304 Chapter 10. Stability of static MHD equilibria
Theycomponent is found by dotting withˆyand then using the vector identity∇·(F×G)=
G·∇×F−F·∇×Gto obtain
γB 1 y=ˆy·∇×[v 1 ×B 0 ]=∇·[(v 1 ×B 0 )׈y]=∇·[v 1 yB 0 ]=ik·B 0 v 1 y.(10.32)Substituting into Eq.(10.29) using Eq.(10.32) and Eq.(10.14) and rearranging the order
gives
∂
∂y{[
γ^2 ρ 0 +1
μ 0
(k·B 0 )^2]
∂v 1 y
∂y}
=k^2{
γ^2 ρ 0 −g∂ρ 0
∂y+
(k·B 0 )^2
μ 0}
v 1 y (10.33)which is identical to the inverted glass of water problem ifk·B 0 =0.
Rather than have an abrupt interface between a heavy and lightfluid as in the glass
of water problem, it is assumed that the magnetofluid fills the container betweeny=0
andy=hand that there is a density gradient in theydirection. This situation is more
appropriate for a plasma which would typically have a continuous density gradient.It is
assumed that rigid boundaries exist at bothy=0andy=hso thatv 1 y=0 at both
y=0andy=h.These boundary conditions mean that rippling is not allowed aty=0,h
but could occur in the interior, 0 <y<h.Equation (10.33) is now impossible to solve
analytically because all the coefficients are functions ofy. However, an approximate
understanding for the behavior predicted by this equation can be found by multiplying the
equation byv 1 yand integrating fromy=0toy=hto obtain
[{
γ^2 ρ 0 +1
μ 0(k·B 0 )^2}
v 1 y
∂v 1 y
∂y]h0−
∫h0[
γ^2 ρ 0 +1
μ 0(k·B 0 )^2](
∂v 1 y
∂y) 2
dy=k^2∫h0{
γ^2 ρ 0 −g∂ρ 0
∂y+
(k·B 0 )^2
μ 0}
v 12 ydy.(10.34)
The integrated term vanishes because of the boundary conditions (which could also have
been∂v 1 y/∂y=0). Solving forγ^2 gives
γ^2 =∫h0dy[
k^2 g∂ρ 0
∂yv^21 y−(k·B 0 )^2
μ 0(
k^2 v 12 y+(
∂v 1 y
∂y) 2 )]
∫h0dyρ 0[
k^2 v^21 y+(
∂v 1 y
∂y) 2 ]. (10.35)
Ifk·B 0 =0and the density gradient is positive everywhere thenγ^2 > 0 so there
is instability. If the density gradient is negative everywhere except at onestratum with
thickness∆y, then the system will be unstable with respect to an interchange at that one
‘top-heavy’ stratum. The velocity will be concentrated at this unstable stratum and so
the integrands will vanish everywhere except at the unstable stratumgiving a growth rate
γ^2 ∼g∆yρ− 01 ∂ρ 0 /∂ywhere∂ρ 0 /∂y is the value in the unstable region. This MHD
version of the Rayleigh-Taylor instability is called the Kruskal-Schwarzschild instability
(Kruskal and Schwarzschild 1954).