## 1. Introduction

Recently, the Boussinesq approximations in ocean models have attracted much attention (e.g., de Szoeke and Samelson 2002; Greatbatch et al. 2001; Huang and Jin 2002; Huang et al. 2001; Lu 2001; McDougall et al. 2002). The approximations, which are commonly employed for computational efficiency in general circulation models and in analytical studies, consist of replacing (i) mass conservation by volume conservation and (ii) the density in temporal and advection operators by a constant reference density (McDougall et al. 2002).^{1}

While these approximations are generally justified for purposes of simulating the ocean circulation with ocean general circulation models (OGCMs), there are many problems in physical oceanography that may require the use of non-Boussinesq OGCMs. Boussinesq models conserve volume; consequently, they cannot recover steric effects. Hence, unless the steric sea level change is explicitly calculated (Greatbatch 1994), one cannot use such models to study global sea level change due to net heating of the ocean at seasonal and longer time scales. Furthermore, changes in the heat and freshwater content of the ocean can have spurious effects on the diagnosed bottom pressure in OGCMs that make the Boussinesq approximations. For example, heating the water column (and neglecting the subsequent adjustment) decreases the density. By volume conservation, decreasing the density reduces the mass and the bottom pressure, which for a real fluid should be unchanged in this case. Therefore, a volume conserving model may be inappropriate to study oceanic mass distribution and bottom pressure in the context of high precision satellite gravity missions such as the Gravity Recovery and Climate Experiment mission (GRACE; Greatbatch et al. 2001). Reducing the mass by heating from above also has dynamical consequences that can result in different adjustment processes and different sea surface elevations of a Boussinesq and a non-Boussinesq model (Huang and Jin 2002).

According to Huang et al. (2001), the Boussinesq approximations also may introduce erroneous energy sources and energy transformation processes, although this issue is under debate (R. Ferrari and A. Adcroft 2002, personal communication). In the ocean, heating the water column from above raises the sea surface and increases the gravitational potential energy. But in a Boussinesq model, the same heating decreases the mass, does not raise the sea level, and thus reduces the gravitational potential energy. Also, neglecting the compressibility in the continuity equation removes the explicit conversion between mechanical and internal energy from the Boussinesq model. This may have an effect on the energy balance in the Boussinesq equations, although the magnitude of these errors is unclear (Huang et al. 2001).

In this paper, we will argue that all of these errors are at the noise level of a coarse resolution OGCM. In particular, they are comparable to, say, errors due to the hydrostatic approximation and uncertainties associated with model parameterizations because the long integration of an OGCM is sensitive to any small perturbation, even at noneddy-permitting resolution.

McDougall et al. (2002) pointed out that—in addition to replacing mass conservation by volume conservation and the density by a constant reference density in temporal and advection operators—when making the Boussinesq approximations, an error in the tracer equation needs to be considered that results from using a divergence-free velocity as the advecting velocity. Greatbatch et al. (2001) and Lu (2001) each suggested a practical solution for accounting for that error in a conventional Boussinesq model. The two solutions differ only slightly: Lu (2001) reinterpreted the model variables as density-weighted averages of the grid cell and added a correction to the vertical advection term in the momentum and tracer balance equations. This correction term is diagnosed from the original continuity equation, which includes time derivatives in density. Greatbatch et al. (2001) reinterpreted the model velocity variables as average mass flux per area normalized by a constant reference density. They arrived at a set of equations in which the density needs to be stepped forward in time.

In a completely different approach, de Szoeke and Samelson (2002) showed that the non-Boussinesq hydrostatic equations in pressure coordinates have a form that is “dual” to the hydrostatic Boussinesq equations. In this way, only the structure of the boundary conditions in the existing Boussinesq ocean model code needs to be modified. The remaining code can be used without any further modifications, provided one replaces depth, vertical velocity, pressure, and scaled density anomaly by pressure, vertical pseudovelocity, Montgomery potential, and the scaled specific volume anomaly, respectively.

In the Massachusetts Institute of Technology General Circulation Model (MITGCM) [Marshall et al. (1997a); the model code is available at http://mitgcm.org], the dynamical core of the model suggested by de Szoeke and Samelson (2002) is already implemented and used as an atmospheric model [see Marshall et al. (2003, manuscript submitted to *Mon. Wea. Rev.*, hereinafter MACH), where atmosphere–ocean isomorphisms and their implementation in the MITGCM are described]. In fact, the MITGCM was originally motivated by an atmospheric model in pressure coordinates (Brugge et al. 1991). By transferring the atmospheric model to the ocean and replacing the equation of state, a fully non-Boussinesq OGCM in pressure coordinates is readily available. Here, this model is integrated in parallel with the Boussinesq height coordinate mode of the MITGCM and the solutions are compared to yield a quantitative assessment of the differences due to the Boussinesq approximation. The MITGCM can also be run as a nonhydrostatic model, which makes it possible to compare the relative impact of the Boussinesq and the hydrostatic approximations and check the conclusion of de Szoeke and Samelson (2002) that the crucial simplification is the latter one.

## 2. Non-Boussinesq pressure coordinate model and Boussinesq height coordinate model: Making the models comparable

Following de Szoeke and Samelson (2002), the hydrostatic, Boussinesq equations of motion in height coordinates have the same form as the hydrostatic, non-Boussinesq equation in pressure coordinates. To obtain a set of non-Boussinesq equations in pressure coordinates from Boussinesq equations in height coordinates, one only has to substitute pressure for height as the vertical coordinate, a pseudovelocity (which is the rate of change of pressure) for the vertical velocity, the geopotential height for hydrostatic pressure, and specific volume for density. MACH described how this isomorphism of the equations is exploited for modeling of the atmosphere and the ocean with the same dynamical kernel code. The description of the isomorphism and its application to a non-Boussinesq pressure coordinate model of the ocean are summarized in appendix A.

In the following sections, the solution of the non-Boussinesq pressure coordinate model is compared with that of the Boussinesq height coordinate model for a coarse resolution configuration with mixed boundary conditions. Although both models use essentially the same dynamical kernel of the MITGCM, details of the implementation and the parameters of the two models render the direct comparison of the two model difficult. Before the models can be compared, the following issues need to be addressed.

