## 1. Introduction

In Sijp et al. (2012; hereafter, Part I) the hysteresis of a climate model of intermediate complexity was examined by equilibration under a fixed anomalous salt flux *H* applied to the North Atlantic for a range of values. This elucidated the domain of existence of the North Atlantic Deep Water (NADW) ON and OFF state. The total salt exchange between the Atlantic and the Southern Ocean *F*_{circ} counters changes in the net Atlantic surface flux *F _{s}* and depends on the volume-averaged upper-ocean Atlantic salinity

*H** where the OFF state exists was calculated from ∂

*F*

_{circ}/∂

*S*= 0. The closed Atlantic salt budget allowed the formulation of a simple evolution equation for

*t*. This equation also describes the evolution of the Antarctic Intermediate Water (AAIW) reverse cell strength

*M*via Eq. (2) of Part I (see also Table 2). The simplicity of this approach allows an analytical solution to the evolution equation to be found. We will compare analytical solutions of the evolution equation to equivalent time-dependent numerical model behavior. Analytical solutions fall into two categories: one that provides stable (that is nonsingular) solutions for certain initial conditions of

*F*close to the critical Atlantic surface flux

_{s}The main terms from Part I. Positive sign of flux terms indicate salt added to ocean.

The main equations from Part I.

Four basic premises allowed us to formulate a simple evolution equation for Atlantic salinity

- the salinities of the NADW formation region and the South Atlantic (between 30°S and the equator) remain linearly related to
under changes in *H*; - the properties of cool fresh AAIW constituting the lower branch of the reverse cell remains relatively unchanged compared to Atlantic surface salinity under changes in
*H*; - the strength of the reverse cell
*M*depends linearly on the density difference Δ*ρ*_{NADW−AAIW}between the NADW and the AAIW formation region; - the term
*F*acts linearly to reduce the salinity difference between the South Atlantic and the Southern Ocean to zero and so causes_{r}to tend toward a certain ; - North Atlantic density changes are dominated by salinity.

We will examine transient behavior arising from our evolution equation. Transient behavior predicted by simplified sets of equations has been studied for instance in box models by Lucarini and Stone (2005a,b), Zickfeld et al. (2004), and Colin de Verdière (2010). Lucarini and Stone (2005a) use the box model of Rooth (1982) for a qualitative study of time-dependent Atlantic Meridional Overturning Circulation (AMOC) behavior in the NADW ON state. In qualitative agreement with general circulation model results (Stocker and Schmittner 1997), they find that fast increases in moisture fluxes to the North Atlantic are more effective in shutting down NADW than slow increases, as with high rates of heat flux increases. Zickfeld et al. (2004) take the box model of Rahmstorf (1996), specify the volumes of the boxes, and determine the remaining parameters by a least squares fit of the hysteresis curve to the output of a coupled climate model of intermediate complexity [Climate and Biosphere Model 2 (CLIMBER-2)]. Weights are used to enforce a closer fit near certain regions of the hysteresis curve, for instance near the bifurcation point. Their approach allows a high number of sensitivity tests, for instance relating to the AMOC sensitivity to modified regional patterns of warming, initial conditions, and the rate of climate change. Their box model qualitatively reproduces the result of Stocker and Schmittner (1997), who demonstrated that the stability of the AMOC is dependent upon the rate of climate change.

The studies of transient behavior by Zickfeld et al. (2004) and Lucarini and Stone (2005a,b) examine transient behavior of the NADW ON state such as collapse and recovery under climate change. In the present study, transient model behavior in the OFF state is examined to validate our equation of motion [Eq. (4) in Part I]. We seek quantitative agreement between certain time-dependent behaviors following from our equation of motion [Eq. (4) in Part I and Table 2] and those exhibited by the three-dimensional coupled numerical model used. Where possible, attempts are made to explain any discrepancies in terms of violations of our basic premises 1–4 (Part I). These tests are intended to further validate the approach described in Part I. Once validated, our framework can be used and expanded in future studies for parameter sensitivity studies to examine the existence of the OFF state under different scenarios.

