Sea level changes resulting from CO2-induced climate changes in ocean density and circulation have been investigated in a series of idealised experiments with the Hadley Centre HadCM3 AOGCM. Changes in the mass of the ocean were not included. In the global mean, salinity changes have a negligible effect compared with the thermal expansion of the ocean. Regionally, sea level changes are projected to deviate greatly from the global mean (standard deviation is 40% of the mean). Changes in surface fluxes of heat, freshwater and wind stress are all found to produce significant and distinct regional sea level changes, wind stress changes being the most important and the cause of several pronounced local features, while heat and freshwater flux changes affect large parts of the North Atlantic and Southern Ocean. Regional change is related mainly to density changes, with a relatively small contribution in mid and high latitudes from change in the barotropic circulation. Regional density change has an important contribution from redistribution of ocean heat content. In general, unlike in the global mean, the regional pattern of sea level change due to density change appears to be influenced almost as much by salinity changes as by temperature changes, often in opposition. Such compensation is particularly marked in the North Atlantic, where it is consistent with recent observed changes. We suggest that density compensation is not a property of climate change specifically, but a general behavior of the ocean.
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 Coastal flooding is already a serious environmental problem in some geographical regions, and any additional future rise in local sea levels will exacerbate it. There is therefore a practical need for projections of regional sea level change due to climate change.
 Physically based three-dimensional coupled atmosphere-ocean climate models are the most credible way to simulate global and regional sea level change for a century or more into the future. Gregory et al.  compared predictions of future sea level change from a range of such climate models, finding in all of them that sea level is predicted to rise over the next century but that the changes will be spatially nonuniform. The range of regional sea level changes was around twice the global-mean value in most of the models considered; however, there was little agreement between the geographical patterns from different models. Consequently, we can presently have little confidence in these patterns, and need to improve our understanding of their physical causes.
Gregory and Lowe  made projections of sea level change for the 21st century using the Hadley Centre atmosphere-ocean general circulation model (HadCM3 [Gordon et al., 2000; Pope et al., 2000]), following a policy-relevant scenario of greenhouse-gas emissions. In this work we analyse in greater detail the mechanisms responsible for global-mean and regional sea level changes in the same model over a period of a few decades for a simpler, idealised, scenario of rapidly increasing greenhouse-gas forcing. Natural forcings, such as from major volcanic eruptions were not included. Only sea level changes associated with variations in ocean density, ocean circulation and atmospheric pressure were considered. The contributions from the melting of land ice or terrestrial water storage were not treated because they are expected to be of secondary importance in the global mean on the time scales of interest here compared to those associated with density changes, while their influence on regional sea level change comes about mainly through effects on the gravitational field and the solid Earth rather than oceanographic change [Church et al., 2001]. Vertical land movements, which occur as a result of various geological and artificial processes, such as glacial isostatic adjustment and extraction of fluid (including groundwater), were also not treated. These can be locally very important [e.g., Church et al., 2001], but they are not a strong function of recent or future climate change simulated by the model on the century time scale.
2. Theoretical Considerations
 Our aim in this section is to discuss how ocean density and circulation are related to sea level, trying to relate various different approximations that are in use for observations and models. This provides a background for interpreting our model results in subsequent sections.
