## 1. Introduction

The thermohaline circulation (THC) of the World Ocean is believed to play a central role in climate variability on timescales from decades to millennia, through both its internal dynamics and its response to external forcing (see, e.g., Weaver and Hughes 1992). In the Atlantic Ocean, as it operates today, the THC has the net effect of transporting a tremendous quantity of heat northward, with significant consequences for the climate of northwest Europe (Rahmstorf 1999). However, it has become clear that the present configuration of the THC may not be the only one it can take. Investigations involving both simple heuristic models (e.g., Stommel 1961; Rahmstorf 1996; Scott et al. 1999; Stone and Krasovskiy 1999; Titz et al. 2001) and complex general circulation models (GCM; e.g., Manabe and Stouffer 1988; Hughes and Weaver 1994; Rahmstorf 1995, 1996; Tziperman 2000) indicate that the THC may display multiple regimes of circulation, transitions between which are very rapid relative to the length of time spent within a regime. Furthermore, evidence of rapid shifts in the climate state abounds in the geological record, with timescales from those of glaciation cycles to higher frequency, millennial scale fluctuations (e.g., Bond et al. 1997; Alley 2000; Stocker and Marchal 2000).

Simple box models have demonstrated a remarkably good ability to reproduce the multiple regimes of circulation demonstrated in full GCMs. They are an attractive conceptual tool because of both their extremely low computational cost and the small number of parameters that govern their dynamics. Considerable attention has been paid to both purely deterministic models (e.g., Stommel 1961; Maas 1994; Rahmstorf 1996; Scott et al. 1999) and to models with a stochastic component (e.g., Stommel and Young 1993; Cessi 1994; Griffies and Tziperman 1995; Lohmann and Schneider 1999; Timmermann and Lohmann 2000). For deterministic models, the natural framework of analysis is dynamical systems theory, and discussion centers on the nature and stability of attractors admitted by the model, and on the nature and location of bifurcations. Analysis of the stochastic class of models proceeds from the theory of stochastic differential equations (Gardiner 1997). Attention in these models has primarily been focused on the nature of the spectrum and on gross features of the stationary distribution.

A study combining perspectives from both stochastic analysis and dynamical systems theory was that of Timmermann and Lohmann (2000), in which changes in the qualitative behavior of the stationary distribution of the meridional salinity gradient with changes in bifurcation parameters were mapped out. In this study, we extend the analysis of Timmermann and Lohmann using a somewhat more general model. We demonstrate that for broad, and physically reasonable, regions of parameter space, although the stationary distribution is technically bimodal, the population of one regime is so much greater than that of the other that the system behaves unimodally for all intents and purposes. This *stabilization* of one regime relative to another can have significant consequences for analyses of the stability of these regimes.

Section 2 below introduces the generalization of Stommel's two-box model, from which we derive a simpler approximation in section 3. The stationary distribution of this simpler model is obtained analytically, and its structure throughout phase space is analyzed. In section 4, the average residence times of the two regimes are calculated for those regions of phase space in which the stationary distribution is bimodal. An important piece of information provided by these residence times is the region of parameter space in which the stationary distribution is relevant for understanding the finite-time behavior of the model. Section 5 applies the above results to an analysis of the hysteresis behavior of the model, and discusses the consequences of these results for discussions of climate stability. A summary and conclusions are presented in section 6.

## 2. The model

*T*and Δ

*S*represent, respectively, the differences between the box-average temperature and salinity of the high-latitude and middle-latitude boxes. These boxes are taken to be of the same volume

*V*and depth

*h.*The strength of the volume-mean flow between the boxes,

*q,*is given by the Stommel (1961) ansatz:

*q*

*c*

*α*

*T*

*β*

*S*

*P*−

*E,*which can in principle depend on the temperature difference Δ

*T,*and the temperature difference is relaxed on the timescale Γ

^{−1}to the “climatological” value Δ

*T*

_{o}. A reference salinity

*S*

_{o}is introduced to convert the freshwater flux into a salinity flux.

*q.*Of course, the temperature and salinity fields of the ocean display considerable spatial structure, and eddy transports (up to gyre-scale circulations) could constitute a nontrivial term in the heat and salinity budgets of each box. We parameterize the eddy transport between the boxes in terms of the mean quantities Δ

*T*and Δ

*S*by introducing an effective eddy diffusivity

*ξ.*We further assume that

*ξ*fluctuates in time, and parameterize it as a stochastic process. This process is

*not*constrained to be positive, so eddy transport between the boxes may be upgradient (e.g., Nakamura and Chao 2000). In what follows, the process

*ξ*will be assumed to be an Ornstein–Uhlenbeck process:

*W*

_{1}is a Wiener process (Gardiner 1997). Note that in the limit

*τ** → 0,

*ξ̇*

_{1}

*Ẇ*

_{1}.

In the interests of model parsimony, we have made three more assumptions that should be noted. First, we have assumed equal eddy diffusivities for temperature and salinity; in principle, the two diffusivities should be represented by separate (although presumably not independent) stochastic processes. As well, we have assumed that *ξ* is Gaussian and of mean zero. Relaxing the last assumption would involve the introduction of a parameter that could tune the relative frequency of upgradient and downgradient eddy fluxes.

*W*

_{2}is a second Wiener process, independent of

*W*

_{1}. More generally, we could consider fluctuations in the freshwater flux with a finite autocorrelation time. However, these fluctuations are meant to represent atmospheric phenomena that presumably evolve on much shorter timescales than those of oceanic variability. In the interest of keeping the model as simple as possible, we will consider only white noise fluctuations in the freshwater flux.

