Differential rotation set up by latitude-dependent heat transport

Abstract

Models of differentially rotating compressible deep spherical shells are computed according to the method of Belvedere and Paternò (1977): the heat transport is entirely convective, small-scale motions are parametrized by a thermal diffusivity and a kinematic viscosity, and the limit of slow rotation and large viscosity is considered.

In order to adapt the resulting differential rotation to the observed equatorial acceleration of the Sun, the heat transport must be more effective in the vicinity of the equator. In all models the latitude dependence of the transport coefficient induces meridional circulation in the form of a large cell, with rising material at high latitudes and sinking material near the equator. On top of this cell, one or two thin countercells develop in a minority of cases. Large pole-equator temperature differences and meridonal velocities at the surface are obtained when the Prandtl number is 1. But values of, say, 1/10 are sufficiently small to allow the models to be applied to the Sun. In general an angular velocity increasing with depth is found, and the surfaces of constant angular velocity are inclined towards greater depth and higher latitude.     

Nonlinear dynamics of boussinesq convection in a deep rotating spherical shell-i

Abstract

Convection in a rotating spherical shell has wide application for understanding the dynamics of the atmospheres and interiors of many celestial bodies. In this paper we review linear results for convection in a shell of finite depth at substantial but not asymptotically large Taylor numbers, present nonlinear multimode calculations for similar conditions, and discuss the model and results in the context of the problem of solar convection and differential rotation. Detailed nonlinear calculations are presented for Taylor number T = 10⁵, Prandtl number P = 1, and Rayleigh number R between 1 |MX 10⁴ and 4 |MX 10⁴ (which is between about 4 and 16 times critical) for a shell of depth 20% of the outer radius. Sixteen longitudinal wave numbers are usually included (all even wave numbers m between 0 and 30) the amplitudes of which are computed on a staggered grid in the meridian plane.

The kinetic energy spectrum shows a peak in the wave number range m = 12–18 at R = 10⁴, which straddles the critical wave number m = 14 predicted by linear theory. These are modes which peak near the equator. The spectrum shows a second strong peak at m = 0, which represents the differential rotation driven by the peak convective modes. As R is increased, the amplitude of low wave numbers increases relative to high wave numbers as convection fills in in high and middle latitudes, and as the longitudinal scale of equatorial convection grows. By R = 3 |MX 10⁴, m = 8 is the peak convective mode. There is a clear minimum in the total kinetic energy at middle latitudes relative to low and high, well into the nonlinear regime, representing the continued dominance of equatorial and polar modes found in the linear case. The kinetic energy spectrum for m > 0 is maintained primarily by buoyancy work in each mode, but with substantial nonlinear transfer of kinetic energy from the peak modes to both lower and higher wave numbers.

For R = 1 to 2 |MX 10⁴, the differential rotation takes the form of an equatorial acceleration, with angular velocity generally decreasing with latitude away from the equator (as on the sun) and decreasing inwards. By R = 4 |MX 10⁴, this equatorial profile has completely reversed, with angular velocity increasing with depth and latitude. Also, a polar vortex which has positive rotation relative to the reference frame (no evidence of which has been seen on the sun) builds up as soon as polar modes become important. Meridional circulation is quite weak relative to differential rotation at R = 10⁴, but grows relative to it as R is increased. This circulation takes the farm of a single cell of large latitudinal extent in equatorial regions, with upward flow near the equator, together with a series of narrower cells in high latitudes. It is maintained primarily by axisymmetric buoyancy forces. The differential rotation is maintained at all R primarily by Reynolds stresses, rather than meridional circulation. Angular momentum transport toward the equator for R = 1–2 |MX 10⁴ maintains the equatorial acceleration while radially inward transport maintains the opposite profile at R = 4 |MX 10⁴.

The total heat flux out the top of the convective shell always shows two peaks for the range of R studied, one at the equator and the other near the poles (no significant variation with latitude is seen on the sun), while heat flux in at the bottom shows only a polar peak at large R. The meridional circulation and convective cells transport heat toward the equator to maintain this difference.

