10.9 Assignments 335
particular, indicate by shading regions whereF< 0 .Do this for bothm=0and for
m=− 1 so as to show the regions of the parameter space where kinks and sausages
are unstable. For a givenkshow on the plot how raising the axial current would lead
to kink or sausage instability. Which happens first (kink or sausage)?
- Are kink instabilities diamagnetic or paramagnetic with respect to the axial field?
To find this, consider the kinked current as a solenoid and determine whether the
orientation of the solenoid is such as to add or subtract from the initialBzfield.
- According to Eq.(10.139) current driven instabilities cause the plasma to relax to a
situation whereμ 0 J=λB.
(a) Show that a class of solutions to this can be found if it is assumed thatBhas the
form
B=λ∇ψ׈z+∇×(∇ψ×zˆ)
provided∇^2 ψ+λ^2 ψ=0.
(b) Suppose thatψ(x,z) =f(z)cos(kx).What is the form off(z)?Show that
solutions which decay inzare only possible for a certain range ofλ^2.
(c) Consider a two dimensional force-free solar coronal loop. Let thez=0plane
denote the solar surface. Calculate the functional form ofBxandBzusing the
solution in part (a) above.
(d) Sketch the shape of the field line going fromx=−π/ktox=+π/kin the
z=0plane. Do this for a sequence of increasing values ofλ^2 .What happens to
the shape of the field line asλ^2 is increased? How is current related toλ?
