1. Introduction
The role of the meridional overturning circulation (MOC) in transporting heat toward the pole in the Atlantic makes it a key player in earth’s climate system (MacDonald and Wunsch 1996; Ganachaud and Wunsch 2000; Trenberth and Caron 2001). It contributes to western Europe’s comparatively mild climate and the collapse of the MOC in the North Atlantic is often proposed as a possible mechanism to account for the rapid onset of ice cover during ice ages (Rahmstorf 2002). However, how local and global processes interact to sustain the MOC’s strength remains unclear.
For a complex numerical model, any extensive sensitivity analysis using traditional perturbation methods is computationally unfeasible, because of the number of parameters involved. Adjoint methods provide a more efficient approach to calculating the impact of changes in all model inputs to a single model output, albeit limited to the context of small perturbations. A system of adjoint equations, developed from a differentiated form of the original model equations, allows the calculation of the linear sensitivity, about the fully nonlinear model trajectory, of the model response or cost function to all model “inputs” (initial conditions, surface boundary fields, equation coefficients, and parameters) in one calculation. In contrast, traditional approaches give the sensitivity of all the model outputs to a single model input in one calculation. The adjoints of numerical ocean models have been shown to give accurate sensitivities for periods lasting a few years. These studies focused on the sensitivity of the ENSO signal (van Oldenborgh et al. 1999; Galanti and Tziperman 2003) or the Atlantic heat transport (Marotzke et al. 1999) to forcing fields, or used the adjoint in data assimilation projects (Stammer et al. 2002, 2003). This article shows that adjoint solutions can be derived and remain reliable over time scales of several hundred of years in coarse-resolution ocean models.
This article uses the adjoint approach to shed a different light on two issues that have been the focus of prior research.
How does the sensitivity of the MOC vary when the model includes wind stress? If wind in the Antarctic Circumpolar Channel draws up the deep water formed in the North Atlantic (Toggweiler and Samuels 1995, 1998), the Drake Passage hypothesis, what is the overturning sensitive to in the absence of wind? Does this sensitivity pattern change significantly when wind is added? Note that a number of authors (e.g., Rahmstorf and England 1997; Tsujino and Suginohara 1999; Klinger et al. 2004) have questioned the robustness of the Drake Passage hypothesis, in particular whether an open channel is required to generate the effect. Wunsch (2002) suggests that mixing generated by winds and tides, represented through diapycnal mixing in this coarse-resolution model, is the rate-limiting factor in setting the intensity of the MOC. How does the sensitivity of the MOC to diapycnal mixing vary in the presence or absence of wind? What is the geographical distribution of the sensitivity of the MOC to diapycnal mixing (Scott and Marotzke 2002; Samelson 1998; Huang 1999)?
How does the MOC’s sensitivity change with the formulation of surface boundary conditions? A number of studies have shown that the type of surface boundary condition can allow or prevent a number of feedback mechanisms, which reinforce or weaken disturbances in the MOC strength (Nakamura et al. 1994; Marotzke and Stone 1995; Rahmstorf and Willebrand 1995; Tziperman and Gildor 2002).
The prognostic model and its adjoint are described in sections 2 and 3. The first simulations, discussed in section 4, neglect wind stress forcing and include only forcing of the surface hydrographic fields (temperature and salinity). The changes induced when wind stress is included are analyzed in the second set of calculations, presented in section 5. The role played by the formulation of the surface hydrography forcing, from restoring the sea surface temperature and salinity toward observations to imposing fixed heat and freshwater fluxes, is examined third, in section 6.
2. Description of the forward model
The ocean model used in this project is the Massachusetts Institute of Technology (MIT) general circulation model (GCM); it is described in detail in Marshall et al. (1997a, b). For this study, the model was configured to operate in hydrostatic form on a single rectangular basin 64° wide in the zonal direction, which extends from 66°S to 70°N. A continuous channel was added in southern latitudes to represent the Antarctic Circumpolar Current, which is 80° wide and extends from 66° to 46°S. The resolution of the configuration is 4° × 4° with 15 layers in the vertical having thicknesses ranging from 50 m near the surface to 690 m at the bottom. The depth of the basin is uniformly 4200 m, except between the “continent” and “Antarctica” where a 2500-m-deep sill is added. The time-stepping procedure is asynchronous (Bryan 1984). The Redi isopycnal diffusion scheme is used in all model runs and the Gent–McWilliams (GM) scheme parameterizes the advective effect of the geostrophic eddies with a “bolus velocity,” which is added to the Eulerian-mean velocity (Danabasoglu and McWilliams 1995). Table 1 summarizes the value of key model parameters. Idealizing the domain geometry and restricting its size, allows an extensive exploration of the sensitivity of the circulation to various surface forcing configurations.
