Wave 33: Tsunami, Satellite Constellation & Volcano

Platform Numbers

Wave 33 Simulations

🌊 Tsunami

The Shallow Water Equations

Tsunamis are fundamentally different from wind-generated ocean waves. Their wavelength — typically 100–500 km — vastly exceeds the ocean depth (average 3.8 km), so the water column oscillates essentially in phase from surface to seafloor. This is the shallow-water regime, and it is governed by the depth-averaged Shallow Water Equations (SWE):

∂η/∂t = −∇·(H·u)

∂u/∂t = −g∇η

Here η is the free-surface elevation, u is the depth-averaged horizontal velocity, H is the undisturbed water depth, and g is gravitational acceleration. Combining these two equations yields the linear wave equation with a spatially variable wave speed c = √(gH): waves travel faster in deep water and slow down abruptly as they approach shallow coasts, causing shoaling and amplification.

Implementation: Leapfrog Scheme on a Staggered Grid

The simulation uses a 220×220 cell grid with a leapfrog (Störmer-Verlet) time-stepping scheme. Surface height η is stored at cell centres; velocity components u and v are stored at the east and north cell faces respectively (a staggered Arakawa C-grid). The update equations are:

• Velocity step: u += −g·dt · (η[i+1] − η[i]) / dx
• Height step: η_next = damp · (2η − η_prev) + dt² · ∇·(c²∇η)

The variable wave speed c = √(gH) is evaluated at each cell face by averaging the depths of the two adjacent cells. This ensures the Laplacian operator correctly handles bathymetry discontinuities. Absorbing boundary conditions damp outgoing waves at all four edges to prevent reflections from the grid boundary contaminating the simulation.

Bathymetry Presets

Four depth profiles illustrate different environments where tsunamis behave distinctively:

• ⛰️ Continent — a continental shelf with a gentle slope transitioning from deep ocean (~4000 m) to shallow coastal waters (~50 m). Click in the deep zone to trigger a subduction-zone-style earthquake.
• 🏝️ Island — a seamount rising from deep ocean, modelling a volcanic island chain. Waves refract around the island and can be amplified on the far side.
• 🌊 Basin — an enclosed basin with shallow edges, producing multiple reflections and resonant standing-wave patterns similar to harbour oscillations (seiches).
• 🏔️ Trench — an oceanic trench flanked by seamounts. The trench traps wave energy and guides it along its length, a phenomenon known as edge-wave trapping.

Earthquake Trigger

Clicking or dragging on the grid adds a Gaussian seafloor uplift with configurable amplitude (1–15 m) and radius (4–30 cells). The initial displacement is added directly to the η field, simulating the impulsive sea-surface deformation caused by a thrust-fault earthquake. The resulting wave packet then propagates outward at speed c = √(gH), visibly slowing from deep-ocean speeds (~200 m/s at 4000 m depth) to shallow-coastal speeds (~22 m/s at 50 m depth).

What to Try

• On the Continent preset, click in the deep ocean and watch the wave front speed up over the deep abyssal plain, then slow and steepen dramatically as it enters the continental shelf.
• On the Basin preset, trigger multiple earthquake sources and observe the interference patterns — constructive amplification and destructive cancellation.
• Lower the absorption slider close to 1.000 to allow boundary reflections and see the wave energy trapped in the domain, building up chaotic interference.
• Set a large Uplift height (15 m) to simulate a megathrust event and watch the maximum surface elevation readout peak as the wave shoals.

🛰️ Satellite Constellation

Walker Delta Constellations

A Walker Delta constellation is described by three parameters: T/P/F — T total satellites, P orbital planes, and F the phasing factor governing the angular offset between planes. Within each plane, satellites are evenly spaced; planes are evenly distributed in right ascension of the ascending node (RAAN) around the equator at the same inclination. This symmetry maximises coverage uniformity for a given total satellite count.

The four presets cover the major operational constellation architectures:

• 🔵 Starlink-class — 6 planes × 10 satellites at 53° inclination, 550 km altitude. Mid-latitude coverage, short orbital period (~95 min), low latency.
• 🟠 GPS — 6 planes × 4 satellites at 55° inclination, 20 200 km altitude (MEO). The classic navigation constellation; ~12 h orbital period, global coverage below 55° latitude.
• 🟡 Galileo — 3 planes × 8 satellites at 56° inclination, 23 222 km altitude. European GNSS constellation in medium-Earth orbit.
• 🟢 Polar — 8 planes × 6 satellites at 90° inclination, 800 km altitude. Sun-synchronous-style polar orbits providing complete global coverage including the poles.

Three.js 3D Rendering

The visualiser uses Three.js r160 with the Keplerian orbital position formula. Each satellite is placed at:

x = a(cosΩcosν − sinΩsinνcosi)
y = a·sinν·sini
z = a(sinΩcosν + cosΩsinνcosi)

where a is the semi-major axis (orbital radius), Ω is the RAAN, ν is the true anomaly, and i is the inclination. Each satellite’s initial true anomaly is offset so satellites within a plane are evenly spaced, and the Walker phasing factor F determines the inter-plane anomaly offset. For a Walker Delta constellation, RAAN for the p-th plane is p·2π/P and the anomaly offset is p·F·2π/(T).

Coverage circles are drawn directly on the Earth sphere as EllipseCurve rings, tilted to align with the satellite ground track. The coverage half-angle ρ is computed from:

ρ = arccos(R_e/a · cosε_min) − ε_min

where ε_min is the minimum elevation angle (5° default) and R_e = 6371 km. Coverage percentage is estimated as the fraction of the Earth sphere covered by the union of satellite footprints.

