The variability of the meridional overturning circulation (MOC) in the upper tropical Atlantic basin is investigated using a reduced-gravity model in a simplified domain. Four sets of idealized numerical experiments are performed: (i) switch-on of the MOC until a fixed value when a constant northward flow is applied along the western boundary; (ii) MOC with a variable flow; (iii) MOC in a quasi-steady flow; and (iv) shutdown of the MOC in the Northern Hemisphere. Results from experiments (i) show that eddies are generated at the equatorial region by shear instability and detached northward; eddies are responsible for an enhancement of the mean flow and the variability of the MOC. Results from experiments (ii) show a transitional behavior of the MOC related to the eddy generation in interannual–decadal time scales as the Reynolds number varies due to the variations in the MOC. In experiments (iii), a critical Reynolds number Rec around 30 is found, above which eddies are generated. Experiments (iv) demonstrate that even after the collapse of MOC in the Northern Hemisphere, eddies can still be generated and carry energy across the equator into the Northern Hemisphere; these eddies act to attenuate the impact of the MOC shutdown on short time scales. The results described here may be particularly pertinent to ocean general circulation models in which the Reynolds number lies close to the bifurcation point separating the laminar and turbulent regimes.
The meridional overturning circulation (MOC) is an important component of earth’s climate system. A mechanically driven heat engine (Munk and Wunsch 1998), the MOC crosses the equator in the upper Atlantic Ocean mainly through the North Brazil Current (NBC) carrying warm water into the higher latitudes. The associated heat transport reaches 1.2 PW (1015W; Ganachaud and Wunsch 2000), and is responsible for moderating climate in northern latitudes (e.g., Vellinga and Wood 2004). While flowing into the Northern Hemisphere, heat is lost to the atmosphere, and the surface limb of the MOC increases its density. In the high latitudes of the Labrador and Greenland–Iceland–Norwegian Seas, these surface waters subduct at the end of winter because of convective processes penetrating up to 1500 m deep to form North Atlantic Deep Water (NADW; e.g., Dickson and Brown 1994). This deep penetration into the ocean interior is a key feature for the uptake of anthropogenic CO2 from the atmosphere (Sabine et al. 2004). These subducted waters then flow southward, closing the MOC cell in the Atlantic Ocean. Variations in the MOC on centennial–millennial time scales may be forced by convection in high latitudes of the North Atlantic (Broecker 1997), wind forcing over the Southern Ocean (Toggweiller and Samuels 1995), or other water-mass transformation processes (e.g., Munk and Wunsch 1998).
Paleoclimatic data suggest that NADW formation may have been interrupted in the past (Sarnthein et al. 1994), consistent with the idea that the present mode of the Atlantic MOC might not be the only one possible. Indeed, multiple equilibrium states of the MOC have been found in a hierarchy of models (e.g., Stommel 1961; Rooth 1982; Marotzke and Willebrand 1991; Manabe and Stouffer 1994), though not in any state-of-the-art ocean–atmosphere model used for climate prediction. Substantial changes may occur over periods as short as 5–10 yr (Schlosser et al. 1991). Furthermore, carbon dioxide emissions into the atmosphere due to human activities have been linked to a slowdown and possible interruption of the MOC (Rahmstorf and Ganopolski 1999); while state-of-the-art climate models suggest that an abrupt collapse of the MOC over the next century is unlikely, most models do show a weakening MOC (Meehl et al. 2007).
Less complex models can help elucidate how anomalies of the MOC propagate throughout the global oceans (see, e.g., Wajsowicz and Gill 1986; Kawase 1987; Huang et al. 2000; Johnson and Marshall 2002a,b, 2004; Cessi et al. 2004; Deshayes and Frankignoul 2005; Liu and Alexander 2007; and references therein). Of particular relevance to the present study is the equatorial buffer mechanism identified by Johnson and Marshall (2002a), which restricts abrupt changes of the MOC to a hemispheric basin at high frequencies. These authors show that in the idealized case of a complete shutdown of the NADW formation at high latitudes, the MOC stops within a few months in the North Atlantic in their simple model, while adjustment in the South Atlantic takes up to several decades.
