A Model of Atlantic Heat Content and Sea Level Change in Response to Thermohaline Forcing

Xiaoming Zhai; Helen L. Johnson; David P. Marshall

doi:10.1175/JCLI-D-10-05007.1

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Abstract
1. Introduction
2. Theory
3. The reduced-gravity model experiment
- a. Model description
- b. Results
  - 1) Periodic forcing
  - 2) Stochastic forcing
4. The MITgcm experiment
- a. Model description
- b. Results
5. Implications for monitoring sea level change
6. Conclusions
Spectrum of Heat Content Change for Stochastic

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Fig. 1.

Illustration of the phase lag and amplitude reduction of Π′ with latitude relative to . The blue line is a sinusoidal curve with a period of 20 yr. The red and black curves are the blue line averaged back in time for 1 and 10 yr, respectively.

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Fig. 2.

The zonally averaged heat content anomaly Π′ divided by AL, at (b) 10°N, (c) 30°N, and (d) 60°N when (a) is stochastic, where A is the constant in Eq. (3) and L is the basin width. (Unit: m.) Note that Figs. 2–5 are integrations of the analytical model.

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Fig. 3.

Spectra of (a) and Π′/(AL) at at (b) 10°N, (c) 30°N, and (d) 60°N from 100 realizations of stochastic . The dashed line in (b) indicates an ω⁻² law.

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Fig. 4.

Both (a) and Π′/(AL), at (b) 10°N, (c) 30°N, and (d) 60°N when T′_N is stochastic. (Unit: m.)

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Fig. 5.

Spectra of (a) and Π′/(AL) at at (b) 10°N, (c) 30°N, and (d) 60°N from 100 realizations of . The dashed line in (a) indicates an ω⁻¹ law.

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Fig. 6.

Anomalies of (a) MOC (Sv) and (b) Π′/(AL) (m) in the reduced-gravity model when forced by with a period of 5 yr and amplitude of 2 Sv. (c) Heat content anomaly predicted by theory.

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Fig. 7.

As in Fig. 6, but forced by sinusoidally varying with a period of 40 yr.

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Fig. 8.

Anomalies of (a) and (b) Π′/(AL) (m) in the reduced-gravity model when forced by , which is stochastic for periods longer than a month.

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Fig. 9.

MOC (Sv) in the Atlantic basin at (a) year 10 and (b) year 30. The (c) new and (d) climatological restoring potential temperature profiles (°C) at 65°N.

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Fig. 10.

MOC anomalies (Sv) in the (a) Atlantic and (b) Pacific.

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Fig. 11.

Heat content anomalies per unit latitude (×10¹⁶ J) in the Atlantic basin integrated zonally and (a) over the whole water depth, (b) in the top 720 m, and (c) below 720 m.

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Fig. 12.

(a) A that decreases linearly from 2 to 1 Sv for the first 40 yr and then is kept at 1 Sv for another 60 yr, with a white noise added. (b) The evolution of Π′/(AL) in the reduced-gravity model averaged over the basin (red), along the eastern boundary (green), along the western boundary (blue), and along the western boundary in the Northern Hemisphere alone (cyan).

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Fig. 13.

(a) The evolution of the sea level change (m) in the MITgcm averaged over the Atlantic basin (red), along the eastern boundary (green), and along the western boundary (blue). (b) As in (a), but for the heat content anomaly per unit area (×10⁸ J m⁻²).

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Fig. 14.

(a) Sea level (m) and (b) heat content anomalies per unit area (×10¹⁰ J m⁻²) in the North Atlantic in the MITgcm 10 yr after the northern boundary condition is switched to excite a stronger MOC.

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Editorial Type:: Article

Article Type:: Research Article

A Model of Atlantic Heat Content and Sea Level Change in Response to Thermohaline Forcing

Xiaoming Zhai

Xiaoming ZhaiAtmospheric, Oceanic and Planetary Physics, University of Oxford, Oxford, United Kingdom

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Helen L. Johnson

Helen L. JohnsonDepartment of Earth Sciences, University of Oxford, Oxford, United Kingdom

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David P. Marshall

David P. MarshallAtmospheric, Oceanic and Planetary Physics, University of Oxford, Oxford, United Kingdom

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Print Publication:: 01 Nov 2011

