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Reduction of Density and Pressure Gradient Errors in Ocean Simulations

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Reduction of Density and Pressure Gradient Errors in Ocean Simulations JULY 2001 NOTES AND CORRESPONDENCE 1915 Reduction of Density and Pressure Gradient Errors in Ocean Simulations JOHN K. DUKOWICZ Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 23 October 2000 and 30 January 2001 ABSTRACT Existing ocean models often contain errors associated with the computation of the density and the associated 21 21 pressure gradient. Boussinesq models approximate the pressure gradient force, r =p, by r 0 =p, where r0 is a constant reference density. The error associated with this approximation can be as large as 5%. In addition, Cartesian and sigma-coordinate models usually compute density from an equation of state where its pressure dependence is replaced by a depth dependence through an approximate conversion of depth to pressure to avoid the solution of a nonlinear hydrostatic equation. The dynamic consequences of this approximation and the associated errors can be signi®cant. Here it is shown that it is possible to derive an equivalent but ``stiffer'' equation of state by the use of modi®ed density and pressure, r* and p*, obtained by eliminating the contribution of the pressure-dependent part of the adiabatic compressibility (about 90% of the total). By doing this, the errors associated with both approximations are reduced by an order of magnitude, while changes to the code or to the code structure are minimal. 1. Introduction approximation to (3) in ®xed vertical coordinate models is Consider the horizontal pressure force in ocean mod- els: r ø r(S, u,p0(z)), (5) 1 where p (z) is a speci®ed depth-to-pressure conversion PGF 5 =p, (1) 0 r function such as that proposed by Fofonoff and Millard (1983), for example. This is done in order to avoid a where the pressure is obtained from the hydrostatic nonlinear solution procedure for density and pressure. equation There are three types of error associated with these ap- ]p proximations. The ®rst error, the one connected with the 52rg, (2) ]z approximation in (4), we will call the Boussinesq error. 23 Assuming that r 0 5 1gcm as in many codes derived and the density from an equation of state from the original Bryan model (Cox 1984; Semtner r 5 r(S, u,p), (3) 1986; Dukowicz and Smith 1994), the Boussinesq error can be as large as 5% based on the typical range of such as the Jackett and McDougall (1995, hereafter densities present in the ocean. The second type of error JMcD) equation, which is a ®t of the commonly ac- is the error in the calculation of density due to the ap- cepted international UNESCO equation of state in terms proximation (5); we will call this the density error. As- of potential temperature u rather than in situ temperature suming a baroclinic displacement of 50 m relative to T. Boussinesq models approximate (1) by the mean represented by p 0(z), the density error is es- 24 23 1 timated to be of order 2 3 10 gcm , which is quite PGF ø =p, (4) small when compared to a typical in situ density de- r 0 22 23 viation, r 2 r 0 ø 3.5 3 10 gcm . However, this where r 0 is a constant reference density. A common can have signi®cant dynamic consequences because the density error implies a corresponding pressure gradient error, which we will call the dynamic error. Dewar et Corresponding author address: Dr. John K. Dukowicz, Los Alamos National Laboratory, Theoretical Division, T-3 Fluid Dynamics, MS al. (1998) have analyzed this error in detail and they B216, Los Alamos, NM 87545. concluded that it can lead to spurious transports of sev- E-mail: [email protected] eral Sverdrups (Sv [ 106 m3 s21) and associated ve- Unauthenticated | Downloaded 09/29/21 03:38 PM UTC 1916 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 locities of several centimeters per second. They there- and the last term dominates for a density difference over fore recommend against the use of approximation (5). the entire column. Therefore, since these errors are di- However, this considerably complicates the computation rectly related to the compressibility, the lower the com- of the baroclinic pressure gradient and entails substantial pressibility (i.e., the ``stiffer'' the equation of state) the code modi®cations. smaller the errors. Let us assume that (2) and (3) may be simultaneously In the following we will show how to transform Eqs. integrated so that the