Martian weather - vorticity and wind direction

This video shows the same simulation as the video • What could the weather... , of a weather simulation on terraformed Mars, but the color gradients show the vorticity and wind direction instead of the density and wind speed.
Recent videos on this channel showed simulations of the weather on the Earth, using a simplified model. The same code can be used to model the weather on other planets having an atmosphere. In this simulation, we assume that Mars has been terraformed, creating an ocean that covers a large part of the northern hemisphere, as well as a sea in the Hellas basin.
The video shows a simulation of the compressible Euler equations on Mars, as a very simplified model for the weather. The main effect of the land masses is that they slow down the wind speed. The initial state consists in 16 different pressure systems spread over the planet, which roughly resemble the pressure and wind distribution on Earth on May 21 2024, as seen on www.windy.com/ . In addition, a ground state made of westerly winds at intermediate latitudes, and easterlies/trade winds near the equator and the poles has been added to the overall wind pattern.
This simulation uses two maps available online, both here in 4,096 x 2048 resolution. The elevation of the planet's surface is based on the digital elevation model (DEM) solarviews.com/cap/mars/marsc... , produced by the USGS from Viking image data, while the coloring stems from the Viking photographic map at astropedia.astrogeology.usgs....
I'm not claiming this simulation to be a realistic representation of the (hypothetical) weather, because many important effects are neglected. However, it does include the Coriolis force, and the pressure systems do rotate in the correct way: High pressure systems rotate clockwise in the northern hemisphere and anticlockwise in the southern hemisphere, while the situation is reversed for low pressure systems. One major limitation is that the density field is too unstable. I suspect this is due to the fact that the speed of sound is way too large in my model equations, and I will try to improve that in future simulations.
The video has four parts, showing the same simulation with two different color gradients and two different representations:
Vorticity, 3D: 0:00
Wind direction, 3D: 1:23
Vorticity, 2D: 2:53
Wind direction, 2D: 4:18
The 2D parts use an equirectangular projection of the sphere. The velocity field is materialized by 2000 tracer particles that are advected by the flow. In parts 1 and 3, the color hue depends on the vorticity of the air, which measures its quantity of rotation. In parts 2 and 4, the color hue depends on the wind direction, while the luminosity depends on its speed. In the 3D parts, the radial coordinate in the oceans also depends on the indicated field, more so for the wind speed. The point of view of the observer is rotating around the polar axis of the sphere at constant latitude.
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.
I will be traveling for a few days, and may not be able to reply to comments before I return.
Render time: Parts 1 and 2 - 1 hour 58 minutes
Parts 3 and 4 - 1 hour 27 minutes
Compression: crf 23
Color scheme: Parts 1 and 3 - Turbo, by Anton Mikhailov
gist.github.com/mikhailov-wor...
Parts 2 and 4 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Music: Heliograph by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 licence. creativecommons.org/licenses/... chriszabriskie.com/divider/Art... chriszabriskie.com/
The simulation solves the compressible Euler equation by discretization.
C code: github.com/nilsberglund-orlea...
#Euler_equation #fluid_mechanics #weather

[@JTolmar]

Would it be possible to simulate rainfall from this?

[@NilsBerglund]

One would have to add features to the model, such as temperature and humidity. This is done for weather models on the Earth, which requires quite a bit of additional work.