How tides could look if the Moon were much closer to Earth @NilsBerglund

Nils Berglund | How tides could look if the Moon were much closer to Earth @NilsBerglund | Uploaded October 2024 | Updated October 2024, 3 minutes ago.
Like in the video youtu.be/DNPFUomUFyw , this simulation uses a shallow water equation as a simplified model for the tides. The main difference with the previous simulation is the initial state, which already takes the influence of the Moon into account.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 parameter tuning this effect has been chosen too large, however, so that in effect we get an impression of what tides would look like if the Moon were much closer to the Earth than it actually is.
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:
Original speed, 3D: 0:00
Original speed, 2D: 1:05
Time lapse, 3D: 2:12
Time lapse, 2D: 2:28
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. The longitude closest to the Moon is indicated by a vertical line in the 2D parts. In the 3D parts, the point of view is slowly rotating around the Earth in a circular orbit. In parts 3 and 4, 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 - 2 hours 29 minutes
2D parts - 1 hour 51 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap

Music: "You're Not Wrong" by roljui@roljui1445

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

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A simplistic model for tides - periodic forcing

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Speed and vorticity of a fluid on a rotating sphere