DOI:: https://doi.org/10.1175/JCLI-D-10-05007.1

Page(s):: 5619–5632

Received:: 08 Dec 2010

Accepted:: 17 May 2011

Published Online:: 01 Nov 2011

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

The response of ocean heat content in the Atlantic to variability in the meridional overturning circulation (MOC) at high latitudes is investigated using a reduced-gravity model and the Massachusetts Institute of Technology (MIT) general circulation model (MITgcm). Consistent with theoretical predictions, the zonal-mean heat content anomalies are confined to low latitudes when the high-latitude MOC changes rapidly, but extends to mid- and high latitudes when the high-latitude MOC varies on decadal or multidecadal time scales. This low-pass-filtering effect of the mid- and high latitudes on zonal-mean heat content anomalies, termed here the “Rossby buffer,” is shown to be associated with the ratio of Rossby wave basin-crossing time to the forcing period at high northern latitudes. Experiments using the MITgcm also reveal the importance of advective spreading of cold water in the deep ocean, which is absent in the reduced-gravity model. Implications for monitoring ocean heat content and sea level changes are discussed in the context of both models. It is found that observing global sea level variability and sea level rise using tide gauges can substantially overestimate the global-mean values.

Corresponding author address: Xiaoming Zhai, Atmospheric, Oceanic and Planetary Physics, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom. E-mail: zhai@atm.ox.ac.uk

Abstract

Corresponding author address: Xiaoming Zhai, Atmospheric, Oceanic and Planetary Physics, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom. E-mail: zhai@atm.ox.ac.uk

Keywords: Meridional overturning circulation; Ocean dynamics; Planetary waves

1. Introduction

The ocean dominates the heat content of the climate system due to its large mass and heat capacity. Observations show that approximately 84% of the heat gained by the earth system (oceans, atmosphere, cryosphere, and continents) over the past 50 years has gone into the oceans (e.g., Levitus et al. 2005). On top of the warming trend, there are also decadal variations in global ocean heat content (Levitus et al. 2005, 2009).

Warming of the oceans is, by no means, uniform. For example, Lozier et al. (2008) find that in the North Atlantic, the tropics and subtropics have warmed but that the subpolar ocean has cooled. Furthermore, in contrast to some climate models (e.g., Bryan et al. 1982), warming of the oceans is not observed to be confined near the surface. Large temperature anomalies in the deep ocean, especially in the North Atlantic, have been observed on decadal time scales, equivalent to vertical displacement of isotherms by several tens of meters (Roemmich and Wunsch 1984; Levitus et al. 2005; Lozier et al. 2008). These observations suggest that warming of the oceans, rather than being a diffusive process, is a consequence of ocean dynamical adjustment to large-scale variability in wind and buoyancy forcing (Lozier et al. 2008). Note that it takes thousands of years for the deep ocean to be warmed diffusively from the surface given observed levels of diapycnal mixing, but that the ocean adjusts dynamically on much shorter time scales.

Poleward heat transport in the Atlantic ocean is mainly achieved by the meridional overturning circulation (MOC), that transports approximately 1 PW of heat northward (Macdonald and Wunsch 1996), contributing to the relatively mild climate of northwestern Europe. Fluctuations in the strength of the MOC and its associated heat transport have been proposed as a major cause of ocean heat content changes in the Atlantic Ocean on decadal time scales (e.g., Dong and Sutton 2002b). The multidecadal variability in surface temperature centered in the North Atlantic Ocean is also believed to be linked to the strength of the MOC (e.g., Delworth and Mann 2000; Knight et al. 2005). Since observations show that the Atlantic Ocean contributes most to the increase in observed ocean heat content (Levitus et al. 2005, 2009), it is important to establish the relationship between the strength of the MOC and ocean heat content change.

The strength of the MOC is closely related to the deep convective activity at high latitudes. The remote ocean response to changes in the MOC at high latitudes has been studied in a hierarchy of models (e.g., Kawase 1987; Döscher et al. 1994; Greatbatch and Peterson 1996; Yang 1999; Huang et al. 2000; Goodman 2001; Johnson and Marshall 2002b; Johnson and Marshall 2002a; Johnson and Marshall 2004; Cessi et al. 2004; Deshayes and Frankignoul 2005). The initial adjustment is via the propagation of boundary waves along the western boundary to the equator, eastward along the equator, and poleward along the eastern boundary. This is followed by Rossby waves radiating from the eastern boundary into the ocean interior. However, an understanding of how ocean heat content responds to natural and/or anthropogenic forcing is still lacking. Here we examine heat content changes in the Atlantic in response to changes in the MOC at high latitudes, following the theory developed by Johnson and Marshall (2002b). We show that, while the transient adjustment of the MOC is asymmetric about the equator, especially on short time scales, the same dynamical mechanisms result in a symmetric response in heat content change, with high-frequency variability confined to the tropics.