*t̂.*In the above,

*W.*Adopting the same parameter values as Timmermann and Lohmann (2000):

*α*= 0.15 K

^{−1},

*c*= 17 × 10

^{6}m

^{3}s

^{−1},

*V*= 2 × 10

^{15}m

^{3}, and Δ

*T*

_{o}= 15 K, we find that

*t*= (1.66 yr)

*t̂.*

Equations (10)–(12) constitute a stochastic differential equation (SDE) for the three-dimensional Markov process (*x,* *y,* *η*) whose associated probability density function satisfies a Fokker–Planck equation (FPE; Gardiner 1997). This system is still too complicated to admit analytic solutions; in particular, the associated FPE is a partial differential equation in one time and three state variables. The following section introduces a simplification of the model that admits analytic results.

## 3. Simplified model

^{3}with five adjustable parameters. The following pair of approximations reduces the system to a one-dimensional SDE with three parameters. First, we will assume that the temperature relaxation timescale is very short,

*γ*≫ 1, so that

*x*does not deviate from unity. In this limit, the equation for

*y*becomes

*ẏ*

*y*

*y*

*ηy*

*μ*

*σ*

_{2}

*Ẇ*

_{2}

*η*is also very short,

*τ*→ 0, so

*η*→

*σ*

_{1}

*Ẇ*

_{1}. Written in differential form, the resulting SDE for

*y*is

*y*∘

*dW*

_{2}is interpreted in the Stratonovich sense [descriptions of the difference between Ito and Stratonovich SDEs are given in Penland (1996) and in Gardiner (1997)]. In the limit that

*σ*

_{1}→ 0, this model is similar to that considered by Cessi (1994).

Equation (20), with *σ*_{2} = 0, was obtained by Timmermann and Lohmann (2000) in a different manner. They did not consider oceanic eddy transports, but assumed that the temperature dynamics was completely described by an Ornstein–Uhlenbeck process, such that in the present notation *x* = 1 + *η.* However, to obtain equation (20), Timmermann and Lohmann were obliged to arbitrarily remove the process *η* from within the absolute value signs. This was necessary to produce a model to which the standard techniques of stochastic analysis could be applied, in the *τ* → 0 limit, but cannot be rigorously justified. In fact, the limiting distribution as *τ* → 0 for the model in which *η* remains within the absolute value operator can be shown to be a trivial delta function at the origin (Peter Imkeller 2000, personal communication). The white noise limit in the original formulation of Timmermann and Lohmann does not produce physically reasonable results.

*y,*

*p*(

*y*), is given by

*p*

_{s}satisfies ∂

_{t}

*p*

_{s}= 0 and is given by the ordinary differential equation

*y*< 1:

*y*> 1:

*N*is a normalization constant. In the limit

*σ*

_{2}→ 0,

*p*

_{s}reduces to the stationary distribution calculated by Timmermann and Lohmann (2000).

*p*

_{s}, we now systematically investigate its behavior in the space of parameters

*μ,*

*σ*

_{1}, and

*σ*

_{2}. We first note that, just as for a range of parameter values the deterministic Stommel model admits two stable fixed points,

*p*

_{s}is bimodal in a certain domain of parameter space. Extrema of

*p*

_{s}occur where its derivative with respect to

*y*vanishes; from (22), this occurs when

*σ*

_{2}plays no role in determining the number or location of extrema in the stationary distribution

*p*

_{s}, as the noise process with which it is associated enters equation (20) additively. Equation (25) has three solutions for

*μ,*

*σ*

_{1}satisfying the inequalities:

*μ*

*σ*

^{2}

_{1}

*μ*

*R*(

*y*

_{±}), there are maxima at

*R*(

*y*

_{±}), the stationary distribution is unimodal. In the regions

*R*(

*y*

_{−}) and

*R*(

*y*

_{+}) only the peaks at

*y*

_{−}and

*y*

_{+}exist, respectively. These three regions are illustrated in Fig. 1. Note that along the line 2

*μ*=

*σ*

^{2}

_{1}

*σ*

_{1}>

*y*

_{+}and

*y*

_{−}coincide with value 1.

Timmermann and Lohmann (2000) concentrated on the behavior of *p*_{s}(*y*) for *σ*_{2} = 0, *μ* = 0.25, and *σ*_{1} > 0.3. A calculation of the average rate at which the mean flow *q* changes sign (not shown) demonstrates that for *σ*_{1} above 0.3, for most of the range of *μ,* sign changes occur on centennial or shorter timescales. Such rapid fluctuations in the sign of *q* are physically unreasonable. Consequently, in this paper, we will concentrate on the lower noise limit, *σ*_{1} < 0.2. Similarly, for the fluctuations in freshwater flux, we will concentrate on the range *σ*_{2} < 0.2.

Within the parameter region *R*(*y*_{±}), the distribution *p*_{s}(*y*) is bimodal. For the range of *σ*_{1}, *σ*_{2}, we are considering, however, throughout much of *R*(*y*_{±}) the relative populations of each regime are very different. Figure 2 displays a plot of *p*_{s}(*y*) as a function of *μ* for *σ*_{1} = 0.1, *σ*_{2} = 0.1. The thick horizontal lines in Fig. 2 correspond to the values of *μ* between which *p*_{s} has two maxima. It is clear that over most of the range for which *p*_{s} is bimodal, only one of the peaks is significantly populated. The stationary density is only effectively bimodal for a small range of *μ* near the value 0.19. At this pair of noise strengths (*σ*_{1}, *σ*_{2}), for most values of *μ,* the stationary behavior of the system will be more like that of a system with a unimodal distribution than with a bimodal. We will use the term *stabilization* to denote the preferred population, reflected in the stationary PDF, of one regime relative to the other. Of course, in the presence of noise, neither regime is strictly speaking stable; they are both metastable. However, a substantial asymmetry in the relative populations of the two regimes reflects a distinct preference of the system for one regime over the other; it is this phenomenon that we denote by the term stabilization.

