Nils Berglund | Vortices in the compressible Euler equation on the sphere with Coriolis force @NilsBerglund | Uploaded April 2024 | Updated October 2024, 6 minutes ago.
This is my first attempt at simulating the compressible Euler equations on a sphere, as opposed to the uncompressible ones that appear in several recent videos. A somewhat similar simulation in the plane appears in the video youtu.be/YXjv21Xojzo . My ultimate goal in these simulations is to be able to see pressure systems evolving on a rotating sphere, as routinely used for weather predictions.
This first simulation suffers from several issues. One of them is the ripple propagating from left to right, which I realized is due to a badly chosen initial state, that does not take the periodicity of longitude into account. Another issue is that there is too much regularization at the poles, making them slightly reflecting.
The video has two parts, showing the same simulation with two representations:
2D: 0:00
3D: 1:06
The 2D part uses an equirectangular projection of the sphere.
The color hue depends on the speed of the fluid. The radial coordinate depends on the vorticity of the fluid, which measures its quantity of rotation. In green areas, the vorticity is zero, meaning that the speed of the fluid does not change when moving laterally with respect to its velocity. In red and blue areas, the vorticity is large. The point of view of the observer is rotating around the polar axis of the sphere at constant latitude.
The compressible Euler equations are partial differential equations for the density and the velocity field of the fluid. The system as such is not closed, because the right-hand side of the velocity equation involves the pressure, which has to be linked to known quantities by a thermodynamic relation. I assumed here that the fluid is an ideal gas, so that the pressure is proportional to the density.
In a sense, the compressible Euler equations are easier to simulate than the incompressible ones, because one does not have to impose a zero divergence condition on the velocity field. However, they appear to be a bit more unstable numerically, and I had to add a smoothing mechanism to avoid blow-up. This mechanism is equivalent to adding a small viscosity, making the equations effectively a version of the Navier-Stokes equations. The equation is solved by finite differences, where the Laplacian and gradient are computed in spherical coordinates. Some smoothing has been used at the poles, where the Laplacian becomes singular in these coordinates.

Render time: Part 1 - 1 hours 6 minutes
Part 2 - 1 hours 12 minutes
Compression: crf 25
Color scheme: Turbo, by Anton Mikhailov
gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "When Gods Pontificate" by Dan Bodan@danbodan

The simulation solves the compressible Euler equation by discretization.
C code: github.com/nilsberglund-orleans/YouTube-simulations

#Euler_equation #fluid_mechanics #vortex