The rest of the paper is arranged as follows. The basic theory of heat content changes is derived in section 2. Model results from a reduced-gravity model and an ocean general circulation model are presented in sections 3 and 4, respectively. Implications for the monitoring of heat content and sea level changes are discussed in section 5. A summary and conclusions are given in section 6.

2. Theory

We first derive the solution for heat content changes in the context of a reduced-gravity model, based on the theory of Johnson and Marshall (2002b). Throughout the derivation, we ignore the heat content of the narrow western boundary region since it is negligible in comparison with the rest of the ocean (<5%).

a. The reduced-gravity model

The reduced-gravity model is intended to represent the upper, warm limb of the MOC, overlying a sluggish abyssal ocean. As argued by Johnson and Marshall (2004), a reduced-gravity model can be expected to provide a useful first-order description, given that the vertical structure of the MOC is dominated by the first baroclinic mode, with a single reversal at the approximate depth of the thermocline. The basic dynamical adjustment process to changes in high-latitude forcing in the reduced-gravity model is a fast Kelvin wave propagation around the perimeter of the basin, followed by the slow radiation of the Rossby waves off the eastern boundary. The wind stress variability and advection by the mean flow are not considered, nor are changes in heat content generated by local buoyancy forcing. Johnson and Marshall (2002b) assume that the divergence of northward transport in their model is balanced by volume fluxes associated with westward- propagating Rossby waves integrated over the meridional extent of the basin:

where R is the radius of the earth,

is the surface layer thickness anomaly on the eastern boundary, c is the long Rossby wave speed, L is the width of the basin, and ϕ is latitude. Both c and L are functions of latitude. The

is the northward transport anomaly prescribed at the northern boundary at latitude ϕ_N, representing the deep-water formation process at high latitudes. The

is the northward transport anomaly at the open southern boundary at latitude ϕ_S south of the equator, representing water mass exchange with other basins. In particular,

is determined by a linearized geostrophic balance:

where H is the mean upper-layer thickness, g′ is the reduced gravity, and f_S is the Coriolis parameter at latitude ϕ_S. We have neglected the western boundary layer thickness anomaly in the southwest corner of the domain in (2), since Johnson and Marshall (2004) have already shown that its effect is very small on time scales shorter than a few decades.

In the reduced-gravity model, the ocean heat content anomaly per unit latitude is given by integrating the heat content anomaly from the eastern boundary to just outside of the western boundary:

where ρ₀ is the upper-layer density, c_p is the specific heat at constant pressure, ΔT is the temperature difference between the upper layer and the stagnant deep water (proportional to g′), λ is longitude, A is a constant, and h′ is the layer thickness anomaly at latitude ϕ and is a function of longitude. In the reduced-gravity model, the heat content depends only on the volume of the thermocline layer since the temperature in the upper layer is fixed. Changes in ocean heat content due to local heating are therefore excluded. Equation (3) states that the heat content anomaly per unit latitude in the reduced-gravity model is proportional to the zonally integrated layer thickness anomaly, which [see (1)] is equivalent to integrating backward in time the eastern boundary layer thickness anomaly over a Rossby wave basin-crossing time:

For a periodic forcing at high latitudes, where

and

, it can be shown (Johnson and Marshall 2004) that

where σ = ωL/c is the ratio of Rossby wave basin-crossing time to the forcing period (divided by 2π) at northern latitudes, and is a function of latitude and forcing period. Substituting (5) into (4), we get

We can then predict the heat content change of the reduced-gravity model in response to periodic thermohaline forcing at northern latitudes with amplitude T₀ and frequency ω.

b. The phase lag between and

For quasigeostrophic dynamics in a rectangular basin, L/c (and hence σ) is independent of latitude so that Rossby wave fronts arrive at the western boundary at the same time across all latitudes (Primeau 2002). Equation (5), in this simple case, becomes

If σ = 2nπ, there is no Rossby wave mass flux divergence and

must be balanced by

at all times.