2.1. Dynamic Topography
 Applying Newton's second law of motion, and following Gill , the equation of motion for the ocean can be written in the form of equation (1):
where u is the horizontal velocity, Pa the atmospheric pressure, η the dynamic topography, i.e., the sea surface height above the geoid (z = 0), ρ0 a reference density (using the Boussinesq approximation), ρ the density field below the geoid, f the Coriolis parameter, k the vertical unit vector, DV the vertical diffusivity of momentum and DH the horizontal diffusivity of momentum. Equation (1) can be simplified by assuming that the ocean responds quickly to atmospheric pressure changes, permitting this sea level response to be calculated separately using the inverted barometer relationship [e.g., Wells, 1997]. Deviations from this relationship do occur but this is only really important on short time scales [Ponte and Gaspar, 1999] and where flow is restricted [Ducet et al., 1999]. This allows us to replace η with η′ = η + Pa/gρ0, so
 In the HadCM3 ocean model used in this work, a rigid lid is employed to suppress external (fast-moving) gravity waves, allowing a longer time step and reducing the computational effort. The pressure on the rigid lid ps replaces gρ0η in the equation of motion used in the model. The model's method of solution eliminates ps as a prognostic quantity but for our purposes we diagnose ∇ps and recover ps using a Poisson equation with gradient boundary conditions [Pinardi et al., 1995] solved by successive over-relaxation [Press et al., 1992]. In earlier work [Gregory et al., 2001] we instead obtained ps as the estimate which gave the best match to the diagnosed ∇ps; this worked reasonably well but did not always converge in the Arctic. The simpler method of integrating ∇ps is not satisfactory because numerical inaccuracy in the model itself means that the diagnosed ∇ps is not an exact gradient, and the result therefore depends on the starting point, making some global optimisation unavoidable. By any of these methods, the solution has an arbitrary additive constant, so the global mean of the field is uniformly subtracted (see section 4). The effect of atmospheric pressure Pa is added to ps to obtain η′.
 Since we are most concerned with mean sea level on decadal and longer averages, the Eulerian rate of change on the left-hand side of equation (2) is negligible. Over most of the ocean the horizontal advection and diffusion of momentum are unimportant, leaving only the surface height gradient, the hydrostatic pressure below the geoid or rigid lid, the Coriolis force and the vertical diffusion of momentum:
2.2. Sea Surface Height in the Surface and Geostrophic Approximations
 At the surface (where u = us and z = 0), the density integral vanishes, so equation (3) can be simplified to:
Assuming the vertical diffusion term is important only in the Ekman layer (thickness hE), within which the wind stress τ is absorbed, by averaging over that layer we obtain
and the average velocity un within the layer can be regarded as the sum of an Ekman velocity associated with balancing the wind stress and a geostrophic velocity associated with balancing the surface slope. These components tend to be orthogonal (Ekman parallel to ∇η′, i.e., up the surface slope and geostrophic normal to ∇η′). At many locations the Ekman velocity dominates and the surface velocity is largely determined by the wind stress.
 We can calculate η′ from equation (5) by using τ to estimate the Ekman velocity and hence the geostrophic velocity as the remainder of un. This method has been used by Niiler et al.  with observational data. For our purposes its drawback is that it does not give us insight into the cause of changes in η′. It is also liable to be somewhat inaccurate because there may be significant ∇ρ within the Ekman layer, so the appropriateness of neglecting this term becomes questionable.
 For the deeper ocean, the density integral in equation (3) is important but, particularly for averages of a year or more, the wind stress can usually be neglected, leaving a geostrophic balance between the pressure gradient and Coriolis terms.
Averaging equation (6) vertically between z = 0 and the ocean floor at z = −H we obtain
This equation shows that the dynamic topography in a geostrophic ocean has a baroclinic component, related to the ocean density structure (the first term), and a barotropic component, related to the depth-mean horizontal circulation (the second term) [Pinardi et al., 1995; Fukumori et al., 1998].
Equations (6) and (7) offer different ways of computing the dynamic topography. Both require knowledge of both density and circulation. Since interior velocity is not well known from measurements, equation (6) is the one generally used with observations, by employing the additional assumption that u = 0 at some “level of no motion” (LNM), typically O(1000) m. This assumption does not help with equation (7), which involves the barotropic velocity, a quantity which is less easy to obtain observationally: Sverdrup balance extended to include bottom pressure torque [e.g., Gregory and Dowrick, 1992] allows us to infer in the gyres, but not in the Antarctic circumpolar current (ACC), and in any case τ is not accurately known. In a model, on the other hand, all quantities are known in principle. Equation (6) can be applied by using the diagnosed velocity at some level rather than by assuming a LNM; this approach has been used to extract η′ in some models [Jackett et al., 2000; Gregory et al., 2001].
