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date: 20 September 2018

Multiple Equilibria in the Climate System

Summary and Keywords

The idea that under the same external forcing conditions, the climate system is able to have several (statistical) equilibrium states is both fascinating and worrying: fascinating because the interaction of different positive and negative feedbacks can then lead to different large-scale reorganizations of the transport of heat (and other properties) over the globe; worrying because perturbations on the current equilibrium state can then unexpectedly cause transitions in large-scale transport properties, with potential disastrous changes in regional weather conditions. In this article, the development of the idea to explain peculiar climate changes using multiple equilibrium states is presented.

Keywords: climate dynamics, multiple equilibria, dynamical systems, bifurcations, transitions, rapid climate change, instabilities and feedbacks, bifurcation behavior

Introduction

Time series of climate observables, such as from ice cores, ocean sediments, and the instrumental record, often display “surprising” or “unexpected” behavior (Mudelsee, Bickert, Lear, & Lohmann, 2014). An important type of peculiar behavior is the occurrence of relatively large-amplitude changes over relatively short timescales compared to the long-term development of the observable. A prominent example within the last 65 My is the relatively rapid transition in global mean temperature (compared to the long-term cooling trend of the Earth) between the Eocene and Oligocene period about 35 My ago (Zachos, Pagani, Sloan, & Thomas, 2001). Apparently, “something” other than the dominant slow timescale forcing (e.g., due to a decrease of atmospheric greenhouse gases) “pushed” the climate state away from its “normal” behavior.

Another type of peculiar behavior is a change in dominant frequency of variability in climate observables. In the Pleistocene (the last 3 My), such a change is found between 1 My and 600 ky in oxygen isotope records of ice cores. Here the frequency of glacial-interglacial variability changes from a dominant 41 ky to 100 ky in the so-called Mid-Pleistocene Transition (Mudelsee & Schultz, 1997). Apparently, “something” other than the dominant slow timescale forcing (again a decrease of atmospheric greenhouse gases) “pushed” the climate state into a new transient equilibrium with the astronomical forcing.

If climate development were only determined by linear processes, one would not expect such transitions to occur because the climate state would closely follow the temporal expression of the forcing (Ghil, 1994). Indeed, the climate system is full of processes that are intrinsically nonlinear, such as radiation, advection, convection, and reaction kinetics. All these nonlinear processes interact on a multitude of scales giving rise to the multi-scale spatial-temporal behavior as observed in the different components of the climate system.

As soon as there are nonlinear processes at work, a multiplicity of equilibria may appear. The simplest example is the quadratic equation (with λ being a real number)

x 2 λ=0.
(1)

There are two real solutions, x=± λ , for λ>0 ; one, for λ=0 ; and there are no real solutions for λ<0 . Hence, if x represents an observable in a climate model and λ a parameter, quadratic interactions can already cause multiple states for the same value of λ.

The idea of multiple equilibrium states in climate can be traced back to the work of Mickael Budyko (1969) and William Sellers (1969), who used energy balance models to study Earth’s global radiation balance. One of the earliest energy balance models was the zero-dimensional model presented by Budyko (1969) for the global mean surface temperature T. The equation is

C T T t = Q 0 ( 1α( T ) )σγ( T ) T 4 ,
(2)

where σ is the Stefan-Boltzmann constant, Cτ is the atmospheric thermal inertia, and Q 0 = 0 /4 , where 0 is the solar constant.

In this energy balance model, the incoming radiation is represented by R i = Q 0 ( 1α( T ) ) and the outgoing radiation by R o =σγ( T ) T 4 . Often a shape of the albedo α( T ) is chosen that mimics that ice (at low temperature) has a higher albedo than ocean (at high temperature), for example, as in Figure 1a. The function γ( T ) represents the effects of greenhouse gases and clouds on the outgoing radiation (Sellers, 1969). A plot of the incoming and outgoing radiation is provided in Figure 1b for typical values of parameters (see, e.g., Dijkstra, 2013). One can see that three equilibrium temperatures (with T/t=0 in (Eq. 2)) are possible as the curves have three intersections. One of the states (at low temperature) represents a totally snow-covered Earth and has helped to shape the Snowball Earth hypothesis (Hoffman, Kaufman, Halverson, & Schrag, 1998) for the Neoproterozoic (730–640 My).

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Figure 1. (a) Albedo α‎ versus temperature T. (b) Plot of the incoming radiation Ri (dashed) and outgoing radiation Ro (drawn) versus temperature T.

Transitions between different equilibrium states have occurred often in the literature as an explanation of peculiar behavior in climate time series. Theories of the change at the Eocene-Oligocene boundary involve changes in Antarctic ice cover induced by critical conditions in atmospheric CO2 (DeConto & Pollard, 2003) possibly in addition to transitions in the ocean circulation (Tigchelaar, von der Heydt, & Dijkstra, 2011). Multiple equilibrium climate states also play an important role in early theories of the Pleistocene Ice Ages (Benzi, Parisi, Sutera, & Vupiani, 1982; Paillard, 1998; Saltzmann, 2001). The switches between the equilibrium states in this case occur due to the time-dependent astronomical Milankovitch forcing, and a positive ice-albedo feedback. This feedback is represented in the energy balance model (Eq. 2) through the temperature dependent albedo. Suppose that through a perturbation the warm state is slightly cooled. As a consequence, the albedo increases and the short wave radiation Ri will decrease which amplifies the original cooling. The model in Equation 2 also contains a negative feedback: when the temperature decreases, the outgoing radiation Ro also decreases, which leads to an increase of temperature. The relative strength of both feedbacks determines the behavior of each equilibrium state to perturbations.