What to Try

• Switch from Starlink to GPS and observe how the coverage circles become much larger at higher altitude, achieving global coverage with far fewer satellites.
• Set inclination to 90° (Polar) and watch the orbital planes lock through the poles; toggle the coverage overlay to confirm complete global coverage.
• Increase time speed to 1000× to watch the constellation complete full orbits in seconds; observe that the Walker phasing ensures no two satellites are ever in the same part of their orbit.
• Drag to rotate, scroll to zoom — the 3D Earth lets you inspect coverage at any latitude, including the polar caps where inclined constellations have gaps.

🌋 Volcano Eruption

Volcanic Eruption Styles

Volcanic eruptions span an enormous range of intensities and styles, governed primarily by magma viscosity and dissolved volatile (gas) content. Low-viscosity basaltic magma allows gas to escape easily, producing mild effusive and mildly explosive eruptions. High-viscosity rhyolitic magma traps gas until pressure drives catastrophic fragmentation. The Volcanic Explosivity Index (VEI) quantifies this on a logarithmic scale; the simulation models four canonical styles:

• 🌋 Stromboli — mild, rhythmic bursts launching incandescent lava bombs and spatter at low velocity. Named after Stromboli volcano in Italy, which has erupted almost continuously for 2000 years.
• 🔥 Hawaiian — sustained lava fountains feeding lava flows. Low viscosity basalt allows quiet effusion punctuated by dramatic fire fountains up to 500 m tall.
• 💥 Vulcanian — discrete, cannon-like explosions ejecting dense blocks and ash. A plug of solidified lava or viscous magma is periodically blasted out as pressure builds below.
• ☁️ Plinian — mega-eruptions sustained by vigorous magma fragmentation. The eruption column rises tens of kilometres into the stratosphere, injecting sulfur aerosols that can cool global climate. Named after Pliny the Elder, who died observing the 79 CE Vesuvius eruption.

Particle System

The simulation maintains a pool of up to 3000 particles, each belonging to one of four physical types:

• Lava bomb (orange-red) — high mass, ballistic trajectory, converts to pyroclastic flow on ground impact. Affected by gravity and wind drift.
• Ash (grey) — buoyant at high altitude, slow settling, wide lateral spread. Drifts with simulated wind and rains down gradually.
• Gas cloud (yellow-green) — rises rapidly, fades as it disperses. Models SO&sub2; and steam plumes.
• Pyroclastic flow (dark orange) — dense hot gas-particle mix that hugs the ground and accelerates downslope. The most lethal volcanic hazard; modelled as a low-angle, high-speed ground-hugging particle stream.

Vertical launch velocities are sampled from a preset-dependent distribution and multiplied by the user’s Eruptive Force slider. A weak horizontal wind field (0.008 px/frame) drifts all particles to one side, producing the characteristic asymmetric ash plume seen in downwind photographs.

Magma Pressure Buildup

Between eruptions, magma pressure builds at a rate set by the Pressure Rate slider. When pressure reaches 100%, the volcano automatically erupts — launching a burst of particles whose count and type distribution depend on the eruption style preset. This automatic mechanism models the cycle of recharge and release in a closed volcanic conduit. During an active eruption, pressure depletes rapidly; it remains at zero while particles are still airborne, then begins rebuilding as the vent clears.

What to Try

• Select Plinian and set Pressure Rate to maximum: the volcano erupts frequently with massive ash columns. Lower Pressure Rate and watch fewer but more intense individual eruptions.
• Switch to Stromboli and set Viscosity high: the eruptions become very brief, throwing only a handful of lava bombs each time — just like the real Stromboli.
• Hit the Erupt button manually in any mode to trigger an on-demand eruption regardless of the current pressure level.
• Watch pyroclastic flows on Vulcanian and Plinian presets — the dense particle streams hug the terrain and accelerate downslope, always concentrated on the steeper side of the cone.

Technical Notes

• Tsunami boundary absorption — the absorbing layer uses a sponge function: cells within 10 cells of each grid edge apply a damping factor damp < 1 per step. The factor is computed as 1 − 0.01 × (1 − d/10)² where d is the distance from the edge, giving a quadratic taper from full absorption at the boundary to no absorption at d = 10.
• Walker Delta phasing — the RAAN of the p-th plane (0-indexed) is p × 2π / P. The initial true anomaly of the s-th satellite in plane p is (s / S) × 2π + (p × F / T) × 2π, where F is the Walker phasing parameter (0 in the Starlink-class preset, 1 for GPS, 0 for Galileo, 1 for Polar). This ensures satellites are evenly distributed in both in-plane position and between planes.
• Volcano particle pool — dead particles (ptype = 0) are recycled to the front of the active particle list to avoid memory allocation during eruption bursts. At 3000 active particles the pool is full; new eruption particles overwrite the oldest active ones, preventing unbounded memory use.
• All Wave 33 simulations ship with full EN + UK pages and are registered in simulations.json.

Tags

Tsunami Shallow Water Equations Wave Propagation Bathymetry Earthquake Satellite Constellation Walker Delta Keplerian Orbits Three.js GPS Starlink Volcano Eruption Pyroclastic Flow Particle Simulation Wave 33

Wave 34 Preview

Three simulations planned for Wave 34:

• Rock Cycle — animated diagram of the geological rock cycle: uplift, erosion, sedimentation, burial, metamorphism, and magmatic melting along interactive timescale controls.
• Hydrothermal Vent — deep-sea black smoker simulation with chemosynthetic ecosystem: mineral chimney growth, superheated fluid convection, and vent fauna responding to temperature gradients.
• MPM Fluid — Material Point Method fluid simulation: a hybrid particle-grid solver that handles large deformations, free surfaces, and fluid-solid interaction in a single unified framework.

All Wave 34 simulations will ship with EN + UK pages on launch day.