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Journal ArticleDOI

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

15 May 2010-Earth and Planetary Science Letters (Elsevier)-Vol. 294, Iss: 1, pp 80-84

AbstractThe 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.

Topics: Subglacial lake (69%), Lake Vostok (60%), Freezing point (58%), Ice shelf (56%), Ice sheet (53%)

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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A comment on the equation of state and the
freezing point equation with respect to
subglacial lake modelling
Malte Thoma
a,b
Klaus Grosfeld
a
Andrew M. Smith
c
Christoph Mayer
b
a
Alfred Wegener Institute for Polar and Marine Research,
Bussestrasse 24, 27570 Bremerhaven, Germany
b
Bayerische Akademie der Wissenschaften, Kommission f¨ur Glaziologie,
Alfons-Goppel-Str. 11, 80539 unchen, Germany
c
British Antarctic Survey, High Cross, Madingley Road, Cambridge, CB3 0ET,
United Kingdom
1
Abstract2
The empirical Equation of State (EoS) allows to calculate the density of water in3
dependence of salinity, temperature, and pressure. The three parameters have a4
complex interdependency on the EoS. Hence, wh ether warmer water parcels sink5
or raise depends on the surrounding salinity and p ressure. The empirical Equation6
of Freezing Point (EoFP) allows to calculate the pressure and salinity dependent7
freezing point of water. Both equations are necessary to model the basal mass b al-8
ance below Antarctic ice shelves or at the ice-water interface of subglacial lakes.9
This article aims three tasks: First we comment on the most common formulations10
of th e EoS and the EoFP applied in numerical ocean and lake models during the11
past decades. Then we describe the impact of the recent and self-consistent Gibbs12
thermodynamic potential-formulation of the EoS and the EoFP on subglacial lake13
modeling. Finally, we show that the circulation regime of subglacial lakes covered14
by at least 3000 m of ice, in principle, is independent of the particular formula-15
tion, in contrast to lakes covered by a shallower ice sheet, like e.g., subglacial Lake16
Ellsworth. However, as mo deled values like the basal mass balance or the distri-17
bution of accreted ice at the ice-lake interface are sensitive to different EoS and18
EoFP, we present updated values for subglacial Lake Vostok and s ubglacial Lake19
Concordia.20
Key words: Subglacial Lakes, Equation of State, Freezing Point Equation,21
Numerical Modelling, Ice-ocean Interaction, Lake Vostok, Lake Concord ia, Lake22
Ellsworth, Antarctica23
Email address: Malte.Thoma@awi.de (Malte Thoma).
Preprint submitted to Elsevier 23 October 2009

1 Introduction24
Water flow within oceans and subglacial lakes is modelled by solving the hydro-25
static primitive equations numerically (e.g., Haidvogel and Beckmann, 1999;26
Griffies, 2004). These equations describe the flow of a fluid on the rotating27
earth by the equation of motion, the conservation laws of temperature and28
salinity, and an equation of state (EoS). Some fundamental differences be-29
tween different models relate to the implementation of t he vertical coordinate,30
which may be orientated planar, terrain- following, or along isopycnals. Well31
known representa t ives for these type of models are the Modular Ocean Model32
(MOM, e.g. Pacanowski and Griffies, 1998; Griffies et al., 2003), the Princeton33
Ocean Model (POM, e.g., Blumberg and Mellor, 1983; Ezer and Mellor, 2004),34
and the Miami Isopycnic Coordinate Ocean Model (MICOM, e.g., Bleck, 1998;35
Holland and Jenkins, 2001), respectively. Other approaches to solve the equa-36
tions on unstructured grids apply spectral formulatio ns (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). The num-39
ber of ocean models originating from these, in particular of those with struc-40
tured horizontal grids, is high. However, each model has to implement the41
EoS. The empirical EoS is a complex nonlinear function t o calculate the42
density as a function of temperature, salinity, and pressure ρ = ρ(T, S, p). For43
the global ocean, it has to cover a wide parameter range in S (0 to 42 psu), T44
(2 t o 40
C), and p (0 to 100 MPa). Subglacial lakes range at the lower bound-45
aries for T and S and the medium pressure range. In this parameter range,46
the slope of the calculated density is at its vertex, which has implications for47
the circulation and basal mass balance within subglacial lakes (Thoma et al.,48
2008b). Models that also include the interaction between ice and water, ad-49
ditionally apply an equation for t he pressure-dependent freezing point of sea50
water (EoFP) T
f
= T
f
(S, p).51
In the following we briefly review different representations of EoS and EoFP52
used in ocean modelling, before we discuss the relevance of their improved53
formulations for the modelling o f subglacial lakes. Finally we present updated54
results of subglacial lake modelling studies, with respect to the revised EoS55
and EoFP.56
1.1 Equation of State ( EoS)57
Early ocean models applied the Knudsen-Ekman equation, which relies on t he58
Boussinesq approximation and linearises the EoS around some reference val-59
ues for temp erature, salinity and pressure (e.g., Fofonoff, 1962; Bryan and Cox,60
1972). Although this approach reduces the computational effort significantly,61
2

