E. Equation__of__State___________
In the original MOM, a third order, local in z, polynomial approximation is constructed
to serve as the equation of state for seawater (Bryan & Cox, 1972). The polynomial
fit closely approximates either the Knudsen or Unesco formula, and the evaluation of
density from this approximation significantly reduces the computational effort over using
the Knudsen and Unesco formulae directly.
The approximation is in potential temperature () and salinity (S), and it is based on
the deviations (anomalies) of salinity, potential temperature, and density (ae) from some
reference profiles. Consequently, it has the following form,
ffiae = c1 ffi + c2 ffiS + c3 ffi2 + c4 ffiffiS + c5 ffiS2 + c6 ffi3 + c7 ffiffiS2 + c8 ffi2 ffiS + c9 ffiS3 (E1)
where
ffiae= ae - aer ;
ffi = - r ;
ffiS= S - Sr :
Here aer , r , and Sr represent the reference profiles for density, potential temperature, and
salinity, respectively. Also, ae is in sigma units (kg=m3 ), is in OC, and S is in model
units defined as 0:001(Sp - 35), where Sp is salinity in parts per thousand. The nine
coefficients actually contain the effects of pressure (depth) on density and, therefore, they
vary with depth. Note that, in Eq. (E1), the influences of salinity, potential temperature
and pressure on density are separated.
Even though the polynomial approximation uses potential temperature, the bounds
(ranges) for the polynomial fit are given in terms of insitu temperature and salinity. This is
due to the fact that the Knudsen and Unesco equations of state require insitu temperature
to compute density. The bounds are specified at 33 depth levels from 0m to 8000m with
250m increments which are different than the actual model level depths. The original
MOM first determines the model level, zk , for which the coefficients are to be evaluated.
Hence, the term local approximation is used. Then, the bounds of the polynomial fit
for the nearest and shallowest depth level are utilized to compute the range averages
of insitu temperature, potential temperature, and salinity which are used as reference
profiles. Finally, the reference density profile is calculated, using either the Knudsen
or Unesco equation of state with the reference insitu temperature and salinity profiles.
Consequently, the original MOM reference profiles are not realistic. Once the reference
profiles are ready, the anomalies are formed and a least squares algorithm is applied to
find the nine coefficients.
In the present work, we replaced the range average reference profiles of the original
MOM with more realistic profiles, corresponding to the annual mean distributions obtained
from Levitus (1982). Levitus (1982) provides only the potential temperature and salinity
distributions. Because the Knudsen and Unesco equations of state necessitate insitu
temperature, a separate code was utilized to compute the reference insitu temperature
E-1
profile as a function of salinity, potential temperature, and pressure (depth). The code
uses the same subroutine as MOM to compute potential temperature and iterates on insitu
temperature until the output potential temperature satisfies the annual mean value within
a specified error (= 10-12 ).
Next, one of the original MOM subroutines, denscoef.F, which computes the nine
coefficients of the polynomial fit and prepares an include file, dncoef.h, was modified to
incorporate the new reference profiles. The reference annual mean insitu temperature and
salinity distributions were defined in data statements at 45 levels (0m to 8000m) with
50m increments down to 300m, 100m increments down to 1500m, and 250m increments
for deeper levels. Wherever necessary, the linear interpolation was applied to obtain the
required values. Because Levitus (1982) data were available only down to 5500m, the values
at 5500m were used for deeper levels. The reference potential temperature and density were
then evaluated for each level, using the potential temperature and the Unesco equation of
state subroutines with the reference insitu temperature and salinity profiles. The vertical
distributions of reference salinity, potential temperature, and density are given in Fig. E1.
For comparison purposes, the original MOM reference profiles (evaluated at the 33 levels)
are also included in the figure.
The vertical distributions of coefficients obtained using the original MOM bounds for
the polynomial fit revealed significant non-smooth behavior. After numerous test runs, we
found that the bounds of the fit had to be changed such that the level averages corresponded
to the specified reference profile values. Consequently, the ranges of insitu temperature
were expanded for depth 1400 and contracted for deeper levels. The ranges of salinity
values were also changed accordingly. However, we found that the ranges of salinity values
did not significantly affect the distributions of coefficients except c5 . Therefore, we adjusted
the bounds of salinity to obtain a smooth distribution for c5 . In Fig. E2, the present bounds
of insitu temperature and salinity are presented. In this and the following figures, the
distributions are given only down to 5500m. The corresponding vertical distributions of
the polynomial approximation coefficients reveal that the coefficients are not very different
than the original MOM coefficients (evaluated at the 33 levels) with the exception of c5
which extends to about -35 in the original MOM.
We next computed the root-mean-square (rms) error which was based on the difference
of the values obtained from the current polynomial approximation and the Unesco equation
of state for seawater. The rms error was calculated at the 45 depth levels, using 5000
equally spaced points over the ranges given in Fig. E2. The vertical distribution of the
rms error is displayed in Fig. E3, revealing a maximum error of about 0.007 which occurs at
the surface. The same figure also includes the corresponding distribution from the original
MOM (evaluated at the 33 levels) for comparison purposes. Due to the wider ranges
of temperature for depth < 1250, we observe slightly larger error values for the present
results. For depth 1250, the observed rms error is much less than the original MOM
distributions, because the bounds of the present polynomial fit are narrower. In Fig. E4,
the contour plot of the deviation of the present results from the Unesco equation of state
at the surface is given, displaying a maximum error of about 0:018kg=m3 . In practice the
density anomaly cannot be determined to better than 0:05kg=m3 (Levitus & Isayev, 1992).
E-2
Therefore, the maximum error of the present results is much less than the accuracy of the
equation of state for seawater.
Bryan, K. & M.D. Cox, 1972: An Approximate Equation of State for Numerical Models
of Ocean Circulation, J. Phys. Oceanogr. 2, 510-514.
Levitus, S. 1982: Climatological Atlas of the World Ocean, Prof. Paper 13, US Department
of Commerce, NOAA, US Government Printing Office, Washington, DC.
Levitus, S. & G. Isayev, 1992: Polynomial Approximation to the International Equation
of State for Seawater, J. Atm. Oce. Tech. 9, 705-708.
E-3
(a) (b) (c)
Figure E1. Vertical distributions of reference (a) salinity (model units), (b) potential
temperature (O C), and (c) density (sigma units) profiles. The solid and dashed lines
represent the present and the original MOM profiles, respectively.
(a) (b)
Figure E2. Vertical distributions of bounds of (a) insitu temperature (O C) and (b)
salinity (parts per thousand) for polynomial fit.
E-4
Figure E3. Vertical distributions of rms error. The solid and dashed lines represent
the present and the original MOM profiles, respectively.
Figure E4. Contour plot of the deviation of the present results from the Unesco
equation of state for seawater at the surface (CI= 0:002). The solid and dashed lines
represent the positive and negative levels, respectively.
E-5