The Atlantic western boundary is a region of intense variability. In the north equatorial Atlantic several eddies (∼5–6 a year) are generated in the surface limb of the MOC because of the North Brazil Current retroflection (Goni and Johns 2003), and their formation and shedding is closely related to the strength of the MOC (Fratantoni et al. 2000), although other processes such as the wind-driven subtropical cells (STCs) also play an important role (Jochum and Malanotte-Rizzoli 2003). Dengler et al. (2004) show that the deep western boundary current (DWBC) also breaks up into eddies in the south equatorial Atlantic, with the eddy kinetic energy (EKE) maximum at about 2-km depth, coincident with the core of the upstream DWBC. Dengler et al. further presents a model simulation that suggests that the DWBC breaks up into eddies at the present intensity of the MOC, whereas for weaker overturning the DWBC may continue as a stable, laminar, boundary current. Nonlinearities are therefore crucial in understanding and predicting the behavior of cross-equatorial flow in the western Atlantic.
The global atmospheric response due to the MOC collapse has been studied in coupled ocean–atmosphere models, and is mediated largely through atmospheric teleconnections. Understanding the sensitivity of flows in the equatorial Atlantic to changes in the MOC is particularly important because the dominant impact of the MOC on the global atmosphere is likely to be communicated through changes in the tropical Atlantic sea surface temperatures (e.g., Dong and Sutton 2002).
The large-scale ocean circulation in the upper tropical Atlantic is dominated by Sverdrup gyres, subtropical cells, and large-scale MOC flows (Goes et al. 2005). All these components are entangled in the observed ocean currents. The disentanglement of the components is a nonlinear problem, and anomalies in the different components can impact each other (Hazeleger and Drijfhout 2006). Hence, simple models can bring helpful insights on how to separate such processes.
Killworth (1991), studying the adjustment in a purely inertial flow, finds that the change in sign of the planetary potential vorticity (PV) as flow crosses the equator is largely balanced by changes in relative vorticity. However, the dissipative terms also play an important role in the boundary layer (Ierley and Young 1991; Cessi and Ierley 1993), no matter how large the Reynolds number, and the dynamics of the oceanic western boundary necessarily involves both inertial and dissipative terms. The transformation of potential vorticity after spinup of an idealized cross-equatorial flow is further studied by Edwards and Pedlosky (1998). With a parameterized dissipation, they define a Reynolds number (Re) to characterize the nonlinearity of the system. Beyond a critical Re the system exhibits a transition to time-dependent motion. In this quasi-steady nonlinear flow, eddies are essential for the transfer of vorticity from the inertial interior to the frictional boundary layer and for the change in the sign of the potential vorticity across the equator.
The present note builds on concepts applied in Johnson and Marshall (2002a) and Edwards and Pedlosky (1998) to study the eddy variability of the MOC in the upper Atlantic Ocean, focusing on the following main questions:
How does a western boundary flow change from laminar to turbulent state in a multilayer primitive equation model?
What is the dynamical response of the upper, warm limb of the MOC, on decadal and shorter time scales, to a change in deep-water formation rate at high latitudes?
2. Model formulation
a. Equations of motion
A 4.5-layer reduced-gravity ocean model is adopted in this study that treats the ocean as four moving layers overlying an infinitely deep lower layer (Fig. 1). Each layer can be regarded as homogeneous, separated by a discontinuity in density.
The model domain is an idealized representation of the tropical Atlantic Ocean, consisting of a rectangular basin extending to 3000 km zonally and 6000 km meridionally. The equator is located in the middle of the basin and the equations are solved using the equatorial β-plane approximation, in which the Coriolis parameter is expressed as f = βy. The uppermost layer is of constant depth of 50 m and is intended to represent both the surface mixed layer and wind-driven surface Ekman layer currents if any wind forcing is applied. The initial layer thicknesses for the remaining layers are shown in Table 1. For simplicity, the layers are defined from one to four, where h1 is the constant depth surface layer, and h2, h3, and h4 are the layers underneath.