In oceans of realistic geometry, the Rossby wave basin-crossing time is a function of latitude, so that

lags

In particular, if the forcing period is much longer than the Rossby wave basin-crossing time (i.e., σ ≪ 1), Rossby wave mass flux divergence becomes inefficient to balance the northern mass source, and

is 180° out of phase with

c. The phase lag and amplitude reduction of Π′ with latitude

Equation (4) states that taking the zonal integral in space of layer thickness in the ocean interior is equivalent to integrating backward in time the eastern boundary layer thickness. Note that the time period to be averaged is the basin-crossing time for Rossby waves, which increases with latitude. If we assume Π′ = Π₀eⁱ^ωt, it is easy to show that

from which we can derive the phase lag and amplitude reduction of Π′ relative to

Generally speaking, a longer time average of a sinusoidal signal results in a larger-amplitude reduction and phase shift. If (or ) varies on decadal time scales, the basin-crossing time for Rossby waves at lower latitudes is much shorter than the forcing period, and the heat content anomaly, Π′, at low latitudes is then only a short time average of the most recent eastern boundary layer thickness anomaly. As a result, there is only a small phase lag and amplitude reduction of Π′ relative to . On the other hand, Rossby wave basin-crossing time is comparable to the forcing period at high latitudes, and the heat content anomaly, Π′, at high latitudes is a longer time average of the previous sinusoidally varying eastern boundary layer thickness anomaly. Consequently, there is a larger phase lag and greater amplitude reduction of Π′ relative to at high latitudes. In other words, the heat content change at low latitudes has a short memory, while that at high latitudes has a long one. The effect of averaging back in time on phase lag and amplitude reduction is illustrated in Fig. 1.

d. Stochastic

In reality, the forcing has a continuous spectrum. For simplicity, we first assume

has a white spectrum: S_h(ω) = S₀ = const. We will show later that the spectrum of

is slightly red in response to stochastic

. Applying a Fourier transformation to (4), we get (see the appendix)

The power spectrum of ocean heat content change is related to the Fourier components by

Here S_Π(ω) has the following features:

a high-frequency −2 slope:

with a modulation of
a flattening at low frequencies (σ ≪ 1, i.e., forcing period much longer than Rossby wave basin-crossing time at all latitudes) toward a level:

The spectrum of ocean heat content change at a given latitude in response to stochastic is similar to that of thermocline variability in response to stochastic basin-scale wind forcing [Figs. 2 and 3; also see Frankignoul et al. (1997) and Cessi and Louazel (2001)].

e. Stochastic

In the ocean,

is more likely to be stochastic due to the deep-water formation process that is in part determined by stochastic atmospheric forcing in the form of surface buoyancy fluxes and wind forcing (Stommel (1979)), while the spectrum of

is to be determined. Furthermore, the strength of the MOC has been observed to vary on all time scales at 26.5°N (e.g., Cunningham et al. 2007). Assuming now that

has a white spectrum so that

and applying Fourier transformation to the relationship between

and

, we get

which reveals that

has a red spectrum in response to stochastic

(see also Deshayes and Frankignoul 2005). Since the layer thickness in the reduced-gravity model can be thought of as a proxy for sea level, (17) suggests that stochastic thermohaline forcing can also cause a red spectrum in sea level variability, for example, at Bermuda, without the need for stochastic wind forcing (Frankignoul et al. 1997). Substituting into (12), we get

In the limit of very low-frequency forcing (σ → 0),

As a result of the low-pass-filtering effect of time integration, the spectrum of Π′ is further reddened compared with that of

(Figs. 4 and 5), when

is stochastic. Note that no effect of the mean flow advection has been taken into account in the theory, but it will be discussed later based on an ocean general circulation model.

