Non-equilibrium of atmospheres stratified transverse to the direction of g
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1.  Consider a gravitational field where g = -gx^.  Show from the full 
ideal fluid eqns that the only possible static equilibria are all 
stratified in the g-direction, ie, the variables p and n (and so T) cannot 
be functions of y and z.  This says that it is not possible to have one 
side of the room hotter than the other side in static equilibrium.

2.  Confirm the above conclusion using the small-g reduced ideal fluid 
eqns.

3.  Suppose initially dp/dx = - nMg (equivalent to p approximately 
constant in small g), ie hydrostatic balance, but T=T(x,y) in a room of 
dimensions L = 5m. Deduce that convection will be driven because of force 
imbalances and find the time scale over which the convection will mix up 
the temperatures and densities.  Do this by scaling the reduced g force 
equation and, assuming time scales tau ~ L/|u|, calculate the 
order-of-magnitude time scale on which convection will mix up the initial 
condition.