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

[Ian Rose]

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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