 Both equations (6) and (7) impose constraints on η′ from density and velocity: Equation (6) requires ∇η′ to balance all of the depth-integral of the density gradient above the LNM, while equation (7) requires it to cancel some of the depth-average of the density gradient, the remainder being balanced by the barotropic velocity , which in a flat-bottomed gyre is independently determined by the wind stress (except in ageostrophic boundary currents and on small scales with strongly curved flow, such as in vortex rings). We might ask how these constraints can be simultaneously met. Suppose that there are no horizontal density gradients below the LNM; in that case the ocean will be motionless below this level as well, and can be ignored, since the integral of ∇ρ down to the LNM has been exactly cancelled by ∇η′. This is reasonable except where there are deep currents. If we regard equation (6) as determining ∇η′, given the vertical integral of ∇ρ, then in order to satisfy equation (7), given , we can use our remaining freedom to adjust the vertical profile of ∇ρ(z), and hence its vertical average, while not altering its vertical integral, i.e., redistributing the mass. If we regard the equations as being simultaneously satisfied, then the degrees of freedom being adjusted are the horizontal density contrast at the surface and the depth over which it decays, both of which are affected and hence coupled by the overturning (Ekman surface drift and pumping/suction) driven by the wind stress. These vertical velocities in turn drive the barotropic circulation as explained, for instance, by Pedlosky . There is thus no contradiction between the appearance of in equation (7), derived assuming geostrophy, and the determination of , through Sverdrup balance, by the wind stress, which is not a component of geostrophy.
 In reality the constraint imposed on ∇η′ by equation (7) is that it must cancel nearly all of the baroclinic term, because the barotropic term is relatively small [Pinardi et al., 1995]. This implies that the main influence of wind stress forcing on sea level is its indirect one through the effect of wind-driven overturning on the density structure, rather than its direct one through the barotropic circulation and the Sverdrup balance; sea level is roughly parallel to barotropic streamlines in the gyres because of the baroclinic term in equation (7).
2.3. Dynamic Topography Change
 Either of equations (6) and (7) could be used to calculate change in dynamic topography between two quasi steady states. If the LNM is unchanged, equation (6) gives
while if the barotropic velocity is unchanged, equation (7) gives
These equations are mutually exclusive. Since is largely determined by τ, the latter assumption is preferable, especially if we are considering sea level change caused by changes in surface buoyancy fluxes (not wind stress), in which case a change in the LNM is likely.
 However, in practice, the RHSs of (8) and (9) will not differ very much if changes in density are restricted to the upper part of the ocean column [Gregory, 1993], which is likely for climate change on decadal timescales. In that case we can use (8), which is a formula for steric (i.e., density-related) sea level change, to approximate the change in the baroclinic term of (7); furthermore, if the change in the barotropic term is small, the steric change will in turn be a good approximation for the change in the dynamic topography. This is useful because it is simple, requiring just the local depth-integral of the density change, which depends on changes in temperature and salinity, whose contributions (called thermosteric and halosteric) can thus be separated. In the depth-integral, increases in density at some depths may offset decreases at other, and increases in temperature (which decrease density) may be offset by increases in salinity (which increase density). Steric sea level change has been used extensively to estimate dynamic topography change [e.g., Antonov et al., 2002; Lombard et al., 2005] for comparison with measurements by tide-gauges and satellite altimetry. It must be kept in mind, however, that it is only an approximation. Furthermore, since η appears only as a gradient, local steric sea level change from equation (8) does not include change in global-mean sea level (see section 4).
 Changes in density depend not only on changes in temperature and salinity, but also on the actual values of in-situ temperature and salinity. For a given addition of heat a warm fresh water mass will expand more than a cold salty one at the same depth. Figure 1 shows the coefficient of thermal expansion for surface waters estimated from a decadal mean from our control simulation. For instance, the relatively small value of the expansion coefficient in cold waters of the Southern Ocean implies that if heat is added to the ocean in this region, the change in density and sea level will be less than if the same amount of heat were added over a similar size of region in a warmer location. This dependence has been discussed and quantified by Gille .
