The onset of magnetoconvection at large Prandtl number in a rotating layer II. Small magnetic diffusion

Abstract

This paper develops further a convection model that has been studied several times previously as a very crude idealization of planetary core dynamics. A plane layer of electrically-conducting fluid rotates about the vertical in the presence of a magnetic field. Such a field can be created spontaneously, as in the Childress-Soward dynamo, but here it is uniform, horizontal and externally-applied. The Prandtl number of the fluid is large, but the Ekman, Elsasser and Rayleigh numbers are of unit order. In Part I of this series, it was also supposed that the ratio thermal diffusivity diffusivity/magnetic diffusivity is O(1), but here we suppose that this ratio is large. The character of the solution is changed in this limit. In the case of main interest, when the layer is confined between electrically-insulating no-slip walls, the solution is significantly different from the solution when the mathematically simpler, illustrative boundary conditions also considered in Part I are employed. As in Part I, attention is focussed on the onset of convection as the temperature difference applied across the layer is increased, and on the preferred mode, i.e., the planform and time-dependence of small amplitude convection.     

Thermal and magnetically driven instabilities in a non-constantly stratified fluid layer

Abstract

The linear hydromagnetic stability of a non-constantly stratified horizontal fluid layer permeated by an azimuthal non-homogeneous magnetic field is investigated for various widths of the stably stratified part of the layer in the geophysical limit q→0 (q is the ratio of thermal and magnetic diffusivities). The choice of the strength of the magnetic field B_o is as in Soward (1979) (see also Soward and Skinner, 1988) and the equations for the disturbances are treated as in Fearn and Proctor (1983). It was found that convection is developed in the whole layer regardless of the width of its stably stratified part. The thermal instability penetrates essentially from the unstably stratified part of the layer into the stably stratified part for A ～ 1 (A characterises the ratio of the Lorentz and Coriolis forces). When the magnetic field is strong (A>1) the thermal convection is suppressed in the stably stratified part of the layer. However, in this case, it is replaced by the magnetically driven instability; which is fully developed in the whole layer. The thermal instabilities always propagate westward and exist for all the modes m. The magnetically driven instabilities propagate either westward or eastward according to the width of the stably and unstably stratified parts and exist only for the mode m=1.     

Thermal convection in a twisted horizontal magnetic field

The onset of convection in a layer of an electrically conducting fluid heated from below is considered in the case when the layer is permeated by a horizontal magnetic field of strength B ₀ the orientation of which varies sinusoidally with height. The critical value of the Rayleigh number for the onset of convection is derived as a function of the Chandrasekhar number Q. With increasing Q the height of the convection rolls decreases, while their horizontal wavelength slowly increases. Potential applications to the penumbral filaments of sunspots are briefly discussed.     

Penetrative convection in rotating fluids: A model for the base of stellar convection zones

Abstract

To model penetrative convection at the base of a stellar convection zone we consider two plane parallel, co-rotating Boussinesq layers coupled at their fluid interface. The system is such that the upper layer is unstable to convection while the lower is stable. Following the method of Kondo and Unno (1982, 1983) we calculate critical Rayleigh numbers R_c for a wide class of parameters. Here, R_c is typically much less than in the case of a single layer, although the scaling R_c～T^2/3 as T → ∞ still holds, where T is the usual Taylor number. With parameters relevant to the Sun the helicity profile is discontinuous at the interface, and dominated by a large peak in a thin boundary layer beneath the convecting region. In reality the distribution is continuous, but the sharp transition associated with a rapid decline in the effective viscosity in the overshoot region is approximated by a discontinuity here. This source of helicity and its relation to an alpha effect in a mean-field dynamo is especially relevant since it is a generally held view that the overshoot region is the location of magnetic field generation in the Sun.     

Nonlinear convection in an imposed horizontal magnetic field

Abstract

Nonlinear two-dimensional magnetoconvection, with a Boussinesq fluid driven across the field-lines, is taken as a model for giant-cell convection in the sun and late-type stars. A series of numerical experiments shows the sensitivity of the horizontal scale of convection to the applied field and to the Rayleigh number R. Overstable oscillations occur in cells as broad as they are deep, but increasing R leads to steady motions of much greater wavelength. Purely geometrical effects can cause oscillation: this work implies that strong horizontal field will in general lead to time-dependent convection.     