### a. Initialization

Because the vertical grids of the pressure coordinate and height coordinate models are different, the models cannot be restarted from a common spunup equilibrium state. Only a start from rest with flat hydrography provides the identical initial conditions that are necessary for the close comparison presented here. Difficulties in interpolating the hydrography onto pressure levels that implicitly depend on temperature and salinity are avoided by starting both models from uniform temperature and salinity fields, where the constant values of *θ* = 3.6°C and *S* = 34.7 are chosen to be approximately the mean temperature and salinity estimated from the Levitus climatologies (Levitus and Boyer 1994; Levitus et al. 1994).

Pressure is a nonlinear function of depth. Therefore, after choosing the depth levels in the height coordinate model, the pressure levels in the non-Boussinesq model and the initial pressure field in the Boussinesq model are determined by integrating the nonlinear hydrostatic equation *dp* = −*gρ*(*p*)*dz.* In the Boussinesq model, the pressure is evaluated at the end of a time step. Then at the beginning of the next time step, the pressure from the previous time step is used in the equation of state to calculate density. Lagging pressure in this way avoids the nonlinear integration at every time step (Griffies et al. 2001). Using an equation of state in which pressure is computed as *p*(*z*) = −*gρ*_{0}*z* can lead to errors of up to a few Sverdrups (1 Sv ≡ 10^{6} m^{3} s^{−1}) in the Gulf Stream region (Dewar et al. 1998) and is therefore explicitly avoided in our comparison. Huang and Jin (2002) used an equation of state that does depend on height and not on pressure in the height coordinate model, which makes definite comparisons between Boussinesq and non-Boussinesq models problematic. In this study, density is computed as a function of pressure in both height coordinate and pressure coordinate model.

### b. Computation of the potential

*ϕ*(

*ϕ*is pressure divided by

*ρ*

_{0}in height coordinates and geopotential height in pressure coordinates) is computed by integrating the generalized buoyancy

*b*starting from the free surface

*r*

_{s}(

*b*is gravitational acceleration times density in height coordinates and specific volume in pressure coordinates; see appendix A for a further explanation of the notation):

*r*is the general vertical coordinate.

*ϕ*must be zero at the ocean–atmosphere interface in both formulations. This condition is easily met in the height coordinate formulation where the air–sea interface is the free surface and

*ϕ*(

*r*

_{s}) =

*p*(

*z*=

*η*)/

*ρ*

_{0}≡ 0. In the pressure coordinate model, on the other hand, the “free surface”

*r*

_{s}is at the ocean floor and appropriate boundary values

*ϕ*(

*r*

_{s}) at the bottom are needed to ensure that, at rest, the geopotential is flat at the air–sea interface. These boundary values are obtained by integrating the initial

*b*from the free surface at rest

*r*

^{0}

_{s}

*r*

_{s}(

*t*= 0) to the fixed surface

*r*=

*R*

_{fixed}(=0 Pa) at the ocean–atmosphere interface:

### c. Natural boundary conditions for freshwater flux

The response to freshwater forcing is anticipated to be one of the major dynamical differences between a Boussinesq and a non-Boussinesq model (Huang and Jin 2002). Hence, the implementation of the natural boundary conditions for freshwater flux requires great care.

In the height coordinate model, adding freshwater locally reduces the salinity of the top layer, but at the same time increases the height of the free surface via an inhomogeneous term in the free surface equation [Eq. (A9) in appendix A]. Note, that in this work, the freshwater flux is balanced globally over the forcing period (1 yr) so that there is no net flux of freshwater into the ocean.

In pressure coordinates, the flux of freshwater is a mass flux at the surface, also balanced to avoid an overall mass drift. The surface mass fluctuations *gρ*_{FW}(*P* − *E*) change the pressure tendency *ω* = *Dp*/*Dt* by a vertically constant value throughout the entire water column. Consequently, *gρ*_{FW}(*P* − *E*) also appears as a forcing term in the bottom pressure equation (A20) in appendix A, which is effectively a mass equation. The salinity, however, is affected in the surface layer. The derivation of the surface boundary conditions for *ω* in pressure coordinates can be found in appendix B.

### d. Free surface versus bottom pressure gradients in the momentum equations

In the height coordinate model the contribution to the momentum equations of the surface pressure gradient is split into *g*∇*η* + *g*∇[(*ρ* − *ρ*_{0})/*ρ*_{0}]*η.* Often, the second term is neglected on the grounds that (*ρ* − *ρ*_{0})/*ρ*_{0} ≪ 1. But in pressure coordinates and with topography, the corresponding geopotential height gradient at the bottom is evaluated at different pressures. Making the above approximation in pressure coordinates introduces larger errors because the specific volume varies with depth. Therefore, terms of order (*ρ* − *ρ*_{0})/*ρ*_{0} in height coordinates and (*α* − *α*_{0})/*α*_{0} in pressure coordinates (*α* = *ρ*^{−1}) are not neglected in this study, thereby reducing the differences between a height coordinate and a pressure coordinate model.

### e. Turbulent diffusion and viscosity

*A*

^{(z)}

_{V}

**u**is the horizontal velocity vector,

*g*is the acceleration due to gravity, and

*ρ*is the in situ density. Therefore only for constant density

*ρ*=

*ρ*

_{0}, does

*A*

^{(p)}

_{V}

*A*

^{(z)}

_{V}

*g*

^{2}

*ρ*

^{2}

_{0}

*A*

^{(p)}

_{V}

*κ*

^{(p)}

_{V}

*ρ*

_{0}= 1035 kg m

^{−3}. Thus, the vertical eddy viscosity and diffusivity coefficients should be scaled by (

*ρ*/

*ρ*

_{0})

^{2}to reduce this error but are not scaled so here. There is a further discrepancy in the lateral diffusion and viscosity terms; in the height coordinate model lateral eddy fluxes are directed downgradient along the horizontal. In pressure coordinates the fluxes are calculated along isobaric surfaces which may be inclined. But the slope of isobaric surfaces is at most of the order 10

^{−4}and this effect is neglected as well. Last, implementing the Gent–McWilliams and Redi schemes (Gent and McWilliams 1990) in pressure coordinates would have incurred the same difficulties, and so we chose to not employ these schemes.