 Although steric sea level change is evaluated from local temperature and salinity changes, these changes result not only from local changes in surface fluxes of heat and freshwater, but also from changes in transport of temperature and salinity by advection, eddies and diffusion. This complicates the study of sea level change mechanisms because local sea level change may be spatially and temporally separated from the changes in surface atmospheric fluxes that ultimately cause it.
3. Experimental Setup and Diagnostics
 A set of five climate model experiments were performed (Table 1) using HadCM3, which has an atmospheric horizontal resolution of 3.75° × 2.5° and an ocean resolution of 1.25° × 1.25° with 20 vertical levels. Flux corrections were not applied. In the control experiment the atmospheric carbon dioxide concentration was set to a pre-industrial value of approximately 290 ppmv. In the baseline experiment the carbon dioxide concentration was increased from the control value at a compound rate of 2% per year for 70 years, reaching 4× the control concentration, enough to cause a significant climate change signal in a relatively short period of time. We calculated a monthly anomalous freshwater flux as the difference between the flux in the baseline experiment and the climatological monthly mean in the control for the same time of year. In the freshwater experiment, carbon dioxide was as in the control and the monthly anomalous freshwater flux was added to the freshwater flux calculated by the model during this experiment. In the 2% minus freshwater experiment carbon dioxide was increased at 2% per year and the anomalous freshwater flux was subtracted from the freshwater flux calculated in the model during this experiment. The freshwater and 2% minus freshwater experiments together allow us to test whether the effects of changes in freshwater and nonfreshwater fluxes combine linearly. Finally, a wind stress experiment was performed in which the anomalous wind stress from the 2% baseline experiment was applied to a climate with control carbon dioxide concentration. We are not able to construct an experiment to test the effect of anomalous heat fluxes independently, because the strong feedback of sea surface temperature on surface heat flux means that such changes cannot be prescribed like those of freshwater and wind stress.
Table 1. Experiments
CO2 Increase, yr−1
Plus freshwater anomaly
2% minus freshwater
Minus freshwater anomaly
Plus wind stress anomaly
 Our procedure of adding an anomalous flux is a method of imposing a time-dependent change, while still allowing the surface fluxes to vary freely in internally generated variability and to react to climate change. It is different from the procedure adopted by Gregory et al. , in which the freshwater flux from one experiment is substituted for the flux computed in another; this procedure suppresses the effect of climate feedbacks on the flux which has been substituted, and could distort internally generated variability, since the flux does not relate to the state variables in the experiment to which it is applied. In our procedure, the anomalous flux includes the imprint of the internal variability in the experiment which it comes from. The experiment to which it is applied has its own independent internal variability. Hence the anomalous flux injects extra variability as well as the time-dependent change. However, we find that the meaning periods we use are sufficiently long for the climate-change signal to be clearly distinguishable from the enhanced noise.
 In all the experiments, two extra tracers, called passive anomaly tracers (PATs), were included to track the anomalous heat and freshwater once it enters the ocean. The surface flux of passive anomalous salinity was the anomalous freshwater flux defined above. The surface flux of passive anomalous temperature was the similarly computed anomalous heat flux. 3D diagnostics were included for the rate of change of the ocean temperature, salinity and PATs resulting from each of the individual transport processes within the ocean, and for the individual terms in the equation of motion solved by the model. These extra tracers and 3D diagnostics added a large overhead to model run time and storage.
4. Global Mean Sea Level Rise
 Global-mean sea level rise due to thermal expansion is calculated from the 3D ocean in-situ temperature fields [e.g., Gregory and Lowe, 2000]. The quantity of relevance of projections is the absolute (i.e., geocentric) local sea level change, obtained by adding the global-mean rise to the spatially varying dynamic topography change [Greatbach, 1994]. However, in the present work, we look at global-mean change and dynamic topography change separately.