it is only appropriate over very narrow ranges of T and S. A more general ap-62
proach is the so- called UNESCO-EoS (Fofono and Millard, 1983), derived63
from the fundamental work of Millero et a l. (1980) and Millero and Poisson64
(1981). It consists o f a set of 15 coefficients, to calculate the ocean’s sur-65
face density ρ
0
(T, S) = ρ(T, S, p = 0) and 26 subsequent coefficients for the66
secant bulk modulus κ to evaluate t he pressure dependence: ρ(T, S, p) =67
ρ
0
(T, S)/(1 p/κ(T, S, p)). This equation is valid over a large parameter range68
2
C < T < 40
C, 0 < S < 42 psu, and 0 < p < 10
8
Pa ( 10 000 m depth),69
and could hence be applied to the global ocean as a whole.70
However, a complication arises from the fact, that the ocean models intrinsic
variable is not the temperature T , but the potential temperature θ, which
excludes temperature changes induced by adiabatic processes. To bypass the
time-consuming conversion of different temperature representations in ocean
models, Jackett and McDougall (1995) published a modified set of coefficients
for the UNESCO-formulation. This allows a straight calculation of the density
from the potential temperature
ρ(θ, S, p) =
ρ
0
(θ, S)
1 p/κ(θ, S, p)
. (1)
The pressure in (1) is calculated from integrating the hydrostatic equation
p
z
= ρg p = g
Z
0
z
ρ(θ, S, p) dz (2)
from the surface t o the depth z. To improve efficiency in numerical ocean mod-71
els solving (1) and (2) iteratively, either the density of a former model-timestep72
has to be used, or another set of coefficients for the UNESCO-formulation of73
the EoS has to be applied, which allows for a depth-dependent density cal-74
culation instead of pressure ρ = ρ(θ, S, z) (Haidvogel and Beckmann, 1999).75
However, this set of coefficients is based on a ho mogeneously stratified standard76
ocean and has significant limits as soon as deviations from this standard strati-77
fication arise. Figure 1 indicates the deviation of the Haidvogel and Beckmann78
(1999) formulation from the Jackett and McDougall (1995) formulation as79
soon as the temperature, salinity and/or depth diverges from the assumed80
reference values, which refer to the mean oceanic properties.81
The most up-to-dat e approach for calculating the density of seawater depends82
on the Gibbs thermodynamic potential (e.g., Feistel, 1993; Feistel and Hagen,83
1995; Feistel, 2003; Jackett et al., 2006). Thermodynamic properties, like den-84
sity, freezing point, heat capacity, and many more, are calculated in a self-85
consistent way by derivatives from this Gibbs potential. The improved density86
algorithm provided by Jackett et al. (2006) shows only minimal adjustments87
with respect to Jackett and McDougall ( 1995). However, because of the consis-88
tency of the derived thermodynamic properties and the significant ly reduced89
3

