M. Kotelnikova $^{a}$ and S. Starchenko $^{b}$
$^{a}$ Institute of Hydrodynamic, Novosibirsk, Russia. $^{b}$ GFO Borok 42-4, Yaroslavskaya oblast, Russia.
It is natural to begin studying heat-mass transfer in cores of planets of Earth's group without taking into account a magnetic field. The solution of this problem give a possibility to investigate a resulting flow for capability to magnetic field generation, that will allow to give for one the answer why Venus and Mars are not capable to support an own magnetic field. The convection of concern exists in a spherical fluid shell which is close to being adiabatically stratified. The fluid in a spherical shell between inner solid core and the mantle has a light and a heavy component. Flows of conducting liquid in the outer core are generated by Archimedean force of the buoyancy arising, basically, due to emersion of the excess light component from boundary with an inner core of planets of Earth's group during growth of the last. Appropriate convection, driving by thermal and compositional effects in quickly rotating spherical shell, nobody has not studied yet. Also down to the present work it was not unequivocally offered the mechanism obviously supporting faster rotation of inner sphere (a solid core) in comparison with outer sphere (mantle). In the given work heat-mass transfer is investigated, supporting differential rotation in decreasing in the size and freezing liquid planet's cores. Spherically symmetrical basic analytical solution of linear heat and diffusion equations was got for the first time. Nonlinear terms, which are responsible for convectional transfer, in those equations were neglected with applicable reason. It was shown that this basic solution is able to support differential rotation without convection if heat flux from the core to the mantle is under adiabatic flux. At performance of this condition growth rate of an inner core linearly grows with increase of the module of a heat flux. (The diagram of dependence of growth rate of an inner core from magnitude of a heat flux and concentration of light component discontinuitie is available at www.mhdintas.ex.ac.uk/PS/5/in_Russian/Fig3drqx.gif.)So for magnitude of a full heat flux $-0.01 W/m^2 $ magnitude of growth rate of an inner core according to estimation made on the basis of the received basic solution gives $ \approx0.27\cdot10 ^ {-11} m/s $ [Starchenko and Kotelnikova (2002), "Symmetrical heat-mass transfer in a rotating spherical shell", JETP, v.121, n.3]. For the comparison, offered by authors [Glatzmaier and Roberts (1996) " An anelastic evolutionary geodynamo simulation driven by compositional and thermal convection ", Physica D 97, 81 - 94.] magnitude is $ \approx0.6\cdot10 ^ {-11} m/s $. And numerical model [Glatzmaier and Roberts (1995) " A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle ", Phys. Earth Planet. Inter. 91, 63 - 75.] was based on age of an inner core approximately 1.3 billion years that gives $ \approx1.3\cdot10 ^ {-11} m/s $. For the general case (random heat flux) homogeneous and non-dimensional system was constructed by the subtraction of basic solution from convectional equations. The last system is controlled by two asymptotically small parameters: Rossby $\epsilon\le 10^{-5}$, which is characteristic of relative magnitude of differential rotation, and generalized Ekman number $E\le 10^{-12}$, which is characteristic of relative influence of viscous and diffusive effects with fast rotation. The main order of the solution at $\epsilon\rightarrow 0$ and after at $\sqrt{E}\rightarrow 0$ with molecular magnitudes of transport coefficients leads to basic flow which is symmetrical relatively rotation axis and has azimuthal direction mainly. This flow is capable to generate asymmetrical magnetic field. Generation is most effective in Ekman boundary layers and alongside the axial cylinder tangential to solid inner core of a planet. The moment of viscous forces in inner Ekman boundary layer (about $E^{1/2}$) provides faster rotation of the solid inner core of Earth's group planets in comparison with a massive mantle due to redistribution of the moment of inertia during growth of a solid core of a planet. Such heat-mass transfer provides the magnitude of differential rotation in Earth core less than one degree per year if one uses molecular transport coefficients and far less if one uses turbulent transport coefficients, with good agreement with geomagnetic and seismic estimations. The appropriate differential rotation symmetrical relatively rotation axis and weak meredional circulation can appear insufficient for excitation of an own magnetic field of a planet, describing a situation similar that exists in liquid nucleus of Mars and Venus. Powerful enough flow, first of all, should raise sharply asymmetrical relatively rotation axis of a planet magnetic field. If this asymmetrical magnetic field be unable to cause significant asymmetrical flow the planetary dynamo remain in a kinematic mode as it, apparently, is observed for Uranus and the Neptune [Ruzmaikin and Starchenko (1991) "On the origin of Uranus and Neptune magnetic fields", Icarus 93, 82 - 87.]. In more realistic for the Earth, Saturn and the Jove a dynamic mode [Starchenko and Jones (2002) "Typical velocities and magnetic field strenghths in planetary interiors", ICARUS - in press] - asymmetrical MHD flow will generate mainly symmetrical relatively rotation axes magnetic field. Thus, the further development of the approaches offered by us here will allow to receive the effective decision of all basic problems of planetary MHD dynamo.