## 3. Comparison of OGCM results

### a. Model parameters and configuration

Both models are integrated for 1000 years. The horizontal resolution is 4°, ranging from 80°S to 80°N with 15 vertical levels. The bottom topography is realistic and derived from ETOPO5 (NOAA 1988). The level thickness ranges from 50 to 690 m in the height coordinate model. Monthly mean wind stress fields by Trenberth et al. (1990) and monthly mean heat flux and climatological freshwater flux by Jiang et al. (1999) force the models at the surface. The surface layer of thickness 50 m is also restored with a time scale of 2 months to monthly mean sea surface temperature (Levitus et al. 1994) to represent an oceanic feedback on the actual heat flux. Basic parameters of both models are summarized in Table 1. The comparison between the pressure coordinate and height coordinate model is carried out by showing the differences of time-averaged fields. Unless indicated otherwise, the averages are taken over the last 100 years of the integration.

The two models operate on different vertical grids. Hence, direct comparison of model variables will in most cases involve the interpolation from one grid to the other, introducing another possible, albeit small, source of differences between the results. To estimate the interpolation error, we took the pressure model state and interpolated it to the height model grid and then back again using linear interpolation. We found the resulting differences (not shown) to be of order 0.01°C and, although these are small, they are still an encumbrance considering the size of the signal (order 0.05°C) that we are expecting to find. The only model variables that evade this interpolation problem are bottom pressure and sea surface elevation. Bottom pressure is a prognostic variable in the pressure coordinate model, but must be diagnosed from the height coordinate model. In contrast, sea surface elevation has to be diagnosed in the pressure coordinate model, but is a prognostic variable in the height coordinate model. Furthermore, the mean bottom pressure in the height coordinate model may contain a false mass drift in time, whereas the mean sea surface elevation in the pressure coordinate model may drift because of steric expansion.^{2} Figure 1 shows the time evolution of the globally averaged bottom pressure of the height coordinate model and the globally averaged sea surface height of the pressure coordinate model. The evolution of the global mean of bottom pressure in the height coordinate model in Fig. 1 is represented in height units after scaling by *gρ*_{0} and reversing the sign. Note that despite the approximate scaling (one could use the vertically averaged in situ density instead of *ρ*_{0}), the mass evolution in the height coordinate model and the volume evolution in the pressure coordinate model are remarkably similar. We thereby confirm the conclusion of Greatbatch (1994) and Mellor and Ezer (1995) who showed that one can recover steric effects by adjusting the sea level by a globally uniform, time-varying correction.

### b. What is the magnitude of the differences?

We now compare sea surface elevations and bottom pressure anomalies (differences from the long time mean). For this comparison, both the time-dependent global averages of the total mass of the height coordinate model and the volume of the pressure coordinate model have been removed.

To gauge how important the observed differences between the Boussinesq and the non-Boussinesq model are, we compare the height coordinate model with the same model with small changes to the dynamics, the parameterizations, and the forcing fields. The most prominent change is the relaxation of the hydrostatic approximation to include the vertical Coriolis terms and nonhydrostatic metric terms that include the vertical velocity *w.* (See appendix D for details.) With these terms, which are generally of the order of 10^{−5} m^{2} s^{−1}, the model is not fully nonhydrostatic, but it has a consistent energy conservation principle. Marshall et al. (1997b) called this model “quasi-hydrostatic.”

We also compare two hydrostatic Boussinesq models that differ only in the choice of the vertical diffusivity parameter for temperature and salinity. This parameter is often tuned to bring the model close to the observations. Therefore it can have values that vary dramatically from one application to another. As an additional test, we compare the results of the height–coordinate model with different implementations of the equation of state and perturbations of the forcing fields at the level of numerical round-off errors.

The mean sea surface of the Boussinesq model is shown in the top panel of Fig. 2. The difference in mean sea surface elevation between the Boussinesq and non-Boussinesq models in the second panel of Fig. 2 reaches 4 cm in the Southern Ocean, otherwise it is small. Such differences will just be detectable when the new high precision geoid models become available, which are expected from the ongoing gravity mission GRACE. The third panel of Fig. 2 shows the difference in mean sea surface height due to quasi-hydrostatic terms. This effect is approximately one-half of the Boussinesq effects. The bottom panel of Fig. 2 compares the Boussinesq model with an experiment in which the vertical diffusivity has been increased by 1% of the standard value. Apparently, this change in diffusivity leads to similar, if not greater, changes in mean sea surface elevation as do the Boussinesq effects.

In general, the differences due to Boussinesq, hydrostatic, or other effects are largest in the Southern Ocean. This region is characterized by large horizontal density gradients and steep isopycnal slopes. Small changes in these large gradients may explain the observed sensitivity to small perturbations. Since these calculations do not have the Gent–McWilliams parameterization (GM), there is more convective activity in the Southern Ocean (ACC) than is normally seen in models with GM. The highly nonlinear nature of the convection parameterization may also be contributing to the sensitivity of the Southern Ocean in our results.

The Ocean Topography Experiment (TOPEX)/Poseidon altimeter mission provided oceanographers with sea surface height anomaly data that have an accuracy of the order of 2 cm (Wunsch and Stammer 1998). The accuracy of these data may serve as the benchmark for the difference in sea surface variability. The square root of the variance of the sea surface height over 100 model years is shown in the top panel of Fig. 3 for the Boussinesq model. The difference in sea surface height variability between Boussinesq and non-Boussinesq model (second panel of Fig. 3) is smaller than that due to quasi-hydrostatic terms (third panel of Fig. 3). It is hardly detectable with today's high precision altimetry.

All experiments so far use the polynomial equation of state derived by Jackett and McDougall (1995) from the United Nations Educational, Scientific, and Cultural Organization (UNESCO) formula. The bottom panel of Fig. 3 shows the difference due to a different equation of state, namely the polynomial published by McDougall et al. (2003). While the difference in the density computed by these different formulas is of the order of 10^{−3} kg m^{−3}, it still causes differences in sea surface variability of the same order as those due to Boussinesq effects.

Bottom pressure gauges and the satellite mission GRACE produce measurements of the bottom pressure fluctuations. We therefore compare the temporal variations of bottom pressure in the models. The top panel of Fig. 4 shows the square root of the bottom pressure variance over a period of 100 years for the Boussinesq model (with the global mean subtracted at every time step). The difference in the bottom pressure variability between the Boussinesq model and the non-Boussinesq model (second panel of Fig. 4) is on the order of 1-cm equivalent sea surface height in the Southern Ocean and much smaller everywhere else. This difference is as large as 30% of the signal and is therefore not negligible. However, the difference in bottom pressure variability due to the quasi-hydrostatic effects (third panel of Fig. 4) has a still larger amplitude.

