2.1 Magnetized fluids and the MHD induction equation

In the interiors of the Sun and most stars, the collisional mean-free path of microscopic constituents is much shorter than competing plasma length scales, fluid motions are non-relativistic, and the plasma is electrically neutral and non-degenerate. Under these physical conditions, Ohm’s law holds, and so does Ampère’s law in its pre-Maxwellian form. Maxwell’s equations can then be combined into a single evolution equation for the magnetic field B, known as the magnetohydrodynamical (MHD) induction equation (see, e.g., Davidson, 2001 ):

$@B-- @t = \~/ × (u × B - j \~/ × B), (1)$

where $2 j = c /4pse$ is the magnetic diffusivity ( $se$ being the electrical conductivity), in general only a function of depth for spherically symmetric solar/stellar structural models. Of course, the magnetic field is still subject to the divergence-free condition $\~/ .B = 0$ , and an evolution equation for the flow field u must also be provided. This could be, e.g., the Navier-Stokes equations, augmented by a Lorentz force term:

$@u 1 r--- + r(u . \~/ )u + 2r(_O_ × u) = - \~/ p + rg + --( \~/ × B) × B + \~/ .t , (2) @t 4p$

where $t$ is the viscous stress tensor, and other symbols have their usual meaning² . In the most general circumstances, Equations (1

) and (2

) must be complemented by suitable equations expressing conservation of mass and energy, as well as an equation of state. Appropriate initial and boundary conditions for all physical quantities involved then complete the specification of the problem. The resulting set of equations defines magnetohydrodynamics, quite literally the dynamics of magnetized fluids.