Fundamentals of Plasma Physics

9.9 Dynamic equilibria:flows 289

form

∇g+Q×∇φ=0. (9.78)

which is the most general form of an axisymmetric partial differential equation involving a
potential. Two distinct scalar partial differential equations can be extracted from Eq.(9.78)
by (i) operating with∇φ·∇×and (ii) taking the divergence. Doing the former it is seen
that Eq.(9.78) becomes

∇φ·∇×(Q×∇φ)=0 (9.79)

or

∇·

(

1

r^2

Qpol

)

=0. (9.80)

Applying this procedure to Eq.(9.76) gives

ρ

2 π

∂

∂t

(

∇·

1

r^2

∇ψ

)

−ρ∇·

(

1

r^2

Uχ

)

−∇·

(

μ 0 I^2

(2πr^2 )^2

zˆ

)

+ρυ∇·

(

1

r^2

∇χ

)

=0(9.81)

or

∂

∂t

(χ

r^2

)

+∇·

(

U

χ

r^2

)

=−

μ 0

4 π^2 ρr^4

∂I^2

∂z

+υ∇·

(

1

r^2

∇χ

)

. (9.82)

Since∇·U=0, this can also be written as

∂

∂t

(χ

r^2

)

+U·∇

(χ

r^2

)

=−

μ 0

4 π^2 ρr^4

∂I^2

∂z

+υ∇·

1

r^2

∇χ (9.83)

which shows that if there is no viscosity and if∂I^2 /∂z=0, then the scaled vorticityχ/r^2
convects with thefluid;i.e., is frozen into thefluid. The viscous term on the right hand
side describes a diffusive-like dissipation of vorticity. The remaining termr−^4 ∂I^2 /∂zacts
as avorticity sourceand is finite only ifI^2 is non-uniform in thezdirection. The vortic-
ity source has a strongr−^4 weighting factor so that axial non-uniformities ofInearr=0
dominate. Positiveχcorresponds to a clockwise rotation in ther,zplane. IfI^2 is an in-
creasing function ofzthen the source term is negative, implying that a counterclockwise
vortex is generated and vice versa. Suppose as shown in Fig.9.12 that the current channel
becomes wider with increasingzcorresponding to a fanning out of the current with in-
creasingz.In this case∂I^2 /∂zwill be negative and a clockwise vortex will be generated.
Fluid willflow radially inwards at smallz,thenflow vertically upwards, and finally radi-
ally outwards at largez.Thefluidflow produced by the vorticity source will convect the
vorticity along with theflow until it is dissipated by viscosity.

Operating on Ohm’s law, Eq.(9.77), with∇φ·∇×gives

∂

∂t

(

I

r^2

)

+U·∇

(

I

r^2

)

=

η

μ 0

∇·

(

1

r^2

∇I

)

(9.84)

showing thatI/r^2 is similarly convected with thefluid and also has a diffusive-like term,
this time with coefficientη/μ 0.