1. Introduction
Reorganizations in the deep-ocean circulation have likely played a major role during past climatic changes and may again play a crucial role in the future (e.g., Brovkin et al. 2007; Lund et al. 2011; Ferrari et al. 2014; Watson et al. 2015; Liu et al. 2017). Yet, despite an increasing number of observations and extensive climate modeling efforts, we still lack a comprehensive mechanistic understanding of changes in the ocean circulation during Earth’s past and future (e.g., Cheng et al. 2013; Marzocchi and Jansen 2017; Liu et al. 2017).
Climate model simulations typically simulate a weakening and shoaling of the Atlantic meridional overturning circulation (AMOC) over the course of the twenty-first century (e.g., Cheng et al. 2013). However, thus far a significant weakening cannot be confirmed by observations (e.g., IPCC 2013). In the absence of direct observational confirmation, our confidence in coupled climate simulations can be strengthened either by proxy data from past climates that can serve as analogs and/or by a solid physical understanding of the ocean’s response to global warming. While proxy data for past warm climates are limited, a more substantial record exists for the Last Glacial Maximum [LGM; 21 kyr before present (1 kyr = 1000 yr)], which was considerably colder than today. Interestingly, the proxies appear to suggest that the AMOC was likely shallower and potentially weaker during the LGM (e.g., Curry and Oppo 2005; Lynch-Stieglitz et al. 2007; Burke et al. 2015), which may appear to be at odds with the expectation that a weaker and shallower AMOC will result from a warmer future climate. The apparent discrepancy may hint at shortcomings in our coupled climate models, which in fact frequently simulate a deeper and stronger AMOC when subjected to LGM-like boundary conditions and forcing (e.g., Otto-Bliesner et al. 2007; Weber et al. 2007; Muglia and Schmittner 2015). However, differences may also arise from a nonmonotonicity in the response of the overturning circulation to climate change, or from distinctions in the boundary conditions that are not directly attributable to changes in atmospheric carbon dioxide (CO2) (e.g., Zhu et al. 2015; Brady et al. 2013). Finally, the apparent discrepancy could be explained by differences between the response to climatic changes on relatively short time scales (for anthropogenic warming) versus much longer time scales (for the last glacial period) (e.g., Stouffer and Manabe 2003; Zhu et al. 2015). In summary, the use of past climates as analogs for anthropogenic warming is limited without a thorough understanding of how the anticipated circulation changes depend on the sign, magnitude, and time scale of the forcing.
Our theoretical grasp of the deep-ocean circulation has improved significantly over the past two decades (Gnanadesikan 1999; Ito and Marshall 2008; Wolfe and Cessi 2009, 2011; Nikurashin and Vallis 2011, 2012; Mashayek et al. 2015), but most of this understanding is based on equilibrium solutions and may not be adequate to understand ocean circulation in a rapidly changing climate. These transient ocean circulation changes play a key role in shaping the climate response to radiative forcing by altering the pattern of local surface temperature change (Manabe et al. 1991) and affecting the magnitude of global-mean surface temperature change through ocean heat uptake and radiative feedbacks (e.g., Winton et al. 2013; Trossman et al. 2016).
A number of authors have recently taken important first steps toward the development of time-dependent theoretical models (Allison et al. 2011; Jones et al. 2011; Samelson 2011; Marshall and Zanna 2014). Allison et al. (2011), Jones et al. (2011), and Samelson (2011) all consider a two-layer model, which can be interpreted as a time-dependent generalization of the model by Gnanadesikan (1999). The focus of these studies is on the ocean’s response to changes in the wind stress over the Southern Ocean. The new models have provided valuable insights into the importance of circulation changes during deep-ocean adjustment, but significant challenges remain to be tackled. When the North Atlantic Deep Water (NADW) formation rate is allowed to vary following the scaling of Gnanadesikan (1999), the characteristic adjustment time scale in the two-layer scaling theories is found to be 
A number of studies consider the time-scale-dependent response of the ocean circulation to global warming in coupled climate models (e.g., Stouffer and Manabe 2003; Stouffer 2004; Zhu et al. 2015; Rugenstein et al. 2016). However, the mechanisms of ocean circulation changes are hard to disentangle in complex coupled GCMs, because of the myriad of changes that occur concurrently. In particular, the oceanic boundary conditions are affected by changes in the temperature, as well as changes in the atmospheric freshwater flux and wind stress.
To bridge the gap between theoretical advancements and comprehensive climate modeling studies, the present manuscript analyzes the response of the deep-ocean overturning circulation to atmospheric warming in a series of idealized ocean general circulation model (OGCM) simulations. The work builds on a recent paper by Jansen (2017), which discusses the equilibrium response of the deep-ocean circulation to global temperature change. The idealized numerical simulations discussed in Jansen (2017) suggest that, in equilibrium, colder climates tend to be associated with stronger abyssal stratification and a shallower AMOC—consistent with proxy evidence for the LGM—while warmer climates would be associated with weaker abyssal stratification and a deeper AMOC. The mechanistic argument is that colder climates are associated with increased sea ice formation around Antarctica, which leads to enhanced brine rejection, and thus density gain. Enhanced density gain during the formation of Antarctic Bottom Water (AABW) then leads to increased abyssal stratification, which in turn is argued to act as a bottom boundary condition for the upper-overturning cell, with stronger stratification leading to a shoaling and slight weakening of the AMOC (Jansen and Nadeau 2016). The opposite is found for the equilibrium response to warming.
