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Unformatted text preview: Phys761/F10lMidterm Take Home due Nov 16, 2010, in class Problem 1: Kelvin Helmholtz Instability (40 points) Instructions:
(1. Please show your work, but ALSO fill in the accompanying answer sheet; the numbers correspond
b. b“ you cant ﬁnd the numbers C and D, you can still continue with the problem
carrying these coeﬁcients thru the rest of the calculation; else, use the C and D that you ﬁnd. Just as an equilibrium in gravity could be unstable to gravitational potential energy
release, resulting in convection cells, a ﬂowing ﬂuid in equilibrium can be unstable to
release of ﬂow kinetic energy in the laminar ﬂow. This is the Kelvin-Helmholtz
instability and it is particularly potentlwhen there is an inﬂexion point in the ﬂow proﬁle. We investigate below the linear theory of the KH. Use the ideal neutral ﬂuid equations.
The problem draws on several ideas from Tepic 6 on the RT instability and the associated HW problem.
Start with an equilibrium with n=n0, p=p0, and u0:V(X)yA, ie, ﬂow along y sheared in x.
I. Show that the above satisﬁes the equilibrium equations. Perturb this equilibrium to look for a low frequency (subsonic) mode, in the same sense
that the RT mode is subsonic. Thus, we are looking for a mode in which p“ is “small”
and the dominant motion is a 2D convection Cell (in the x—y plane). Thus, start by taking
the 2 component of the curl of the momentum equation. We will assume perturbations of the form iw = tw (X) exp[iky — icot]. 2. Write down the equation obtained by taking the 2 component of the curl of the
linearized momentum equation. 3. Write down the linearized version of the pressure equation [use the form below].
-.J ~J —J \3
274% vet/trey tb’lbmu :0) 325/3 Suppose we suspect that the pm term in this equation is small. Accordingly, throw this
term away and state the condition that p” must satisjy in order that you can neglect it.
Upon making this ansatz, what is the resulting equation? Now check that this equation
and the equation from 2 aboveform a closed set. 4. Use the Eqﬂom 3 in the qurom 2 and obtain the eigenvalue equation satisﬁed by ax“.
Show that this equation is of the form given below and speciﬁ/ the coeﬁ‘icient C. 50% -—- ~— (anew/38¢ ) r72; aZ/axih where (o_bar = a) - kV(x). Now assume that V(X) is of the form shown in the Figure.
Note that at x=+a or x=—a there is a singularity in V’ ’. We will thus need jump
conditions. Obtain these as follows: i kg, , ‘ Vin)
ac '2. 5. Integrate over it the approximate Eqﬁom 3 about a small interval overlapping each of
the points Ix! =a and so obtain a jump conditions on ux’". You may assume that u},~ is at
worst piecewise discontinuous. 6. Rewrite the eigenvalue equation above so that all terms containing d/dx (on both
linear and equilibrium quantities) are on the LHS and all others on the RHS. Show that
the LHS can be manipulated and written as a total derivative, of the form (d/dx)[ ....... ]
:RHS. Thus, show that a jump condition involving the derivative 0}"ch~ is given by ._v\/ iot+e _- . [m ictH—e .2. ._._ D/l-t DQV‘R]
[Cally , _ ( ) X 1‘0?“— 5-
where prime means d/dx, ie, f ’ = df/dx. Speciﬁ/ the constant D. To solve the eigenproblern, one must now ﬁnd solutions separately in the 3 regions x < —a, —a < x < +a, +a < x. There-will be 4 unknown coefﬁcients needed in choosing
the solutions with the appropriate boundary conditions 7 you must use A, B, C, D for
these coefﬁcients. One must then apply the 4 jump conditions. This will result in 4
equations for the 4 unknowns. Do all this and ﬁnd the dispersion relation. For partial
credit, show the following results. 7. Apply the 2 jump conditions obtained in 5. 8. Apply the 2 jump conditions obtained in 6. 9. Obtain the final dispersion relation involving (0(k, V0). Plot (02 vs it (you may sketch
this or use Mathematica, etc) and identiﬁ} a range of instability. l 0. Find an analytic expression forthe growth rate for when kL << 1. (Beware: there ’s
cancellations; keep enough terms to. find theﬁrst non—zero term). ll. Check the condition that pH is small, as we did in part 3 above. Under what
conditions on V0 is the approximation valid? Problem 2: Firehose Instability (20 points) Collisionless plasmas in a strong magnetic ﬁeld can be shown, under certain conditions,
to be governed by the Chew—Goldbergeerow Equations. These are given below. The
CGL equations are characterized by the fact that a pressure tensor is needed and, while
the tensor only has diagonal terms nonzero, it is anisotropic in that the parallel (to B) pressure, p“, is different from the perpendicular pressure, p L. The full pressure tensor can
be written as P = n I + (le - puma/B2 where 1 is the unit tensor and BB is a dyadic. Alternatively, the tensor can be expressed
in Cartesian components as ij = pi5jk t (Pu - PilBjBk/BZ Note that the tensor is deﬁned anisotropic with respect to the local and instantaneous B,
ie, B = B(x,t). Consider an equilibrium with n=no iconstant, p i0 = pno : p0 = constant, and B = Bozn, B0
= constant. Subject this equilibrium to small perturbations, with the wavevector k only
in the 2A direction and a velocity perturbation n; being nonzero. Note that this
perturbation would give a shear Alfven wave (Topic 15) if we were dealing with the
usual ideal MHD equations. Find the frequency of oscillation for this wavemode. Show that under certain conditions, the mode can be unstable. State this instability condition. ...
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- Fall '09
- Plasma Physics