Topology of Rayleigh–Bénard convection and magnetoconvection in plane layer

ABSTRACT

The present study aims to link the dynamics of geophysical fluid flows with their vortical structures in physical space and to study the transition of these structures due to the control parameters. The simulations are carried in a rectangular box filled with liquid gallium for three different cases, namely, Rayleigh–Bénard convection (RBC), magnetoconvection (MC) and rotating magnetoconvection (RMC). The physical setup and material properties are similar to those considered by Aurnou and Olson in their experimental work. The simulated results are validated with theoretical results of Chandrasekhar and experimental results of Aurnou and Olson. The results are also topologically verified with the help of Euler number given by Ma and Wang. For RBC, the onset is obtained at Ra greater than 1708 and at this Ra, the symmetric rolls are orientated in/along a horizontal axis. As the value of Ra increases further, the width of the horizontal rolls starts to amplify. It is observed that these two-dimensional rolls are nothing but the cross-sections of three-dimensional (3D) cylindrical rolls with wave structures. When the vertically imposed magnetic field is added to RBC, the onset of convection is delayed due to the effect of Lorentz force on the thermal buoyancy force. The presence of 3D rectangular structures is highlighted and analysed. When the magnetically influenced rectangular box rotates about vertical axis at low rotation rates in magnetoconvection model, the onset of convection gets further delayed by magnetic field, which is in general agreement with the theoretical predictions. The critical Ra increases linearly with magnetic field intensity. Coherent thermal oscillations are detected near the onset of convection, at moderate rotation rates.     

The rapid dissipation of magnetic fields in highly conducting fluids

Abstract

This paper treats the dynamical conditions that obtain when long straight parallel twisted flux tubes in a highly conducting fluid are packed together in a broad array. It is shown that there is generally no hydrostatic equilibrium. In place of equilibrium there is a dynamical nonequilibrium, leading to neutral point reconnection and progressive coalescence of neighboring tubes (with the same sense of twisting), forming tubes of larger diameter and reduced twist. The magnetic energy in the twisting of each tube declines toward zero, dissipated into small-scale motions of the fluid and thence into heat.

The physical implications are numerous. For instance, it has been suggested that the subsurface magnetic field of the sun is composed of close-packed twisted flux tubes. Any such structures are short lived, at best.

The footpoints of the filamentary magnetic fields above bipolar magnetic regions on the sun are continually shuffled and rotated by the convection, so that the fields are composed of twisted rubes. The twisting and mutual wrapping is converted directly into fluid motion and heat by the dynamical nonequilibrium, so that the work done by the convection of the footpoints goes directly into heating the corona above. This theoretical result is the final step, then, in understanding the assertion by Rosner, Tucker, and Valana, and others, that the observed structure of the visible corona implies that it is heated principally by direct dissipation of the supporting magnetic field. It is the dynamical nonequilibrium that causes the dissipation, in spite of the high electrical conductivity. It would appear that any bipolar magnetic field extending upward from a dense convective layer into a tenuous atmosphere automatically produces heating, and a corona of some sort, in the sun or any other convective star.     

Distribution of magnetic energy in αΩ-dynamos,I: The method

Abstract

In this paper a method for solving the equation for the mean magnetic energy <BB> of a solar type dynamo with an axisymmetric convection zone geometry is developed and the main features of the method are described. This method is referred to as the finite magnetic energy method since it is based on the idea that the real magnetic field B of the dynamo remains finite only if <BB> remains finite. Ensemble averaging is used, which implies that fields of all spatial scales are included, small-scale as well as large-scale fields. The method yields an energy balance for the mean energy density ε ≡ B ²/8π of the dynamo, from which the relative energy production rates by the different dynamo processes can be inferred. An estimate for the r.m.s. field strength at the surface and at the base of the convection zone can be found by comparing the magnetic energy density and the outgoing flux at the surface with the observed values. We neglect resistive effects and present arguments indicating that this is a fair assumption for the solar convection zone. The model considerations and examples presented indicate that (1) the energy loss at the solar surface is almost instantaneous; (2) the convection in the convection zone takes place in the form of giant cells; (3) the r.m.s. field strength at the base of the solar convection zone is no more than a few hundred gauss; (4) the turbulent diffusion coefficient within the bulk of the convection zone is about 10¹⁴cm²s^?1, which is an order of magnitude larger than usually adopted in solar mean field models.     

