-14 author:"Durney, Bernard R." ------------------------------------------------------------------------ Title: The Hawking Effect for Massive Particles Authors: Durney, Bernard R. Bibcode: 2014IJAA....4...11D Altcode: No abstract at ADS Title: Magnetically Preferred Solar Longitudes: Reality? Authors: Henney, C. J.; Durney, B. R. Bibcode: 2005ASPC..346..381H Altcode: 2007astro.ph..1118H The observed persistence of specific periodicities detected in time series associated with solar surface magnetic activity over several solar cycles has led to numerous papers supporting the existence of preferred longitudes. Recent analysis of the past 120 years of sunspot number data showed that no observed periodicity remained coherent for durations greater than two 11-year solar cycles. Here we address the question: Could the observed periodicities of solar magnetic signals on time scales of two decades be the result of a purely stochastic process? We begin to answer this by comparing phase coherence between observed periodic signals and signals from a model using longitudinally random eruptions. A surprisingly non-negligible likelihood is found, approximately 1 in 3, that observed periodicities from integrated full-disk solar parameters are a chance occurrence for time series on the order of 20 years in duration. Title: Solar Subsurface Fluid Dynamics Descriptors Derived from Global Oscillation Network Group and Michelson Doppler Imager Data Authors: Komm, R.; Corbard, T.; Durney, B. R.; González Hernández, I.; Hill, F.; Howe, R.; Toner, C. Bibcode: 2004ApJ...605..554K Altcode: We analyze Global Oscillation Network Group (GONG) and Michelson Doppler Imager (MDI) observations obtained during Carrington rotation 1988 (2002 March 30-April 26) with a ring-diagram technique in order to measure the zonal and meridional flow components in the upper solar convection zone. We derive daily flow maps over a range of depths up to 16 Mm on a spatial grid of 7.5d in latitude and longitude covering +/-60° in latitude and central meridian distance and combine them to make synoptic flow maps. We begin exploring the dynamics of the near-surface layers and the interaction between flows and magnetic flux by deriving fluid dynamics descriptors such as divergence and vorticity from these flow maps. Using these descriptors, we derive the vertical velocity component and the kinetic helicity density. For this particular Carrington rotation, we find that the vertical velocity component is anticorrelated with the unsigned magnetic flux. Strong downflows are more likely associated with locations of strong magnetic activity. The vertical vorticity is positive in the northern hemisphere and negative in the southern hemisphere. At locations of magnetic activity, we find an excess vorticity of the same sign as that introduced by differential rotation. The vertical gradient of the zonal flow is mainly negative except within 2 Mm of the surface at latitudes poleward of about 20°. The zonal-flow gradient appears to be related to the unsigned magnetic flux in the sense that locations of strong activity are also locations of large negative gradients. The vertical gradient of the meridional flow changes sign near about 7 Mm, marking a clear distinction between near-surface and deeper layers. GONG and MDI data show very similar results. Differences occur mainly at high latitudes, especially in the northern hemisphere, where MDI data show a counter cell in the meridional flow that is not present in the corresponding GONG data. Title: The Contribution of Sound Waves and Instabilities to the Penetration of the Solar Differential Rotation Below the Convection Zone Authors: Durney, Bernard R. Bibcode: 2004SoPh..219..231D Altcode: The response of a layer to a horizontal shear flow at its top the surface was studied numerically as an initial value problem. The geometry was Cartesian and the conservation equations were solved with the help of the Zeus-3D code. In the initial state, the pressure, p, and density, ρ, of the layer were assumed to be related by a polytropic equation of index 1.14, which best approximates the solar values in the region of interest. The values of p and ρ at the lower boundary of the layer, namely r=Rl=0.4 R⊙, were taken to be the solar values. The upper boundary was chosen to be the base of the solar convection zone, r=Rc=0.7 R⊙. The shear flow at the surface, vφ(Rc), was proportional to the solar differential rotation, and acoustical oscillations were present in the layer. Title: The Energy Equation in the Lower Solar Convection Zone Authors: Durney, Bernard R. Bibcode: 2003SoPh..217....1D Altcode: As a consequence of the Taylor-Proudman balance, a balance between the pressure, Coriolis and buoyancy forces in the radial and latitudinal momentum equations (that is expected to be amply satisfied in the lower solar convection zone), the superadiabatic gradient is determined by the rotation law and by an unspecified function of r, say, S'Ω(r), where r is the radial coordinate. If the rotation law and S'Ω(r) are known, then the solution of the energy equation, performed in this paper in the framework of the MLΩ formalism, leads to a knowledge of the Reynolds stresses, convective fluxes, and meridional motions. The MLΩ-formalism is an extension of the mixing length theory to rotating convection zones, and the calculations also involve the azimuthal momentum equation, from which an expression for the meridional motions in terms of the Reynolds stresses can be derived. The meridional motions are expanded as Ur(r,θ)=P2(cosθ)ψ2(r)/r2ρ+P4(cosθ)ψ4(r)/r2 ρ+..., and a corresponding equation for Uθ(r,θ). Here θ is the polar angle, ρ is the density, and P2(cosθ), P4(cosθ) are Legendre polynomials. A good approximation to the meridional motion is obtained by setting ψ4(r)=−Hψ2(r) with H≈−1.6, a constant. The value of ψ2(r) is negative, i.e., the P2 flow rises at the equator and sinks at the poles. For the value of H obtained in the numerical calculations, the meridional motions have a narrow countercell at the poles, and the convective flux has a relative maximum at the poles, a minimum at mid latitudes and a larger maximum at the equator. Both results are in agreement with the observations. Title: Temporal Variation of Angular Momentum in the Solar Convection Zone Authors: Komm, R.; Howe, R.; Durney, B. R.; Hill, F. Bibcode: 2003ApJ...586..650K Altcode: We derive the angular momentum as a function of radius and time with the help of the rotation rates resulting from inversions of helioseismic data obtained from the Global Oscillation Network Group (GONG) and the Michelson Doppler Imager (MDI) and the density distribution from a model of the Sun. The base of the convection zone can be identified as a local maximum in the relative angular momentum after subtracting the contribution of the solid-body rotation. The angular momentum as a function of radius shows the strongest temporal variation near the tachocline. This variation extends into the lower convection zone and into the radiative interior and is related to the 1.3 yr periodicity found in the equatorial rotation rate of the tachocline. In the upper convection zone, we find a small systematic variation of the angular momentum that is related to torsional oscillations. The angular momentum integrated from the surface to a lower limit in the upper convection zone provides a hint that the torsional oscillation pattern extends deep into the convection zone. This is supported by other quantities such as the coefficients of a fit of Legendre polynomials to the rotation rates as a function of latitude. The temporal variation of the coefficient of P4, indicative of torsional oscillations, suggests that the signature of these flows in the inversion results extend to about r~0.83Rsolar. With the lower limit of integration placed in the middle or lower convection zone, the angular momentum fluctuates about the mean without apparent trend, i.e., the angular momentum is conserved within the measurement errors. However, when integrated over the layers slightly below the convection zone (0.60-0.71Rsolar), the angular momentum shows the 1.3 yr period and hints at a long-term trend that might be related to the solar activity cycle. Title: Temporal variation of angular momentum in the convection zone Authors: Komm, R.; Howe, R.; Durney, B. R.; Hill, F. Bibcode: 2003ESASP.517...97K Altcode: 2003soho...12...97K We derive the angular momentum as a function of radius and time with the help of the rotation rates resulting from inversions of helioseismic data obtained from the Global Oscillation Network Group (GONG) and the Michelson Doppler Imager (MDI) and the density distribution from a model of the Sun. The angular momentum as a function of radius shows the strongest temporal variation near the base of the convection zone. This variation extends into the lower convection zone and into the radiative interior and is related to the 1.3-yr periodicity found in the equatorial rotation rate of the tachocline. In the upper convection zone, we find a small systematic variation of the angular momentum that is related to torsional oscillations. The angular momentum integrated from the surface to a lower limit in the upper convection zone provides a hint that the torsional oscillation pattern extends deep into the convection zone. With the lower limit of integration placed in the lower half of the convection zone, the angular momentum fluctuates about the mean without apparent trend, i.e. the angular momentum is conserved within the measurement errors. However, when integrated over the layers slightly below the convection zone, the angular momentum shows the 1.3-yr period and hints at a long-term trend which might be related to the solar activity cycle. Title: Temporal Variation of Angular Momentum in the Solar Convection Zone Authors: Komm, R.; Howe, R.; Durney, B.; Hill, F. Bibcode: 2002AAS...200.0404K Altcode: 2002BAAS...34Q.644K We present the temporal variation of the solar angular momentum derived from helioseismic observations. In the absence of `true' angular momentum inversions, we use the rotation rates resulting from rotation inversions of GONG data and the density distribution from a model of the Sun. We focus especially on the layers near the base of the convection zone and the layers near the solar surface. We derive the angular momentum as a function of depth and the corresponding solid-body rotation. The angular momentum decreases with increasing radius following essentially the product of density times the fourth power of radius. The tachocline can be identified as a local maximum in the radial gradient of the angular momentum and as a local maximum in the relative angular momentum after subtracting the contribution of the solid-body rotation. The angular momentum shows the strongest temporal variation near the tachocline. This variation is reminiscent of the 1.3-yr periodicity found in the equatorial rotation rate of the tachocline, which is