This is a side-on cross-section of a real, uneven lake floor — not a flattened copy of the 3D tank. Each bubble's fate is still the classic viscous-rise vs. dissolution competition, but here the water column depth itself varies along the transect, and a horizontal thermocline current shears rising plumes sideways, so where a bubble is born on the undulating floor changes both how far it must climb and which way it drifts:
Rise speed: v(r) = min( 2g(ρw−ρg)r²/(9μ) , v_plateau ) — Stokes below ~0.39mm, capped above it
Radius: dr/dt = −k/r ⇒ closed form r(t) = √(r0² − 2kt)
Floor: depth(x) = D0 + 0.3·D0·sin(2πx/λ) — an undulating lake-bottom, not a flat tank
Drift: dx/dt = U0·sign(sin 2πx/λ)·exp(−((z−z_therm)/σ)²) — a flat thermocline shear layer
- Below about 0.39mm radius a bubble is still in the slow, size-squared Stokes regime; above it, rise speed saturates near the turbulent-wake plateau (~25 cm/s) and further growth barely changes speed — only the dissolution reserve keeps growing, which is why initial radius mostly buys survival time, not extra speed, once past the transition.
- Mean water depth sets the bathymetry: basins are ~30% deeper than the mean, shelves ~30% shallower, so a seep's local floor depth (not just the slider) decides how much dissolution reserve a bubble needs.
- Thermocline current is a flat shear band roughly 40% of the way up the water column — alternating direction over each bathymetric basin/shelf pair, like real internal-wave-driven circulation cells — that bends rising plumes into the S-shaped drift you see in the trails, independent of the vertical rise physics.
- The inset curve is not decorative: it is computed live by bisecting the same rise/dissolve ODE for each depth to find the exact radius that just barely survives — the dot shows whether your current depth/radius setting sits above (survives) or below (dissolves) that boundary.
- Winter ice cover seals the flat surface; survivors freeze onto its underside near wherever the current carried them, not necessarily above the seep that made them.
Real-world relevance: this depth-and-current-dependent survival boundary is exactly why field researchers can map thermokarst-lake seep fields by the geometry of frozen bubble towers in the ice — the towers mark not the seep locations themselves but where the shear layer happened to carry the survivors.