Mechanism for Potential Strengthening of Atlantic Overturning Prior to Collapse

The Atlantic meridional overturning circulation (AMOC) carries large amounts of heat into the North Atlantic influencing climate regionally as well as globally. Palaeo-records and simulations with comprehensive climate models suggest that the positive salt-advection feedback may yield a threshold behaviour of the system. That is to say that beyond a certain amount of freshwater flux into the North Atlantic, no meridional overturning circulation can be sustained. Concepts of monitoring the AMOC and identifying its vicinity to the threshold rely on the fact that the volume flux defining the AMOC will be reduced when approaching the threshold. Here we advance conceptual models that have been used in a paradigmatic way to understand the AMOC, by introducing a density-dependent parameterization for the Southern Ocean eddies. This additional degree of freedom uncovers a mechanism by which the AMOC can increase with additional freshwater flux into the North Atlantic, before it reaches the threshold and collapses: an AMOC that is mainly wind-driven will have a constant upwelling as long as the Southern Ocean winds do not change significantly. The downward transport of tracers occurs either in the northern sinking regions or through Southern Ocean eddies. If freshwater is transported, either atmospherically or via horizontal gyres, from the low to high latitudes, this would reduce the eddy transport and by continuity increase the northern sinking which defines the AMOC until a threshold is reached at which the AMOC cannot be sustained. If dominant in the real ocean this mechanism would have significant consequences for monitoring the AMOC.


