Relaxation of Topography Benchmark

Given an infinitely deep purely viscous medium with an infinitesimal initial sinusoidal height profile, the topography will decay exponentially with the timescale [Folds]

[;t_r = \frac{4\pi\eta}{gL},;]

where [;\eta;] is the viscosity, [;g;] is the gravitational constant, and [;L;] is the wavelength of the initial sinusoid.

In our case, we simulate a medium with finite depth and finite height. The internal fields decay exponentially with depth with a length scale of [;L/{2\pi};]. The error in the solution due to a finite height is of order [;(2\pi{A}/L)^2;], where [;A;] is the amplitude of the sinusoid. We use [;L=1;] and [;A=0.01;], giving errors of order 0.02% and 0.4%.

The file input/benchmarks/sinusoid/README explains how to run this benchmark. Figure 1 shows the results of a low-resolution run. Even this run is not particularly small (128 × 256), because we need fairly high resolution to be able to accurately resolve the small (1%) height difference. Also note that we use symmetry to only simulate half of the wavelength.

Figure 1. Strain rate and velocities for a sinusoidal topography relaxing under gravity.

Running the code with multiple resolutions and measuring the error in the height in the trough gives Figure 2. Scaling the error with resolution gives Figure 3. The error decreases linearly with increasing resolution, giving us confidence in our ability to accurately track topography.

Figure 2. Error in the height at the trough

Figure 3. As in Figure 2, but with the error scaled with [;h;]. So the medium-resolution error is multiplied by 2 and the high-resolution error is multiplied by 4.