3. The reduced-gravity model experiment

We now go on to illustrate the predicted heat content change in response to high-latitude thermohaline forcing using a reduced-gravity model. Results from an ocean general circulation model, the Massachusetts Institute of Technology (MIT) general circulation model (MITgcm) are given in section 4.

a. Model description

The nonlinear reduced-gravity model used in this study is similar to that described in (Johnson and Marshall 2002b). The model domain is an idealized sector ocean 50° wide and stretching from 45°S to 65°N, with vertical sidewalls and a resolution of 0.25°. The background model layer thickness is 750 m, with a reduced gravity of 0.02 m² s⁻¹. No-slip and no-normal flow boundary conditions are applied. A sponge region ramps up over the southernmost 1000 km of the domain to damp out any waves approaching the southern boundary.

b. Results

1) Periodic forcing

We first examine heat content changes in the reduced-gravity model when is periodic, with an amplitude of 2 Sv (1 Sv ≡ 10⁶ m³ s⁻¹) and period of 5 yr. As shown by Johnson and Marshall (2002a), the MOC anomaly is largely confined to the Northern Hemisphere because of the low-pass-filtering effect of the equator (Fig. 6a). The heat content anomaly, on the other hand, is symmetric about the equator and confined to low latitudes (Fig. 6b). The pattern of heat content anomaly in the reduced-gravity model is similar to that predicted by the theory (i.e., there is an amplitude reduction and phase lag with latitude), although the magnitude is underestimated in the model (Figs. 6b,c). This underestimate could be due to numerical damping in the model but not in the theory, or (as noted by a reviewer) a reduction in the zonal propagation speed of Rossby waves in the model associated with reduced meridional wavelength when Rossby waves are sufficiently tilted in the horizontal.

When the forcing period increases to 40 yr, the MOC anomaly is transmitted into the Southern Hemisphere (Fig. 7a; also see Johnson and Marshall 2002a). The heat content anomaly now extends to the whole basin (Fig. 7b). The amplitude reduction and phase lag with latitude of the heat content anomaly again agree with the theoretical prediction, although damping of anomalies and slower Rossby wave propagation speed in the model introduces additional phase shift and amplitude reduction with latitude [Figs. 7b,c; Eq. (10)].

Note that one needs to exercise caution when viewing the patterns in Figs. 6 and 7. The phase lag with latitude in both MOC and heat content anomaly is associated with anomalies propagating zonally, rather than meridionally. The basic explanation of the heat content response lies in Fig. 1. When the model is forced with a period of 5 yr, the Rossby wave basin crossing time is much longer than the forcing period at mid- and high latitudes. As a result, there are heat content anomalies with alternating signs in the ocean interior at mid- and high latitudes, which, after zonal integration, largely cancel out. Therefore, the lack of signal at mid- and high latitudes in Fig. 6b does not mean there are no heat content anomalies at any particular longitude and latitude. Recall that the initial boundary wave adjustment occurs within a month. At low latitudes, on the other hand, the Rossby wave basin-crossing time is less than (or comparable to) the forcing period. As such, the heat content anomaly is of the same sign across the basin, with no cancellation after zonal integration, so that the heat content anomaly has a large amplitude at low latitudes (Fig. 6b). When the model is forced with a period of 40 yr, the Rossby wave basin-crossing time is shorter than the forcing period over the whole basin (i.e., the wavelength of heat content anomaly is larger than the basin width everywhere). The cancellation of the anomalies after zonal integration is thus small, even at high latitudes, and the heat content change extends to the whole basin (Fig. 7b).

2) Stochastic forcing

Now we force the model with a stochastic forcing, that is, is stochastic for periods longer than a month (Fig. 8a). Figure 8b shows the evolution of the heat content anomalies at different latitudes. Consistent with the theory, the high-frequency heat content anomaly forced by the high-frequency variability of is confined to low latitudes, and only the low-frequency heat content changes extend to mid- and high latitudes. Interestingly, for MOC anomalies, the equator acts as a low-pass filter (not shown; see Johnson and Marshall 2002b), confining high-frequency MOC anomalies to the hemisphere in which they are generated. Conversely, for heat content anomalies, the slower Rossby wave speed at mid- and high latitudes reduces the zonal-mean heat content variability at mid- and high latitudes in both hemispheres. We term this low-passing-filtering effect of the mid- and high latitudes on ocean heat content change the “Rossby buffer.” Figure 8b also suggests that the equatorial heat content is sensitive to high-latitude forcing at all frequencies (Dong and Sutton 2002a; Ivchenko et al. 2004).