Forcing fields are derived from the zonal, annual average of the wind stress, heat flux, and net precipitation fields over the Atlantic basin (Schmitt et al. 1989; Trenberth and Solomon 1994; Trenberth et al. 1989; Jiang et al. 1999). The impact of an annual cycle in the forcing fields on the sensitivity patterns was left to a later analysis, although it would likely have the greatest impact on high-latitude sensitivities. The heat flux and freshwater forcing fields are adjusted to ensure zero net input of energy or mass into the system. The net precipitation field [evaporation − precipitation (E − P)] is imposed as a virtual salt flux. A restoring toward zonally averaged temperature and salinity climatologies is used in some experiments (Levitus and Boyer 1994; Levitus et al. 1994).
3. The adjoint model
The output of an adjoint model calculation gives the sensitivity of the cost function to the initial conditions as well as to the model’s physical parameters. The cost function can be any scalar function of the model output, as long as it remains differentiable.
The reader is referred to Morse and Feshbach (1953), Lanczos (1961), and Hall et al. (1982) for a derivation of the general principles of adjoint construction, both in the continuous and discrete time domains.
Equation (2) illustrates the fact that the adjoint model is integrated backward in time: ∇yN R is calculated first before being multiplied with 𝗖*N, 𝗖*N−1 back to 𝗖*1.
The derivation of the adjoint of the MIT general circulation model is performed automatically by the Tangent Linear and Adjoint Model Compiler (TAMC). A detailed description of the construction of the adjoint code as well as an example of its implementation can be found in Marotzke et al. (1999).
The adjoint model provides the sensitivity of a diagnostic to all model parameters and boundary conditions in a single integration. While this is a very efficient way of providing a complete picture of climate sensitivity, this approach still suffers from a number of limitations.
The first restriction is purely computational. It is not practical, at this point, to integrate the adjoint model, even in a coarse resolution, for more than a few hundred years. The sensitivity fields seem to have equilibrated by years 200–300. This does not perfectly guarantee that the ocean’s slowest processes, such as slow upwelling in the Pacific, would not have an impact on the meridional overturning on longer time scales (Danabasoglu et al. 1996). The integrations presented in this paper have been carried out for 400 yr, with the assumption that the dominant signals have become visible by that point.
The second restriction relates to uncertainties in the surface boundary conditions, such as wind stress, and in the model parameters, such as diapycnal mixing. A quantitative comparison of the effect of perturbations in the parameters on the overturning’s intensity requires the multiplication of the sensitivities obtained with the adjoint model with a perturbation in the field or parameter [e.g. ΔψMAX = (∂ψMAX/∂τx)Δτx]. While the uncertainty and the natural variability in fields such as wind stress may be quantifiable, other fields are much less precisely known.
This leads to the third caveat. While the underlying model trajectory may be nonlinear, the adjoint method does require the perturbation growth to be linear. This, theoretically, restricts the validity of the analysis to small perturbations in the underlying fields. The validity of the linearity assumption may also be location dependent (e.g., a 10% perturbation in the heat flux that leads to a linear model response in the Tropics can change the convection pattern in the Labrador Sea, with strongly nonlinear consequences). This also limits the accuracy of the adjoint model to coarse resolution, non-eddy-resolving ocean models (Lea et al. 2000). The accuracy of the adjoint solutions was validated by perturbing the forward model with a Δs = 1% perturbation to a single parameter s at a single grid point (or water column in the case of diapycnal mixing) and comparing the resulting streamfunction after 400 yr to the reference run {[ψMAX(s + Δs) − ψMAX(s)]/Δs}. The difference between adjoint and finite difference sensitivities is on average 2.6% and never worse than 8% in the 22 random cases tested. All adjoint fields were validated with this approach.
a. Cost function
4. Adjoint sensitivities under pure buoyancy forcing
The first set of simulations relies on buoyancy and diapycnal mixing alone to drive the overturning, the wind stress was set to zero. Both the sea surface temperature and the sea surface salinity were relaxed toward the Atlantic basin mean climatological values (Levitus and Boyer 1994; Levitus et al. 1994) on a 60-day time scale.
a. Circulation
Because of the absence of wind, the strong surface western boundary currents extend to the northern boundary in the Northern Hemisphere and to the northern limit of the circumpolar channel in the Southern Hemisphere. The flow then converges to two primary areas of downwelling: the northeast corner and the southernmost point along the eastern boundary. While the former point is the source of the abyssal waters of the basin, the latter is the source of a shallow overturning cell, which returns toward the western boundary no deeper than 500–700 m. The 3 Sv (1 Sv ≡ 106 m3 s−1) of Northern Hemisphere Deep Water that penetrate into the Southern Hemisphere are forced to upwell underneath this shallow cell and return to the surface north of the equator (Fig. 1, bottom).
The maximum intensity of the meridional streamfunction is 10.5 Sv, close to what was obtained when the wind stress is turned on (11.5 Sv).>
b. Diapycnal mixing
Figure 2 shows the two-dimensional pattern of sensitivity to diapycnal mixing, it represents the response of a perturbation applied throughout each individual water column. Three regions play an important role: the eastern and western boundaries and the Tropics.
