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Almost adiabatic states are typical for the deep convective interiors of all known planets and their moons, e.g., the deviations from the adiabatic state in the Earth's outer core and in the MHD dynamo region of Jupiter are about or less than 10 À5 %. We approximated the equations governing convection in the deep interiors of planets and their moons to obtain a system, which is more accurate than the traditional Boussinesq equations. Fortunately, our system, which adopts almost uniform entropy instead of the temperature as the basic thermodynamic states, can still be investigated by standard methods. We considered the marginal stability of wellmixed almost adiabatic states in rapidly rotating thick spherical shells, whose inner to outer radius ratio does not exceed that of the modern Earth. The critical Rayleigh-type numbers, frequencies and solution structures of the marginal states were determined by both analytical and numerical methods. Our new estimates differ from those obtained previously using the Boussinesq equations, suggesting that the earlier Boussinesq results for convection in the deep planetary interiors should be re-assessed. The small molecular Prandtl number limit was adopted to model the marginal stability of thermal planetary convection. It was found that the critical Rayleigh number for convection sharply diminished as the radius of the inner rigid core is increased. We modelled the instability of the combined compositional-thermal turbulent geo-convection for Prandtl number unity. When thermal convection is in opposition to compositional convection, extremely large critical Rayleigh numbers are possible. This might happen for a terrestrial planet during its later stage of evolution. Pure compositional convection has been investigated in the large compositional Prandtl number limit, for which the critical Rayleigh number is rather large and the variations of all critical parameters are small. The large size of the critical Rayleigh number ensures that the actual values used in numerical dynamo experiments are only moderately supercritical.

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UDC 532.516 + 538.4 M. S. Kotel'nikova and B. A. LugovtsovThe region of instability of the Hill-Shafranov viscous MHD vortex with respect to azimuthal axisymmetric perturbations of the velocity field is determined numerically as a function of the Reynolds number and magnetization in a linear formulation. An approximate formulation of the linear stability problem for MHD flows with circular streamlines is considered. The further evolution of the perturbations in the supercritical region is studied using a nonlinear analog model (a simplified initial system of equations that takes into account some important properties of the basic equations). For this model, the secondary flows resulting from the instability are determined.Introduction. We consider steady-state axisymmetric (poloidal) viscous flow in a bounded axisymmetric region that is sustained by the field of mass forces (axisymmetric, poloidal) and (or) by moving (in an appropriate manner) boundaries, on which the attachment condition is satisfied. In cylindrical coordinates,Small axisymmetric perturbations (generally arbitrary) with nonzero azimuthal (v ϕ = v) velocity component are introduced into this flow. The mass-force field and the boundary conditions are not perturbed. If the perturbations damp or their amplitude does not increase, the flow is steady-state and swirling does not occur. In the case of instability, the perturbations grow. If the evolution of the initial perturbations, by virtue of the exact nonlinear equations, results in flow (steady-state, periodic, unsteady random, turbulent) in which the average azimuthal velocity component is finite:and if the energy of rotational motion around the symmetry axis is comparable to the energy of the initial poloidal flow, we shall speak of the occurrence of spontaneous swirling. The problem of spontaneous swirling was first formulated in [1] as follows: can rotationally symmetric flow occur in the absence of obvious external sources of rotation, i.e., under conditions where axisymmetric irrotational motion is possible a priori?At present, there at least two points of view concerning the occurrence of spontaneous swirling. One of them is described above, and the second is consists of the following. Self-similar axisymmetric (conical) flows ofFor the function f (θ) in the first case and function g(z) in the second, ordinary differential equations are derived; for small Reynolds numbers, they have solutions only with poloidal components, and for Reynolds numbers exceeding a certain critical value, solutions with v ϕ = 0 appear [2]. This mathematical fact is treated as the occurrence of spontaneous swirling or autorotation. However, it is obvious that in the flows Lavrent'ev Institute