Fundamentals of Plasma Physics

296 Chapter 9. MHD equilibria

Shafranov equation describes axisymmetric equilibria and so can be used to charac-

terize an axisymmetric magneticflux tube. Suppose thatψ=ψ(r,z)and thatψis a

monotonically increasing function ofr.Assume that both the currentIand pressure

Pare linear functions ofψso that

μ 0 I = λψ

P = P 0 (1−ψ/ψ 0 )

whereψ 0 is theflux surface at whichPvanishes andP 0 is the pressure on thezaxis.

Show that the Grad-Shafranov equation can be written in the form

r^2 ∇·

(

1

r^2

∇ψ ̄

)

+μ^20 λ^2 ψ ̄=

4 π^2 r^2 μ 0 P 0

ψ^20

whereψ ̄=ψ(r,z)/ψ 0 .Show that if

P 0 =

μ^20 λ^2 ψ^20

4 π^2 μ 0 a^2

whereais the radius at which the pressure vanishes atz=0,theflux tube must

be axially uniform, i.e.,ψcannot have anyzdependence. Express this condition in

terms of the pitch angle of the magnetic fieldBφ/Bzmeasured atr=a. Hint: note

that the Grad-Shafranov equation has a both homogeneous part (left hand side) and

inhomogeneous part (right hand side). Show thatψ ̄=r^2 /a^2 is a particular solution

and then consider the form of the general solution (homogenous plus inhomogeneous

solution).

5. Lawson Criterion: Of the many possible thermonuclear reactions, the deuterium-
   tritium (DT) reaction stands out as being most attractive because ithas the largest
   reaction cross-section at accessible temperatures. The DT reactionhas the form

D+T−→n+He^4 +17.6MeV.

The output energy consists of 3.5 MeV neutron kinetic energy and 14.1 MeV alpha

particle kinetic energy. Fusion reaction cross-sections have a very strong dependence

on the impact energy of the reacting ions because of the Coulomb barrier. In order for

a controlled fusion reaction to be economically useful, more energy must begenerated

than is invested. The break-even condition is determined by equating theenergy input

per volume to the energy output for the reaction duration. The time duration that this

condition is maintained is called theenergy confinementtimeτEof the configuration,

a measure of the insulating capability of the plasma confinement device. Let all tem-

peratures and energies be measured in kilovolts (keV) so that the Boltzmann constant

isκ=1. 6 × 10 −^16 and consider a volumeVof reacting deuterium, tritium and asso-

ciated neutralizing electrons (the plasma needs to be overall neutral,otherwise there

will be enormous electrostatic forces).

(a) Show that if the electrons and ions have the same temperatureTthe energy

required to heat this volume isEin∼ 3 neVκTwherene is the electron density

and equal densities of deuterium and tritium are assumed so thatnD=nT=