Also regarding future climate change, potentially abrupt or irreversible changes have received much attention in recent years. In the last IPCC AR5 report, abrupt climate change is defined as a large-scale change in the climate system that takes place over a few decades (or less), persists for at least a few decades, and causes substantial disruptions in human and natural systems (Collins et al., 2013). In Figure 2, a number of components in the climate system are shown where critical transitions may play a role to cause abrupt climate change. For most of these components, relatively simple conceptual models have indicated that such transitions may exist. Only for a few of these phenomena, however, such as the Atlantic Meridional Overturning Circulation (MOC) collapse, are there more detailed models that possess multiple equilibria.

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Figure 2. Components in the climate system with possible abrupt change. (From Collins et al., 2013, where more explanation is provided.)

Here we will provide a basic theory on why multiple equilibria can appear in climate models due to an interplay of different feedbacks and what consequences this can have on transient behavior (e.g., under transient forcing or the presence of noise). To illustrate the theory, we first focus on the Atlantic MOC collapse and then come back to other components of the climate system later. The MOC case is also strongly motivated by another important example of peculiar behavior in climate time series—the Dansgaard-Oeschger events.

Peculiar Climate Change

The reconstruction of past temperatures (and other properties) from ice cores has provided much information on climate variability on timescales of centuries to millennia (Jouzel et al., 2007).

The Dansgaard-Oeschger Events

Isotope analyses from ice cores on Greenland provide information on the local temperatures over the last 100 ky. The local oxygen isotope anomalies (δ18O) from the NGRIP ice core are plotted in Figure 3. Slow variations are associated with the development of the last ice age of which the extremum occurred around 25 ky. The relatively rapid transitions (e.g., between 50 ky and 20 ky) with an equivalent peak-to-peak amplitude of about 10°C are called the Dansgaard-Oeschger events (Oeschger et al., 1984). There has been an extensive discussion on the dominant timescale of these Dansgaard-Oeschger events (Wunsch, 2000). After careful analysis of the GISP2 (Stuiver & Grootes, 2000) record, Schultz (2002) concludes that between 46 and 13 ky, the onset of Dansgaard-Oeschger events was paced by a fundamental period of ∼1,470 years. Before 50 ky, the presence of such a dominant period is unclear due to dating uncertainties in the ice core record.

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Figure 3. Oxygen isotope anomaly (δ‎18O) from the NGRIP ice core (Andersen et al., 2004). Peak-to-peak temperature changes between 50 ky and 20 ky are about 10°C.

Proxy data indicate that there have been large-scale reorganizations of both the atmosphere and ocean associated with Dansgaard-Oeschger events (a review is provided in Clement & Peterson, 2008). In the subpolar North Atlantic, Dansgaard-Oeschger events were matched with corresponding sea surface temperature changes of at least 5°C. Of special interest is that the temperature anomalies on Antarctica are about 180° out of phase with those on Greenland. As discussed in Clement and Peterson (2008), several different views have been proposed to explain the millennial climate variability during the last glacial period. Leading theories all involve changes in the Atlantic Ocean circulation.

The Atlantic Meridional Overturning Circulation

On a large scale, ocean circulation is driven by momentum fluxes (by the wind), the tides, and fluxes of heat and freshwater at the ocean–atmosphere interface. The buoyancy fluxes affect the surface density of the ocean water and, through mixing and advection, density differences are propagated horizontally and vertically. In the North Atlantic, the Gulf Stream transports relatively warm and saline waters northward. Part of the heat is taken up by the atmosphere, making the water denser. In certain areas (e.g., the Labrador Sea), when there is strong cooling in winter, the water column becomes unstably stratified, resulting in strong convection (Marshall & Schott, 1999). The interaction of this convection with boundary currents (Spall, 2003) eventually leads to the formation of deepwater, which overflows the various ridges present in the topography and enters the Atlantic basin.

This deepwater is transported southward at a depth of about 2 km, where it enters the Southern Ocean. Through upwelling in the Atlantic, Pacific, and Indian Oceans, the water is slowly brought back to the surface (Talley, 2008). To close the mass balance the water eventually is transported back in the upper ocean to the sinking areas in the North Atlantic. In the Southern Ocean, bottom water is also formed which has a higher density than that from the northern North Atlantic and therefore appears in the abyssal Atlantic and Pacific. In the North Pacific no deepwater is formed.

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Figure 4. Volume transport of the Atlantic MOC at 26°N as measured by the RAPID-MOCHA array from April 2004 to April 2014 (Smeed et al., 2014).

The Atlantic Meridional Overturning Circulation (MOC) is the zonally integrated volume transport, characterized by the meridional overturning stream function. This transport is mainly responsible for the meridional heat transport in the Atlantic. The strength and spatial pattern of the MOC are determined by density differences which set up pressure differences in the Atlantic. There are no observations available to reconstruct the pattern of the MOC, but its strength at 26°N in the Atlantic is now routinely monitored by the RAPID-MOCHA array (Cunningham et al., 2007; Srokosz & Bryden, 2015). The currently available time series of the MOC strength shown in Figure 4 indicates a mean of about 19 Sv, a standard deviation of 5 Sv, and a decreasing trend of about 1.6 Sv over the last decade (Smeed et al., 2014). At 26°N the heat transport associated with the Atlantic MOC is estimated to be 1.2 PW (Johns, Baringer, & Beal, 2011). This heat is transferred to higher latitudes, leading to a relatively mild climate over Western Europe compared to similar latitudes on the eastern Pacific coast.