In the present manuscript we use the idealized ocean–sea ice model setup of Jansen (2017) to better understand the ocean’s transient adjustment to surface warming. In particular, we are interested in changes in the AMOC on time scales from centuries to millennia. Unlike the circulation change in equilibrium, the transient response to surface warming is found to be a weakening and shoaling of the overturning, consistent with global warming experiments in coupled climate models. To further disentangle the mechanisms leading to the time-scale-dependent circulation changes, the ocean–ice model simulations are complemented by a series of further stripped-down simulations using a linear equation of state and no sea ice. This idealized model hierarchy helps us to isolate the mechanisms leading to the time-scale-dependent circulation changes, and allows us to explicitly attribute the response to temperature and sea ice changes, as all other boundary conditions are held fixed.
2. Simulated response to abrupt surface warming
a. Model setup
The numerical simulations use the Massachusetts Institute of Technology General Circulation Model (MITgcm; Marshall et al. 1997), in a hydrostatic Boussinesq configuration. The primary set of simulations discussed here employs a setup identical to the one described in Jansen (2017). The idealized domain contains a single basin between 48°S and 64°N, which opens to a circumpolar channel between 69° and 48°S. The domain covers 70° in longitude and has a flat, 4-km-deep bottom, except for a 1°-wide sill below 3000 m in the channel. The horizontal resolution is 1° × 1° and the vertical resolution ranges from 20 m near the surface to 200 m in the deep ocean, with a total of 29 levels. The ocean is forced by prescribed evaporation minus precipitation, atmospheric temperatures, and winds, using simplified bulk formulas. The boundary conditions are constant in time and zonally symmetric and have idealized meridional structures chosen to crudely reproduce present-day conditions [see Jansen (2017) for details].
The ocean model is coupled to a dynamic sea ice model employing a viscous-plastic rheology (Hibler 1979; Zhang and Hibler 1997; Losch et al. 2010). All simulations use a Gent and McWilliams (1990) parameterization (GM parameterization) with a variable eddy transfer coefficient to represent mesoscale eddy fluxes and a prescribed vertical profile for the diapycnal diffusivity with a minimum of about 2 × 10−5 m2 s−1 in the thermocline region and increasing by about an order of magnitude toward the bottom of the ocean. Computational needs for the long simulations were reduced by the use of an accelerated tracer time-stepping, following Bryan (1984). The tracer acceleration scheme distorts the physics during the transient adjustment phase (effectively reducing the gravity wave speed), but additional shorter simulations performed without tracer acceleration show that this has virtually no effect on the results discussed in this manuscript. All warming simulations are initialized from an equilibrated present-day-like reference simulation, which has been integrated for over 10 000 yr and has negligible drifts.
In addition to the ocean–ice model simulations, we discuss results from a stripped-down model, using a linear equation of state (EOS) with a single density variable, and without an explicit representation of sea ice. The domain, resolution, and mixing coefficients are identical to the simulations above. The primary difference is that the effect of surface heat and salt fluxes is parameterized by a linear restoring of the surface density, combined with a fixed flux around the southern boundary. The fixed flux is a crude representation of the effect of brine rejection from sea ice formation around Antarctica (see Jansen and Nadeau 2016). Additional details of the stripped-down linear EOS model setup are provided in appendix A.
b. Simulated response to large-amplitude surface warming in the ocean–ice model
In this section, we discuss the response of the ocean state and circulation to a polar-amplified surface warming, reaching from a 2°C warming in the tropics to 6°C at the highest latitudes, with a domain-averaged surface warming of just over 3°C—as may be expected for a doubling of CO2 concentration (IPCC 2013). Unlike CO2-forced climate change in coupled models, the surface air temperatures are here increased instantaneously at t = 0, and all other boundary conditions remain unchanged, which allows us to isolate the effect of warming alone.
The simulated overturning circulation in the present-day-like reference climate shows the familiar two-cell structure [see Fig. 1a, left, reproduced from Jansen (2017)]. The upper cell resembles the AMOC and is associated with the formation of deep water in the north of the basin (which we will loosely refer to as NADW). NADW flows southward at middepth and returns to the upper ocean via both along-isopycnal upwelling in the channel and diapyncal upwelling at lower latitudes. Below the upper cell lies an abyssal cell, associated with the formation of bottom water near the southern boundary of the domain (which we will loosely refer to as Antarctic Bottom Water). AABW upwells diffusively to middepth in the basin before returning to the surface in the channel. Notice that the zonally integrated overturning streamfunction shown in Fig. 1 appears to suggests that the abyssal cell is not connected to the surface around “Antarctica” (i.e., the southern boundary of the domain). However, analysis of the isopycnal overturning streamfunction (not shown) clearly reveals the connection, and the difference can readily be attributed to the important role played by standing meanders in the channel (see also Jansen and Nadeau 2016).