Introduction
The Atlantic meridional overturning circulation (AMOC) is being considered as one of the tipping elements of the climate system (Lenton et al., 2008).While the definition by (Lenton et al., 2008) is based on the idea that tipping elements re-35 spond strongly to a small perturbation, the AMOC might also be a tipping element in the dynamic sense of the word (Levermann et al., 2012).That is to say that a small external perturbation induces a self-amplification feedback by which the circulation enters a qualitatively different state.This self-40 amplification is due to the salt-advection feedback (Stommel, 1961;Rahmstorf, 1996) and has been found in a number of comprehensive ocean as well as coupled climate models (Manabe and Stouffer, 1993;Rahmstorf et al., 2005;Stouffer et al., 2006b;Hawkins et al., 2011).A cessation of the 45 AMOC would have far-reaching implications for global climate (Vellinga and Wood, 2002) which include (1) a strong reduction of northern hemispheric air and ocean temperatures (Manabe and Stouffer, 1988;Mignot et al., 2007), (2) a reduction in European precipitation and (3) its wind pattern the vertical density structure (Gnanadesikan, 1999).Stommel's model captures the salt-advection feedback in a pure form by resolving only the advection of the active tracers in two fixed-size boxes representing the northern downwelling and southern upwelling regions.The overturning strength is assumed to be proportional to the meridional density difference which was found to be valid in a number of ocean and climate models (e.g.Griesel and Morales-Maqueda (2006); Rahmstorf (1996); Schewe and Levermann (2010)).The Stommel-model is however missing a representation of the energy-providing processes for the overturning, such as the Drake-Passage effect and low-latitudinal mixing (Kuhlbrodt et al., 2007) as well as the influence of the Southern Ocean eddy circulation.
These processes are captured in a conceptual way by the model of Gnanadesikan (1999) which links the overturning to the vertical density profile as represented by the pycnocline depth.It was shown that this kind of model is not consistent with the fact that the meridional density gradient indeed changes with changing overturning in a number of different climatic conditions (Levermann and Griesel, 2004;Griesel and Morales-Maqueda, 2006).By construction it does not capture the salt-advection feedback and can thereby not be used to study the possibility of a threshold behaviour of the overturning.
There have been a number of attempts to combine these two approaches and thereby to comprise the horizontal tracer-advection with the vertical one (Marzeion and Drange, 2006;Johnson et al., 2007;Fürst and Levermann, 2011).
Here we advance the simplest of the suggested models (Fürst and Levermann, 2011) by introducing an additional parametrisation for the Southern Ocean eddy flux.As found in a comprehensive coarse resolution ocean model (Levermann and Fürst, 2010) the horizontal scale of the Southern upwelling region can change and neglecting this change leads to a misrepresentation of the circulation within the Gnanadesikan (1999) framework.We attempt to complement the conceptual model in order to correct for this shortcoming.
To this end we add a variable, meridional density difference in the southern Atlantic ocean in the scaling of the eddyinduced return flow.As will be shown, this allows for a qualitatively different response of the AMOC under freshwater forcing compared to earlier studies: a growth of the northern deep water formation with increasing freshwater flux from low-to high northern latitudes within the Atlantic before the threshold is reached and no AMOC in the modelled sense can be sustained.The threshold behaviour found here is consistent with the salt-advection feedback in the sense of a netsalinity transport by the overturning as suggested by Rahmstorf (1996) and recently verified for a number of climate models and observations by Huisman et al. (2010).
This paper is structured as followed: Firstly we describe the parametrisation of the transport processes, pycnocline dynamics and salinity dynamics, i.e. horizontal density distribution (section 2).The transport processes include the two fun-Fig.1.Schematic of the conceptual model as suggested in Fürst and Levermann (2011) and used here.The depth of the pycnocline D is determined by the balance between the northern deep water formation mN , the upwelling in the low-latitudes mU in response to downward mixing, the Ekman upwelling mW and the eddy-induced return flow mE.Salinity is advected along with these transport processes and determines together with a fixed temperature distribution the horizontal density differences.The differences are between lowlatitudinal box and northern box, ∆ρ, and low-latitudinal and southern box, ∆ρSO, respectively.The density differences, in turn, determines the northern sinking, mN ∝ D 2 ∆ρ, and the eddy-induced return flow, mE ∝ D∆ρSO.
damental driving mechanism (Kuhlbrodt et al., 2007) which are low-latitudinal upwelling (Munk, 1966;Munk and Wunsch, 1998;Huang, 1999;Wunsch and Ferrari, 2004) and 120 wind-driven upwelling in southern latitudes (Toggweiler andSamuels, 1995, 1998).In order to examine the behaviour of the model we derived governing equations for the two driving mechanisms separately as well as for the full case.The threshold behaviour, as described by Stommel (1961) is 125 caused by the salinity advection.For simplicity we keep the temperatures fixed through-out the paper (section 3).Section 4 discusses the change in the AMOC with increasing freshwater flux into the North Atlantic for the wind-driven case and the full case.We conclude in section 5.  (Gnanadesikan, 1999).The northern and southern boxes are fixed in volume while the low-latitudinal boxes vary in size according to the dynamically computed pycnocline depth.The four meridional tracer transport pro-140 cesses between the boxes control the horizontal and vertical density structure and the overturning.The density structure, in turn, determines the transport processes.Changes in the vertical density structure are described by variations in the pycnocline depth.The horizontal density structure is expressed by a southern and a northern meridional density difference.They depend on the dynamics of the active tracers,temperature, T, and salinity, S. For simplicity we assume a linear density function ∆ρ = ρ 0 (β S ∆S − α T ∆T ) (Stommel, 1961).In order to capture the main feedback for a threshold behaviour while keeping the model legible, we include salinity advection and neglect changes in temperature.The simplification further is justified because the temperature in the upper layers is strongly coupled to atmospheric temperature which is to first order determined by the solar insulation.We thus assume, the ocean temperature in the upper layers to be constant.The high-latitudinal boxes represent strong out-cropping regions which homogenizes the water column and extends the argument to depth.In steady state, the fourth box, deeper low-latitudes ocean, is determined by the three other boxes.That means the approximation is valid for the whole model in equilibrium and temperature is used as an external parameter.
The base of our work is the model by Fürst and Levermann (2011).We use the same parametrisations for the northern deep water formation and the upwelling processes.For the eddy return flow we introduce a different scaling by implementing southern meridional density difference which has strong influences on the behaviour of the model (sections 3 and 4).

