Magnetohydrodynamics for a multicomponent ionic mixture

Damping and dispersion of oscillating modes of a multicomponent ionic mixture in a magnetic field

“…The continuity equation for species s reads in first order of the perturbations [7] As an independent set we will choose (2.2) in combination with (2.1) for sa1.…”

“…The time dependence of the barycentric motion is given by the equation of motion, which up to first order in the perturbations reads [7] m v q qt vðr; tÞ ¼ À rdpðr; tÞ À r Á dPðr; tÞ þ q v ½Eðr; tÞ þ c À1 vðr; tÞ^B…”

“…These quantities can be expressed in terms of thermodynamic forces by means of phenomenological relations [13]. The forces that drive the heat flow and the diffusion flows are [7] X ¼ r 1 T , (2.5)…”

“…Using the above decomposition of (3.8) in zeroth order of k and combining the resulting three equations with the zeroth-order approximation to (3.3) one gets four coupled equations from which the oscillating modes a ð0Þ lr and the associated frequencies z ð0Þ lr , with l ¼ AE1 and r ¼ AE1, can be obtained [7]. The time dependence of the modes is determined by the eigenvalue equation The coefficients b, b 0 and c are defined as…”

“…Their labels are chosen such that one has z à lr ¼ Àz l;Àr . The four mode amplitudes can be written as [7] a ð0Þ…”