Nils Berglund | Weather on the Earth, with 17 pressure systems @NilsBerglund | Uploaded May 2024 | Updated October 2024, 1 hour ago.
Like the recent video youtu.be/CDuMPM0jRKM , this one shows a simulation of the compressible Euler equations on the Earth, 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 17 different pressure systems spread over the Earth, which roughly resemble the pressure and wind distribution on May 21 2024, as seen on windy.com
I'm not claiming this simulation to be a realistic representation of the 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:
Density, 2D: 0:00
Wind speed, 2D: 1:23
Density, 3D: 2:54
Wind speed, 3D: 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 density of the fluid, which is related to its pressure. In parts 2 and 4, the color hue depends on the fluid's 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.
Render time: Parts 1 and 2 - 1 hour 28 minutes
Parts 3 and 4 - 1 hour 58 minutes
Compression: crf 23
Color scheme: Parts 1 and 3 - Viridis, by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap
Parts 2 and 4 - Parula, originally from Matlab
mathworks.com/help/matlab/ref/colormap.html
Music: "Sunspots" by Jeremy Blake@RedMeansRecording
The simulation solves the compressible Euler equation by discretization.
C code: github.com/nilsberglund-orleans/YouTube-simulations
#Euler_equation #fluid_mechanics #weather
Like the recent video youtu.be/CDuMPM0jRKM , this one shows a simulation of the compressible Euler equations on the Earth, 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 17 different pressure systems spread over the Earth, which roughly resemble the pressure and wind distribution on May 21 2024, as seen on windy.com
I'm not claiming this simulation to be a realistic representation of the 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:
Density, 2D: 0:00
Wind speed, 2D: 1:23
Density, 3D: 2:54
Wind speed, 3D: 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 density of the fluid, which is related to its pressure. In parts 2 and 4, the color hue depends on the fluid's 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.
Render time: Parts 1 and 2 - 1 hour 28 minutes
Parts 3 and 4 - 1 hour 58 minutes
Compression: crf 23
Color scheme: Parts 1 and 3 - Viridis, by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap
Parts 2 and 4 - Parula, originally from Matlab
mathworks.com/help/matlab/ref/colormap.html
Music: "Sunspots" by Jeremy Blake@RedMeansRecording
The simulation solves the compressible Euler equation by discretization.
C code: github.com/nilsberglund-orleans/YouTube-simulations
#Euler_equation #fluid_mechanics #weather