Figure 3 contours *p*_{s}(*y*) as a function of *σ*_{2} for *μ* = 0.205 and *σ*_{1} = 0.1. Note that while *σ*_{2} does not affect the positions of the extrema of *p*_{s}, as discussed above, it does affect the relative population of the two regimes, and in a complicated manner. For the smallest values of *σ*_{2}, the regime around *y*_{−} is more populated than that around *y*_{+}. As *σ*_{2} increases, the mass shifts to the regime around *y*_{+}. However, as *σ*_{2} increases further, the population of the regime around *y*_{−} begins to increase again.

The nontrivial dependence of the stationary PDF, *p*_{s}, on the parameters (*μ,* *σ*_{1}, *σ*_{2}) is succinctly illustrated through consideration of the first two moments of the distribution. Figures 4a and 4b contour, respectively, the mean and standard deviation of *p*_{s} as a function of *σ*_{1} and *σ*_{2} for *μ* = 0.19. For these parameter values *y*_{−} ≃ 0.25 and *y*_{+} ≃ 1.16. It is clear from Figs. 4a and 4b that for small values of *σ*_{1}, the regime around *y*_{+} is favored, for most values of *σ*_{2} considered. Conversely, for small values of *σ*_{2}, it is the regime around *y*_{−} that is more populated for most of the range of values of *σ*_{1} under consideration. It is interesting to note the counterintuitive result that there are regions in parameter space in which increasing either *σ*_{1}, *σ*_{2}, or both, can actually lead to a *decrease* in the size of the fluctuations of *y,* as one regime becomes stabilized.

Figure 5 contours the mean and standard deviation of *y* for *μ* = 0.15. For this value of *μ,* it is clear that for the range of *σ*_{1}, *σ*_{2} considered, the PDF remains effectively unimodal around *y*_{−}. As well, the mean and variance of *y* are much more sensitive to *σ*_{2} than they are to the value of *σ*_{1}. The dependence of the mean and standard deviation of *y* on *σ*_{1} and *σ*_{2} for *μ* = 0.23 is displayed in Fig. 6. Here, for a broad range of values of *σ*_{1} and *σ*_{2}, the distribution is effectively unimodal and concentrated around *y*_{+}. In contrast to the behavior at *μ* = 0.15, the mean and standard deviation are more or less equally dependent on the values of *σ*_{1} and *σ*_{2}.

*μ*below (above) a certain window, the population of the regime around

*y*

_{−}(

*y*

_{+}) is stabilized. Within this window, the populations of the two regimes are of comparable size. We can characterize the location of this window of effective bimodality by defining the surface

*μ*

_{0.5}(

*σ*

_{1},

*σ*

_{2}) as the value of

*μ,*for a given (

*σ*

_{1},

*σ*

_{2}), for which the stationary probability of

*y*being in either regime is equal; that is,

*σ*

_{1}and

*σ*

_{2}in Fig. 7a. For the range of

*σ*

_{1},

*σ*

_{2}considered,

*μ*

_{0.5}falls between 0.17 and 0.22. For

*μ*<

*μ*

_{0.5}, the regime centered about

*y*

_{−}will dominate that about

*y*

_{+}, and conversely for

*μ*>

*μ*

_{0.5}. Figure 7b characterizes the width of the window around

*μ*

_{0.5}within which the probabilities of being in either regime differ by less than a factor of 10. Note that this window is not symmetric about

*μ*

_{0.5}. Clearly, the smaller the noise level, the sharper is the change in

*p*

_{s}as

*μ*passes through

*μ*

_{0.5}. Even for intermediate noise strength, the change may be rather abrupt, as is demonstrated in Fig. 2.

In summary, for the range of noise strengths *σ*_{1}, *σ*_{2} under consideration, there exists a range of values of the parameter *μ* for which the regimes centered around *y*_{−} and *y*_{+} are both significantly populated, as described by the stationary PDF *p*_{s}, and the distribution is effectively bimodal. As the noise strengths decrease, this window shrinks to a point. For values of *μ* below this window, almost all of the mass of *p*_{s} is concentrated around *y*_{−}, and the stationary PDF is effectively unimodal. For *μ* above this window, the stationary PDF is also essentially unimodal, with all of its mass concentrated around *y*_{+}. The phenomenon of stabilization provides an example of the manner in which fluctuations can be important in determining the mean state of a nonlinear stochastic system.

*ẋ*

_{x}

*V*

*σẆ,*

*x*

_{−}and

*x*

_{+}, then the PDF will have peaks at these points. The ratio of the peaks of the PDF is determined by the difference in depth of the two potential wells:

*σ*decreases, as was noted by Molteni and Tibaldi (1990) and Gardiner (1997).

Stabilization is clearly not limited to potential systems driven by additive noise; the model (20) is not driven just by additive white noise, so the above argument does not apply. Furthermore, numerical investigations (not shown) involving finite noise autocorrelation time *τ,* or temperature relaxation timescale *γ*^{−1}, for which the deterministic part of the dynamics does not correspond to the gradient of a potential, indicate that the phenomenon of stabilization is qualitatively unchanged.

So far we have considered only the stationary PDF *p*_{s}, which, starting from the initial condition *y* = *y*(0) at *t* = 0, describes the distribution of *y* as *t* → ∞. For finite times, the stationary distribution may not provide a complete description of the system, and must be supplemented by a consideration of the residence times within each of the regimes. This is discussed in the following section.