The kinematics of hexagonal magnetoconvection

Abstract

Calculations are presented for the evolution of a magnetic field which is subject to the effect of three-dimensional motions in a convecting layer of highly conducting fluid with hexagonal symmetry. The back reaction of the field on the motions via the Lorentz force is neglected. We consider cases where the imposed field is either vertical or horizontal. In the former case, flux accumulates at cell centres, with subsidiary concentrations at the vertices of the pattern. In the latter, topological asymmetries between up- and down-moving fluid regions generate positive flux at the base of the layer and negative flux at the top, though the system is actually an amplifier rather than a self-excited dynamo. Spiral field lines form in the interiors of the cells, and the phenomenon of “flux expulsion” found in two-dimensional solutions is somewhat altered when the imposed field is horizontal. Applications for stellar magnetic fields include a possible mechanism for burying flux at the base of a convection zone.     

Residual fields from extinct dynamos

Abstract

The paper explores some of the many facets of the problem of the generation of magnetic fields in convective zones of declining vigor and/or thickness. The ultimate goal of such work is the explanation of the magnetic fields observed in A-stars. The present inquiry is restricted to kinematical dynamos, to show some of the many possibilities, depending on the assumed conditions of decline of the convection. The examples serve to illustrate in what quantitative detail it will be necessary to describe the convection in order to extract any firm conclusions concerning specific stars.

The first illustrative example treats the basic problem of diffusion from a layer of declining thickness. The second adds a buoyant rise to the field in the layer. The third treats plane dynamo waves in a region with declining eddy diffusivity, dynamo coefficient, and large-scale shear. The dynamo number may increase or decrease with declining convection, with an increase expected if the large-scale shear does not decline as rapidly as the eddy diffusivity. It is shown that one of the components of the field may increase without bound even in the case that the dynamo number declines to zero.     

A model of thermal convection in the major planets with a strongly density-stratified atmospheric layer

Abstract

As an extension of a model by Busse (1983a), a two-layer model of thermal convection in the self-gravitating rotating spherical fluid is considered. The upper layer with arbitrary vertical distributions of density and potential temperature representing the atmospheric layer of major planets is imposed on the spherical Boussinesq fluid. The Prandtl number P and the ratio of the mass of the upper layer to that of the lower layer are used as small expansion parameters. The modification of the critical Rayleigh number by imposing the upper layer are clearly separated into two parts, proportional to (1) the mass of the upper layer and to (2) an integral representing a measure of convective instability of the upper layer. Some implications for atmospheric dynamics of the major planets are also presented.     

Magnetic Rossby waves in a stably stratified layer near the surface of the Earth's outer core

Abstract

The stratification profile of the Earth's magnetofluid outer core is unknown, but there have been suggestions that its upper part may be stably stratified. Braginsky (1984) suggested that the magnetic analog of Rossby (planetary) waves in this stable layer (the ‘H’ layer) may be responsible for a portion of the short-period secular variation. In this study, we adopt a thin shell model to examine the dynamics of the H layer. The stable stratification justifies the thin-layer approximations, which greatly simplify the analysis. The governing equations are then the Laplace's tidal equations modified by the Lorentz force terms, and the magnetic induction equation. We linearize the Lorentz force in the Laplace's tidal equations and the advection term in the magnetic induction equation, assuming a zeroth order dipole field as representative of the magnetic field near the insulating core-mantle boundary. An analytical β-plane solution shows that a magnetic field can release the equatorial trapping that non-magnetic Rossby waves exhibit. A numerical solution to the full spherical equations confirms that a sufficiently strong magnetic field can break the equatorial waveguide. Both solutions are highly dissipative, which is a consequence of our necessary neglect of the induction term in comparison with the advection and diffusion terms in the magnetic induction equation in the thin-layer limit. However, were one to relax the thin-layer approximations and allow a radial dependence of the solutions, one would find magnetic Rossby waves less damped (through the inclusion of the induction term). For the magnetic field strength appropriate for the H layer, the real parts of the eigenfrequencies do not change appreciably from their non-magnetic values. We estimate a phase velocity of the lowest modes that is rather rapid compared with the core fluid speed typically presumed from the secular variation.     