not too surprising since the angular momentum of a spherical shell is heavily weighted toward the equator. We discuss the extension of this variation into the convection zone and into the radiative interior. In addition, we fit the rotation rates as functions of latitude with Legendre polynomials to cross-validate the numerical results and to draw conclusions about the zonal flows (`torsional oscillations') in the upper convection zone. This work was supported by NASA Grant S-92698-F. Title: Approximate Isocontours For The Solar Angular Velocity In The Convection Zone Authors: Durney, Bernard R. Bibcode: 2001SoPh..202..201D Altcode: The angular velocity, Ω, in the solar convection zone (SCZ) is expanded in Legendre polynomials, Pn(cosθ), and the values for Ω at the equator are assumed to be given by Kosovichev's helioseismic data; here, r, θ, and φ, label the radial, latitudinal and longitudinal coordinates, respectively. The isocontours for Ω are calculated for the following two cases. (i) The angular momentum of a thin spherical shell of radius r is identical to the shell's angular momentum for solid body rotation, i.e., rotation just distributes in latitude the angular momentum of each layer. (ii) Considerations based on the Taylor-Proudman balance (a balance between the pressure, Coriolis and buoyancy forces which is expected to be amply satisfied in the SCZ), require that the radial component of the superadiabatic gradient be strongly dependent on latitude unless the coefficients in the expansion for Ω defined above satisfy a first-order differential equation, DE. The isocontours for the angular velocity determined from DE, compare remarkably well with the helioseismic data, whereas for case (i) there is a marked difference at high latitudes. The radial and latitudinal balance of angular momentum are studied. The meridional motions are determined mainly (but not entirely) by the radial balance of angular momentum, and they depend principally on the Reynolds stress, «ur uφ ». Concerning the latitudinal balance, ∂Ω/∂θ increases until the transport of angular momentum toward the poles by the meridional motions is able to balance the transport of angular momentum towards the equator by « uθ uφ » ( = « uθ uφ »0 - νt sinθ∂Ω/∂θ)). Here the subscript 0 stands for solid body rotation, and νt is a turbulent viscosity coefficient. In contrast to the radial balance, the viscosity term plays a fundamental role in the latitudinal balance of angular momentum. Title: The Energy Lost by Differential Rotation in the Generation of the Solar Toroidal Magnetic Field Authors: Durney, Bernard R. Bibcode: 2000SoPh..197..215D Altcode: The integrals, Ii(t) = ∫GL uij × Bidv over the volume GL are calculated in a dynamo model of the Babcock-Leighton type studied earlier. Here, GL is the generating layer for the solar toroidal magnetic field, located at the base of the solar convection zone (SCZ); i=r, θ, φ, stands for the radial, latitudinal, and azimuthal coordinates respectively; j = (4π)-1∇ × B, where B is the magnetic field; ur,uθ are the components of the meridional motion, and uφ is the differential rotation. During a ten-year cycle the energy ∫cycle Iφ(t)dt needs to be supplied to the azimuthal flow in the GL to compensate for the energy losses due to the Lorentz force. The calculations proceed as follows: for every time step, the maximum value of |Bφ| in the GL is computed. If this value exceeds Bcr (a prescribed field) then there is eruption of a flux tube that rises radially, and reaches the surface at a latitude corresponding to the maximum of |Bφ| (the time of rise is neglected). This flux tube generates a bipolar magnetic region, which is replaced by its equivalent axisymmetric configuration, a magnetic ring doublet. The erupted flux can be multiplied by a factor Ft, i.e., by the number of eruptions per time step. The model is marginally stable and the ensemble of eruptions acts as the source for the poloidal field. The arbitrary parameters Bcr and Ft are determined by matching the flux of a typical solar active region, and of the total erupted flux in a cycle, respectively. If E(B) is the energy, in the GL, of the toroidal magnetic field Bφ = B sin θ cos θ, B (constant), then the numerical calculations show that the energy that needs to be supplied to the differential rotation during a ten-year cycle is of the order of E(Bcr), which is considerably smaller than the kinetic energy of differential rotation in the GL. Assuming that these results can be extrapolated to larger values of Bcr, magnetic fields ≈104 G, could be generated in the upper section of the tachocline that lies below the SCZ (designated by UT). The energy required to generate these 104 G fields during a cycle is of the order of the kinetic energy in the UT. Title: On the Differences Between Odd and Even Solar Cycles Authors: Durney, Bernard R. Bibcode: 2000SoPh..196..421D Altcode: It is proposed that the observed differences between odd and even solar cycles are a consequence of the nonlinear interactions that provide the stabilizing mechanism for the cycle's amplitude. If, for example, the magnetic field is larger than average for a given cycle (say odd), the nonlinear feedback mechanism can generate a magnetic field that is smaller than average for the next cycle (even), and then one that is larger than average for the following cycle (odd), etc. As a consequence the odd cycles have larger amplitudes than even cycles. A very simply model having a nonlinear interaction that reproduces this behavior is discussed. Title: On The Torsional Oscillations In Babcock-Leighton Solar Dynamo Models Authors: Durney, Bernard R. Bibcode: 2000SoPh..196....1D Altcode: The torsional oscillations at the solar surface have been interpreted by Schüssler and Yoshimura as being generated by the Lorentz force associated with the solar dynamo. It has been shown recently that they are also present in the upper half of the solar convection zone (SCZ). With the help of a solar dynamo model of the Babcock-Leighton type studied earlier, the longitudinal component of the Lorentz force, Lφ, is calculated, and its sign or isocontours, are plotted vs. time, t, and polar angle, θ (the horizontal and vertical axis respectively). Two cases are considered, (1) differential rotation differs from zero only in the tachocline, (2) differential rotation as in (1) in the tachocline, and purely latitudinal and independent of depth in the bulk of the SCZ. In the first case the sign of Lφ is roughly independent of latitude (corresponding to vertical bands in the t,θ plot), whereas in the second case the bands show a pole-equator slope of the correct sign. The pattern of the bands still differs, however, considerably from that of the helioseismic observations, and the values of the Lorentz force are too small at low latitudes. It is all but certain that the toroidal field that lies at the origin of the large bipolar magnetic regions observed at the surface, must be generated in the tachocline by differential rotation; the regeneration of the corresponding poloidal field, Bp has not yet been fully clarified. Bp could be regenerated, for example, at the surface (as in Babcock-Leighton models), or slightly above the tachocline, (as in interface dynamos). In the framework of the Babcock-Leighton models, the following scenario is suggested: the dynamo processes that give rise to the large bipolar magnetic regions are only part of the cyclic solar dynamo (to distinguish it from the turbulent dynamo). The toroidal field generated locally by differential rotation must contribute significantly to the torsional oscillations patterns. As this field becomes buoyant, it should give rise, at the surface, to the smaller bipolar magnetic regions as, e.g., to the ephemeral bipolar magnetic regions. These have a weak non-random orientation of magnetic axis, and must therefore also contribute to the source term for the poloidal field. Not only the ephemeral bipolar regions could be generated in the bulk of the SCZ, but many of the smaller bipolar regions as well (at depths that increase with their flux), all contributing to the source term for the poloidal field. In contrast to the butterfly diagram that provides only a very weak test of dynamo theories, the pattern of torsional oscillations has the potential of critically discriminating between different dynamo models. Title: Meridional Motions and the Angular Momentum Balance in the Solar Convection Zone Authors: Durney, Bernard R. Bibcode: 2000ApJ...528..486D Altcode: The solar angular velocity, Ω, and meridional motions in the solar convection zone (SCZ) are expanded in Legendre polynomials. If the velocity correlations <uruφ>,<uθuφ> and the angular velocity are known, then the azimuthal momentum equation determines the meridional flow; here u stands for the turbulent convective velocities and the bracket denotes an appropriate average; θ and φ are the polar angle and longitude.

The velocity correlation <uruφ> transports angular momentum to the inner regions of the SCZ. This angular momentum can either spin-up the inner regions, or be removed by a meridional motion that rises at the equator and sinks at the poles; the stream function for this motion will be designated by ψ2. For slowly rotating stars, the inner regions must spin-up. As the angular velocity increases, a transition must take place to the second option: in the Sun the angular velocity does not increase sharply with depth. This transition should occur at a value for Ω at which the Taylor-Proudman balance (a balance between the pressure, Coriolis, and buoyancy forces) becomes valid. In the SCZ, this balance determines the latitudinal variations of the superadiabatic gradient (\b.nabla ΔT) from the rotation law, and it provides, therefore, a link between the energy equation and the azimuthal momentum equation.

The solar meridional motion also has a component, with stream function ψ4, that rises at the equator and poles and sinks at midlatitudes; its contribution to the removal of angular momentum from the inner regions of the SCZ is negligible. In the Sun, ψ2 depends mainly on <uruφ> and ψ4~-4ψ2/3 (this expression for ψ4 is not as robust as that of ψ2, which is an excellent approximation). Therefore, the meridional motions are essentially determined by <uruφ>. However, the ψ2-meridional circulation transports angular momentum toward the polar regions of the Sun which must be balanced by <uθuφ> and ψ4. Globally, the conservation of angular momentum in the latitudinal direction requires that the sum of the terms in <uruφ> and in <uθuφ> of an integral over the entire SCZ cancels. For this to be the case, <uθuφ> must be positive since <uruφ> is negative (which is a very robust result). For stars satisfying the Taylor-Proudman balance, a fast rotating equator appears to be an unavoidable necessity.

An equation is derived that clarifies the reasons for the existence of the relation ψ4~-4ψ2/3 and for the weak dependence of ψ4 on <uθuφ>.