1) Tropical dynamics
The pattern in Fig. 2 clearly shows that the sensitivity of the overturning to mixing increases from the poles toward the Tropics and peaks at 6°–10°N. The only region of negative sensitivity is found north of 60°N.
The vertical temperature gradient is everywhere positive and the diapycnal mixing coefficient is constant in the vertical direction. This leaves only two possible scenarios: the downward mixing of heat (Tzz > 0) must be balanced by upward motion (wTz > 0) or the upward mixing of heat (Tzz < 0) by downward motion (wTz < 0). One can a priori expect the sensitivity to diapycnal mixing to be positive where it induces upward motion and negative where it induces downward motion, except perhaps in the latitudes where the cost function is diagnosed. Scaling the advection–diffusion equation yields an upwelling velocity proportional to the diapycnal mixing and inversely proportional to the thermocline depth δ, [W ∝ (κd/δ)]. Diapycnal mixing is most efficient in the Tropics because of the large stratification in that region and the large amount of heat, which diffuses into the thermocline.
Without wind forcing to force a ventilated thermocline, upwelling takes place in the entire tropical region, with only a narrow region of downwelling near the surface driven by mass convergence along the equator. This allows each point in a broad region from 10°S to 20°N to have an almost equally large impact on the overturning. The maximum sensitivity is found in a band at 10°N where the relaxation temperatures, and hence the surface temperatures, are the warmest.
The depth dependency of the sensitivity pattern, shown for a sample column at 14°N in the middle of the basin (right-hand panel in Fig. 3), indicates that the overturning will respond most vigorously to perturbations applied between 200- and 500-m depth. An analysis of the vertical heat budget, the left-hand panel of the same figure, confirms that the maximum diffusive heat flux is taking place at the depths where the maximum sensitivities are observed. This is the depth of transition between the relatively uniform near-surface temperature gradients and the very small abyssal gradients.
2) Eastern boundary dynamics
The source of the sensitivity to diapycnal mixing along the eastern boundary (shown in the right panel in Fig. 4) follows the boundary between the downwelling region and the upwelling, which lies beneath it (the left-hand panel shows the vertical velocity). Vertical diffusion peaks just below the depth of the zero crossing of the vertical velocity, it balances part of the upward advection below this region. This process is most efficient between 45° and 60°N where large vertical velocities are observed. A similar process can be observed along the downwelling region in the Southern Hemisphere.
The role played by the eastern boundary in determining the intensity of the meridional overturning was outlined by Scott and Marotzke (2002): downward diffusion of heat along the eastern wall increases the zonal temperature gradient, which in thermal wind balance equilibrates with a northward flow at the surface and a deep southward return flow. The results obtained with the adjoint model suggest that the role of mixing in setting the east–west temperature contrast is limited to the region located at the base of the thermocline. The strength of the vertical shear in the north–south flow may also be determined by the quasi-adiabatic warming of the eastern boundary associated with downwelling.
3) Western boundary dynamics
The contrast between the warm waters, which are advected northward by the western boundary current, and the upwelling of cold water from the deep western boundary current leads to large vertical temperature and density gradients. Note that the heat budget along the western boundary is by no means in vertical advective–diffusive balance, it is largely a balance between northward and vertical heat advection: υTy ≈ −wTz, with the diabatic terms playing a smaller, yet not insignificant role (not shown).
The sensitivity of the overturning to diapycnal mixing along the western boundary (right-hand panel in Fig. 5) is significant down to the base of the thermocline (500 m). A localized increase in diapycnal mixing induces upward motion and cooling throughout the depth of the water column below the level of the perturbation. A substantial fraction of this upwelling is compensated by increased downwelling along the eastern boundary. Much of the associated mass transport is in the zonal direction, but a small fraction projects onto the meridional plane and contributes to enhancing the overturning. This meridional transport is also a necessary consequence of the warming associated with enhanced downwelling along the eastern boundary and cooling associated with upwelling along the western boundary. The increase in the zonal temperature gradient must, namely, be compensated for by an increase in the meridional shear of the flow.
c. Relaxation temperature and salinity
The sensitivity to the relaxation temperatures and salinities in Fig. 6 have a maximum over the densest spot in the model, the northwest corner. This coincides with the site of “convection.” While the model has a convection scheme, the GM isopycnal diffusion replaces convection in the regions of steeply sloping isopycnals during the model’s spinup. We will refer to this process as convection since it achieves a similar homogenization of the water column while offering the advantage of remaining differentiable for the adjoint code. Note that this effect was diagnosed while trying to understand how the adjoint solutions could remain reliable for such a long integration. A model with a seasonal cycle would, in all likelihood, have episodic convection, which may limit the validity of the adjoint solutions.
