Thermohaline Circulation — Two-Box Model
Stommel bistability — freshwater forcing can collapse the Atlantic overturning
q = — | T_E−T_P = — | S_E−S_P = —
Stommel (1961) two-box model: equatorial (E) and polar (P) boxes.
Temperature: dT_E/dt = λ(T*_E − T_E) − |q|ΔT; dT_P/dt = λ(T*_P − T_P) + |q|ΔT.
Salinity: dS_E/dt = F − |q|ΔS; dS_P/dt = −F + |q|ΔS.
Overturning: q = k(α·ΔT − β·ΔS) where α (thermal) and β (haline) are expansion coefficients.
Two stable states: ON (thermally driven, warm-salty pole sinks) and OFF (haline-driven or collapsed).
Freshwater forcing F weakens the salinity contrast driving collapse via hysteresis.
Left: system state in phase space. Right: bifurcation diagram. Top: box visualization.