## 2. The model and experimental design

We use the intermediate complexity coupled model described in detail in Weaver et al. (2001). This consists of an ocean general circulation model [Geophysical Fluid Dynamics Laboratory Modular Ocean Model (GFDL MOM) Version 2.2; Pacanowski 1995] coupled to a simplified one-layer energy–moisture balance model for the atmosphere and a dynamic–thermodynamic sea ice model of global domain and horizontal resolution 3.6° longitude by 1.8° latitude. Heat and moisture advection takes place via advection and Fickian diffusion. Air–sea heat and freshwater fluxes evolve freely in the model, yet a noninteractive wind field is employed. The wind forcing is taken from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis fields (Kalnay et al. 1996), averaged over the period 1958–97 to form a seasonal cycle from the monthly fields. Vertical mixing in the control case is represented using a diffusivity that increases with depth, taking a value of 0.3 cm^{2} s^{−1} at the surface and increasing to 1.3 cm^{2} s^{−1} at the bottom. We use version 2.8 of the University of Victoria (UVic) model. The effect of subgrid-scale eddies on tracer transport is modeled by the parameterizations of Gent and McWilliams (1990). We will refer to this model as “the numerical model” to distinguish it from our conceptual model. A detailed outline of the experimental procedure used here is contained in Part I of this paper.

## 3. A test for the existence of the attraction basin

In Part I two equilibrium solutions *S*_{unst} directly in the numerical model. However, their values may be found via transient numerical experiments. The model should spend a particularly long time near this unstable flux-equilibrium state after an excitation by a salt pulse of the right magnitude. This is because small salinity perturbations grow or decay approximately exponentially around any average Atlantic salinity where the salt fluxes are balanced (see Part I and below). This growth is initially small for values very close to a particular salinity where salt fluxes are balanced, as in *S*_{unst} studied here.

To examine the role of the unstable OFF branch, we conduct experiments where an 80-year-long anomalous salt flux to the North Atlantic is applied for a certain *F _{s}* to obtain an initial condition close to that of the unstable OFF branch salinity at the chosen

*F*. The pulse consists of a simple triangular profile as described in Sijp and England (2006, 2008), peaking at year 40. It is not a priori clear that this approach is valid, as the salt pulse will likely cause a violation of Eqs. (1)–(3) in Part I. Therefore, to track how well some of our premises are satisfied at each time during transient behavior, an error indicator

_{s}*M*computed from

*M*

_{GCM}is the AAIW reverse cell strength in the model. We refer to this as the

*M*-

*S*error indicator and take it as a measure of the validity of our basic Eq. (2) (Part I), which encapsulates a linear dependence of

*M*on

*ρ*

_{NADW}−

*ρ*

_{AAIW}and our assumption about a linear relationship between the average Atlantic salinity and the average salinity at the NADW formation sites. Note that the connection between the average Atlantic salinity and the gyre salt flux

*F*[Eq. (3)] is not incorporated in this error indicator. Additional errors here may arise from the time it takes for a salinity anomaly in the North Atlantic to travel to the South Atlantic.

_{r}To isolate the component of the *M-S* error indicator arising from a violation of the linear relationship between *M* and *ρ*_{NADW} − *ρ*_{AAIW} on our error indicator, an “*M*-rho” error indicator (*M*(Δ*ρ*)−*M*_{GCM})/*M*_{GCM} is also computed, where *M*(Δ*ρ*) is the reverse cell strength *M* computed from Δ*ρ* = *ρ*_{NADW} − *ρ*_{AAIW} as shown in Fig. 6 of Part I. When conducting our pulse experiments, the error indicators are tracked to see how well our OFF state premises hold, and so judge whether conclusions can be drawn from our transient experiments.

Figure 1 shows the evolution in response to the short salt pulse of the Atlantic salinity and the AAIW reverse cell strength *M* in the model as a function of time, along with the two error indicators. We use the anomalous salt flux *H* = 0.0312 Sv (1 Sv ≡ 10^{6} m^{3} s^{−1}) salt equivalent, a value between *H* = 0 and the critical *H** (see Fig. 12 of Part I) where ^{−3} (Fig. 5a in Part I) for about 8000 years (not visible because of figure scale), before eventually collapsing back to the OFF state. As expected, the error indicators (Fig. 1c) are large during the initial perturbation, but settle down below 0.1 by year 500. The initial *M*-rho error is significantly lower than the *M*-*S* error, and close scrutiny reveals that the initial *M*-rho error appears to incorporate a time lag of M behind Δ*ρ* of around 30 years. The strong responsiveness of the AAIW reverse cell strength to NADW formation region density is remarkable.