The sign of the sensitivity pattern is such that an increase in the surface density increases the meridional overturning. Because of the relaxation time scale, which effectively erases the effect of surface temperature and salinity perturbations after a year at most, the sensitivity pattern is limited to high latitudes where perturbations can have a direct impact on the buoyancy of the basin’s waters on very short time scales. The northwest corner controls the temperature of the abyssal waters, −0.5°C in this basin. More specifically, by controlling the temperature of the deep western boundary current (see, e.g., Blanke et al. 2002 for a description of Lagrangian trajectories of water parcels in the “cold limb” of the North Atlantic overturning), the convecting corner can have an impact on the zonal temperature gradient below the thermocline and on the strength of the meridional overturning. The effect of salinity is smaller, but also works through its effect on the density of the deep water.
There is no reversal in the sign of the sensitivity of the meridional overturning to the restoring temperature and salinity between high latitudes and the Tropics. This result is surprising, a number or studies having pointed out that the overturning’s strength scales as a positive power of the north–south temperature gradient (Zhang et al. 1999; Huang 1999). The results presented here indicate that the overturning would only increase if high-latitude temperatures cooled; increasing low-latitude temperatures would have virtually no impact. For constant tropical temperatures, the overturning scales as a function of the temperature in the northern part of the basin.
5. The role of wind forcing
This set of simulations adds a zonally averaged wind to the forcing fields (Trenberth et al. 1989). The changes in the sensitivity patterns are analyzed.
a. Circulation
The meridional overturning peaks in high latitudes at a value of 11.5 Sv (Fig. 1, top) and the heat transported northward in the basin amounts to 0.7 PW, nearly all transported by the MOC. As is often the case in similar models (Colin De Verdière 1988; Marotzke and Scott 1999), the actual downward mass transport signature of the MOC is taking place along the eastern wall from 40° to 68°N, with peak downwelling velocity in the northeast corner. Convection takes place in the northwest corner below the surface’s coldest spot.
The presence of gyres and the separation of the western boundary current from the coast at 50°N are a consequence of wind forcing. There is a strong eastward westerly current in the circumpolar channel and 2–3 Sv of deep water upwell in the open channel. The thermocline structure is now the product of a wind-driven ventilated thermocline as well as a diffusive internal thermocline (Samelson and Vallis 1997). The Northern Hemisphere overturning remains quite similar in intensity and structure to the no wind case, convection takes place in the northwest corner and downwelling in the northeast. There is no overturning cell in the Southern Hemisphere.
b. Diapycnal mixing
The two most notable differences to the no-wind case are in the eastern equatorial region and along the western boundary. The patterns are otherwise quite similar, notably the dynamics along the eastern boundary.
1) Equatorial dynamics
The sensitivity maximum to diapycnal mixing (Fig. 7) and zonal wind stress (Fig. 8) are both located in the eastern equatorial region. The easterlies shoal the thermocline on the eastern side of the basin and deepen it on the western side (Pond and Pickard 1983). The divergence of the Ekman transport at the equator and its associated upwelling further reduce the depth of the thermocline in that region and increase the efficiency of the downward diffusion of heat: the κd · Tzz term in the equatorial heat budget is close to being one order of magnitude larger than it was in the no-wind simulation (Fig. 3). The maximum sensitivity is confined to the near surface (not shown), confirming that it is related to the shallow Ekman suction process. The overturning is less sensitive to mixing in the regions of Ekman pumping (north of 15°N and south of 5°S) than in the no-wind case.
Wind also determines much of the thermocline’s stratification (Samelson and Vallis 1997; Vallis 2000). The region sandwiched between the upper ventilated thermocline and the lower internal thermocline at a depth of approximately 500 m exhibits very large curvatures in the temperature profile. It corresponds to the location of the secondary maximum in sensitivity to diapycnal mixing.
2) Western boundary dynamics
The sensitivity at the western boundary is greatly enhanced and dominates when wind forcing is added to the model.
A large fraction of the 11 Sv of bottom water formed in the northern part of the basin upwells in the Northern Hemisphere; only about 5 Sv cross the equator (Fig. 1). The circulation at the western wall shows that intense upwelling is taking place between 20° and 60°N, thereby strengthening the western boundary current through vertical mass convergence as well as through horizontal recirculations (Colin De Verdière 1988). It is therefore not a surprise to see large sensitivities in that region, they are associated with sharp temperature gradients and an increase in the effectiveness of diapycnal mixing. It is, however unclear how much of this effect is an artifact of the mixing scheme. Böning et al. (1995) discussed the “excessive” upwelling often observed along the western boundary in ocean models, and the effective shortcut, which this upwelling provides for the meridional overturning. Much of this spurious upwelling was attributed to the use of x–y–z mixing schemes, and the introduction of the GM scheme (Gent and McWilliams 1990), which is used in all of the present simulations, appears to have resolved much of this issue. Huck et al. (1999) analyzed the influence of the parameterization of lateral boundary layers on the thermohaline circulation. They concluded that more sophisticated frictional closures further reduced the upwelling along the western boundary by allowing horizontal recirculations of geostrophic currents impinging along the coasts.