To assess the extent to which the above responses are an innate property of the system (i.e., does any small perturbation lead to large changes?), the height coordinate model run is repeated with slightly different forcing fields: all forcing fields are perturbed by random noise with a relative amplitude of 2.22 × 10^{−16}. This amounts to changing the last digit of a 64-bit (double precision) real number and simulates the differences in round-off error that one encounters when changing compilers or computing platforms. The bottom panel of Fig. 4 shows the resulting differences in bottom pressure variability. As with the use of a different equation of state and a perturbed vertical diffusivity, the effect of these tiny perturbations on the variability is of the same order of magnitude as the Boussinesq effects. This again suggests, that our model is particularly sensitive in the Southern Ocean, where the flow is geostrophically balanced by large density gradients.

The most likely explanation for the observed sensitivity is the choice of mixed boundary conditions. The combination of restoring conditions for temperature and flux boundary conditions for salinity has been reported to make OGCMs unstable (Power and Kleeman 1994; Rahmstorf 1996). Repeating the above experiments with restoring boundary conditions for both temperature and salinity constrains the height coordinate and the pressure coordinate model to stay on very similar trajectories. However, this does not change the qualitative result: when the models are restored to the same surface fields, the solutions of the various models become more similar and all differences are reduced by one order of magnitude. But the relative sizes of the differences remain the same (not shown).

Figure 5 summarizes the previous comparisons as a function of scale. Plotted is the square root of the degree variances _{m} |*c*_{nm}|^{2}*c*_{nm}. To bring the experiments into the context of the current high precision satellite gravity mission GRACE, the expected error of a geoid model that will be derived from the GRACE gravity data (Balmino et al. 1998) is also drawn as a function of spherical harmonic degree. (It is assumed that the bottom pressure measurement errors, which are unknown at this time, are comparable to the errors of the mean geoid. In this sense, the geoid errors provide only a rough estimate of the measurement errors.) All differences in bottom pressure variability are larger than the geoid errors on the very large scales; degree 10 corresponds to a wavelength of approximately 4000 km, degree 15 to approximately 2670 km. On shorter scales the geoid errors dominate. For most coefficients, that is spatial scales, the differences due to Boussinesq effects are smaller than those due to both nonhydrostatic effects and round-off noise in the forcing fields.

Last, we found that the differences in the interior between experiments are small (not shown). For example, the global root-mean-square (rms) differences between the height coordinate and the pressure coordinate model is 0.08°C, the differences due to hydrostatic effects, different vertical viscosity, different equation of state, and perturbed forcing are 0.03°, 0.05°, 0.04°, and 0.05°C, respectively. Note that to calculate the rms temperature difference between Boussinesq and non-Boussinesq models interpolation is used, which incurs an error of at least 0.01°C in the interpolated temperature and which accounts for a significant portion of the rms difference. Also the seasonal variance of temperature has an rms value of order 0.5°C. With this in mind, we consider all the rms differences between the models to be essentially of the same magnitude. But there is a slight indication that the Boussinesq approximations might affect the temperature distribution more than the other small effects.

In summary, comparison of solutions obtained using Boussinesq and non-Boussinesq models shows significant differences in the time mean and variance of surface elevation and in the variance of bottom pressure. Comparison of the hydrostatic model to the quasi-hydrostatic model indicates larger differences than those due to the non-Boussinesq approximations in sea-surface height and bottom-pressure variances though a considerably smaller change in mean sea surface height. Thus the relative effect of the two approximations seems comparable. Changing the vertical diffusivity by 1% (actually, we know that both vertical viscosity and diffusion can potentially vary by 2–3 orders of magnitude) the form of the equation of state, or adding truncation level noise to the forcing leads to equally large changes.

## 4. Are the differences between height coordinate and pressure coordinate models really due to the Boussinesq approximations?

After showing that the differences between the Boussinesq and the non-Boussinesq models are generally of the same order of magnitude as those due to relaxing the hydrostatic approximation or changing uncertain parameters slightly, it is still not clear whether the observed differences between the models are really due to the Boussinesq approximation or simply due to the numerical difference introduced by the different coordinate systems. These may arise because in a physically identical fluid the gradient operators act along different surfaces (pressure vs height surfaces) so that numerical truncation may lead to different trajectories. The nonlinear free surface causes an additional numerical difference: in the height coordinate model the surface layer has a variable thickness, while in the pressure coordinate model it is the bottom layer that can vary in time and space (Campin et al. 2003).

To test how much of the differences between the height coordinate model and the pressure coordinate model are due to the numerical discretization, any dependence on the Boussinesq approximations is removed by replacing the pressure dependent density with a constant in both the height coordinate model and the pressure coordinate model. The system is forced by wind stress only, and there is no buoyancy flux. Then any pressure level inclination is solely barotropic and due to sea surface elevation changes.

Only two differences remain between the pressure coordinate and the height coordinate model: changes in layer thickness due to the nonlinear free surface (at the bottom in pressure coordinates and at the top in the height coordinate model) and the way the horizontal gradients in the nonlinear advection terms and the horizontal viscosity terms are evaluated (strictly along pressure surfaces and along horizontal surfaces, which form a small angle with the pressure surfaces). Note that in this configuration, *A*^{(p)}_{V}*A*^{(z)}_{V}*g*^{2}*ρ*^{2}_{0}

Figure 6 illustrates the contributions of the effects due to the nonlinear free surface. Shown is the difference in sea surface height variability. Because of its barotropic nature the model equilibrates very quickly and the averaging period is the 10th year of integration. Although both the height coordinate and the pressure coordinate model describe the same fluid of constant density, the answers are slightly different due to the different truncation errors implied by the different formulation (top panel of Fig. 6). The difference between pressure and height coordinate model is three orders of magnitude smaller than in the full model of section 3. Replacing the nonlinear free surface by a rigid lid (bottom panel of Fig. 6), decreases the difference in sea surface height variability further by two orders of magnitude (and the differences in mean sea surface elevation and bottom pressure variability as well, not shown).