Multiple Equilibria of the MOC

By the early 1960s, basic theories of the Atlantic wind-driven ocean circulation (e.g., the Gulf Stream) had been developed (Munk, 1950; Stommel, 1948) and researchers started to develop theories for the Atlantic MOC.

A Conceptual Model of the Atlantic MOC

It was realized that density differences are essential for the deep circulation and that temperature T and salinity S affect density in an opposite way, which can be seen from the linear equation of state

ρ= ρ 0 ( 1 α T ( T T 0 )+ α S ( S S 0 ) ),
(3)

where the subscript “0” refers to the reference values and α T and α S are the positive (and constant) thermal expansion and haline contraction coefficients, respectively.

In addition, there is a nonlinear coupling between the MOC and the density field: the MOC advects density anomalies but the density field affects the MOC. In a seminal paper, Stommel (1961) studied the consequences of this nonlinear coupling of the temperature, salinity and MOC in its simplest form using a two-box model. A sketch of a variant of this model (Cessi, 1994) is shown in Figure 5.

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Figure 5. A Stommel-type two-box model of the Atlantic MOC as formulated by Cessi (1994).

A polar box (with temperature T p and salinity S p ) and an equatorial box (with temperature T e and salinity S e ) having the same volume V 0 are connected by an overturning flow and exchange heat and fresh water with the atmosphere. The heat and salt balances are (Cessi, 1994)

d T e dt = 1 t r ( T e ( T 0 + θ 2 )) 1 2 Q(Δρ)( T e T p ),
(4a)

d T p dt = 1 t r ( T p ( T 0 θ 2 )) 1 2 Q(Δρ)( T p T e ),
(4b)

d S e dt = F S 2H S 0 1 2 Q(Δρ)( S e S P ),
(4c)

d S p dt = F S 2H S 0 1 2 Q(Δρ)( S P S e ),
(4d)

where F S is the fresh water flux H the ocean depth, t r is the surface temperature restoring timescale and θ‎ is the equator-to-pole atmospheric temperature difference. In Cessi (1994), the transport function Q is chosen as

Q( Δρ )= 1 t d + q 0 V 0 ( Δρ ρ 0 ) 2 ,
(5)

where q 0 is a transport coefficient, t d a diffusion time and Δρ= ρ p ρ e .

Subtracting (Eq. 4b) from (Eq. 4a) and (Eq. 4d) from (Eq. 4c) and introducing ΔT= T e T p and ΔS= S e S p leads to

dΔT dt = 1 t r ( ΔTθ )Q( Δρ )ΔT,
(6a)

dΔS dt = F S H S 0 Q( Δρ )ΔS.
(6b)

When non-dimensional quantities x and y are introduced according to ΔT=xθ , ΔS=y α T θ/ α s and time is scaled with t d , the non-dimensional system of equations (6) becomes

dx dt =α( x1 )x( 1+μ ( xy ) 2 ),
(7a)

dy dt =Fy( 1+μ ( xy ) 2 ),
(7b)

where α= t d / t r and

μ= q 0 t d ( α T θ ) 2 V 0 ,
(8)

is the ratio of the diffusion timescale t d and the advective timescale t a = V o /( q o ( α T θ ) 2 ) . Finally, the dimensionless freshwater flux parameter F is given by

F= α S S 0 t d α T θH F S .
(9)

Typical values of the non-dimensional parameters as motivated in Cessi (1994) are α=130 , F=1.1 , and μ=6.2 . The volume V0 is based on the area of transport near the western boundary in the Atlantic Ocean, and q0 is determined from the strength of the southward branch of the MOC.

Multiple Steady States

The mathematical model (Eq. 7) is an example of an autonomous dynamical system with two degrees of freedom (x and y) and several parameters, i.e., of the form

dx dt =f( x,p )
(10)

where x=( x,y ) is the state vector and p=( F,α,μ ) is the parameter vector. One of these parameters, i.e., F, is of particular interest as it monitors the strength of the freshwater forcing and is considered the control parameter.

As seen in the previous section, the value of α is large, and hence one can approximate x1 ; Equation 7b then becomes

dy dt =F-y( 1+μ ( 1y ) 2 ).
(11)

When the freshwater flux is time independent, say F= F ¯ , this equation can be written as a gradient system

dy dt = V ( y ),
(12)

for the potential function

V(y)= F ¯ y+ y 2 2 +μ( y 4 4 2 y 3 3 + y 2 2 ),
(13)

with the prime in (Eq. 12) indicating differentiation to y.

Using the formulation in Equation 12, the equilibrium solutions are now determined by the equation V ( y )=0 and hence are the extrema of the potential function V( y ) . For μ=6.2 , this potential is plotted in Figure 6 for three values of F ¯ . For F ¯ =1.1 a typical so-called double-well potential is found with minima at y a =0.24 and y c =1.07 and a maximum at y b =0.69 . For the other two values of F ¯ the potential has only one extremum. The geometric shape of a potential hence indicates that for some values of the freshwater forcing F ¯ , in particular F ¯ =1.1 , there are multiple steady states.

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Figure 6. Potential V(y) as in Equation 13 for μ=6.2 and three values of F ¯ (drawn: F ¯ = 1.1, dashed: F ¯ =0.8, dotted: F ¯ =1.3).

The next issue is, however, which of these equilibrium states are realizable for large time, starting from an arbitrary initial condition. Let an equilibrium state by indicated by y ¯ , then the evolution of infinitesimally perturbations y ˜ is considered, such that y= y ¯ + y ˜ . Substitution of this expression in Equation 12 and linearizing in y ˜ gives

d y ˜ dt = V ( y ¯ ) y ˜ .
(14)

This equation has exponential solutions with a growth rate V ( y ¯ ) at the equilibrium solutions. For a concave shape of V, with V ( y ¯ )<0 , the equilibrium y ¯ is unstable as perturbations will grow. For a convex shape of V, the equilibria are stable, and hence for F ¯ =1.1 , ya and yc are linearly stable steady states and yb is an unstable steady state.