Tracer transport processes
Different scaling for the deep water formation (as summarized in Fürst and Levermann (2011)) have been suggested.Here we use a parametrisation suggested by Marotzke (1997) and apply a β-plane-approximation to it.The resulting northern sinking scales linearly with the meridional density difference and quadratically with the pycnocline depth following geostrophic balance and vertical integration.
Because all values are external parameters (table 1) except the meridional density difference ∆ρ = ρ N − ρ U and the pycnocline depth D, the parameters are comprised into one constant C N .In contrast to previous approaches (e.g.(Rahmstorf, 1996)) the meridional density difference does not span the whole Atlantic but instead is taken between low and high northern latitudes in accordance with the geostrophic balance between the meridional density difference and the North Atlantic Current.The low-latitudinal upwelling follows a vertical advectiondiffusion balance (Munk and Wunsch, 1998).That is to say, downward turbulent heat flux is balanced by upward advection.This balance with a constant diffusion coefficient for the full upwelling region yields an inverse proportionality between upward volume transport and pycnocline depth.Again all external parameters are expressed by one constant C U to obtain The southern upwelling term is considered to be independent of the pycnocline depth and results from the so-called Drake-Passage effect (Toggweiler and Samuels, 1995): The eddy return flow is parametrised following Gent and McWilliams (1990) which yields a tracer transport proportional to the slope of the outcropping isopycnals.In the formulation of Gnanadesikan (1999) this is represented by a lin-205 ear dependence on the pycnocline depth divided by a horizontal scale for the outcropping region which is taken to be constant.The assumption of a constant horizontal scale for the outcropping region is not consistent with freshwater hosing experiments in a comprehensive though coarse resolution 210 ocean model (Levermann and Fürst, 2010).Here we attempt to capture variations in the meridional horizontal length scale of the outcropping region by the meridional density difference between the low-latitude ocean and the Southern Ocean, ∆ρ SO = ρ S − ρ U .We thus use the parametrisation As before, all quantities except D and ∆ρ SO are external parameters and compressed into one constant C E .

Pycnocline and salinity dynamics
The temporal evolution of the pycnocline is determined by 220 the tracer transport equation following Marzeion and Drange (2006).
Salinity equations for each box are derived from the advection in and out of the box, conserving salinity, as well as 225 the surface fluxes, F N and F S which represent atmospheric freshwater transport as well as the horizontal gyre transport.The advection scheme follows the arrows shown in figure 1.In computing the temporal changes in total salinity the changes in volume due to the pycnocline dynamics needs to 230 be accounted for.
With finite difference method applied to equations (1-6), we made numerical simulations which reached in equilibrium the values shown in table 2 with the parameters given in table 1.

Governing Equation
Here we derive an equation for the steady-state solution of equations (1-6) by comprising them into one equation of the oceanic pycnocline, D. We derive governing equations for the full case as well as for the analytically solvable cases of a purely mixing-and a purely wind-driven cases.The model is limited to positive and real solutions for the pycnocline as 1000 500 0 500 1000 1500 2000

Pycnocline Depth D [m]
Variable Scale full wind−driven mixing−driven Fig. 2. Trend of the governing equation for the full case (red line), the wind-driven case (mU = 0, blue line) and the mixingdriven case (mW = mE = 0, green line).The intersections with zero (black dashed line) are solutions of the polynomial but those in the grey shadowed area correspond to a negative pycnocline depth.Therefore they are not physical.In all three cases there are two positive solutions, a lower stable, physical one D and a higher unstable or non-physical one D.In the wind-driven case the non-physical solution is out of range of the pycnocline but is shown in figure 5.For the full case the solutions are D=616m and D =1342m, for the wind-driven case they are D=523m and D =6190m and for the mixing-driven case they are D=446m and D =1985m.
well as for non-negative tracer transport values.A parameter combination that does not allow for a solution of this kind is 250 thereby inconsistent with an overturning circulation as represented by this model.We denote a parameter region for which no such a physical solution exists as an "AMOC-offstate-region".As in the earlier version of the model (Fürst and Levermann, 2011) we find a threshold behaviour with respect to an increase of the freshwater flux, F N , for all three cases.The focus of this study is not to show the existence of such a threshold of all parameter values, but to present a mechanism by which the overturning can increase before the threshold is reached and no AMOC can be sustained.260

Full case
In the full case the governing equation is a polynomial of 10th order in the pycnocline depth (appendix A1, equation A7).Thus solutions can only be found numerically.Of the 10 mathematical roots, two are positive and real but of two 265 adjacent solutions only one can be stable.Numerical solutions were obtained in two ways.First by finding the roots of the polynomial (appendix A1, equation A7) and second by time forward integration of the original set of equations (1-6) with different initial conditions.The time integration naturally selects the stable solutions.Though this is not a proof by any means, we feel confident to say that the solution with D=616m is the stable of the two physical solutions (figure 3a).The corresponding tracer transport values are provide in figure 3b.The northern sinking decreases with increasing 275 freshwater forcing for the parameter set of table1.The equation for the northern sinking as it results from the scaling (equation 1) and the salinity equations: was also valid in the earlier version of the model (Fürst and 280 Levermann, 2011).Rahmstorf (1996) provides a similar solution for the northern deep water formation with k as proportionality factor between the northern sinking and the northsouth density difference: 285 In these earlier models only positive roots of the solution yield stable equilibria.That differs from our model where for certain amounts of freshwater forcing the negative sign of the root in equation ( 7) (respectively equation 8) needs to be considered, as for example in the wind-driven case dis-290 cussed below.Similar to the wind-driven case, the threshold of the overturning is reached when the eddy return flow becomes negative (figure 3b, grey shaded area).

