Nils Berglund | Video #1300: What if the Earth's oceans were much shallower? @NilsBerglund | Uploaded September 2024 | Updated October 2024, 4 seconds ago.
This is the 1300th video published on this channel, not counting some videos published multiple times because of corruption issues. Thanks again to all you viewers for your fidelity, comments and suggestions.
This is a first simulation of the shallow water equations on a model for the Earth. The wave speed in these equations becomes larger when the water is shallower, which can lead to build-up of the waves in coastal and other shallow regions.
The most important parameter in shallow water equations is the ratio between the size of depth variation and the average depth. This videos uses two different values of this parameter, the last two parts having a larger ratio. In both cases, this ratio is quite large compared to what applies to the Earth, making the effect of shallower parts more important.
The shallow water equations are nonlinear equations, which give a better description of the motion of water than the linear wave equation. In particular, unlike the linear wave equation, they conserve the total volume of water. The linear equation gives an approximation of the solutions, when the wave height remains close to its average over space.
The equations used here include viscosity and dissipation, as described for instance in
en.wikipedia.org/wiki/Shallow_water_equations#Non-conservative_form , including the Coriolis force.
One difficulty is to model the wetting boundary, which separates regions that are under water and those which are not. This difficulty has been circumvented here by replacing the continents by a repulsive force field, directed downslope, instead of a sharp boundary.
The video has four parts, showing simulations with two different parameter values and two different visualizations:
Smaller ratio, 3D: 0:00
Smaller ratio, 2D: 0:48
Larger ratio, 3D: 1:36
Larger ratio, 2D: 2:24
The color hue and radial coordinate show the height of the water, on an exaggerated radial scale. The 2D parts use a projection in equirectangular coordinates. In the 3D parts, the point of view is slowly rotating around the Earth in a plane containing its center. The position of the sphere is represented by a line starting above the north pole, and pointing in a fixed direction.
The velocity field is materialized by 2000 tracer particles that are advected by the flow.
Render time: Part 1 - 1 hour 30 minutes
Part 2 - 1 hour 14 minutes
Part 3 - 1 hour 8 minutes
Part 4 - 1 hour 36 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap
Music: "Gymnopédie No. 1" by Erik Satie
See also
https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ for more explanations (in French) on a few previous simulations of wave equations.
The simulation solves the 2D shallow water equation by discretization (finite differences).
C code: github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code!
#shallowwater #waves #Earth
This is the 1300th video published on this channel, not counting some videos published multiple times because of corruption issues. Thanks again to all you viewers for your fidelity, comments and suggestions.
This is a first simulation of the shallow water equations on a model for the Earth. The wave speed in these equations becomes larger when the water is shallower, which can lead to build-up of the waves in coastal and other shallow regions.
The most important parameter in shallow water equations is the ratio between the size of depth variation and the average depth. This videos uses two different values of this parameter, the last two parts having a larger ratio. In both cases, this ratio is quite large compared to what applies to the Earth, making the effect of shallower parts more important.
The shallow water equations are nonlinear equations, which give a better description of the motion of water than the linear wave equation. In particular, unlike the linear wave equation, they conserve the total volume of water. The linear equation gives an approximation of the solutions, when the wave height remains close to its average over space.
The equations used here include viscosity and dissipation, as described for instance in
en.wikipedia.org/wiki/Shallow_water_equations#Non-conservative_form , including the Coriolis force.
One difficulty is to model the wetting boundary, which separates regions that are under water and those which are not. This difficulty has been circumvented here by replacing the continents by a repulsive force field, directed downslope, instead of a sharp boundary.
The video has four parts, showing simulations with two different parameter values and two different visualizations:
Smaller ratio, 3D: 0:00
Smaller ratio, 2D: 0:48
Larger ratio, 3D: 1:36
Larger ratio, 2D: 2:24
The color hue and radial coordinate show the height of the water, on an exaggerated radial scale. The 2D parts use a projection in equirectangular coordinates. In the 3D parts, the point of view is slowly rotating around the Earth in a plane containing its center. The position of the sphere is represented by a line starting above the north pole, and pointing in a fixed direction.
The velocity field is materialized by 2000 tracer particles that are advected by the flow.
Render time: Part 1 - 1 hour 30 minutes
Part 2 - 1 hour 14 minutes
Part 3 - 1 hour 8 minutes
Part 4 - 1 hour 36 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap
Music: "Gymnopédie No. 1" by Erik Satie
See also
https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ for more explanations (in French) on a few previous simulations of wave equations.
The simulation solves the 2D shallow water equation by discretization (finite differences).
C code: github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code!
#shallowwater #waves #Earth
