Fundamentals of Plasma Physics

9.9 Dynamic equilibria:flows 293

hand ifJpolwere not parallel toBpol,these surfaces would not be coincident and there
would be a forceJpol×Bpolin the toroidal direction would tend to cause a change in the
toroidal velocity, i.e., an acceleration or deceleration ofUφ.We argue thatJpol×Bpolmust
be a transient force with zero time average because any current whichflows perpendicular
to the poloidalflux surfaces must be the polarization current as given by Eq.(3.145). This
polarization current results from the time derivative of the electricfield perpendicular to
the poloidalflux surfaces and this electric field in turn is proportional to the timerate of
change of the poloidal current as determined from Faraday’s law. Thus, once thecurrent
is in steady state, it is necessary to haveI=I(ψ)and so the surfaces of constantI(r,z)
in Fig.9.12 can also be considered as surfaces of constantψ(r,z).Another way to see this,
is to realize that a steady current perpendicular toflux surfaces would cause a continuous
electrostatic charging of theflux surfaces which would then violate the stipulation that the
plasma is quasi-neutral.
The initial magnetic force is thus the same as for the situation of a purely toroidal
magnetic field since
J×B=Jtor
︸︷︷︸
zero

×Bpol+Jpol×Btor. (9.96)

The remaining forceJpol×Btor=(Jrrˆ+Jzzˆ)×Bφφˆ=JrBφzˆ−JzBφˆrhas components
in both therandzdirections. If the plasma axial length is much larger than its radius (i.e.,
it is long and skinny) then the plasma will develop a local radial pressure balance

(−∇P+Jpol×Btor)·ˆr=0. (9.97)

However, this local radial pressure balance precludes the possibility ofan axial pressure
balance if ∂ψ/∂z=0because, as discussed earlier,∇P cannot balanceJ×Bif the
latter has a finite curl. Suppose that the radius of the current channel and the poloidalflux
are both given bya=a(z)and sinceI=I(ψ)let us assume a simple linear dependence
where μ 0 I(r,z)=λψ(r,z).Thus,∇Iis parallel to∇ψandIis just the current perflux.
We assume a parabolic poloidalflux

ψ(r,z)

ψ 0

=

(

r

a(z)

) 2

(9.98)

forr≤a(z)andψ 0 is theflux atr=a;this is the simplest allowed form for theflux (if
ψdepended linearly onr,there would be infinite fields atr=0). The local radial pressure
balance is essentially a local version of the Bennett pinch relation

∂P

∂r

= −JzBφ

= −

λ^2

8 π^2 μ 0 r^2

∂ψ^2

∂r

. (9.99)

On substitution of Eq.(9.98) this becomes

∂P

∂r

= −

μ 0 λ^2 ψ^20

8 π^2 r^2

∂

∂r

(

r

a(z)

) 4

= −

λ^2 ψ^20 r

2 π^2 μ 0 a^4

. (9.100)