## 4. Residence times and the deterministic limit

*σ*

_{1}and

*σ*

_{2}become very small, the behavior of the stochastic model (20) should converge in some sense to that of the deterministic model

*ẏ*

*y*

*y*

*μ.*

*μ*< 0.25, whose positions are given, respectively, by Eqs. (27)–(29) with

*σ*

_{1}= 0. In a deterministic system with multiple stable fixed points, the asymptotic state of the system as

*t*→ ∞ is determined entirely by the basin of attraction in which the initial conditions lie. For this simple system, the boundary between the basins of attraction of

*y*

_{−}and

*y*

_{+}is the unstable fixed point

*y*

_{0}.

*σ*

_{1}and

*σ*

_{2}, within the range of values of

*μ*for which the distribution is bimodal, there is only a narrow window in which the populations of the two regimes are of comparable size. In the limit that

*σ*

_{1}and

*σ*

_{2}go to zero, this window, centered on

*μ*

_{0.5}, shrinks to a point: for

*μ*<

*μ*

_{0.5}, the entire mass of

*p*

_{s}is concentrated around

*y*

_{−}, and for

*μ*>

*μ*

_{0.5}around

*y*

_{+}, independent of the initial conditions. Thus, the

*t*→ ∞ description of the stochastic system for small

*σ*

_{1}and

*σ*

_{2}is completely different than that of the deterministic system. Compounding the problem is that the limiting value of

*μ*

_{0.5}for vanishing noise strengths is not unique, but depends on how the limit is taken. In particular,

*T*(

*y*

_{−})〉 and 〈

*T*(

*y*

_{+})〉 the respective average residence times of the system in the regimes around

*y*

_{−}and

*y*

_{+}, a standard result yields

*B*

*y*

*σ*

^{2}

_{1}

*y*

^{2}

*σ*

^{2}

_{2}

*T̃*such that if the average escape time for a regime is much greater than

*T̃,*and if the initial conditions lie in this regime, then the system will for all practical considerations remain in this regime. If the average residence times of both regimes are much larger than

*T̃,*then, practically speaking, the system will for the vast majority of realizations remain within the domain of attraction in which it started and the asymptotic behavior of the system will be described by only that part of the stationary distribution within this regime. The entire stationary distribution

*p*

_{s}is of relevance only if the system has time enough to sample the complete range of state space to which it theoretically has access. This will not be the case if the average escape times are much longer than any physically relevant timescale. As

*σ*

_{1}and

*σ*

_{2}go to zero, the average residence times of both regimes become infinite. It is in this sense that the deterministic limit arises as the noise strength vanishes. This can be expressed formally by the inequality:

Figures 8–10 contour 〈*T*(*y*_{−})〉 and 〈*T*(*y*_{+})〉 (in nondimensional time units) as functions of *σ*_{1} and *σ*_{2} for *μ* = 0.19, *μ* = 0.15, and *μ* = 0.23, respectively. Remembering that one nondimensional time unit corresponds to 1.66 yr, these figures may be essentially read as being in years. For millennial-scale variability of the THC, reasonable values of *T̃* are on the order of 10^{4}–10^{5} yr. For *μ* = 0.19, there are broad regions of (*σ*_{1}, *σ*_{2}) space for which 〈*T*(*y*_{−})〉 and 〈*T*(*y*_{+})〉 are of about the same size, and both are less than *T̃.* Within such regions, the distribution of *y* will be realizably bimodal, independent of the initial conditions, and consistent with *p*_{s}. In other parts of (*σ*_{1}, *σ*_{2}) space, while both residence times are less than *T̃,* one is very much larger than the other. In these regions, the system will behave in an effectively unimodal fashion, as described by the stationary PDF, and independent of initial conditions. Finally, for smaller values of *σ*_{1} and *σ*_{2}, the residence times of both regimes are much larger than *T̃.* For these parameter values, the stationary PDF (23)–(24) will be essentially irrelevant, and the system will fluctuate for all time (practically speaking) within the basin of attraction in which it started.

For *μ* = 0.15 (Fig. 9), 〈*T*(*y*_{−})〉 is everywhere larger than 〈*T*(*y*_{+})〉, usually by at least an order of magnitude. For sufficiently large *σ*_{1}, *σ*_{2}, 〈*T*(*y*_{−})〉 is less than *T̃,* and transitions to the regime centered around *y*_{+} will occur. However, the time spent near *y*_{+} will be typically much less than that spent near *y*_{−}, so at finite times the regime around *y*_{−} will be stabilized relative to that around *y*_{+} at these noise levels. For *μ* = 0.23, as is demonstrated in Fig. 10, 〈*T*(*y*_{−})〉 is everywhere much smaller than 〈*T*(*y*_{+})〉, and it is the regime around *y*_{+} that is stabilized relative to that around *y*_{−} for sufficiently large *σ*_{1}, *σ*_{2}. For both *μ* = 0.15 and *μ* = 0.23, for small enough *σ*_{1}, *σ*_{2}, 〈*T*(*y*_{−})〉 and 〈*T*(*y*_{+})〉 are both much greater than *T̃* and the description of the system by the stationary distribution will not be relevant for finite times.

Thus, for a broad range of values of *σ*_{1}, *σ*_{2}, and *μ,* the average residence times 〈*T*(*y*_{−})〉 and 〈*T*(*y*_{+})〉 are sufficiently different and sufficiently small that, on millennial timescales, one regime is stabilized relative to the other, irrespective of the initial conditions. The inclusion of noise, strong enough that the residence times are sufficiently small that the stationary distribution is relevant on finite timescales, produces a substantially different description of the system than the deterministic model, for which the system is bistable over a broad range of values of *μ.* The inclusion of fluctuations dramatically reduces this region of bistability. This difference in descriptions has a direct bearing on the issue of the stability of the THC, to which we now turn.