The complete momentum equation is
where un = (u, v) is the horizontal fluid velocity, ζn = ∇ · un is the relative vorticity, and Bn is the Bernoulli potential defined as
where g′ is the reduced gravity. The horizontal diffusion (AH) is introduced to allow the representation of the western and equatorial boundary layers. The term between brackets on the left side of Eq. (1) is an entrainment term, which only acts between layers 1 and 2. This term is sign dependent, defined by the Heaviside function ℋ(w), where w is defined by the divergence of the first layer:
To close the system we have the continuity equation
The model parameters used are shown in Table 1.
b. Discretization and boundary conditions
The equations are spatially discretized on an Arakawa C-grid and are stepped forward in time using a leapfrog scheme; a Robert–Asselin time filter is applied to prevent divergence of alternate time steps. The horizontal resolution is 25 km and the time step, Δt, is 1000 s. A flux correction transport (FCT) algorithm (Salezak 1979) is applied to solve the continuity equation [Eq. (4)].
The no-slip boundary condition is applied at the meridional boundaries. A sponge layer is applied to the open boundaries on the south and north of the domain. Therefore each open boundary is in fact a closed wall, along which the sponge layer is found wherein the desired layer thickness is relaxed to a prescribed value, h0, by the introduction of a restoring term in the continuity equation (e.g., Bryan and Holland 1989). The sponge layer is applied to Eq. (4) above as follows:
where γ is an inverse relaxation time scale and varies as a hyperbolic tangent from 0 within 2000 km of the equator through 3 × 10−6 s−1 on the meridional edges (Fig. 1b). The sponge layer parameterizes advective exchanges across the open boundaries through a local diffusive source/sink of the desired property (Haidvogel and Beckmann 1999). The sponge also damps Kelvin and gravity waves in the northern and southern boundaries, preventing them from reentering the domain.
c. The MOC experiments
To represent the northward branch of the Atlantic meridional overturning circulation, a mass flux is imposed entering the southern part of the basin and leaving the basin through the north (Fig. 1a). A new longitudinally dependent restoring factor h0(x) is thus established for the sponge layer in Eq. (5) instead of the initial constant term h0, which will allow a background flow along the western boundary. The factor h0(x) varies linearly within d = 150 km of the western boundary, according to the geostrophic equation
and is equal to the initial h0 otherwise (Figs. 1d,e). The value of γ [Eq. (5)] is then calibrated on the western boundary (Fig. 1c) to generate a T = 6 Sv flow in the interior of the basin in the linear case (see SEMOC1 experiment below). Results are not sensitive to the shape of the sponge. They are slightly sensitive to the relaxation time scale of the sponge (the shorter the time scale the stronger the flow) and to the thickness value on the western corners of the domain.
Even though the velocities are assumed to be infinitely small in the abyssal layer, the transports are not since the abyssal layer is assumed to be infinitely deep; indeed there must be a finite southward transport associated with the deep limb of the MOC, which is carried by the abyssal layer. Nevertheless, the present model is incapable of describing the instabilities of the DWBC (discussed by Dengler et al. 2004), and instead we focus exclusively on the upper limb of the MOC, which we henceforth refer to as the MOC in our model.
In the AMOC experiment, a northward flow simulating the MOC at the western boundary is switched on for 20 yr. The flow has an intensity of approximately 6 Sv, consistent with the observational values found in the upper Atlantic Ocean (as in Johns et al. 1998).
In the MOCVAR experiment, the prescribed northward flux is allowed to vary sinusoidally in both the Northern and Southern Hemispheres from 0 to 6 Sv, with a period of 8 yr.
The GROUP experiments contain three pairs of simulations. In the first member of each pair, the MOC is increased smoothly from zero to a certain value during the first 5 yr; the final MOC is then kept constant for a further 10 yr. In the second member of each pair, the MOC is increased from 0 to 6 Sv during the first 5 yr, then kept constant at 6 Sv for 10 yr, and then the MOC is dropped abruptly to a smaller value matching the first member of the pair.
