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

Mon Sep 8 11:30:06 PDT 2014

Hi,

This is a very interesting discussion. I agree with much of what has been
said, but just wanted to add a few comments.

Computing a Rayleigh number can indeed be tricky for cases where parameters
vary in space, and may in fact be cumbersome
for some numerical applications, leading to difficulties for users.
However, I would maintain that the Rayleigh number still holds value in
that, if computed correctly, it provides a valid description for the
average system behavior, i.e. it does exactly what it's meant to do.

There is quite some value in trying to work out what "correctly" means, as
we know, for example, from looking at temperature and/or stress dependent
convection cases in the late 80ies. Average system behavior (in terms of,
say, Nusselt number, or RMS velocities) of complex models can be captured
by rheologically more simple models, if the Rayleigh number is adjusted
properly (e.g. by log averaging viscosity before plugging it into the
Rayleigh number, which may or may not work, depending on loading).
Re-establishing such dynamic similarity is important, and worth the effort,
I would argue, because it allows one to see which aspects of a complex
model's dynamics are truly emergent, and which can simply be attributed to
doing something that, on average, has larger or smaller convective vigor.

This argument speaks to the variations in expansivity, density, viscosity,
diffusivity, total temperature difference, while g should not deviate much
from the 1D average. The fact that the length scale is sometimes causing
confusion is also inherent in the approach. If you want to know the
convective vigor of the system, use the box height. If you're zooming into
the boundary layer, you can get helpful estimates if adjusting length
scales (and see, why, for example the top boundary layer will be thicker
than the bottom for temperature dependent rheologies), but in general the
approach brakes down, as for all such non-dimensional numbers (high
Rayleigh number means advection wins over conduction in the bulk of the
domain, but conduction is important in the boundary layers). There's of
course other options of defining a length scale, e.g. using the radius of
the Earth rather than mantle thickness, but those mainly come down to
converting non-dimensionalization conventions to physically relevant scales
for comparability.

Cheers

T

Thorsten W Becker - USC
geodynamics.usc.edu <http://geodynamics.usc.edu/~becker>

On Mon, Sep 8, 2014 at 11:07 AM, Magali Billen <mibillen at ucdavis.edu> wrote:

> Wolfgang, I completed agree - for the types of problems people are working
> on now, non-dimensional
> calculations make no sense.
>
> I've basically been using CitcomS "dimensionally", but with constant
> coefficients(alpha, k, g) since my post-doc.
> I always design a dimensional problem, choose "reference" values and then
> calculate the effective Rayleigh number
> and Dissipation number to input into the code. When I first starting doing
> this, I struggled with which value to use
> for the reference viscosity in the Rayleigh number, but then I just stop
> reporting a Rayleigh number (or dissipation number)
> in my papers because they are meaningless, and I just reported the table
> of values and the equations defining how
> these parameters varied (viscosity, density).
>
> Magali
>
> On Sep 7, 2014, at 3:08 PM, Wolfgang Bangerth <bangerth at tamu.edu> wrote:
>
> On 09/04/2014 08:10 PM, Katrina Arredondo wrote:
>
> Here at Davis the ASPECT discussion group has repeatedly asked ourselves
> what
> is the best way to translate the dimensional terms in ASPECT into the
> nondimensional Rayleigh and Dissipation numbers during postprocessing.
>
>
> That is of course the question at the heart of the matter: if you don't
> know how to translate from physical quantities to nondimensional ones, then
> the other direction is equally ill-defined.
>
> To give an example, the Rayleigh number is defined as
>  Ra = alpha g dT L^3 / eta k
> In realistic cases, every single one of these physical quantities is
> spatially variable as they depend on temperature, pressure and, in the case
> of gravity, on basically everything everywhere. For the temperature
> difference dT, one may ask where the upper and lower temperatures should be
> taken if you consider the Earth (top of the crust, bottom of the crust?) or
> a regional model. Finally, what exactly is the length scale L? If you have
> a regional model with a subducting slab, is L the horizontal extent of your
> domain? The vertical extent? Maybe the size of the convecting wedge above
> the subducting slab and the overriding plate?
>
> What then *is* the Rayleigh number? Even if there was a good way to define
> dt and L, it would be different at every point of the domain, of course.
> What I mean to say by this is that dimensionless quantities are only
> defined in simple cases with constant coefficients -- the things we have
> known how to do for a long time in geodynamics and that we are now trying
> to move beyond. These were my considerations when I thought about how one
> describes cases in ASPECT when I was designing the code: would I stick with
> how it has been done in the past but that is no longer adequate when you
> want to use complex cases, or should I "break backward compatibility" and
> force everyone to describe things in physical quantities and, if they
> wanted to, let them compute nondimensional quantities as a postprocess
> using some appropriate "reference viscosity", "reference thermal expansion
> coefficient", etc. I continue to believe that the way we chose to do it was
> the right way forward.
>
> Best
> Wolfgang
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth               email:            bangerth at math.tamu.edu
>                                www: http://www.math.tamu.edu/~bangerth/
>
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>
> -----------------------------------------------------------
> Professor of Geophysics & UCD Chancellor Fellow
> Chair, Geology Graduate Program
> Earth & Planetary Sciences Dept., UC Davis
> Davis, CA 95616
> Room 2129 Earth & Physical Sciences Bldg.
> Office Phone: (530) 752-4169
> http://mygeologypage.ucdavis.edu/billen/
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