## 5. Hysteresis loops and climate stability

As mentioned above, the deterministic equation (34) has one unstable and two stable fixed points for 0 ≤ *μ* ≤ 0.25. At *μ* = 0, the larger stable fixed point, *y*_{+}, merges with the unstable fixed point; for *μ* < 0 only the smaller stable fixed point, *y*_{−}, remains. At *μ* = 0.25, the smaller stable fixed point merges with the unstable fixed point in another bifurcation, and above *μ* = 0.25 only the larger stable fixed point remains. The bifurcation diagram is displayed in Fig. 11a. In the deterministic limit, if *μ* is varied from below 0 to above 0.25 and back down to below zero, the hysteresis response illustrated in Fig. 11b is obtained. The qualitative character of this deterministic hysteresis loop is generic to all simple models of the thermohaline circulation that display multiple stable fixed points, and has been well-discussed in the literature (e.g., Rahmstorf 1996). In this deterministic loop, transitions between the two equilibria occur exactly at the bifurcation points where one equilibrium disappears.

As discussed above, a qualitative difference between the stochastic and deterministic models is the potential stabilization of one regime by the fluctuations. For the range of (*σ*_{1}, *σ*_{2}) considered, there is a range of *μ,* well below the bifurcation point at *y* = 0.25, within which the stationary distribution favors the regime around *y*_{+}. Thus, if the residence time of the regime around *y*_{−} is less than the timescale of the circuit around the loop, *T̃,* then the jump to the upper branch will occur at a random value of *μ,* on average well before the bifurcation point at *μ* = 0.25. Similarly, moving down the hysteresis loop, as *μ* drops below *μ*_{0.5}, the stationary distribution *p*_{s} will be dominated by the regime around *y*_{−}. If 〈*T*(*y*_{+})〉 is smaller than *T̃,* then the jump down to the lower branch will most often occur at some random value of *μ* before the bifurcation point at *μ* = 0. It follows that the hysteresis loop, which is now random, will be smaller in the stochastic case than in the deterministic limit. Such an effect was noted in the study of Wang et al. (1999), although it was not systematically investigated. The difference between the deterministic and stochastic hysteresis loops will be determined by the values of *σ*_{1} and *σ*_{2}, which control the residence times of the two regimes, and by the rate at which *μ* is changed in the circuit around the loop.

Figure 12 displays sample hysteresis loops obtained by numerical solution of equation (20) with a time-varying value of *μ,* for four different pairs of noise strengths (*σ*_{1}, *σ*_{2}). The numerical simulations were carried out using a simple forward-Euler discretization (Kloeden and Platen 1992). The parameter *μ* is moved from −0.1 to 0.3 over 2500 nondimensional time units, and back down to −0.1 at the same rate. To emphasise the differences induced by changing the noise levels alone, the same realizations of the Wiener processes *W*_{1} and *W*_{2} were used in all four simulations. Even for the fairly small noise levels (*σ*_{1}, *σ*_{2}) = (0.05, 0.05), the hysteresis curve is substantially narrower than in the deterministic limit. As the strength of the noise increases, the hysteresis curves become narrower, on average. Figure 13 displays the random hysteresis curves obtained with four different realizations of the stochastic processes *W*_{1} and *W*_{2}, at the fixed noise level (*σ*_{1}, *σ*_{2}) = (0.075, 0.075). This figure demonstrates that, even for moderate noise levels, there is considerable variability in the values of *μ* at which the transitions occur. Figure 14 displays estimates of the PDFs of the values of *μ* for which the jumps between regimes occur, for (*σ*_{1}, *σ*_{2}) = (0.075, 0.075), based on 1000 realizations of the hysteresis loop. For both transitions, the most likely value of *μ* at which the jump occurs is well separated from the deterministic bifurcation point. For smaller noise strengths, these distributions tighten and the most likely values of *μ* for the jumps approach the deterministic bifurcation points. For *σ*_{1}, *σ*_{2} sufficiently small, the residence times of the regimes are for all *μ* larger than the timescale of the circuit around the loop, and the deterministic limit is recovered. On the other hand, for sufficiently large noise strengths, the residence times will be so much smaller than *T̃* that more than two jumps between the regimes will be made, and the hysteresis loop structure is lost.

Multiple equilibria in the THC appear not only in simple box models, but also in sophisticated general circulation models (e.g., Manabe and Stouffer 1988; Hughes and Weaver 1994; Rahmstorf 1995, 1996, 1999; Tziperman 2000). Rahmstorf (1995, 1996) evaluated the hysteresis loop of a full ocean GCM coupled to a simple atmosphere, and demonstrated that it could be fit rather well to the hysteresis loop produced by a simple box model. He was then able to estimate the position of the present climate on the box model hysteresis curve, and thereby estimate its proximity to the bifurcation point. However, we have demonstrated that in the presence of noise of even moderate strength, representing internal and external high-frequency variability, transitions between climate regimes may occur some distance away from the deterministic bifurcation points. Consequently, the distance between the present climate state and the deterministic bifurcation point may overestimate the stability of the present climate. Evidence for sensitivity of the hysteresis structure of a GCM to both noise realization and strength is given in Wang et al. (1999).

Furthermore, the results above suggest a potential problem with using even complex models to assess the stability of the THC, assuming that our stochastic parameterizations are appropriate representations of high-frequency variability. First, more than a single circuit of the hysteresis loop by the simulation would need to be carried out in order to estimate the distribution of the jump points. Admittedly, for models of intermediate complexity such as in Rahmstorf (1999), or for a full GCM, it is computationally very expensive to carry out multiple circuits around the hysteresis loop. However, this could potentially be important to determine the variability in the values of the control parameters at which the transitions occur. Second, even complex models tend to be biased toward low levels of internal variability. This bias can arise from either the finite resolution of the model or from the use of an uncoupled ocean model. No current ocean GCMs used for diagnosis of the stability of the thermohaline circulation resolve mesoscale eddies, and 2D ocean models do not resolve even gyre-scale circulations. Their internal noise levels are then presumably lower than those of the real ocean, perhaps substantially so. Furthermore, many studies of the stability of the thermohaline circulation use uncoupled ocean models, or ocean models coupled to toy atmospheres. It is a generic feature of uncoupled ocean or atmosphere models that they systematically underestimate the variability observed in nature, as discussed by Barsugli and Battisti (1998). Our simple stochastic box model suggests that an underestimate of the noise level in a more complex ocean model could lead to systematic errors in the range of control parameters at which the model is found to switch between climate equilibria. These systematic errors would presumably be removed through improved model resolution or coupling to an atmospheric model, or potentially through the introduction of stochastic subgrid-scale parameterizations (Palmer 2001).