4. The MITgcm experiment

a. Model description

The ocean model used in this section is the MITgcm in a global configuration from 78°S to 74°N, with realistic bathymetry and a horizonal resolution of 1° × 1°. There are 33 geopotential levels whose thickness increases with depth, ranging from 10 m at the surface to 250 m at the bottom. The model is driven by climatological monthly-mean forcing obtained from National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis and freshwater fluxes from the Ocean Model Intercomparison Project (OMIP) forcing dataset. Exchange with the Nordic Seas, which lies outside of the model domain, is crudely taken into account by restoring the model temperature and salinity fields near the northern boundary toward the observed, climatological values. In addition, sea surface temperature and salinity are restored to climatological values to prevent model drift. There is no explicit treatment of sea ice. A similar model setup has been used by Czeschel et al. (2010) to investigate the oscillatory sensitivity of Atlantic overturning to high-latitude forcing.

After the model is spun up for 900 yr, we relax the model temperature in the northern restoring zone (north of 65°N) toward 2 different sets of temperature section data, following Döscher et al. (1994). As shown by Döscher et al. (1994), the climatological data are heavily smoothed and do not resolve the deep boundary current south of the Denmark Strait. After relaxing the model temperature south of the Denmark Strait to a more realistic temperature field, Döscher et al. (1994) found that the meridional overturning and northward heat transport in the North Atlantic, too weak in the case with climatological boundary conditions, increased significantly to more realistic levels. Here we use the same methodology to examine ocean heat content change in response to MOC changes forced by different relaxation northern boundary conditions. We switch the temperature field to which the northern boundary relaxes abruptly between climatological values (CLIM; Fig. 9d) and more realistic values (NEW; Fig. 9c) every 20 yr, for a total additional model run of 100 yr. Despite these abrupt transitions in boundary relaxation conditions, the MOC responds in a smooth oscillatory manner.

b. Results

In comparison with the experiment CLIM (Fig. 9b), the North Atlantic Deep Water cell widens and extends all the way to the bottom in experiment NEW (Fig. 9a). The maximum transport increases from 17 to 23 Sv at about 40°N due to enhanced downwelling in the northern relaxation zone. The Antarctic Bottom Water cell is pushed back farther south in experiment NEW. The changes in the MOC in our experiment are broadly consistent with the findings of Döscher et al. (1994).

Figure 10a shows the MOC anomalies in the Atlantic, where the MOC strength varies every 20 yr in accord with the forcing at the northern relaxation zone. Note that the maximum and minimum MOC occur about 10 yr after the forcing is switched on, instead of at the end of each 20-yr forcing period. This seems to be associated with the oscillatory behavior of the MOC in response to high-latitude forcing found in an adjoint model study by Czeschel et al. (2010). In comparison, the response of the MOC in the Pacific is very weak, even though there is a hint of 20-yr variability. This is consistent with the prediction of Johnson and Marshall (2004) that, on decadal time scales, the MOC anomaly due to high-latitude thermohaline forcing in the North Atlantic is largely confined to the Atlantic basin.

The heat content anomaly is calculated using

where ρ, c_p, and θ′ are the potential density, specific heat at constant pressure, and potential temperature anomaly, respectively. Here Δx, Δy, and Δz are grid length in the x, y, and z directions, respectively. The heat content anomaly integrated zonally and over the whole water column has a complex pattern, with southward propagation at mid- and high latitudes and an indication of poleward propagation at low latitudes (Fig. 11a). However, if we consider the heat content anomaly in the upper ocean and deep ocean separately, a clear picture emerges.

The depth separating the upper and deep oceans is set to be 720 m, which corresponds approximately to the depth of the maximum MOC. Note that the depth of the maximum MOC varies slightly; being several tens of meters deeper when the MOC is switched to a strong phase. Apart from north of 40°N where the mean flow advection becomes important, the upper-ocean heat content (Fig. 11b) behaves in a similar way to that in the reduced-gravity model (Fig. 7b), confirming that the wave adjustment process plays a key role in establishing the upper-ocean heat content anomaly. The time lag between the onset of the heat content anomaly at the equator and the MOC strength at the northern latitude seems also be to in line with the results from the reduced-gravity model. When the MOC is strong, there is a negative heat content anomaly that first develops in the equatorial region and subsequently spreads to higher latitudes. The stronger tilting of heat content anomaly with latitude may be due to either a wider basin or a slower Rossby wave speed in the MITgcm than in the reduced-gravity model. The deep-ocean heat content anomaly, on the other hand, spreads equatorward from the northern latitudes. This is associated with the advective spreading of anomalous cold water in the deep ocean. Note that similar patterns as Fig. 11 are obtained when using buoyancy instead of heat content. There is density compensation between temperature and salinity, especially at low and midlatitudes, but the contribution from temperature dominates the buoyancy.