The helicity of the convection plus the differential rotation produced by it suggest the system may be capable of driving a field reversing dynamo, but the toroidal field may migrate with lime in each cycle toward the poles and equator, rather than just toward the equator as apparently occurs on the sun.

We finally outline additions to the physics of the model to make it more realistic for solar application.     

Convection in rotating annular channels heated from below. Part 1. Linear stability and weakly nonlinear mean flows

Convection in a Boussinesq fluid in an annular channel rotating about a vertical axis with lateral rigid sidewalls, stress-free top and bottom, uniformly heated from below is investigated. The sidewalls are assumed to be either perfectly insulating or conducting. Three different types of convection are identified when the channel is rotating sufficiently fast: (i) global oscillatory convection preferred for small Prandtl numbers in channels with intermediate or large aspect ratios (width to height ratio), (ii) wall-localized oscillatory convection representing the most unstable mode for moderate or large Prandtl numbers in channels with intermediate or large aspect ratios and (iii) global stationary convection preferred in channels with sufficiently small aspect ratios regardless of the size of the Prandtl number. The corresponding weakly nonlinear problem describing differential rotation and meridional circulation is also examined, showing that geostrophic, multiple-peaked (two prograde and two retrograde) differential rotation can be maintained by the Reynolds stresses in wall-localized convective eddies in a rapidly rotating channel.     

Magnetic tracers,a probe of the solar convective layers

Abstract

A meridional circulation of sunspots has been measured through the digital analysis of the Meudon spectroheliograms from 1978 to 1983. Old and young sunspots follow a zonal meridional circulation, in several bands of latitude, in which two adjacent bands have opposite motions. This meridional circulation pattern is time-dependent. Using the H _α filaments as magnetic field tracers, a large-scale magnetic pattern has been found that was also obtained independently by direct measurement of the magnetic field (Hoeksema, 1988).

The coincidence of a large-scale magnetic pattern with a zonal meridional circulation suggests the existence of azimuthal rolls below the surface, and these azimuthal rolls can explain a number of properties of the solar cycle. New rolls occur with increasing proximity to the Equator, thereby indicating the direction of propagation of the dynamo wave. The occurrence of rolls is very favorable to the emergence of the magnetic regions. The rolls also influence the magnetic complexity of the active regions. They modulate the surface rotation through the Coriolis force, which accelerates or decelerates the fluid particles. They therefore offer a plausible explanation of the torsional oscillation pattern.

There are a number of problems raised by such an unexpected circulation pattern: for example, the coexistence of axisymmeric rolls with hypothetical giant cells, the location of the dynamo source below or within the convective zone, and the coupling of the radiative interior and the convective layers. To resolve these important issues, continuous observational studies are needed of the manifestation of solar activity, as well as of radius and luminosity variations. So, we have aimed our paper at an audience of theoreticians in the hope that they take up the challenges we describe.     

Meridional circulation from differential rotation in an adiabatically stratified solar/stellar convection zone

Meridional circulation in stellar convection zones is not generally well observed, but may be critical for the workings of MHD dynamos operating in these domains. Coriolis forces from differential rotation play a large role in determining what the meridional circulation is. Here, we consider the question of whether a stellar differential rotation that is constant on cylinders concentric with the rotation axis can drive a meridional circulation. Conventional wisdom says that it can not. Using two related forms of the governing equations that respectively estimate the longitudinal components of the curl of the meridional mass flux and the vorticity, we show that such differential rotation will drive a meridional flow. This is because to satisfy anelastic mass conservation, non-spherically symmetric pressure contours must be present for all differential rotations, not just ones that depart from constancy on cylinders concentric with the rotation axis. Therefore, the fluid is always baroclinic if differential rotation is present. This is because, in anelastic systems, the perturbation pressure must satisfy a Poisson type equation, as well as an equation of state and a thermodynamic equation. We support our qualitative reasoning with numerical examples, and show that meridional circulation is sensitive to the magnitude and form of departures from rotation constant on cylinders. The effect should be present in 3D global anelastic convection simulations, particularly those for which the differential rotation driven by global convection is nearly cylindrical in profile. For solar-like differential rotation, Coriolis forces generally drive a two-celled circulation in each hemisphere, with a second, reversed flow at high latitudes. For solar like turbulent viscosities, the meridional circulation produced by Coriolis forces is much larger than observed on the Sun. Therefore, there must be at least one additional force, probably a buoyancy force, which opposes the meridional flow to bring its amplitude down to observed values.     

