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Journal ArticleDOI

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

TL;DR: In this paper, the most common formulations of the EOS and the EOFP applied in numerical ocean and lake models during the past decades are discussed and the impact of the recent and selfconsistent Gibbs thermodynamic potential formulation on subglacial lake modelling.
About: This article is published in Earth and Planetary Science Letters.The article was published on 2010-05-15 and is currently open access. It has received 16 citations till now. The article focuses on the topics: Subglacial lake & Lake Vostok.

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
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Journal ArticleDOI
TL;DR: In this paper, the authors summarize the scientific protocols and methods being developed for the exploration of Ellsworth Subglacial Lake in West Antarctica, planned for 2012-2013, which they offer as a guide to future subglacial environment research missions.
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.

153 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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Journal ArticleDOI
TL;DR: Woodward et al. as mentioned in this paper used geophysical data and modeling information to confirm that Lake Ellsworth is ideal for direct access and propose an optimal drill site, and the likelihood of dissolved gas exchange between the lake and the borehole is also assessed.
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.

115 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]....

    [...]

Journal ArticleDOI
TL;DR: In this article, an improved three-dimensional full-Stokes ice flow model with a nonlinear rheology was employed to simulate the basal mass balance at the lake-ice interface, and a newly developed coupler to exchange boundary conditions between the two individual models.
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.

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…...

    [...]

  • ...…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....

    [...]

Journal ArticleDOI
TL;DR: In this article, repeated measurements of surface height profiles around Vostok station using kinematic GNSS observations on a snowmobile allow the quantification of surface surface height changes at 308 crossover points.
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.

30 citations

Journal ArticleDOI
TL;DR: In this paper, the authors simulate the Vostok Subglacial Lake area with a coupled full Stokes 3D ice-flow model and a 3D lake-circulation model and 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.
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.

28 citations

References
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Journal ArticleDOI
TL;DR: A preconditioner is used which, in the hydrostatic limit, is an exact integral of the Poisson operator and so leads to a single algorithm that seamlessly moves from nonhydrostatic to hydrostatic limits, competitive with the fastest ocean climate models in use today.
Abstract: The numerical implementation of an ocean model based on the incompressible Navier Stokes equations which is designed for studies of the ocean circulation on horizontal scales less than the depth of the ocean right up to global scale is described. A "pressure correction" method is used which is solved as a Poisson equation for the pressure field with Neumann boundary conditions in a geometry as complicated as that of the ocean basins. A major objective of the study is to make this inversion, and hence nonhydrostatic ocean modeling, efficient on parallel computers. The pressure field is separated into surface, hydrostatic, and nonhydrostatic components. First, as in hydrostatic models, a two-dimensional problem is inverted for the surface pressure which is then made use of in the three-dimensional inversion for the nonhydrostatic pressure. Preconditioned conjugate-gradient iteration is used to invert symmetric elliptic operators in both two and three dimensions. Physically motivated preconditioners are designed which are efficient at reducing computation and minimizing communication between processors. Our method exploits the fact that as the horizontal scale of the motion becomes very much larger than the vertical scale, the motion becomes more and more hydrostatic and the three- dimensional Poisson operator becomes increasingly anisotropic and dominated by the vertical axis. Accordingly, a preconditioner is used which, in the hydrostatic limit, is an exact integral of the Poisson operator and so leads to a single algorithm that seamlessly moves from nonhydrostatic to hydrostatic limits. Thus in the hydrostatic limit the model is "fast," competitive with the fastest ocean climate models in use today based on the hydrostatic primitive equations. But as the resolution is increased, the model dynamics asymptote smoothly to the Navier Stokes equations and so can be used to address small- scale processes. A "finite-volume" approach is employed to discretize the model in space in which property fluxes are defined normal to faces that delineate the volumes. The method makes possible a novel treatment of the boundary in which cells abutting the bottom or coast may take on irregular shapes and be "shaved" to fit the boundary. The algorithm can conveniently exploit massively parallel computers and suggests a domain decomposition which allocates vertical columns of ocean to each processing unit. The resulting model, which can handle arbitrarily complex geometry, is efficient and scalable and has been mapped on to massively parallel multiprocessors such as the Connection Machine (CM5) using data-parallel FORTRAN and the Massachusetts Institute of Technology data-flow machine MONSOON using the implicitly parallel language Id. Details of the numerical implementation of a model which has been designed for the study of dynamical processes in the ocean from the convective, through the geostrophic eddy, up to global scale are set out. The "kernel" algorithm solves the incompressible Navier Stokes equations on the sphere, in a geometry as complicated as that of the ocean basins with ir- regular coastlines and islands. (Here we use the term "Navier Stokes" to signify that the full nonhydrostatic equations are being employed; it does not imply a particular constitutive relation. The relevant equations for modeling the full complex- ity of the ocean include, as here, active tracers such as tem- perature and salt.) It builds on ideas developed in the compu- tational fluid community. The numerical challenge is to ensure that the evolving velocity field remains nondivergent. Most

2,315 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)....

    [...]

Journal ArticleDOI
TL;DR: In this article, a spectral element method was proposed for numerical solution of the Navier-Stokes equations, where the computational domain is broken into a series of elements, and the velocity in each element is represented as a highorder Lagrangian interpolant through Chebyshev collocation points.

2,133 citations

DOI
01 Jan 1983
TL;DR: In this article, the APSO Joint Panel on Oceanographic Tables and Standards (APSO) and SCOR Working Group (SCOR) have published a joint report on oceanographic tables and standards.
Abstract: Endorsed by Unesco/SCOR/ICES/lAPSO Joint Panel on Oceanographic Tables and Standards and SCOR Working Group 51

1,534 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)....

    [...]

  • ...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)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the Navier Stokes model on the sphere has been used to model the global circulation of the ocean, from the convective scale to the global scale, and a solution strategy has been proposed to deal with small-scale phenomena which are not in hydrostatic balance.
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.

1,302 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)....

    [...]

Journal ArticleDOI
01 Jun 1981
TL;DR: In this article, the authors proposed a new 1-atm equation of state for seawater that has been suggested for use by the United Nations Educational, Scientific and Cultural Organization (UNESCO) joint panel on oceanographic tables and standards.
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.

599 citations

Frequently Asked Questions (1)
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.