Our results demonstrate that both wave adjustment and advective spreading are important for ocean heat content changes in response to high-latitude thermohaline forcing, consistent with Goodman (2001). The reduced-gravity model, which only has one active layer, captures the upper-ocean heat content change associated with the wave adjustment process, but is unable to simulate the heat content change associated with the advective spreading in the deep ocean. The comparison between the reduced-gravity model and the MITgcm shows how simple models are useful both when they work and when they are wrong, since they can highlight the missing processes.

Additional model runs with the northern relaxation boundary conditions switching every 2 yr rather than every 20 yr reveal that the MOC anomaly is confined–arrested north of the Gulf Stream (not shown), in contrast to the reduced-gravity model. The Gulf Stream appears to act as a barrier for high-frequency variability generated to the north of it, although it is not clear what the exact mechanisms are. Again, these mechanisms are missing from the theoretical model described in section 2. If confirmed, this would suggest that higher-frequency fluctuations in heat content near the equator are unlikely to be the result of the MOC changes in the north, but more likely to be locally generated.

5. Implications for monitoring sea level change

Traditionally, global sea level change over the past century has been measured from tide gauges, assuming that sea level change near the coast is representative of the global mean. By comparing the sea level trend estimated from global temperature data with that from tide gauges, Cabanes et al. (2001) show that the twentieth-century sea level rise estimated from tide gauge records may have been overestimated. However, the underlying cause of this overestimate by tide gauges was not explained. Here we investigate this matter using the reduced-gravity model and MITgcm. In particular, we are interested in the feasibility of using tide gauges to monitor heat content change and sea level rise in response to variability in the deep-water formation process in the North Atlantic.

After spinning up the reduced-gravity model to equilibrium with a constant of 2 Sv, we first decrease linearly from 2 to 1 Sv over a period of 40 yr and then keep for another 60 yr, with a white noise added over this 100-yr period (Fig. 12a). Figure 12b shows the heat content (sea level) change averaged along the western boundary, along the eastern boundary and over the whole basin. Recall that the sea level anomaly is proportional to the heat content anomaly in the reduced-gravity model. There are three noticeable features. 1) There is large variability at the western boundary, which is greatly reduced at the eastern boundary, and almost absent when averaged over the basin. As a result, trends in sea level and heat content changes over a few years at the western boundary will be hard to interpret, whereas there is a much larger signal to noise ratio at the eastern boundary, or when averaged over the basin. 2) The heat content and sea level continue to rise for at least another three decades after stops declining at the end of year 40, demonstrating the memory in the system associated with the slow Rossby wave adjustment process. 3) The trend in heat content and sea level averaged along the western boundary is almost double that along the eastern boundary and the basin mean. Furthermore, the variability and trend averaged along the western boundary in the Northern Hemisphere alone is even bigger. Similarly, in the MITgcm, the sea level variability averaged along the western boundary in the Atlantic Ocean is larger than that along the eastern boundary, and much larger than that of the basin average (Fig. 13a). Figures 12b and 13a highlight the potential danger of rapid sea level rise on the east coast of the United States and Canada in the future due to a weakening of the MOC in the Atlantic Ocean, a point recently made by Yin et al. (2009) using a set of state-of-the-art climate models (see also Bingham and Hughes 2009 and Frankcombe and Dijkstra 2009). Observing global sea level rise using tide gauges can substantially overestimate the true value. Rather than spatial variation in thermal expansion, the overestimate in the models is due to dynamical adjustment. The inaccuracy of using averaged coastal sea level rise to represent averaged global sea level rise has also been pointed out in the past by Hsieh and Bryan (1996).

