288 Chapter 9. MHD equilibria
zIr,zconst.Figure 9.12: Current channelI(r,z).The magnetic force is thereforeJ×B=
1
2 π(∇I×∇φ)×μ 0
2 πI∇φ=−μ 0
(2πr)^2∇
(
I^2
2
)
. (9.73)
It is seen that there is an axial force ifI^2 depends onz;this is the essential condition that
produces axialflows in MHD arcs (Maecker 1955) and plasma guns(Marshall 1960).
Using the identities∇U^2 /2=U·∇U+U×∇×Uand∇×∇×U=−∇^2 U=∇χ×
∇φthe equation of motion Eq.(9.64) becomes
ρ(
∂U
∂t+∇
U^2
2
−U×χ∇φ)
=−
μ 0
(2πr)^2∇
(
I^2
2
)
−∇P−ρυ∇χ×∇φ (9.74)or
ρ(
∂U
∂t+∇
U^2
2
−U×χ∇φ)
=−
μ 0
(2π)^2∇
(
I^2
2 r^2)
+
μ 0 I^2
(2π)^2∇
(
1
2 r^2)
−∇P−ρυ∇χ×∇φ. (9.75)Every term in this equation is either a gradient of a scalar or else can be expressed as a
cross-product involving∇φ.The equation can thus be regrouped as
∇
(
ρU^2
2+
μ 0
(2π)^2I^2
2 r^2+P
)
+
(
ρ
2 π∇
∂ψ
∂t−ρUχ−μ 0 I^2
(2πr)^2ˆz+ρυ∇χ)
×∇φ=0.(9.76)
Similarly, the MHD Ohm’s lawE+U×B=ηJcan be written as
−
∂A
∂t−∇V+U×
μ 0
2 πI∇φ=
η
2 π∇I×∇φ (9.77)whereVis the electrostatic potential. SinceBis purely toroidal,Ais poloidal and so is
orthogonal to∇φ.Thus, both the equation of motion and Ohm’s law are equations of the