The average Atlantic salinity ^{−3} (Fig. 1a) The dashed line in Fig. 1a indicates the unstable equilibrium OFF branch salinity predicted by Eq. (5) (Part I), and is in very good agreement with the model behavior. The associated behavior of the AAIW reverse cell strength *M* (Fig. 1b) exhibits a similarly good agreement, with a slight offset from the calculated unstable branch *M*, perhaps because of the *M*-*S* error shown in Fig. 1c.

## 4. Equilibration from one stable equilibrium to another

Using an OFF state equilibrium for one fixed value of *H* and then changing *H* from one value to another within the range of stable OFF states leads to transient behavior suitable for examination. Atlantic salinity *H*. The evolution equation [Eq. (4), Part I] offers a prediction for this transient behavior. Here, the agreement between the evolution equation and the numerical model with respect to this transient behavior is examined.

*t*=

*t** when the denominator is 0. That is,

*t*→

*t**. For this equation to be valid, the second term needs to be positive (as the exponential term is always positive). In other words, either

*t**. This behavior is of course in agreement with our considerations in Part I. For all initial conditions

*e*

^{−αt}= 0 in Eq. (8) to find the asymptotic end state

To examine how well the evolution to the stable OFF branch described by Eq. (1) agrees with the behavior of the model, a stable OFF state equilibrium can be taken in the model at *H* = 0 Sv and apply a constant *H* = 0.0160 Sv from thereon to examine the transient behavior as it equilibrates toward its new steady OFF state. The transient behavior should be described by Eq. (1), where initial and final conditions *β* = 0.25 (see Fig. 3, Part I) should be taken into account, where an initial increase *δH* in *H* eventually leads to an increase *δF _{s}* = (1 +

*β*)

*δH*in the total net Atlantic salt flux. This affects the rate of evolution but not the final state. The climate amplification effect can be incorporated into the coefficient

*α*of time

*t*. We use an averaged term

*H*= 0.0160 Sv. It can be shown that this is expected to lead to an error in equilibration rate (and time) of less than 6%. Furthermore,

*H*= 0),

*F*= 0.0160 Sv, see Part I) and

_{s}*a*=

*υ*

_{1}

*d*

_{1}= 0.4404 m

^{6}kg

^{−1}. The transient behavior of Atlantic salinity

*M*[computed via Eq. (2)]. The

*M*-

*S*error indicator (see explanation of Fig. 1) shown in Fig. 2c takes a maximum value around 0.015 in the first few hundred years, but is generally negligible. This indicates that basic Eq. (2) of Part I holds well, and that the source of the deviation in the transient behavior of the Atlantic salinity is likely caused by a violation of Eq. (1) and/or Eq. (3) of Part I. The change in surface salt flux

*H*occurs in the North Atlantic, and the discrepancy seen in Fig. 2a may be related to the time it takes for the associated salinity anomaly to travel to the South Atlantic and influence

*F*and

_{r}*F*. Nonetheless, the model clearly behaves in a very similar fashion to that described by Eq. (1), and predictions can be made using this equation.

_{m}## 5. Duration of transition from the OFF state to the ON state

The coefficient *F _{s}* approaches

*δF*

_{circ}and

*F*

_{circ}/∂

*S*, where

*F*

_{circ}evolves very slowly to match

*F*(after increasing

_{s}*F*from a lower

_{s}*F*close to

_{s}To examine the emergence of the ON state for *H**) is equilibrated with respect to a new unstable *H* > *H**. Figure 3 shows the time development of several key parameters in one of these numerical experiments where *H* is close to *H**. The AAIW reverse cell strength *M* in the model (Fig. 3a) shows a very slow initial decline in *M* accompanied by a slow increase in Atlantic salinity (Fig. 3c), followed by a rapid collapse of the AAIW reverse cell toward the end. The rapid collapse is accompanied by the sudden emergence of the NADW cell (Fig. 3b). The NADW value shown is measured as the maximum in the meridional streamfunction in the North Atlantic below the wind-driven surface layers. The error indicator (shown in Fig. 3d, see above for explanation) remains remarkably small throughout the process, where the error is negligible up to year 45 000. After that, a kink occurs in *M* leading into a regime of noticeably faster decline in *M* (Figs. 3a,c). However, the error indicator remains small throughout the rest of the integration period, indicating that the kink around year 45 000 does not yet herald a violation of the relationship between *M* and *S* of Eq. (2) (Part I). Subsequently at approximately year 50 000, the error indicator inflates rapidly when strong NADW formation appears near the end of the integration period. The relationship between *M* and