Elementary Transitions

As an alternative to the potential plot in Figure 6, one can also plot the steady solutions y ¯ of Equation 11, i.e., solutions of

Fy(1+ μ 2 (1y) 2 )=0,
(15)

versus F ¯ . This so-called bifurcation diagram, where the dimensionless volume transport q=1+μ ( 1y ) 2 is used, is shown in Figure 7. It shows that there is an interval of values F ¯ for which there are multiple equilibria. Here the dashed states are unstable and the drawn states are stable. The interval of multiple states is bounded by two so-called saddle-node bifurcation points L1 and L2.

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Figure 7. Bifurcation diagram of the model in Equation 11 for μ=6.2 , showing q=1+μ ( 1y ) 2 versus.

Here, a connection is made with the elementary theory of qualitative changes of solutions of nonlinear differential equations when parameters are changed as developed from the late 1800s, starting with Henri Poincaré (1892). This theory provides a geometric framework to analyze the behavior of dynamical systems such as Equation 10 capturing the whole solution structure in parameter space (Guckenheimer & Holmes, 1990; Strogatz, 1994). Bifurcation theory addresses changes in the qualitative behavior of a dynamical system as one or several of its parameters vary. The results of this theory permit one to follow systematically the behavior from the simplest kind of model solutions to the most complex, from single to multiple equilibria, and from periodic, chaotic to fully turbulent solutions. The statistics of the behavior of the solutions is dealt with in ergodic theory (Lasota & Mackey, 1994).

Suppose that the autonomous dynamical system (Eq. 10) has a steady solution (also called a fixed point) x ¯ at a certain single parameter value p ¯ . For the linear stability of this fixed point, the transient development of small perturbations y is considered. Substituting the solutions y( t )= e σt y ^ , an eigenvalue problem results:

J( x ¯ , p ¯ )y ^ =σ y ^ .
(16)

Here J( x ¯ , p ¯ ) is the Jacobian matrix and σ= σ r +i σ i is the complex growth factor. Fixed points for which there are eigenvalues of J with σ r >0 are unstable since the associated perturbations are exponentially growing, whereas fixed points for which σ r <0 are linearly stable. In the situation where σ i 0 the associated eigenmodes will be oscillatory with frequency σ i , i.e., with a characteristic period of 2π/ σ i .

Bifurcations occur at parameter values for which σ r =0 . Normal forms of bifurcations are the simplest dynamical systems in which a particular type of bifurcation occurs. The strength of dynamical systems theory is that there is a classification of bifurcation points when only one parameter is varying. The elementary bifurcations giving rise to multiple equilibria are described by many sources (Strogatz, 1994). The three relevant ones for which multiple equilibria appear, with the simplest dynamical system in which such a bifurcation occurs, are shown in Figure 8.

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Figure 8. Supercritical (a) and subcritical (b) saddle-node bifurcation with normal form dx/dt=p± x 2 . Supercritical (c) and subcritical (d) of the transcritical bifurcation with normal form dx/dt=px± x 2 . Supercritical (e) and subcritical (f) of the pitchfork bifurcation with normal form dx/dt=px± x 3 . The solid (dash-dotted) branches indicate stable (unstable) solutions.

Inspection of the model equations that represent dominant balances of momentum, heat, freshwater, and other properties in the physical system may already a priori indicate what type of elementary bifurcations can be expected.

  1. 1. Symmetry. When symmetry is present in the model equations, a restriction will be put on the type of elementary bifurcations that can occur. For example, consider the presence of a reflection symmetry in a physical model defi by dx/dt=f( x,p ) , such as

f(x,p)=f(x,p).
(17)

Hence, when a bifurcation occurs as a parameter p is varied, one expects pitchfork bifurcations rather than transcritical bifurcations, because the normal form of the latter bifurcation does not satisfy the requirement (Eq. 17). This is only a very elementary example of the constraints put on bifurcation diagrams through symmetry (Golubitsky et al., 1988). For systems of equations that have no symmetry, only transcritical and saddle-node bifurcations are expected to cause multiple equilibria.

  1. 2. Special solutions. Of particular importance are solutions which remain a solution for all values of a particular control parameter p. Note that when bifurcations occur from these solutions, the requirement on the dynamical system will be

f( x ¯ ,p)=0.
(18)

for the steady state x ¯ and for every value of p. This excludes the occurrence of saddle-node bifurcations on this branch of solutions.

Many more of these examples exist that demonstrate that a priori information on the solution structure can be derived.

Transitions of the MOC

A regime of multiple equilibria implies that even when steady states are stable against infinitesimal perturbations, larger perturbations may induce a transition from one state to another. Furthermore, transient changes in the forcing of the system may cause peculiar transitions on a timescale much faster than that of forcing. Finally, the interaction of noise and a periodical forcing may lead to counterintuitive synchronization phenomena. These general results are illustrated using the Stommel model of “A Conceptual Model of the Atlantic MOC” as an example.