The remaining differences in sea surface height variability are largely due to the fact that sea surface elevation in the pressure coordinate model is a diagnostic variable, whereas it is prognostic in the height coordinate model. In a barotropic model, sea surface height and bottom pressure are equivalent. Comparing the sea surface height of the height coordinate model and the bottom pressure anomaly of the pressure coordinate model reveals that, after proper scaling, they are the same in the case of a rigid lid to working precision. The differences in horizontal velocities are also smaller than 10^{−14} for the barotropic experiment with rigid lid, and the two models give essentially the same result. That is, the angles between surfaces of constant height and surfaces of constant pressure are so small that the horizontal gradients do not cause any differences that exceed the noise level. We can assume that in the case of variable density, the differences in numerical discretization are also mainly due to the nonlinear free surface. Although there may be a larger impact of the free surface in the full baroclinic model because of the vertical advection of buoyancy near the surface, the numerical effects due to different coordinate systems are still much smaller than the Boussinesq effects.

## 5. Discussion and conclusions

Conventional OGCMs make a number of approximations that influence their solution, such as the hydrostatic approximation and the Boussinesq approximations. We find that relaxing the hydrostatic approximation has a larger impact on the variability of a coarse resolution global model than do Boussinesq effects. Non-Boussinesq effects lead to larger changes in mean sea surface elevation than do nonhydrostatic effects and this is consistent with switching from conserving volume to conserving mass, as discussed in Huang and Jin (2002). Note, however, that we have not used the MITGCM in a fully nonhydrostatic mode. The fully nonhydrostatic model involves an additional algorithmic step to compute the nonhydrostatic pressure contribution, which is complicated when using a nonlinear free surface. The changed algorithm represents a further perturbation and will surely lead to additional deviations from the solution of the hydrostatic model.

There is also convincing evidence that the changes due to Boussinesq effects are smaller than the errors introduced by other approximations and parameterizations generally made by ocean general circulation models.

Greatbatch et al. (2001) showed that the differences between a Boussinesq model and a non-Boussinesq model are reassuringly on the order of a few percent in the mean fields. Here we confirm that the differences between the non-Boussinesq pressure coordinate model and the Boussinesq height model are small in the time mean. But even the coarse resolution experiments are sensitive to small differences in the dynamics or parameterizations. These small perturbations lead to detectable changes in the computed circulation, particularly in the variability of bottom pressure and sea surface height.

It is particularly interesting to point out that all results shown here were obtained by using the same compiler on the same platform. Use of a different compiler or platform changes the results due to differing numerical floating point operations, and gives an additional clue about the order of magnitude of errors the Boussinesq approximations introduce: the numerical noise level. This conclusion is supported by the experiment that shows the differences due to round-off noise in the forcing fields. Such differences are as large as those due to Boussinesq effects.

McDougall et al. (2002) showed that the steady-state geostrophic Boussinesq equations are equivalent to the non-Boussinesq equations under the same dynamical regime. Therefore, at the coarse resolution used here, we can expect only small differences between a Boussinesq and a non-Boussinesq ocean model because much of the simulation should be geostrophically balanced. With increasing realism of the simulations (increased resolution, higher frequencies, etc.) that leads to sufficient ageostrophic and time-dependent motion, one expects the Boussinesq effects to become more important. But at the same time, hydrostatic effects will become more prominent as well. Further, the model solutions of chaotic (eddy resolving) systems will be highly sensitive to noise in the initial fields and the boundary conditions. Whether the Boussinesq effects will be discernible under these circumstances remains questionable.

Still, models that do not make the Boussinesq approximation are recommended over the conventional height coordinate models when they are not more expensive to integrate. This point of view is taken for the puristic reason of abandoning an approximation. If, however, there is a considerable additional effort involved in integrating these mass conserving models, the minor effects of the Boussinesq approximation do not justify that cost. Instead, efforts to improve the parameterization of unresolved physics such as mixing will be more beneficial to OGCM solutions that relaxing the Boussinesq approximations.

One of the conclusions of this study is only indirectly related to the Boussinesq approximations. In the configuration of this study with mixed boundary conditions for temperature and salinity, the OGCM yields robust results on the large scale. But at the same time, it is so sensitive to small, in fact, almost indetectable changes in the parameterizations, largely unknown forcing fields, or approximations made that one cannot expect two models that use a different coordinate system to stay on almost identical trajectories for an infinite time (of integration). The results presented here possibly depend on this detail. It is particularly important to compare the model variances while the two models are on a similar trajectory. Once the models have diverged, there is little hope of recovering the results shown.

When the property of exact mass conservation is required for a particular study, non-Boussinesq models must be preferred. Also when working with hydrographic data, a pressure coordinate model seems to have the advantage that the data are originally measured in that coordinate. However, current Boussinesq models in height coordinates still have their benefits. For instance, at eddy resolving scales, nonhydrostatic effects are believed to be important, and a nonhydrostatic model is much easier to formulate in height coordinates than in pressure coordinates. [On this note, the approach by Greatbatch et al. (2001) may allow the inclusion of non-Boussinesq effects into the nonhydrostatic version of the MITGCM and make the comparison of the different height coordinate models simpler. However, we suspect that the Greatbatch et al. approach also introduces artificial differences due to the different structure of the equations—new terms—while the height-pressure isomorphic equations have the same structure.] Also, the representation of atmospheric pressure load in a pressure coordinate model is slightly more complicated than in a height coordinate model because it involves two moving surfaces. Further, while the representation of bottom pressure in the pressure coordinate model is more natural, the bottom pressure diagnosed from the height coordinate model appears accurate enough in the light of other approximations made, namely, the hydrostatic approximation.

Constraining pressure coordinate OGCMs with bottom pressure data is more straightforward in the same way that height coordinate models are more easily constrained by sea surface elevation data. Although we found high sensitivities to dynamical formulation under mixed boundary conditions, constrained models may show less sensitivity, as was the case with restoring boundary conditions. If this is the case, then concerns about assimilation of bottom pressure even in height coordinate OGCMs are unwarranted, particularly since we have demonstrated that the evolution of net ocean mass can be quite accurately accounted for in such a model.

## Acknowledgments

The authors thank Carl Wunsch and John Marshall for initiating this work and for many valuable comments. Raffaele Ferrari helped to improve the manuscript. We also thank Steve Griffies and one anonymous reviewer for their helpful comments. This work is a contribution to the ECCO (Estimating the Circulation and Climate of the Ocean) project. Author ML was funded by JPL/NASA Contract 1205624; AA was funded by ONR/NOPP N00011-99-1-1050. JMC was funded by the MIT Climate Modeling Initiative.

