Characterising meridional overturning bistability using a minimal set of state variables

Abstract

A close approximation of key state variables and salt fluxes for both the North Atlantic Deep Water (NADW) “on” and “off” states in a General Circulation Model (GCM) is constructed, yielding a natural stability condition. Here, stability is linked to the effect of feedbacks on infinitesimal salinity anomalies on the average Atlantic salinity. The stability condition simply states that the total advective salt feedback must be negative in each steady state, ensuring stability by damping the growth of infinitesimal salinity perturbations. However, a decomposition of the salt feedback into three components shows that only the interaction between the mean salinity and infinitesimal perturbations of the meridional flow have the potential to render a state unstable, holding the key to state transitions. In contrast, the interaction between the mean meridional flow and infinitesimal salinity perturbations yields a negative (stabilising) component feedback. Similarly, the gyre salt flux also stabilises the overturning states. Furthermore, the nodes limiting the “on” and “off” state regimes in the GCM can be accurately computed based on linear fits of basic state variables and the gyre salt flux. It is shown that the NADW “on” state closest to collapse must be contained within a neighbourhood of fresh water exporting states. Finally, the role of temperature in the bistability structure is elucidated.

Appendix: Linear relationships

Local salinities, such as that of the North Atlantic, are assumed to be linearly related to the average Atlantic salinity in the hysteresis parameter experiment. Here, a simplified theoretical argument is offered to suggest that this is plausible. Consider the Atlantic divided into two meridionally adjacent boxes of the depth of the overturning and each spanning the entire basin width, and having a volume V _i . Denote the combined volume of the Atlantic boxes by V. One could represent the subtropical Atlantic, and the other the high northern latitudes. Let the southern box have salinity S ₂ and surface flux F _s,2, while the northern box has salinity S ₁ and surface flux F _s,1. Also assume a constant salinity S ₃ for the Southern Ocean, which is in contact with the southern box. There is no further northward exchange from the northern box. Assume an anomalous flux H is applied to F _s , consisting of component fluxes over each box i of H _i  = c _i H for constants c _i (that is, the anomalous flux over each box is proportional to the total flux). Denote the change in S _i associated to H _i (compared to H = 0) by δS _i .

First, assume only a simple diffusive exchange with coefficients r _i between the boxes i and i + 1 (mimicking the gyres). Then the exchange between the SO box and the southern box in the Atlantic is r ₂(S ₃ − S ₂) and r ₁(S ₂ − S ₁) between box 2 and 1. Flux balance with surface fluxes and re-arrangement of the terms gives: $S_{1} = S_{2} + r_{1}^{-1}F_{s,1}$ and $S_{2} = S_{3} + r_{2}^{-1}F_{s}.$ Therefore, $\delta S_{1} = r_{1}^{-1}c_{1}H + \delta S_{2}$ and $\delta S_{2} = r_{2}^{-1}H$ as S ₃ is constant. Also, $\delta \bar{S} = V^{-1}(V_{1}\delta S_{1} + V_{2}\delta S_{2} ) = (\frac{V_{1}}{V} r_{1}^{-1} c_{1} +r_{2}^{-1} )H$ As a result, $\frac{\delta S_{i}}{\bar{S}}$ are constant and the relationships between S _i and $\bar{S}$ is linear. This process can be generalised to more boxes. In conclusion, the linearities between local salinities and the basin-wide average salinity is easy to show for a simple diffusive box model.

We now incorporate the zeroth order effect of the AMOC, and estimate the error in the coefficients arising from this approximation. Let the gyre transport from the southern box to the northern box be γ G(S ₂ − S ₁), where G is the gyre volume transport and γ some constant incorporating basin width, depth and other factors. Similar, considering the choice of box depth, let the MOC transport be γ M(S ₂ − S ₁). Then S ₁ can be expressed as S ₁ = S ₂ + γ⁻¹(α + M)⁻¹ F _s,1. Taylor expansion of the second term on the right in terms of a perturbation M = M ₀ + δM around some average M ₀ yields

$$ \gamma^{-1}(\alpha +M_{0})^{-1}F_{s,1} - \gamma^{-1}(\alpha +M_{0})^{-2}F_{s,1} \delta M + \gamma^{-1} (\alpha- M_{0})^{-3}F_{s,1} \delta M^{2} - ... $$

Using a gyre strength G = 40 Sv, overturning M ₀ = 10 Sv and perturbation δM = 5 Sv, an estimate of the first order term (second term) yields 10% of the zeroth order term (first), while the the second order term (third term) is 1%. We accept the 10% error and use a zeroth order approximation, yielding:

$$ S_{1} = S_{2} + \gamma^{-1}(\alpha +M_{0})^{-1}F_{s,1} $$

Renaming the coefficient r ₁ ≡ γ(α + M ₀) and r ₂ ≡ γ′(α′ + M ₀) , where γ′ and α′ are the equivalent parameters to γ and α but appropriate to the interface between the Southern Ocean and the southern box in the Atlantic, we recognise the diffusive model described above, where the linearities hold. We saw that we can expect a 10% error in the terms $r_{i}^{-1}.$ In practice, this leads to a smaller error in the quotient $\frac{S_{1}}{\bar{S}},$ as the r _i terms can be divided out if r ₁ ≈ r ₂ . Note that the reverse circulation of the OFF state should yield a smaller error, as δM is smaller. In summary, it is reasonable to expect the linear relationships between local salinity of a box and the basin-wide average salinity under changes in the control parameter H in a gyre-dominated case (assuming unchanging gyres), and deviations from this linearity due to the AMOC may be small. Note that this is contrary to the speculation about the ON state stated in Sijp et al. (2011a).