Response to a sudden atmospheric warming in the idealized ocean–ice GCM simulation. (a) Zonally averaged potential density (referenced to 2-km depth; contours; kg m−3) and overturning transport (shading, with interval of 1 Sv; 1 Sv ≡ 106 m3 s−1) shown (left) before the surface warming is applied, (center) averaged over 90–100 yr after the initial warming, and (right) at the end of the simulation. The overturning streamfunction includes the zonally integrated Eulerian mean flow, as well as, the parameterized eddy transport. [The equilibrium solutions in the left and right panels of (a) are reproduced from Jansen (2017).] (b) Hovmoeller plot of the overturning transport at 50°N (contour interval of 1 Sv). Shown are consecutive 10-yr averages for the first 500 yr, 50-yr averages between 500 and 1000 yr, and 200-yr averages thereafter. Notice the different time scales on the horizontal axis for the first 500 yr and the remainder of the simulation.
Citation: Journal of Climate 31, 13; 10.1175/JCLI-D-17-0797.1
Examining the circulation 100 years after a sudden atmospheric warming, the upper-overturning cell is much shallower and weaker, with NADW not extending below 1-km depth (see Fig. 1a, center). The weakening and shoaling of the overturning circulation is in qualitative agreement with the response to increased atmospheric CO2 concentration in coupled climate simulations (e.g., Stouffer and Manabe 2003; Cheng et al. 2013; Zhu et al. 2015) and can here be attributed exclusively to the effect of surface warming. At 30°N, the maximum overturning weakens by about 30%, which is on the higher side of climate model projections for the twenty-first century under the representative concentration pathway 4.5 (RCP4.5) scenario (Cheng et al. 2013)—consistent with the fact that the imposed surface warming here is at the high end of the simulated twenty-first-century warming under RCP4.5. At 50°N, the response is even stronger, with the maximum overturning transport decreasing by over 50%. The response of the overturning transport is also roughly consistent with the 20%–30% weakening at 40°–50°N found by Zhang and Vallis (2013) in response to a sudden uniform warming of 2°C. (The sensitivity to polar-amplified versus uniform warming will be discussed in the next section.)
The equilibrium response of the overturning circulation to warming, however, is opposite to the transient response! The right panel in Fig. 1a shows the density and meridional overturning circulation at the end of the 14 000-yr-long simulation, which is reasonably well equilibrated. The upper overturning deepens to cover the entire depth of the basin and slightly strengthens [by O(10%), depending on the exact definition] in the north of the basin. AABW formation and the associated abyssal cell disappear1 and the abyssal stratification becomes vanishingly small.
The time evolution of the overturning circulation in the north of the basin (hereinafter referred to as the “AMOC”) is summarized in Fig. 1b. The initial weakening and shoaling occurs on a decadal time scale. The circulation then recovers partially on a centennial time scale. Indeed, a 500-yr-long time series (typical for coupled climate simulations) could easily be interpreted to suggest that the overturning circulation is near equilibrium. However, a slower strengthening and deepening of the AMOC continues for millennia, eventually leading to a slightly stronger and much deeper AMOC compared to the present-day-like reference climate.
c. Simulated response in the limit of small-amplitude surface warming
The simulation discussed above includes a substantial high-latitude warming, which leaves a large body of abyssal water unventilated, forced to adjust diffusively. Faster adjustment processes may instead be dominant for smaller or more gradual changes in the boundary conditions. Moreover, the polar-amplified warming changes not only the mean surface temperature but also the equator-to-pole temperature gradient, which may affect the circulation both in the transient and equilibrium states.
To simplify the problem, we here analyze the ocean’s response to a spatially constant small-amplitude atmospheric warming of 0.1°C. The response to a small-amplitude cooling was also explored, but will not be discussed in detail as the results are approximately equal and opposite to the warming case—consistent with the expectation that the response is linear for small-amplitude forcing.
The qualitative aspects of the AMOC response to warming on time scales from decades to millennia are found to be independent of the amplitude and detailed structure of the warming perturbation. Figure 2 compares the AMOC response for a spatially constant small-amplitude warming to the polar-amplified large-amplitude warming case discussed above. In both cases, the AMOC rapidly weakens, followed by a slow recovery, which eventually leads to a strengthening of the AMOC at depth. The amplitude of the AMOC response in the small-amplitude warming experiment is reduced by about a factor of 40–50, which reflects the difference in the amplitude of warming at mid-to-high latitudes. The recovery and eventual strengthening of the AMOC at depth takes multiple millennia in both the small- and large-amplitude cases, with the anomaly changing sign after about 5000 yr in the large-amplitude warming case and after about 3000 yr in the small-amplitude limit. Full equilibration takes more than 10 000 yr in both cases. Some differences can be seen in the vertical structure. In the large-amplitude warming simulation, the maximum anomaly (both negative and positive) gradually shifts downward, roughly following the increasing depth of the upper-overturning cell (cf. Fig. 1). In the small-amplitude limit, the total depth of the cell interface changes only very little and the largest anomalies are generally near the bottom of the upper cell at around 1500–2000-m depth.
Hovmoeller plot of the anomalous overturning transport at 50°N, relative to an unperturbed control simulation, for (a) large-amplitude, polar-amplified warming simulation shown in Fig. 1 and (b) simulation with small-amplitude, uniform warming of 0.1°C. The contour interval is 1 Sv in (a) and 0.02 Sv in (b). Notice the factor of 50 difference in the contour interval and color bar range reflects the difference in the prescribed surface warming at mid-to-high latitudes.