## REFERENCES

Adcroft, A., C. Hill, and J. Marshall, 1997: Representation of topography by shaved cells in a height coordinate ocean model.

,*Mon. Wea. Rev.***125****,**2293–2315.Balmino, G., F. Perosanz, R. Rummel, N. Sneeuw, H. Sünkel, and P. Woodworth, 1998: European views on dedicated gravity field missions: GRACE and GOCE, An Earth Sciences Division Consultation Document. ESA, ESD-MAG-REP-CON-001, 66 pp.

Brugge, R., H. L. Jones, and J. C. Marshall, 1991: Non-hydrostatic ocean modeling for studies of open-ocean deep convection.

*Deep Convection and Deep Water Formation in the Ocean,*P. C. Chu and J. C. Gascard, Eds., Elsevier Oceanography Series, Vol. 57, Elsevier, 325–340.Campin, J-M., A. Adcroft, C. Hill, and J. Marshall, 2003: Conservation of properties in a free-surface model.

*Ocean Modell.,*in press.de Szoeke, R. A., and R. M. Samelson, 2002: The duality between the Boussinesq and non-Boussinesq hydrostatic equations of motion.

,*J. Phys. Oceanogr.***32****,**2194–2203.Dewar, W. K., Y. Hsueh, T. J. McDougall, and D. Yuan, 1998: Calculation of pressure in ocean simulations.

,*J. Phys. Oceanogr.***28****,**577–588.Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models.

,*J. Phys. Oceanogr.***20****,**150–155.Greatbatch, R. J., 1994: A note on the representation of steric sea level in models that conserve volume rather than mass.

,*J. Geophys. Res.***99**(C6) 12767–12771.Greatbatch, R. J., Y. Lu, and Y. Cai, 2001: Relaxing the Boussinesq approximation in ocean circulation models.

,*J. Atmos. Oceanic Technol.***18****,**1911–1923.Griffies, S. M., R. C. Pacanowski, M. Schmidt, and V. Balaji, 2001: Tracer conservation with an explicit free surface method for

*z*-coordinate models.,*Mon. Wea. Rev.***129****,**1081–1098.Haltiner, G. J., and R. T. Williams, 1980:

*Numerical Prediction and Dynamical Meteorology*. 2d ed. John Wiley and Sons, 477 pp.Huang, R. X., and X. Jin, 2002: Sea surface elevation and bottom pressure anomalies due to thermohaline forcing. Part I: Isolated perturbations.

,*J. Phys. Oceanogr.***32****,**2131–2150.Huang, R. X., X. Jin, and X. Zhang, 2001: An oceanic general circulation model in pressure coordinates.

,*Adv. Atmos. Sci.***18****,**1–22.Jackett, D. R., and T. J. McDougall, 1995: Minimal adjustment of hydrographic profiles to achieve static stability.

,*J. Atmos. Oceanic Technol.***12****,**381–389.Jiang, S., P. H. Stone, and P. Malanotte-Rizzoli, 1999: An assessment of the Geophysical Fluid Dynamics Laboratory ocean model with coarse resolution: Annual-mean climatology.

,*J. Geophys. Res.***104**(C11) 25623–25645.Levitus, S., and T. Boyer, 1994:

*Temperature*. Vol. 4,*World Ocean Atlas 1994,*NOAA Atlas NEDSIS 4, 117 pp.Levitus, S., R. Burgett, and T. Boyer, 1994:

*Salinity*. Vol. 3,*World Ocean Atlas 1994,*NOAA Atlas NEDSIS 3, 99 pp.Lu, Y., 2001: Including non-Boussinesq effects in Boussinesq ocean circulation models.

,*J. Phys. Oceanogr.***31****,**1616–1622.Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997a: A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers.

,*J. Geophys. Res.***102**(C3) 5753–5766.Marshall, J., C. Hill, L. Perelman, and A. Adcroft, 1997b: Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling.

,*J. Geophys. Res.***102**(C3) 5733–5752.McDougall, T. J., R. J. Greatbatch, and Y. Lu, 2002: On conservation equations in oceanography: How accurate are Boussinesq ocean models?

,*J. Phys. Oceanogr.***32****,**1574–1584.McDougall, T. J., D. R. Jackett, D. G. Wright, and R. Fiestel, 2003: Accurate and computationally efficient algorithms for potential temperature and density of seawater.

,*J. Atmos. Oceanic Technol.***20****,**730–741.Mellor, G. L., and T. Ezer, 1995: Sea level variations induced by heating and cooling: An evaluation of the Boussinesq approximation in ocean models.

,*J. Geophys. Res.***100**(C10) 20565–20577.NOAA, 1988: Digital relief of the surface of the earth. National Geophysical Data Center Data Announcement 88-MGG-02, Boulder, CO.

Power, S. B., and R. Kleeman, 1994: Surface heat flux parameterization and the response of ocean general circulation models to high-latitude freshening.

,*Tellus***46A****,**86–95.Rahmstorf, S., 1996: Comments on “Instability of the thermohaline circulation with respect to mixed boundary conditions: Is it really a problem for realistic models?.”.

,*J. Phys. Oceanogr.***26****,**1099–1105.Roullet, G., and G. Madec, 2000: Salt conservation, free surface, and varying levels: A new formulation for ocean general circulation models.

,*J. Geophys. Res.***105**(C10) 23927–23942.Spiegel, E. A., and G. Veronis, 1960: On the Boussinesq approximation for a compressible fluid.

,*Astrophys. J.***131****,**442–447.Trenberth, K. M., J. G. Olson, and W. G. Large, 1990: The mean annual cycle in global ocean wind stress.

,*J. Phys. Oceanogr.***20****,**1742–1760.Wunsch, C., and D. Stammer, 1998: Satellite altimetry, the marine geoid, and the oceanic general circulation.

,*Annu. Rev. Earth Planet. Sci.***26****,**219–253.

## APPENDIX A

### Isomorphic Pressure and Height Coordinate Formulation of the Primitive Equations

To familiarize the reader with an ocean model in pressure coordinates, the symmetry between the pressure and the height model formulations as pointed out by de Szoeke and Samelson (2002) and MACH is reviewed.

