## 1. Introduction

*ρ*

*ρ*

*S,*

*θ,*

*p*

*θ*rather than in situ temperature

*T.*Boussinesq models approximate (1) by

*ρ*

_{0}is a constant reference density. A common approximation to (3) in fixed vertical coordinate models is

*ρ*

*ρ*

*S,*

*θ,*

*p*

_{0}

*z*

*p*

_{0}(

*z*) is a specified depth-to-pressure conversion function such as that proposed by Fofonoff and Millard (1983), for example. This is done in order to avoid a nonlinear solution procedure for density and pressure. There are three types of error associated with these approximations. The first error, the one connected with the approximation in (4), we will call the Boussinesq error. Assuming that

*ρ*

_{0}= 1 g cm

^{−3}as in many codes derived from the original Bryan model (Cox 1984; Semtner 1986; Dukowicz and Smith 1994), the Boussinesq error can be as large as 5% based on the typical range of densities present in the ocean. The second type of error is the error in the calculation of density due to the approximation (5); we will call this the density error. Assuming a baroclinic displacement of 50 m relative to the mean represented by

*p*

_{0}(

*z*), the density error is estimated to be of order 2 × 10

^{−4}g cm

^{−3}, which is quite small when compared to a typical in situ density deviation,

*ρ*−

*ρ*

_{0}≈ 3.5 × 10

^{−2}g cm

^{−3}. However, this can have significant dynamic consequences because the density error implies a corresponding pressure gradient error, which we will call the dynamic error. Dewar et al. (1998) have analyzed this error in detail and they concluded that it can lead to spurious transports of several Sverdrups (Sv ≡ 10

^{6}m

^{3}s

^{−1}) and associated velocities of several centimeters per second. They therefore recommend against the use of approximation (5). However, this considerably complicates the computation of the baroclinic pressure gradient and entails substantial code modifications.

*p*

*F*

*ρ*

*S,*

*θ,*

*p*

*z*

*p*

*F*

*ρ*

*S,*

*θ,*

*p*

_{0}

*z*

*z*

*x,*

*y,*

*z,*

*t*) unless explicitly indicated. The errors in the above approximations may now be evaluated as follows. The pressure gradient error is

*κ*is the adiabatic compressibility and

*c*is the speed of sound in seawater, we observe that the errors

*E*

_{2}and

*E*

_{3}are directly related to the compressibility (or, alternatively, to the sound speed). Similarly, the Boussinesq error

*E*

_{1}is largely associated with compressibility, since any vertical density difference in the water column may be written as

In the following we will show how to transform Eqs. (1)–(3) into equivalent forms that effectively make use of a much stiffer equation of state before making the approximations (4)–(5), thereby resulting in much smaller errors.

## 2. Modified density, pressure, and equation of state

*κ*

*κ*

^{(p)}

*δκ,*

*κ*

^{(p)}is dependent on pressure only and

*δκ*is the residual, termed the thermobaric compressibility. As seen in Sun et al. (1999),

*δκ*is at least an order of magnitude smaller than

*κ*

^{(p)}or

*κ.*There is some arbitrariness in choosing

*κ*

^{(p)}, however. In view of (11) and (13), the density may be written as

*p*

_{r}is an arbitrary reference pressure, and

*A*(

*p*) is a function of pressure only, which we will subsequently use to appropriately modify the factor

*r*(

*p*). Thus, the density may be expressed as the product of two factors, a factor

*r*(

*p*) that contains most of the pressure dependence and a factor

*ρ** that is only weakly dependent on pressure. As a result,

*ρ** will be very nearly independent of depth, much more so than the density

*ρ*itself. In Sun et al. (1999),

*ρ** is called the “virtual potential density,” although here we prefer to call it the thermobaric density due to its direct dependence on the thermobaric compressibility

*δκ.*

*p**, through the relationship

*p**(

*p*) is invertible in general so that we can alternatively write

*p*=

*p*(

*p**). Given (14) and (16), an ocean model may alternatively be expressed in terms of

*ρ** and

*p**, rather than

*ρ*and

*p.*Equations (1)–(3) then become

*ρ*

*ρ*

*S,*

*θ,*

*p*

^{*}

_{0}

*z*

*δκ*is at least an order of magnitude smaller than

*κ,*the modified equation of state (19) is at least an order of magnitude stiffer with respect to changes in thermobaric density than is the original equation of state (3) with respect to changes in in situ density. Therefore, the error associated with the above approximations will also be at least an order of magnitude smaller. Also, note that we can use the degree of freedom provided by

*A*to make

*κ** vanish along some curve in the “phase space” of

*S,*

*θ,*and

*p,*thereby further reducing the compressibility in some desired region of phase space. We will make good use of this possibility in what follows.

