Physics 218B
Problem Set IV:
1.)
a.)
Plasma Physics
Winter 2009
Due Thursday, March 19
There are six problems. Choose three.
Extra credit will be given for extra problems.
An internal wave is a gravity wave which is the stable counterpart of the
interchange.
For g = !gzˆ , g > 0 and motion combined to two dimension s( x, z ) with ! " V = 0 ,
show that the internal wave dispersion relation is:
!2 = "
g d # 0 $ k 2x '
&
).
# 0 dz % k 2z + k 2x (
Derive the vertical and horizontal group velocities. Discuss your result. What are
the effects of viscosity and diffusion on the frequency?
b.)
Assume that wave-wave interactions maintain a spectrum of IW's against some
excitation rate ! k . Take N ( k,t ) as evolving slowly. Derive the mean-field
equation for the horizontal mean flow V x ( z ) . This should determine the mean
flow as evolving from a competition between wave Reynolds stress V˜ z V˜ x and
viscosity. Relate the Reynolds stress to the wave population density N .
c.)
As with the Langmuir turbulence problem, calculate the modulational stability of
a weak test shear. Derive the growth rate ! q . What conditions on N are
required for growth of the shear flow? [Hint: Modulate the Reynolds stress, like
we modulated the radiation pressure in Langmuir turbulence.]
2.)
A hot surface produces thermal convection above it. The convection is turbulent,
and causes a fixed net heat flux Q, upward.
a.)
By balancing turbulent dissipation with buoyant production, estimate a typical
2
2
turbulent velocity V T . Calculate V T ( z ) and T˜ ( z ) .
b.)
Using mixing length arguments, determine the vertical profile of the mean
temperature T ( z ) .
1
Physics 218B
Plasma Physics
Winter 2009
3.)
Kulsrud, Chapter 5, Problem 2.
4.)
In familiar quasilinear theory, the most physically appealing proof of energy
conservation is that which balances resonant particle kinetic energy density with
wave energy density.
a.)
State the counterpart at this theorem for Langmuir turbulence. Assume a broad
spectrum of sound waves is present. Carefully explain what the 'waves' and
'resonant particles' correspond to in this case.
b.)
Prove the theorem you stated in Part a.).
5.)
Consider magnetic buoyancy interchange instabilities as discussed in class.
Assume entropy stratification is neutral, so d S 0 dz = 0 . Take ! small, but nonzero.
a.)
Use quasilinear theory to calculate the vertical flux of magnetic intensity. Since,
! ~ " # z !n( B $ ) , show that ! may be written as
! = "D
# B
+V B .
#z
Calculate D, V. Interpret your result. For ! = ! 0 ( z ) . What profile corresponds to
the zero flux state.
b.)
What is the origin of the pinch velocity V? Explain its significance.
c.)
As a related example, consider evolution of the particle density according to
!n !t + " # ( nV) = 0 .
Take n0 = n0 ( x ) , B = B0 ( x )zˆ and V = ! ( v"xzˆ ) B0 ( x ) .
Show that density evolution can be related to the incompressible advection of the
field n B :
!n !t + Veff " #( n B) = 0
2
Physics 218B
Plasma Physics
Winter 2009
where ! " Veff = 0 .
Show that the mean field equation for n obeys:
!
!
! " n %
n =
D $
'
!t
!x !x # B &
where we took n B ! n
B . Discuss the zero flux state here. What are its
implications for the density profile?
Re-write the mean field equation as
! n
! #
! n
&
= " % "D
+ V n (.
!t
!x $
!x
'
Relate D and V, here. Under what circumstances will V be inward, i.e. up the
density gradient?
d.)
Relate the results of parts b.), c.) here. What is the lesson?
Congratulations! You have just developed the basics of TEP pinch theory!
6.)
A passive scalar c is advected in a turbulent flow with outer scale v0 , !0 ,
according to
!c
+ v " # c $ D #2 c = s˜ .
!t
Here s˜ is some random source of c, with correlation time ! s .
a.)
Compute the self-similarity range spectrum
!c ( !) . Calculate the
2
dissipation scale of c, and compare it to !d .
b.)
Discuss the dynamics of !c ( !)
2
spectrum look like in each case?
3
for D < ! , D > ! .
What does the