#### Boussinesq hydrostatic ocean model in height coordinates

*z*coordinates after making the Boussinesq approximation can be cast as

**u**is the horizontal velocity and

*w*is the vertical velocity;

*p*is the full hydrostatic pressure;

**F**is the frictional force;

*Q*and

*Q*

_{S}represent the source terms of potential temperature

*θ*and salinity

*S,*for example, atmospheric fluxes of heat and freshwater, respectively. The subscript of ∇

_{z}indicates that the gradient is taken along surfaces of constant

*z*; the substantial time derivative (rate of change) operator is

*z.*At the free surface,

*z*=

*η*(

*x,*

*y,*

*t*), we assume that the pressure is constant (taken to be zero), which leads to the boundary conditions

*P*−

*E*) is the freshwater flux into the ocean (precipitation minus evaporation).

*η*is obtained by integrating (A3) and applying (A7) and (A8):

_{z}· (

**u**

*η*) in the surface boundary condition for

*w*(Roullet and Madec 2000).

#### Non-Boussinesq hydrostatic ocean model in pressure coordinates

*gz*is the geopotential and ∇

_{p}is the gradient along pressure surfaces; the substantial time derivative (rate of change) operator is

*ω,*the pressure tendency, is defined by

*α*is given by the equation of state

*ρ*

^{−1}

*α*

*α*

*S,*

*θ,*

*p*

At this point, it is interesting to point out that the form of the continuity equation (A12), albeit resembling the incompressibility statement of Boussinesq models in *z* coordinates, is a consequence of the hydrostatic assumption (A11) and does not require the neglect of dilation *ρ*^{−1}*Dρ*/*Dt* (de Szoeke and Samelson 2002). Hence, a non-Boussinesq model with continuity equation (A12) conserves mass in contrast to a Boussinesq model, which conserves volume.

*ω*(see appendix B). The flux is scaled by the gravitational acceleration

*g*and the density of freshwater

*ρ*

_{FW}= 999.8 kg m

^{−3}:

*z*= −

*H*(

*x,*

*y*), the boundary condition is expressed in terms of bottom pressure

*p*

_{b}(

*x,*

*y,*

*t*) as

#### Symmetry of the primitive equations and the isomorphism in the MITGCM

*r*coordinates:

*r*replaces

*p*and

*z,*the vertical velocities

*ω*and

*w*become

*ṙ*=

*Dr*/

*Dt,*the general potential

*ϕ*takes the place of Φ and

*p*/

*ρ*

_{0}, and the generalized buoyancy

*b*is substituted for the specific volume −

*α*and the scaled density −

*gρ*/

*ρ*

_{0}. The substantial time derivative (rate of change) operator in this formulation is

*r.*

*r*=

*r*

_{s}is the free surface;

*r*

_{s}is the bottom pressure

*p*

_{b}in pressure coordinates and the sea surface elevation

*η*in height coordinates; and

*r*= −

*R*

_{fixed}is the topography of the fixed surface, that is, 0 in pressure coordinates and depth

*H*in height coordinates. So with the additional substitutions for the boundary conditions

*γ*(

*P*−

*E*) has to be added to the boundary condition (A28) or (A29), whichever is the one at the ocean–atmosphere interface. The scale factor

*γ*converts the freshwater flux into pressure or height units, that is, mass flux (

*γ*=

*gρ*

_{FW}) or volume flux (

*γ*= −1), respectively. The prognostic equation for the free surface

*r*

_{s}is again the same for both coordinate systems:

## APPENDIX B

### Natural Boundary Conditions in Pressure Coordinates

*ω*in pressure coordinates (or

*w*in height coordinates) has to be modified. Without a freshwater flux,

*ω*= 0 at the surface. Including the flux leads to the boundary condition (A18), which may appear inconsistent at first, because the surface

*p*= 0 is a coordinate plane. Integrating the non-Boussinesq, hydrostatic continuity equation in height coordinates for

*w*over the top pressure layer with thickness Δ

*p*yields

*η*and −

*z*

_{1}are the respective heights of the surface pressure and the base of the layer. Multiplying through with

*g,*using Leibniz's rule, and using the boundary condition (A7) for

*w,*we get

_{p}

*η*and ∇

_{p}

*z*

_{1}are gradients along surfaces of constant pressure. With the layer-averaged velocity

**u**

*p*is constant (∂Δ

*p*/∂

*t*= 0),

*p*→ 0, one recovers the differential form, and the implied boundary condition for

*ω*becomes

## APPENDIX C

### Finite-Volume Discretization of the Hydrostatic Equation

*k*be the index of the grid point at the center between the vertical cell interfaces

*k*− 1/2 and

*k*+ 1/2. If the distance between these vertical cell interfaces is called Δ

*r*

_{k}, a finite-difference discretization of the hydrostatic equation (A23) that yields the potential

*ϕ*at the center between two grid planes is

*ϕ*

_{k+1/2}

*ϕ*

_{k−1/2}

*r*

_{k}

*b*

_{k}

## APPENDIX D

### Quasi-Hydrostatic Equations

*D*/

*Dt*= ∂/∂

*t*+

**v**· ∇. Here,

**v**= (

*u,*

*υ,*

*w*) is the three-dimensional velocity and

*R*is the radius of the earth, and

*ϕ*is the latitude;

*F*

_{u,υ,w}stand for all friction terms. The so-called metric terms are the terms in the first set of braces after the equal sign in all equations; the Coriolis terms are in the second set of braces after the equal signs. In the quasi-hydrostatic approximation the terms that are underlined twice are dropped. Making the full hydrostatic approximation amounts to dropping the total time derivative and the friction term

*F*

_{w}in the vertical momentum equation and all terms involving

*w*or cos

*ϕ*in the horizontal equations, that is, all terms that are underlined once or twice (Marshall et al. 1997b). Note that for the quasi-hydrostatic approximations the classical hydrostatic equation is augmented by a metric term and a Coriolis term.