How feasible is it to make such a change in variables? Most ocean codes make use of the Boussinesq approximation. A Boussinesq code is particularly simple because it is unchanged if *ρ* is replaced by *ρ** and *p* by *p**, except for the equation of state (and possibly in some of the parameterizations where in situ density or pressure may be required, but notably not in the convective adjustment parameterization). The equation of state must be changed from (3) or (5) to (19) or (21). However, this is a very minor change that entails no change in the structure of the code. If the in situ density or pressure is required for diagnostic or parameterization purposes, or in the continuity equation of a non-Boussinesq code, then it is a simple matter to make the conversion using (14) and (16a).

## 3. Specific example

*z*-coordinate code based on the Bryan–Cox model (Dukowicz and Smith 1994; further information available online at http://www.acl.lan.gov/climate/models/pop). We will optimize the transformation described previously around the global mean climatology of Levitus et al. (1994a,b). Figures 1a,b show the vertical profiles of global mean in situ temperature and salinity from this climatology, as a function of depth in the depth range from 0 to 5500 m. These profiles (or a “cast”) are defined as

*T*(

*z*) and

*S*(

*z*), respectively. Because in situ temperature is provided, we use the Bryden (1973) algorithm to convert to potential temperature. The corresponding potential temperature is shown in Fig. 1a. Given these profiles, self-consistent density and pressure as a function of depth were obtained by integrating the hydrostatic equation (2) using

*Mathematica.*The integration is carried out with g = 9.806 m s

^{−2}, the JMcD equation of state, the Bryden (1973) algorithm, and assuming that pressure equals zero at the surface. The resulting profiles are denoted by

*ρ*(

*z*) and

*p*(

*z*), and are plotted in Figs. 2a,b respectively. We note that the pressure is nearly proportional to the depth so that it is always possible to invert this functional relationship and express depth as a function of pressure; that is,

*z*=

*z*(

*p*). This means that we may alternatively express any quantity along the cast as a function of pressure rather than depth, that is,

*T*(

*p*),

*S*(

*p*),

*ρ*(

*p*),

*θ*(

*p*). Because the cast is representative of the entire ocean, it is convenient to take

*p*

_{0}(

*z*) =

*p*(

*z*) as the depth-to-pressure conversion function. A simple fit to this function is

*r*(

*p*) so that

*κ** is as small as possible in order to minimize the approximation errors, as discussed previously. One way of doing this is to enforce

*κ** = 0 along the global mean cast since then departures will be minimized. According to (22), this will be true if

*S*=

*S*(

*p*) and

*θ*=

*θ*(

*p*). This is an ordinary differential equation that is easily integrated using

*Mathematica*and the JMcD equation of state. Since

*r*(

*p*) is undetermined to within a constant factor, we have chosen to normalize it so that

*r*(

*z*= 3000) =

*ρ*(

*z*= 3000)/

*ρ*

_{0}, where

*ρ*

_{0}= 1 gm cm

^{−3}, in order that the thermobaric density be equal to

*ρ*

_{0}at a depth of 3 km. The scaling factor

*r*(

*p*) is plotted in Fig. 3, but as a function of depth instead of pressure for convenience in comparing it with density. It is apparent that it effectively captures the bulk of the effect of compressibility on the density. The scaling factor may be fitted by

*p*is in bars, with an error in the range {−1 × 10

^{−4}↔ 2 × 10

^{−4}} over the pressure range from 0 to 560 bars. Note that both fits, (23) and (25), were chosen so that the error is constrained not to grow excessively outside the fitted range. Equations (23) and (25) are all that we need to implement both the transformation and the approximate conversion of pressure to depth.

*ρ*

_{0}= 1 gm cm

^{−3}with an error of about 0.5%, about an order of magnitude smaller than previously. The error is largely concentrated within the thermocline, that is, within the upper one or two kilometers. Figure 4b shows the thermobaric density error, calculated as

*ρ*

_{JMcD}represents the density calculated with the JMcD equation of state and

*θ*

_{B}represents the potential temperature calculated from the Bryden conversion algorithm. We note that the maximum error is less than about 3 × 10

^{−7}gm cm

^{−3}. This is more than an order of magnitude smaller than the rms error in the international equation of state or the maximum error of the JMcD fit (5 × 10

^{−6}and 6.7 × 10

^{−6}gm cm

^{−3}, respectively, according to JMcD). It is also about three orders of magnitude smaller than the density error in current models estimated in the introduction.