S = 35 psu
0
1000
2000
3000
4000
5000
6000
Depth (m)
−5 0 5 10 15 20 25 30 35 40
Temperature (°C)
0
1000
2000
3000
4000
5000
6000
Depth (m)
−5 0 5 10 15 20 25 30 35 40
Temperature (°C)
1028
1032
1032
1036
1036
1040
1040
1044
1048
S = 0 psu
0
1000
2000
3000
4000
5000
6000
Depth (m)
−5 0 5 10 15 20 25 30 35 40
Temperature (°C)
0
1000
2000
3000
4000
5000
6000
Depth (m)
−5 0 5 10 15 20 25 30 35 40
Temperature (°C)
1000
1000
1004
1004
1008
1008
1012
1012
1016
1016
1020
1024
Jackett & McDougall (1995)
Haidvogel & Beckmann (1999)
Jackett et al. (2006)
0 1 2 3
Difference (kg/m
3
)
Fig. 1. Density (kg/m
3
) as a function of depth and potential temperature for
oceanic water masses (left) and fresh water (right). The blue and green lines, which
are quite close together, refer to Jackett and McDougall (1995) and Jackett et al.
(2006), respectively, while the red lines refers to Haidvogel and Beckmann (1999).
The background color indicates the increasing difference between the pressure-
and depth-dependent density according to Jackett and McDougall (1995) and
Haidvogel and Beckmann (1999).
computational effort, the implementation of the Gibbs-potential algor ithms in90
ocean models is the preferred formulation.91
1.2 Equation of freezing point ( EoFP)92
For a n adequate treatment of the ice-water interaction the equations for the
conservation of temperature and salinity are complemented by an equation to
calculate the pressure- and salinity-dependent f r eezing point of water (EoFP,
e.g., Holland and Jenkins, 1999)
T
f
= T
f
(S, p) αS + β + γp, (3)
where α = 0.057
C/psu, β = 0.0939
C, and γ = 7.64 · 10
4
C/dbar. For93
an analytic solution o f the complete set of the three equations a linearized94
version of the EoFP is needed as indicated on the right hand side of (3). This95
set of coefficients dating back to Foldvik and Kvinge (1974) is still in use in96
models dealing with ice-water interaction a nd has not always been replaced97
by a linearised version of the more precise (but higher order) formulatio n98
of Fofonoff and Millard (1983). One drawback of (3) is the need for regular99
4

temperature conversions between T and the models intrinsic variable θ. Also,100
the EoFP (3) was not designed for the high-pressure, low-salinity environ-101
ment within subglacial lakes, which are covered by several thousand meters102
of ice (Feistel, 2003, 2008). Jackett et al. (2006) present an algorithm to cal-103
culate the freezing point in terms of the potential temperature θ
f
= θ
f
(S, p),104
based on the Gibbs-potential considerations of Feistel ( 2003). This for mu-105
lation of the EoFP is also valid for high-pressure environments found in106
subglacial lakes. To make this formulation applicable with the analytic so-107
lution of the three-equation formulation, it has to be linearised with respect108
to the specific environmental needs (S mean-salinity-at-ice-water-interface,109
p mean-interface-depth). For subglacial Lake Vostok (with S = 0 psu and110
p 3700 m) the adjusted linearized equation (3) is indicated by the r ed line111
in Figure 2, while the original freezing point line (according to Jackett et a l.,112
2006) is drawn in black.113
2 Relevance for subglacial lake modelling114
In former studies of subglacial lake circulation, different formulations of the115
EoS have been applied. In the first three-dimensional numerical model studies116
of Lake Vostok, the simplistic Knudsen-Ekman equation was used (Williams,117
2001; Mayer et al., 2 003). Lat er studies dealing with Lake Vostok and Lake118
Concordia (Thoma et al., 2007, 2008a,b, 2009) applied the improved depth-119
dependent EoS after Haidvogel and Beckmann (1999). However, Figure 1 in-120
dicates that in the fresh-water regime of subglacial lakes the application of this121
convenient approach is questionable. Although the absolute densities are quite122
similar (Figure 1), the different vertical gradient and in particular the resulting123
significantly different isopycnal-vertices determine the cha r acteristics of flow124
and basal mass balance within subglacial lakes. The line of maximum density125
(LoMD) connects the vertices of the isopycnals, indicated as a dashed line in126
Figure 2. The LoMD determines if warming leads to rising of water masses127
or sinking. By using the improved Gibbs-potent ial formulation, the LoMD is128
moved to a greater depth compared to the Haidvogel and Beckmann (1999)129
approach. However, as long as the a lake’s depth below the ice surface remains130
well below the LoMD in Figure 2, t he principle circulation regime doesn’t131
change (Thoma et al., 2008 b).132
3 Updated subglacial lake model results133
The most up-to date model t o simulate the three-dimensional flow regime and134
the basal mass balance within subglacial lakes is Rombax (Thoma et al., 2007,135
5

Figures (13)
Citations
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Abstract: Antarctic subglacial lakes are thought to be extreme habitats for microbial life and may contain important records of ice sheet history and climate change within their lake floor sediments. To find whether or not this is true, and to answer the science questions that would follow, direct measurement and sampling of these environments are required. Ever since the water depth of Vostok Subglacial Lake was shown to be >500 m, attention has been given to how these unique, ancient, and pristine environments may be entered without contamination and adverse disturbance. Several organizations have offered guidelines on the desirable cleanliness and sterility requirements for direct sampling experiments, including the U.S. National Academy of Sciences and the Scientific Committee on Antarctic Research. Here we summarize the scientific protocols and methods being developed for the exploration of Ellsworth Subglacial Lake in West Antarctica, planned for 2012–2013, which we offer as a guide to future subglacial environment research missions. The proposed exploration involves accessing the lake using a hot-water drill and deploying a sampling probe and sediment corer to allow sample collection. We focus here on how this can be undertaken with minimal environmental impact while maximizing scientific return without compromising the environment for future experiments.

