Vortex dynamics on the sphere in a strong Coriolis force

Описание: Like the video    • Vortices in the compressible Euler equatio...   , this one shows a simulation of the compressible Euler equations on a sphere. The difference is that the Coriolis force is much larger: it is five times as large as in the previous simulation. This would occur on a quickly rotating planet.
The initial state of this simulation consists in two vortices at antipodal points on the sphere, rotating in opposite directions. I have fixed a problem that caused ripples to appear in the previous simulation, because the initial state was not periodic in the longitudinal direction.
The video has two parts, showing the same simulation with two representations:
3D: 0:00
2D: 0:58
The 2D part uses an equirectangular projection of the sphere. It also shows the motion of 1000 tracer particles that are transported by the velocity field. One can see how large the Coriolis force is, since many particles move on small loops.
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. The point of view of the observer is rotating around the polar axis of the sphere at constant latitude. The white bar above the sphere points away from the polar axis in a fixed direction, to indicate the position of points with constant longitude on the sphere.
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 2 minutes
Part 2 - 1 hours 1 minute
Compression: crf 25
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-wor...

Music: "Light Years Away" by Doug Maxwell‪@dougmaxwell9118‬

The simulation solves the compressible Euler equation by discretization.
C code: https://github.com/nilsberglund-orlea...

#Euler_equation #fluid_mechanics #vortex