9.10 Assignments 295can be imagined as a sort of “zipper” which collimates theflux surfaces. Collimation is
often seen in current-carrying magneticflux tubes (e.g., see Hansen and Bellan (2001) and
is an important property of solar coronal loops (Klimchuk 2000) and of astrophysical jets
(Livio 1999)).
9.10 Assignments
- Show that if a Bennett pinch equilibrium has aJzindependent ofr,then the az-
imuthal magnetic field is of the formBθ(r)= Bθ(a)r/a. Then show that if the
temperature is uniform, the pressure will be of the formP =P 0 (1−r^2 /a^2 )where
P 0 =μ 0 I^2 / 4 π^2 a^2. Show that this result is consistent with Eq.(9.29). - Show that it is impossible to confine a spherically symmetric pressure profile using
magnetic forces alone and therefore show that three dimensional magnetostatic equi-
libria cannot be found for arbitrarily specified pressure profiles. To do thissuppose
there exists a magnetic field that satisfiesJ×B=∇PwhereP=P(r)andris the
radius in spherical coordinates{r,θ,φ}.
(a) Show that neitherJnorBcan have a component in theˆrdirection. Hint: assume
thatBris finite and write out the three components ofJ×B=∇Pin spher-
ical coordinates. By eliminating eitherJrorBrfrom the equations resulting
from theθandφcomponents, obtain equations of the formJr∂P/∂r=0and
Br∂P/∂r=0.Then use the fact that∂P/∂ris finite to develop a contradiction
regarding the original assumption regardingJrorBr.
(b) By calculatingJθandJφfrom Ampere’s law (use the curl in spherical coordi-
nates as given on p.523), show thatJ×B=∇PimpliesBθ^2 +Bφ^2 =B^2 (r).Ar-
gue that this means that the magnitude ofB=Bθˆθ+Bφφˆmust be independent
ofθandφand thereforeBθandBφmust be of the formBθ=Bsin[η(θ,φ,r)]
andBφ=Bcos[η(θ,φ,r)]whereηis some arbitrary function.
(c) UsingJr=0,Br=0 and∇·B=0(see p. 523 for the latter) derive a pair
of coupled equations for∂η/∂θand∂η/∂φ.Solve these coupled equations for
∂η/∂θand∂η/∂φusing the method of determinants (the solutions for∂η/∂θ
and∂η/∂φshould be very simple). Integrate the∂η/∂φequation to obtainηand
show that this solution forηis inconsistent with the requirements of the solution
for∂η/∂θ, thereby demonstrating that the original assumption of existence of a
spherically symmetric equilibrium must be incorrect. - Using numerical methods solve for the motion of charged particles in the Solov’ev
field,B=∇ψ×∇φwhereψ=B 0 r^2 (2a^2 −r^2 −4(αz)^2 )/a^4 whereB 0 ,aand
αare constants. Plot the surfaces of constantψand show there are open and closed
surfaces. Select appropriate characteristic times, lengths, and velocities and choose
appropriate time steps, initial conditions, and graph windows. Plot thex,yplane and
thex,zplane. What interesting features are observed in the orbits. How do theyrelate
to theflux surfacesψ=const.What can be said aboutflux conservation? - Grad-Shafranov equation for a current-carryingflux tube:The Grad-