Citation: Journal of Climate 31, 13; 10.1175/JCLI-D-17-0797.1
d. Isolating warming and ice processes
In the ocean–ice model used in the previous two subsections, the ocean circulation responds to two changes in the ocean’s boundary conditions: 1) surface warming and 2) reduced (or eliminated) sea ice formation around Antarctica. The results are further complicated by nonlinearities in the equation of state, which lead to inhomogeneous density forcing, even in the case of spatially constant warming (e.g., de Boer et al. 2007). To isolate these effects, we performed a number of simulations with a further stripped-down version of the MITgcm, using a linear EOS and no sea ice model. We hypothesize that the initial shoaling of the AMOC is due to the surface warming, while the eventual deepening of the AMOC instead results from a modification of surface density fluxes from ice formation around Antarctica [following the mechanism discussed by Shin et al. (2003a,b), Liu et al. (2005), Jansen and Nadeau (2016), and Jansen (2017)].
To test our hypothesis, we use the stripped-down linear EOS model, where density fluxes around Antarctica can be decoupled from the effect of warming and thus density reduction over the rest of the ocean. The meridional overturning circulation and density structure of a present-day-like reference simulation with the stripped-down linear EOS model is qualitatively similar to the ocean–ice model shown in Fig. 1, although the AMOC is somewhat deeper in the linear EOS model (Fig. 3a). From this equilibrium solution, we impose a uniform 3°C surface warming and simultaneously eliminate the prescribed surface density flux around Antarctica. The elimination of surface density gain around Antarctica mimics the effect of changes in the sea ice cover. In the ocean–ice simulation discussed in section 3b, Antarctic sea ice disappears entirely over the first decade, which has a profound effect on surface density fluxes, eliminating density gain around Antarctica. As found in the ocean–ice model, the initial response to the perturbed surface boundary conditions in the stripped-down model is a rapid weakening and shoaling of the AMOC. The AMOC then recovers first somewhat more rapidly, then slowly and continuing over multiple millennia, eventually leading to a much deeper and slightly stronger AMOC with no abyssal cell (Fig. 3).
As in Fig. 1, but showing the response to a uniform 3°C surface warming and elimination of surface density gain around Antarctica in the stripped-down GCM with a linear EOS and no explicit sea ice.
Citation: Journal of Climate 31, 13; 10.1175/JCLI-D-17-0797.1
To isolate the response of the ocean circulation to surface warming from that of variations in the prescribed flux around Antarctica, we perform two additional simulations. A “warming only” simulation, where the surface temperature is uniformly increased by 3°C while keeping fixed the surface density gain around Antarctica, and an “ice loss only” simulation, where the surface temperature is held fixed while eliminating surface density gain around Antarctica. Results are shown in Fig. 4. In the warming-only experiment, a rapid weakening of the AMOC is simulated initially, followed by a complete recovery over multiple millennia. The eventual equilibrium state is identical to the reference state, save for a constant 3°C warming, which has no effect on the dynamics. In the ice-loss-only experiment, a slow strengthening of the AMOC at depth occurs.
Hovmoeller plot of the anomalous overturning transport at 50°N, relative to an unperturbed control simulation for three different perturbations: (a) uniform 3°C increase (warming only), (b) eliminating surface density gain around Antarctica (ice loss only), and (c) uniform warming combined with elimination of surface density gain around Antarctica. Results are obtained with the stripped-down GCM with a linear EOS.
Citation: Journal of Climate 31, 13; 10.1175/JCLI-D-17-0797.1
These results confirm that the weakening of the AMOC, as well as the eventual recovery to its original strength can be understood as a direct response to surface warming, while the eventual deepening of the AMOC (relative to the colder reference climate) is attributable to sea ice–driven changes in the density gain around Antarctica.
3. Mechanistic interpretation
a. Mechanisms controlling AMOC depth and strength
The simulation results here, together with their broad agreement with coupled climate simulations (Cheng et al. 2013), point toward a robust weakening and shoaling of the AMOC in response to surface warming on decadal–centennial time scales. The AMOC is found to recover completely on millennial time scales, with the ocean–ice model of sections 2a and 2b simulating an even deeper and stronger equilibrium AMOC in a warmer climate. In this section, we address the mechanisms responsible for the simulated AMOC response, as well as their characteristic time scales. We will focus the discussion around the large-amplitude warming simulation from section 2a. Other simulations are discussed only where the results differ significantly and additional insight can be gained.
Two isopycnal surfaces in the north of the basin, from the reference simulation using the ocean–ice model. The red surface shows σ2 = 1036.6 kg m−3, which is near the maximum of the upper-overturning cell, and the blue surface shows σ2 = 1037.04 kg m−3, which is at the bottom of the upper cell.
Citation: Journal of Climate 31, 13; 10.1175/JCLI-D-17-0797.1
To test the relationship in Eq. (1) quantitatively, we diagnose the density profiles through the course of the 14 000-yr warming simulation discussed in section 2b and attempt to recreate the time-dependent overturning circulation using Eq. (1) together with the boundary conditions that Ψ = 0 at the surface and bottom of the ocean. We define ρnorth as the zonal-mean density at the northernmost grid point and ρbasin as a horizontal average over the rest of the basin (reaching southward to the channel–basin boundary at 48°S). The Coriolis parameter is set to f = 1.2 × 10−4 s−1.