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  • ...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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Abstract: Ellsworth is 14.7 km ×3 .1 km with an area of 28.9 km 2 . Lake depth increases downlake from 52 m to 156 m, with a water body volume of 1.37 km 3 . The ice thickness suggests an unusual thermodynamic characteristic, with the critical pressure boundary intersecting the lake. Numerical modeling of water circulation has allowed accretion of basal ice to be estimated. We collate this physiographic and modeling information to confirm that Lake Ellsworth is ideal for direct access and propose an optimal drill site. The likelihood of dissolved gas exchange between the lake and the borehole is also assessed. Citation: Woodward, J., A. M. Smith, N. Ross, M. Thoma, H. F. J. Corr, E. C. King, M. A. King, K. Grosfeld, M. Tranter, and M. J. Siegert (2010), Location for direct access to subglacial Lake Ellsworth: An assessment of geophysical data and modeling, Geophys. Res. Lett., 37, L11501, doi:10.1029/ 2010GL042884.

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Abstract: . Subglacial lakes in Antarctica influence to a large extent the flow of the ice sheet. In this study we use an idealised lake geometry to study this impact. We employ a) an improved three-dimensional full-Stokes ice flow model with a nonlinear rheology, b) a three-dimensional fluid dynamics model with eddy diffusion to simulate the basal mass balance at the lake-ice interface, and c) a newly developed coupler to exchange boundary conditions between the two individual models. Different boundary conditions are applied over grounded ice and floating ice. This results in significantly increased temperatures within the ice on top of the lake, compared to ice at the same depth outside the lake area. Basal melting of the ice sheet increases this lateral temperature gradient. Upstream the ice flow converges towards the lake and accelerates by about 10% whenever basal melting at the ice-lake boundary is present. Above and downstream of the lake, where the ice flow diverges, a velocity decrease of about 10% is simulated.

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  • ...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)....

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  • ...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…...

    [...]

  • ...…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)....

    [...]

  • ...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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Journal ArticleDOI
Abstract: Several hundred subglacial lakes have been identified beneath Antarctica so far. Their interaction with the overlying ice sheet and their influence on ice dynamics are still subjects of investigation. While it is known that lakes reduce the ice-sheet friction towards a free-slip basal boundary condition, little is known about how basal melting and freezing at the lake/ice interface modifies the ice dynamics, thermal regime and ice rheology. In this diagnostic study we simulate the Vostok Subglacial Lake area with a coupled full Stokes 3-D ice-flow model and a 3-D lake-circulation model. The exchange of energy (heat) and mass at the lake/ice interface increases (decreases) the temperature in the ice column above the lake by up to 10% in freezing (melting) areas, resulting in a significant modification of the highly nonlinear ice viscosity. We show that basal lubrication at the bottom of the ice sheet has a significant impact not only on the ice flow above the lake itself, but also on the vicinity and far field. While the ice flow crosses Vostok Subglacial Lake, flow divergence is observed and modelled. The heterogeneous basal-mass-balance pattern at the lake/ice interface intensifies this divergence. Instead of interactive coupling between the ice-flow model and the lake-flow model, only a single iteration is required for a realistic representation of the ice/water interaction. In addition, our study indicates that simplified parameterizations of the surface temperature boundary condition might lead to a velocity error of 20% for the area of investigation.

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Abstract: Height changes of the ice surface above subglacial Lake Vostok, East Antarctica, reflect the integral effect of different processes within the subglacial environment and the ice sheet. Repeated GNSS (Global Navigation Satellite Systems) observations on 56 surface markers in the Lake Vostok region spanning 11 years and continuous GNSS observations at Vostok station over 5 years are used to determine the vertical firn particle movement. Vertical marker velocities are derived with an accuracy of 1 cm/yr or better. Repeated measurements of surface height profiles around Vostok station using kinematic GNSS observations on a snowmobile allow the quantification of surface height changes at 308 crossover points. The height change rate was determined at 1 ± 5 mm/yr, thus indicating a stable ice surface height over the last decade. It is concluded that both the local mass balance of the ice and the lake level of the entire lake have been stable throughout the observation period. The continuous GNSS observations demonstrate that the particle heights vary linearly with time. Nonlinear height changes do not exceed ±1 cm at Vostok station and constrain the magnitude of spatiotemporal lake-level variations. ICESat laser altimetry data confirm that the amplitude of the surface deformations over the lake is restricted to a few centimeters. Assuming the ice sheet to be in steady state over the entire lake, estimates for the surface accumulation, on basal accretion/melt rates and on flux divergence, are derived.