A simple model for the velocity correlations is studied. In this simple model, if the latitudinal differential rotation increases, <uθuφ> must decrease for the integral relation defined above to remain valid. This dependence of <uθuφ> on ∂Ω/∂θ agrees with what can be inferred from physical considerations. Title: The Taylor-Proudman Balance and the Solar Rotational Data Authors: Durney, Bernard R. Bibcode: 1999ApJ...511..945D Altcode: For the Sun (and because of the presence of buoyancy), the implications of the Taylor-Proudman balance (TPB, a balance between the pressure, Coriolis, and buoyancy forces in the radial and latitudinal momentum equations) differ fundamentally from those of an incompressible fluid. The TPB now only determines the latitudinal variations of the solar entropy in terms of the rotation law and known functions of r. As a consequence of the TPB, the energy equation is in fact an equation for the angular velocity, Ω(r, θ)=Ω0(ω0(r)+ω2(r)P2(cosθ)), where P2(cosθ) is the second-order Legendre polynomial. In agreement with data from the Solar Oscillations Investigations project (SOI) Michelson Doppler Imager (MDI) on board SOHO, we assume that ω0(r) is constant with r, and solve the equation for ω2(r) with a simple, heuristic expression for the convective flux [if ω0(r) is constant, then ∂Ω/∂r vanishes for θc=54.7d, P2(cosθc)=0, in remarkable agreement with Kosovichev's results inferred from isocontours for Ω(r, θ)]. For values of the meridional motions that are not too large, solutions for ω2(r) exist that agree with these isocontours. These solutions are such that the latitudinal variation of the convective flux, arising from the latitudinal variations of the entropy, required by the TPB, are significantly reduced. Title: On the Power in the Legendre Modes of the Solar Radial Magnetic Field Authors: Durney, Bernard R. Bibcode: 1998SoPh..180....1D Altcode: The power in the different ℓ modes of an expansion of the solar radial magnetic field at the surface in terms of Legendre polynomials,Pℓ , is calculated with the help of a solar dynamo model studied earlier. The model is of the Babcock-Leighton type, i.e., the surface eruptions of the toroidal magnetic field - through the `tilt angle', γ, formed by the magnetic axis of a bipolar magnetic region with the east-west line - are the sources for the poloidal field. In this paper it is assumed that the tilt angle is subject to fluctuations of the form, γ = γ'(σ)+ <γ> where <γ> is the average value and γ'(σ) is a random normal fluctuation with standard deviation σ which is taken from Howard's observations of the distribution of tilt angles. For numerical considerations, negative values of γ were not allowed. If this occurred, γ was recalculated. The numerical integrations were started with a toroidal magnetic field antisymmetric across the equator, large enough to generate eruptions, and a negligible poloidal field. The fluctuations in the tilt angle destroy the antisymmetry as time increases. The power of the antisymmetric modes across the equator (i.e., odd values of ℓ) is concentrated in frequencies, νp, corresponding to the cycle period. The maximum power lies in the ℓ=3 mode with considerable power in the ℓ=5 mode, in broad agreement with Stenflo's results who finds a maximum power at ℓ=5. For the symmetric modes, there is considerable power in frequencies larger than νp, again in broad agreement with Stenflo's power spectrum. Title: On a Babcock-Leighton Solar Dynamo Model with a Deep-seated Generating Layer for the Toroidal Magnetic Field. IV. Authors: Durney, Bernard R. Bibcode: 1997ApJ...486.1065D Altcode: The study is continued of a dynamo model of the Babcock-Leighton type (i.e., the surface eruptions of toroidal magnetic field are the source for the poloidal field) with a thin, deep seated layer (GL), for the generation of the toroidal field, Bφ. The partial differential equations satisfied by Bφ and by the vector potential for the poloidal field are integrated in time with the help of a second order time- and space-centered finite different scheme. Axial symmetry is assumed; the gradient of the angular velocity in the GL is such that within this layer a transition to uniform rotation takes place; the meridional motion, transporting the poloidal field to the GL, is poleward and about 3 m s-1 at the surface; the radial diffusivity ηr equals 5 × 109 cm2 s-1, and the horizontal diffusivity ηθ is adjusted to achieve marginal stability. The initial conditions are: a negligible poloidal field, and a maximum value of |Bφ| in the GL equal to 1.5 × Bcr, where Bcr is a prescribed field.

For every time step the maximum value of |Bφ| in the GL is computed. If this value exceeds Bcr, then there is eruption of a flux tube (at the latitude corresponding to this maximum) that rises radially to the surface. Only one eruption is allowed per time step (Δt) and Bφ in the GL is unchanged as a consequence of the eruption. The ensemble of eruptions is the source for the poloidal field, i.e., no use is made of a mean field equation relating the poloidal with the toroidal field. For a given value of Δt, and since the problem is linear, the solutions scale with Bcr. Therefore, the equations need to be solved for one value of Bcr only.

Since only one eruption is allowed per time step, the dependence of the solutions on Δt needs to be studied. Let Ft be an arbitrary numerical factor (= 3 for example) and compare the solutions of the equations for (Bcr, Δt) and (Bcr, Δt/3). It is clear that there will be 3 times as many eruptions in the second case (with the shorter time step) than in the first case. However, if the erupted flux in case one is multiplied by 3, then the solutions for this case become nearly identical to those of case two (Δt is shorter than any typical time of the system, and the difference due to the unequal time steps is negligible). Therefore, varying the time step is equivalent to keeping Δt fixed while multiplying the erupted flux by an appropriate factor. In the numerical calculations Δt was set equal to 105 s. The factor Ft can then be interpreted as the number of eruptions per 105 s. The integration of the equations shows that there is a transition in the nature of the solutions for Ft ~ 2.5. For Ft < 2.5, the eruptions occur only at high latitudes, whereas for Ft > 2.5, the eruptions occur for θ greater than ~ π/4, where θ is the polar angle. Furthermore, for Ft < 2.5, the toroidal field, |Bφ|, in the GL can become considerably larger than Bcr, while this ceases to be the case for Ft > 2.5.

The factor Ft is an arbitrary parameter in the model and an appeal to observations is necessary. We set Bcr = 103 G. In the model, the magnetic flux of erupting magnetic tubes, is then about 3 × 1021 G, of the order of the solar values. For this value of Bcr and for the value of Ft (~2.5) at which the transition takes place, the total erupted flux in 10 years is about 0.85 × 1025 Mx in remarkable agreement with the total erupted flux during a solar cycle. Concerning the dynamo models studied here, a major drawback encountered in previous papers has been the eruptions at high latitudes, which entail unrealistically large values for the radial magnetic field at the poles. The results of this paper provide a major step forward in the resolution of this difficulty. Title: On the Influence of Gradients in the Angular Velocity on the Solar Meridional Motions Authors: Durney, Bernard R. Bibcode: 1996SoPh..169....1D Altcode: If fluctuations in the density are neglected, the large-scale, axisymmetric azimuthal momentum equation for the solar convection zone (SCZ) contains only the velocity correlations and where u are the turbulent convective velocities and the brackets denote a large-scale average. The angular velocity, Ω, and meridional motions are expanded in Legendre polynomials and in these expansions only the two leading terms are retained (for example, where θ is the polar angle). Per hemisphere, the meridional circulation is, in consequence, the superposition of two flows, characterized by one, and two cells in latitude respectively. Two equations can be derived from the azimuthal momentum equation. The first one expresses the conservation of angular momentum and essentially determines the stream function of the one-cell flow in terms of : the convective motions feed angular momentum to the inner regions of the SCZ and in the steady state a meridional flow must be present to remove this angular momentum. The second equation contains also the integral indicative of a transport of angular momentum towards the equator. Title: On a Babcock-Leighton Dynamo Model with a Deep-Seated Generating Layer for the Toroidal Magnetic Field, II Authors: Durney, Bernard R. Bibcode: 1996SoPh..166..231D Altcode: In a previous paper (Paper I), we studied a dynamo model of the Babcock-Leighton type (i.e., the surface eruptions of toroidal magnetic field are the source for the poloidal field) that included a thin, deep seated, generating layer (GL) for the toroidal field, Bφ. Meridional motions (of the order of 12 m s−1 at the surface), rising at the equator and sinking at the poles were essential for the dynamo action. The induction equation was solved by approximating the latitudinal dependence of the fields by Legendre polynomials. No solutions were found with Φp = Φf where Φp and Φf are the fluxes for the preceding and following spot, respectively. The solutions presented in Paper I, had Φp = −0.5 Φf, were oscillatory in time, and large radial fields, Bτ, were present at the surface. Title: On a Babcock-Leighton dynamo model with a deep-seated generating layer for the toroidal magnetic field Authors: Durney, Bernard R. Bibcode: 1995SoPh..160..213D Altcode: A dynamo model of the Babcock-Leighton type having the following features is studied. The toroidal fieldBφ is generated in a thin layer (the GL), located at the lower solar convection zone, by a shear in the angular velocity acting on the poloidal fieldBp(= ∇ × [0, 0,Aφ].) If, in this layer, and for a certain value of the polar angle,θ, |BØ | exceeds a critical field,Bcr, then the eruption of a flux tube occurs. This flux tube, which is assumed to rise radially, generates, when reaching the surface, a bipolar magnetic region (BMR) with fluxes Φp and Φf for the preceding and following spot respectively. For the purpose of the numerical calculations this BMR is replaced by its equivalent axisymmetrical magnetic ring doublet. The ensemble of these eruptions acts as the source term for the poloidal field. This field, generated in the surface layers, reaches the lower solar convection by transport due to meridional motions and by diffusion. The meridional motions are the superpositions of a one-cell velocity field that rises at the equator and sinks at the poles and of a two-cell circulation that rises at the equator and poles and sinks at mid latitudes. The toroidal field andAØ were expanded in Legendre polynomials, and the coupled partial differential equations (int andr; time and radial coordinate) satisfied by the coefficients in these expansions were solved by a finite difference method. In the expansions, Legendre polynomials up to order thirty were included. Title: On a Babcock-Leightom dynamo model with a thin, deep seated generating