The results of applying Eq. (1) to the diagnosed GCM profiles is encouraging in that the estimated Ψ is similar to the GCM-simulated Ψ (cf. Figs. 6 and 1b). The approximation not only captures the main qualitative features of the AMOC evolution discussed in section 2, but quantitatively reproduces the time-dependent depth and magnitude of the circulation. The most significant deviation between the estimate of Eq. (1) and the diagnosed streamfunction in Fig. 1 occurs during the first few decades. Indeed, the use of Eq. (1) is somewhat dubious on such a short time scale, which is of similar order as the time for Rossby wave propagation and other processes that can communicate signals between the northern deep convection region and the western boundary.
Hovmoeller plot of the estimated overturning transport (Sv) in the north of the basin from solving Eq. (1) for each time slice of the large-amplitude warming simulation. The variables ρbasin and ρnorth are diagnosed from the simulation, with ρnorth taken as the zonal average along the northern boundary and ρbasin taken as the horizontal average over the rest of the basin. Compare to Fig. 1b.
Citation: Journal of Climate 31, 13; 10.1175/JCLI-D-17-0797.1
The relationship in Eq. (1) is diagnostic, as the density profiles ρbasin and ρnorth are diagnosed from the simulations, but its success simplifies the mechanistic interpretation of our results. Figure 7 shows the evolution of the density profiles in the basin and northern sinking region. The sudden increase in the prescribed atmospheric temperature first leads to a warming of water masses in immediate contact with the surface (i.e., the thermocline and NADW). While warming is thus initially limited to the upper few hundred meters in the basin (Fig. 7, top left), it reaches much deeper in the northern deep-water formation region (Fig. 7, bottom), leading to a reduced density contrast between the basin and the north at middepth (at around 500–1500 m). The density anomaly then gradually penetrates further down in the basin over the next few centuries, acting to restore the north–south gradient at middepth. The abyssal density decreases much more slowly over multiple millennia. Unlike the initial density reduction in the upper 1 km of the basin, which propagates downward from the surface (Fig. 7, top left), the abyssal density decrease is strongest near the bottom of the ocean and appears almost simultaneously throughout the abyss (Fig. 7, top right).
Density anomaly relative to the reference simulation (kg m−3; with red shading denoting a density reduction, indicative of warming) as a function of depth and time. (top) Horizontal average of the entire basin and (bottom) zonal average at the northern end of the basin.
Citation: Journal of Climate 31, 13; 10.1175/JCLI-D-17-0797.1
Figure 6 shows that the density changes in Fig. 7, together with Eq. (1), explain the time evolution of the AMOC to a good approximation. However, an intuitive interpretation of the relationship between the density change and the overturning circulation is not straightforward, because of the nontrivial form of Eq. (1). Notice in particular that Eq. (1) provides a boundary value problem for the streamfunction whose solution at any level can depend on changes in the buoyancy field throughout the whole water column. To help with the interpretation, Fig. 8 shows the potential density σ2 profiles together with the overturning streamfunction at three times during the simulation.2 We start by considering the reference simulation, which serves as initial condition for our warming experiment (Fig. 8, black lines), and we will focus on the question of what controls the depth of the AMOC. Nikurashin and Vallis (2012) estimate the depth extent of the upper cell based on the level where σnorth matches σbasin (an intuitive estimate for the expected depth range of convection). Equation (1), however, suggests that the depth where σbasin = σnorth (and thus ρbasin ≈ ρnorth) provides an estimate for the turning point of the streamfunction and thus the maximum southward flow, rather than the bottom of the overturning cell. This inference is confirmed by the results in Fig. 8. Near the bottom of the convective region in the north lies a region where σbasin > σnorth such that Ψ has positive curvature and the southward transport goes smoothly to zero at the bottom of the cell.
Potential density averaged over the basin (solid lines) and at the northern end of the basin (dashed lines), diagnosed overturning streamfunction at 50°N (dash-dotted lines), and predicted overturning based on the density difference between the basin and the north (dotted lines). The black lines denote the initial conditions (i.e., reference simulation), red lines show results at 90–100 yr after the large-amplitude warming is applied, and blue lines show results at the end of the large-amplitude warming simulation.
Citation: Journal of Climate 31, 13; 10.1175/JCLI-D-17-0797.1
After 100 years (Fig. 8, red lines), the thermocline and NADW have warmed and lightened. As the warming signal extends deeper in the northern deep-water formation region, σnorth decreases more strongly than σbasin at depth. This leads to a shoaling of the depth where σbasin = σnorth and a strongly positive σbasin − σnorth below. As a result, the core of NADW shoals and the transport decreases rapidly below. Notice that the rapid termination below the core of NADW is associated with a large density “jump” at the bottom of the convective range in the north, which in turn follows directly from the fact that the abyssal density (i.e., below the extent of the upper cell) remains virtually unchanged on centennial time scales (Fig. 7). From the perspective of Eq. (1) the large density jump enables a sudden increase in ρbasin − ρnorth at the bottom of the deep convection region, which in turn leads to a large curvature in Ψ and thus a rapid termination of the southward transport.
