Physics 218B
Problem Set III:
Plasma Physics
Winter 2009
You should complete this assignment by March 2.
1.)
Consider a fluid in hydrostatic equilibrium with a vertical entropy gradient
!
!S !z < 0 . Take g = !gz .
a.)
Starting from the basic equations, derive the growth rate of ideal RayleighBernard instability. You will find it helpful to relate the density perturbation
!! ! 0 to the temperature perturbation T! T 0 by exploiting the fact that the
instability develops slowly in comparison to the sound transit time across a cell.
Relate your result to the Schwarzschild criterion discussed in class.
b.)
Now, include thermal diffusivity
(!)
and viscosity ( ! ) in your analysis.
Calculate the critical temperature gradient for instability, assuming ! ~ " .
Discuss how this compares to the ideal limit. What happens if ! >" ?
2.)
Now again, consider the system of Problem 2, now immersed in a uniform
!
magnetic field B = B 0z .
a.)
Assuming ideal dynamics, use the Energy Principle to analyze the stability of a
convection cell of vertical wavelength k z . Of course, k z L p >> 1 , where L p is a
mean pressure scale length. What is the effect of the magnetic field? Can you
estimate how the growth rate changes?
b.)
Now, calculate the growth rate using the full MHD equations. You may
assume ! " V = 0 . What structure convection cell is optimal for vertical transport
of heat when B 0 is strong? Explain why. What happens when B 0 ! " ?
Congratulations - you have just derived a variant of the Taylor-Proudman
theorem!
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Physics 218B
Plasma Physics
Winter 2009
c.)
Compare and contrast this interchange instability to an incompressible RayleighTaylor instability. Make a table showing the detailed correspondences.
4.)
Kulsrud, Chapter 5, Problem 2
5.)
a.)
Derive the quasilinear equation for the evolution of the pressure profile in
a plasma which supports resistive interchange turbulence. Express your
2
answer in terms of the radial displacement spectrum !˜ r k .
b.)
c.)
Assuming localized heating with on-axis central deposition, derive an
expression for the stationary pressure gradient in terms of the intensity
profile of the resistive interchange displacement spectrum.
Using the wave-breaking estimate for saturation level (!˜ r ~ k r"1) , calculate
how ! scales with input power, assuming fixed density. When does this
scaling fail? Why?
6.)
This problem asks you to explore the Current Convective Instability (CCI) in a
homogeneous medium and its sheared field relative, the Rippling Instability.
a.)
Consider first a current carrying plasma in a straight magnetic field
B = B0 zˆ - i.e. ignore the poloidal field, etc. Noting that the resistivity ! is
a function of temperature (ala' Spitzer - c.f. Kulsrud 8.7), calculate the
electrostatic resistive instability growth rate, assuming T evolves
according to:
!T
+ v "#T $ % || ! 2z T $ % & #2& T = 0
!t
and the electrostatic Ohm's Law is just
! B0 " z # =
1 d$ ˜
T ($ J 0 ) .
$ dt
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Physics 218B
b.)
Plasma Physics
Winter 2009
Thoroughly discuss the physics of this simple instability, i.e.
- what is the free energy source?
- what is the mechanism?
- what are the dampings and how do they restrict the unstable spectrum?
- how does spectral asymmetry enter?
- what is the cell structure?
c.)
Use quasilinear theory and the wave breaking limit to estimate the heat
flux from the C.C.I.
d.)
Now, consider the instability in a sheared magnetic field.
i.)
ii.)
What difficulties enter the analysis?
Resolve the difficulty by considering coupled evolution of
vorticity, Ohm's Law (in electrostatic limit but with temperature
fluctuations) and electron temperature. Compute the growth rate in
the limit ! || , " ! # 0 . Compute the mode width. Discuss how
asymmetry enters here. Explain why.
e.)
Noting that ! || >> ! " (why? - see Kulsrud 8.7), estimate when parallel
thermal conduction becomes an important damping effect. Can ! || alone
ever absolutely stabilize the rippling mode?
f.)
7.)
Calculate the quasilinear heat flux and use the breaking limit to estimate
its magnitude.
Read K.V. Roberts and J.B. Taylor, Physics of Fluids, Vol. 8, pg 315, 1965. Write
a short essay (including cartoons and schematic equations) explaining the "point"
of this paper. Contrast the "twisted slicing modes" it describes with the resistive
interchange mode discussed in class.
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Physics 218B
8.)
Plasma Physics
Winter 2009
Taylor in Flatland
Taylor awakes one morning, and finds himself in Flatland, a 2D world. Seeking
to relax, he sets about reformulating his theory for that planar universe.
a.)
Write down the visco-resistive 2D MHD equations, and show that three
quadratic quantities are conserved, as ! " 0 , ! " 0 .
b.)
Which of these is the most likely to constrain magnetic relaxation? Argue
that
i.)
ii.)
the local version of this quantity is conserved for an 'flux circle', as
! " 0,
the global version is the most "rugged", for finite ! .
c.)
Formulate a 2D Taylor Hypothesis - i.e. that magnetic energy is
minimized while the quantity you identified from b.) ii.) is conserved.
What equation describes this state? Show that the solution is force-free.
What quantity is constant in Flatland? Hence, what is the endstate of
Taylor relaxation in 2D?
d.)
Consider the possibility that ! >> " in Flatland. Derive the mean field
evolution equation for mean magnetic potential. Discuss!
9.)
e.)
Optional - Extra Credit - Describe the visit of the Terrifying Torus to
Flatland. How would 2D Taylor perceive this apparition?
N.B. You may find it useful to consult Flatland, by E. Abbott.
a.)
Carefully calculate the radial diffusion coefficient Dm for stochastic
magnetic fields, as discussed in class. Be sure your answer includes the:
i.)
identification of resonances,
ii.)
identification of all length scales,
iii.)
a clear statement of all dependencies.
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Physics 218B
b.)
Plasma Physics
Winter 2009
Recover the result from a.) in a limiting case of the radial quasilinear
diffusion coefficient for
!f
B
+ v||
" # f = 0,
!t
B0
where B = B0 + B˜ .
c.)
Discuss the consequences of stochastic fields for Taylor relaxation theory
- why is it plausible to associate the validity of the Taylor hypothesis with
magnetic stochasticity?
d.)
By analogy with the calculation of the shear dispersion rate, calculate the
length for dispersion of two magnetic field lines. Note q = q( r ) couples
radial scattering with dispersion on angle. Show that the length is:
1
~ ( k!2 Dm L 2s ) ,
1/3
!d
where q! = dq dr , 1
e.)
Ls
= rq ! q 2 R .
What happens when particle motion along the (stochastic) magnetic field
line is diffusive, i.e. when !mfp < !d ? Show the outcome in as simple a
way as possible. You may find the article by Rechester and Rosenbluth
(posted) to be useful.
10.)
a.)
Show that fluid helicity is conserved for inviscid incompressible
hydrodynamics, i.e.
d 3
! d x[ v " # ] = 0 .
dt
b.)
How does finite viscosity affect fluid helicity balance?
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Physics 218B
Plasma Physics
Winter 2009
c.)
Give a pictorial interpretation of fluid helicity in terms of linked vortex
lines.
d.)
What kind of state would develop if enstrophy were to decay for constant
global fluid helicity? In what sense is this vortex "force free"? Discuss
your result.
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