Non-linear marginal convection in a rotating magnetic system

Abstract

An inviscid, electrically conducting fluid is contained between two rigid horizontal planes and bounded laterally by two vertical walls. The fluid is permeated by a strong uniform horizontal magnetic field aligned with the side wall boundaries and the entire system rotates rapidly about a vertical axis. The ratio of the magnitudes of the Lorentz and Coriolis forces is characterized by the Elsasser number, A, and the ratio of the thermal and magnetic diffusivities, q. By heating the fluid from below and cooling from above the system becomes unstable to small perturbations when the adverse density gradient as measured by the Rayleigh number, R, is sufficiently large.

With the viscosity ignored the geostrophic velocity, U, which is aligned with the applied magnetic field, is independent of the coordinate parallel to the rotation axis but is an arbitrary function of the horizontal cross-stream coordinate. At the onset of instability the value of U taken ensures that Taylor's condition is met. Specifically the Lorentz force, which results from marginal convection must not cause any acceleration of the geostrophic flow. It is found that the critical Rayleigh number characterising the onset of instability is generally close to the corresponding value for the usual linear problem, in which Taylor's condition is ignored and U is chosen to vanish. Significant differences can occur when q is small owing to a complicated flow structure. There is a central interior region in which the local magnetic Reynolds number, R_m , based on U is small of order q and on exterior region in which R_m is of order unity.     

Effect of a uniform magnetic field on nonlinear magnetocenvection in a rotating fluid spherical shell

Abstract

Dynamic interaction between magnetic field and fluid motion is studied through a numerical experiment of nonlinear three-dimensional magnetoconvection in a rapidly rotating spherical fluid shell to which a uniform magnetic field parallel to its spin axis is applied. The fluid shell is heated by internal heat sources to maintain thermal convection. The mean value of the magnetic Reynolds number in the fluid shell is 22.4 and 10 pairs of axially aligned vortex rolls are stably developed. We found that confinement of magnetic flux into anti-cyclonic vortex rolls was crucial on an abrupt change of the mode of magnetoconvection which occurred at Δ = 1 ～ 2, where A is the Elsasser number. After the mode change, the fluid shell can store a large amount of magnetic flux in itself by changing its convection style, and the magnetostrophic balance among the Coriolis, Lorentz and pressure forces is established. Furthermore, the toroidal/poloidal ratio of the induced magnetic energy becomes less than unity, and the magnetized anti-cyclones are enlarged due to the effect of the magnetic force. Using these key ideas, we investigated the causes of the mode change of magnetoconvection. Considering relatively large magnetic Reynolds number and a rapid rotation rate of this model, we believe that these basic ideas used to interpret the present numerical experiment can be applied to the dynamics in the Earth's and other planetary cores.     

The effect of time-dependent γ-pumping on buoyant magnetic structures

ABSTRACT

In this paper, we explore for the first time the interactions of the net downward, time-dependent, γ-pumping overlying an imposed layer of magnetic fluid, in a polytropic atmosphere. Our calculations show that an equipartition of energy, between the magnetic and kinetic components, must be reached for buoyancy-driven magnetic structures to rise into the pumping region. However, structures do not rise unhindered, as in a previous investigation. We show that the evolution and other features of the emerging magnetic flux structures are significantly affected by the temporal variation of the γ-pumping. The rate of emerging structures, the strength of magnetic concentrations and the extent to how far magnetic field can travel were all found to depend on the timescale of the γ-pumping.     

