Manuel Baptista
,
University of Porto
We are considering perturbations of MHD steady states in a horizontal
rotating layer, which satisfy (i) the Navier-Stokes equation with
the Lorentz force and thermal buoyancy force taken into account in
the Boussinesq approximation, (ii) the magnetic induction equation
and (iii) the heat transfer equation, with production of heat by
electric
currents (Joule effect) taken into account (though this restricts
the symmetries consistent with the steady state).
~
Linearised equations for small perturbations of thermoconvective
magneto-hydrodynamic
steady states of an electrically conducting fluid in a layer are
analysed
by multiple-scale expansion methods [1,2,3,4,5}.
Periodic boundary conditions are imposed in horizontal directions.
On horizontal boundaries temperature is assumed to be constant,
stress-free
boundary conditions are imposed for the flow and perfect conductor
boundary conditions for the magnetic field.
~
An eigenvalue problem for perturbations is analysed. The eigenmodes
and the eigenvalues are supposed to depend on both small- (fast) and
large-scale (slow) spatial variables. They are expanded in power series
of the scale ratio and a hierarchy of equations is derived. Solvability
of the zeroth and first order problems in this hierarchy is assured
by symmetry arguments. Solvability conditions at the second order
yield a closed set of equations for the leading terms in the expansions
of the dominant mode and its growth rate.
~
Simulations are done with our code employing pseudo-spectral methods.
[1] V.A. Zheligovsky, O.M. Podvigina. Generation of multiscale magnetic
field by parity-invariant time-periodic flows. Geophys. Astrophys.
Fluid Dynamics} 97, 2003, 225-248
{[}
http://xxx.lanl.gov/abs/physics/0207112{]}.
[2] V.A. Zheligovsky. On the linear stability of spatially periodic
steady
magnetohydodynamic systems with respect to long period perturbations.
Izvestiya, Physics of the Solid Earth 39 N5, 2003, 409-418.
[3] V.A. Zheligovsky, O.M. Podvigina, U. Frisch. Dynamo effect in
parity-invariant
flow with large and moderate separation of scales.
Geophys. Astrophys. Fluid Dynamics 95, 2001, 227-268 {[}
http://xxx.lanl.gov/abs/nlin.CD/0012005{]}.
[4] A. Lanotte, A. Noullez, M. Vergassola, A. Wirth.
Large-scale dynamoproduced by negative magnetic eddy
diffusivities.
Geophys. Astrophys. Fluid Dynamics 91, 1999, 131-146.
[5] B\'ereng\`ere Dubrulle, Uriel Frisch. Eddy viscosity of
parity-invariant flow. Physical Review A, Vol. 43, No.10, 1991.
