||In problem 6-6, we are told the
ions are a "fixed background". Does this mean that v_i0 = v_i1 =
0 ? I was assuming that v_i0 = 0 , but that the ions had a
nonzero velocity perturbation. This doesn't seem to match up with
what Chen did in the back-of-the-book solution.
||Fixed means exactly that.
||I'm still very confused on
problem (2). I got Pmhd - (Pi + Pe) into a
relatively decent-looking form that includes things like u_-,i*u_-,j and u_-,i*u_+,j But now I'm not sure what to do!
First, I can't remember if there's anything special I should know about what to do when you have things like b_i*a_j since I've never had a good reference for how to do index notation -- I've only ever learned bits of it on the fly, like if it were b_i*a_i, I'd know that was a dot product.
Second, I THINK I want to take the gradient of Pmhd - (Pi + Pe) because that IS what appears in the equation of motion. and compare that to the equation of motion (or of continuity?), but I don't have a lot of confidence there. Am I on the right track?
|You want to show that the MHD
equation of motion is valid for arbitrary (including large) speeds.
See Lea Ch 1.
Review the "formal derivation of the fluid equations" especially pages 7 and 8 of the fluids notes.
Look at page 1 of the MHD notes. We added two eqations for electrons and ions to get the single-fluid MHD eqn of motion. But Pe + Pi is NOT QUITE Pmhd because the average velocity with respect to which the random velocity is computed is not quite the same for each fluid. And we also have to deal with the non-linear (v dot grad)v terms. Your goal is to show that when everything is done correctly we get equation 7 (in MHD notes) EXACTLY.
||In question 2, do we need to take into account recombination andionization for the continuity equation? If not, it seems like the continuity equation in part (a) is just dn/dt = D * grad^2(n). Am I underthinking this?||No
Start with the basic equation in terms of v, and use the equation of motion including E to get v.
||Problem 2 tells us to assume T=0. Should we also assume that in problem 1 for the plasma oscillations?||No. Here you can use the
full relation for Langmuir waves. Do take T=0 for the EM waves
||For this w-k plot:
Pg 145 summarizes the waves we have studied, and includes the dispersion relations. However, there are tons of variables in them besides k... plasma frequencies, alfven velocities, thermal velocities. How do I plot these? I can call them all 1 and plot only the qualitative shape of w(k) but I doubt that is what you are looking for.
|You will need to take some
values for ratios such as wp/wc. Actually
I think that is the only value you need. The speeds like vA
and vs are less critical since they relate to slopes of
lines in the w(k) plot. Here you can be qualitative.
||I'm confused about the signs of
the x-hat and y-hat components of E inthe equations at the bottom of
page 25 of the notes.
As we discussed in your office today, if ER and EL both point in the x-hat direction at t=0, a short time after that, Re(ER) should have positive x-hat and y-hat components, while Re(EL) should have a positive x-hat component and a negative y-hat component. That seems to imply that
ER = E0R * (+x-hat - i*y-hat) * exp(i*kR*z)
Re(ER) = E0R * [x-hat*cos(kR*k) - i*y-hat*(i*sin(kR*z))]
= E0R * [x-hat*cos(kR*k) + y-hat*sin(kR*z)]
EL = E0L * (+x-hat + i*y-hat) * exp(i*kL*z)
Re(EL) = E0L * [x-hat*cos(kL*k) - y-hat*sin(kL*z)]
1) That's opposite what you have in your notes
2) If I do it that way, I ultimately end up with a negative rotation angle which is not what I expect since v_phi is faster for the R wave than for the L wave
|You are forgetting that the
phase is actually kz MINUS omega t , so as t imcreases at fixed z the
phase decreases and so the E vector rotates counter-clockwise.
||In 4.10, a<>re we assuming
_transverse_ EM waves for this problem, since that's what section 4.12
I'm also confused in part b about what type of ion waveswe're supposed to find. I'm presuming ion waves perpendicular to B as in section 4.10, but the problem is not clear.
|Well, what does he ask you to
do? Note that (a) and (b) are different.
||I'm not quite sure what's being
asked in problem 2-16. I'm guessing
that we want to prove that w_c << w, but I'm not quite sure why.
change will be adiabatic if the heating itself is slow compared to
w_c, but I guess I'm not quite sure why it would be OK to directly
relate the frequency of the heating wave to the speed of heating since
it seems like that would depend on other factors, like the density.
||Remember that I discussed in
class how the word "adiabatic" is used in plasma physics?
Adiabatic invariance holds when the system changes slowly. So the important
question here is whether the applied
magnetic field is changing slowly. For the 1st adiabatic
invariant (the magnetic moment) the relevant timescale for comparison
is the gyration period. Context is everything! Even though
the system is being heated, we are not using adiabatic in the
||In the first problem, are the energies given just kT or are they the kinetic energies of the particles (3/2 kT)?||The wording is certainly ambiguous, but I would take it to be the total KE.|
||Chen 2-13. I'm
uncomfortable here with the presence of the dv_parallel/dt term,
since it seems like, from the previous problem, we're treating the
changes in v_parallel as instantaneous at the bounce, so v_// is
really a step function and the derivatives won't be nice. Are we
assuming here that v_parallel is changing gradually?
Also, more basically, my first instinct was to say T = L/v_parallel, but I didn't like that since v_parallel is changing in time (so is L, but less dramatically), so dimensionally I think I want T = v_parallel / (dv_parallel/dt) = L / (dL/dt) but I'm having a hard time justifying it beyond "it seems to have fewer problems than the 1st expression."
|Aah, the joys of
estimation! But this is a realy valuable skill to learn.
From dJ/dt you get an
expression that equals 0. L
and dL/dt are
straightforward. You are right to focus on the vparallel
terms as the problems. Since vparallel is changing,
you will need to pick an appropriate average value. Similarly,
we'll need an appropriate average for the derivative too. You are
right that the actual derivative is actually zero (most of the
time) or very large for a very short time. But check the
definition of average acceleration in LB Ch 2 to see how to get a
reasonable estimate of this quantity. Somewhere in all this the
time T that you want will
appear, and so you can solve for it.
||In 2.8a, are we meant to compute
the velocities in the equatorial
plane only, or as a function of the polar angle?
||The given information is "in the
equatorial plane", so that's all you can reasonably be expected to do.
||Chen 1.5 . Are we able to assume that e*phi/kT << 1?||Yes. The derivation should
closely follow what we did in class (but this time in 1-D as in Chen
||P1.3. We can approximate
n~ne for plasma, right?
Are we to plot n vs kt or just use them as x and y ranges? I'm confused because I can't get Lambda D and ND in the same plotting window. My values for Lambda D are in the order of 10^17 and for ND, 10^50s. The expressions for Lamda D and ND are straight forward (pg. 10 &11). I'm using the n and kt value ranges given (intervals of 10).
The axes of the plot will be n and kT. Given the huge range of values, what kind of plot will it be? Each plasma will be a point in this plot. Lines of constant lambdaD will be lines in your plot. Your values for ND and lambdaD are much too big. Make sure that you are using the correct units.
||P1.3. For this question,
it asks us to "satisfy ourselves" that these things
are plasmas. I can look at the condition ln(N_D) >> 1 easily enough,
but lambda_D << L is harder since the length scales are not provided.
I can guess for some of them, but I don't even know what a Cs plasma
or a pinch fusion experiment is! How should I handle this?
|Well, it should be clear that these are lab systems, and that should be enough.|