Elsevier

Ocean Modelling

Volume 90, June 2015, Pages 29-43
Ocean Modelling

Accurate polynomial expressions for the density and specific volume of seawater using the TEOS-10 standard

https://doi.org/10.1016/j.ocemod.2015.04.002Get rights and content

Highlights

  • The new standard TEOS-10 should be implemented in every ocean models.

  • Polynomial approximations designed for ocean models are proposed here.

  • Both Boussinesq and non-Boussinesq versions with 55 terms are presented.

  • A 75 term polynomial expression, with accurate sound speed, is also given.

Abstract

A new set of approximations to the standard TEOS-10 equation of state are presented. These follow a polynomial form, making it computationally efficient for use in numerical ocean models. Two versions are provided, the first being a fit of density for Boussinesq ocean models, and the second fitting specific volume which is more suitable for compressible models. Both versions are given as the sum of a vertical reference profile (6th-order polynomial) and an anomaly (52-term polynomial, cubic in pressure), with relative errors of ∼0.1% on the thermal expansion coefficients. A 75-term polynomial expression is also presented for computing specific volume, with a better accuracy than the existing TEOS-10 48-term rational approximation, especially regarding the sound speed, and it is suggested that this expression represents a valuable approximation of the TEOS-10 equation of state for hydrographic data analysis. In the last section, practical aspects about the implementation of TEOS-10 in ocean models are discussed.

Introduction

The equation of state is an empirical thermodynamic relationship linking seawater density to a number of state variables, most typically temperature, salinity and pressure. Because density gradients control the pressure gradient force through the hydrostatic balance, the equation of state provides a fundamental bridge between the distribution of active tracers and the fluid dynamics.

Recently, TEOS-10 (International Thermodynamic Equation Of Seawater – 2010, IOC et al. (2010)) has been released as the official replacement for the previous EOS-80 standard. Apart from an improved accuracy of the formulation owing to considerations of a greater number of refined laboratory measurements, this standard introduced two new concepts to be used as the conservative state variables: Absolute Salinity (unit: g/kg, notation: SA) and Conservative Temperature (unit: oC, notation: Θ). All thermodynamical variables, including seawater density, are derived analytically from the seawater Gibbs function Feistel (2008), and are given as a function of SA, Θ and the gauge pressure p (unit: dbar).

Because the exact expression for the TEOS-10 seawater density (or equivalently, specific volume) is complex and rather computing expensive, it is not well suited for practical applications such as hydrographic data analysis or ocean modeling. This is why an approximation has been proposed in TEOS-10, which consists of a 48-term rational function, with an accuracy better than field measurement accuracy over the whole oceanographic range of (SA,Θ, p) values.

Although this expression is very well suited for most practical applications, it is not so well adapted for ocean modeling applications because it remains difficult to compute analytically its partial derivatives or integrals. In particular, two quantities are key and need to be computed at least as often as the seawater density in an ocean model, namely the thermal expansion coefficient α and the haline contraction coefficient β, which are proportional to the Θ and SA partial derivatives, respectively, α(SA,Θ,p)=1vvΘ|SA,p=1ρρΘ|SA,p,β(SA,Θ,p)=1vvSA|Θ,p=1ρρSA|Θ,p,where v is the specific volume, and ρ = 1/v is the density.

The thermal expansion and haline contraction coefficients are useful to estimate the pressure gradient force term (through computation of the isobaric gradient of specific volume, ∇pv = (∂v/∂x, ∂v/∂y) at constant p), and the buoyancy frequency N, pv=v[αpΘβpS],N2=g[αΘzβSAz].These two coefficients are also important in isoneutral and vertical mixing parameterizations.

The equation of state is also important to determine the sound speed (unit: ms1), which is a function of the local compressibility, cs=(ρP|SA,Θ)1,where P is the absolute pressure, P = Po + p (preferably expressed in Pa = 10−4 dbar), with Po = 101, 325 Pa. Finally, one quantity of great interest for the analysis of ocean energetics is enthalpy, h(SA,Θ,P)=cpoΘ+h,where the first right hand side (r.h.s.) term is the potential enthalpy (McDougall, 2003), and the second is dynamic enthalpy (Young, 2010) defined as h=PoPv(SA,Θ,P)dP. Potential enthalpy is an excellent measure of heat content. Dynamic enthalpy is central to the ocean potential energy budget, as it represents the amount of total potential energy that would be needed to lift the water parcel adiabatically to the level P = Po.