Mixing-driven case 295
The purely mixing-driven case is defined by C E = C W = 0.In this case the pycnocline dynamics in steady state (equation 5) reduces to m N = m U = C U /D.As the eddy return flow is eliminated from the equation, this case has not changed compared to the model of Fürst and Levermann (2011): The gov-300 erning equation is a polynomial of fourth order in pycnocline depth and has one physical solution which decreases with increasing freshwater forcing (figure 4a).The overturning decreases until a threshold level (figure 4b) which is reached when the pycnocline and therefore the tracer transport pro-305 cesses become complex.The critical northern freshwater flux can be calculated by zero-crossing of the discriminant of the polynomial.

Wind-driven case 310
The purely wind-driven circulation is defined by C U = 0. Thus the tracer-transport balance in steady state (equation (b) ance (equation 6d).The emerging governing equation is a third order polynomial of the pycnocline depth D which we solve analytically.
Ehlert and Levermann: Minimal overturning model 7 The the solutions depend on the sign of the discriminant of the polynomial d = (q/2) 2 + (p/3) 3 with p and q defined as: A polynomial of third order has either one root (appendix A2, equation A9) if the discriminant is positive, or three roots (appendix A2, equation A10) if the discriminant is negative 330 which is the case for the parameters of table 1 near the threshold (figure 5).Only one of the three mathematical roots is a physical solution of equilibrium state of the model because one root is negative (figure 5 The salinity difference contains no variables.As the temperature dynamics are not considered in this model, the horizontal density difference between these two boxes is constant for a fixed set of parameters. The critical eddy return flow is equal to zero.Using the definition of the flow (equation 4) and the fact that the critical pycnocline depth is far in the positive range, equation ( 10) can be set to zero at the threshold level.The critical freshwater flow becomes: The critical northern freshwater flow depends linearly on the 360 southern temperature difference and on the southern wind stress (via C W ) and a higher southern freshwater flux would lower the critical northern freshwater flow.Please note that this is a significant difference to previous approachers (Fürst and Levermann, 2011;Rahmstorf, 1996), where the critical 365 freshwater flow is in first or higher order (equation 9) sensitive to the northern temperature difference which has no influence onto the critical freshwater flux in this case.

Freshwater-induced MOC strengthening
The introduction of the southern density difference as a vari-370 able changing the eddy return flow results in a mechanism that, to our knowledge, has not been reported before: an increasing overturning under northern freshwater forcing the threshold beyond no AMOC as described by these equations can be sustained.The mechanism is simple: A freshwater 375 flux from low-latitudes into the high northern latitudes reduces the eddy return flow.If this reduction is not compensated by a reduction in mixing-driven upwelling (as for example in a mainly wind-driven AMOC) then due to continuity northern sinking has to increase since Southern Ocean 380 upwelling is constant.The mechanism is always dominant in the wind-driven case which we will proof at the end of this section.In the full case the mechanism takes not effect for the parameters of table 1 but it emerges if the Southern Ocean temperature difference is changed in such a way as to 385 make the mixing less relevant (figure 6).

Full case
In order to gain a better understanding of this behaviour, the tracer transport processes balance in steady state (equation 5 equal to zero) is differentiated with respect to the northern 390 freshwater flux.That gives an equation for the derivative of northern sinking: Using the parametrisations of the eddy return flow (equation 4) and low-latitudinal upwelling (equation 2), equation ( 11) The polynomial consists of two terms of opposing sign: The first term on the left depends on the change of pycnocline depth (representing the vertical density structure) with 400 increasing freshwater flux.Since this derivative, ∂D ∂F N , is generally positive the full term is negative.The second term is positive since the horizontal density difference in the Southern Ocean declines when F N is increased.The sign of the derivative of the northern sinking is determined by the ratio between the two terms.Thus strong increasing pycnocline depth, i.e. strong positive changes in vertical density structure, shift the overturning to a deceasing threshold behaviour.If the southern meridional density difference decreases stronger (in absolute values), then the overturning rises under freshwater forcing.The crucial point is that the absolute value of pycnocline is present in the term with the derivative of southern meridional density difference.That means rising pycnocline depth also amplifies the term that depends on horizontal density structure and vice versa for the meridional density difference.A stronger statement can be derived for the purely wind-driven case.

