Fundamentals of Plasma Physics

9.8 Static equilibria 285

field does not contribute to confinement ifI′=0.IfI′is finite, plasma diamagnetism

or paramagnetism will affectP(ψ).Although the toroidal field does not directly pro-

vide confinement, it can affect the rate of cross-field particle and energy diffusion and

hence the functional form ofP(ψ).For example, the frequency of drift waves will

be a function of the toroidal field and drift waves can cause an outward diffusion of

plasma acrossflux surfaces, thereby affectingP(ψ).

7. The poloidalflux surfaces are related to the surfaces of constant canonical angular
   momentum sinceBpol=∇×(ψ∇φ/ 2 π)=∇×

(

ψφ/ˆ 2 πr

)

=∇×Aφˆφimplies

Aφ=ψ/ 2 πr.Thus, the canonical angular momentum can be expressed as

pφ = mr^2 φ ̇+qrAφ

= mrvφ+

qψ

2 π

. (9.60)

Surfaces of constant canonical angular momentum correspond to surfacesof constant

poloidalflux in the limitm→ 0 .Because the system is toroidally symmetric (axisym-

metric) the canonical angular momentum of each particle is a constant of the motion

and so, to the extent that the particles can be approximated as having zero mass, parti-

cle trajectories are constrained to lie on surfaces of constantψ.Sincepφis a conserved

quantity, the maximum excursion a finite-mass particle can make from a poloidalflux

surface can be estimated by writing

δpφ=δ(mrvφ+qrAφ)=0. (9.61)

Thus,

vφ

r

δr+δvφ+

q

m

δr

1

r

∂

∂r

(rAφ)=0 (9.62)

or

δr=−

rδvφ

vφ+rωc,pol

(9.63)

whereωc,pol =qBpol/mis the cyclotron frequency measured using the poloidal
magnetic field. Sincevφis of the order of the thermal velocity or smaller, if the
poloidal field is sufficiently strong that the poloidal Larmor radiusrlpol∼vφ/ωc,polis
much smaller than the radiusr,then the first term in the denominator may be dropped.
Sinceδvφis also of the order of the thermal velocity or smaller, the particle cannot
deviate from its initialflux surface by more than a poloidal Larmor radius. Since
poloidal field is produced by toroidal current, it is clear that particle confinement to
axisymmetricflux surfaces requires the existence of a toroidal current.
Tokamaks, reverse field pinches, spheromaks, and field reversed configurations are all
magnetic confinement configurations having three dimensional axisymmetric equilibria
similar to this Solov’ev solution and all involve a toroidal current to produce a set of nested,
closed poloidalflux surfaces which link a magnetic axis. Stellarators are non-axisymmetric
configurations which have poloidalflux surfaces without toroidal currents;the advantage
of current-free operation is offset by the complexity of non-axisymmetry.