Nils Berglund | Trying to model tides with a shallow water equation @NilsBerglund | Uploaded October 2024 | Updated October 2024, 1 hour ago.
This is an attempt at modeling tides with a shallow water equation. A version using the linear wave equation previously appeared in the video youtu.be/SqwvZX20Q2Y . The initial state has been chosen somewhat arbitrarily, just to have some currents in the oceans. Then the effect of the Moon is modeled by a force acting on the water height, which is maximal at the position of the Moon and at its antipode. The longitude closest to the Moon is indicated by a vertical line in the 2D parts.
The main difference between the nonlinear equation and the linear wave equation is that the wave speed becomes larger when the water is shallower, which can lead to build-up of the waves in coastal and other shallow regions. This causes problems in the simulation, because land masses can lead to blow-up of the solution. This problem is circumvented here by replacing the continents by a repelling force field, plus a dissipative term.
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 at two different speeds and with two different visualizations:
Time lapse, 3D: 0:00
Time lapse, 2D: 0:17
Original speed, 3D: 0:35
Original speed, 2D: 1:45
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 circular orbit. In parts 1 and 2, the animation has been speeded up by a factor 4.
The velocity field is materialized by 2000 tracer particles that are advected by the flow.
Render time: 3D parts - 1 hour 58 minutes
2D parts - 2 hours 42 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap
Music: "5 Star" by Causmic@Causmic
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 #tsunami
This is an attempt at modeling tides with a shallow water equation. A version using the linear wave equation previously appeared in the video youtu.be/SqwvZX20Q2Y . The initial state has been chosen somewhat arbitrarily, just to have some currents in the oceans. Then the effect of the Moon is modeled by a force acting on the water height, which is maximal at the position of the Moon and at its antipode. The longitude closest to the Moon is indicated by a vertical line in the 2D parts.
The main difference between the nonlinear equation and the linear wave equation is that the wave speed becomes larger when the water is shallower, which can lead to build-up of the waves in coastal and other shallow regions. This causes problems in the simulation, because land masses can lead to blow-up of the solution. This problem is circumvented here by replacing the continents by a repelling force field, plus a dissipative term.
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 at two different speeds and with two different visualizations:
Time lapse, 3D: 0:00
Time lapse, 2D: 0:17
Original speed, 3D: 0:35
Original speed, 2D: 1:45
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 circular orbit. In parts 1 and 2, the animation has been speeded up by a factor 4.
The velocity field is materialized by 2000 tracer particles that are advected by the flow.
Render time: 3D parts - 1 hour 58 minutes
2D parts - 2 hours 42 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap
Music: "5 Star" by Causmic@Causmic
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 #tsunami