Response to Transient Freshwater Changes

Due to atmospheric processes, the freshwater flux can exhibit a slow trend or other time dependence, in general represented by F= F ¯ ( 1+g( t ) ) . This transient effect changes the shape of the potential in Equation 13 through

V(y)= F ¯ (1+g(t)y+ y 2 2 +μ( y 4 4 2 y 3 3 + y 2 2 ).
(19)

In Figure 9 the case of a slowly varying function g( t )=t for different small values of ∈ is considered. On the y-axis the dimensionless volume transport q=1+μ ( 1y ) 2 is shown. In the case =0.001 (Fig. 9a), the change is the freshwater forcing is not able to induce a transition, whereas in the second case ( =0.005 ) it is. The precise rate and timing of the transition is dependent on the rate of change in the forcing (Berglund & Gentz, 2006).

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Figure 9. (a) Realization of the system with the potential (Eq. 19) starting at y a =0.24 for F ¯ =1.1 , μ=6.2 , g( t )=t , for =0.001 . (b) Same but for =0.005 . The dashed line shows the change in the freshwater forcing F.

A second example mimics the “water hosing” numerical simulations that have been carried out for a hierarchy of ocean-climate models (Weaver et al., 2012). First the freshwater forcing is increased up to a time t0, after which it is decreased at the same rate. Hence, the forcing can be chosen as

g(t)=t,t[0, t 0 ]
(20a)

g(t)=(2 t 0 t),t[ t 0 ,3 t 0 ]
(20b)

for a given t0 and . In Figure 10, typical hysteresis behavior is found where the volume transport q collapses and recovers later in time.

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Figure 10. (a) Plot of the volume transport q versus F= F ¯ ( 1+g( t ) ) for F ¯ =1.1 , μ=6.2 and the forcing (Eqs. 20) for =0.005 and t o =100 . The direction of the change of F is indicated by the arrows.

When the forcing is changing much more slowly than the internal response timescales of the system, this is called a fast-slow system (Berglund & Gentz, 2006). For the system in Equation 19, we can introduce the variable z=g( t ) , and then it can be written as

dy dt = F ¯ (1+z)y(1+μ (1y) 2 ),
(21a)

dz dt = g (t).
(21b)

Assume, for example, that g( t )=t as in Figure 9, and introduce a slow timescale τ=t . Then, with dy/dt=dy/dτ , we find the equivalent system to Equations 21 as

dy dτ = F ¯ ( 1+z )y( 1+μ ( 1y ) 2 ),
(22a)

dz dτ =1,
(22b)

In the limit =0 , these equations reduce to

0= F ¯ ( 1+z )y( 1+μ ( 1y ) 2 ),
(23a)

dz dτ =1,
(23b)

which are referred to as the fast (Eq. 23a) and the slow (Eq. 23b) flow. Hence, when z develops on the timescale τ‎, the equilibrium solutions of (Eqs. 23) are followed.

Such fast-slow systems are the prototype dynamical systems for studying the popular notion of a so-called tipping point, where small changes in parameters can lead to a large response over a relatively short time interval. Critical transitions (Kuehn, 2011) can only occur for a saddle-node bifurcation (Fig. 11a), the subcritical pitchfork bifurcation (Fig. 11b), and the transcritical bifurcation (Fig. 11c). In Figure 9, such critical transitions are caused by the appearance of the two saddle-node bifurcations.

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Figure 11. Fast subsystem bifurcation diagrams for three bifurcations that are critical transitions. Double arrays indicate the flow of the fast subsystem: (a) saddle-node bifurcation, (b) subcritical pitchfork bifurcation, and (c) transcritical bifurcation. (Redrawn from Kuehn, 2011.)

At this point one may ask, going back to the original box model, why the MOC is so sensitive to changes in freshwater forcing. Here, this sensitivity connects to the existence of a positive feedback, the salt-advection feedback (Fig. 12). If there is a freshwater perturbation in the northern North Atlantic (e.g., through melting of the Greenland Ice Sheet or additional rainfall), the water will locally become less dense. This leads to a decrease in deepwater formation and hence a weaker MOC. The MOC then transports less salt northward and hence the original freshwater perturbation is amplified. The MOC also transports less heat northward (a negative feedback), but this anomaly is quickly dissipated by the atmospheric damping.

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Figure 12. Salt-advection feedback: dashed arrows indicate the MOC after the freshwater perturbation has been applied.

Response to Freshwater Noise

Consider next that the freshwater flux has a stochastic component, say F= F ¯ + F ˜ , with F ˜ being additive white noise with amplitude η, then Equation 12 becomes

dy dt = V (y)+ηζ( t ),
(24)

where ζ represents a white noise process (with mean zero and a delta function co-variance).

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Figure 13. (a) Realization of the system in Equation 24 starting at y a =0.24 for F ¯ =1.1 , μ=6.2 , and η=0.1 . (b) Same as (a) but for η=0.25 .

Starting at y a =0.24 , two transient solutions of Equation 24 for F ¯ =1.1 , μ=6.2 are plotted in Figure 13, one for η=0.1 and one for η=0.25 . The transient solution for η=0.1 does not make any transition and stays near the equilibrium y a =1.07( q a =4.58 ) . As can be seen, the trajectory for η=0.25 undergoes transitions between the two stable states ( y c and y a ). Clearly the transitions in Figure 13b are noise-induced transitions as the value of F ¯ is still in the multiple equilibrium regime and smaller than the F ¯ value at the saddle-node bifurcation L1 (Fig. 7).

Consider now the situation of a slowly varying noisy freshwater forcing, i.e., Equation 24 with a potential (Eq. 19) for again the function g( t )=t . For small-amplitude noise (Fig. 14a), the MOC does not undergo a transition before the saddle-node bifurcation. However, for larger-amplitude noise (Fig. 14b) a rapid transition occurs and is often referred to as noise-induced transition. The value of F= F ¯ ( 1+g( t ) ) is plotted as the dashed curve in Figure 14.