layer for the toroidal field. Authors: Durney, Bernard R. Bibcode: 1994AAS...185.9201D Altcode: 1994BAAS...26Q1472D The following dynamo model will be discussed and hopefully numerical results will be presented. Let A be the vector potential for the axisymmetric poloidal field, and B, the toroidal field. B is generated by a shear in the angular velocity acting on A in a thin layer located in the lower solar convection zone. If in this layer B exceeds a critical value for a certain value of theta (the polar angle), eruption occurs. The flux tube is assumed to rise radially and to surface as a magnetic ring doublet. The rates of eruption of the ensemble of these doublets constitute the source term of the equation for partial A / partial t that regenerates the poloidal field. The poloidal field generated in the solar surface layers reaches the lower solar convection by transport due to meridional motions and by diffusion. The meridional motions being considered are the superposition of a one-cell velocity field that rises at the equator and sinks at the poles and of a two-cell motion that rises at the equator and poles and sinks at mid latitudes. Meridional motions of this type have a strong theoretical and observational support. Title: On the Generation of the Largescale and Turbulent Magnetic Fields in the Solar Type Stars Authors: Durney, Bernard R.; De Young, David S.; Roxburgh, Ian W. Bibcode: 1993SoPh..145..207D Altcode: It is thought that the large-scale solar-cycle magnetic field is generated in a thin region at the interface of the radiative core (RC) and solar convection zone (SCZ). We show that the bulk of the SCZ virogoursly generates a small-scale turbulent magnetic field. Rotation, while not essential, increases the generation rate of this field. Title: On the Solar Differential Rotation: Meridional Motions Associated with a Slowly Varying Angular Velocity Authors: Durney, Bernard R. Bibcode: 1993ApJ...407..367D Altcode: The paper calculates the meridional flow in the solar convection zone from the azimuthal momentum equation and compares the results with the existing surface observations. The dominant flows are found to contain only a few cells per hemisphere. Other than the meridional flow, the azimuthal momentum equation contains the velocity correlations (ur uphi) and (utheta uphi), as well as the angular velocity, Omega. The stream functions psi2 and psi4 define meridional motions with one and two latitudinal cells, respectively. The resultant meridional flow has a one-cell component rising at the equator and a two-cell component sinking at midlatitudes, in agreement with observations. Title: Observational Constraints on Theories of the Solar Differential Rotation Authors: Durney, Bernard R. Bibcode: 1991ApJ...378..378D Altcode: The dependence of the inner solar angular velocity (ISAV) on the spatial coordinates is examined through helioseismic observation. If the ratios, designated RR, are specified and ISAV is known from the observations, then the azimuthal momentum equation determines the meridional motions. For all realistic values of RR, the stream function of a meridional motion is negative, i.e., the flow rises at the equator and sinks at the poles. Such flows together with the fact that the turbulent convective velocities are negative suggest a natural explanation for the helioseismic observations near the solar surface (with increasing depth, the angular velocity first increases and then decreases). It is proposed that flows with few cells per hemisphere will dominate. Restrictions of this type imposed on the azimuthal momentum equation circumscribe the values of RR. In the simple case under consideration, the eddies take on a slablike appearance elongated along the axis of rotation. Title: On the Generation of the Solar Magnetic Field in a Region of Weak Buoyancy Authors: Durney, Bernard R.; De Young, David S.; Passot, Thierry P. Bibcode: 1990ApJ...362..709D Altcode: The possibility that the cyclic magnetic field of the sun can be generated in a layer of weak buoyancy at the lower boundary of the solar convection zone (SCZ) is addressed using an eddy-damped quasi-normal Markovian closure model of the turbulent MHD equations. It is concluded that only models with kinetic energy of turbulent motion larger than roughly 10 to the 6th g/cm/s are viable. The action of differential rotation on this poloidal field generates a toroidal field exceeding the equipartition value which is sufficiently strong to interfere with the transport of heat and is amplified further in the SCZ. Title: On the Numerical Calculation of the Solar Rotational Splitting Coefficients Authors: Durney, Bernard R. Bibcode: 1990ApJ...351..682D Altcode: An alternative method for numerically calculating the solar rotational splitting coefficient is developed. The efficacy of the method is illustrated using South Pole data. The robustness of a(2i+1) is shown to be excellent for i = 0 and to decrease with increasing i, whereas the values of b(2i+1) appear to be sensitive to the method of calculation. Title: Some Controversial Issues in Theories of the Solar Differential Rotation and Dynamo Authors: Durney, Bernard R. Bibcode: 1989SoPh..123..197D Altcode: The following points are discussed: The dependence of the angular velocity, , on the spatial coordinates near the lower boundary, Rc, of the solar convection zone (SCZ) can be obtained from an integration with respect to r of a sound approximation to the azimuthal equation of motion. Here P2 (cos θ) is the second-order Legendre polynomial and θ is the polar angle. Estimates of ω'0, ω'2 (the primes denote derivatives with respect to r), based on the best available values for the Reynolds stresses and anisotropic viscosity coefficients, suggest that ω'0 < 0, ω'2 ≈ 0 for r = Rc. Since a reliable theory of anisotropic turbulent coefficients does not exist at present, positive values of ω'0 are conceivable. Title: On the Behavior of the Angular Velocity in the Lower Part of the Solar Convection Zone Authors: Durney, Bernard R. Bibcode: 1989ApJ...338..509D Altcode: The solar angular velocity is expanded in Legendre polynomials. The meridional motions are restricted to one or two cells per hemisphere, and an approximation to the azimuthal equation of motion is integrated with respect to r with the help of the boundary condition at r = Rc, the lower boundary of the solar convection zone (SCZ). The Reynolds stresses appearing in the equation are estimated for the lower SCZ, and approximate expressions are derived for the turbulent viscosity coefficients (which are due to the influence of the mean flow on the turbulent velocities). It is shown that the assumption of isotropic viscosity is always open to criticism. An order-of-magnitude estimate of the different terms in the E(r) and E(theta) equations suggests that the Reynolds and viscous stresses are important only near the boundaries of the SCZ. Away from the boundaries, in the Taylor-Proudman region, the suggested balance is between Coriolis forces, pressure gradients, and buoyancy forces. Title: Book-Review - the Internal Solar Angular Velocity - Theory Observations and Relationship to Solar Magnetic Fields Authors: Durney, B. R.; Sofia, S.; Gough, D. Bibcode: 1988JBAA...98..261D Altcode: No abstract at ADS Title: Book-Review - the Internal Solar Angular Velocity - Theory Observations and Relationship to Solar Magnetic Fields Authors: Durney, B. R.; Sofia, S. Bibcode: 1988S&T....75Q.498D Altcode: No abstract at ADS Title: On the Expansion of the Rotational Eigenfrequencies in Legendre Polynomials Authors: Durney, Bernard R.; Hill, Frank; Goode, Philip R. Bibcode: 1988ApJ...326..486D Altcode: In the context of helioseismology, it has become customary to fit data using Δv(n, l, m) ≡ v(n, l, m) - v(n, l) = L ΣN i=0 ai Pi(-m/L) (Duvall, Harvey, and Pomerantz) where v is the frequency of the nth p-mode averaged over m, the Pi are Legendre polynomials and L = [(l + 1)l]1/2. It is shown here that, instead, it is advantageous to use the following expansion for v(n, l, m) - v(n, l): v(n, l, m) - v(n, l) = m Σ N i=0 bi Pi (m/L). In this case the bi's are simply related to the coefficients which determine the angular velocity, leading to the expectation that we can more accurately determine the internal rotation of the Sun from the extant helioseismological data. Title: Book-Review - the Internal Solar Angular Velocity - Theory Observations and Relationship to Solar Magnetic Fields Authors: Durney, B. R.; Sofia, S. Bibcode: 1988Sci...239..926D Altcode: No abstract at ADS Title: A simple dynamo model and the anisotropic alpha-effect Authors: Durney, B. R. Bibcode: 1988A&A...191..374D Altcode: The α-term in dynamo theory is evaluated in the quasi-linear approximation with no assumption concerning the isotropy of the turbulent motions. The resulting expression for α is discussed in the framework of a dynamo model depending only on time and having a simple buoyancy term. It is argued that the main cyclic solar magnetic field is amplified in the lower boundary layer of the solar convection zone whereas the convection zone proper generates a weak, stochastic, background, magnetic field. Title: Book-Review - the Internal Solar Angular Velocity Authors: Durney, B. R.; Sofia, S. Bibcode: 1988ApL&C..27R.286D Altcode: 1988ApL....27R.286D No abstract at ADS Title: Book-Review - the Internal Solar Angular Velocity - Theory Observations and Relationship to Solar Magnetic Fields Authors: Durney, B. R.; Sofia, S. Bibcode: 1987JBAA...98Q..48D Altcode: No abstract at ADS Title: On the Solar Angular Velocity in the Lower Solar Convection Zone Authors: Durney, Bernard R. Bibcode: 1987BAAS...19Q.934D Altcode: No abstract at ADS Title: The Generalization of Mixing Length Theory to Rotating Convection Zones and Application to the Sun Authors: Durney, Bernard R. Bibcode: 1987ASSL..137..235D Altcode: 1987isav.symp..235D The consequences of a balance between the Coriolis forces, pressure gradients and buoyancy forces in a compressible medium are investigated (the Taylor-Proudman theorem). A simple proof is given that if this balance holds, then the latitudinally dependent part of the superadiabatic gradient (∇ΔT) is determined by the angular velocity, Ω, and it is of the order of 2Ω20T/7g for rotation laws other than Ω constant along cylinders (it vanishes in this case). Here Ω0 is the average angular velocity, T the temperature and g gravity. In the lower part of the solar convection zone, 2Ω20T/7g is of the order of ∇ΔTr, itself, i.e., very large. Title: The internal solar angular velocity. Theory, observations and relationship to solar magnetic fields Authors: Durney, Bernard R.; Sofia, Sabatino Bibcode: 1987ASSL..137.....D Altcode: 1987isav.symp.....D The conference presents papers on observations of solar p-mode rotational splittings, observations of surface velocity fields, the equatorial rotation rate in the solar convective zone, chromospheric activity in open clusters, and solar rotation variations from sunspot group statistics. Other topics include adiabatic nonradial oscillations of a differentially rotating star, a spherical harmonic decomposition technique for analyzing steady photospheric flows, turbulent transport in the radiative zone of a rotating star, and the generation