Wind-driven case
Upwelling in the lower latitudes amplifies the decreasing of (equation 10) the derivative of the northern sinking emerges.
Now, solely the term depending on the negative southern 430 density difference could diminish the derivative.For the values given in table 1, ∂D ∂F N 100m 0.1Sv , and D 1000m, the derivative is far in the positive range ( ∂m N ∂F N 5000).In order to calculate the critical derivative, we use again the fact that the southern density difference equals zero at the threshold.
The emerging critical derivative depends only on positive constants and the positive critical pycnocline depth, i.e. the overturning always increases close to the threshold.This result is not surprising in light of the heuristic explanation given above, but it is not trivial due to the still complex vertical and horizontal density dynamics.

Conclusion and Discussion
The conceptual model of the Atlantic overturning presented here builds on a previous model (Fürst and Levermann, 2011) and advances the model by the introduction of a dynamic southern ocean density difference for the eddy return flow as imposed by comparison with comprehensive ocean model results (Levermann and Fürst, 2010).As a first result the model reproduces the qualitative result that a threshold behaviour is a robust feature that is independent of the driving mechanism, i.e. it is present in a mixing-, a wind-driven as well as in a combined case.The regime of existence of a solution for the overturning for a specific parameter combination is defined by the simultaneous compliance of a number of conditions, e.g.positive volume fluxes and pycnocline depth.In the presented model the threshold is generally reached when the eddy return flow becomes negative.Similar to the predecessor of the model also here the threshold is associated with the salt-advection feedback.As suggested by Rahmstorf

460
(1996), a threshold thus only exists when the salinity in the low-latitude box is higher than in the northern box.This is the case here (see table 2).Whether the real ocean is in a bistable regime and thereby exhibits a threshold behaviour is of yet unclear.According to a diagnostic by Rahmstorf (1996), an 465 overturning is bistable if the overturning carries a net salinity transport at 30S.This diagnostic was confirmed to be valid in a comprehensive climate model (Dijkstra, 2007) and is discussed in depth by Hofmann and Rahmstorf (2009).Following this diagnostic most climate models do not show a thresh-470 old behaviour, while observational data indicates that the real ocean is in a bistable regime (Drijfhout et al., 2010;Huisman et al., 2010).
The main result is the observation that the overturning can increase prior to its collapse in response to a freshwa-475 ter flux from low-latitudes to high northern latitudes.Previous models including the base models (Johnson et al., 2007;Marzeion and Drange, 2006;Fürst and Levermann, 2011) show the opposite behaviour, similar to the bifurcation in the initial model of Stommel (1961).The emergence of the ef-480 fect depends on the inclusion of Southern Ocean winds as a driving-mechanism for the overturning and the inclusion of a dynamic southern ocean horizontal density difference.It thus does not include in the mixing-driven case.Thus our model has opposite behaviour prior to reaching the thresh-485 old depending on whether the circulation is wind-or mixingdriven.
This has strong implications for potential monitoring systems that aim to detect the vicinity to the threshold.Methods that depend on the decline of the overturning prior to 490 the threshold for example in order to detect an increase in variability might not be suitable in a situation (Lenton, 2011;Scheffer et al., 2009) in which the presented mechanism is relevant.
Whether the mechanism described here is dominant in 495 the real ocean is beyond the scope of this paper.This study presents the physical processes which need to be investigated with comprehensive quantitative models and verified against observation in order to assess its relevance.Though a large number of so-called water hosing experiments have been car-500 ried out (e.g.Manabe and Stouffer (1995); Rahmstorf et al. (2005); Stouffer et al. (2007)), few studies have focussed on freshwater transport from low-to high-latitudes.Such experiments are needed in order to find whether the mechanism is indeed relevant for the real ocean.
The salinity balance of the southern box combined with equation (A4) results into an equation for the southern salinity difference.
The scaling of the eddy return flow (equation 4), the linear density function for southern meridional density difference (∆ρ SO = β∆S SO −α∆T SO ), and equation (A5) can be collapsed into a quadratic equation for m E .
It has the solution: The governing equation of the pycnocline depth emerges by using equation (A6) and replacing the eddy return flow by m E = m U + m W − m N , m 2 N by equation (A2), and the 540 upwelling transport processes, m U and m W , by their scaling (equations 2 and 3).The solutions of the polynomial depend on the sign of the discriminant d = (q/2) 2 + (p/3) 3 with p and q defined as: If the disciminant is positive the governing equation has one 565 real solution.
For a negative discriminant there are three real solutions.
We use a standard inter hemispheric model with four varying boxes (figure1): (1) a northern box representing the northern North Atlantic with deep water formation, (2) an upper lowlatitudinal box and (3) a deeper low-latitudinal box below the 135 pycnocline, (4) a southern box with southern upwelling and eddy return flow