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Figure 14. (a) Realization of the system in Equation 24 starting at y a =0.24 for F ¯ =1.1 , μ=6.2 for =0.001 , and η=0.05 . (b) Same as (a) but for η=0.15 .

Recently, much attention has been paid to the development of early warning indicators of transitions such as in Figure 14 (Scheffer, Bascompte, Brock, & Brovkin, 2009). Here, critical slowdown has been a key phrase in the development of such early warning indicators. Early warning signals of MOC collapse have so far mostly been based on the analysis of single time series (Lenton, 2011). Critical slowdown induces changes in variance and lag-1 autocorrelation in the time series that can be connected to the distance to the saddle-node bifurcation point, and hence these quantities can serve as early warning indicators (Scheffer et al., 2009). It is thereby important that all relevant indicators of critical slowdown are considered (Ditlevsen & Johnsen, 2010).

Stochastic Resonance

With noise added to the freshwater flux, the transitions between the on- and off-states are not regular and cannot serve as a prototype to explain the specific temporal variability as in the Dansgaard-Oeschger events. The box model (Eqs. 7) was therefore extended to include a weak periodic forcing (Velez-Belchi, Alvarez, Colet, Tintore, & Haney, 2001) by choosing g(t) in Equation 19 as

g( t )=Asin(2π t τ e ).
(25)

The periodic forcing has an amplitude A in the freshwater flux and a dimensionless period τ e (the dimensional value can be obtained by multiplying with the diffusion timescale t d ).

The time series of the dimensionless transport q from this model for τ e =200 and A=0.1 are shown in Figure 15a for η=0.0125 . The periodic change in F is shown as the dashed curve. Although transitions are possible, they occur irregularly and do not synchronize with the period of the forcing. The results in Figure 15b for η=0.0175 show that the transitions between the stable states are clearly influenced by the periodic forcing and now become more regular. Apparently the very small periodic signal is amplified by the noise in the nonlinear system having a multiple equilibrium regime; such an amplification is called stochastic resonance. A characteristic of stochastic resonance is the signal-to-noise ratio Σ‎, as measured by

SR =10 log 10 S( ω 0 ) B ,
(26)

where ω 0 =2π/ τ e is the forcing frequency, S( ω 0 ) the power associated with that frequency, and B the background spectrum of the noise. In stochastic resonance, the value of SR first increases with the noise variance, reaching a maximum value corresponding to the maximum cooperation between the periodic forcing and the noise. For large values of the noise, the transitions will be noise dominated, which is shown as a decay in SR . The amplification of the signal happens at small noise amplitude and for a large range of forcing frequencies.

Multiple Equilibria in the Climate SystemClick to view larger

Figure 15. (a) Time series (drawn curve) of the transport q between the basins for the periodically and stochastically forced model (Eqs. 7), for τ e =200 and A=0.1 and η=0.0125 . (b) Same as (a) but for η=0.0175 .

Multiple Equilibria in the Climate SystemClick to view larger

Figure 16. The potential V(y) for three values of τ=0 (dash-dotted), τ=π/( 2Ω ) (dotted), and τ=π/Ω (solid).

To understand the stochastic resonance mechanism in more detail, consider a sinusoidal temporal deterministic forcing in the problem with a potential V (x) given by

V(y)= 1 2 y 2 + 1 4 y 4 AycosΩτ.
(27)

For A=0.1 , this potential is plotted for three values— τ=0 , τ=π/( 2Ω ) , and τ=π/Ω —in Figure 16. Although the fixed point locations (where V ( y )=0 ) do not depend strongly on τ‎, the actual values of V( y ) at these fixed points do.

Central to stochastic resonance is the relative magnitude of the mean transition times with respect to the periodic forcing. The transition time to go from one of the stable states ( ( y=±1 ) ) to another depend on the potential difference between that stable state and the unstable state y = 0 and is given by (Gardiner, 2002)

t 11 =2π 1 V ( 0 ) V ( 1 ) e 2( V( 0 )V( 1 ) ) η 2 ,
(28a)

t 11 =2π 1 V ( 0 ) V ( 1 ) e 2( V( 0 )V( 1 ) ) η 2 ,
(28b)

where η‎ is the noise amplitude.

The transition times for the potential (Eq. 27), with V( 0 )=0 , V( 1 )=1/4+AcosΩτ , V( 1 )=1/4AcosΩτ , V ( ±1 )=2 and V ( 0 )=1 finally are given by

t -11 = π 2 e 1 2 η 2 (14AcosΩτ)) ,
(29a)

t 11 = π 2 e 1 2 η 2 (1+4AcosΩτ)) ,
(29b)

The important element is that the transition times vary with τ‎ as the depth of the potential wells deepens and shallows. Suppose now that at τ=0 , the state of the system is near y = 1. At that time, the transition time t 11 is maximal (as the potential well is deepest), but it decreases with time until it becomes a minimum at τ=π/Ω . Consequently, over the time interval [ 0,π/Ω ] the probability to exit the potential well near y=1 increases. When the transition time t 11 at τ=0 is on the order of the timescale of the periodic forcing and the transition time at τ=π/Ω is much smaller than the timescale of the periodic forcing, then the system will surely exit the well near y=1 at τπ/Ω . Once in the other well, the same reasoning can be applied using the transition time t 11 .

Since the variance in the transition time is much smaller than the transition time itself, the transition occurs over a small, well-defined time interval. Consequently, the Fourier spectra of the sample paths have a strong peak at the forcing frequency Ω‎. Hence, the noise amplifies the periodic signal by establishing a coherent transition from one well to the other. Stochastic resonance provides a mechanism by which a small-amplitude periodic signal and noise work together to induce well-defined transitions between different states. Neither noise nor the periodic forcing can do this by itself.