Solar dynamo and geomagnetic activity

The correlation between geomagnetic activity and the sunspot number in the 11-year solar cycle exhibits long-term variations due to the varying time lag between the sunspot-related and non-sunspot related geomagnetic activity, and the varying relative amplitude of the respective geomagnetic activity peaks. As the sunspot-related and non-sunspot related geomagnetic activity peaks are caused by different solar agents, related to the solar toroidal and poloidal fields, respectively, we use their variations to derive the parameters of the solar dynamo transforming the poloidal field into toroidal field and back. We find that in the last 12 cycles the solar surface meridional circulation varied between 5 and 20 m/s (averaged over latitude and over the sunspot cycle), the deep circulation varied between 2.5 and 5.5 m/s, and the diffusivity in the whole of the convection zone was ～10⁸ m²/s. In the last 12 cycles solar dynamo has been operating in moderately diffusion dominated regime in the bulk of the convection zone. This means that a part of the poloidal field generated at the surface is advected by the meridional circulation all the way to the poles, down to the tachocline and equatorward to sunspot latitudes, while another part is diffused directly to the tachocline at midlatitudes, “short-circuiting” the meridional circulation. The sunspot maximum is the superposition of the two surges of toroidal field generated by these two parts of the poloidal field, which is the explanation of the double peaks and the Gnevyshev gap in sunspot maximum. Near the tachocline, dynamo has been operating in diffusion dominated regime in which diffusion is more important than advection, so with increasing speed of the deep circulation the time for diffusive decay of the poloidal field decreases, and more toroidal field is generated leading to a higher sunspot maximum. During the Maunder minimum the dynamo was operating in advection dominated regime near the tachocline, with the transition from diffusion dominated to advection dominated regime caused by a sharp drop in the surface meridional circulation which is in general the most important factor modulating the amplitude of the sunspot cycle.     

Turbulent transport of angular momentum and differential rotation

Abstract

With the help of simplifying approximations, we have derived expressions for the non-diffusive fluxes of the angular momentum which are brought about by the action of Coriolis forces on the convective motion. The original turbulence, which is not perturbed by the Coriolis forces, is considered given and weakly anisotropic, the anisotropy having a preferred radial direction. The eddy viscosities are evaluated. Hence, a closed equation for the angular velocity is derived, and then solved for the case of slow rotation. It is shown that the differential rotation is generated most effectively in the case of moderate rotation when the Rossby number is of order unity. At small Rossby numbers, the rotation differentiality is inhibited. A negative eddy viscosity is suggested for the case of rapid rotation. Some implications for the Sun and other astrophysical objects are discussed.     

Transitions to chaotic thermal convection in a rapidly rotating spherical fluid shell