In the reduced-gravity model, change in heat content is proportional to change in sea level. Therefore, observing heat content changes at the coast will again overestimate the basin-average in this model. However, in the MITgcm, the heat content change per unit area averaged along the western boundary is very similar to that averaged along the eastern boundary and averaged over the basin (Fig. 13b). To understand this peculiar feature, we plot the sea level and heat content anomalies 10 yr after we switch the northern relaxation boundary condition to a stronger MOC (Fig. 14). In response to changes in the strength of the North Atlantic deep-water formation, the maximum warming and cooling occur along the route of the deep western boundary current (Fig. 14b). In contrast, the heat content change on the continental shelf is small due to the shallow-water column. The same is true for steric sea level changes. The sharp steric sea level gradient across the shelf break cannot be balanced by geostrophic currents, leading to mass redistribution onto the continental shelf and therefore large sea level change at the coast [Fig. 14a; see also Yin et al. (2009)]. However, it is not clear whether Fig. 13b is a robust feature in response to high-latitude thermohaline forcing.

6. Conclusions

We have examined ocean heat content changes in the Atlantic in response to thermohaline forcing at high latitudes. A solution has been derived for heat content change in the context of the reduced-gravity model, based on the theory developed by Johnson and Marshall (2002b), and then extended to ocean general circulation models. In particular, the solution predicts that the ocean heat content change is confined to low latitudes in both hemispheres when the high-latitude MOC changes rapidly, but extends to mid- and high latitudes when the high-latitude MOC varies on decadal or multidecadal time scales. This low-pass-filtering effect of the mid- and high latitudes is associated with the ratio of Rossby wave basin-crossing time to the forcing period at high northern latitudes. We term this effect the “Rossby buffer.”

Results from the reduced-gravity model and MITgcm confirm many aspects of the theory. There is a clear separation with latitude of the heat content change at different frequencies when the reduced-gravity model is forced by stochastic forcing. While there is high-frequency heat content change at low latitudes, there is only multidecadal variability in heat content at mid- and high latitudes. When the MITgcm is forced by alternating northern boundary conditions every 20 yr, the upper-ocean heat content change largely behaves in a similar way as the reduced-gravity model and the theory predicts.

However, there are significant differences. First, in the upper ocean, the effect of advection by the horizontal gyre circulation, which is absent in the reduced-gravity model, plays a role at mid- and high latitudes. Second, the deep ocean heat content anomaly in the MITgcm, which is associated with equatorward advective spreading of cold water, is absent in the reduced-gravity model. Our results demonstrate that both wave adjustment and advective spreading are important for ocean heat content changes in response to high-latitude thermohaline forcing.

Implications for monitoring ocean heat content and sea level changes have also been investigated. We find that the trend and variability of sea level change are much larger along the western boundary than the basin-average due to dynamical adjustment in both the reduced-gravity model and MITgcm. As a consequence, observing global sea level rise using tide gauges can substantially overestimate the true global-mean values. The same is true for heat content change in the reduced-gravity model, where heat content change is proportional to sea level change. On the other hand, in the MITgcm, the heat content change per unit area averaged along the boundaries is very similar to that of the basin average. This peculiar feature is shown to be associated with the fact that the heat content change is large where the ocean is deep, but small on the continental shelf due to the shallow-water column there.

For about two decades, sea level has been monitored by satellite altimeter every 10 days with nearly global coverage, permitting detection of the trend in global-mean sea level. Can heat content change also be monitored from space? Figure 14 highlights the difficulty. Even though there are similarities between the sea level change and heat content change, some of the heat content changes do not project well onto the sea surface height. The feasibility of monitoring ocean heat content change using satellite altimetry certainly merits further investigation.

We have focused here on ocean heat content change in response to high-latitude thermohaline forcing. Variability in the wind forcing will, of course, induce additional changes in ocean heat content, and may indeed dominate in some regions and/or at certain frequencies. The question of the relative importance of the wind forcing and high-latitude thermohaline forcing is beyond the scope of this paper, and is left for a future study.

Acknowledgments

We are grateful for funding from the U.K. Natural Environment Research Council. The numerical calculations were performed at the Oxford Supercomputing Centre (OSC). DPM acknowledges additional support from the Oxford Martin School. We thank the two anonymous reviewers for their detailed and insightful comments.

APPENDIX

Spectrum of Heat Content Change for Stochastic

Applying a Fourier transformation to (4), we get

Integrating by parts gives

Now substituting the expression for

(t), we get

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A Model of Atlantic Heat Content and Sea Level Change in Response to Thermohaline Forcing

Abstract

Abstract

1. Introduction