## 6. Summary and conclusions

In this study, we have used a simple box model to consider the effects on the thermohaline circulation of both internal fluctuations, associated with unresolved internal eddy variability, and external fluctuations, associated with atmospheric processes. Because of the extreme simplicity of the model, a closed-form, analytic expression for the stationary probability density function *p*_{s} could be obtained from the associated Fokker–Planck equation. This stationary density was found to have a complicated dependence on the noise strengths *σ*_{1} and *σ*_{2}, and on the parameter *μ* describing the differential freshwater forcing between the boxes. In particular, it was demonstrated that for much of the parameter space in which *p*_{s} is *technically* bimodal, in that it can be shown to have two extrema, it is *effectively* unimodal, in that one regime is very much more populated than the other. We denote this enhanced population of one regime as *stabilization.* For a broad range of noise levels within that part of parameter space in which the distribution is technically bimodal, the populations of the two regimes are comparable only for a small window of *μ* centered on the value *μ*_{0.5}. Figure 7a illustrates that *μ*_{0.5} itself lies within a small range of values of *μ.* We see then that in a nonlinear stochastic system, the fluctuations may play a substantial role in determining the mean state of the system.

The stationary distribution *p*_{s} describes the behavior of the system in the infinite time limit. For finite times, it will only be relevant if the residence times of the two regimes are small compared to the the relevant timescale dictated by the physical process under consideration. Here, this process is the THC and the timescale is on the order of 10^{4}–10^{5} yr. The simplicity of the model allows us to easily evaluate these residence times. For very small noise levels, the residence times are so large that the finite-time behavior of the system shares the feature with the deterministic system that the regime in which the system resides over the timescale of interest is determined by the initial conditions. However, there exists a broad range of parameter space for which the stationary distribution describes the behavior of the system and stabilization by fluctuations is relevant.

Stabilization of the system by fluctuations has a significant impact on the stability of the THC with regards to changes in freshwater forcing. In the deterministic limit, this may be described by a hysteresis loop, in which rapid transitions between regimes occur precisely at bifurcation points of the system. When fluctuations are added to the description, the hysteresis loop becomes random, and for intermediate noise strengths the transitions between regimes generally occur well away from the deterministic bifurcation points. This result suggests that fitting GCM output to simple deterministic models, as in Rahmstorf (1995, 1996), may overestimate the stability of the present climate. Furthermore, as GCMs generally are biased toward lower noise levels than those of the real world, this result also has bearing on GCM estimations of climate sensitivity.

These results provide another simple illustration of the idea put forward by Palmer (1999) that the effect of small perturbations on the climate system, such as anthropogenic forcing, may be not to change the structure of the existing regimes of circulation, but their occupation statistics (assuming the system is sufficiently far from a bifurcation point). Each of the parameters *μ,* *σ*_{1}, *σ*_{2} considered in this study could conceivably be altered by anthropogenic forcing, with consequent changes in the populations of the regimes, especially for *μ* in the neighborhood of *μ*_{0.5}.

It is also worth noting that this study demonstrates the importance of accurate discretization schemes for the numerical solution of nonlinear SDEs. Inaccurate discretizations can lead to substantial systematic errors in the strength of the noise, as in Lagerloef (1995). The qualitative behavior of the model considered in this study is sensitive to the strengths of the stochastic fluctuations, and simulations carried out using an inaccurate discretization scheme could consequently be qualitatively inaccurate. The numerical solution of SDEs is described in detail in Kloeden and Platen (1992).

The analysis presented in this study also demonstrates the effects that stochastic fluctuations can have on the bifurcation structure of a dynamical system. The stationary PDF displays bifurcation points at which it changes from bimodal to unimodal; these bifurcations are the distributional equivalents of the deterministic bifurcations at which a stable fixed point disappears. For the range of noise strengths considered here, the bifurcations of the stationary PDF and the deterministic system occur at not very different values of *μ.* However, because of stabilization, the bifurcation behavior of individual realizations is generally quite different than in the deterministic case. An interesting analog occurs in the effect of stochastic perturbations on Hopf bifurcations. In a deterministic forward Hopf bifurcation from a fixed point attractor, the asymptotic behavior of the system changes from one in which there is no characteristic oscillatory timescale to one characterized by a dominant oscillatory timescale. In the stochastic setting, the Hopf bifurcation is associated with changes in the stationary PDF and Lyapunov exponents (Arnold et al. 1996), but individual realizations possess oscillatory timescales that change smoothly through the bifurcation point. In effect, the presence of fluctuations destroys the bifurcation in the power spectrum present in the deterministic limit, a result that is of significance to theories of ENSO dynamics (Penland et al. 2000) and of interdecadal THC variability (Rivin and Tziperman 1997). These results demonstrate that stochastic fluctuations can have significant effects on important aspects of the bifurcation structure of climatic dynamical systems.