Abstract

Numerical simulations of thermal convection in a rapidly rotating spherical fluid shell heated from below and within have been carried out with a nonlinear, three-dimensional, time-dependent pseudospectral code. The investigated phenomena include the sequence of transitions to chaos and the differential mean zonal rotation. At the fixed Taylor number T _a =10⁶ and Prandtl number Pr=1 and with increasing Rayleigh number R, convection undergoes a series of bifurcations from onset of steadily propagating motions SP at R=R _c = 13050, to a periodic state P, and thence to a quasi-periodic state QP and a non-periodic or chaotic state NP. Examples of SP, P, QP, and NP solutions are obtained at R = 1.3R _c , R = 1.7 R _c , R = 2R _c , and R = 5 R _c , respectively. In the SP state, convection rolls propagate at a constant longitudinal phase velocity that is slower than that obtained from the linear calculation at the onset of instability. The P state, characterized by a single frequency and its harmonics, has a two-layer cellular structure in radius. Convection rolls near the upper and lower surfaces of the spherical shell both propagate in a prograde sense with respect to the rotation of the reference frame. The outer convection rolls propagate faster than those near the inner shell. The physical mechanism responsible for the time-periodic oscillations is the differential shear of the convection cells due to the mean zonal flow. Meridional transport of zonal momentum by the convection cells in turn supports the mean zonal differential rotation. In the QP state, the longitudinal wave number m of the convection pattern oscillates among m = 3,4,5, and 6; the convection pattern near the outer shell has larger m than that near the inner shell. Radial motions are very weak in the polar regions. The convection pattern also shifts in m for the NP state at R = 5R _c , whose power spectrum is characterized by broadened peaks and broadband background noise. The convection pattern near the outer shell propagates prograde, while the pattern near the inner shell propagates retrograde with respect to the basic rotation. Convection cells exist in polar regions. There is a large variation in the vigor of individual convection cells. An example of a more vigorously convecting chaotic state is obtained at R = 50R _c . At this Rayleigh number some of the convection rolls have axes perpendicular to the axis of the basic rotation, indicating a partial relaxation of the rotational constraint. There are strong convective motions in the polar regions. The longitudinally averaged mean zonal flow has an equatorial superrotation and a high latitude subrotation for all cases except R = 50R _c , at this highest Rayleigh number, the mean zonal flow pattern is completely reversed, opposite to the solar differential rotation pattern.     

Rotational and magnetic instability in the diffusive tachocline

In this article we study the linear instability of the two-dimensional strongly stratified model for global MHD in the diffusive solar tachocline. Gilman and Fox [Gilman, P.A. and Fox, P., Joint instability of the latitudinal differential rotation and toroidal magnetic fields below the solar convection zone. Astrophys. J., 1997, 484, 439–454] showed that for ideal MHD, the observed surface differential rotation becomes more unstable than is predicted by Watson's [Watson, M., Shear instability of differential rotation in stars. Geophys. Astrophys. Fluid Dyn., 1981, 16, 285–298] nonmagnetic analysis. They showed that the solar differential rotation is unstable for essentially all reasonable values of the differential rotation in the presence of an antisymmetric toroidal field. They found that for the broad field case B _φ～sinθcosθ, θ being the co-latitude, instability occurs only for the azimuthal m?=?1 mode, and concluded that modes which are symmetric (meridional flow in the same direction) about the equator onset at lower field strengths than the antisymmetric modes. We study the effect of viscosity and magnetic diffusivity in the strongly stably stratified case where diffusion is primarily along the level surfaces. We show that antisymmetric modes are now strongly preferred over symmetric modes, and that diffusion can sometimes be destabilising. Even solid body rotation can be destabilised through the action of magnetic field. In addition, we show that when diffusion is present, instability can occur when the longitudinal wavenumber m?=?2.     

Heat transport by variable viscosity convection and implications for the Earth's thermal evolution