c. Wind stress
With the exception of the eastern equatorial region and Ekman upwelling, the sensitivity pattern to wind stress is concentrated in high latitudes (Fig. 8), in both hemispheres. A significant fraction of the deep water upwells in the latitudes of the southern channel (see Fig. 1). The Deacon cell is the product of wind stress divergence in that region. The wind stress peaks around 50°S (Fig. 14), upwelling therefore takes place poleward and downwelling equatorward of that latitude, this cell closes below the level of the sill, here set at a depth of 2500 m (Toggweiler and Samuels 1995; Gnanadesikan 1999). Although much of the Deacon cell is canceled by the eddy-induced bolus velocity of the GM scheme, the remaining circulation is sufficient to upwell between a third and a quarter of the deep water formed in the Northern Hemisphere, notably because it can draw up the water still lying below 2500 m. Peak sensitivities coincide with peak upward velocities along the sill and along the northern edge of the channel. This confirms Klinger et al. (2003, 2004), who showed that the meridional overturning is more sensitive to wind perturbations in the southern channel than elsewhere. Sensitivities to the sea surface temperature and salinity fields (Fig. 9) are analyzed jointly with other types of surface forcing in section 6.
6. The role of heat and freshwater forcing
This section seeks to illustrate and explain the differences in the sensitivity maps, which are obtained by imposing the same effective surface forcing, but formulating that forcing in a variety of ways, summarized in Table 2. All simulations were performed with the same wind stress forcing.
Under “mixed” boundary conditions, the net water flux at the ocean–atmosphere [P − E] interface is derived from a zonally averaged climatology. The heat flux boundary condition contains both the observed heat flux as well as a term relaxing temperatures toward climatology (Jiang et al. 1999).
The heat flux field under flux boundary conditions (Qdiag) was diagnosed from the effective exchange of heat taking place between the atmosphere and the ocean at the end of the spinup of the model under mixed boundary conditions (5000 yr), it is therefore different from the zonally averaged field used under mixed boundary conditions (Qobs). Because the effective heat flux forcing at the ocean’s surface remains constant, the flow differs very little from what is observed under mixed boundary conditions.
As the constraints are relaxed, the ocean goes from having quasi-fixed surface temperatures and salinities under restoring boundary conditions, to allowing first the salinity (mixed boundary conditions) then the temperature (flux boundary conditions) to evolve forced by precipitation and heat flux fields. The only quantities held fixed under flux boundary conditions are the amount of heat and freshwater entering the basin.
The mixed boundary conditions are thought to be the most realistic of the three formulations: the sea surface salinity has no influence on the atmospheric hydrological cycle but the difference between the ocean and the atmosphere’s temperatures does, however, influence the heat exchange at the ocean’s surface. By providing the two extreme formulations, the idea was to determine which sensitivity patterns depended on the nature of the surface forcing. Note that restoring boundary conditions are often used in ocean-only model studies.
a. Diapycnal mixing
The sensitivities to diapycnal mixing observed under mixed or flux boundary conditions are plotted in Fig. 10. Sensitivities are greater by close to an order of magnitude under flux boundary conditions when compared to relaxation boundary conditions, yet the three cases all have the largest sensitivities in the Tropics.
1) Tropical dynamics
Near-surface tropical sensitivities are, in all cases, concentrated in the 0°–10°N latitude band where Ekman suction is taking place. This confirms the hypothesis outlined by Samelson and Vallis (1997) that the MOC is effectively insulated from buoyancy forcing by the ventilated thermocline in the region of the Ekman pumping.
The local maximum remains located at the eastern boundary because of wind-induced coastal upwelling. It is however, twice as large under flux or mixed boundary conditions as in the restoring case. This is associated with the lack of constraint on the sea surface temperature and salinity under flux boundary conditions. Considering an extreme scenario in which the surface density is held fixed, which is nearly the case under restoring boundary conditions, an increase in diapycnal mixing can only reduce the vertical density gradients, thereby reducing the efficiency of the vertical advective–diffusive process. Under mixed boundary conditions, only the surface temperature is constrained to remain close to climatology. Under flux boundary conditions, the surface density is no longer fixed. This allows the density to decrease above the level of a localized increase in diffusivity, while the buoyancy gradients in the rest of the water column remain unchanged. This maintains the efficiency of vertical diffusion. To test this hypothesis, the diapycnal mixing coefficient was increased by 10% (Δκd = 3 × 10−6 m2 s−1) just below the surface at 2°N along the eastern boundary. Table 3 summarizes the resulting surface temperature and salinity perturbations. The surface buoyancy perturbation is smallest under restoring boundary conditions. Salinity contributes approximately two-thirds of the overall buoyancy perturbation under mixed boundary conditions, but less than a third under flux boundary conditions.
2) Western boundary dynamics
The maximum sensitivity at 40°N and 200 m is 4–5 times larger under flux boundary conditions. We hypothesize that the advection of surface temperature perturbations toward the site of convection explains this pattern.
