A tide simulation with a more realistic lunar forcing @NilsBerglund

Nils Berglund | A tide simulation with a more realistic lunar forcing @NilsBerglund | Uploaded October 2024 | Updated October 2024, 2 hours ago.
The simulation youtu.be/QYqjmapPKsc tried to reproduce tides on the Earth, but the lunar forcing was way too strong. In this variant, the lunar forcing has been reduced to about 40% of its value in the previous video, with a more realistic result.
The effect of the Moon is modeled here by a force acting on the water height, which is maximal at the part of Earth closest to the Moon and at its antipodal point. The motion of the water is modeled by a shallow water equation, taking the varying ocean depth into account.
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: 1:13
Time lapse, 3D: 2:27
Time lapse, 2D: 2:46
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. A modification compared to previous simulations is that the particles are "respawned" at random times, meaning that their location is changed randomly. This avoids the concentration of tracers seen in some previous simulations.

Render time: 3D parts - 2 hours 12 minutes
2D parts - 2 hour 46 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
github.com/BIDS/colormap

Music: "Boom Bap Flick" by Quincas Moreira@QuincasMoreira

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

$Cherenkov radiation of two particles moving in opposite directions This video was suggested in comments to a previous video on Cherenkov radiation. It features two particles moving in opposite directions in a medium with varying refractive index. Cherenkov radiation is produced when a fast moving particle enters a medium in which the speed of light is smaller than the speed of the particle. This does not contradict special relativity, because only the speed of light in a vacuum cannot be reached by a massive particle, while the speed of light in a medium is in general smaller than in a vacuum. In this simulation, two particles move at constant speed, one from left to right and the other one from right to left, emitting pulses at regular time intervals. The lower particle emits waves of smaller amplitude, to avoid that its radiation completely masks the radiation of the upper particle. The speed of light decreases linearly, dropping to 0 at the right boundary. When the speed of light in the medium is smaller than the speed of the particle, a shock waves appears, which is the origin of Cherenkov radiation. No shock wave is produced when the particle moves in a region where the speed of light is larger than its own speed. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:16 In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the bottom shows the signal along a horizontal line, located halfway between the paths of the two particles. Render time: 28 minutes 59 seconds Compression: crf 20 Color scheme: Part 1 - Twilight by Bastian Bechtold Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: The Coldest Shoulder by The 126ers@hutchinw See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on a few previous simulations of wave equations. The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf C code: https://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! #wave #Cherenkov #shock_wave$

Martian weather - vorticity and wind direction

$Waves escaping a sunflower spiral of obstacles #art #waveequation #diffraction #fibonacci$

Simulation of an asteroid impact between Ireland and Newfoundland

The asymmetric rock-paper-scissors-lizard-Spock equation on the sphere, with improved polar behavior

$Filming a moving source with a gradient index lens Like the video https://youtu.be/KB_Roqs38F8 , this one shows a simulation of light from a nearby source crossing a gradient index lens. The difference is that here, the source is moving on a path perpendicular to the axis of the lens. Note that the speed of the source is very large, about 33% of the speed of light, but the simulation does not take any relativistic effects into account. The refractive index along the central axis of the lens is quite large, with a value of 1.82, in order to obtain the small focal distance required by the simulation. Im not quite sure what causes the echo of the transmitted waves, which was not present in other simulations with a static source. Is might be due to light reflecting on the horizontal boundaries of the lens. If someone has a better explanation, please feel free to leave a comment. Lenses focus incoming rays of light by delaying them more near the center of the lens than at its periphery. This is often done with a material of constant index of refraction, by making the lens thicker near the center, as shown for instance in the simulation https://youtu.be/rrJJBh9ubUE . However, one can also build lenses of constant thickness, by making the index of refraction of their material depend on the location in the lens. In this simulation, the index decreases like sqrt(n0² - a*r²), where r is the distance to the axis of symmetry. This results in the incoming waves being focused on the (estimated) focal plane, marked by a vertical line. The plot to the right shows a time-averaged value of the field along that plane. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:39 In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows the signal along the focal plane, which is indicated by a vertical line. Render time: 51 minutes 32 seconds Compression: crf 23 Color scheme: Part 1 - Twilight by Bastian Bechtold https://github.com/bastibe/twilight Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Runner by Silent Partner See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on a few previous simulations of wave equations. The simulation solves the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf C code: https://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! #wave #lens #gradient_index$ $A gradient index lens with quadratic refractive index exposed to two light sources$

Winnowing: Separating particles by size using wind

$3D representation of a gradient index lens with quadratic refractive index$