 Thermal expansion time series for the four anomaly experiments listed in Table 1 are shown in Figure 2; from each a quadratic fit [Gregory and Lowe, 2000] to the control simulation has been subtracted, to remove the effect of drift in the deep ocean due to insufficient spin-up. The size of the global mean thermal expansion drift was around −0.0125 cm/year. In the 2% baseline experiment the sea level rise over 70 years (the time taken to quadruple atmospheric carbon dioxide concentrations) was 24 cm. In the 2% minus freshwater experiment the rise was 22 cm, showing that omitting the freshwater flux has a small but noticeable effect on the thermal expansion. The anomalous freshwater flux can alter the amount of thermal expansion by changing the upper ocean density structure and stability which, in turn, can alter the vertical heat transport, the sea surface temperature and hence the flux of heat from the atmosphere. In addition, it can modify ocean horizontal transports so that heat is redistributed from a region with a small thermal expansion coefficient to one where the coefficient is larger.
 The wind stress anomaly acting alone produces a negligible global mean thermal expansion, indicating that it is globally ineffective at redistributing heat to regions of higher thermal expansivity or increasing the surface heat flux.
 The global mean curves for the 2% minus freshwater and the freshwater experiments combine to give a curve that is almost identical to the 2% baseline result. The difference between this combined curve and the 2% curve is less than the internally generated variability. Recalling that the global mean thermal expansion caused by the anomalous wind stress is negligible, it appears that the results of the freshwater flux and heat flux combine linearly in the global mean.
Antonov et al.  computed global-mean sea level change due to salinity change (halosteric) over recent decades, finding it to be about 10% of the thermal expansion (thermosteric change). We would say that this global halosteric effect should not be counted as a separate contribution to global-mean sea level; unlike thermal expansion, it does not occur with constant mass, but is included in the increase in ocean volume due to adding freshwater (shown in Appendix A [cf. Munk, 2003]), which causes the salinity change, as Antonov et al. point out. Since we are not considering ocean mass change, we therefore exclude this term by using the initial 3D field of salinity from the control in all global-mean calculations. There could be a global halosteric contribution due to redistribution of salinity within the ocean, while not changing its volume integral, because the equation of state is nonlinear in salinity, but this effect is negligible [Gregory and Lowe, 2000].
 The global-mean sea level contribution from change in atmospheric pressure (inverted barometer effect) is negligible, even though there may be a nonzero average pressure change over the ocean, since water is practically incompressible. For instance, a uniform atmospheric pressure increase of 1 hPa would depress sea level by about 0.2 mm cf. 10 mm locally by the inverted barometer effect.
5. Influences on Dynamic Topography Change
 The regional distributions of dynamic topography change (i.e., sea level change with global mean subtracted) with respect to the control for the final decade of the four anomaly experiments listed in Table 1 are shown in Figure 3 and summarised in Table 3. A linear function of time was fitted to the control result at each point, then subtracted from the anomaly experiments in order to remove the drift. The spatial standard deviation of the control drifts in the dynamic topography was around 0.027 cm/year. The inverted barometer component was calculated separately and added to the oceanographic changes.
 In the baseline experiment large positive rises occur around Greenland, north of Canada, in the western North Pacific, the South Pacific Convergence Zone (SPCZ), and along the north of the ACC. The largest negative changes occur in the Arctic around 180°E, and throughout the Southern Ocean, especially in the Ross Sea area. A fall in dynamic topography in the Southern Ocean is a common feature of AOGCM climate-change experiments [Gregory et al., 2001]; since the dynamic topography change has zero global mean by construction, a relative fall can be interpreted as a lack of sea level rise. In the North Atlantic there is a dipole, with a decrease in dynamic topography south of around 40°N and a slight increase to the north. The spatial standard deviation of dynamic topography change is 10 cm, which is 40% of the global-mean thermal expansion (Table 2).
Table 2. Statistics of Dynamic Topography Changes
Standard Deviation SD, m
Ratio of SD to Global Mean
Maximum, m (Global Mean Subtracted)
Minimum, m (Global Mean Subtracted)
Correlation Coefficient With Baseline
2% minus freshwater
 In the 2% minus freshwater experiment many of the same large-scale features are present, but they differ somewhat in their detail. A significant difference occurs in the North Atlantic, where the dipole is reversed relative to the baseline case.
