# A comment on the Equation of State and the freezing point equation with respect to subglacial lake modelling

Abstract: The empirical Equation of State (EOS) allows the calculation of the density of water in dependence of salinity, temperature, and pressure. Water density is an important quantity to determine the internal structure and flow regime of ocean and lakes. Hence, its exact representation in numerical models is of utmost importance for the specific simulation results. The three parameters namely salinity, temperature, and pressure have a complex interdependency on the EOS. Whether warmer water parcels sink or rise, therefore depends on the surrounding salinity and pressure. The empirical Equation of Freezing Point (EOFP) allows to calculate the pressure- and salinity-dependent freezing point of water. Both equations are necessary to model the basal mass balance below Antarctic ice shelves or at the ice-water interface of subglacial lakes. This article aims three tasks: first we comment on the most common formulations of the EOS and the EOFP applied in numerical ocean and lake models during the past decades. Then we describe the impact of the recent and selfconsistent Gibbs thermodynamic potential formulation of the EOS and the EOFP on subglacial lake modelling. Finally, we show that the circulation regime of subglacial lakes covered by at least 3000 m of ice, in principle, is independent of the particular formulation, in contrast to lakes covered by a shallower ice sheet, like e.g., Subglacial Lake Ellsworth. However, as modelled values like the freezing and melting patterns or the distribution of accreted ice at the ice-lake interface are sensitive to different EOS and EOFP, we present updated values for Subglacial Lake Vostok and Subglacial Lake Concordia. (C) 2010 Elsevier B.V. All rights reserved.

## Summary (1 min read)

### 1 Introduction

- In the following the authors briefly review different representations of EoS and EoFP used in ocean modelling, before they discuss the relevance of their improved formulations for the modelling of subglacial lakes.
- Finally the authors present updated results of subglacial lake modelling studies, with respect to the revised EoS and EoFP.

### 1.1 Equation of State ( EoS)

- Water depth and potential temperature dependence of isopycnals (Feistel, 2003; Jackett et al., 2006) .
- The black solidus line shows the depth-dependent freezing point of fresh water (Feistel, 2003; Jackett et al., 2006) , the red solidus line indicates the linearized form of the freezing point equation adjusted for Lake Vostok.

### The

- The dashed line connects the isopycnal's vertices and indicates the line of maximum density (LoMD).
- Here the authors only present the updated results with respect to the revised EoS and EoFP with otherwise identical configurations.
- In Table 1 the authors present updates of the most relevant results and their uncertainties for Lake Vostok and Lake Concordia published in the aforementioned studies.
- Malte.Thoma@awi.de (Malte Thoma), also known as Email address.

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##### Citations

148 citations

### Cites background from "A comment on the Equation of State ..."

...However, below the LOMD, where overburden pressure is “high” relative to the Pc, any heated water will rise through buoyancy [Thoma et al., 2010]....

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112 citations

### Cites methods from "A comment on the Equation of State ..."

...The model has recently been improved by an updated equation of state and a revised equation for the freezing point temperature, according to the Gibbs thermodynamic potential [Thoma et al., 2010]....

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38 citations

### Cites background or methods from "A comment on the Equation of State ..."

...The horizontal resolution (0.025◦×0.0125◦, about 0.7× 1.4 km), the number of vertical layers (16), as well as the horizontal and vertical eddy diffusivities (5 m2/s and 0.025 cm2/s, respectively) are adopted from a model of subglacial Lake Concordia (Thoma et al., 2009a)....

[...]

...Previous subglacial lake simulations of Lake Vostok (Thoma et al., 2007, 2008a; Filina et al., 2008), Lake Concordia (Thoma et al., 2009a), or Lake Ellsworth (Woodward et al., 2009) used a prescribed average heat conduction into the ice (QIce = dT /dz×2.1 W/(K m)), based on borehole temperature…...

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...…spherical coordinates and has been applied successfully to ice-shelf cavities (e.g.,Grosfeld et al., 1997; Williams et al., 2001; Thoma et al., 2006) as well as to subglacial lakes (Williams, 2001; Thoma et al., 2007, 2008a,b; Filina et al., 2008; Thoma et al., 2009a,b,c; Woodward et al., 2009)....

[...]

...The horizontal resolution (0.025◦×0.0125◦, about 0.7× 1.4 km), the number of vertical layers (16), as well as the horizontal and vertical eddy diffusivities (5 m2/s and 0.025 cm2/s, respectively) are ad pted from a model of subglacial Lake Concordia (Thoma et al., 2009a)....

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...The strength of the mass transport is between those modelled for Lake Vostok and those for Lake Concordia (Thoma et al., 2009a), and hence reasonable for subglacial lakes....

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27 citations

25 citations

##### References

2,015 citations

### "A comment on the Equation of State ..." refers background in this paper

...Other approaches to solve the equa-36 tions on unstructured grids apply spectral formulations (SEOM, e.g., Patera,37 1984), finite volumes (MITgcm, e.g., Marshall et al., 1997a,b), or finite ele-38 ments (COM, e.g., Danilov et al., 2004; Timmermann et al., 2009)....

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1,997 citations

1,483 citations

### "A comment on the Equation of State ..." refers background in this paper

...A more general ap-62 proach is the so-called UNESCO-EoS (Fofonoff and Millard, 1983), derived63 from the fundamental work of Millero et al. (1980) and Millero and Poisson64 (1981)....

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...This95 set of coefficients dating back to Foldvik and Kvinge (1974) is still in use in96 models dealing with ice-water interaction and has not always been replaced97 by a linearised version of the more precise (but higher order) formulation98 of Fofonoff and Millard (1983)....

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1,189 citations

### "A comment on the Equation of State ..." refers background in this paper

...Other approaches to solve the equa-36 tions on unstructured grids apply spectral formulations (SEOM, e.g., Patera,37 1984), finite volumes (MITgcm, e.g., Marshall et al., 1997a,b), or finite ele-38 ments (COM, e.g., Danilov et al., 2004; Timmermann et al., 2009)....

[...]

544 citations