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Hi all,<br>
Juliane and myself had a quite lengthy discussion about this issue
(it started yesterday and it is actually not finished yet ;-)) but I
just wanted you to know what would be our opinion up to now (to save
double effort in thinking). <br>
We think Ian had an interesting point here, although Thomas is right
about the correctness of the currently used term in a way as well.
Ian's term is derived from the momentum equation and exactly
balances out the viscous dissipation (since it is derived that way).
However, there is no guarantee that this term actually is(!) the
adiabatic heating in all material models / approximations, except
that it seems to be the right term for the boussinesq / anelastic
liquid approximation with a simple material model. <br>
The term that is currently used in Aspect on the other hand is
derived from the actual thermodynamic equations (dS/dt = ... the
root of our equation of conservation of energy as stated in "Mantle
convection in the Earth and Planets") and is therefore consistent,
but it uses at least one approximation, namely that the pressure is
only lithostatic and there is no dynamic pressure. The right term
(in a thermodynamic sense) would not be:<br>
<br>
<div style=""> Q_a = ( velocity * gravity ) * alpha * density *
temperature (1)<br>
<br>
but rather:<br>
Q_a = ( velocity * pressure_gradient ) * alpha *
temperature (2)<br>
<br>
The gravity and density in the currently used term (1) just came
out of the assumption that p = rho * g * h, and by the way, this
justifies the use of the real rho instead of the reference_rho in
the equation (1) although it does not seem to fit with the viscous
heating. The use of the above mentioned formula (2) reduces the
error in a test case (attached) from around 3% to 0.2% ... that is
not perfect considering the fit of Ian's term of 1e-13 or so.
Still I would currently rather use term (2) since it also makes us
independent of the choice of a right reference density and it is
also consistent in the compressible case (because it makes no
assumptions on the material model at all, it just needs the real
alpha und the local solution variables). Additionally with this
term we also capture the heating of material that moves laterally
along a pressure gradient and therefore also cools/heats up
adiabatically.<br>
Nevertheless, the misfit of 0.2% seems too large to be satisfied.
I first thought that the inconsistency between the definition of
alpha = - (1 / rho) * drho / dT = const and the definition of rho
= rho_ref (1 - alpha * (T - T_ref)) leads to the misfit and tried
a definition of rho = rho_ref * exp (- alpha * (T - T_ref)), which
is the solution of the definition of alpha above. But this does
not seem to make a large difference in the error. So currently we
are thinking in two ways:<br>
1. Find the reason for the misfit between thermodynamically
derived adiabatic heating term and viscous dissipation that is
computed out of the stokes velocity field.<br>
2. If we can derive that Ian's term actually is the adiabatic
heating term in a general sense (perhaps with an additional or
without an currently used approximation in term (2)), then of
course that term would be more precise, although then we end up
asking how to use this in a compressible model.<br>
<br>
Currently, I am a bit sceptical about the possibility of point 2.
therefore I am thinking of point 1.. The next part is speculation
but I find it a bit intriguing that Leng and Zhong (2008) also end
up with a difference of the order of 0.2 % although they do not
consider the formula (2) but rather Ian's term extended by an
compressible correction. Could it be that the remaining 0.2%
originate from an approximation that is not bound to the
formulation of the adiabatic heating term but rather something
influencing the velocity field (and therefore the viscous
dissipation)?<br>
<br>
<br>
</div>
By the way during this discussion we came across the formulation of
compressible stokes flow in aspect and think there is an
inconsistency as well. In the right-hand side of the stokes equation
we have the term rho*g as well as an additional compressible term.
Considering a compressible term in the compressible case seems
reasonable if the first term just covers the temperature dependency
of rho, but since in Aspect the material model takes care of all the
influences on the density, this term should be moved to any material
model that wants to be capable of doing compressible convection. For
example a material model could take the density values directly from
a thermodynamic calculation rather than a simplified equation of
state. In that case the current formulation would double-count the
pressure effect (without any chance for the user to know this, since
he thinks the material model need to take care of all the
influences).<br>
<br>
We are continuing working on this, but please let us know, if you
have an opinion or suggestion on this (and also in case something in
this is wrong ;-)),<br>
<br>
Cheers,<br>
Rene<br>
<br>
<br>
<pre class="moz-signature" cols="72">--
Rene Gassmoeller
2.5/Geodynamic Modelling
Tel.: +49 (0)331/288-28744
Fax: +49 (0)331/288-1938
Email: <a class="moz-txt-link-abbreviated" href="mailto:rengas@gfz-potsdam.de">rengas@gfz-potsdam.de</a>
___________________________________
Helmholtz Centre Potsdam
German Research Centre for Geosciences GFZ
Telegrafenberg, 14473 Potsdam</pre>
<br>
<br>
<div class="moz-cite-prefix">On 02/13/2013 01:21 AM, Magali Billen
wrote:<br>
</div>
<blockquote
cite="mid:C013EDB1-48F9-401C-ACA1-5B9B274B459C@ucdavis.edu"
type="cite">Hello All,
<div><br>
</div>
<div>Each of the different approximations of the equations have
specific sets of terms that drop in or out together,</div>
<div>so you need to be careful adding back in just one term or
another without taking into account which approximation</div>
<div>has leads to that term being assumed small. When people state
Boussinesq approximation in mantle convection </div>
<div>calculations, this has historically meant an approximation
that does not include adiabatic heating. Also, I agree with </div>
<div>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.</div>
<div><br>
</div>
<div>Speaking for myself, I don't have the specific approximations
memorized, however, I've found Chapter 6 of</div>
<div>"Mantle Convection in the Earth and Planets" by Schubert,
Turcotte and Olson, very helpful when trying to understand</div>
<div>the origin and loss of terms for boussinesq,
extended-boussinesq and TALA approximations - it goes through</div>
<div>each of these approximation in detail and explains the
specific assumptions for each approximation and which</div>
<div>terms drop out. I worked through all these once about a year
ago and I think it might help with this particular question.</div>
<div><br>
</div>
<div>Magali</div>
<div><br>
<div>
<div>On Feb 12, 2013, at 4:09 PM, Ian Rose wrote:</div>
<br class="Apple-interchange-newline">
<blockquote type="cite">
<div dir="ltr">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).
