Meridional overturning circulation: stability and ocean feedbacks in a box model

Abstract

A box model of the inter-hemispheric Atlantic meridional overturning circulation is developed, including a variable pycnocline depth for the tropical and subtropical regions. The circulation is forced by winds over a periodic channel in the south and by freshwater forcing at the surface. The model is aimed at investigating the ocean feedbacks related to perturbations in freshwater forcing from the atmosphere, and to changes in freshwater transport in the ocean. These feedbacks are closely connected with the stability properties of the meridional overturning circulation, in particular in response to freshwater perturbations. A separate box is used for representing the region north of the Antarctic circumpolar current in the Atlantic sector. The density difference between this region and the north of the basin is then used for scaling the downwelling in the north. These choices are essential for reproducing the sensitivity of the meridional overturning circulation observed in general circulation models, and therefore suggest that the southernmost part of the Atlantic Ocean north of the Drake Passage is of fundamental importance for the stability of the meridional overturning circulation. With this configuration, the magnitude of the freshwater transport by the southern subtropical gyre strongly affects the response of the meridional overturning circulation to external forcing. The role of the freshwater transport by the overturning circulation (M _ov ) as a stability indicator is discussed. It is investigated under which conditions its sign at the latitude of the southern tip of Africa can provide information on the existence of a second, permanently shut down, state of the overturning circulation in the box model. M _ov will be an adequate indicator of the existence of multiple equilibria only if salt-advection feedback dominates over other processes in determining the response of the circulation to freshwater anomalies. M _ov is a perfect indicator if feedbacks other than salt-advection are negligible.

Appendix: Derivation of E a | 0 M_ov and \(\Updelta_E\)

To compute the E _a value needed to bring M _ov to zero with q _S  > 0, the system (10) has to be solved.

In the non diffusive limit κ → 0 Eq. (10e) reduces to q _S  − q _N  = 0, a second order equation in D, which can be solved giving as positive solution:

$$ \lim_{\kappa \to 0}D|_{M_{ov}}^0=- \frac{1}{2} \frac{1}{\left(T_{ts}-T_n\right)\alpha \eta} \frac{A_{GM} L_{xA}}{L_y} \left[1-\sqrt{1+4 q_{Ek}\left(\frac{L_y}{A_{GM} L_{xA}}\right)^2\left(T_{ts}-T_n\right)\alpha \eta} \right]. $$

This result can be substituted into Eqs. (10a–10d), and salinity can be eliminated giving:

$$ E_s \left[r_N - r_S + \left(T_{ts}-T_n\right) \left(D|_{M_{ov}}^0\right)^2 \alpha \eta\right] = E_a \left[r_N + r_S + \left(T_{ts} - T_n\right) \left(D|_{M_{ov}}^0\right)^2 \alpha \eta\right]. $$

(13)

The latter is solved for E _a giving E _a | ⁰ _{M_ov} in the case of κ → 0:

$$ \lim_{\kappa \to 0}E_a|_{M_{ov}}^0= E_s\frac{r_N - r_S + \left(T_{ts}-T_n\right) \left(D|_{M_{ov}}^0\right)^2 \alpha \eta}{r_N + r_S + \left(T_{ts}-T_n\right) \left(D|_{M_{ov}}^0\right)^2 \alpha \eta} =E_s\frac{r_N - r_S +q_{N}|_{M_{ov}}^0}{r_N + r_S +q_{N}|_{M_{ov}}^0}. $$

(14)

(14) can be combined with Eq. (9) to give $\Updelta_E$ in the non diffusive limit, Eq. (11).

The method outlined for the non diffusive limit can be used for the model including vertical diffusion as well. Also in this case, the mathematics involved is very simple on conceptual grounds, but the large expressions obtained render the problem tedious with pencil and paper. The final result is obtained with Mathematica software. Eq. (10d), in this case a third order algebraic equation, can be solved for pycnocline depth, and the positive solution reads:

$$ \begin{aligned} D|_{M_{ov}}^0 &\quad = \frac{A_{GM} L_{xA}}{3 L_y \left(T_n-T_{ts}\right) \alpha \eta}\\ &-\frac{\left(1+i \sqrt{3}\right) \left(A_{GM}^2 L_{xA}^2+3 L_y^2 q_{Ek} \left(T_{ts}-T_n\right) \alpha \eta \right)} {\left[ \begin{array}{l} 3\sqrt[3]{4}L_y \left(T_n-T_{ts}\right) \alpha \eta \left(2 A_{GM}^3 L_{xA}^3 + 9 A_{GM} L_{xA} L_y^2 q_{Ek} \left(T_{ts}-T_n\right) \alpha \eta \right.\\ -3 L_y^{3/2} \left(T_n-T_{ts}\right) \alpha \eta \left(9 A L_y^{3/2} \left(T_n-T_{ts}\right)^2 \alpha \eta \kappa - \sqrt{3}\left(T_n-T_{ts}\right)\left(A_{GM}^2 L_{xA}^2 L_y q_{Ek}^2 \right. \right.\\ +4 A A_{GM}^3 L_{xA}^3 \kappa - 18 A A_{GM} L_{xA} L_y^2 q_{Ek} \left(T_n-T_{ts}\right) \alpha \eta \kappa\\ \left. \left. \left. - L_y^3 \left(T_n-T_{ts}\right) \alpha \eta \left(4 q_{Ek}^3 + 27 A^2 \left(T_n-T_{ts}\right) \alpha \eta \kappa^2 \right)\right)^{1/2}\right)\right)^{1/3} \end{array}\right]}\\ &\quad +\frac{1}{6 \sqrt[3]{2}L_y \left(T_n-T_{ts}\right) \alpha \eta } \left(i\sqrt{3} -1\right) \left\{2 A_{GM}^3 L_{xA}^3+9 A_{GM} L_{xA} L_y^2 q_{Ek} \left(T_{ts}-T_n\right) \alpha \eta \right.\\ &\quad -3L_y^{3/2} \left(T_n-T_{ts}\right) \alpha \eta \left[ 9 A L_y^{3/2} \left(T_n-T_{ts}\right) \alpha \eta \kappa +\sqrt{3} \left(-A_{GM}^2 L_{xA}^2 L_y q_{Ek}^2 \right. \right.\\ &\quad -4 A A_{GM}^3 L_{xA}^3 \kappa +18 A A_{GM} L_{xA} L_y^2 q_{Ek} \left(T_n-T_{ts}\right) \alpha \eta \kappa\\ &\quad \left. \left. \left. + L_y^3 \left(T_n-T_{ts}\right) \alpha \eta \left(4 q_{Ek}^3 +27 A^2 \left(T_n-T_{ts}\right) \alpha \eta \kappa ^2\right)\right)^{1/2}\right]\right\}^{1/3} \end{aligned} $$

After eliminating salinity, Eqs. (10a–10d) reduce again to (13), and E _a | ⁰ _{M_ov} is still given by Eq. (14), but with D| _{M_ov} ⁰ and q _N | ⁰ _{M_ov} for the finite vertical diffusion case.

The difference between E _a | ⁰ _{q_N} and E _a | ⁰ _{M_ov} then gives $\Updelta_E$, which can be written as:

$$ \begin{aligned} \Updelta_E= &\frac{2 E_s r_S}{r_N+r_S+ \left(T_{ts}-T_n\right)\left(D|_{M_{ov}}^0\right)^2 \alpha \eta}\\ &\quad - \frac{A_{GM} D|_{q_N}^0 L_{xA} \left[A_{GM} D|_{q_N}^0 L_{xA} +L_y\left( r_S -q_{Ek}\right)\right]\left[r_N \left(T_n-T_{ts}\right)\alpha +2 E_s S_0 \beta \right]}{\left[\left(A_{GM} D|_{q_N}^0 L_{xA}\right)^2+L_y^2 q_{Ek} r_N +A_{GM} D|_{q_N}^0 L_{xA} L_y \left(r_S -q_{Ek}\right)\right] S_0 \beta}\\ = & \frac{2 E_s r_S}{r_N +r_S+q_{N}|_{M_{ov}}^0}\\ &\quad - \frac{q_{e}|_{q_N}^0\left(q_{e}|_{q_N}^0 +r_S -q_{Ek}\right)\left[r_N \left(T_n-T_{ts}\right) \alpha+2 E_s S_0 \beta\right]}{\left[\left(q_{e}|_{q_N}^0\right)^2+ q_{Ek}r_N+ q_{e}|_{q_N}^0\left(r_S-q_{Ek}\right)\right] S_0 \beta}. \end{aligned} $$

In the limit of no vertical diffusion, q _e | ⁰ _{q_N}  = q _Ek if q _N  = 0, and thus $\Updelta_E$ reduces to:

$$ \lim_{\kappa \to 0}\Updelta_E= r_S \left[ \frac{2 E_s}{r_N + r_S + q_{N}|_{M_{ov}}^0} - \frac{2 E_s}{r_N +r_S} + \frac{r_N \left(T_{ts}-T_n\right) \alpha}{\left(r_N+r_S\right) S_0 \beta}\right], $$

which can be written in the form of Eq. (11) when q _N | ⁰ _{M_ov} is written explicitly.