[aspect-devel] Adiabatic conditions

[Wolfgang Bangerth]

All,
I need some help from geophysicists. Before the first time step, we 
compute adiabatic pressures and temperatures. I'm a bit confused by the 
equations we need to solve. Would someone mind taking a look at the 
current version of the manual here:
   http://dealii.org/svn/aspect/trunk/aspect/doc/manual.pdf
Given the basic equations in section 2.1, I can derive equations for 
adiabatic conditions by assuming a volume of rock that is sent to depth 
at a velocity u=[0,0,-1]. I then get

   rho C_p  dT/dz  =  drho/dT T g_z

These are now in section 2.4. Is this in principle the correct formula?


On the other hand, the original equation does have a term of the form
   2 eta (eps(u) - 1/3(div u)1) : (eps(u) - 1/3(div u)1)

For my assumed velocity field, eps(u)=0, but because
    div(rho u) = 0
I have
    div u = -1/rho u.grad rho
and therefore
    div u = 1/rho (drho/dp dp/dz + drho/dT dT/dz)
which I could enter into the extra term in the temperature equation to 
get something of the kind

   rho C_p  dT/dz  =  drho/dT T g_z
                      + 2/3 eta/rho^2  (drho/dp dp/dz + drho/dT dT/dz)^2

for the adiabatic temperature field.

So my question: what is the correct approach?

Best
  Wolfgang
-- 
------------------------------------------------------------------------
Wolfgang Bangerth               email:            bangerth at math.tamu.edu
                                 www: http://www.math.tamu.edu/~bangerth/