The case is presented that the efficiency of variable viscosity convection in the Earth's mantle to remove heat may depend only very weakly on the internal viscosity or temperature. An extensive numerical study of the heat transport by 2-D steady state convection with free boundaries and temperature dependent viscosity was carried out. The range of Rayleigh numbers (Ra) is 10⁴?10⁷ and the viscosity contrast goes up to 250000. Although an absolute or relative maximum of the Nusselt number (Nu) is obtained at long wavelength in a certain parameter range, at sufficiently high Rayleigh number optimal heat transport is achieved by an aspect ratio close to or below one. The results for convection in a square box are presented in several ways. With the viscosity ratio fixed and the Rayleigh number defined with the viscosity at the mean of top and bottom temperature the increase of Nu with Ra is characterized by a logarithmic gradient β = ?ln(Nu)/? ln(Ra) in the range of 0.23–0.36, similar to constant viscosity convection. More appropriate for a cooling planetary body is a parameterization where the Rayleigh number is defined with the viscosity at the actual average temperature and the surface viscosity is fixed rather than the viscosity ratio. Now the logarithmic gradient β falls below 0.10 when the viscosity ratio exceeds 250, and the velocity of the surface layer becomes almost independent of Ra. In an end-member model for the Earth's thermal evolution it is assumed that the Nusselt number becomes virtually constant at high Rayleigh number. In the context of whole mantle convection this would imply that the present thermal state is still affected by the initial temperature, that only 25–50% of the present-day heat loss is balanced by radiogenic heat production, and the plate velocities were about the same during most of the Earth's history.     

Patterns of thermal convection in slowly rotating,weakly magnetic fluid shells

The effects of rotation and a toroidal magnetic field on the preferred pattern of small amplitude convection in spherical fluid shells are considered. The convective motions are described in terms of associated Legendre functions P_l^|m| (cos θ). For a given pair of Prandtl number P and magnetic Prandtl number $\text{P}$_m the physically realized solution is represented either by m = 0 or |m| = l depending on the ratio of the rotation rate Λ to the magnetic field amplitude H. The case of m = 0 is preferred if this ratio ranges below a critical value, which is a function of the shell thickness, and |m| = l otherwise.     

Specific features of development of two-dimensional convection in a rotating compressible conducting fluid

An axisymmetric model of convection in a rotating cylinder in an external uniform magnetic field has been considered. In the considered model, the meridional circulation is created by a nonuniform rotation of the lower boundary relative to the other boundaries. In the considered model, the time of formation of the stationary regime in the magnetic field considerably increases if the vertical density (compressibility) inhomogeneity is taken into account for Ekman numbers of E = E _M = 3 × 10⁻³. This example shows that the compressibility of a medium should be taken into account in the convection and dynamics of the magnetic field when the magnetohydrodynamics of the Earth is analyzed.     

Large aspect ratio cells in two-dimensional thermal convection

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 × 10⁵ 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.     

On the onset of convection in rotating spherical shells

Abstract

The linear problem of the onset of convection in rotating spherical shells is analysed numerically in dependence on the Prandtl number. The radius ratio η=r _i/r _o of the inner and outer radii is generally assumed to be 0.4. But other values of η are also considered. The goal of the analysis has been the clarification of the transition between modes drifting in the retrograde azimuthal direction in the low Taylor number regime and modes traveling in the prograde direction at high Taylor numbers. It is shown that for a given value m of the azimuthal wavenumber a single mode describes the onset of convection of fluids of moderate or high Prandtl number. At low Prandtl numbers, however, three different modes for a given m may describe the onset of convection in dependence on the Taylor number. The characteristic properties of the modes are described and the singularities leading to the separation with decreasing Prandtl number are elucidated. Related results for the problem of finite amplitude convection are also reported.     

Meridional circulation in the sun and stars

Mean-field hydrodynamics advanced to clear explanations for the origin and properties of the global meridional flow in stellar convection zones. Qualitative arguments and analysis of basic equations both show that the meridional circulation is driven by non-conservative centrifugal and buoyancy forces and results from a slight disbalance between these two drivers. The deviations from the thermal wind balance are relatively large near the boundaries of convection zones. Accordingly, the meridional flow attains its largest velocities in the boundary layers and decreases inside the convection zone. This picture, however, is neither supported nor dismissed by the conflicting results of recent helioseismic soundings or 3D numerical experiments. The relevant physics of the differential temperature and its possible relation to the solar oblateness are briefly discussed.     

