6.2 Thermohaline Circulation

Step 1: Hydrostatic Pressure in Two Columns

Consider two water columns (equatorial and polar) connected at depth. Each column is in hydrostatic balance. The pressure at depth $z$ in column $i$ is:

$$p_i(z) = p_{atm} + g\int_z^0 \rho_i(z')\,dz'$$

Step 2: Linear Equation of State

The density of seawater depends on temperature and salinity through the linearized equation of state:

$$\rho = \rho_0\bigl[1 - \alpha(T - T_0) + \beta(S - S_0)\bigr]$$

where $\alpha \approx 2\times10^{-4}$ K$^{-1}$ is the thermal expansion coefficient and $\beta \approx 7.6\times10^{-4}$ PSU$^{-1}$ is the haline contraction coefficient. Cold, salty water is denser.

Step 3: Horizontal Pressure Gradient at Depth

If the polar column is denser than the equatorial column ($\rho_{polar} > \rho_{eq}$), a horizontal pressure difference develops at depth. At the surface, sea level adjusts so that pressure is nearly equal. At depth $-H$:

$$\Delta p = g H\,\Delta\rho = g H\,\rho_0\bigl[\alpha\,\Delta T - \beta\,\Delta S\bigr]$$

Step 4: Geostrophic Overturning Flow

This pressure gradient drives a deep flow from high-latitude (high pressure at depth) toward low-latitude. In geostrophic balance on a rotating Earth, the thermal wind relation connects vertical shear to horizontal density gradients:

$$f\frac{\partial v}{\partial z} = -\frac{g}{\rho_0}\frac{\partial\rho}{\partial x}$$

Integrating vertically gives the baroclinic transport, which is the fundamental driver of the thermohaline overturning.

Step 5: Overturning Transport Scaling

A scaling argument for the MOC strength relates the overturning to the meridional density contrast and the depth of the thermocline $D$:

$$\Psi_{MOC} \sim \frac{g\,\Delta\rho\,D^2}{\rho_0\,f}$$

For $\Delta\rho/\rho_0 \sim 2\times10^{-3}$, $D \sim 1000$ m, and $f \sim 10^{-4}$ s$^{-1}$, this gives $\Psi_{MOC} \sim 20$ Sv, consistent with RAPID observations of ~18 Sv.

Step 1: Non-Dimensionalize the Box Model

Define anomaly variables $\Delta T = T_1 - T_2$ and $\Delta S = S_1 - S_2$. Subtracting the box equations and non-dimensionalizing time by the restoring timescale $1/\gamma$:

$$\frac{d(\Delta T)}{dt} = 1 - \Delta T - |\Delta T - \delta\,\Delta S|\,\Delta T$$

$$\frac{d(\Delta S)}{dt} = \mu - \Delta S - |\Delta T - \delta\,\Delta S|\,\Delta S$$

where $\delta = \beta/\alpha$ is the salinity-to-temperature density ratio and $\mu$ is the dimensionless freshwater forcing.

Step 2: Steady-State Condition

At steady state, setting the time derivatives to zero and defining the flow $q = \alpha\,\Delta T - \beta\,\Delta S$, the system reduces to a single equation for $q$. Using $\Delta T = \Delta T^*/(1 + |q|)$ and $\Delta S = \Delta S^*/(1 + |q|) + F_w/(1 + |q|)$:

$$q = \frac{\alpha\,\Delta T^* - \beta\,\Delta S^*}{1 + |q|} - \frac{\beta\,F_w}{1 + |q|}$$

Step 3: Cubic Equation for Flow Strength

Rearranging for the thermally-dominated regime ($q > 0$) and the salinity-dominated regime ($q < 0$) separately, one obtains a cubic equation in $|q|$:

$$|q|(1 + |q|) = \alpha\,\Delta T^* - \beta(\Delta S^* + F_w) \quad (q > 0)$$

$$|q|(1 + |q|) = \beta(\Delta S^* + F_w) - \alpha\,\Delta T^* \quad (q < 0)$$

For certain ranges of $F_w$, both equations have positive solutions, meaning two stable equilibria coexist.

Step 4: Linear Stability Analysis

Perturbing around a steady state $(\overline{\Delta T}, \overline{\Delta S})$ and linearizing, the Jacobian matrix $\mathbf{J}$ determines stability. The eigenvalues are:

$$\lambda_{1,2} = -\frac{1}{2}\text{tr}(\mathbf{J}) \pm \frac{1}{2}\sqrt{\text{tr}(\mathbf{J})^2 - 4\det(\mathbf{J})}$$

A steady state is stable when both eigenvalues have negative real parts. The system loses stability when $\det(\mathbf{J}) = 0$ (saddle-node bifurcation), which defines the critical freshwater forcing $F_w^{crit}$.

Step 5: Saddle-Node Bifurcation and Hysteresis

At the critical freshwater forcing, the thermally-dominated equilibrium (strong AMOC) collides with an unstable equilibrium and both vanish. The system jumps discontinuously to the salinity-dominated state (collapsed AMOC):

$$F_w^{crit} = \frac{\alpha\,\Delta T^*}{4\beta} \quad \text{(for the simplified symmetric case)}$$

Crucially, reducing $F_w$ below $F_w^{crit}$ does not restore the AMOC -- the reverse bifurcation occurs at a lower $F_w$, creating a hysteresis loop. This irreversibility is why AMOC collapse is considered a climate tipping point.

Step 6: Temperature vs Salinity Feedback

The key asymmetry is that temperature is restored toward the equilibrium by air-sea heat flux (negative feedback), while salinity responds to a fixed freshwater flux (no restoring feedback). This means:

$$\text{Temperature: } \frac{d(\Delta T)}{dt} \propto -\gamma\,\Delta T \quad \text{(self-correcting)}$$

$$\text{Salinity: } \frac{d(\Delta S)}{dt} \propto F_w \quad \text{(externally forced, no restoring)}$$

This mixed boundary condition (restoring for $T$, flux for $S$) is the fundamental reason for the existence of multiple equilibria in the thermohaline circulation.