Convection in a rotating magnetic system and Taylor's constraint

Abstract

A system is considered in which electrically conducting fluid is contained between two rigid horizontal planes and bounded laterally by a circular cylinder. The fluid is permeated by a strong azimuthal magnetic field. The strength of the field increases linearly with distance from the vertical axis of the cylinder, about which the entire system rotates rapidly. An unstable temperature gradient is maintained by heating the fluid from below and cooling from above. When viscosity and inertia are neglected, an arbitrary geostrophic velocity, which is aligned with the applied azimuthal magnetic field and independent of the axial coordinate, can be superimposed on the basic axisymmetric state. In this inviscid limit, the geostrophic velocity which occurs at the onset of convection is such that the net torque on geostrophic cylinders vanishes (Taylor's condition). The mathematical problem which describes the ensuing marginal convection is nonlinear, and was discussed previously for the planar case by Soward (1986). Here new features are isolated which result from the cylindrical geometry. New asymptotic solutions are derived valid when Taylor's condition is relaxed to include viscous effects.     

Finite amplitude convection and magnetic field generation in a rotating spherical shell

Abstract

Finite amplitude solutions for convection in a rotating spherical fluid shell with a radius ratio of η=0.4 are obtained numerically by the Galerkin method. The case of the azimuthal wavenumber m=2 is emphasized, but solutions with m=4 are also considered. The pronounced distinction between different modes at low Prandtl numbers found in a preceding linear analysis (Zhang and Busse, 1987) is also found with respect to nonlinear properties. Only the positive-ω-mode exhibits subcritical finite amplitude convection. The stability of the stationary drifting solutions with respect to hydrodynamic disturbances is analyzed and regions of stability are presented. A major part of the paper is concerned with the growth of magnetic disturbances. The critical magnetic Prandtl number for the onset of dynamo action has been determined as function of the Rayleigh and Taylor numbers for the Prandtl numbers P=0.1 and P=1.0. Stationary and oscillatory dynamos with both, dipolar and quadrupolar, symmetries are close competitors in the parameter space of the problem.     

Thermal and magnetic instabilities in a rapidly rotating fluid sphere

Abstract

A linear analysis is used to study the stability of a rapidly rotating, electrically-conducting, self-gravitating fluid sphere of radius r ₀, containing a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance from the rotation axis. The Lorentz force is of a magnitude comparable with that of the Coriolis force and so convective motions are fully three-dimensional, filling the entire sphere. We are primarily interested in the limit where the ratio q of the thermal diffusivity κ to the magnetic diffusivity η is much smaller than unity since this is possibly of the greatest geophysical relevance.

Thermal convection sets in when the temperature gradient exceeds some critical value as measured by the modified Rayleigh number R_c. The critical temperature gradient is smallest (R_c reaches a minimum) when the magnetic field strength parameter Λ ? 1. [R_c and Λ are defined in (2.3).] The instability takes the form of a very slow wave with frequency of order κ/r ² ₀ and its direction of propagation changes from eastward to westward as Λ increases through Λ _c ? 4.

When the fluid is sufficiently stably stratified and when Λ > Λ_m ? 22 a new mode of instability sets in. It is magnetically driven but requires some stratification before the energy stored in the magnetic field can be released. The instability takes the form of an eastward propagating wave with azimuthal wavenumber m = 1.     

Superhelicity,helicity and potential vorticity

Abstract

‘‘Helicity'’ density H ≡ u · ω and other pseudo-scalar fields such as P ≡ ω · Vlnρ (which is related to Ertel potential vorticity) are useful quantities in theoretical fluid dynamics and magneto-fluid dynamics. Here u denotes the Eulerian flow velocity relative to the chosen frame of reference, ω ≡ V × u is the corresponding relative vorticity and ρ the mass density of the fluid. A general expression is readily obtained for ?H/?t (where t denotes time) in terms of P and the ‘‘superhelicity'’ density S ≡ ω · V × ω which, in fluids of low viscosity, has its highest values in boundary layers. One need for such a relationship became evident during an attempt to interpret the findings of laboratory experiments on thermal convection in rotating fluids in containers of various geometrical shapes and topological characteristics.

In electrodynamics an analogous expression can be found relating the time rate of change of ‘‘magnetic helicity'’ A · B to ‘‘magnetic superhelicity'’ B · ? × B (where B · ? × A is the magnetic field) and a scalar quantity analogous to P which involves non-Ohmic contributions to the relationship between the electric current density and the electric field.