Revisiting meridional overturing bistability using a minimal set of state variables: stochastic theory

Abstract

A stochastic analytical model of the Atlantic meridional overturning circulation (AMOC) is presented and tested against climate model data. AMOC stability is characterised by an underlying deterministic differential equation describing the evolution of the central state variable of the system, the average Atlantic salinity. Stability of an equilibrium implies that infinitesimal salinity perturbations are damped, and violation of this requirement yields a range of unoccupied salinity states. The range of states is accurately predicted by the analytical model for a coupled climate model of intermediate complexity. The introduction of climatic noise yields an equation describing the evolution of the probability density function of the state variable, and therefore the AMOC. Given the hysteresis behaviour of the steady AMOC versus surface freshwater forcing, the statistical model is able to describe the variability of the AMOC based on knowledge of the variability in the forcing. The method accurately describes the wandering between AMOC-On and AMOC-Off states in the climate model. The framework presented is a first step in relating the stability of the AMOC to more observable aspects of its behaviour, such as its transient response to variable forcing.

Appendices

Appendix 1: Calculation of diffusivity κ

In Sect. 4 we add a stochastic salt flux H _rnd to the surface salt flux of the numerical model and developed a stochastic version of the deterministic theory, involving a Fokker Planck equation, Eq. 4, described in Sect. 3 to determine the validity of that theory. The equation requires an estimate of the effective diffusivity associated with the spreading effect of H _rnd on the state S. Here, we present an attempt at calculating this value based on the magnitude of H _rnd by considering the smearing effect of the stochastic salt flux H _rnd on S. We note that Monahan and Culina (2011) offer a systematic method of stochastic averaging, not used here, to mathematically express the effect of the fast variables on the evolution of the longer time scale variables and that works also with non-Gaussian and auto-correlated noise.

The random values employed in H _rnd have a uniform distribution between limits −L < H _rnd  < L, where we have chosen L = 34.0 × 10⁻⁸ kg/m²/s in the numerical model (all model salt fluxes are in units of kg of salt per m² per s). This choice is to allow AMOC state transitions but retain distinguishable peaks for the two states. H _rnd is updated every 10 years by drawing a new random number, and is added uniformly in time and space over each entire period of 10 years and over the entire Atlantic surface. This is to cause minimal disruption to the ocean model, allow smooth numerical integration and to keep calculation times low in our long model integrations.

To consider the problem of associating a diffusivity κ in S-space with H _rnd , we will consider the immediate net effect of the application of H _rnd on the change in S in isolation from the long time scale responsive forcing encapsulated in F(S) that comes into play after S has changed more significantly after successive 10 year intervals where H _rnd is updated. We use the resulting κ value as the diffusivity in the Fokker-Planck equation, Eq. 4 in the main text.

Recall that we consider the Atlantic basin to include the Arctic and bounded only at approximately 30° south, as Bering Strait is closed in our model. The cumulative change $\Updelta S_i$ in S due to H _rnd alone from the beginning to the end of each 10 year interval, labelled with index i, is $\Updelta S_i = \frac{A}{V}(H_{rnd})\times10\times365\times24\times3,600\,\hbox{kg}/\hbox{m}^3.$ Here, A = 7.765 × 10¹³ m² is the Atlantic surface area and V = 2.492 × 10¹⁷ m³ is the Atlantic volume in the numerical model. However, during the 10 year period only a portion 0 < α < 1 of the anomaly $\Updelta S_i$ remains in the Atlantic as some of it has leaked out of the basin. The net combined effect is an effective change in S of $\alpha\Updelta S_i.$

To associate a diffusivity κ with the stochastic perturbations in the numerical model, we examine the discrete time development of S at 10 year time steps t _i for ${i \in {\mathbb{Z}}.}$ These new time steps greatly exceed the numerical model time step but are small compared to the millennial time scale of the AMOC processes under investigation (see Sect. 4) They coincide with each time the stochastic perturbation value is updated with a new random number and the state S has attained the value S _i . With this interpretation and choice of step sizes, the isolated system exhibits a random walk of S _i taking steps of varying sizes $\alpha\Updelta S_i.$ Approximating the fraction α by a constant value (see below), the step sizes $\alpha\Updelta S_i$ are uniformly distributed between $\pm \frac{\alpha AL}{V} \times10\times365\times24\times3,600\,\hbox{kg}/\hbox{m}^3.$ The standard deviation σ of the distribution of each $\Updelta S_i$ is $\sigma = 3^{-\frac{1}{2}} \frac{\alpha AL}{V} \times10\times365\times24\times3,600\,\hbox{kg}/\hbox{m}^{3}/10\,\hbox{year}.$

We can calculate an effective diffusivity κ to help approximate the average squared displacement during the random walk at 10 year step intervals and with varying step sizes in S _i . It is related to the expected displacement of S after i steps (assuming no displacement at i = 0 where we choose S ₀ = 0) via <S ² _i  >  = 2κ t _i . Here, κ has units (kg/m³)²/10 year as it represents diffusion in S-space and we model the system at 10 year time steps. However, as the values of H _rnd are uncorrelated at the 10 year steps, <S ² _i > also equals σ² t _i as it is the sum of the variances of the independent changes in S at each step. Therefore, κ = σ²/2 with units (kg/m³)²/10 year. In units of (kg/m³)²/s, diffusion κ in S space arising from H _rnd is $\frac{1}{2} 10\times365\times24\times3,600\,\sigma^2 = 5.9\times10^{-13} \alpha^2.$