(top) Mean sea surface elevation (cm) of the hydrostatic, Boussinesq model; contour interval is 20 cm. (second) Sea surface height difference due to Boussinesq effects; contour interval is 1 cm. (third) Sea surface height differences due to quasi-hydrostatic effects; contour interval is 1 cm. (bottom) Difference in sea surface height variability due to a change of 1% in vertical diffusivity; contour interval is 1 cm

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

(top) Mean sea surface elevation (cm) of the hydrostatic, Boussinesq model; contour interval is 20 cm. (second) Sea surface height difference due to Boussinesq effects; contour interval is 1 cm. (third) Sea surface height differences due to quasi-hydrostatic effects; contour interval is 1 cm. (bottom) Difference in sea surface height variability due to a change of 1% in vertical diffusivity; contour interval is 1 cm

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

(top) Mean sea surface elevation (cm) of the hydrostatic, Boussinesq model; contour interval is 20 cm. (second) Sea surface height difference due to Boussinesq effects; contour interval is 1 cm. (third) Sea surface height differences due to quasi-hydrostatic effects; contour interval is 1 cm. (bottom) Difference in sea surface height variability due to a change of 1% in vertical diffusivity; contour interval is 1 cm

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

(top) Sea surface height variability (square root of the variance in meters); contour interval is 1 cm. (second) Difference in sea surface height variability due to Boussinesq effects; contour interval is 0.5 cm. (third) Difference in sea surface height variability due to quasi-hydrostatic effects; contour interval is 0.5 cm. (bottom) Sea surface height differences due to a changed equation of state; contour interval is 0.5 cm

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

(top) Sea surface height variability (square root of the variance in meters); contour interval is 1 cm. (second) Difference in sea surface height variability due to Boussinesq effects; contour interval is 0.5 cm. (third) Difference in sea surface height variability due to quasi-hydrostatic effects; contour interval is 0.5 cm. (bottom) Sea surface height differences due to a changed equation of state; contour interval is 0.5 cm

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

(top) Sea surface height variability (square root of the variance in meters); contour interval is 1 cm. (second) Difference in sea surface height variability due to Boussinesq effects; contour interval is 0.5 cm. (third) Difference in sea surface height variability due to quasi-hydrostatic effects; contour interval is 0.5 cm. (bottom) Sea surface height differences due to a changed equation of state; contour interval is 0.5 cm

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

(top) Bottom pressure variability (square root of the variance); contour interval is 40 Pa ≈ 4 mm equivalent sea surface height. (second) Difference in bottom pressure variability due to Boussinesq effects; contour interval is 10 Pa ≈ 1 mm. (third) Difference in bottom pressure variability due to quasi-hydrostatic effects; contour interval is 10 Pa ≈ 1 mm. (bottom) Difference in bottom pressure variability due to numerical noise in the forcing fields; contour interval is 10 Pa ≈ 1 mm

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

(top) Bottom pressure variability (square root of the variance); contour interval is 40 Pa ≈ 4 mm equivalent sea surface height. (second) Difference in bottom pressure variability due to Boussinesq effects; contour interval is 10 Pa ≈ 1 mm. (third) Difference in bottom pressure variability due to quasi-hydrostatic effects; contour interval is 10 Pa ≈ 1 mm. (bottom) Difference in bottom pressure variability due to numerical noise in the forcing fields; contour interval is 10 Pa ≈ 1 mm

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

(top) Bottom pressure variability (square root of the variance); contour interval is 40 Pa ≈ 4 mm equivalent sea surface height. (second) Difference in bottom pressure variability due to Boussinesq effects; contour interval is 10 Pa ≈ 1 mm. (third) Difference in bottom pressure variability due to quasi-hydrostatic effects; contour interval is 10 Pa ≈ 1 mm. (bottom) Difference in bottom pressure variability due to numerical noise in the forcing fields; contour interval is 10 Pa ≈ 1 mm

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

The difference in bottom pressure variability as a function of scale. Shown is the square root of the degree variances _{m} |*c*_{nm}|^{2}*c*_{nm}. All approximations and errors give rise to differences in bottom pressure variability that exceed the estimated errors of a geoid derived from GRACE (Balmino et al. 1998) at large scales. However, the hydrostatic approximation (QH), small changes in the vertical diffusivity (KAPPA), small differences in the equation of state (EOS), and numerical noise in the forcing fields (NOISE) seem to be as important as Boussinesq effects (NB)

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

The difference in bottom pressure variability as a function of scale. Shown is the square root of the degree variances _{m} |*c*_{nm}|^{2}*c*_{nm}. All approximations and errors give rise to differences in bottom pressure variability that exceed the estimated errors of a geoid derived from GRACE (Balmino et al. 1998) at large scales. However, the hydrostatic approximation (QH), small changes in the vertical diffusivity (KAPPA), small differences in the equation of state (EOS), and numerical noise in the forcing fields (NOISE) seem to be as important as Boussinesq effects (NB)

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

The difference in bottom pressure variability as a function of scale. Shown is the square root of the degree variances _{m} |*c*_{nm}|^{2}*c*_{nm}. All approximations and errors give rise to differences in bottom pressure variability that exceed the estimated errors of a geoid derived from GRACE (Balmino et al. 1998) at large scales. However, the hydrostatic approximation (QH), small changes in the vertical diffusivity (KAPPA), small differences in the equation of state (EOS), and numerical noise in the forcing fields (NOISE) seem to be as important as Boussinesq effects (NB)

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

Difference of the square root of the sea surface variance over an averaging period of 1 yr for the model with constant density: (top) with nonlinear free surface (contour interval is 1 × 10^{−6} m); and (bottom) with rigid lid (contour interval is 0.02 × 10^{−6} m)

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

Difference of the square root of the sea surface variance over an averaging period of 1 yr for the model with constant density: (top) with nonlinear free surface (contour interval is 1 × 10^{−6} m); and (bottom) with rigid lid (contour interval is 0.02 × 10^{−6} m)

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

Difference of the square root of the sea surface variance over an averaging period of 1 yr for the model with constant density: (top) with nonlinear free surface (contour interval is 1 × 10^{−6} m); and (bottom) with rigid lid (contour interval is 0.02 × 10^{−6} m)

Citation: Journal of Physical Oceanography 34, 1; 10.1175/1520-0485(2004)034<0306:HSACGC>2.0.CO;2

Summary of model parameters

^{1}

Spiegel and Veronis (1960) summarized the Boussinesq approximations as follows: “(1) The fluctuations in density which appear with the advent of motion result principally from thermal (as opposed to pressure) effects. (2) In the equations for the rate of change of momentum and mass, density variations may be neglected except when they are coupled to the gravitational acceleration in the buoyancy force.”

^{2}

Diagnosing bottom pressure in the height coordinate model or surface elevation in the pressure coordinate model is only consistent with the model discretization if the hydrostatic equation is integrated using a finite-volume discretization as opposed to finite-difference discretization. The finite-volume discretization of the hydrostatic equation is described in appendix C.