Fundamentals of Plasma Physics

334 Chapter 10. Stability of static MHD equilibria

10.9 Assignments

1. Interchange instabilities and volume per unitflux- Another way to consider inter-
   change instabilities is to calculate the consequence of interchanging twoflux tubes
   having the same magneticflux. Doing so will not change the magnetic energy since
   the magnetic field is unchanged by this interchange. However, if theflux tubes con-
   tain finite-pressure plasma and the volumes of the twoflux tubes differ, then the inter-
   change will result in compression of the plasma in theflux tube which initially had the
   larger volume and expansion of plasma in theflux tube which initially had the smaller
   volume. The former requires work on the plasma and the latter involves workby the
   plasma. If the net work must be done on the plasma to effect the interchange, then
   the interchange is stable and vice versa. In a magnetic confinement configuration, the
   high pressure is by assumption on the inside of the configuration and the low pressure
   is on the outside. Thus, the question is whether interchanging a high pressure,inner
   regionflux tube with a low pressure out regionflux tube requires positive or negative
   work.
   (a) Show that the volume per unitflux in aflux tube is given by

V′=

∮

dl

B

where the contour is over the length of theflux tube. Hint: Consider thatBA=

const.on aflux tube.

(b) Show that instability corresponds toV′increasing on going outwards.

(c) Consider the magnetic field external to a current-carrying straight wire. How

doesV′scale with distance from the wire and would a plasma confined by such

a magnetic field be stable or unstable to interchanges. Is this result consistent

with the concepts of good and bad curvature?

2. Work through the algebra of the magnetic energy principle and verify Eqs.(10.113),
   (10.114), (10.115), and (10.116).


3. Show that the force on a plasma in anarched magnetic fieldwith fixed ends tends
   to push the plasma towards the shape of a vacuum magnetic field arch havingthe
   same boundary conditions. Under what circumstances will the force (i) increase the
   magnetic field arch major radius, (ii) decrease the magnetic field archmajor radius,
   (iii) leave the magnetic field arch major radius unchanged. Show that, incontrast, the
   force on a plasma containing anarched currentalways tends to increase the major
   radius of the arched current.


4. Show that if the wall radiusb→∞,then Eq.(10.174) reduces to the condition

F(x,B ̄ 0 vz,P ̄ 0 )=xB ̄pvz^2

[

Im(x)

I′m(x)

]

−

(m+xB ̄ 0 vz)^2

x

Km(x)

Km′(x)

− 1 > 0 for stability

wherex=ka.Assume thatB ̄pvz=B ̄ 0 vzand use a computer to evaluate the modified

Bessel functions numerically. Make a plot of this expression in a parameter space

where the vertical axis isxand the horizontal axis isB 0 vθ/B 0 vz=1/B ̄ 0 vzand in