In order to roughly estimate the fraction α of a typical anomaly that remains in the Atlantic for all simulations, we have performed a 10 year experiment where a passive tracer is added uniformly at the Atlantic surface in an identical manner and magnitude to a typical addition of H in the DRY configuration of our numerical model only (and without stochastic forcing). Here, we examine the fraction of the time rate of change in average Atlantic passive tracer concentration due to the passive tracer surface flux over its total rate of change (Fig. 10). The cumulative fraction of passive tracer that remains in the Atlantic in our numerical model simulation is α = 0.89, yielding an effective S change of $0.89\Updelta S_i.$ Therefore, the effective diffusivity κ for the non-isolated case must be κ = 0.89² × 5.9 × 10⁻¹³ (kg/m³)²/s = 0.47 × 10⁻¹³ (kg/m³)²/s. This is the value we have used for κ to generate the figures in the main text.

Fig. 10

Time series of the rate of (time) change of the average Atlantic passive tracer concentration divided by the rate of change expected from the passive tracer surface flux alone. This is used to estimate the fraction of tracer remaining inside the basin after a 10 year period

In the absence of the longer timescale dynamics encapsulated by F _H (S), the application of H _rnd would yield a diffusive process acting on ψ(S) with the diffusion coefficient κ calculated above, where S becomes increasingly delocalised and no steady ψ(S) is attained. Incorporating longer time scale oceanic adjustments that act on larger salinity changes, attempting to maintain S near its equilibrium values in the face of the above described random walk tendency (e.g. an AMOC adjustment responding to a net excursion in S arising from the cumulative effect of H _rnd ), yields the evolution equation for ψ(S) (Eq. 4) and elucidates how the state S remains in the vicinity of the attractors (AMOC-On and AMOC-Off).

Appendix 2: The glacial change in the disequilibrium flux

The glacial experiments require the application of a lower constant additional fresh water flux H to the climate model to maintain the same salinity states S compared to their Holocene counterparts (Fig. 2). This is equivalent to a lower disequilibrium salt flux $F_{H_0}(S)$ in the glacial experiments. To briefly examine this further, we will compare the Atlantic budget in the glacial to the Holocene. We shall decompose the Atlantic salt divergence minus the anomalous salt flux associated with H. First, the overturning salt transport component flux F _m (e.g. see Rahmstorf 1996) associated with the AMOC, sometimes referred to as the zonal component, is the vertical integral of the zonally integrated velocity times the zonally averaged salinity, evaluated at 32 °S. In addition, the subtropical gyre in the South Atlantic is responsible for a certain salt transport F _g across 32 °S, the azonal component. This is simply calculated by subtracting F _m from the total ocean salt import, F _ocn (a term that is easily diagnosed from models). The total oceanic salt divergence (salt flux into the Atlantic) is then F _ocn  = F _g  + F _m . Finally, the Atlantic salt budget contains a term associated with fresh water exchange at the air-sea interface, −F _atm . These components are thoroughly described in Sijp (2012), and are shown for our experiments in Fig. 11.

Fig. 11

Decomposition of the Atlantic salt divergence minus the anomalous salt flux associated with H into components related to the gyre, F _g (dots), the AMOC, F _m (crosses) and the air-sea interface, −F _atm (triangles). Note that F _atm appears with a minus sign, as F _atm denotes a fresh water flux into the ocean. Glacial climate model equilibria are indicated in blue, and the Holocene in red

The glacial F _m is reduced with respect to the Holocene in the AMOC-On states (branch on the right), whereas the descrepancy in this term is smaller and reversed in sign for the AMOC-Off states (left branch). In contrast, the glacial NADW outflow as a function of S is greater than in the Holocene in the AMOC-On states (figure not shown). This should not be confused with the outflow for a given H (i.e. the glacial and Holocene outflow in experiments with the same value of H), where the glacial rate is less. In contrast to what we find, the increased glacial outflow for a given S would suggest an increased F _m . But this holds only if the weighted vertical salinity gradient at 32°S remains unchanged (Sijp 2012). Therefore, it is the interaction between the spatial distribution of Atlantic salinity and the AMOC that differs between the glacial and Holocene experiments so as to reduce F _m in the glacial.

In addition to the glacial AMOC having a more freshening influence on the Atlantic for a given S, the glacial Atlantic is less evaporative than the Holocene, particularly for the AMOC-Off states, as −F _atm is positive in both experiments and greater in the Holocene. This may be due to the cooler Atlantic conditions that prevail in the glacial experiments (figures not shown). The gyre term remains relatively similar for the glacial and Holocene climates in the AMOC-On states. In the AMOC-Off states this term is somewhat greater in the glacial. As a result, F ^glac _atm (S) < F ^Hol _ocn (S) and F ^glac _atm (S) > F ^Hol _atm (S) (where the superscript “glac” indicates the glacial and “Hol” the holocene flux). Using F _ocn (S) − F _atm (S) = H (Eq. 1), we find that a lower fresh water flux H for S in equilibrium with H is required to to maintain the same glacial average salinity S as in the Holocene experiments. As a result, the glacial curve lies below the Holocene curve. This is indeed what we see in Fig. 2. In summary, smaller values of the fresh water flux H are required to maintain identical salinity states S in the glacial. This is because the Atlantic in the glacial experiments is less evaporative, particularly in the AMOC-Off states, and the different interaction between the AMOC and the spatial salinity distribution yields a reduced net influx of salt.