In the final set called the SEMOC experiments, the MOC is initially set to 6 Sv as in the AMOC experiment. The prescribed flow is then switched off in the northern part of the basin after 10 yr of integration, simulating an abrupt change of the convection at the high latitudes of the Northern Hemisphere. The SEMOC experiments are divided into SEMOC1 and SEMOC2, whose difference is that in SEMOC1 the advective terms are removed from the momentum equation [Eq. (1)]. Therefore, SEMOC1 represents a version of the model in which the momentum equation is linearized.
a. Characteristics of eddies in the model
In the AMOC experiment, a mean flow of 6 Sv is introduced along the western boundary in layers 3 and 4. This flow is forced by the sponge layer [Eq. (5)] as described in the previous section. The model adjusts the flow hydrostatically, and the pressure gradient in Eq. (2) is then transmitted through the four layers, spreading the total flow between the layers. As the model starts its adjustment from rest, the flow increases in the western part of the basin and eddies begin to be generated in the equatorial region. Eddies are generated in the four layers and shed northward into the Northern Hemisphere (Fig. 2). The eddies have a mean diameter of 100–150 km, which is in fairly good agreement with the deformation radius in the equatorial region (100–250 km) (Chelton et al. 1998). The eddies are anticyclonic, evident from their negative relative vorticity (Figs. 2a,c), because of their Southern Hemispheric origin.
Southern Hemisphere waters carry negative potential vorticity northward across the equator, shifting the PV isolines northward along the western boundary (Figs. 2b,d). As the eddies propagate northward they start to dissipate and their PV alters to match the PV of the environment (i.e., the background planetary PV).
b. Variation of the MOC with the Reynolds number
The generation of eddies is related to the strength of the northward flow and also to the diffusion coefficient AH. A Reynolds number can be defined (Edwards and Pedlosky 1998) that represents the relative importance of the advection and diffusion of momentum:
where V and L are the meridional velocity and length scale. Here we define a separate Reynolds number in each layer through
where Sn is the meridional transport and hn is the initial thickness of each layer n. An integrated Reynolds number for all the layers together can then be calculated as
This nondimensional number measures the degree of nonlinearity of the system. Viscous instability arises from horizontal shear between the flow and the western boundary (Ierley and Young 1991), and is more likely to be generated when the Reynolds number is large. In other words, if the intensity of the northward flow varies in the model, a critical Reynolds (Rec) number can be found above which eddies are generated.
Figure 3 shows the Reynolds number (Ren) calculated for each model layer separately over the equator for the MOCVAR experiment (e.g., Table 2); Ren is calculated according to Eq. (8), with AH and hn obtained from Table 1. After reaching a critical Reynolds number (Rec), the high-frequency variability shown in Fig. 3 develops in the western boundary (Fig. 2) as the flow becomes turbulent at that latitude. For layers 1, 2, and 4, Rec ranges from 35 to 40, and for layer 3, Rec is approximately 50. According to Fig. 3, even when the MOC starts decreasing and Ren falls below Rec, eddies are still present in the western part of the basin as a legacy of the earlier instability. This suggests that under decadal or higher-frequency variability of the MOC, eddies may continue to transport layer thickness and heat northward even when the MOC is subcritical; indeed in the simulation shown in Fig. 3, the eddies vanish only when the prescribed MOC returns to zero.
This behavior can be further illustrated by showing the variation of the eddy kinetic energy with Re while in MOCVAR the strength of the MOC increases, passing a supercritical state, and then returns to a state of rest. The EKEn is calculated by partitioning the velocity field into mean (U, V) and eddy (u′, v′) components for each layer, or un = Un + u′n and vn = Vn + v′n. The mean velocity is calculated by averaging the velocities over the Northern Hemisphere, and then applying a 1-month temporal filter to avoid high-frequency oscillations. From the eddy velocities, the EKE can be then calculated using
For each layer there is a critical value of Ren = Rec in which the EKE starts to differ from 0 (Fig. 4). This bifurcation point is located, consistent with Fig. 3, at about Rec = 35–50. Considering that a steady flow is never reached in the MOCVAR experiment, there may be errors associated with the inertia of the system, which may push Rec to slightly higher values, and also errors associated with the calculation of the mean flow that may account for the differences between Rec for the layers. Nonetheless, for the latter, it can be noted that the values of Rec for each layer are broadly consistent with each other. A sensitivity test was produced for different horizontal viscosity coefficients, one higher (800 m2 s−1) and one lower (100 m2 s−1) than the original one: the results were not sensitive to these changes in viscosity, and the values for Rec remained the same (not shown).
c. Hysteresis behavior?
