[CIG-LONG] CIG-LONG Digest, Vol 47, Issue 2

[Walter Landry]

Nicolas RIEL <nicolas.riel at free.fr> wrote:
> Hi Walter,
> I attached you a basic input file with temperature diffusivity problems.
> The model is made of one sphere in middle of a square of 1000m².
> Diffusivity is set at 1e20, temperature of the sphere is 500K whereas in the
> square temperature is 300K.

The short answer is that I think you need to set the viscosity to
something more realistic.  Right now you have it set at 1e-2, which is
a bit small for rock.

The long answer is that there is a mismatch in what the timestep
should be.

Gale chooses a time step based on the maximum velocity from the Stokes
solve and the grid spacing.  In this case, the max velocity is about
4.5e7.  You can see where this velocity comes from by doing some
dimensional analysis

  Force from temperature perturbation ~ viscosity * v/(length * length)

=>  alpha * dT * density * gravity ~ viscosity * v/(length * length)

Putting in some numbers

  3e-5 * 200 * 2000 * 9.81 ~ 1e-2 * v /(1000 * 1000)

which implies that v ~ 4e9.  Higher, but in the same ballpark.

The grid spacing is 1000/100=10.  So that gives a time step
of about 10/4.5e7 ~ 2e-7.

However, given a diffusivity of 1e20, that means that, in the time of
2e-7, the temperature spreads about sqrt(1e20*2e-7) ~ 4e6.  This is
much larger than the grid spacing of 10.  This makes the time stepping
unstable, giving rise to the effects that you see.

If you increase the viscosity to something more reasonable
(e.g. 1e20), then the implied velocity would be 22 orders of magnitude
less.  That, in turn, would reduce the diffusion length scale by 11
orders of magnitude, making the simulation stable again.

Cheers,
Walter Landry
walter at geodynamics.org