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

[Thomas Geenen]

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)
> >
> >
> > _______________________________________________
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> > 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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