10.3 The MHD energy principle 311becauseǫis small the spatial dependence ofU 1 can be ignored when evaluating the time
integral, i.e., terms of orderξ·∇U 1 are ignored since these are of orderǫ^2.
The kinetic energy associated with the mode is
δT=∫
d^3 rρ 0 U 12
2(10.65)
which clearly is of orderǫ^2 .Since total energy is conserved it is necessary to have
δT+δW=0 (10.66)leading to the important conclusion that the perturbed potential energyδW must also be
of orderǫ^2.
The perturbed pressure and magnetic field can be found to first-order inǫby integrating
Eqs.(10.42) and (10.44) respectively to obtain
B 1 =∇×(ξ×B 0 ) (10.67)and
P 1 =−ξ·∇P 0 −γP 0 ∇·ξ (10.68)
showing that the first-order pressure and first-order magnetic fields are linear functions of
ξ.However, since it was shown above thatδWscales asǫ^2 ,only terms second-order inξ
can contribute toδW, and so all first-order terms must average to zero when integrating
over the volume. To specify thatδWdepends only to second-order inξ, we write
δW=δW(ξ,ξ) (10.69)where the double argument means thatδW is a bilinear function,i.e.,δW(aξ,bη)=
abδW(ξ,η)for arbitraryξ,η.The time derivative ofδWis thus
δW ̇ =δW(ξ ̇,ξ)+δW(ξ,ξ ̇). (10.70)Sinceξ ̇is algebraically independent ofξ, Eq.(10.70) means thatδW ̇ is self-adjoint (i.e.,
δW ̇ is invariant when its two arguments are interchanged). Self-adjointness is a direct
consequence of the existence of an energy integral.
10.3.3Formal solution for perturbed potential energy
The self-adjointness property can be exploited by explicitly calculatingthe time derivative
of the perturbed potential energy. This is done using the linearized equation ofmotion
ρ 0∂^2 ξ
∂t^2=F 1 (10.71)
where
F 1 =J 0 ×B 1 +J 1 ×B 0 −∇P 1 (10.72)
results from linear operations onξsinceB 1 ,μ 0 J 1 =∇×B 1 andP 1 are all result from
linear operations onξ.