The model we have investigated in this study is highly simplified, even by the standards of box modeling. Because of the extreme simplicity of the model, no attempt has been made to fit the parameters *μ,* *σ*_{1}, or *σ*_{2} to data from observations or GCMs. Preliminary numerical investigations of the model (10)–(12) indicate that the stabilization phenomenon is qualitatively unchanged for nonzero *γ*^{−1} and *τ.* A physically meaningful quantitative examination of stabilization of THC regimes by noise would require more sophisticated models. A more dynamically significant, yet computationally efficient, class of models is the zonally averaged 2D ocean circulation models (e.g., Wright et al. 1998). This class of models could potentially be used for more quantitative, albeit numerical, studies of THC stabilization by noise.

Simple box models represent the most coarsely discretized extreme of models of ocean dynamics, and should be interpreted accordingly. While their *quantitative* features cannot be expected to be reliable, their *qualitative* behavior can be quite instructive. The remarkable correspondence between the general character of the behavior of box models and that of complicated GCMs demonstrated by Rahmstorf (1995, 1996) suggests that their apparent simplicity belies their ability to capture essential features of ocean dynamics. The stochastic model considered above introduces the phenomenon of stabilization that has important consequences for the dynamics of the model. The results of this simple model suggest an interesting experiment, in which a more complex model is run through as many hysteresis loops as computing power allows. If the stochastic parameterization of eddy variability is appropriate, the model should demonstrate random hysteresis loops, and this experiment would allow the assessment of their distribution. Such an experiment could be carried out using different model resolutions in order to vary the strength of the internal noise in the system. Evidence of a nondeterministic character of the hysteresis loops would have important consequences for any discussion of the stability of the THC.

## Acknowledgments

The author would like to thank Peter Imkeller, Till Kuhlbrodt, Cécile Penland, Sven Titz, Stefan Rahmstorf, Lionel Pandolfo, Richard Greatbatch, Gerrit Lohmann, and Axel Timmermann, for their suggestions and helpful comments. The original manuscript was also greatly improved by the comments and suggestions of two anonymous referees. This work was supported by DFG Schwerpunktprogramm “Inter-agierende stochastische Systeme von hoher Komplexität.”

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Contour plot of stationary distribution *p*_{s}(*y*) as a function of *μ* for *σ*_{1} = 0.1, *σ*_{2} = 0.1. Contours: 0.5, 1, 1.5, … , 6. The thick horizontal lines denote the boundaries between which *p*_{s} has two maxima

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Contour plot of stationary distribution *p*_{s}(*y*) as a function of *μ* for *σ*_{1} = 0.1, *σ*_{2} = 0.1. Contours: 0.5, 1, 1.5, … , 6. The thick horizontal lines denote the boundaries between which *p*_{s} has two maxima

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Contour plot of stationary distribution *p*_{s}(*y*) as a function of *μ* for *σ*_{1} = 0.1, *σ*_{2} = 0.1. Contours: 0.5, 1, 1.5, … , 6. The thick horizontal lines denote the boundaries between which *p*_{s} has two maxima

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Contour plot of stationary distribution *p*_{s}(*y*) as a function of *σ*_{2} for *μ* = 0.205, *σ*_{1} = 0.1. Contours: 0.5, 1, 1.5, … , 10.5

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Contour plot of stationary distribution *p*_{s}(*y*) as a function of *σ*_{2} for *μ* = 0.205, *σ*_{1} = 0.1. Contours: 0.5, 1, 1.5, … , 10.5

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Contour plot of stationary distribution *p*_{s}(*y*) as a function of *σ*_{2} for *μ* = 0.205, *σ*_{1} = 0.1. Contours: 0.5, 1, 1.5, … , 10.5

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Contour plot of (a) *E*(*y*) and (b) {*E*[(*y* − *E*(*y*))]^{2}}^{1/2,} where *E*( · ) denotes the expectation operator with respect to *p*_{s}, for *μ* = 0.19. For these parameter values, *y*_{−} ≃ 0.25 and *y*_{+} ≃ 1.16.

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Contour plot of (a) *E*(*y*) and (b) {*E*[(*y* − *E*(*y*))]^{2}}^{1/2,} where *E*( · ) denotes the expectation operator with respect to *p*_{s}, for *μ* = 0.19. For these parameter values, *y*_{−} ≃ 0.25 and *y*_{+} ≃ 1.16.

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Contour plot of (a) *E*(*y*) and (b) {*E*[(*y* − *E*(*y*))]^{2}}^{1/2,} where *E*( · ) denotes the expectation operator with respect to *p*_{s}, for *μ* = 0.19. For these parameter values, *y*_{−} ≃ 0.25 and *y*_{+} ≃ 1.16.

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 4 but for *μ* = 0.15. For these parameter values, *y*_{−} ≃ 0.18 and *y*_{+} ≃ 1.13

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 4 but for *μ* = 0.15. For these parameter values, *y*_{−} ≃ 0.18 and *y*_{+} ≃ 1.13

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 4 but for *μ* = 0.15. For these parameter values, *y*_{−} ≃ 0.18 and *y*_{+} ≃ 1.13

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 4 but for *μ* = 0.23. For these parameter values, *y*_{−} ≃ 0.36 and *y*_{+} ≃ 1.19

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 4 but for *μ* = 0.23. For these parameter values, *y*_{−} ≃ 0.36 and *y*_{+} ≃ 1.19

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 4 but for *μ* = 0.23. For these parameter values, *y*_{−} ≃ 0.36 and *y*_{+} ≃ 1.19

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

(a) Contour plot of *μ*_{0.5}, the value of *μ* such that the system is equally likely to be in either regime, as a function of *σ*_{1} and *σ*_{2}. (b) Contour plot of the window in *μ* around *μ*_{0.5} for which the ratio of the stationary probabilities of being in one regime is less than a factor of 10 of being in the other