<div>
<br>
</div>
<div style="">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.</div>
<div style=""><br>
</div>
<div style="">Cheers,</div>
<div style="">Ian</div>
<div style=""><br>
</div>
</div>
<div class="gmail_extra"><br>
<br>
<div class="gmail_quote">On Tue, Feb 12, 2013 at 3:28 AM,
Thomas Geenen <span dir="ltr"><<a
moz-do-not-send="true"
href="mailto:geenen@gmail.com" target="_blank">geenen@gmail.com</a>></span>
wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="ltr">he Timo,
<div><br>
</div>
<div>there is no such thing as adiabatic heating in
the incompressible Boussinesq case Di (alpha*g/cp)
is assumed zero .</div>
<div>for extended Boussinesq there should also be no
problem since there is no density in the net
adiabatic heating term.</div>
<div><br>
</div>
<div>setting thermal diffusion, viscous dissipation
and internal heating to zero (dS/dt=0) we end up
with </div>
<div>rhocp(dT/dt) - alphaTdP/dt=0 </div>
<div>or <br>
</div>
<div>
rho*cp*(dT/dt) - alpha*rho*g*u_r*T=0</div>
<div><br>
</div>
<div>this will give for an adiabatic temperature
profile</div>
<div>T(r) = T_0*exp(alpha*g*r/cp) </div>
<div><br>
</div>
<div>iow the density does not play a role since its
devided out of the equation.</div>
<div><br>
</div>
<div>this also holds for the compressible case i
would say.</div>
<div><br>
</div>
<div>cheers</div>
<span class="HOEnZb"><font color="#888888">
<div>Thomas</div>
<div><br>
</div>
</font></span></div>
<div class="HOEnZb">
<div class="h5">
<div class="gmail_extra"><br>
<br>
<div class="gmail_quote">
On Tue, Feb 12, 2013 at 5:57 AM, Timo Heister
<span dir="ltr"><<a moz-do-not-send="true"
href="mailto:heister@math.tamu.edu"
target="_blank">heister@math.tamu.edu</a>></span>
wrote:<br>
<blockquote class="gmail_quote"
style="margin:0 0 0 .8ex;border-left:1px
#ccc solid;padding-left:1ex">
Hey everyone,<br>
<br>
Ian approached me about this and I asked him
to write it down here.<br>
Does anyone have any feedback about this,
especially (assuming this is<br>
correct), what to do in the compressible
case?<br>
<div>
<div><br>
On Wed, Feb 6, 2013 at 6:33 PM, Ian Rose
<<a moz-do-not-send="true"
href="mailto:ian.rose@berkeley.edu"
target="_blank">ian.rose@berkeley.edu</a>>
wrote:<br>
> Hi Aspect folks,<br>
><br>
> I was working through some tests
with Aspect and came across what I
believe<br>
> is an inconsistency in the
governing equations.<br>
><br>
> For incompressible Boussinesq flow,
the global viscous dissipation should<br>
> exactly cancel the global adiabatic
heating. This can be seen by<br>
> multiplying the momentum equation
by velocity and integrating over the<br>
> domain.<br>
><br>
> As it stands in assembly.cc, the
formula used for calculating adiabatic<br>
> heating is different from that you
would get by integrating the momentum<br>
> equation. I wrote a simple
postprocessor that compares the two
integrated<br>
> quantities which I am attaching.
The difference is quite a lot for the<br>
> current formula.<br>
><br>
> Put another way, this is the
formula that is currently used:<br>
><br>
> Q_a = ( velocity * gravity ) *
alpha * density * temperature<br>
><br>
> The density at this point however,
has already been adjusted for<br>
> temperature, so we are in effect
double counting the thermal expansion.<br>
> Instead, I believe it should be<br>
><br>
> Q_a = ( velocity * gravity ) * (
density - reference_density )<br>
><br>
><br>
> The compressible case, too, should
require some thought, though I have not<br>
> gone through the paces there.<br>
><br>
> Thoughts?<br>
><br>
> Best,<br>
> Ian<br>
><br>
> PS, for some details on the
derivations, I refer you to Leng and
Zhong<br>
> (2008)<br>
><br>
><br>
</div>
</div>
>
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<br>
--<br>
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<a moz-do-not-send="true"
href="http://www.math.tamu.edu/%7Eheister/"
target="_blank">http://www.math.tamu.edu/~heister/</a><br>
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