By the end of the simulation (Fig. 8, blue lines) the abyssal density reduction exceeds the density change at middepth. The abyssal ocean becomes very weakly stratified and σbasin − σnorth is vanishingly small below about 1-km depth. The result that σbasin ≈ σnorth ≈ σNADW throughout the deep ocean can readily be explained by the elimination of the positive density flux region around Antarctica, which implies that the abyssal density in equilibrium cannot significantly exceed the surface density in the deep-water formation region in the north of the basin. Consistent with Eq. (1), the streamfunction then has no significant curvature at depth, and the southward return flow of NADW extends almost uniformly to the bottom of the ocean.
To summarize, we find that the initial shoaling and weakening of the AMOC results from a rapid warming of NADW, which at middepth leads to a reduction of the density contrast between the northern sinking region and the basin. A partial recovery begins after a few decades as the density reduction penetrates into the basin’s lower thermocline. The eventual full recovery and deepening of the AMOC instead follows the reduction of the abyssal density.
A number of previous studies have attributed the transient weakening of the AMOC in climate models to changes in surface freshwater fluxes in the NADW formation region (e.g., Stouffer and Manabe 1999, 2003), which cannot explain the results of our simulations, where surface freshwater fluxes are held fixed. Indeed more recent studies have pointed toward the dominant role of heat flux changes (Gregory et al. 2005; Weaver et al. 2007; Zhu et al. 2015) to explain the decadal to centennial time scale response of the AMOC in global warming simulations. Our results are qualitatively consistent with these more recent studies and provide a mechanistic interpretation.
An eventual strengthening of the AMOC in response to warming has previously been attributed to effects associated with the nonlinearity in the equation of state (Stouffer and Manabe 2003; de Boer et al. 2007). We notice that in both of these studies the equilibrium AMOC in a warm climate is stronger but shallower, whereas we find a somewhat stronger and much deeper AMOC. A stronger but shallower AMOC is consistent with an increased density contrast between the northern sinking and low-latitude upwelling region, following scaling laws for the diffusive-overturning circulation (Welander 1971). The deeper and stronger equilibrium AMOC in our simulations instead is associated with a reduced abyssal stratification driven by changes in brine rejection around Antarctica [see also Jansen (2017) and Jansen and Nadeau (2016)]. Whether a similar result will be found in coupled climate models will likely depend on the model’s representation of sea ice processes around Antarctica [see also Marzocchi and Jansen (2017)]. Notice, however, that an adequate analysis of the long-term AMOC response to climate change requires multimillennial climate simulations, which are unfortunately rare.
b. Adjustment time scales
Figures 1 and 7 suggest that we may crudely distinguish three time scales during the adjustment process. The initial weakening and shoaling of the AMOC happens on a decadal time scale, associated with the rapid warming of NADW. The AMOC then recovers partially over the first couple of centuries, accompanied by a warming of the lower thermocline. A slow strengthening and deepening of the AMOC continues for millennia, associated primarily with an adjustment of the abyssal density.
Following Eq. (1) and the discussion in section 3a, the initial AMOC weakening is caused by the warming of the northern convection region. A crude estimate of the adjustment time scale can be found based on the heat capacity of the O(1)-km-deep water column and a surface heat flux sensitivity to surface temperature on the order of 10 W m−2 K−1. Dividing the heat capacity by the surface flux sensitivity yields a time scale of O(10) yr, which is consistent with the results.
Figure 7 shows that the partial recovery on a centennial time scale is associated with a reduction in the thermocline density in the basin, primarily in the depth range between 500 and 1500 m. A centennial time scale for the adjustment of the thermocline is of similar order to what has been proposed by Allison et al. (2011), Jones et al. (2011), and Samelson (2011), who consider a two-layer model and solve for the time-dependent thermocline depth in the basin. The adjustment in these models is controlled by an imbalance between vertical advection and diffusion, where the former is governed by the difference in the overturning transport in the north of the basin and in the Southern Ocean. Figures 1–4 show that changes in the upwelling are here dominated by the response of the overturning circulation in the north of the basin, which is in qualitative agreement with the scaling arguments in Allison et al. (2011) and Jones et al. (2011). Using a two-layer model similar to Allison et al. (2011) and Jones et al. (2011), we find an advective adjustment time scale for the thermocline of around 50 yr (see appendix B for details). A 50-yr time scale is somewhat faster but within an order-of-one factor of the initial upper-ocean recovery in the idealized GCM simulations.
However, the AMOC only partially recovers on this relatively fast time scale, with a much slower deepening and strengthening continuing for millennia. Figures 7 and 8 show that the long time-scale change in the AMOC is associated with an adjustment of the abyssal density.
The very slow adjustment of the abyssal density suggests a diffusive time scale. In the large-amplitude warming simulation, a considerable abyssal water mass becomes entirely isolated from the surface during the first few thousand years, as its density is larger than any surface waters. This abyssal water mass is then forced to adjust by diffusion only. But the situation is less obvious in the simulation with a small-amplitude warming, or in the stripped-down, warming-only simulation, where dense water formation persists around Antarctica. We will argue that diffusive adjustment nevertheless governs the equilibration time in all cases, as the abyssal cell is fundamentally diffusive.
Left-hand side (blue) and right-hand side (red) of Eq. (3) estimated from results of (a) the large-amplitude warming experiment using the ocean–ice model, (b) the small-amplitude warming experiment using the ocean-ice model, and (c) the warming-only simulation using the stripped-down GCM with linear EOS. In all cases it was assumed that zint = −2 km, and correspondingly habyss = zint + 4 km = 2 km. Horizontal averages are taken over the entire basin north of 48°S.
