Rayleigh Taylor in exponential profile
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Consider an equilibrium density striation given by n(x) = n0 exp(x/L), 
where n0, L are constants.   There is a gravitational field g pointing in 
the negative x direction.   For the purposes of this problem, L may be 
positive or negative.  Assume that g/L << (cs/L)^2, cs being the sound 
speed.

Investigate the stability of this profile to the RT instability. In 
solving for the eigenfunction, you may assume that the perturbed 
quantities do not have to be localized in x and may be nonzero at |x| 
large (provided they do not blow up).  Find the dispersion relation. 
From this, answer the following:

1.  Suppose the convection cells are approximately circular in some 2D 
plane but that the radius of the cells is << L, ie, kL >> 1 where k is the 
typical wavenumber.  Under what conditions on L is the system unstable? 
What is the approximate growth rate?

2.  Suppose the cells are still approximately circular but L is small in 
comparison, ie, kL << 1.  Under what conditions on L is the system 
unstable?  What is the approximate growth rate?  Is there anything else 
peculiar about the time behaviour of the instability here?

3.  Suppose L > 0.  Draw a picture of the turbulence scales you might 
expect to see on a scale including L.  Depict how this picture changes if 
L < 0?  (Assume there is always some short wavelength cutoff.)