Criteria to be in a Multiple Equilibrium Regime?

The Atlantic MOC is sensitive to freshwater perturbations due to the existence of the salt advection feedback (Bryan, 1986; Stocker, 2000; Stommel, 1961). The advective transport of salt by the circulation, the density dependence on salinity, and the fact that the circulation is driven by density differences cause this to be a robust feedback in the Atlantic Ocean circulation. In several ocean-only (Huisman, den Toom, Dijkstra, & Drijfhout, 2010) and ocean-climate models (Hawkins et al., 2011), this feedback causes the existence of a multiple equilibrium regime and hence simultaneous existence of an on-state and off-state of the MOC under the same atmospheric forcing conditions.

Can we determine from observations whether the present-day MOC is in a multiple equilibrium regime or not? One direction of research to answer this question has been to develop indicators for the multiple equilibrium regime of the MOC. Based on earlier work of Rahmstorf (1996) and de Vries and Weber (2005), a scalar indicator Σ‎ was shown (Dijkstra, 2007) to provide an accurate measure of the presence of a multiple equilibrium regime. Σ‎ is based on the freshwater budget over the Atlantic basin and is given by

= M ov ( θ s ) M ov ( θ n ),
(30)

where Mov is the freshwater transport (in Sv) due to the MOC and θs and θn are the southern and northern latitude of the Atlantic basin boundaries.

When Σ‎ < 0, the MOC transports freshwater out of the Atlantic basin. Hence when the MOC is weakened due to a freshwater perturbation, the freshwater export decreases and consequently the Atlantic becomes fresher and amplifies the original perturbation. In this way, Σ‎ can be seen as a measure of an integral salt-advection feedback. In Huisman et al. (2010), it was shown that the sign of Σ‎ can be connected to the stability of the MOC if the perturbation MOC pattern (due to freshwater perturbations) has the same spatial structure (but opposite sign) as the mean MOC.

When Σ‎ is computed for the equilibrium solutions of typical Global Climate Models (GCMs) used in the CMIP5 effort (Drijfhout, Weber, & van der Swaluw, 2010), it is found that for nearly all models Σ‎ > 0. Based on the indicator, this would mean that the MOC in these models is not in a multiple equilibrium regime. This is consistent with the fact that it is difficult to determine an off-state in these models. However, the reason that Σ‎ > 0 in these models is that they have a large bias in the freshwater budget in the Atlantic mainly related to too strong evaporation over the Atlantic. Hence the MOC has to import freshwater transport at its southern boundary to close the freshwater balance.

On the other hand, when Σ‎ is determined from observations, all estimates so far provide Σ <‎ 0 and hence indicate that the present-day Atlantic MOC is in a multiple equilibrium regime. An observational estimate of Weijer, De Ruijter, Dijkstra, and Van Leeuwen (1999) follows from an inversion of WOCE data and provide the most negative value of 0.2Sv . The estimate from Huisman et al. (2010) is directly from one WOCE section and gives 0.1Sv . The one from Bryden, King, and McCarthy (2011) is at 24°S and gives a mean of 0.13Sv . The values from the different reanalysis projects give values between 0.0 and -0.2 Sv and hence are consistent with the observational estimates (Hawkins et al., 2011).

In many models it turns out that | M ov ( θ n ) | is smaller than | M ov ( θ s ) | , and hence M ov ( θs ) , with θs 35 S has often been used as indicator of the multiple equilibrium regime of the MOC. However, the importance of Σ‎ for the collapse of the MOC (instead of only M ov ( θ S ) ) was recently demonstrated by Liu, Xie, Liu, and Zhu (2017). When they bias-corrected a state-of-the-art climate model (the CESM) to create an equilibrium state with Σ <‎ 0, the MOC collapsed 300 years after a CO2 doubling. The original version (with Σ >‎ 0) did not collapse after the same perturbation.

Summary and Discussion

The possible existence of multiple equilibria in the climate system has fascinated scientists for decades. Many studies have appeared in which it was shown that conceptual climate models exhibit multiple equilibria and associated transient behavior, such as hysteresis, jumps, and stochastic resonance (Benzi et al., 1982). In most of these conceptual models, the multiple equilibria appear through the three elementary bifurcations: the saddle-node, the transcritical, or the pitchfork bifurcation. A necessary condition for such bifurcations to occur is that the conceptual model captures at least one positive feedback process caused by underlying nonlinear processes such as radiation, advection, and reaction kinetics.

For the transitions in the Atlantic Meridional Overturning Circulation (MOC), box models serve as such conceptual models as they capture the salt-advection feedback. This particular case of multiple equilibria was used to illustrate the behavior of time series of observables, which typically can be expected. Characteristics of multiple equilibria regimes are hysteresis behavior, bimodal probability density functions, and the possible occurrence of stochastic resonance. Transitions between different MOC states, possibly due to stochastic resonance (Ganopolski & Rahmstorf, 2002), are a plausible scenario for Dansgaard-Oeschger events, although the precise mechanisms are still under much discussion (Clement & Peterson, 2008).

