Stommel Box Model: Thermohaline Circulation Bistability

Forcing

Freshwater forcing η₂ 1.30

Thermal forcing η₁ 3.00

Simulation speed 1.0×

Circulation State

— warming up —

Flow q = x − y

0.00

Illustrative Sv

0.0

ΔT index (x)

0.00

ΔS index (y)

0.00

Model time elapsed

0 yr

How it works

This is the classic Stommel two-box model (Stommel 1961; simplified form after Cessi 1994) of the thermohaline "conveyor belt". A polar box and an equatorial box exchange heat and salt with a flow driven by their density difference:

dx/dt = (η1 − x(1+|x−y|)) / ε     (pole↔equator temperature contrast x)
dy/dt =  η2 − y(η3+|x−y|)          (pole↔equator salinity contrast y)
q = x − y                          (density-driven overturning flow)

Here x is the temperature difference and y the salinity difference between the boxes; both are restored toward forcing values η1 (thermal) and η2 (freshwater) but eroded by the flow itself. ε ≪ 1 (fixed at 0.1) makes x relax in years while y — controlled by slow freshwater/evaporation input — drifts over centuries, exactly like the real ocean's heat vs. salt memory.

q > 0 ("ON") — temperature dominates density: cold, salty water sinks at the pole, driving a strong overturning cell (today's real AMOC, ~15–20 Sv).
q < 0 ("OFF / reversed") — enough fresh water dilutes the polar box that salinity dominates instead, sinking stops or reverses, and heat transport to high latitudes collapses.
For a range of η2 the ON and OFF states are both stable at the same forcing (bistability) — a big enough freshwater pulse (the button) can flip the system permanently, even if you never change η2 again. This hysteresis is the toy-model analogue of real concerns that Greenland/Arctic meltwater could trigger an irreversible AMOC weakening.

The 3D loop shows water parcels advected along the conveyor at a speed and direction set by q; colour shifts from warm (surface) to cold (deep) as they travel, and the loop reverses direction when the circulation flips to the OFF state.

Warm surface water Cold deep water Equator box Polar box

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