Figure 11 shows the surface temperature response, under mixed (left) and flux (right) boundary conditions, to a near-surface perturbation in κd located at 38°N at the western wall (the location is marked with a box in Fig. 11). The perturbation is advected by the western boundary current and farther northward into the polar basin. This process is, however, inhibited under mixed boundary conditions when compared to flux boundary conditions. The key measure, which determines the magnitude of the response of the meridional streamfunction appears to be the magnitude of the temperature perturbation in the northwest corner, above the convection site. This allows the cooling to extend through the water column and forces enhanced clockwise recirculation along the northern boundary, with downwelling in the northeast corner. Although surface salinity perturbations are also observed, the effect of temperature perturbations is clearly dominant (Table 4), which explains why the sensitivities in the western boundary current are so much larger under flux boundary conditions.
3) Midlatitude dynamics
Figure 12 shows the effect on the surface temperature of a perturbation in κd imposed at 42°N, 8°E of the western boundary (the location is shown with a box in Fig. 10) at a depth of 500 m, under both mixed and flux boundary conditions. Under mixed boundary conditions the overturning’s intensity decreases; it increases under flux boundary conditions. This was predicted in both cases by the adjoint model: (∂ψMAX/∂κd) > 0 under mixed boundary conditions, (∂ψMAX/∂κd) < 0 under flux boundary conditions. In the first case, the response is largely damped before the perturbation reaches the surface. Under flux boundary conditions, the perturbation extends to the surface and is advected to the northeast corner, where it affects downwelling in high latitudes. The northwest corner is in fact warmed under flux boundary conditions, while it experiences a slight cooling under mixed boundary conditions.
b. Wind stress
While the sensitivity to the wind forcing is concentrated in high latitudes under relaxation boundary conditions (Fig. 8), the Tropics dominate under mixed or flux boundary conditions (Fig. 13). Overall sensitivities are one order of magnitude larger under mixed or flux boundary conditions.
Given the role played by Ekman pumping already highlighted within the context of the equatorial sensitivity to diapycnal mixing, it is not surprising to see that the latitudes at which the sensitivity bands change sign correspond to the zero crossings of the wind stress curl (shown for reference on the right in Fig. 14). The polarity of the pattern is such that increasing the upwelling increases the overturning while enhancing the downwelling has the opposite effect. The effect decays toward the pole as the influence of shallow upwelling or downwelling on the structure of the thermocline decreases.
The 40°–50°N latitude band is important under flux boundary conditions. This feature is associated with the advection of buoyancy perturbations and is discussed in the next section.
c. Precipitation and heat flux
The pattern of sensitivity to the observed heat flux under mixed boundary conditions (upper-left-hand panel in Fig. 15) does not differ significantly from the pattern of sensitivity to the restoring sea surface temperature calculated under relaxation boundary conditions (left-hand panel in Fig. 9). The restoring time scale in either case is sufficiently short to prevent any perturbation distant from the site of convection from being effective.
Because the effects of increased rainfall or evaporation can be advected by the surface circulation under mixed boundary conditions instead of being dissipated, the sensitivity to the net water flux field shows some interesting features (Fig. 15, lower-left-hand panel). An increase in precipitation anywhere in the basin would decrease the strength of the overturning. With the exception of the northwest corner, increases in the moisture flux in the Tropics have the greatest impact. The importance of the northwest corner hints to an important role for the advection of salinity perturbations toward the site of convection. Because that corner sets the buoyancy of the bottom water, or at the very least that of the deep western boundary current, it determines in part the zonal density gradient below the thermocline.
The analysis of Zhang et al. (1999), which reproduces the hysteresis behavior of the meridional overturning under increasing freshwater forcing observed by Rahmstorf (1995), used mixed boundary conditions. Their results show a decrease in the overturning strength for increasing freshwater forcing, which in their model corresponds to increased precipitation in high latitudes and increased evaporation in the Tropics, the two effects are not isolated from one another. The adjoint result indicates that only the former effect weakens the overturning, the latter intensifies it. Under mixed boundary conditions, the dominant mechanism is the increase in evaporation in the Tropics. The adjoint result agrees with Wang et al. (1999a, b) and Wiebe and Weaver (1999), who noted that the equilibrated response to a climate change type of perturbation with increased evaporation in the Tropics was in fact a net increase in the overturning’s strength. The weakening of the circulation is a transient phenomenon, which does not last beyond the point when the forcing ceases to increase. Note that the results obtained with the adjoint model are a linearization around a steady state, as such they can only be expected to be valid over a very small portion of the hysteresis curve shown in either Zhang et al. (1999) or Rahmstorf (1995).
Under flux boundary conditions, on the right in Fig. 15, the role played by the buoyancy forcing (both heat and moisture) is no longer constrained to the polar region. Two separate mechanisms determine these sensitivity patterns: the direct interaction of the heat and moisture fluxes with downwelling at the eastern boundary, and the influence of heat and precipitation on the density of abyssal waters set in the northwest corner. The elongated band of sensitivity at 40°N is aligned with the northern edge of the subtropical gyre and the strong eastward current in that region. A positive perturbation in the heat flux in that band will be advected toward the boundary, where it downwells quasi adiabatically. A warming or enhanced precipitation along the eastern boundary will result in an increase of the overturning by intensifying zonal density gradients and the vertical shear in the flow.