of magnetic fields in the sun. Consideration is also given to magnetic fields and the rotation of the solar convection zone, the hydrostatic adjustment time of the solar subconvective layer, models for a differentially rotating solar-convection zone, and horizontal Reynolds stress and the radial rotation law of the sun. Title: On theories of rotating convection zones Authors: Durney, B. R. Bibcode: 1985ApJ...297..787D Altcode: It is shown that the time rate of change (brought about by the turbulent convective motions) in the angular momentum of a thin spherical shell of radius r is such as to increase the angular velocity of the lower part (τΩ0 > 1; τ is the dominant eddy's lifetime) of the solar convection zone (SCZ) and to decrease the angular velocity of the upper part (τΩ0 < 1). A tentative model of rotation in the SCZ is proposed: (1) the (τΩ0 > 1)-region is in weaker differential rotation than the surface and not constrained by the Taylor-Proudman theorem, (2) the observed solar differential rotation at the surface is then generated as the SCZ relaxes from the (τΩ0 > 1)-state in the lower part to the (τΩ0 < 1)-state at the surface. In the upper and lower layers of the SCZ, angular-momentum conservation between the turbulent motions and viscous stresses leads to an angular velocity increasing inward. Title: A search for long-lived velocity fields at the solar poles Authors: Durney, B. R.; Lytle, D. M.; Cram, L. E.; Guenther, D. B.; Keil, S. L. Bibcode: 1985ApJ...292..752D Altcode: A search has been made in the polar regions of the sun for large-scale (50-200 Mm) velocity fields with lifetimes of the order of the solar rotation period (approximately equal to or greater than 30 days). The observations show that any such large-scale, long-lived velocity patterns in the polar regions must have an amplitude less than 5 m/s. Marginally significant detections (at the 2-3 sigma level) were made of two kinds of structures with amplitudes of order 3 m/s. One has a rotation period approximately 38 days (close to the polar rotation period at the sun's surface), and a scale approximately 150 Mm; the other has a period approximately 24 days and a scale approximately 100 Mm. Tentatively, the first structure is interpreted as being of supergranular origin. The second structure is interpreted as the overshooting of the dominant convective mode of the lower solar convection zone - the giant granulation. Title: On the Generalization of the Mixing Length Theory to Rotating Convection Zones Authors: Durney, B. R. Bibcode: 1985BAAS...17..644D Altcode: No abstract at ADS Title: On the influence of turbulent motions on non-radial oscillations. Authors: Durney, B. R. Bibcode: 1984sses.nasa..325D Altcode: 1984sss..conf..325D The effect of turbulent motions on oscillations is studied, considering only the coupling between turbulent and oscillatory velocities. In this case, the turbulence affects the oscillations through the Reynolds stresses in the momentum equation for the pulsations. A simple model of turbulence is adopted to evaluate these Reynolds stresses and the perturbed eigenfrequencies are expressed as a function of certain averages of the turbulent velocities. Title: On the rotation rate of polar features in the sun Authors: Durney, B. R.; Lytle, D. M.; Keil, S. L. Bibcode: 1984ApJ...281..455D Altcode: The authors evaluate the rotation rate of solar features in the vicinity of the poles with the help of a correlation procedure. The average rotation rates for both poles are systematically smaller than those predicted by Howard and Harvey's formula, but not in serious disagreement with their results. Title: On the large-scale dynamics of rapidly rotating convection zones Authors: Durney, B. R. Bibcode: 1983ApJ...269..671D Altcode: The fact that the values of the eight basic waves present in turbulent flows in the presence of rotation prohibit a tilt of eddy towards the axis of rotation is incorporated into a formalism for rapidly rotating convection zones. Equations for turbulent velocities are defined in a rotating coordinate system, assuming that gravity and grad delta T act in a radial direction. An expression is derived for the lifetime of a basic wave and then for the average velocity vector. A real convective eddy is formulated and the wave vectors are calculated. The velocity amplitude and the stress tensor amplitude are integrated over the eddy domain. Applied to the solar convective zone, it is found that the convective cells are aligned along the axis of rotation at the poles and at the equator, a model that conflicts with nonrotating mixng length theory predictions. Title: On the first-order smoothing expression for the alpha-effect in dynamo theory Authors: Durney, B. R. Bibcode: 1983ApJ...267..822D Altcode: The term regenerating the poloidal magnetic field from the toroidal one (the alpha-effect) plays a central role in the generation of magnetic fields by a dynamo process. Using the so-called first order smoothing approximation, Steenbeck and Krause (1969) derived an expression for alpha, taking into account the case of homogeneous, isotropic motions. An equation, which is valid in the Boussinesq approximation, was found for alpha by making use of the velocity distribution employed by Durney and Spruit (1979) in the evaluation of the Reynolds stresses generating the solar differential rotation. It is important to derive an expression for alpha which is compatible with this equation, because little is known about the structure of the dominant convective eddy in the lower solar convection zone. The present investigation is concerned with the derivation of such an expression, taking into account a disappearance of alpha in the Boussinesq approximation if curvature effects are neglected. Title: Preliminary observations of velocity fields at the solar poles Authors: Cram, L. E.; Durney, B. R.; Guenther, D. B. Bibcode: 1983ApJ...267..442C Altcode: Using the 13 m Littrow spectrograph at Sacramento Peak Observatory, the Doppler shift of Fe I 5863 A in the polar regions of the sun over a 20 day interval is studied. The daily observations were assembled into a polar projection of the line-of-sight velocity field. The projection shows a very clear pattern of supergranulation. When a low-pass spatial filter is run over the data, a pattern of large-scale (80-100 Mm) velocity features can be seen. Cross-correlation studies show that the supergranular pattern rotates with a synodic period of 35 days, while there is evidence that the larger features rotate with a shorter period of about 30 days. At present, it is not possible to say whether the large-scale patterns represent a new scale of convection (possibly related to the dominant convective eddy in the lower solar convection zone) or to the low-wavenumber tail of a distribution of supergranular cells. Title: Observations of Polar Velocity Fields Authors: Durney, B. R.; Lytle, D. M.; Cram, L. E.; Guenther, D. B.; Keil, S. L. Bibcode: 1983BAAS...15..716D Altcode: No abstract at ADS Title: On the generation of magnetic fields in late-type stars - A local time-dependent dynamo model Authors: Robinson, R. D.; Durney, B. R. Bibcode: 1982A&A...108..322R Altcode: We assume that the magnetic field of late-type stars is generated in the lower part of the star's convection zone and study this generation mechanism with the help of local (in latitude) dynamo equations. For the spectral types GO, GS, KO, KS, MO, M2, and M5 we evaluate the magnetic field and period of the cycles as a function of rotation and compare them with the available observational data. Title: On an estimate of the dynamo-generated magnetic fields in late-type stars. Authors: Durney, B. R.; Robinson, R. D. Bibcode: 1982ApJ...253..290D Altcode: The principal objective of the present investigation is related to a prediction of the variation of magnetic fields with stellar type and the role of pertinent variables (such as rotation and differential rotation) with respect to the field properties. This is accomplished by estimating a typical amplification time for the magnetic field, and a typical 'time of rise' for the magnetic field due to magnetic buoyancy. It is assumed that the magnetic field is generated principally in the lower part of the stellar convection zone. Local (in latitude) dynamo equations are considered. The selected approach consists basically in an estimate of the typical magnitude of the magnetic field as predicted by the local dynamo equations. The employed approach constitutes only a first step towards the evaluation of magnetic fields in stars other than the sun. Title: A preliminary interpretation of stellar chromospheric CA II emission variations within the framework of stellar dynamo theory. Authors: Durney, B. R.; Mihalas, D.; Robinson, R. D. Bibcode: 1981PASP...93..537D Altcode: Recent stellar chromospheric Ca II emission data are analyzed and interpreted within the framework of simple concepts of dynamo theory. From an examination of the rotation rates and B-V indexes of 26 stars as presented by Vaughn at el. (1981) and the background flux values derived by Wilson (1978) for 18 reference stars, an empirical relation is derived between dynamo number, calculated from the B-V index and rotation rate, and stellar chromospheric emission flux. The Ca-emission cycle morphology of the sample stars is then examined, and differences between the four morphological classes identified are explained in terms of the correlation of large dynamo numbers with the presence of several interfering magnetic modes of different spatial scales, which do not exhibit a marked cyclic behavior, and small numbers with the excitation of only a single mode. The gap noted by Vaughn and Preston (1980) in the relation between the log of the emission flux with (B-V) is then interpreted as representing a transition from a multiple-mode dynamo to a single-mode dynamo as the dynamo number decreases. Title: On an Estimate of the Dynamo-Generated Magnetic Fields in Late-Type Stars Authors: Durney, B. R.; Robinson, R. D. Bibcode: 1981BAAS...13..791D Altcode: No abstract at ADS Title: On Theories of Rotating Convection Zones Authors: Durney, B. Bibcode: 1981siwn.conf....1D Altcode: No abstract at ADS Title: On an Estimate of the Dynamo-Generated Magnetic Fields in Late-Type Stars Authors: Durney, B. R.; Robinson, R. D. Bibcode: 1981BAAS...13..906D Altcode: No abstract at ADS Title: Observations of Velocity Fields at the Solar Poles Authors: Guenther, D.; Cram, L.; Durney, B. Bibcode: 1981BAAS...13Q.906G Altcode: No abstract at ADS Title: On a model of a slowly rotating solar convection zone Authors: Durney, B. R. Bibcode: 1981ApJ...244..678D Altcode: Numerical solutions are evaluated of the equations governing the large-scale motions of rotating stellar convection zones, as derived by Durney (1976) and Spruit (1977) (DS). With reference to the solar convection zone, these equations were solved by a perturbation method with the uniformly rotating convection zone as the unperturbed state (approximated by a polytrope). The calculations suggest that (1) large pole-equator differences in flux in the lower part of the convection zone are entirely compatible with negligible pole-equator differences in flux at the surface; (2) in realistic models of the rotating