 In the freshwater experiment the fall in dynamic topography in the North Atlantic south of 40°N is similar to the baseline experiment, but there are big differences elsewhere. In general the changes are of a lower magnitude than in the baseline simulation (Table 2).
 In the wind stress experiment many of the Pacific ocean features are represented, as are some of the features along the north of the ACC. The Atlantic sea level changes are small and dissimilar to those in the baseline experiment. The wind stress forcing explains more than half the spatial variance of the baseline experiment (Table 2).
 By comparison of Figures 5b and 5d we can infer that the positive changes in the western North Pacific and the generally negative changes in the Southern Ocean are mostly induced by heat flux changes. There is some similarity with the results of Fukumori et al. , who found for sea level variability on timescales less than two years that wind stress variability was generally more important than heat flux variability and explained features in the low-latitude Pacific, while heat flux variability caused changes in the western North Pacific and the North Atlantic. It is interesting that our North Atlantic pattern is dominated by freshwater rather than heat flux changes, since this is the reverse of the conclusion for factors leading to weakening of the Atlantic meridional overturning circulation [Gregory et al., 2005].
 The frequency distribution (Table 2 and Figure 4) of sea level change in the baseline results is skewed towards negative changes. The 2% minus freshwater experiment is qualitatively and quantitatively similar to the baseline experiment, with which it has a large pattern correlation coefficient (Table 2), despite difference of detail. The frequency distributions of the wind stress anomaly and freshwater anomaly experiments are less skewed towards negative values than the baseline results and they both have a lower range and standard deviation. The pattern correlation coefficients confirm that the wind stress anomaly experiment is more similar than the freshwater anomaly experiment to the baseline case.
6. Mechanisms of Dynamic Topography Change
6.1. Dynamical Changes
 The change in dynamic topography in the 2% baseline experiment (Figure 3a) can be decomposed (Figure 5 and Table 3) into baroclinic and barotropic components, as defined by equation (7), and the inverted barometer effect of Pa. The baroclinic component (Figure 5a) clearly dominates the pattern of dynamic topography change, with deviations of up to ±30 cm from the global mean value. In the tropics, where the change in dynamic topography is almost entirely baroclinic, we note from Figure 3 that the surface heat and freshwater fluxes, and the change in wind stress all lead to sizeable baroclinic changes.
Region and sign of features of sea level change in the 2% baseline experiment are listed. The Flux column attributes positive and negative contributions, most important first, to changes in surface fluxes of heat H, freshwater F and momentum M. The C/T column describes the change as baroclinic C or barotropic T. The T/S column decomposes baroclinic change (in regions where well approximated by steric change) into contributions from temperature T and salinity S. The A/R column accounts for temperature changes as due to local anomalous heat input A or redistribution of heat content R.
Pacific N of 45°N
W Pacific 40°N
H+ F+ M−
W Pacific 10°N
E Pacific 15–30°S
Arctic around 180°
H− M− F−
H+ M+ F+
Atlantic N of 45°N
W Indian 30°S
N margin of ACC
H+ M+ F+
S Ocean S of ACC
 The barotropic contribution (Figure 5b) provides several additional centimetres of change at mid- and high latitudes, with notable contributions in the storm track regions of both hemispheres. Barotropic sea surface height changes are related to changes in (equation (7)) and therefore driven by the same factors that alter the gyre circulation. Because of the latitudinal variation of the Coriolis parameter, for a given change in the stream function, sea level changes nearer the equator are smaller, so the barotropic contribution is minor at low latitudes, as found by Fukumori et al.  for sea level variability. Consistent with the main influence on barotropic flow being change in wind stress, we note that Figure 3d (wind stress experiment) explains barotropic changes around the ACC and in the Pacific subpolar gyre.
 The inverted barometer term makes a positive contribution at northern high latitudes, caused by a reduction in air pressure, and negative contributions poleward of 40° in the North and South Pacific, caused by increases in air pressure.
6.2. Steric Changes
 The baroclinic component of dynamic topography change is very similar to steric sea level change, except where the heat penetrates a large fraction of the water column (section 2.3). Since the baroclinic component is generally dominant, that means we can explain dynamic topography change in the baseline experiment over most of the world in terms of temperature and salinity variations (Figure 6). Areas (principally in the Arctic) where the baroclinic and steric results deviate significantly (by more than 5 cm) are not considered in this section.