With the exception of the eastern boundary, the sign of the sensitivity is everywhere such that an increased loss of heat by the ocean or a buoyancy loss through evaporation would enhance the overturning. This pattern is set by advection, which is the most rapid and efficient way for buoyancy perturbations to affect the overturning. The region of greatest sensitivity is bounded by the region of the subtropical gyre that feeds directly into the western boundary current, the current region itself and its outflow into the high-latitude region. Since the western boundary current controls the exchange of water between the tropical and subtropical regions, and hence the heat exchange between south and north, this result is not surprising. It appears that heating or rainfall in these key regions control the density of the abyssal waters. Since much of the deep southward return flow takes place in the deep western boundary current, cooling it or increasing its salinity also increases the east to west density contrast, which balances the meridional overturning.
7. Summary and discussion
This article seeks to use the results of 400-yr simulations with the adjoint to the MIT GCM to offer a new viewpoint on issues tied to the sensitivity of the meridional overturning circulation.
The first question is the relative role played by wind forcing and diapycnal mixing in sustaining the MOC.
In the absence of wind forcing, diapycnal mixing across isopycnals is the only mechanism that can return deep water to the surface (Sandström 1908; Jeffreys 1925; Munk and Wunsch 1998; Wunsch 2002). The highest sensitivities to diapycnal mixing are in the Tropics, where the stratification is largest and the downward diffusion of heat is most efficient. Note that this “diffusion” is only a crude parameterization for small-scale mixing thought to be related to the breaking of internal gravity waves (Gregg 1987; Polzin et al. 1997). The geographic distribution of sensitivity calculated with and without wind are not dissimilar and the peak sensitivities are of the same order or magnitude (480 Sv m−2 s in the absence of wind, 550 Sv m−2 s when wind is included). While the overturning is stronger by a few Sverdrups in the presence of wind, this suggests that diapycnal mixing, not wind, may largely be responsible for closing the meridional overturning.
As observed by Scott and Marotzke (2002) in a model forced by buoyancy alone, enhancing the downward mixing of heat along either boundary in midlatitudes will sustain a more intense overturning than enhancing it in the middle of the basin. The eastern boundary sustains the overturning through downwelling and adiabatic warming above the thermocline and diffusive warming below it. Recirculation in the western boundary current and the cooling associated with the upwelling of the cold deep western boundary current water set the sensitivity in that region. The zonal temperature gradient set up between the two boundaries is required to sustain the vertical shear of the flow associated with the meridional overturning.
Three major changes are observed when wind forcing is added. The first is the increased sensitivity to diapycnal mixing in the eastern equatorial region, in direct relation with equatorial Ekman upwelling and the role of the easterlies in setting thermocline depths in that region. The second finding is the increase in sensitivity to mixing in the western boundary current region because of greater stratification there. The third change is the sensitivity to the wind itself in the open channel region. This effect is related to the presence of the Deacon cell and the upwelling associated with it, confirming Toggweiler and Samuels (1995).
The second issue we seek to address is the role played by the formulation of surface buoyancy forcing on the sensitivity of an ocean model.
The overturning is much more sensitive to surface forcing and diapycnal mixing under mixed or flux boundary conditions than under relaxation boundary conditions. The relaxation formulation allows the “atmosphere” to have infinite capacity to suppress ocean heat and salinity anomalies. That constrains the sensitivity to buoyancy forcing to the region that sets the temperature of the abyssal waters along the western boundary. It is the ability of the flow to advect buoyancy perturbations toward high latitudes under flux boundary conditions that allows high sensitivities to heat and freshwater fluxes to extend throughout the basin. Both high and low latitudes play an important role under mixed boundary conditions, the case thought to be the most realistic. Sensitivities to wind stress are also much greater under mixed or flux boundary conditions. Contrary to the restoring case, which had the highest sensitivities in the Antarctic Circumpolar Channel, the highest sensitivities are in the Tropics where they enhance the role played by diapycnal mixing by compressing isopycnals. This is also an indication that, in this simplified framework and within the limitations of the linearization, diapycnal mixing is the key to setting the meridional overturning’s strength. The sensitivity to wind stress in the Southern Oceans is linked to the formulation of the surface boundary conditions as restoring terms, as noted by Rahmstorf and England (1997). The physical mechanisms responsible for setting these sensitivity patterns are explored further in more realistic geography models in Bugnion et al. (2006)
The fundamental differences in the sensitivity patterns obtained under different surface boundary conditions highlights the importance of a very careful formulation of the surface forcing terms in ocean modeling. A model that appears to have a realistic circulation under restoring boundary conditions will respond very differently to perturbations from a model with a similar circulation but forced with mixed or flux boundary conditions. By providing complete sensitivity maps, the adjoint approach provides new validation to a fact often noted in previous studies.