Fig. 3 .Fig. 4 .
Fig. 3.In steady state only one real stable solution of governing equation of the full exists which increases under freshwater forcing (diagram a).The tracer transport processes show different behaviours (diagram b).The eddy return flow mE decreases (diagram b, green line) until it becomes negative and the break down of circulation is reached (grey shaded area).Also the density difference between the southern box and the low-latitudinal box, ∆ρSO, crosses zero at the threshold level (diagram c, green line).
, solution 1) and the other solution has a negative northern sinking and the pycnocline values are out of range of the ocean depth (figure 5, solution 0).No physical solution exists, when the eddy return flow becomes negative.At this threshold the discriminant of the governing equation has a negative pole which can be used to calculate the critical freshwater flux.In the following we describe a more straight forward way to give dependencies of the critical freshwater flux.Assuming steady state for the salinity balance of the upper low-latitudinal box (equation 6b equal to zero, with m U = 0) and for the tracer transport balance (m E + m N = m W = C W ), the salinity difference between the Southern Ocean and the upper low-latitudes emerges:

Fig. 5 .
Fig.5.In steady state only one physical solution of governing equation for the wind-driven case exists.There are three real solutions before the circulation breaks down (diagram a, white area) because the discriminant is negative (diagram e).The physical branch is solution 2 (red line).The threshold (grey shaded area) is reached when the eddy return flow becomes negative (diagram c, red line) and the discriminant of the governing equation has a negative pole (diagram e).The zero crossing of the discriminant, which was in the parent model(Fürst and Levermann, 2011) the indicator for a cessation of the circulation, does not appear within the range applicability of our model.Within that range the northern sinking always increases (diagram b, red line) and its derivative is positive (diagram d, red line).

420Fig. 6 .
Fig.6.The derivative of the northern sinking with respect to freshwater forcing in full case.The derivative is positive before the circulation collapses (white area).This behaviour is caused by a change in the Southern Ocean temperature from TS = 7 • C to TS = 5 • C.
-driven overturning the upwelling in the lower latitudes is zero by setting C U = 0. Thus the tracer transport balance in steady state (5 equal to zero) reduces to m W = m N + m E .Differences in salinity are defined as in the full problem and the salinity balance in the northern box 550 is the same as in the full problem.Therefore equations (A1-A3) are valid.Using the salinity balance of the southern box, in this case the southern salinity difference reduces to:∆S SO = − S 0 (F N + F S ) m WFor the eddy return flow it follows:555m E = −C E Dβ S 0 (F N + F S ) C W − α∆T SO C E D (A8)Replacing the northern sinking by equation (A3) and the eddy return flow by equation (A8) in the tracer transport balance the governing equation of the pycnocline depth emerges.560D 3 C E C N α∆T [ βS 0 (F N + F S ) C W + α∆T SO ] + D 2 [C N F N S 0 β + C N C W α∆T β(F N + F S ) + C W α∆T SO ) 2 ] + D2C E [βS 0 (F N + F S ) + C W α∆T SO ] + C 2 W = 0

Table 1 .
Physical parameters for used for the model 3 psu) Product of ρ0 and βS C 0.1 − Constant accounting for geometry and stratification External Forcing βN 2 • 10 −11 1/(ms) Coefficient for β-plane approximation in the North Atlantic fDr −7.5 • 10 −5 1/s Coriolis parameter in the Drake Passage τDr 0.1 N/m 2 Average zonal wind stress in the Drake Passage FN 0.1 Sv Northern meridional atmospheric freshwater transport FS 0.1 Sv Southern meridional atmospheric freshwater transport TN 5.0 C • Temperature of the northern box TU 12.5 C • Temperature of the tropical surface box TS 7.0 C • Temperature of the southern box

Table 2 .
Numerical solution of the model by applying finite difference method on equations (1-6).Equilibrium state is obtained after 2000 years with a time step of 14 days and the starting conditions: Salinities set to 35psu and the pycnocline depth set to 500m.