solution for the pressure may be (1)±(3) into equivalent forms that effectively make use formally expressed as of a much stiffer equation of state before making the approximations (4)±(5), thereby resulting in much p 5 F[r(S, u,p), z], (6a) smaller errors. and therefore the corresponding approximate pressure is 2. Modi®ed density, pressure, and equation of state p95F[r(S, u,p0(z)), z], (6b) where all variables are functions of (x, y, z, t) unless Following Sun et al. (1999), we observe that com- explicitly indicated. The errors in the above approxi- pressibility may be split into two terms: mations may now be evaluated as follows. The pressure k 5 k (p) 1 dk, (13) gradient error is where k (p) is dependent on pressure only and dk is the 11 =p 2 =p9 residual, termed the thermobaric compressibility. As ((rr0 seen in Sun et al. (1999), dk is at least an order of (p) EPGF55E 11 E 3, (7) magnitude smaller than k or k. There is some arbi- \PGF\ trariness in choosing k (p), however. In view of (11) and where (13), the density may be written as p 11 r 5 r(S, u, p ) exp k dp9 2 =p r E (12rr0 ( r pr E1 5 ø 1 2 (8) pp \PGF\ ((r 0 5 r(S, u, p ) exp k(p) dp9 exp dk dp9 r EE is the Boussinesq error, and pprr ]F ]r 5 r(p)r*(S, u, p), (14) = (p 2 p (z)) 0 where p is an arbitrary reference pressure, and 1 \=(p 2 p9)\ ( 12]r ]p ( r E 5 ø (9) p 3 r \PGF\\=p9\ 0 r(p) 5 A expE k(p) dp9, (15a) is the dynamic error considered by Dewar et al. (1998). pr The density error is simply p r*(S, u, p) 5 A21r(S, u, p ) exp dk dp9, (15b) ]r r E (p 2 p0(z)) pr \r 2 r9\ ((]p E2 5 ø . (10) where A(p) is a function of pressure only, which we \r\\r\ will subsequently use to appropriately modify the factor Noting that r(p). Thus, the density may be expressed as the product of two factors, a factor r(p) that contains most of the ]r 1 pressure dependence and a factor r* that is only weakly 5 rk 5 , (11) 2 dependent on pressure. As a result, r* will be very 12]pc,S u nearly independent of depth, much more so than the where k is the adiabatic compressibility and c is the density r itself. In Sun et al. (1999), r* is called the speed of sound in seawater, we observe that the errors ``virtual potential density,'' although here we prefer to E 2 and E3 are directly related to the compressibility (or, call it the thermobaric density due to its direct depen- alternatively, to the sound speed). Similarly, the Bous- dence on the thermobaric compressibility dk. sinesq error E1 is largely associated with compressibil- We now de®ne an associated quantity, called the ther- ity, since any vertical density difference in the water mobaric pressure p*, through the relationship column may be written as p dp9 p*(p) 5 , (16a) ]r ]r E r(p9) Dr 5DS 1Du 1 rkDp, (12) 0 ]p ]p 12u,pS,p 12 which results from integrating Unauthenticated | Downloaded 09/29/21 03:38 PM UTC JULY 2001 NOTES AND CORRESPONDENCE 1917 FIG. 1. Global mean in situ temperature and salinity from the Levitus et al. (1994a,b) climatology, and the associated potential temperature from the Bryden (1973) algorithm. dp*1 The approximations corresponding to (4) and (5) may 5 . (16b) dp r(p) now be introduced to obtain We may expect that the pressure function p*(p) is in- 1 vertible in general so that we can alternatively write p PGF ø =p* (20) r 0 5 p(p*). Given (14) and (16), an ocean model may alternatively be expressed in terms of r* and p*, rather and than r and p. Equations (1)±(3) then become r* ø r*(S, u,p*(0 z)). (21) 1 PGF 5 =p*, (17) We have already shown that the error due to these ap- r* proximations is directly related to the magnitude of the ]p* adiabatic compressibility. The effective compressibility 52r*g, (18) associated with the equation of state (19) is ]z and ]r*1]rrdr dp r*k* 55 2 ]p* r ]p r2 dp dp* r* 5 r(S, u, p(p*))/r(p*) 5 r*(S, u, p*), (19) 12u,S [] 12u,S where this is the modi®ed equation of state. Note that d lnA these equations retain their original form, although the 5 rdk2 # rdk, (22) dp functional relationship of the equation of state is mod- 12 i®ed. In fact, as we will see, this modi®cation serves to where we have made use of (11) and (15a). Noting that ``stiffen'' the equation of state. The equations are exact dk is at least an order of magnitude smaller than k, the at this point; there is no approximation. modi®ed equation of state (19) is at least an order of Unauthenticated | Downloaded 09/29/21 03:38 PM UTC 1918 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 FIG.
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