The ability to run adjoint simulations on centennial time scales provides a new and efficient way of approaching questions related to climate sensitivity. The adjoint gives the complete geographical distribution of the sensitivity of a diagnostic to model parameters and forcing fields. This could be helpful to research that seeks to narrow the uncertainty in these parameters and fields by focusing it where it can have the greatest impact. This approach is, however, limited to models that can be accurately linearized, primarily non-eddy-resolving ocean models (Köhl and Willebrand 2002).
Acknowledgments
The research presented here was partially supported with funds from the U.S. Department of Energy.
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Meridional overturning streamfunction (Sv) in the single basin model: (top) mixed boundary conditions (wind) and (bottom) relaxation boundary conditions (no wind). The rectangular box encloses the area used as cost function for some of the calculations.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Sensitivity of the maximum value of the streamfunction ψMAX to the diapycnal mixing: (∂ψMAX/∂κd) Sv m−2 s); relaxation boundary conditions (no wind). This figure represents the response of a perturbation applied throughout each water column.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
(left) Vertical profile of the heat budget at 14°N in the middle of the basin, all curves are in °C m−1 s (solid line: diapycnal diffusion, dashed line: vertical advection). (right) Vertical profile of the sensitivity to diapycnal mixing (∂ψMAX/∂κd) (Sv m−3 s) at 14°N in the middle of the basin. Both plots represent the flow under relaxation boundary conditions with no wind forcing.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
(left) North–south cross section along the eastern wall of the vertical velocity (10−6 m s−1). (right) North–south cross section along the eastern wall of the sensitivity to diapycnal mixing (∂ψMAX/∂κd) (Sv m−3 s). Both plots were calculated under relaxation boundary conditions with no wind forcing.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
(left) North–south cross section along the western wall of the vertical gradient of the vertical temperature gradient (Tzz; °C m−2). (right) North–south cross section along the western wall of the sensitivity to diapycnal mixing (∂ψMAX/∂κd) (Sv m−3 s). Both plots were calculated under relaxation boundary conditions with no wind forcing.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Sensitivity of the maximum value of the streamfunction ψMAX to the (left) sea surface temperature: (∂ψMAX/∂SST) (Sv °C−1) and (right) the relaxation sea surface salinity (∂ψMAX/∂SSS) (Sv psu−1). Relaxation boundary conditions (no wind).
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Sensitivity of the maximum value of the streamfunction ψMAX to the diapycnal mixing: (∂ψMAX/∂κd) (Sv m−2 s). Relaxation boundary conditions (wind). This figure represents the response of a perturbation applied throughout each water column.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Sensitivity of the maximum value of the streamfunction ψMAX to the zonal wind stress: (∂ψMAX/∂τx) (Sv N−1 m2). Relaxation boundary conditions (wind).
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Sensitivity of the maximum value of the streamfunction ψMAX to the relaxation (left) SST and (right) salinity: (∂ψMAX/∂SST) (Sv °C−1) (∂ψMAX/∂SSS) (Sv psu−l). Relaxation boundary conditions (wind).
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Sensitivity of the streamfunction maximum to diapycnal mixing (∂ψMAX/∂κd) (Sv m−2 s): (left) mixed boundary conditions; (right) flux boundary conditions. The two boxes at 38° and 42°N mark the location of the perturbations applied in Figs. 11 and 12.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Equilibrated response (ΔT; °C) to a perturbation in diapycnal mixing imposed at 38°N at the western boundary at a depth of 95 m: Δκd = 3.10−6 m2 s−1: (left) mixed boundary conditions; (right) flux boundary conditions.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Same as in Fig. 11, but at 42°N, 8°E of the western boundary at a depth of 500 m.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Sensitivity of the streamfunction maximum to the zonal wind stress (∂ψMAX/∂τx) (Sv N−1 m2): (left) mixed boundary conditions; (right) flux boundary conditions.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
(left) Observed zonal mean wind stress (τx) (in N m−2). (right) Meridional gradient of the zonal wind stress (∂τx/∂y), (N m−3).
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Sensitivity of the streamfunction maximum to (top) the heat flux (∂ψMAX/∂Q) (Sv W−1 m2) and (bottom) the freshwater flux (∂ψMAX/∂E − P) (Sv m−1 s): (left) mixed boundary conditions; (right) flux boundary conditions.
Citation: Journal of Climate 19, 15; 10.1175/JCLI3787.1
Model parameters.
Formulation of the surface buoyancy forcing terms for mixed, restoring, and flux boundary conditions (BCs). The bar refers to a zonal averaging over the Atlantic basin, obs refers to a field derived from observations (Levitus and Boyer 1994; Levitus et al. 1994; Jiang et al. 1999), ref refers to a reference salinity (35 psu), and diag refers to a field, which has been diagnosed from the spinup under mixed boundary conditions.
Sea surface temperature, salinity, and density perturbations induced by a perturbation in diapycnal mixing (Δκd = 3 × 10−6 m2 s−1) applied one level below the surface at 2°N along the eastern boundary. The density perturbation was estimated by using the linearized equation of state.