 The largest positive contribution made by temperature change to sea level rise is in the North Atlantic, and the largest negative in the western Pacific around 10°N and in the Southern Ocean (Figure 6a). In the North Atlantic, 64% of the change in ocean heat content occurs in the upper 530 m of the water column, and 16% occurs in the upper 110 m. Upper level warming occurs over much of the North Atlantic, whereas the deeper changes occur south of Greenland and along the east coast of North America, possibly transported by the deep southward moving western boundary current.
 The largest positive contribution made by salinity change to sea level rise is in the western Pacific around 10°N, and the largest negative in the North Atlantic (Figure 6b). In the North Atlantic, 72% of the change in volume weighted salinity occurs in the upper 530 m of the water column, and 10% occurs in the upper 110 m. Deeper changes are mostly positive and the pattern of change is similar to that of temperature, with the largest increases in the north and along the western boundary.
 Thermosteric and halosteric sea level changes generally occur in similar regions but often with opposite sign so that they partly or wholly offset each other. Thus, although density changes are dominated by temperature changes in the global mean, salinity variations are as important as temperature variations in determining the spatial pattern of the steric sea level change. Some compensation occurs for almost all the notable features of sea level change in the baseline experiment (Table 3); exceptions include the rise along the north of the ACC and the fall in the Southern Ocean, which are associated only with temperature change. Levitus et al.  find that in observations for recent decades there is a strong compensation in the North Atlantic, with thermosteric rise and halosteric fall south of 45°N. The same is evident in the model results. They find the reverse north of 45°N, where the model shows only small changes in each component. The model might also be in accord with some of the zonal features they note in the Indian and Pacific Oceans.
 The phenomenon could be explained by an anticorrelation between local changes in surface freshwater and heat fluxes, but the evidence for this in the model results is weak; moreover, density changes are not closely related to local surface flux changes (see next section). A more likely explanation is such compensation is not a property of climate change specifically but is widespread in the ocean, tending to reduce the range of density and the contrasts between adjacent water masses, for instance across fronts. This comes about because density anomalies are dynamically active and promote their own dispersal. Widespread density compensation means that any change in water mass boundaries or advection tends to produces partially cancelling thermosteric and halosteric anomalies. In the case of the North Atlantic, the model changes are consistent with a reduction of the meridional overturning, since the northward-flowing water is warm and salty, the southward cold and fresh.
6.3. Heat Input and Heat Transport
 Using the passive anomaly temperature tracer it is possible to further decompose the thermosteric changes of Figure 6a into those caused by the addition of anomalous (extra) heat from the atmosphere since the start of the experiment and those caused by a redistribution of existing heat (Figure 7 and Table 3). In the global mean the anomalous heat accounts for the entire steric sea level rise; however, redistribution does have a sizeable regional pattern. For instance, in the western Pacific this term is larger in magnitude but opposite in sign to the anomalous heat contribution, which it outweighs.
 In the North Atlantic there is very large anomalous heat uptake, which is partially offset by export along the eastern boundary, while along the western boundary there is additional steric sea level rise, probably because of reduced southward advection of cold water in the deep boundary current. In the low-latitude Atlantic there is a positive contribution to sea level from heat redistribution, presumably because of reduced northward heat transport, but this is outweighed by reduced heat input (Figure 7a) and increased salinity (Figure 6b). It is interesting to note that maximum anomalous heat uptake in the North Atlantic is also the dominant feature of the pattern of uptake of anthropogenic carbon dioxide [Sabine et al., 2004], which is a passive tracer, like passive anomalous temperature.
 The features in the western Pacific at 10°N, in the SPCZ and along the north of the ACC are all accounted for by heat redistribution. These features were found in section 5 to be partly or wholly wind-driven, and they are probably associated with changes in subduction or the movement of fronts.
