Nils Berglund | Weather on the Earth with a random initial state @NilsBerglund | Uploaded June 2024 | Updated October 2024, 2 hours ago.
In a comment to a previous video of the weather on Earth, it was asked what would happen if the initial state were random. This simulation tests this, by choosing an initial condition that is essentially white noise for the velocity components, and a constant plus white noise for the density. The compressible Euler equations simulated here use some smoothing, which quickly turn the noise into a less singular colored noise. While the density of the air retains a pretty random aspect, one can see some structures appearing in the wind patterns.
The video 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.
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, which would make pressure systems rotate in the correct direction.
The video has four parts, showing the same simulation with two different color gradients and two different representations:
Density, 3D: 0:00
Wind speed, 3D: 1:00
Density, 2D: 2:06
Wind speed, 2D: 3:06
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 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 on a circular orbit.
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 24 minutes
Parts 3 and 4 - 1 hour 3 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: "Fall Colors" by ann annie@annannie
The simulation solves the compressible Euler equation by discretization.
C code: github.com/nilsberglund-orleans/YouTube-simulations
#Euler_equation #fluid_mechanics #weather
In a comment to a previous video of the weather on Earth, it was asked what would happen if the initial state were random. This simulation tests this, by choosing an initial condition that is essentially white noise for the velocity components, and a constant plus white noise for the density. The compressible Euler equations simulated here use some smoothing, which quickly turn the noise into a less singular colored noise. While the density of the air retains a pretty random aspect, one can see some structures appearing in the wind patterns.
The video 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.
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, which would make pressure systems rotate in the correct direction.
The video has four parts, showing the same simulation with two different color gradients and two different representations:
Density, 3D: 0:00
Wind speed, 3D: 1:00
Density, 2D: 2:06
Wind speed, 2D: 3:06
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 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 on a circular orbit.
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 24 minutes
Parts 3 and 4 - 1 hour 3 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: "Fall Colors" by ann annie@annannie
The simulation solves the compressible Euler equation by discretization.
C code: github.com/nilsberglund-orleans/YouTube-simulations
#Euler_equation #fluid_mechanics #weather
