Thermohaline circulation - 2D Stommel box model

Stommel's two-box ocean

Stommel (1961) modelled the ocean as an equatorial box and a polar box. Let y = Teq - Tpole and x = Seq - Spole. The pole is denser when αy > βx, and the overturning rate is r = κ(αy - βx) (1/yr; r > 0 sinks at the pole). The two contrasts obey dy/dt = (y0 - y)/τ - |r| y and dx/dt = F - |r| x: the atmosphere restores temperature, freshwater/evaporation forcing F sets the salt contrast, and the flow itself mixes both away.

Why it can tip

Heat relaxes quickly but salt has no restoring force, only the flow. When F grows, the flow weakens, the salt contrast stops being flushed, and x grows: a positive feedback. Steady states satisfy the cubic τr3 + r2 + (κβFτ - καy0) r + κβF = 0 (and its mirror for reversed flow). For a range of F three roots exist: stable "on", unstable, and stable "reversed/off". Ramping F up collapses the circulation at a saddle-node; ramping it down recovers it at a smaller F (hysteresis).

Units

Rates are scaled so one turnover of the polar box per year is about 150 Sv (assumed volume 4.7e15 m³); the default state gives roughly 13 Sv, comparable to the Atlantic overturning of about 15-18 Sv. Salinity forcing is scaled as F S0/35.

Honest simplifications: two well-mixed boxes, no wind-driven gyres, no geometry, no Southern Ocean, linear equation of state, constants α = 2e-4 /K, β = 8e-4 /permille, κ = 40 /yr, τ = 10 yr tuned for illustration. Real-world tipping thresholds are uncertain and this model cannot predict them.

• Set F = 0.06 on the default preset, then add a freshwater pulse: the circulation collapses even though F did not change.
• Use "Ramp F up then down" to trace the hysteresis loop on the bifurcation diagram.
• Cool the equator (smaller y0) and see the tipping point move.