*F*is increased above

_{s}*F*just below

_{s}*F*

_{circ}= 0), the Atlantic salinity evolves aswhere

*t*is

*t** needed to reach the first singularity of the tan function, it can be shown that the reverse cell collapse times are given byFigure 4 shows these integration times as a function of the added anomalous salt flux Δ

*H*relative to an stable equilibrium OFF state close to

*H**. Also shown are the model OFF-state-collapse times as a function of Δ

*H*, where the time to an OFF-state collapse is defined as the time it takes for the reverse cell strength to become 0. Here, the climatic feedback is ignored, which may introduce errors between 0.2 and 0.4.

The times needed for the reverse cell to disappear agree very well with Eq. (3) for Δ*H* > 1.5 × 10^{−3} Sv, and deviate more significantly close to the singularity. The deviations in integration times may arise from additional dynamical effects overlooked by our approach. These unknown factors appear to smear the predicted *t** curve in the direction of *H*. Nonetheless, Eq. (3) captures the general behavior of the strongly increasing integration times around *H** and so provides insight into the reason behind these long evolution times.

Equation (3) suggests that *H** can be approached arbitrary closely, leading to arbitrary large decay times. However, an exceedingly small change *δF _{s}* in

*F*is not meaningful because of the background noise level in

_{s}*F*in the model. Inevitable perturbations of the OFF state at

_{s}*H** should render its decay time finite. We attempted to compute the collapse time

*H*with a normally distributed

*H*+

*δH*around

*H*in Eq. (3). A standard deviation in

*H*similar to that of

*F*corresponds to a 46 000 year maximal decay time. Decay times in excess of 54 000 years were found in the model. The main point here is that noise in the model should provide an upper bound to integration times. Note that here

_{s}## 6. Summary and conclusions

In Part I the closed Atlantic salt budget *F*_{circ} as a function of the average upper Atlantic salinity *F _{s}*. This equation describes the time-dependent behavior of

*S*

_{unst}that mark the upper boundary of the OFF state attraction basin are elucidated by the transient behavior shown here. Nonsingular solutions to the evolution equation offer an excellent description of the transient equilibration from one stable NADW OFF state to another in response to a change in the anomalous salt flux

*H*. We have also incorporated the effect of the climate amplification described in Part I in this experiment. Finally, unstable solutions to the evolution equation offer a good description of the distribution of decay times for the NADW OFF state around the critical surface flux

*H**. These long times to collapse for

*H*close to

*H** arise from the very small slope of

*F*

_{circ}with respect to

*H*is close to

*H** therefore implies very slow initial growth for small positive salinity anomalies

*H** in hysteresis experiments where the anomalous flux is slowly varied to take the system through a bistable trajectory.

Note that our experiments have dealt with perturbations to the NADW OFF state alone. It is unclear whether the unstable equilibria *M* and the density contrast between the AAIW and the NADW formation regions is better in our OFF state than the analogous case for the NADW formation rate in ON, where it appears that a more crude linear relationship holds (figure not shown).

Our aim has been an understanding of the dynamics responsible for the existence of the NADW OFF state in models where the AAIW reverse cell has a significant influence on the Atlantic salt budget. In this case, the OFF state meridional circulation has a significant influence on AMOC bistability and the transient behavior of the model. We found that we can accurately calculate the maximal Atlantic salinity and salt flux where OFF states exist. We could do this because of a set of simplifying properties of the OFF state that also allowed us to derive simple and testable analytical solutions to the evolution equation of the system in the OFF state. The numerical experiments shown here indicate that this predicted transient behavior also corresponds well with the numerical general circulation model.

We thank the University of Victoria staff for support in usage of the their coupled climate model. This research was supported by the Australian Research Council and the Australian Antarctic Science Program. We thank Marc d’Orgeville for an internal review and Bruce Henry from the School of Mathematics at the University of New South Wales for help in deriving the time-dependent solutions [Eqs. (1) and (2)] to the evolution equation shown in Part II.

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