In this paper, we present a set of TEOS-10 polynomial approximations which are specifically designed for ocean models. The proposed TEOS-10 approximations are suitable for ocean models using z- or sigma-vertical coordinates (e.g. Griffies, 2004), and could potentially be adapted to isopycnic coordinate models following standard procedures (Adcroft et al., 2008). A fundamental distinction must be made however between Boussinesq and non-Boussinesq models, as the former requires the fast computation of density and its partial derivatives, while the latter rather depends on specific volume. Various TEOS-10 approximations for Boussinesq or compressible (i.e. non-Boussinesq) models are proposed in this paper. The accuracy of these approximations is shown to be comparable to the TEOS-10 rational function approximation, but the polynomial expressions have simpler and more computationally efficient expressions for their derived quantities which make them more adapted for use in ocean models. Also, a high precision form is presented, that we suggest as a replacement of the TEOS-10 rational function approximation for hydrographic data analysis.

The paper is organized as follows. Elements related to the implementation of Boussinesq ocean models will be provided in Section 2, then polynomial approximations for Boussinesq models Section 3 and compressible models (Section 4) will be described. In Section 5, practical aspects on the implementation of TEOS-10 in ocean models will be discussed, and conclusions will finally be drawn in Section 6.

Section snippets

The Boussinesq approximation in ocean models

Currently, a majority of ocean general circulation models (OGCMs) are making both the hydrostatic and Boussinesq approximations (e.g. Vallis, 2006). The Boussinesq approximation is well adapted to the ocean case because seawater is nearly incompressible, and all dynamical effects of specific volume variations can be safely neglected except for the contribution to buoyancy variations.

Ocean dynamics is sensitive at leading order to the horizontal gradient of density instead of density itself,

polyTEOS10-bsq: fitting the density anomaly

The existing form of the TEOS-10 expression for density that is written as a function of Conservative Temperature is a 48-coefficient rational function with the numerator quadratic in the depth z and the denominator cubic in z. This form is an approximation of the expression derived from the Gibbs function for seawater of Feistel (2008) within the range of possible (SA,Θ, z) oceanic values. Here we present another approximation, referred to as polyTEOS10-bsq, which is particularly well suited

polyTEOS10-55t: a polynomial fit of specific volume

In the compressible case, the important variable is specific volume, instead of density for the Boussinesq case. Here we apply exactly the same procedure to obtain a polynomial fit of TEOS-10, but applied to v.

Hence, specific volume is decomposed into a vertical reference profile (function of gauge pressure, unit: dbar) and the residual, called the specific volume anomaly, v(SA,Θ,p)=vo(p)+δ(SA,Θ,p).

Similarl to the density fit, the vertical reference profile is taken as the 6th-order polynomial

About the implementation of TEOS-10 in a Boussinesq model

In a Boussinesq model, the choice of the ρo value is important, yet it varies significantly among OGCMs, as it is a matter of personal preference. A usual value is 1035 kgm3, the ocean average density. This choice minimizes the error of the pressure gradient at mid-depth (i.e. around 2000 m depth). Another sensible choice would be to take 1030 kgm3, which is a typical value of density in the thermocline, minimizing the mean error of the thermal wind. Finally, values around 1025 kgm3 allow

Conclusions and discussion

The TEOS-10 equation of state represents the new standard, and replaces the previous EOS-80 standard for all oceanographic applications. This includes both hydrographic data analysis and ocean modeling applications. Apart from the necessary compliance to official standard, all ocean models should progressively switch to the TEOS-10 standard because (1) it is more accurate, being based on an updated database of laboratory measurements, and (2) it uses Conservative Temperature and Absolute

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at 10.1016/j.ocemod.2015.04.002.

Supplementary materials

Acknowledgments

We would like to thank the three anonymous reviewers for their helpful and constructive comments. This work has been funded by the Bolin Centre for Climate Research, research area “Oceans-atmosphere dynamics and climate”. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at NSC, Linköping.

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