312 Chapter 10. Stability of static MHD equilibria
Multiplying Eq.(10.71) by the perturbed velocityξ ̇and integrating over the volume of
the magnetofluid and vacuum region gives
∫
d^3 rρ 0∂
∂t(
̇ξ^2
2)
=
∫
d^3 rξ ̇·F 1 (ξ). (10.73)Although the integration includes the vacuum region, both the right and left hand sides of
this equation vanish in the vacuum region because no density, current, or pressure exist
there.
The left hand side of Eq.(10.73) is just the time derivative of the kinetic energyδT ̇.
SinceδT ̇+δW ̇ =0,it is seen that
δW ̇ =−∫
d^3 rξ ̇·F 1 (ξ). (10.74)However, sinceδW ̇ was shown to be self-adjoint, Eq.(10.74) can also be written asδW ̇ =
−
∫
d^3 rξ·F 1 (ξ ̇).Combining these last two equations provides an integrable form forδW ̇,
namely
δW ̇ = −1
2
(∫
d^3 rξ·F 1 (ξ ̇)+∫
d^3 rξ ̇·F 1 (ξ))
= −
1
2
∂
∂t(∫
d^3 rξ·F 1 (ξ))
. (10.75)
Performing the time integration gives the desired result, namely thechange in system po-
tential energy as a function of thefluid displacement is
δW=−1
2
∫
d^3 rξ·F 1 (ξ). (10.76)Standard techniques of normal mode analysis can be invoked by assuming that thedis-
placement has the form
ξ=Re(
̃ξe−iωt)
(10.77)
so the equation of motion can be written as
−ω^2 ρ ̃ξ=F 1 ( ̃ξ). (10.78)Multiplication by ̃ξ
∗
and integrating over the volume gives−ω^2∫
d^3 rρ| ̃ξ|2
=∫
d^3 r ̃ξ∗
·F 1 ( ̃ξ). (10.79)In the earlier discussion of self-adjointness, it was noted thatξ ̇is essentially an arbitrary
function for a givenξ,i.e., anyξ ̇could be obtained by choosing a suitable time dependence
forξ.Thus,ξ ̇ could be chosen to be proportional to ̃ξ
∗
and so the right hand side of
Eq.(10.79) is also self-adjoint. Equation (10.79) can therefore be recast as
−ω^2∫
d^3 rρ| ̃ξ|2
=1
2
[∫
d^3 r ̃ξ∗
·F 1 ( ̃ξ)+∫
d^3 r ̃ξ·F 1 ( ̃ξ∗
)