[aspect-devel] Thermodynamic consistency of Aspect's temperature and momentum equations

Tue Feb 12 16:21:48 PST 2013

Hello All,

Each of the different approximations of the equations have specific sets of terms that drop in or out together,
so you need to be careful adding back in just one term or another without taking into account which approximation
has leads to that term being assumed small. When people state Boussinesq approximation in mantle convection 
calculations, this has historically meant an approximation that does not include adiabatic heating. Also, I agree with 
Thomas that it doesn't really make sense to have such a term in an incompressible convection case, since without compression there is no physical cause for an adiabatic gradient.

Speaking for myself, I don't have the specific approximations memorized, however, I've found Chapter 6 of
"Mantle Convection in the Earth and Planets" by Schubert, Turcotte and Olson, very helpful when trying to understand
the origin and loss of terms for boussinesq, extended-boussinesq and TALA approximations - it goes through
each of these approximation in detail and explains the specific assumptions for each approximation and which
terms drop out. I worked through all these once about a year ago and I think it might help with this particular question.

Magali

On Feb 12, 2013, at 4:09 PM, Ian Rose wrote:

> Hmm, I am not sure I agree.  Di is frequently assumed to be zero in mantle convection problems, but that is not a result of the Boussinesq approximation.  That is to say, the "work done by gravity" term in the kinetic energy equation arises just fine with Boussinesq (there are just more terms that come from the div velocity terms in the compressible case).  
> 
> Even though this term (along with viscous dissipation) are likely to be smallish, I see no reason not to allow them to be turned on and off with flags as they are now.  But if it is turned on, it should be consistent with what you get from integrating the momentum equation.
> 
> Cheers,
> Ian
> 
> 
> 
> On Tue, Feb 12, 2013 at 3:28 AM, Thomas Geenen <geenen at gmail.com> wrote:
> he Timo,
> 
> there is no such thing as adiabatic heating in the incompressible Boussinesq case Di (alpha*g/cp) is assumed zero .
> for extended Boussinesq there should also be no problem since there is no density in the net adiabatic heating term.
> 
> setting  thermal diffusion, viscous dissipation and internal heating to zero (dS/dt=0) we end up with 
> rhocp(dT/dt) - alphaTdP/dt=0 
> or 
> rho*cp*(dT/dt) - alpha*rho*g*u_r*T=0
> 
> this will give for an adiabatic temperature profile
> T(r) = T_0*exp(alpha*g*r/cp) 
> 
> iow the density does not play a role since its devided out of the equation.
> 
> this also holds for the compressible case i would say.
> 
> cheers
> Thomas
> 
> 
> 
> On Tue, Feb 12, 2013 at 5:57 AM, Timo Heister <heister at math.tamu.edu> wrote:
> Hey everyone,
> 
> Ian approached me about this and I asked him to write it down here.
> Does anyone have any feedback about this, especially (assuming this is
> correct), what to do in the compressible case?
> 
> On Wed, Feb 6, 2013 at 6:33 PM, Ian Rose <ian.rose at berkeley.edu> wrote:
> > Hi Aspect folks,
> >
> > I was working through some tests with Aspect and came across what I believe
> > is an inconsistency in the governing equations.
> >
> > For incompressible Boussinesq flow, the global viscous dissipation should
> > exactly cancel the global adiabatic heating.  This can be seen by
> > multiplying the momentum equation by velocity and integrating over the
> > domain.
> >
> > As it stands in assembly.cc, the formula used for calculating adiabatic
> > heating is different from that you would get by integrating the momentum
> > equation.  I wrote a simple postprocessor that compares the two integrated
> > quantities which I am attaching.  The difference is quite a lot for the
> > current formula.
> >
> > Put another way, this is the formula that is currently used:
> >
> >    Q_a = ( velocity * gravity ) * alpha * density * temperature
> >
> > The density at this point however, has already been adjusted for
> > temperature, so we are in effect double counting the thermal expansion.
> > Instead, I believe it should be
> >
> >   Q_a = ( velocity * gravity ) * ( density - reference_density )
> >
> >
> > The compressible case, too, should require some thought, though I have not
> > gone through the paces there.
> >
> > Thoughts?
> >
> > Best,
> > Ian
> >
> > PS, for some details on the derivations, I refer you to Leng and Zhong
> > (2008)
> >
> >
> > _______________________________________________
> > Aspect-devel mailing list
> > Aspect-devel at geodynamics.org
> > http://geodynamics.org/cgi-bin/mailman/listinfo/aspect-devel
> 
> 
> 
> --
> Timo Heister
> http://www.math.tamu.edu/~heister/
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> 
> 
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