Citation: Journal of Climate 31, 13; 10.1175/JCLI-D-17-0797.1
To estimate the diffusive adjustment time scale in our models, we assume a depth of the abyssal cell of around habyss = 2000 m. With a vertical diffusivity at this depth of about 6 × 10−5 m2 s−1, Eq. (4) predicts a diffusive adjustment time scale of about 
Notice that our result for the time scale of complete AMOC adjustment differs substantially from that of Allison et al. (2011) and Jones et al. (2011). These studies find that even isopycnals that outcrop only in the Southern Ocean can adjust via changes in eddy-driven circulation on a much faster time scale than vertical diffusion. The difference here stems from different assumptions for the surface boundary conditions in the AABW formation region. Allison et al. (2011) and Jones et al. (2011) assume a prescribed isopycnal outcrop location (i.e., fixed surface density), allowing for arbitrary (and potentially unphysically large) density transformation rates during the formation of AABW. Instead, we here assume a fixed surface flux Q. In the stripped-down linear EOS simulations discussed in section 2d, a fixed surface flux boundary condition has been prescribed around Antarctica, making our assumption accurate by construction. The situation is less straightforward in the ocean–ice simulations in sections 2b and 2c, where surface density transformation is affected by both heat fluxes (calculated based on bulk formulas) and freshwater fluxes, which arise from (prescribed) evaporation–precipitation and brine rejection from sea ice formation (determined by the sea ice model). However, a fixed-flux condition appears to be more representative than a prescribed surface density in the bottom water formation region at the southern end of the basin, where freshwater fluxes play a major role.
4. Discussion
The idealized simulations discussed in this study allow us to relatively easily disentangle the main mechanisms responsible for the time-scale-dependent response of the overturning circulation to global temperature change. However, the simplicity of the models of course also limits what the simulations can and cannot tell us about the response of the ocean circulation to global climate change.
First and foremost, we here use an ocean-only model, where instantaneous changes in the surface restoring temperatures are prescribed, and all other boundary conditions are held fixed. While this allows us to cleanly isolate the response of the deep-ocean circulation to surface warming, important feedbacks are ignored. In the real world, atmospheric and oceanic surface temperatures coevolve as a result of coupled dynamics and thermodynamics, with ocean heat uptake delaying atmospheric warming. Climatic changes are also associated with adjustments in evaporation and precipitation and surface winds, which would further modify the ocean’s response. Our results should therefore be viewed as a “null model” that isolates only the most direct (and likely most robust) effect of climate warming on the deep-ocean circulation.
Our model uses an idealized basin geometry and topography, where interbasin interactions are missing and the specific processes by which deep and bottom waters form are likely to differ from the real ocean. However, as found by previous studies (e.g., Wolfe and Cessi 2009, 2011; Nikurashin and Vallis 2012) the model does reproduce the main features of the present-day ocean overturning circulation. Moreover, the simulated transient shoaling and weakening of the AMOC is consistent with climate model predictions for the twenty-first century (Cheng et al. 2013), and the attribution of the circulation changes to high-latitude warming is qualitatively consistent with the sensitivity studies of Gregory et al. (2005) and Weaver et al. (2007). The consistency with results from climate models with realistic geometries provides some confidence in the robustness of our main results.
A number of studies have highlighted the potentially important role of a salt-advection feedback on the AMOC’s response to climate warming (e.g., Drijfhout et al. 2011; Liu et al. 2017). A comparison of the results from our ocean–ice model (which includes salt) to the stripped-down model (which does not include salt) suggests that a salt-advection feedback does not play a dominant role in our model. This is further confirmed by analysis of the buoyancy budget of the northern deep-water formation region (not shown). Rahmstorf (1996) and a number of studies since have suggested that AMOC stability is affected by the advection of freshwater into the Atlantic basin by the Eulerian mean overturning circulation. As most coupled climate models, our simulations are in a regime where the overturning circulation advects freshwater into the basin, indicative of a monostable state (Drijfhout et al. 2011). This result appears almost inescapable in our single-basin configuration, where a “warm route” of relatively salty thermocline waters entering from the Indian Ocean is naturally missing, leaving the northward branch of the overturning circulation fed entirely by relatively fresh intermediate waters formed in the southern channel. It has been argued that Earth’s ocean may instead be in a bistable regime, where a positive salt-advection feedback could fuel more dramatic and irreversible circulation changes (e.g., Bryden et al. 2011; Drijfhout et al. 2011; Liu et al. 2017). This possibility cannot be explored with our idealized model setup.
Our results highlight the potential importance of Antarctic sea ice on the long-term evolution of the overturning circulation. While the simplified geometry and lack of atmospheric coupling clearly leads to an oversimplified representation of sea ice processes in our model, the long-term response is argued to result directly from a reduction of Antarctic sea ice formation in a warmer climate, which is likely to be a robust response to long-term warming. Previous coupled modeling studies have shown that sea ice in the North Atlantic can also play an important role in modifying deep-water formation (e.g., Bitz et al. 2007; Zhu et al. 2015; Sévellec et al. 2017). Since the northern basin is ice free in our simulations, these ice–ocean interactions are missing in our study.