solar convection zone the energy carried by radiation should be included; and (3) in the lower part of the convection zone the solar convection velocities could differ substantially from those evaluated in the absence of rotation. Title: Sacramento Peak Observatory, Sunspot, New Mexico 88349. Report. Authors: Zirker, J. B.; Durney, B. R. Bibcode: 1981BAAS...13..389Z Altcode: No abstract at ADS Title: On the Effect of Rotation on Solar Convection Authors: Durney, B. R. Bibcode: 1980BAAS...12..895D Altcode: No abstract at ADS Title: A Formalism for Differential Rotation Authors: Durney, B. R.; Spruit, H. C. Bibcode: 1980HiA.....5..121D Altcode: No abstract at ADS Title: On the Dynamics of the Solar Convection Zone Authors: Durney, B. R.; Spruit, H. C. Bibcode: 1980LNP...114...15D Altcode: 1980IAUCo..51...15D; 1980sttu.coll...15D No abstract at ADS Title: On the dynamics of stellar convection zones - The effect of rotation on the turbulent viscosity and conductivity Authors: Durney, B. R.; Spruit, H. C. Bibcode: 1979ApJ...234.1067D Altcode: We derive expressions for the turbulent viscosity and turbulent conductivity applicable to convection zones of rotating stars. We assume that the relative dimensions of the dominant convective cell are known and derive a simple distribution function for the turbulent convective velocities under the influence of rotation. From this distribution function (which includes, in particular, the stabilizing effect of rotation on convection) we calculate in the mixing-length approximation: (i) the turbulent Reynolds stress tensor and (ii) the expression for the heat flux in terms of the superadiabatic gradient. The contributions of the turbulent convective motions to the mean momentum and energy equation (which determine the large-scale motions in stellar convection zones) are treated consistently, and assumptions about the turbulent viscosity and heat transport are replaced by assumptions about the turbulent flow itself. The free parameters in our formalism are the relative cell dimensions and their dependence on depth and latitude. Title: A comparison of numerical simulations of eddy generation performed with a two- and a three-layer quasi-geostrophic model of oceanic mesoscale eddies Authors: Durney, Bernard Bibcode: 1978GApFD..10..275D Altcode: The generation of eddies by a large-scale flow over mesoscale topography is studied with the help of two- and three-layer nonlinear quasi-geostrophic models of the open ocean. The equations are integrated forward in time with no eddies present initially. For a given time, the displacement of the interface between layers two and three () tends to a well-defined limit (function of the horizontal spatial coordinates) as 3- 2 0 (r is the density of layer r). Even for values of α[= (ρ3 - ρ2)/(ρ2 - ρ1)] as small as 0.01 the potential energy due to ζ is not negligible and it can reach, in some cases, a considerable fraction of the total eddy energy. Title: On the angular momentum loss of late-type stars. Authors: Durney, B. R.; Latour, J. Bibcode: 1978GApFD...9..241D Altcode: The observed surface angular velocity of main-sequence stars shows a sharp decrease at about spectral type F6. It is suggested that stars more massive than F6 cannot experience an appreciable angular-momentum loss because their convection zones cannot sustain a magnetic dynamo: without a magnetic field the angular-momentum loss is very small. The influence of rotation on the convective motions is essential for the existence of a solar-type dynamo. Rotation can influence these convective motions only if the typical convective time is larger than the rotation time. For main-sequence stars of different masses and chemical compositions the dimensionless parameter (convective velocity/sum's angular velocity times mixing length in the lower part of the convection zone) is evaluated. It is shown that this parameter increases very sharply for stars whose mass exceeds that defined by the relation log(star mass/solar mass) is of the order of 0.1. Thus even for large angular velocities, magnetic dynamos are not feasible if log(star mass/solar mass) appreciably exceeds 0.1. Title: On the angular momentum loss of late-type stars Authors: Durney, B. R.; Latour, J. Bibcode: 1977GApFD...9..241D Altcode: The observed surface angular velocity of main-sequence stars shows a sharp decrease at about spectral type F6. We suggest that stars more massive than F6 cannot experience an appreciable angular momentum loss because their convection zones cannot sustain a magnetic dynamo: without a magnetic field the angular momentum loss is very small. The influence of rotation on the convective motions is essential for the existence of a solar type dynamo. Rotation can influence these convective motions only if the typical convective time is larger than the rotation time, i.e., if l/uc > 1/, where uc and l are typical values of the convective velocity and mixing length in the lower part of the convection zone and is the star's angular velocity. For main-sequence stars of different masses and chemical compositions we evaluate the dimensionless parameter uc/Ω⊙ l and show that it increases very sharply for stars whose mass, M, exceeds that defined by log(M /M⊙ ) eDot 0.1 (Ω⊙, and M⊙, are the sun's angular velocity and mass, respectively). Thus even for large angular velocities, magnetic dynamos are not feasible if log(M/M⊙) appreciably exceeds 0.1. Title: The influence of mesoscale topography on the stability and growth rates of a two-layer model of the open ocean Authors: Durney, Bernard R. Bibcode: 1977GApFD...9..115D Altcode: The influence of mesoscale topography on the baroclinic instability of a two-layer model of the open ocean is considered. For westward velocities in the top layer (U), and for a sinusoidal topography independent of x or longitude (a cross-stream topography), the critical value of U (Uc) leading to instability is the same as when there is no topography. The wavelength of the unstable perturbation corresponding to Uc is shortened. For a given wavevector (k) of the perturbation the system becomes stable (as also in the absence of topography) for large values of |U|. The minimum value of the shear leading to stability is, however, significantly reduced by the topography. For sufficiently large values of the height of the topographic features, instabilities appear which are localized within a narrow range of the shear. These instabilities are studied for a topography that depends both on x and y.

For a cross-stream topography the growth rates are somewhat smaller than those without topography and they depend only weakly on ky. For the topographies considered here which depend both on x and y, perturbations with different values of ky can again have roughly the same growth rate.

In the case of stable oscillations, variations in the eddy energy with very long periods are made possible by the coexistence of topographic modes with closely lying periods. Title: On Theories of Solar Rotation Authors: Durney, B. R. Bibcode: 1976IAUS...71..243D Altcode: No abstract at ADS Title: Summary of the Final Discussion on August 29 Authors: Durney, B. R.; Gilman, P. A.; Stix, M. Bibcode: 1976IAUS...71..479D Altcode: No abstract at ADS Title: A comparison of the meridional flows in the sun's convection zone, predicted by theories of the solar dynamo and differential rotation. Authors: Durney, B. R. Bibcode: 1975ApJ...199..761D Altcode: Recently Yoshimura has evaluated the gradient of the Sun's angular velocity (d /Jr) necessary to give a good fit to the observed solar activity cycle. We estimate the meridional velocities in the convection zone, implied by this value of . These meridional velocities are in good agreement with those necessary to explain the Sun's surface differential rotation. Subject headings: hydromagnetics - interiors, solar - rotation, solar Title: On Coronal Streamers with T-Type Neutral Points Authors: Durney, B. R. Bibcode: 1975SoPh...41..233D Altcode: The gas-magnetic field interaction of an isothermal axisymmetric corona is considered. A method is suggested for solving the MHD equations in the case when a uniform gas pressure and the radial component of the magnetic field (as in a dipole) are specified at the Sun's surface. The flux of open field lines (φ) can be given arbitrarily, and no reconnection or opening of field lines can take place. If configurations in hydrostatic equilibrium between the regions of open and closed field lines can be found, then the method of solution converges. The equation of hydrostatic equilibrium at the neutral point (assumed to be of the T-type) is written in a simple form, and it is shown that if φ is smaller than a certain φmin, this equation cannot be satisfied. Configurations in hydrostatic equilibrium between the regions of open and closed field lines are expected to exist for any value of φ larger than φmin. Title: A Comparison of the Meridional Flows in the Sun's Convection Zone Predicted by Theories of the Solar Dynamo and Differential Rotation Authors: Durney, B. R. Bibcode: 1975BAAS....7Q.364D Altcode: No abstract at ADS Title: Solar-Interplanetary Modeling: 3-D Solar Wind Solutions in Prescribed Non-Radial Magnetic Field Geometries Authors: Durney, B. R.; Pneuman, G. W. Bibcode: 1975SoPh...40..461D Altcode: A model is presented which describes the 3-dimensional non-radial solar wind expansion between the Sun and the Earth in a specified magnetic field configuration subject to synoptically observed plasma properties at the coronal base. In this paper, the field is taken to be potential in the inner corona based upon the Mt. Wilson magnetograph observations and radial beyond a certain chosen surface. For plasma boundary conditions at the Sun, we use deconvoluted density profiles obtained from synopticK-coronameter brightness observations. The temperature is taken to be 2 × 106 K at the base of closed field lines and 1.6 x 106K at the base of open field lines. Title: On the Sun's Differential Rotation. Implications of the Difference in Angular Velocity between the Sunspots and Photosphere Authors: Durney, B. R. Bibcode: 1974SoPh...38..301D Altcode: It is assumed that the meridional motions (U) and angular velocity (Ω) in the surface layers of the convection zone are given by simple expressions of the form: Ur= 2ψ(r) P2(cosθ)/ϱr2, U0 = −ψ'(r) sinθ cosθ/ϱr, and Ω = Ω0[(1 + ω0(r) + ω2(r) P2(cosθ)]. Here ψ(r) is the stream function, P2(cosθ) the second order Legendre polynomial, and θ the polar angle. Allowance is made for a possible difference in the rate of momentum exchange between the directions parallel and perpendicular to gravity by introducing an anisotropic turbulent viscosity coefficient, μ, which is assumed furthermore to be proportional to the density, ϱ;μ = ϱν, and νθθ= νφφ= sνrr. It is shown that if the sunspots give an indication of the Sun's angular velocity at a depth h(∽ 3 × 104 km) then the turbulent viscosity is necessarily anisotropic. The radial variation introduced by this anisotropy seems to explain well the sunspot data if we assume that the sunspots act as tracers of the Sun's angular velocity. Title: On the Sun's Differential Rotation: its Maintenance by Large-Scale Meridional Motions in the Convection Zone Authors: Durney, Bernard