 In total, 4.7 × 109 J/m2 of anomalous heat are added to the ocean. To account for the pattern of change in Figure 7b there must be a redistribution of around 1.7 × 109 J/m2 of heat, i.e., that is the magnitude of the area-integral of either the positive or the negative regions, which must be equal. Thus, both the pattern of surface heat fluxes and ocean heat transport are important.
 Changes in the passive anomaly salinity tracer are similar to changes in salinity. This implies that effects involving the redistribution of existing freshwater are less important than the anomalous flux.
 The sea level changes caused by ocean density variations and changes in ocean circulation in the Hadley Centre coupled climate model have been investigated using a series of idealised model experiments. In the global mean, sea level rise due to density change is dominated by the thermal expansion of the ocean; including salinity variations in the density calculations has a negligible effect. However, changes in surface freshwater flux resulting from increasing atmospheric greenhouse gas concentrations account for around 8% of the global-mean thermal expansion, by causing the surface heat flux to change. Changes in the surface momentum flux (wind stress) have a negligible effect on simulated global-mean thermal expansion.
 As in previous simulations, sea level change is projected to be spatially inhomogeneous with a range of more than ±100% of the global mean thermal expansion. Significant and distinct contributions to the pattern can be attributed to each of the surface fluxes (heat, freshwater and momentum), with momentum being the most important.
 In the baseline experiment (CO2 increased by 2% per year compounded) the sea level changes can be split, in order of decreasing importance, into baroclinic changes (associated with density variations), barotropic changes (associated with circulation changes) and changes in the pattern of atmospheric pressure over the ocean. The density changes can be further decomposed into temperature and salinity components. The relatively small sea level rise in the Southern Ocean – a feature common to many climate models - is predominantly due to small thermal expansion there. The spatial distribution of density changes is found in general to be influenced as much by salinity variations as by changes in temperature, the two often having opposing local effects on sea level, notably in the North Atlantic. The patterns are complex and further work is required to understand them in terms of ocean transport processes and water mass changes.
 Since the patterns of sea level change from different climate models are not similar in most regions, we recommend performing similar experiments and analyses with other models. In addition, it would be useful to apply the anomalous surface fluxes from one model to the ocean components of others. Taken together, these experiments would allow us to compare the relative importance of atmospheric changes and ocean transport mechanisms in explaining the differences in patterns among models, which is currently an obstacle to making confident projections of local sea level change.
 Consider a mass M of sea water with volume V and salinity S, and a small mass δM of freshwater with volume δV to be added to it. Their total volume is obviously V + δV. When they are mixed, they share the original mass MS of salt, so the new salinity will be S′ = MS/(M + δM) ≈ S(1 − δM/M). For thermal expansion caused by temperature change δT, the volume change is calculated as Vα δT, where α is the thermal expansitivity, i.e., the fractional increase of volume with temperature in a linear equation of state. The global halosteric sea level change of Antonov et al.  is calculated by analogy as Vβ (S′-S), where β is the increase of specific volume with salinity. Since S′ < S and β is negative, the halosteric term is positive. It is the difference in volume between the original M and an equal mass M of the mixed water, the latter being larger because the salinity has been diluted. But this is not the whole story, because after mixing the remaining mass δM also has a different volume from the original δM of freshwater, which had zero salinity. This difference is δV β (S′-0), which is negative. If we add these volume differences together, we obtain the change in volume of the total mass upon mixing, as Vβ (S′-S) + δV β (S′-0) = βS(−V δM/M + δV) = 0 since δV/V = δM/M to first order in small quantities. That is, the “halosteric expansion” of M is cancelled by the “contraction” of δM; the total volume is V + δV still and the sea level change is given by δV alone. To include an additional Vβ (S′-S) term is double-counting.
 This work was supported by the UK Department for the Environment, Food and Rural Affairs under contract PECD7/12/37 and by the Government Meteorological Research and Development Programme. We thank Helene Banks and Robert Thorpe for running several of the HadCM3 model experiments and for useful discussion on the passive anomaly tracers, Jürgen Willebrand for discussion of the halosteric effect, Rémi Tailleux for valuable comments, and the referees for their reviews.