Nils Berglund | Video #1200: Weather on the Earth @NilsBerglund | Uploaded May 2024 | Updated October 2024, 2 hours ago.
This is the 1200th video published on this channel (not counting a handful of videos I released multiple times, due to compression issues). Once again, thanks to all of you for visiting this channel and providing new ideas and interesting thoughts in the comments.
As usual, for a milestone I try to do something a little bit new or otherwise out of the ordinary. This one I have been preparing in a number of recent simulations of the compressible Euler equations on the sphere. The new feature here is that the Earth's land masses have been added to the sphere, in order to provide a simplified model for the weather on our planet. The main effect of the land masses is that they slow down the wind speed. The initial state consists in five different pressure systems spread over the oceans.
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.
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: 1:16
Density, 3D: 2:37
Wind speed, 3D: 3:59
The 2D parts use an equirectangular projection of the sphere. The velocity field is materialized by 1000 tracer particles that are advected by the flow. In parts 1 and 3, the color hue depends on the density of the air, which is related to its pressure. In parts 2 and 4, the color hue depends on the wind 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 21 minutes
Part 3 - 1 hour 42 minutes
Part 4 - 1 hour 16 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: "Pixelated Autumn Leaves" 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
This is the 1200th video published on this channel (not counting a handful of videos I released multiple times, due to compression issues). Once again, thanks to all of you for visiting this channel and providing new ideas and interesting thoughts in the comments.
As usual, for a milestone I try to do something a little bit new or otherwise out of the ordinary. This one I have been preparing in a number of recent simulations of the compressible Euler equations on the sphere. The new feature here is that the Earth's land masses have been added to the sphere, in order to provide a simplified model for the weather on our planet. The main effect of the land masses is that they slow down the wind speed. The initial state consists in five different pressure systems spread over the oceans.
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.
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: 1:16
Density, 3D: 2:37
Wind speed, 3D: 3:59
The 2D parts use an equirectangular projection of the sphere. The velocity field is materialized by 1000 tracer particles that are advected by the flow. In parts 1 and 3, the color hue depends on the density of the air, which is related to its pressure. In parts 2 and 4, the color hue depends on the wind 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 21 minutes
Part 3 - 1 hour 42 minutes
Part 4 - 1 hour 16 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: "Pixelated Autumn Leaves" 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