R. Bibcode: 1974ApJ...190..211D Altcode: It is shown that if the observed differential rotation of the Sun is generated in the lower, Boussinesq part of the convection zone (where the interaction of rotation with convection is important), a large pole-equator difference in flux would also have to be present. The Sun's angular velocity is evaluated as a function of depth and latitude under the assumption that the main effect of the interaction of rotation with convection is the generation of a small pole-equator difference in temperature in the lower part of the convection zone, which drives a meridional motion over the entire convection zone. The pole-equator difference in flux associated with this meridional circulation (which gives rise to the Sun's differential rotation) is negligible. Subject headings: interiors, solar - rotation, solar Title: The expansion of a low-density solar corona: A one-fluid model with magnetically modified thermal conductivity Authors: Durney, B. R.; Hundhausen, A. J. Bibcode: 1974JGR....79.3711D Altcode: A one-fluid model of the coronal expansion, including the reduction in radial heat conduction produced by a spiral interplanetary magnetic field, is extended to the low coronal densities that may occur in the regions of open diverging magnetic field lines, or ‘coronal holes,’ that are regarded as probable sources of the solar wind. At such densities, the ‘cutoff’ in heat conduction at very large heliocentric distances (where the magnetic field becomes nearly azimuthal) has a profound effect on the nature of the expansion. The corona becomes nearly isothermal out to the distance where the flow of plasma dominates the transport of energy. This outward extension of high coronal temperatures leads to large solar wind speeds, approaching those given by Parker's original isothermal model as the coronal density becomes vanishingly small. The model predicts expansion speeds as high as 500 km s-1, with densities in agreement with those observed near the orbit of earth, for a reasonable set of coronal densities and temperatures (e.g., with coronal temperatures no higher than 2.1×106 °K). However, the temperatures (or pressures) predicted at the orbit of earth are substantially higher than those observed; this deficiency of the model could only be removed by incorporation of additional physical effects or processes. Title: On the Energetics and Momentum Balance of Pole-Equator Temperature Differences in the Sun Authors: Durney, Bernard Bibcode: 1973ApJ...183..665D Altcode: It is suggested that, as a consequence of the action of magnetic fields, the acoustic energy heating the chromosphere could be deposited at different heights at the equator than at the poles. The resultant pole-equator difference in pressure can be balanced by tilted sinusoidal motions [in the (r, 0)-plane] having some resemblance to horizontal Rossby waves. Subject heading: atmospheres, solar Title: Solar-Wind Properties at the Earth as Predicted by the Two-Fluid Model Authors: Durney, B. R. Bibcode: 1973SoPh...30..223D Altcode: The two-fluid equations for the solar wind are written down in a simplified form, similar to that suggested by Roberts (1971) for the one-fluid model. The equations are shown to depend only on one parameter, K = GMκemp (ɛ&inftyT0)3/2/4k2 Fe, where G is the gravitational constant, M the mass of the star, κe the thermal electron conductivity, mp the proton mass, k the Boltzman constant, kɛ∞T0 the residual energy per particle at infinity and Fe the electron-particle flux. For a variety of values of the density and temperature at the base of the corona we compute the solutions of the two-fluid solar wind model and compare the predicted and observed solar wind parameters at the Earth. Title: On the Energetics and Momentum Balance of Pole-equator Temperature Differences in the Sun Authors: Durney, B. R. Bibcode: 1973BAAS....5V.271D Altcode: No abstract at ADS Title: Solar wind properties at the Earth as predicted by the one-fluid model with helioclassical thermal electron conductivity Authors: Durney, B. R. Bibcode: 1973JGR....78.7229D Altcode: The solar wind properties at the earth are computed for a variety of values of the density and temperature at the base of the corona. For the electron thermal conductivity the expression derived by Perkins is adopted. Good agreement with observations is obtained for values of the density and temperature at the base of the corona equal to ∼9 × 107 cm-3 and ∼1.7 × 106 °K, respectively. Title: On the Sun's Differential Rotation and Pole-Equator Temperature Difference Authors: Durney, B. Bibcode: 1972SoPh...26....3D Altcode: The Sun's differential rotation can be understood in terms of a preferential stabilization of convection (by rotation) in the polar regions of the lower part of the convection zone (where the Taylor number is large). A significant pole-equator difference in flux (Δℱ) can develop deep inside the convection zone which would be unobservable at the surface, because ℱ can be very efficiently reduced by large scale meridional motions rising at the poles and sinking at the equator. This is the sense of circulation needed to produce the observed equatorial acceleration of the Sun. Differential rotation is generated, therefore, in the upper part of the convection zone (where the interaction of rotation with convection is small) and results as the convection zone adjusts to a state of negligible Taylor number. Title: Polytropic Subsonic Stellar Winds with Magnetic Fields Authors: Durney, B. Bibcode: 1972Ap&SS..17..489D Altcode: In any complex magnetic field configuration it is to be expected that there will be not only regions of no flow (closed magnetic field lines) and of supersonic flow, but also regions of subsonic flow. Subsonic stellar winds could also be of importance in stars of different type than the Sun. In the present paper the equations for the stellar wind are examined in the case of a polytropic relation between pressure and density and for small values of the parameter ɛ=Ω2 r a 2/u a 2. The radial distance (r) and the velocity at the Alfvénic point (4πϱu 2/B 2=1) are denoted byr a andu a , and Ω is the angular velocity. It is shown that: solar breeze solutions (that is, solutions that are subsonic for values ofr such thatrnot ≫ r_a) exist only if the dimensionless energy flux is larger thantfrac{3}{2}\varepsilon ^{{raise0.5exhboxriptstyle 2kern-0.1em/kern-0.15emlower0.25exhboxriptstyle 3}}. If ɛ→0 the flow with magnetic field tends to the flow without magnetic field forr<R(R→∞ as ɛ→0). Forr→∞ the velocity tends always to a finite value; this does not introduce, however, a singularity in the equations. Title: The Effect of Radiative Equilibrium on the Photospheric Angular Velocity. Authors: Durney, B. R. Bibcode: 1972ApJ...172..479D Altcode: The photospheric angular velocity [co = w(r)] is evaluated in the range of optical depths 0.05 < r 0.8under the assumption of radiative equilibrium and vanishing von Zeipel currents. The angular velocity decreases outward, and significant pole-equator temperature differences develop The von Zeipel currents are estimated for the case of uniform rotation. Title: On Stellar Activity Cycles Authors: Durney, B. R.; Stenflo, J. O. Bibcode: 1972Ap&SS..15..307D Altcode: The relation between the average magnetic fieldB, the angular velocity Ω, and the periodP of stellar activity cycles is studied. For the calculations we have used Leighton's (1969) model for the solar cycle with the additional assumption that the differential rotation and the cyclonic turbulence (Parker, 1955) (that is the ‘sunspot tilt’ or the ‘α-effect’) are both proportional to Ω. We then find thatB is roughly proportional to Ω and thatP decreases with increasing Ω. The period of the solar cycle increases therefore with the age of the Sun. Title: On the Domains of Existence of the Three Types of Supersonic Solutions of the Inviscid Solar-Wind Equations Authors: Durney, B. R.; Werner, N. Bibcode: 1972ApJ...171..609D Altcode: The approximate energy equation for radial distances larger than the critical point is solved with "critical point" boundary conditions. It is shown that if E = (41/24) (12/5)5/3 (35/2A)2/3 = e "' (where E is the residual energy per particle at infinity, A = 5.8 X 106/C, and C is the mass flow) then the equation has the solution T . If 6 < , the asymptotic behavior of the temperature is whereas if e > 6 "' then T for large values of r. This clarifies the domains of existence of the Parker, Whang and Chang, and supersonic solutions of the solar-wind equations. Title: Transition From a Supersonic to a Subsonic Solar Wind Authors: Durney, B. Bibcode: 1972NASSP.308..232D Altcode: 1972sowi.conf..232D No abstract at ADS Title: Solar-wind properties at the Earth as Predicted by One-Fluid Models Authors: Durney, B. R. Bibcode: 1972JGR....77.4042D Altcode: The spiraling magnetic field of the sun reduces the electron conductivity κ by the factor cos²θ, where θ is the spiral field angle. For a variety of values of the density and temperature at the base of the corona, we compute one-fluid solar-wind models for thermal conductivities equal to κ and κ cos²θ. For both cases, the values of the computed solar-wind parameters at the earth are compared with observed properties. Title: Evidence for Changes in the Angular Velocity of the Surface Regions of the Sun and Stars - Comments Authors: Durney, B. Bibcode: 1972NASSP.308..282D Altcode: 1972sowi.conf..282D No abstract at ADS Title: On the solar oblateness: The combined effect of a pole-equator difference in effective temperature and mechanical heating Authors: Durney, B. R.; Werner, N. E. Bibcode: 1971SoPh...21...21D Altcode: With the help of a model atmosphere of the Sun we evaluate the pole-equator difference in flux (as measured by Dicke and Goldenberg) assuming the following type of pole-equator temperature difference (ΔT=Te−Tp): (a) ΔT ≈ 2K for τ > τ0 (τ0 ≈ 0.05); (b) ΔT ≈ 10K for τ < τ0. Title: On the Theory of Stellar Winds Authors: Durney, B. R.; Roberts, P. H. Bibcode: 1971ApJ...170..319D Altcode: It has recently been shown by Roberts that solutions of the stellar wind equations depend essentially on a single parameter IC = 12A where E is the (dimensionless) residual energy per particle at infinity and A is a nondimensional constant proportional to the reciprocal of the mass flux C. This transformation makes it a comparatively simple matter to examine solutions for a wide variety of A and t . The calculations reported below are for the range 75 <K < 2000, which covers many cases of astrophysical interest. Title: A New Type of Supersonic Solution for the Inviscid Equations of the Solar Wind Authors: Durney, B. Bibcode: 1971ApJ...166..669D Altcode: The transition from a supersonic to a subsonic corona was investigated by increasing the density N0 (initially N0 = 9.3 X 10 cm-3) at the base of the corona while keeping the temperature T0 there constant (T0 = 2.1 X 108 K). As the density was increased, the energy flux at infinity due to thermal conductivity, t , steadily decreased and vanished for N0 1.17 X 108 . However, the total energy flux at infinity 8 remained different from zero. From somewhat higher values of the density up to N0 3 X 108 a new type of supersonic solution was found; the temperature behaving