The MOCVAR experiments are forced by an MOC varying on interannual–decadal time scales. It is not clear whether the different eddying behaviors in the opposite phases of the cycle (MOC increasing and MOC decreasing) are indicative of hysteresis behavior, or are simply a legacy of Rec having been exceeded in the earlier part of the cycle. To resolve this issue, the integrated Reynolds number Re for the GROUP experiments is presented in Fig. 5. These experiments are divided into three pairs of simulations shown in Figs. 5a, 5c, and 5e.
The black curves in Figs. 5a, 5c, and 5e show a strong turbulent pattern as soon as the MOC reaches 6 Sv at around day 1500, at which point Re exceeds Rec, or about 40, as in the MOCVAR experiment. The prescribed decrease in the MOC at the boundaries after 15 yr lowers the EKE of the system, as seen by the decreased amplitudes of oscillation. However, in all experiments, after 15 yr, the black curves still present considerable eddy energy remaining from their previous states.
In comparison to the black curves, the gray ones take longer to develop eddies, but eddying still occurs when Re exceeds Rec by about 30 in this case (Figs. 5a,c); however, in the third case in which Re peaks at 25 (Fig. 5e), Re remains subcritical and eddies are not generated. In all cases, the black curves present a higher Re variability (Figs. 5b,d,f), which represents a higher EKE. A bifurcation is seen from the spread of the probability distribution functions (pdf) of the inverse chi-squared distribution of the standard deviations (e.g., see Lee 2004, section 2.12) for each pair of runs, but no evidence of hysteresis behavior in these integrations is seen. Rather, we see a persistence of the system to remain in a turbulent state after returning to a subcritical state.
d. Relationship between the eddy generation and the shutdown of the MOC
The SEMOC experiments represent the response to an abrupt collapse of the MOC from an initial state consistent with the AMOC experiment. In its initial condition the MOC starts at 6 Sv and then the MOC at the northern edge of the domain is abruptly terminated.
Immediately after the MOC shutdown, Kelvin waves are triggered along the western boundary in the Northern Hemisphere. The anomalies take about 25 days to arrive at the equatorial region, traveling at an average speed of about 1 m s−1, in good agreement with the theoretical group velocity of baroclinic Kelvin waves in a viscous fluid (e.g., see Fig. 12c in Hsieh et al. 1983). In turn, equatorial Kelvin waves are triggered and take about 25 days to cross the basin onto the eastern side. Figure 6 shows zonal velocity anomalies propagating along the equator in layers 2 and 3 after the MOC shutdown (SEMOC1). Thus, 50 days after imposition of the transport anomaly at the northern margin of the domain, Rossby waves are formed and propagate westward.
The evolution of the mass transport at 4°N (black line) and 4°S (gray line) for the SEMOC1 and SEMOC2 experiments is shown in Figs. 7a and 7b, respectively. For the first 10 yr of integration an MOC of 6 Sv is prescribed at the southern and northern boundaries. Initially there is a disturbance in the model due to an adjustment to the imposition of the 6-Sv MOC. After that, SEMOC1 (Fig. 7a) has a well-behaved linear state. This does not occur in SEMOC2 (Fig. 7b). In this case, the Northern Hemisphere MOC adds on average 1 Sv to its transport because of the nonlinear advection terms. The eddy contribution is also responsible for an MOC standard deviation of σ = ±1.4 Sv around its mean value.
