Hysteresis diagrams of the Atlantic meridional overturning circulation (AMOC) under freshwater forcing from climate models of intermediate complexity are fitted to a simple model based on the Langevin equation. A total of six parameters are sufficient to quantitatively describe the collapses seen in these simulations. Reversing the freshwater forcing results in asymmetric behaviour that is less well captured and appears to require a more complicated model. The Langevin model allows for comparison between models that display an AMOC collapse. Differences between the climate models studied here are mainly due to the strength of the stable AMOC and the strength of the response to a freshwater forcing.

The Atlantic meridional overturning circulation (AMOC) is an important circulation in the Atlantic Ocean.
It is also an important part of the climate system overall due to the heat it transports from the South Atlantic to the North Atlantic

Palaeoclimate records of the last glacial period show a rapid switching of temperature, which might be associated with the presence or absence of a vigorous AMOC as it exists today

The Langevin equation has been posited before as being suitable to capture the essential dynamics of an AMOC collapse

Though the Langevin equation has played a role in the conceptual picture of bistability and tipping points in the climate, it has not been used to actually fit the parameters to a (simulated) AMOC collapse.
Here, we attempt to construct a simple model based on the Langevin equation and fit its dynamics to salt-advection-driven collapse trajectories of the AMOC seen in climate models

Section

An increase in surface air temperatures or an increased surface freshwater flux by changes in precipitation minus evaporation will decrease the buoyancy in the shallow layer of the deep water formation regions in the North Atlantic subpolar gyre.
The deep water formation is reduced, and the southward meridional flow is also reduced.
In principle, this mechanism can reduce the AMOC to zero gradually if fully buoyancy-driven.
A salt-advection feedback mechanism that leads to a bimodal AMOC was proposed by

Example bifurcation diagram of the AMOC (

Figure

Below we will derive a model based on the Langevin equation that captures the essential dynamics of a bimodal AMOC under a freshwater forcing

The conceptual picture of the AMOC being a zero-dimensional variable that is driven by stochastic forces trapped in a potential is similar to that of a particle's motion described by Langevin dynamics

The double well potential seen in Fig.

AMOC bistability has, however, been studied quantitatively in, e.g.

The potential function takes the following form

The two parameters

To fit the model trajectories we need to find expressions for

In Fig.

Sample potentials (right) and their derivatives (left) for (top to bottom) the three possible varieties of bimodal state (

In Fig.

Discriminant determining the stability and number of critical points. The splitting factor

Our aim is to arrive at a description that matches a series of

With a varying

Solving for

For

The value of

This constrains the values of

With the deterministic framework in place, the stochastic nature can be reintroduced.
The potential function can be replaced by a distribution that is the stationary distribution in the asymptotic limit (i.e. the long-term behaviour of repeated sampling of the hysteresis loop).
The potential (a fourth-order polynomial) gives the following probability distribution

The factor

Example trajectory with corresponding distribution. Parameterised by

An example bifurcation diagram with corresponding distribution is shown in Fig.

The distributions in Fig.

We are now in a position to apply the above to collapse trajectories from climate models.

We describe how to find an optimal solution under the framework described in the previous section.
Using a Bayesian optimisation procedure, estimated values of

The parameters

Bayes' rule tells us the probability of a given observation

Sampling different values from the parameters' prior distributions will give corresponding values for the posterior distributions.
A Bayesian sampler chooses successive values that tend towards greater likelihood of the model, given the observed trajectory, and will converge towards an optimal fit.
Conceptually, this is what an MCMC (Markov chain–Monte-Carlo) optimiser does

The sampling process is time consuming because the evaluation of the potential (to calculate

The prior distribution of a parameter represents all the information known about that parameter before confrontation with the observed values

We are nonetheless still faced with infinite support on the coefficients of the expansion of the parameters (

where

To exclude parameter values that lead to intersections of

An AMOC collapse was induced in models of intermediate complexity in

Overview of models used. Each data point is independent of the others because each is the result of a quasi-steady-state run. The number of data points for each model was regridded onto a uniform freshwater forcing range consisting of 300 points. The summary of the type of model component and references are taken from

Overview of models, the estimated standard deviation with the upper branch fitted to a linear function (note that the original trajectories had already been smoothed), the ranges of

The trajectories are from the numerical Earth System Models (EMICs)

Absolute values of numerical derivatives (left) from the trajectories of AMOC strength as function of freshwater forcing to the right (taken from

If no other mechanisms apart from the salt advection are important, we expect the bifurcation points to lie beyond the observed transition points because a noise-induced transition pushes the AMOC into the off state sooner.
Note that although the collapse points are expected to lie before these peaks, low levels of noise will obscure this effect.
The dashed lines indicate the regions where we will search for the optimum values of

Before fitting, the upper and lower branches were extended to the left and right to fill the space of

Our main goal is to model the transition from the on branch to the off branch, i.e. the upper right half of the hysteresis curve, and not so much the dynamics that govern the lower branch. In addition, because of this assumption, other dynamics govern the lower branch, and our simple model has to be extended to account for those dynamics. We ignore the data on the lower branch before the collapse point so that the fits would not be influenced by these points. We expect the remaining points of the trajectory to be dominated by the salt-advection mechanism.

We start by identifying some characteristic points in the trajectories in Table

In Fig.

Estimated distributions under changing

Mean values and standard deviations of parameters corresponding to the fitted functions in Fig.

The fits with a linear series through the

We derived a simple model of AMOC collapse based on Langevin dynamics (Eq.

Any process that allows two stable states with rapid transitions between them and an asymmetric response to the forcing could in principle be described by our method.
Other such geophysical processes can include ice sheet mass loss (e.g.

The resurgences of the AMOC seen in the hysteresis diagrams behave differently from the collapses. The Langevin model is too simple to capture both processes. It is, however, possible to fit the change in the upper branch of the AMOC – the on state – as it moves towards a critical point and the dominant salt-advection feedback mechanism breaks down.

We note that

However, another possible explanation is that (two) separate mechanisms are responsible for the upper and lower branch dependency on

A third explanation is that deep water formation is a local process, and as a result an asymmetry is to be expected between the two branches.
Local convection can, however, be subject to global controls and be associated with a sinking branch which occurs in conjunction with deep convection, but is not directly driven by it, see

Furthermore, the methodology used in this paper comes with difficulties in the numerical implementation.
The fit procedure requires the normalisation of each distribution in the

As stated in

Finally, the fitted collapse trajectories were done on an ensemble of EMICs, which arguably are not sufficiently representative of the real climate.
As noted by

The data can be obtained from the first author at jellevandenberk@gmail.com upon request.

SD conceived the original idea. JvdB developed the theoretical formalism, performed the calculations, and prepared the figures. JvdB, SD, and WH contributed to the final version of the manuscript.

The authors declare that they have no conflict of interest.

The authors thank the two anonymous referees and editor Gerrit Lohmann for their valuable comments and suggestions that have improved the manuscript greatly. The authors also thank Stefan Rahmstorf for providing the original data.

This research has been supported by the European Commission's 7th Framework Programme (grant no. 282672).

This paper was edited by Gerrit Lohmann and reviewed by two anonymous referees.