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

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Abstract: A spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible Navier-Stokes equations. In the spectral element discretization, the computational domain is broken into a series of elements, and the velocity in each element is represented as a high-order Lagrangian interpolant through Chebyshev collocation points. The hyperbolic piece of the governing equations is then treated with an explicit collocation scheme, while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle. The implementation of the technique is demonstrated on a one-dimensional inflow-outflow advection-diffusion equation, and the method is then applied to laminar two-dimensional (separated) flow in a channel expansion. Comparisons are made with experiment and previous numerical work.

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Abstract: Ocean models based on consistent hydrostatic, quasi-hydrostatic, and nonhydrostatic equation sets are formulated and discussed. The quasi-hydrostatic and nonhydrostatic sets are more accurate than the widely used hydrostatic primitive equations. Quasi-hydrostatic models relax the precise balance between gravity and pressure gradient forces by including in a consistent manner cosine-of-latitude Coriolis terms which are neglected in primitive equation models. Nonhydrostatic models employ the full incompressible Navier Stokes equations; they are required in the study of small-scale phenomena in the ocean which are not in hydrostatic balance. We outline a solution strategy for the Navier Stokes model on the sphere that performs efficiently across the whole range of scales in the ocean, from the convective scale to the global scale, and so leads to a model of great versatility. In the hydrostatic limit the Navier Stokes model involves no more computational effort than those models which assume strict hydrostatic balance on all scales. The strategy is illustrated in simulations of laboratory experiments in rotating convection on scales of a few centimeters, simulations of convective and baroclinic instability of the mixed layer on the 1- to 10-km scale, and simulations of the global circulation of the ocean.

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)....

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Journal ArticleDOI
01 Jun 1981
Abstract: The density measurements by Millero, Gonzalez and Ward (1976, Journal of Marine Research,34, 61–93) and Poisson, Brunet and Brun-Cottan (1980, Deep-Sea Research, 27, 1013–1028), from 0 to 40°C and 0.5 to 43 salinity, have been used to determine a new 1-atm equation of state for seawater. The equation is of the form (t°C; S; ϱ kg m−3) ρ=ρ 0 +AS+BS 3 2 +CS , where A=8.24493×10 −1 −4.0899×10 −3 t+7.6438×10 −5 t 2 −8.2467×10 −7 t 3 +5.3875×10 −9 t 4 B=−5.72466×10 −3 +1.0227×10 −4 t−1.6546×10 −6 t 2 C=4.8314×10 −4 and ϱ0 is the density of water ( Bigg , 1967, British Journal of Applied Physics, 8, 521–537). ρ 0 =999.842594+6.793952×10 −2 t−9.095290× −3 t 2 +1.001685×10 −4 t 3 −1.120083×10 −6 t 4 +6.536336×10 −9 t 5 . The standard error of the equation is 3.6 × 10−3 kg m−3. This equation will become the new 1-atm equation of state of seawater that has been suggested for use by the UNESCO (United Nations Educational, Scientific and Cultural Organization) joint panel on oceanographic tables and standards.

544 citations


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Q1. What are the contributions in "A comment on the equation of state and the freezing point equation with respect to subglacial lake modelling" ?

This article aims three tasks: Then the authors describe the impact of the recent and self-consistent Gibbs 12 thermodynamic potential-formulation of the EoS and the EoFP on subglacial lake 13 modeling. Finally, the authors show that the circulation regime of subglacial lakes covered 14 by at least 3000 m of ice, in principle, is independent of the particular formula15 tion, in contrast to lakes covered by a shallower ice sheet, like e. g., subglacial Lake 16 Ellsworth. However, as modeled values like the basal mass balance or the distri17 bution of accreted ice at the ice-lake interface are sensitive to different EoS and 18 EoFP, the authors present updated values for subglacial Lake Vostok and subglacial Lake 19 Concordia.