Nils Berglund | An asteroid impact in the Indian Ocean, with tsunami-induced flooding @NilsBerglund | Uploaded September 2024 | Updated October 2024, 1 hour ago.
This is an attempt at modeling an asteroid impact in the Indian ocean using a nonlinear wave equation. This model allows for some flooding of land areas. The vertical scale of both mountains and waves has been exaggerated. A mistake in the code has been corrected, which incorrectly computed the influence of the non-constant ocean depth - the east-west component of that force was using the north-south gradient of the depth, instead of the east-west gradient.
A version using the linear wave equation appeared previously on this channel, see youtu.be/DRsKobzGCVk The main difference in the nonlinear 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 initial state features velocities radiating outward from the impact point of the asteroid. 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: 0:13
Original speed, 3D: 0:27
Original speed, 2D: 1:21
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. 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 - 1 hours 50 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap
Music: "Mast" by Silent Partner
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 an attempt at modeling an asteroid impact in the Indian ocean using a nonlinear wave equation. This model allows for some flooding of land areas. The vertical scale of both mountains and waves has been exaggerated. A mistake in the code has been corrected, which incorrectly computed the influence of the non-constant ocean depth - the east-west component of that force was using the north-south gradient of the depth, instead of the east-west gradient.
A version using the linear wave equation appeared previously on this channel, see youtu.be/DRsKobzGCVk The main difference in the nonlinear 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 initial state features velocities radiating outward from the impact point of the asteroid. 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: 0:13
Original speed, 3D: 0:27
Original speed, 2D: 1:21
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. 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 - 1 hours 50 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap
Music: "Mast" by Silent Partner
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
