Fundamentals of Plasma Physics

314 Chapter 10. Stability of static MHD equilibria

and

δWP 1 = −

1

2

∫

Vmf

d^3 rξ·∇[ξ·∇P 0 +γP 0 ∇·ξ]

=

1

2

∫

Vmf

d^3 r

[

−ξ·∇(ξ·∇P 0 )+γP 0 (∇·ξ)^2

]

−

1

2

∫

Smf

ds·ξγP 0 ∇·ξ.

(10.89)

Since bothJ×Band∇P are perpendicular toBat the surface of the magnetofluid,
the force acting on the magnetofluid at the surface is perpendicular toBand so the dis-
placementξof the magnetofluid at the surface is perpendicular toB, i.e., at the surface
ξ=ξ⊥.The surface integral in Eq.(10.88) can be expanded

1

2 μ 0

∫

Smf

ds·B 1 ×(ξ×B 0 )=

1

2 μ 0

∫

Smf

ds·ξ⊥B 0 ·B 1 (10.90)

sinceds·B 0 =0on the magnetofluid surface. This latter condition is true because the
magnetic field was initially tangential to the magnetofluid surface (i.e.,J 0 ×B 0 =∇P 0
implies∇P 0 is perpendicular toB 0 ) and must remain so since the field is frozen into the
magnetofluid.

Recombining and re-ordering these separate contributions gives

δW =

1

2

∫

Vmf

d^3 r

{

γP 0 (∇·ξ)^2 +

B^21

μ 0

−ξ·[J 0 ×B 1 +∇(ξ·∇P 0 )]

}

+

1

2 μ 0

∫

Smf

ds·ξ⊥[B 0 ·B 1 −μ 0 γP 0 ∇·ξ]. (10.91)

The substitutionds·ξ=ds·ξ⊥has been made for the pressure contribution to the surface
term on the grounds thatdsmust be perpendicular toB 0 .Further simplification is obtained
by considering the dot product withB 0 of the term in square brackets in Eq.(10.91), namely

B 0 ·[J 0 ×B 1 +∇(ξ·∇P 0 )]= −∇P 0 ·B 1 +B 0 ·∇(ξ·∇P 0 )

= −∇P 0 ·∇×(ξ×B 0 )+B 0 ·∇(ξ·∇P 0 )

= ∇·{∇P 0 ×(ξ×B 0 )+(ξ·∇P 0 )B 0 }

= ∇·[ξB 0 ·∇P 0 ]

=0 (10.92)

where Eq.(10.61) has been used to obtain the last line. Thus,ξ·[J 0 ×B 1 +∇(ξ·∇P 0 )]=
ξ⊥·[J 0 ×B 1 +∇(ξ·∇P 0 )]since Eq.(10.92) shows that the factor in square brackets has
no component parallel to the equilibrium magnetic field.

The potential energy variationδW is now decomposed into its magnetofluid volume
and surface components,

δWF′=

1

2

∫

Vmf

d^3 r







γP 0 (∇·ξ)^2 +

B 12 ⊥

μ 0

+

B^21 ‖

μ 0

−ξ⊥·J 0 ×B 1 −ξ⊥·∇⊥(ξ⊥·∇P 0 )







(10.93)