Unfortunately, it is not possible to easily evaluate the improvement in the dynamic error, *E*_{3}, which is problem dependent. However, as is obvious from the definitions in the introduction, both the dynamic and the density errors are directly related to the magnitude of the density perturbation and, therefore, if the density error vanishes then so does the dynamic error. Furthermore, since both these errors are directly related to the magnitude of the adiabatic compressibility, it is clear that if the compressibility is reduced by a constant factor then both errors are reduced by the same factor, all other things being equal. It is to be expected, therefore, that if the density error is reduced because compressibility is made smaller, then the dynamic error will be correspondingly reduced.

*ρ*=

*ρ*(

*S,*

*θ,*

*p*

_{0}(

*z*)), rather than the thermobaric density. Thus, this routine is unchanged except for the use of (23) to convert depth to pressure. This is done to minimize changes in the code and also in case the in situ density is required for other purposes. The biggest change is to the hydrostatic equation, which is solved in the form

*p*is never calculated because it is not used elsewhere in the code.

## 4. Summary

Many existing ocean codes make certain simplifying approximations based on the fact that the adiabatic compressibility of seawater is rather low. However, the error associated with these approximations is not negligible and can have significant dynamic consequences, as detailed in Dewar et al. (1998). We demonstrate that it is possible to greatly reduce the error by transforming to an equation of state with a much smaller compressibility, expressed in terms of new state variables termed the thermobaric density and pressure, before making these approximations. The error in the thermobaric density obtained from the transformed equation of state due to these approximations is within the uncertainties in the equation of state itself. The dynamic error studied by Dewar et al. (1998) is directly related to the density error, and it should be reduced by at least an order of magnitude by means of the present method. The present method is particularly simple for a Boussinesq model and the resulting code changes are minimal.

## Acknowledgments

This work was made possible by the support of the DOE CCPP (Climate Change Prediction Program) program.

## REFERENCES

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Dewar, W. K., Y. Hsueh, T. J. McDougall, and D. Yuan, 1998: Calculation of pressure in ocean simulations.

,*J. Phys. Oceanogr***28****,**577–588.Dukowicz, J. K., and R. D. Smith, 1994: Implicit free-surface method for the Bryan–Cox–Semtner ocean model.

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The self-consistent density and pressure associated with the global mean Levitus et al. (1994a,b) climatology, obtained using the Jackett and McDougall equation of state and the Bryden algorithm

Citation: Journal of Physical Oceanography 31, 7; 10.1175/1520-0485(2001)031<1915:RODAPG>2.0.CO;2

The self-consistent density and pressure associated with the global mean Levitus et al. (1994a,b) climatology, obtained using the Jackett and McDougall equation of state and the Bryden algorithm

Citation: Journal of Physical Oceanography 31, 7; 10.1175/1520-0485(2001)031<1915:RODAPG>2.0.CO;2

The self-consistent density and pressure associated with the global mean Levitus et al. (1994a,b) climatology, obtained using the Jackett and McDougall equation of state and the Bryden algorithm

Citation: Journal of Physical Oceanography 31, 7; 10.1175/1520-0485(2001)031<1915:RODAPG>2.0.CO;2

The density scaling factor *r*(*p*) plotted as a function of depth, and the in situ density, overlaid for comparison

Citation: Journal of Physical Oceanography 31, 7; 10.1175/1520-0485(2001)031<1915:RODAPG>2.0.CO;2

The density scaling factor *r*(*p*) plotted as a function of depth, and the in situ density, overlaid for comparison

Citation: Journal of Physical Oceanography 31, 7; 10.1175/1520-0485(2001)031<1915:RODAPG>2.0.CO;2

The density scaling factor *r*(*p*) plotted as a function of depth, and the in situ density, overlaid for comparison

Citation: Journal of Physical Oceanography 31, 7; 10.1175/1520-0485(2001)031<1915:RODAPG>2.0.CO;2

Thermobaric density as a function of depth, and the associated density error due to the use of the approximate depth-to-pressure conversion function, for six representative temperature and salinity profiles from the Levitus (1994) climatology

Citation: Journal of Physical Oceanography 31, 7; 10.1175/1520-0485(2001)031<1915:RODAPG>2.0.CO;2

Thermobaric density as a function of depth, and the associated density error due to the use of the approximate depth-to-pressure conversion function, for six representative temperature and salinity profiles from the Levitus (1994) climatology

Citation: Journal of Physical Oceanography 31, 7; 10.1175/1520-0485(2001)031<1915:RODAPG>2.0.CO;2

Thermobaric density as a function of depth, and the associated density error due to the use of the approximate depth-to-pressure conversion function, for six representative temperature and salinity profiles from the Levitus (1994) climatology

Citation: Journal of Physical Oceanography 31, 7; 10.1175/1520-0485(2001)031<1915:RODAPG>2.0.CO;2