Nonlinear internal waves in the upper atmosphere

To investigate the mechanism of mixing in oscillatory doubly diffusive (ODD) convection, we truncate the horizontal modal expansion of the Boussinesq equations to obtain a simplified model of the process. In the astrophysically interesting case with low Prandtl number (traditionally called semiconvection), large-scale shears are generated as in ordinary thermal convection. The interplay between the shear and the oscillatory convection produces intermittent overturning of the fluid with significant mixing. By contrast, in the parameter regime appropriate to sea water, large-scale flows are not generated by the convection. However, if such flows are imposed externally, intermittent overturning with enhanced mixing is observed.     

三维地震波速结构约束下的变黏度地幔对流及其动力学意义

建立了三维黏度扰动下的变黏度地幔对流模型,并提供了在引入地幔的三维地震波速度结构下相应的求解方法. 依此反演了瑞利数Ra = 106时,两种不同边界条件下的极、环型场对流图像,这有助于深化对地幔物质流动和大地构造运动的深部动力学过程的认识和理解. 研究结果表明,不但地幔浅部的极型场对流图像显示出了与大地构造运动的相关性并揭示了其深部动力学过程,更重要的是,地幔浅部的环型场对流图像首次为我们认识和理解板块构造的水平与旋转运动提供了重要的信息：环型场速度剖面中在赤道附近存在一条大致南东东—北西西向的强对流条带,可能与环赤道附近大型剪切带的形成相关,进而表明可能是该带强震发生的深部动力学背景;在南北半球存在的旋转方向相反的对流环表明它们整体上可能存在差异旋转.     

Modeling of the solar activity double cycle using dynamical systems

The generation and evolution of the Sun’s magnetic field and other stars is usually related to the dynamo mechanism. This mechanism is based on the consideration of the joint influence of the α effect and differential rotation. Dynamo sources can be located at different depths of the convection zone and can have different intensities. Based on such a system, the dynamical system in the case of the stellar dynamo in a two-layer medium has been constructed with regard to meridional fluxes in order to model the double cycle that corresponds to the simultaneous presence of 22-year and quasi-biennial magnetic field oscillations. It has been indicated that the regime of mixed oscillations can originate because a dynamo wave moves oppositely to the meridional flows in the upper layer of the convection zone. This results in the deceleration of the toroidal field propagation and in the generation of slow oscillations. In deeper layers, the directions of a dynamo wave and meridional flows coincide with each other, as a result of which fast magnetic fields originate. Therefore, the total contribution of two oscillations with different frequencies corresponds to the appearance of quasi-biennial cycles against 22-year cycles. It has been indicated that the beating regime, which can be related to the secular oscillations of solar magnetic activity, originates in the system when the meridional flows are weak.     

Thermal convection in differentially rotating systems

Abstract

The onset of convection in a cylindrical fluid annulus is analyzed in the case when the cylindrical walls are rotating differentially, a temperature gradient in the radial direction is applied, and the centrifugal force dominates over gravity. The small gap approximation is used and no-slip conditions on the cylindrical walls are assumed. It is found that over a considerable range of the parameter space either convection rolls aligned with the axis of rotation or rolls in the perpendicular (azimuthal) direction are preferred. It is shown that by a suitable redefinition of parameters, results for finite amplitude Taylor vortices and for convection rolls in the presence of shear can be applied to the present problem. Weakly nonlinear results for transverse rolls in a Couette flow indicate the possibility of subcritical bifurcation for Prandtl numbers P less than 0.82. Heat and momentum transports are derived as functions of P and the problem of interaction between transverse and longitudinal rolls is considered. The relevance of the analysis for problems of convection in planetary and stellar atmospheres is briefly discussed.     

Nonlinear free thermal convection in a spherical shell:A variable viscosity model

Introduction The velocity field of surface plate motion can be split into a poloidal and a toroidal parts.At the Earth′s surface,the toroidal component is manifested by the existence of transform faults,and the poloidal component by the presence of convergence and divergence,i.e.spreading and subduc-tion zones.They have coupled each other and completely depicted the characteristics of plate tec-tonic motions.The mechanism of poloidal field has been studied fairly clearly which is related to …