Finally, as with any numerical modeling study, questions arise as to the effect of limited model resolution, which here requires us to parameterize the effects of mesoscale eddy fluxes. These eddy fluxes are of particular importance in modulating the overturning transport in the Southern Ocean, where they can, for example, counteract the effect of modified wind stress (Hallberg and Gnanadesikan 2006). However, it has also been found that the effect of resolution on the meridional overturning circulation’s response to varying buoyancy forcing is less pronounced than the response to varying winds (e.g., Zhang and Vallis 2013). Here, we argue that the abyssal adjustment time scale is ultimately set by the diffusive time scale in the basin, which is independent of eddy dynamics in the channel. We therefore believe that our results are unlikely to depend qualitatively on the representation of turbulent eddy fluxes, although some quantitative differences in the adjustment process would likely be found if eddies were resolved explicitly.
5. Conclusions
A series of idealized simulations illustrates the strong time-scale dependence of the AMOC response to surface temperature change. While a surface warming initially leads to a shoaling and weakening of the AMOC, the equilibrium response may be the opposite. The initial shoaling and weakening of the AMOC establishes within O(10) yr, which is fast compared to typical forced surface climatic changes. Partial recovery of the AMOC happens on a centennial time scale. However, full equilibration takes many millennia.
The AMOC evolution on different time scales is attributed to distinct physical mechanisms. The transient shoaling of the AMOC is attributed to a rapid warming of NADW, which reduces the subsurface density gradient between the basin and the northern sinking region. The AMOC partially recovers on a centennial time scale, associated with the adjustment of the density in the lower thermocline, primarily between 500- and 1500-m depth. This thermocline adjustment is likely dominated by reduced upwelling and is crudely consistent with the theoretical derivations of Allison et al. (2011), Jones et al. (2011), and Samelson (2011). However, full equilibration of the AMOC is dependent on the adjustment of the abyssal density, which occurs diffusively and takes multiple millennia. The equilibrium response to warming is associated with a deeper and slightly stronger AMOC, which is consistent with the opposite response for a cold LGM-like climate discussed in Jansen (2017), and attributed to changes in brine rejection around Antarctica [see also Jansen and Nadeau (2016)].
Our results have important implications for the interpretation of GCM results as well as past climate proxies. The millennial time scale for full equilibration is much longer than the typical integration time of coupled climate model simulations, suggesting that the results of these models cannot be interpreted in terms of equilibrium theory. The distinction between the transient versus equilibrium response of the ocean circulation to changes in the climate is also crucial when attempting to use past climates as analogs for the coming decades or centuries. In particular, we argue that the apparent conundrum posed by the suggestion of a shallower and weaker AMOC both during the cold LGM and under anthropogenic warming is most easily explained by the expected difference between the transient and equilibrium response to surface temperature change.
Acknowledgments
Computational resources for this project were generously provided by the University of Chicago Research Computing Center. M. F. J. acknowledges support from NSF Award OCE-1536454. The MITgcm is freely available at http://mitgcm.org/, and the specific configuration files used for this study are available upon request from the authors.
APPENDIX A
Forcing and Parameters for the Stripped-Down Linear EOS Model
The stripped-down version of the model uses a linear equation of state that depends only on temperature, ρ = ρ0(1 − αθθ′), with a constant thermal expansion coefficient αθ = 2.0 × 10−4 K−1. Temperature and density are thus equivalent in this model configuration. Instead of representing the surface flux of momentum using bulk formulas, wind stress is directly prescribed at the surface using the profile shown in Fig. A1. An approach similar to Jansen and Nadeau (2016) is used to account for the effect of sea ice on the surface density flux. An “ice layer” of 2° in latitude is added just north of Antarctica under which a fixed density gain is prescribed in the form of an equivalent heat loss of 20 W m−2. North of this ice layer, surface density is instead restored to the profile shown in Fig. A1. The small asymmetry between the northernmost and southernmost prescribed density was adjusted to crudely match the overturning circulation of the present-day-like reference simulation in the ice-ocean model (Fig. 1a, left).
(left) Surface wind stress and (right) equilibrium surface density for restoring in stripped-down simulations with linear EOS.
Citation: Journal of Climate 31, 13; 10.1175/JCLI-D-17-0797.1
APPENDIX B
Two-Layer Scaling for Thermocline Adjustment
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Figure 1 shows a small counterclockwise abyssal overturning cell near the southern end of the basin in the large-amplitude warming simulation. A more detailed analysis shows that this circulation occurs in a virtually unstratified part of the water column and is generated by the GM parameterization, which becomes poorly defined for vanishingly small stratification. This overturning cell does not connect to the surface layer around Antarctica, and thus appears to be an inconsequential numerical artifact.
We here discuss potential density profiles rather than in situ density. Potential density profiles have the advantage that they also provide an estimate of stratification and allow us to identify regions of convection. The potential density contrast between the interior basin and in the north is not identical to the in situ density contrast relevant for the dynamics, but it provides a good enough approximation for a qualitative mechanistic interpretation.
Time scales are calculated by fitting an exponential function of the form σ(t) = σ∞ + dσ exp(−t/τ) to the evolution of the average basin density below 2-km depth. Note that σ∞, dσ, and τ are all treated as free parameters and the rms error was minimized for the period between 500 yr and the end of the simulation.