Add arbitrary topography on P4estMeshCubedSphere

[MarcoArtiano]

Other than baroclinic test case with real horography, some idealized test cases with horography are really interesting and I will add them in this PR (at least 1 for having 1 case with real horography and 1 idealized one):
https://sites.google.com/umich.edu/dcmip-2025/dcmip-2025-test-cases/test-case-2-mountain-triggered-meso-scale-flow-phenomena.

The visualization has been realized by clipping the data in paraview with radius = 6371300.

With the horography, two functions are implemented to adapt the vertical size elements to smoothly restore the cubed sphere.

Vortex shedding generated by a gaussian hill on the sphere (the hole on the left is the island). Clipped radius for visualization: 318650 (this is a small earth size, 20 times smaller)

ps: I have removed the simulation with real topography, because until we have a good imex scheme in the main, it is prohibitively to run that simulation.

High-resolution vortex shedding

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 97.14%. Comparing base (8ebf7f6) to head (95e3635).

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #187      +/-   ##
==========================================
+ Coverage   97.11%   97.14%   +0.03%     
==========================================
  Files          38       39       +1     
  Lines        5128     5186      +58     
==========================================
+ Hits         4980     5038      +58     
  Misses        148      148              

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.

[benegee]

Just a very quick reply to say this looks really cool!

I assume, the usual mapping argument of P4estMesh (used in the 2D Schär mountain test case) does not work here.

[MarcoArtiano]

Just a very quick reply to say this looks really cool!

I assume, the usual mapping argument of P4estMesh (used in the 2D Schär mountain test case) does not work here.

No, it does not work. Here I'm just changing the the radial coordinate of the cubed sphere mapping, and then everything gets translated accordingly. So at the end you would have something that looks like a sphere but with some bumps (the topography is propagated also in the vertical layer). To avoid that we apply a smoothing function in the vertical direction. Two famous ones are Gal-Chen and Sleve, so that given a surface topography, we adjust the element size, so that we get again the perfect sphere after some vertical elements.

The Schär mountain can be redefined also here. The vortex shedding example shows how to define a simple Gaussian hill, you have to choose the spherical coordinate and then provide a function $$z(\lambda, \phi)$$, where $$\lambda$$ and $$\phi$$ are the longitude and latitude, respectively.

[benegee]

Like!