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

(a) Contour plot of *μ*_{0.5}, the value of *μ* such that the system is equally likely to be in either regime, as a function of *σ*_{1} and *σ*_{2}. (b) Contour plot of the window in *μ* around *μ*_{0.5} for which the ratio of the stationary probabilities of being in one regime is less than a factor of 10 of being in the other

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

(a) Contour plot of *μ*_{0.5}, the value of *μ* such that the system is equally likely to be in either regime, as a function of *σ*_{1} and *σ*_{2}. (b) Contour plot of the window in *μ* around *μ*_{0.5} for which the ratio of the stationary probabilities of being in one regime is less than a factor of 10 of being in the other

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Base 10 logarithm of mean residence times (a) 〈*T*(*y*_{−})〉, (b) 〈*T*(*y*_{+})〉, in nondimensional time units, as a function of *σ*_{1} and *σ*_{2} for *μ* = 0.19

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Base 10 logarithm of mean residence times (a) 〈*T*(*y*_{−})〉, (b) 〈*T*(*y*_{+})〉, in nondimensional time units, as a function of *σ*_{1} and *σ*_{2} for *μ* = 0.19

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Base 10 logarithm of mean residence times (a) 〈*T*(*y*_{−})〉, (b) 〈*T*(*y*_{+})〉, in nondimensional time units, as a function of *σ*_{1} and *σ*_{2} for *μ* = 0.19

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 8 but for *μ* = 0.15

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 8 but for *μ* = 0.15

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 8 but for *μ* = 0.15

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 8 but for *μ* = 0.23

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 8 but for *μ* = 0.23

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 8 but for *μ* = 0.23

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

(a) Plot of the unstable fixed point *y*_{o} and the two stable fixed points *y*_{−} and *y*_{+} of the deterministic system (34) as functions of *μ.* There are bifurcations at *μ* = 0 and *μ* = 0.25. (b) Hysteresis curve for the deterministic model (34) obtained by increasing *μ* from −0.1 to 0.3 and then decreasing back to −0.1. The arrowheads illustrate the sense of movement around the curve

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

(a) Plot of the unstable fixed point *y*_{o} and the two stable fixed points *y*_{−} and *y*_{+} of the deterministic system (34) as functions of *μ.* There are bifurcations at *μ* = 0 and *μ* = 0.25. (b) Hysteresis curve for the deterministic model (34) obtained by increasing *μ* from −0.1 to 0.3 and then decreasing back to −0.1. The arrowheads illustrate the sense of movement around the curve

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

(a) Plot of the unstable fixed point *y*_{o} and the two stable fixed points *y*_{−} and *y*_{+} of the deterministic system (34) as functions of *μ.* There are bifurcations at *μ* = 0 and *μ* = 0.25. (b) Hysteresis curve for the deterministic model (34) obtained by increasing *μ* from −0.1 to 0.3 and then decreasing back to −0.1. The arrowheads illustrate the sense of movement around the curve

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Simulated hysteresis loops for the stochastic system (20) with noise levels (*σ*_{1}, *σ*_{2}): (a) (0.05, 0.05), (b) (0.1, 0.05), (c) (0.05, 0.1), (d) (0.1, 0.1). The thick curve is the deterministic hysteresis curve, plotted for comparison. The same realizations of the noise processes *W*_{1} and *W*_{2} were used in each simulation

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Simulated hysteresis loops for the stochastic system (20) with noise levels (*σ*_{1}, *σ*_{2}): (a) (0.05, 0.05), (b) (0.1, 0.05), (c) (0.05, 0.1), (d) (0.1, 0.1). The thick curve is the deterministic hysteresis curve, plotted for comparison. The same realizations of the noise processes *W*_{1} and *W*_{2} were used in each simulation

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Simulated hysteresis loops for the stochastic system (20) with noise levels (*σ*_{1}, *σ*_{2}): (a) (0.05, 0.05), (b) (0.1, 0.05), (c) (0.05, 0.1), (d) (0.1, 0.1). The thick curve is the deterministic hysteresis curve, plotted for comparison. The same realizations of the noise processes *W*_{1} and *W*_{2} were used in each simulation

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 12 but with (*σ*_{1}, *σ*_{2}) = (0.075, 0.075) for four different realizations of *W*_{1} and *W*_{2}

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 12 but with (*σ*_{1}, *σ*_{2}) = (0.075, 0.075) for four different realizations of *W*_{1} and *W*_{2}

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

As in Fig. 12 but with (*σ*_{1}, *σ*_{2}) = (0.075, 0.075) for four different realizations of *W*_{1} and *W*_{2}

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Gaussian kernel estimates of the PDF of the value of *μ* at which the transition occurs from the regime around *y*_{−} to that around *y*_{+} as *μ* increases (thin line), and the regime around *y*_{+} to that around *y*_{−} as *μ* decreases (thick line), for *σ*_{1} = 0.075, *σ*_{2} = 0.075. The estimates are based on 1000 realizations of the hysteresis loop

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Gaussian kernel estimates of the PDF of the value of *μ* at which the transition occurs from the regime around *y*_{−} to that around *y*_{+} as *μ* increases (thin line), and the regime around *y*_{+} to that around *y*_{−} as *μ* decreases (thick line), for *σ*_{1} = 0.075, *σ*_{2} = 0.075. The estimates are based on 1000 realizations of the hysteresis loop

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2

Gaussian kernel estimates of the PDF of the value of *μ* at which the transition occurs from the regime around *y*_{−} to that around *y*_{+} as *μ* increases (thin line), and the regime around *y*_{+} to that around *y*_{−} as *μ* decreases (thick line), for *σ*_{1} = 0.075, *σ*_{2} = 0.075. The estimates are based on 1000 realizations of the hysteresis loop

Citation: Journal of Physical Oceanography 32, 7; 10.1175/1520-0485(2002)032<2072:SOCRBN>2.0.CO;2