Many other components in the climate system, containing different feedbacks, may also introduce transition behavior in observables. Bathiany et al. (2016) review the potential for multiple equilibria to explain abrupt changes in the Arctic sea ice, ice sheets, vegetation in semi-arid areas (the Green Sahara), monsoons, rainforests and savannas, coral reefs, and permafrost. At least three criteria are important to attribute a transition to the presence of multiple equilibria: (1) conceptual low dimensional models of the phenomenon show multiple equilibria; (2) a clear feedback mechanism causing the different equilibria is identified; and (3) the characteristics of multiple equilibrium regimes (e.g., hysteresis) are found in time series from high-dimensional (more detailed) models. It would also be desirable that an indicator, such as Σ‎ for the MOC, of the presence of such a regime be available.

For most of the cases described in Bathiany et al. (2016), the first two criteria are well satisfied. For example, energy balance models display transitions between ice-free and ice-covered states through the occurrence of saddle-node bifurcations with the ice-albedo feedback being responsible (Eisenman & Wettlaufer, 2009). Another example is the local pattern formation in models of vegetation cover in semi-arid areas, where multiple equilibria appear through a transcritical bifurcation associated with an infiltration feedback (Dijkstra, 2011; Rietkerk & van de Koppel, 2008). There are cases where also criterion (3) is met as hysteresis behavior is found in high-dimensional models. An example is the different vegetation states associated with the Green Sahara transition (Claussen, Kubatzki, Brovkin, & Ganopolski, 1999). Also the hysteresis effects of the sea-ice albedo feedback can be found in quite sophisticated climate models (Lucarini, Fraedrich, & Lunkeit, 2010).

Also in global ocean models, the saddle-node bifurcations associated with the MOC transition can be found (Dijkstra & Weijer, 2005). Also in early GCMs, signatures of multiple equilibrium behavior have been found (Manabe & Stouff, 1988, 1999). The most sophisticated GCM where multiple equilibria of the Atlantic MOC have been found is the FAMOUS model (Hawkins et al., 2011). In fully coupled ocean-atmosphere aqua-planet models (Ferreira, Marshall, & Rose, 2011), multiple equilibria with different sea-ice extensions are found, which are due to different meridional patterns of the oceanic heat transport. In the state-of-the-art climate models, such as those used in the CMIP5 intercomparison, hysteresis effects have not been found, yet but it is fair to say that the necessary computations have not yet been performed. Note that many results on changes in the MOC due to global warming (Hu, Meehl, Han, Lu, & Strand, 2013), including the more recent results by Liu et al. (2017), do not demonstrate the existence of a multiple equilibrium regime as no hysteresis is found. Many of the models used in CMIP5 may actually be configured in a unique regime (Drijfhout et al., 2010), as evaluated by the multiple equilibrium regime indicator Σ‎ described in “Criteria to Be in a Multiple Equilibrium Regime”.

These criteria are important as often multiple equilibrium explanations have been proposed erroneously for transition behavior. For example, Wu, Anderson, and Davey (1993) have argued that the transitions between El Niño and La Niña states are due to the presence of multiple equilibria. Indeed in early conceptual flux-corrected models, multiple equilibria occurred due to transcritical bifurcations. However, it was shown later (Neelin & Dijkstra, 1995) that such multiple states were an artifact of the flux correction procedure. It is now well accepted that multiple equilibria do not play any role in the El Niño/Southern Oscillation phenomenon.

Another case is the transition behavior in mid-latitude atmospheric flows. In a conceptual model (Charney & DeVore, 1979) transitions between zonal and blocked flows are found due to the presence of multiple equilibria (Ghil & Childress, 1987). The dominant nonlinear process here is advection of momentum. However, the blocked-zonal flow transitions, which are seen in regime changes in more detailed models, cannot be easily related to those causing the multiple equilibria in the Charney–deVore model (Crommelin, Opsteegh, & Verhulst, 2004; Sterk, Vitolo, Broer, Simo, & Dijkstra, 2010; Tantet, van der Burgt, & Dijkstra, 2015). This illustrates that the connection between multiple equilibria in idealized conceptual models and “intransitive” (Lorenz, 1970) behavior (i.e., displaying more than one statistical equilibrium state) in more detailed models is often difficult.

A convincing demonstration that multiple equilibrium climate states would exist in CMIP5 models or even ones with a higher spatial resolution would have a profound impact on how to interpret the proxy record. Although there appear no direct indications from proxy data for the existence of multiple climate states in the geological past, the record is full of jumps (Mudelsee et al., 2014) and trends. Up until now, there are also no results from state-of-the-art paleoclimate models that illustrate multiple climate states under specified solar forcing and fixed greenhouse gas concentrations, but again not enough effort has been put into this search.

A demonstration of multiple equilibrium climate states in CMIP5 models would also affect our view on future climate changes. So far there is no evidence from the instrumental record that there are multiple equilibria, but the possible existence of such multiple states is both fascinating and worrying. Under such a regime, extreme atmospheric perturbations, such as due to an extreme phase of the North Atlantic Oscillation, might induce a spontaneous collapse of the Atlantic MOC. This would have substantial consequences for the temperature in Western Europe and could lead to a 50 cm sea level increase over a few decades in the eastern North Atlantic, which would be hard to adapt to. If the present climate system is in a multiple equilibrium regime, unexpected and bumpy climate change may lie ahead.

Acknowledgments

Data from the RAPID-WATCH MOC monitoring project (used in Fig. 2) are funded by the Natural Environment Research Council and are freely available from www.rapid.ac.uk/rapidmoc.

Further Reading

Dijkstra, H. A. (2013). Nonlinear climate dynamics. Cambridge: Cambridge University Press.Find this resource:

Ghil, M., & Childress, S. (1987). Topics in geophysical fluid dynamics. New York: Springer.Find this resource:

Strogatz, S. H. (1994). Nonlinear dynamics and chaos. New York: Perseus Books.Find this resource:

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