as (1/r)413 for large distances. It is expected that this type of supersonic solution will exist up to a value of the density for which subsonic solutions are possible (N0 > 10 cm-3). Thus for the chosen value of T0, there is not a direct transition from the Parker-type supersonic solution to the subsonic solution (as N0 increases from its initial value); instead, there is first a transition to a supersonic flow characterized by a (l/r)413 asymptotic behavior for the temperature. Title: Differential Rotation, Meridional Velocities, and Pole-Equator Difference in Temperature of a Rotating Convective Spherical Shell Authors: Durney, B. Bibcode: 1971ApJ...163..353D Altcode: A rotating, convective, spherical layer of fluid is considered in the Boussinesq approximation. The coupled equations for the axisymmetric modes of the velocity and temperature fields are solved in the steady state with the low-order "Legendre components" of the fluctuating self-interactions (evaluated in the quasilinear approximation) as the driving terms It is found that (a) the angular velocity increases inward; (b) there is pole-equator differential rotation with equatorial acceleration; (c) beneath the surface the equator is hotter than the poles (at the surface the temperature is given as a boundary condition); (d) two types of meridional circulation are compatible with the observed differential rotation of the Sun. For example, in the northern hemisphere, in one case the flow comprises two cells in the radial direction, and in the other case each radial cell is divided latitudinally into two subcells Title: Inhomogeneous convection and the equatorial acceleration of the sun. Authors: Durney, B. R.; Roxburgh, I. W. Bibcode: 1971BAAS....3S.260D Altcode: No abstract at ADS Title: Inhomogeneous Convection and the Equatorial Acceleration of the Sun Authors: Durney, B. R.; Roxburgh, I. W. Bibcode: 1971SoPh...16....3D Altcode: The interaction of rotation and turbulent convection is assumed to give rise to an inhomogeneous, but isotropic, latitude dependent turbulent energy transport, which is described by a `convective conduction coefficient κc' which varies with latitude. Energy balance in the convective zone is then possible only with a slow meridian circulation in the outer convective zone of the sun. The angular momentum transported by this circulation is balanced in a steady state by turbulent viscous transport down an angular velocity gradient. A detailed model is constructed allowing for the transition from convective transport to radiative transport at the boundaries of the convective zone, by using a perturbation analysis in which the latitude variation of κc is small. The solution for a thin compressible shell gives equatorial acceleration and a hotter equator than pole, assuming that the convection is preferentially stabilised at the equator. For agreement with the sun's equatorial acceleration the model predicts an equatorial temperature excess of 70 K and a surface meridional velocity of 350 cm/sec from pole to equator. Title: The Interaction of Convection with Rotation Authors: Durney, B. R. Bibcode: 1970BAAS....2S.310D Altcode: No abstract at ADS Title: Nonaxisymmetric Convection in a Rotating Spherical Shell Authors: Durney, B. Bibcode: 1970ApJ...161.1115D Altcode: The problem of a rotating, convective spherical shell is considered in the Herring approximation. The relevant equations are integrated in time as an initial-value problem. The main results are: 1. The most unstable modes correspond to convective cells stretching from pole to pole. 2. The calculations of the Reynolds stresses show transport of angular momentum toward the equator. That is, differential rotation sets in with equatorial acceleration. 3. The convective transport of heat is maximum at the equator. This would give rise to an equatorpole difference in flux. 4. If convection is nonaxisymmetric (as in the most unstable modes), then there are no time-independent solutions. The time-dependence is oscillatory and of the form A cos (cot + m ) + B sin (cot + m . Title: Models of close and contact binary stars 1.Polytropic models Authors: Durney, B. R.; Roxburgh, I. W. Bibcode: 1970MNRAS.148..239D Altcode: Polytropic models of close and contact binary stars are constructed using a combination of perturbation techniques and a Laplace approximation previously applied to uniformly rotating stars. Synchronism between orbital and intrinsic angular velocity is assumed. Models are constructed including the effects of distortion for polytropes with indices fl = 1, , 2, 3 and 4. The conditions for the two stars to be just in contact are determined and contact models with a mass ratio of unity are constructed, right up to the limiting case when the stars fill all the available space inside the critical Roche surface surrounding the two stars. When the two stars are built on the same polytropic model contact stars with mass ratios different from unity are not possible. Title: The Interaction of Rotation with Convection Authors: Durney, B. R. Bibcode: 1970stro.coll...30D Altcode: 1970IAUCo...4...30D No abstract at ADS Title: Decrease in the Period of Pulsar PSR 0833-45 Authors: Durney, B. Bibcode: 1969Natur.222.1260D Altcode: A DECREASE of 196 ns in the period of pulsar PSR 0833-45 has recently been observed1,2; I suggest that this decrease results from the addition of mass to the pulsar. A remnant of the supernova explosion forming the pulsar may not have escaped the gravitational field and may now have fallen back on the pulsar. Title: Model Atmosphere Calculation of the Solar Oblateness Authors: Durney, B. R. Bibcode: 1969Natur.221..646D Altcode: Dicke and Goldenberg1 measured the difference between the polar and the equatorial flux coming from the limb of the Sun (ΔF), and inferred that the surface of equal potential at the limb is oblate by 35 km. The Sun thus has a quadrupole moment due to a rapidly rotating interior, producing a perihelion shift of Mercury of 3.4 s of arc century-1. Agreement between the value predicted by general relativity and the observed perihelion shift of Mercury is thus destroyed. One of us2 has criticized the Dicke and Goldenberg interpretation and has suggested that the flux difference is due to a stronger stabilization of convection, by rotation, at the pole. Title: Pulsation Periods of Rotating White Dwarfs Authors: Durney, B. R.; Faulkner, J.; Gribbin, J. R.; Roxburgh, I. W. Bibcode: 1968Natur.219...20D Altcode: When uniform rotation is included, the periods of pulsation for white dwarfs can become as small as 0.9 s. With non-uniform rotation, periods as short as 0.1 s may be possible. Title: Non-Radial Oscillations of Slowly Rotating Polytropes Authors: Durney, B.; Skumanich, A. Bibcode: 1968ApJ...152..255D Altcode: The linearized equations for non-radial adiabatic oscillations of slowly rotating polytropes are studied for both stable and marginally stable states. For oscillations in the stable state, the eigenfrequencies are a continuous function of the parameter = (~ - F)/(m~)2, which measures the ratio of buoyant to gyroscopic forces; here T = 1 + 1/n, where n is the polytropic index, ~y is the ratio of the specific heats, w the non-dimensional angular velocity, and m determines the azimuthal dependence. The structures of these oscillations, which could be called gravitational gyroscopic waves, are determined from two coupled first-order partial differential equations. In the marginally stable state one finds the solutions to be oscillatory, thus indicating overstability; the parameter yi takes on discrete negative values which indicate the stabilizing influence of rotation Title: Rotating Massive Stars in General Relativity Authors: Durney, B. R.; Roxburgh, I. W. Bibcode: 1967RSPSA.296..189D Altcode: Equilibrium models of uniformly rotating massive stars are investigated, using a weak field, slow rotation approximation, which is shown to be adequate for all cases of interest. The fate of radial perturbations about these equilibrium configurations is investigated using a linearized stability analysis to determine the oscillation frequency σ in a peturbation propto e1σ t. An eigenvalue equation for σ^2 is obtained which can be made self adjoint with respect to the spatial metric, and a variational principle to determine σ^2 is derived. Numerical determinations of σ^2 have been carried out for a variety of masses, radii and rotational velocities, and these results are incorporated in a simple formula that gives the dependence of σ^2 on these quantities. The condition for instability, σ^2 negative, is determined, and it is found that for large masses and maximum rotation velocity, so that when centrifugal force balances gravity at the surface, a massive star becomes unstable when its radius is 208 times the Schwarzschild radius 2GM/c^2. Title: The effect of a toroidal magnetic field on the radial oscillations of stars Authors: Roxburgh, I. W.; Durney, B. R. Bibcode: 1967MNRAS.135..329R Altcode: The internal structure of a polytrope n =3 containing a toroidal magnetic field is investigated. For static equilibrium configurations the general solution for the structure of the field is given and a particular solution Ht rp sin 0 is investigated in detail. The linearized equations for small radial motion about the equilibrium configuration are presented and with a time dependence ei these equations reduce to an eigenvalue equation for 2 A variational principle for determiing is derived and 2 is estimated using this principle as well as by direct numerical iteration, for values of the ratio of specific heats of the gas F = 4/3,413+ , and 5/3. Results are given for different field strengths. For F =4/3 the star is neutrally stable whether or not there is a magnetic field, whereas for the other values of F the magnetic field decreases the value of a as compared to the non-magnetic values. Title: Structure, Oscillations and Stability of Rotating White Dwarfs Authors: Roxburgh, I. W.; Durney, B. R. Bibcode: 1966ZA.....64..504R Altcode: No abstract at ADS Title: Stability of Rotating Massive Stars in General Relativity Authors: Durney, B.; Roxburgh, I. W. Bibcode: 1965Natur.208.1304D Altcode: THE suggestion by Hoyle and Fowler1 that stars with masses of 106-1010 Msolar may provide the energy for radio sources, and the subsequent discovery of quasars, has stimulated considerable interest in the structure of very massive stars2. Iben3, using a binding-energy argument, showed that within the framework of general relativity a spherical massive star becomes unstable long before it has contracted to the stage at which nuclear reactions become important. A similar conclusion was obtained by Chandrasekhar4, using a detailed stability analysis on the spherically symmetric relativistic equations and calculating the relaxation oscillations from a variational principle. Similar results have been obtained by Fowler5 using a virial theorem approach. Title: Stresses Induced in a Purely Elastic Earth Model under Various Tectonic Loads Authors: Durney, B. Bibcode: 1965GeoJ...10..163D Altcode: 1965GeoJI..10..163D No abstract at ADS Title: Distorted Wave Approximation in the Reaction P + PT Authors: Durney, B. R. Bibcode: 1958RSPSA..71..654D Altcode: 1958RSLPS..71..654D No abstract at ADS