[aspect-devel] Help needed with setting up an EBA benchmark

[Katrina Arredondo]

I study EBA and BA in CitcomS. EBA should be a strictly defined set of
equations so we simply need a dimensional form of the nondimensional
equations defined in the benchmark. Why do we need them in non-dimensional
from?

EBA is essentially the same as TALA but incompressible (I think?). If there
are other differences, they would be well defined in the Schubert and
Turcotte book "Mantle Convection in the Earth and Planets."

Starting from a dimensional BA, you would add three terms to the energy
equation to form EBA: the adiabatic heating, viscous dissipation and latent
heat terms. A simple switch should be fine.

I can help with this if needed. Is the problem translating the
nondimensional terms into a dimensional form that can be used in ASPECT?

 - Katrina


On Thu, Sep 4, 2014 at 4:39 PM, Wolfgang Bangerth <bangerth at tamu.edu> wrote:

>
> All,
> I need a bit of help from people who understand the nondimensionalization
> procedure in geodynamics better than I do. I want to set up the Davies et
> al. benchmark 3.1 outlined in
>
> https://github.com/bangerth/aspect/blob/more-davies-
> benchmarks/benchmark/davies_et_al/Cylindrical_Benchmark.pdf?raw=true
>
> It uses the extended Boussinesq approximation (EBA) that includes
> adiabatic heating and friction heating terms. The temperature equation
> stated in this document (equation (17), p. 3) looks like this:
>
> [1]
>   cp DT/Dt  -  Di alpha g . u (T+T0) - div(k grad T) = Di/Ra Phi
>
> where
>   Phi = 2 eta eps(u):eps(u)
> is the friction heating term. The problem, of course and as always, is
> that the benchmark is stated in terms of Dissipation and Rayleigh numbers
> Di and Ra, instead of the physical quantities we use in ASPECT. In
> particular, the equation we use has the form
>
> [2]
>   rho cp DT/Dt - alpha T u . grad p - div(k grad T) = Phi
>
> So I need to finagle [1] into form [2]. I believe that the authors of [1]
> have in mind that in their form cp=alpha=|g|=k=1. They prescribe Ra=10^4
> and Di=0.25.
>
> Then, if I use these values for the physical quantities and multiply their
> equation by Ra/Di, I get
>
> [1']
>   Ra/Di DT/Dt  -  Ra g . u (T+T0) - div(Ra/Di grad T) = Phi
>
> So, to put this into form [2], I do
>   g=10^10
>   alpha=10^-6
>      --> these two guarantee that rho=1 to good accuracy because T=0..1
>   cp=Ra/Di=4.10^4
>   k=Ra/Di=4.10^4
> With this, I can match the first and third terms of [1'] and [2]. However,
> I'm stuck at the second term. I do have that
>   grad p  =  -rho g
> (I assuming that Davies et al. define g to point upward, whereas we define
> it as pointing downward) so the terms look like this:
>
> [adiabatic heating in 1' -- Davies et al.]
>   - Ra g . u (T+T0)               (g points upward)
> [adiabatic heating in 2  -- ASPECT]
>   + alpha rho g . u T             (g points downward)
>
> With the values I have chosen above, alpha*rho*g=Ra, so this part is good.
> But how do I get the factor T0 into my equations? The benchmark defines it
> as T0=0.091 (page 4). This term simply doesn't appear anywhere in ASPECT.
> Anyone's got an idea how to finagle it into ASPECT without having to change
> the code?
>
> Thanks in advance
>  Wolfgang
>
>
> PS: Model setup file is here:
> https://github.com/bangerth/aspect/blob/more-davies-
> benchmarks/benchmark/davies_et_al/case-3.1.prm
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth               email:            bangerth at math.tamu.edu
>                                 www: http://www.math.tamu.edu/~bangerth/
>
> _______________________________________________
> Aspect-devel mailing list
> Aspect-devel at geodynamics.org
> http://lists.geodynamics.org/cgi-bin/mailman/listinfo/aspect-devel
>



-- 

Katrina Arredondo
Ph.D. Candidate
Geodynamics/Subduction Zones
University of California, Davis
Department of Geology
One Shield Avenue, Davis CA 95616
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.geodynamics.org/pipermail/aspect-devel/attachments/20140904/c602994f/attachment.html>