Notes on Mantle Convection Benchmarks

(1) Ra = 3x10**6 (I think this is high enough to give some real time dependence without pushing available resolution very much.

(2) Constant properties (thermal expansivity, thermal diffusivity, thermal conductivity, density, gravity, viscosity, internal heat generation) to keep things very simple.

(3) Free slip upper and lower boundaries.

(4) Radius ratio = 0.546 (cmb/surface radius)

(5) purely internally heated

(6) insulating at cmb, constant temperature at surface

(7) model resolution: 65 nodes (64 layers) radially, with some packing of nodes near the top and bottom boundaries. (We'll send you the actual radii we use, assuming you can vary them at will.)

(8) initial diagnostics: (basically, these are just to get started and see if we're in the same universe)

• (a) Nu vs. time (this should square with the internal heating in a time-average sense)
• (b) Radial temperature profile vs. time - this is effectively a measure of the efficiency of heat transfer, or equivalent of Nu for bottom heated cases.
• (c) Spherical harmonic expansion of temperature field at all radial levels at beginning and ending time (see below)
• (d) peak velocity and peak temperature in each radial layer vs. time
• (e) for now, let's ignore dynamic topography, since it's derived from primitive results

(9) Initial conditions and run time: This is a bit thorny, so here's a proposal. We can run TERRA to equilibrium under the specified model conditions. Equilibrium is where Nu has settled down to fluctuations about a steady mean value. At some point, call it time = 0.0, we'll stop the code and output the full temperature field in the form of a spherical harmonic expansion up to degree 128, which corresponds to the highest model resolution. We can then restart both TERRA and CitcomS using this spherical harmonic expansion (NOT the full temperature field at each node, since this would prejudice things with regard to the particular horizontal discretization.) Then both codes can run for a defined amount of model time, keeping track of Nu, peak T, and peak V as a function of time as indicated above. At the end of this time, or at several times along the way, we can output spherical harmonic representations of T at each layer for comparison.

I added the following comments:

1) We use some analytic expressions for initial conditions (e.g., some radial profile superimposed with a small perturbation of a given harmonic function. In this way, others, if they want to benchmark their codes, do not need to get the Terra output. Also in case some summary report comes out of this effort, we can simply write down the initial conditions.

2) We aim to produce four benchmark cases in steady of just one. The four cases at the moment in my mind can be: three constant property cases with purely basal heating at Ra=1e5 (case 1) and Ra=1e6 (case 2), and purely internal heating at Ra=1e6 (case 3), and one temperature dependent viscosity and purely basal heating at Ra=1e6 (case 4).

Case 1 will likely reach to a steady state, which is always a good thing for benchmark. Cases 2 and 3 are almost identical to what you have suggested recently, and they are most likely time-dependent. The 1e6 Ra is smaller than what you suggested today but is consistent with your earlier suggestion. With Ra=1e6, we may not need grid refinement, which is also good for benchmark purpose (again, others can do it later). Case 4 is obviously of interest too.