After 10 yr of integration (t = 3650 days), the MOC is shut down at the northern margin of the basin. In SEMOC1 (Fig. 7a), the MOC shows a gradual decrease in the Southern Hemisphere, which equilibrates in about 12 yr. In the Northern Hemisphere, the MOC quickly drops to zero after a few days. After this initial fast adjustment, the MOC in the Northern Hemisphere gains strength and equilibrates at the same rate as in the Southern Hemisphere. This latter feature is due to the nature of the forcing, which forces the MOC to equilibrate ultimately to roughly half its initial value (∼3 Sv). Because of the imbalance of the MOC into the northern and out of the southern sponges, respectively, the layer thicknesses increase until equilibrium is restored. At the end of the simulation 8%–10% is added to the total volume of the moving layers. This feature does not invalidate the results, since we can still compare both SEMOC1 and SEMOC2 runs, which pile up buoyant fluid in the same way.
As depicted in Fig. 7, the decay of the flow in SEMOC2 after the collapse presents a different pattern than the one in SEMOC1. There is still a strong influence of eddies generated along the western boundary in the regions near the equator even after the interruption of the MOC north of the equator. This is evident in the oscillatory behavior of the flow seen in Fig. 7b.
The thicknesses of layer 2 in both experiments in Fig. 8 show a baroclinic Rossby adjustment pattern, with the long Rossby waves propagating westward, most rapidly at low latitudes. This causes the zonal gradient of the layer thickness to increase poleward. Comparing the panels of Fig. 8, it is seen that the inclusion of nonlinearities in the model strengthens the meridional pressure gradient off the equatorial region, in accordance with the increased mean transport in Fig. 7, since the eddy energy can be converted into mean energy for the flow.
4. Concluding remarks
In this study the behavior of a cross-equatorial flow, representing the upper limb of the MOC in the tropical Atlantic, has been investigated in an idealized model under different MOC change scenarios. Particular attention has been paid to the mesoscale contribution to the MOC variability. In reality, forcing occurs on all time scales. Thus, the results here indicate that the high-frequency forcing of the MOC could be a significant source of variability in the tropical Atlantic in numerical ocean models, in which the Reynolds number often lies close to the regimes described here.
The MOC is imposed as a 6-Sv northward flow along the western boundary. Eddies are generated in the equatorial region and flow toward the northern side of the domain. In an MOC shutdown experiment, the Northern Hemisphere responds quickly (approximately one month) to the anomalous changes in transport as the pressure gradient is quickly destabilized. At the end of the run the pressure gradient restabilizes partially in the Northern Hemisphere with half of its strength. This is due to the remaining mass source in the Southern Hemisphere, which responds in a much smoother way to those anomalies, consistent with the results of Johnson and Marshall (2002a). In comparison to a linear run, the nonlinear run is characterized by a higher meridional pressure gradient, representing a contribution from the eddy field to the mean northward flow. Eddy energy is responsible for an enhancement of the northward flow and accounts for a variability greater than 2 Sv.
In a quasi-steady state, eddies are produced when the Reynolds number exceeds a critical value Rec of about 30 in our integrations, in accordance with Edwards and Pedlosky (1998). However, when the MOC varies on decadal or shorter time scales, eddies can persist even when the critical Reynolds number is not exceeded, as a legacy of an earlier instability.
The present work does not account for other forcings that are important for the MOC variability on seasonal to interannual time scales. Large variability in the tropical Atlantic Ocean can be generated by wind stress variability, heat, and freshwater forcing, which are induced primarily through atmospheric feedbacks (Haarsma et al. 2008). In the ocean, Ekman transports, wave and eddy propagation, and density-related mechanisms are the large sources of variability from seasonal to interannual time scales, which can add a variability of 8 Sv in the overturning. In long-term changes, compensation may take place decreasing the importance of the high-frequency variability (Kanzow et al. 2007).
This work was sponsored by FAPESP Process 01315-0, CAPES Process 2931/03-6, and CNPq. Ilana Wainer thanks the INCT-Criosfera/CNPq-MCT 573720/2008-8. David P. Marshall is supported by the U.K. Natural Environment Research Council (NERC). The authors want to thank Dr. Rein Haarsma for valuable discussions and the anonymous reviewers who greatly improved the manuscript.
Corresponding author address: Marlos Goes, Department of Geosciences, The Pennsylvania State University, 411 Deike Building, University Park, PA 16802. Email: email@example.com