Year 0Overturning 0 SvΔT 0 KΔS 0 psuΦ 0 Sv—FPS 0

Cut-away of an ocean basin: pole and ice shelf on the left, tropics on the right. Glowing streaks trace the overturning loop: warm surface water flows poleward, cools, sinks and cascades down the slope as a dense overflow, returns along the bottom and upwells in the tropics. Streak speed follows the model flow q; if q turns negative the loop runs backwards.

Stommel Two-Box Ocean: Brine Plume, Overflow and Overturning Collapse

Add freshwater at the pole and watch the density-driven overturning weaken, collapse and flip, then try to bring it back.

Forcing

Time

Live readouts

Overturning flow q

Density terms

Experiment: hysteresis

Headless slow ramp of the freshwater input from 0 to 1.6 Sv and back (60 000 model years per leg), starting from the thermally driven state with your other settings.

Independent numerical checks

Controls

Drag or one finger: orbit; wheel or pinch: zoom. Camera buttons jump to the whole basin, the sinking region, the tropical upwelling or a top-down look at the seabed.
Φ is the freshwater input (ice melt plus rain surplus) to the polar box. Presets change Φ without resetting the ocean, so you can watch the transition. "Auto ramp" sweeps Φ up and down slowly.
Time scale is model years per real second; the flow is slow, so thousands of years pass in a minute.

The science

Two boxes. Stommel (1961): a warm, salty low-latitude box E and a cold polar box P, each of volume V = 1.4×10¹⁷ m³, exchanging water at rate |q|. Density: ρ = ρ₀(1 − αT + βS), α = 2×10⁻⁴ K⁻¹, β = 7.6×10⁻⁴ psu⁻¹. The flow follows the density contrast: q = k (αΔT − βΔS), ΔT = T_E − T_P, ΔS = S_E − S_P, k = 3.4×10⁹ m³ s⁻¹ (chosen so that q ≈ 16 Sv at Φ = 0; 1 Sv = 10⁶ m³ s⁻¹). Positive q means surface water flows to the pole and sinks there.

Budgets. dT_E,P/dt = (T*_E,P − T_E,P)/τ + (|q|/V)(T_P,E − T_E,P) (surface heat restored to T* = 15 ± ΔT*/2 °C in time τ) and dS_E/dt = ΦS₀/V + (|q|/V)(S_P − S_E), dS_P/dt = −ΦS₀/V + (|q|/V)(S_E − S_P), S₀ = 35 psu. Salt is conserved: S_E + S_P never changes.

Steady states. With ΔT = ΔT*/(1 + 2|q|τ/V) and ΔS = ΦS₀/|q|, steady flow solves q = k(αΔT(q) − βΔS(q)). Freshening the pole lowers the density contrast; with ΔT held fixed, the thermally driven branch exists only while Φ ≤ k(αΔT)²/(4βS₀) (about 0.7 Sv here). Beyond it the loop collapses and settles in a weak salt-driven state with reversed flow. Coming back needs much less freshwater than going out needed: that gap is hysteresis. The panel lists all steady roots.

Dense overflow. Cold, salty water formed at the pole spills down the continental slope as a reduced-gravity current: U = √(g′ h sinθ /(C_D + E)), g′ = g(αΔT − βΔS), h = 50 m, C_D = 0.003, entrainment E = 0.1. It sets the readout of plume speed only; the drawn streak speed follows q.

Simplifications: two well-mixed boxes (no latitude structure, no wind-driven gyres, no Coriolis balance, no mixing in the deep ocean), constant α and β (no thermobaricity or cabbeling), freshwater input as a constant rate. The volumes and the 0.7 Sv threshold are illustrative of the mechanism, not predictions for the real Atlantic circulation. The streak colours are schematic. Assets built with Blender; scene by Three.js.