Ocean models of the Southern Ocean

From ACCE
Revision as of 15:33, 6 August 2014 by Maintenance script (Talk)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This page is part of the topic Models of the physical and biological environment of the Antarctic

Introduction

All numerical models involve a compromise between cost and physical realism. This is especially true for ocean models because the computational cost is always high, and can be hundreds of times that of a comparable atmospheric model.

This underlying conflict arises because the scale of ocean features is small compared with the size of the ocean and because the computer effort required is proportional to the cube of the horizontal resolution. However, the time step of integration may be longer for the ocean models because velocities are smaller (stability condition depends on speed). Thus, although ocean models need to resolve smaller scales, they save on time-step compared to atmosphere models.

Horizontal scales in the ocean are usually determined by the Rossby radius - a measure of how far the lowest internal wave mode can travel before being affected by the Earth's rotation. In the sub-tropics where ocean models were first developed, the Rossby radius is typically 25 km (in the atmosphere the Rossby radius is nearer 250 km). In polar regions, where the ocean is less stratified, this can drop to 8 km or less. The weak stratification also means that the influence of bottom topography is much stronger in polar regions, so when steep topography is involved a fine horizontal resolution is again required.

The lack of sufficient synoptic data for initialising and validating ocean models is also an issue because it means that any estimate of skill has to be based largely on qualitative judgements. Even so, because we have a good theoretical understanding of the problems involved, long model runs usually show up gross errors. We also have satellites, which provide good data on the surface fields, and the Argo programme is starting to provide a good data set for the subsurface and near surface layers. Detailed studies of density currents, the sea ice field and other key regions provide further local checks on model performance.

The major model choices

Most ocean models represent the spatial structure of the ocean by storing the model variables on a regular horizontal and vertical grid. A few (Iskandarani et al., 2003[1]) split the ocean into finite elements within which the spatial variation is represented by linear or higher order functions. The latter approach is widely used by engineers modelling steady state structures, but it has not been widely adopted by oceanographers, possibly because of cost. If the finite elements are not on a regular grid there are also problems with spurious reflection and refraction.

Arakawa (1966[2]) investigates the five standard ways that velocity and other model variables can be arranged on a regular horizontal grid. Of these he shows that two, the Arakawa B and C grids, are most accurate at representing the large-scale circulation of the ocean and atmosphere. The B-grid is slightly better at representing geostrophic flows within the ocean, i.e. flows in which the main balance is between the horizontal pressure gradient and the Coriolis force. For this reason the B-grid is used by many large-scale ocean models such as MOM (Griffies et al., 2004[3]) and OCCAM (Coward and de Cuevas, 2005[4]; Webb and de Cuevas, 2007[5]).

The C-grid is slightly better at representing gravity waves and so is usually used for models near the coast where tides are important or where turbulent effects are large, so that flows are not geostrophic. It is also used for some large-scale models (Penduff et al., 2007[6]), where the improved representation of gravity waves makes it easier to add sea ice. On the B-grid, gravity waves on the 'black' and 'white' sub-grids (to use a chess analogy) are only weakly coupled. In regions where internal waves are active this can produce an apparently noisy temperature field in the surface layers, which, if left uncorrected, affects the sea ice field.

In the vertical dimension, three standard schemes are used. Many, following the original Bryan and Cox model, use fixed horizontal layers (Bryan and Cox, 1968[7]; Semtner, 1974[8]; Cox, 1984[9]). Together with the horizontal grid this generates a 3-D array of grid-boxes, each model variable nominally placed at the centre of each box defining the mean value of the variable within the box. Modern versions usually extend the vertical grid to include a free upper surface, allowing tides and other waves, and a variable thickness bottom box in each column, for better representation of topography.

The one major problem of the horizontal level scheme arises because most density surfaces within the ocean are gently sloping and there is little mixing between water masses of different density. The level scheme only allows fluxes through the horizontal and vertical faces of each box and this produces spurious numerical mixing between water masses of different density. To overcome this, isopycnal models have been developed (Bleck et al., 1992[10]; Hallberg, 1997[11]; Chassignet et al., 2007[12]) in which the layers correspond to constant potential density surfaces within the ocean. The scheme works well in removing numerical mixing. It can also handle overflows well. There are problems in handling mixed layers (Gnanadesikan et al., 2007[13]), in handling the non-linear effect of temperature on compressibility (so really a unique constant potential density surface does not exist) and in handling regions where weak stratification means that only a few model layers are present. However the gain from reduced numerical mixing is often more important.

A third scheme, used most often near coastlines, is to split the water column into a fixed number of layers (Haidvogel et al., 1991[14]; Song and Haidvogel, 1994[15]; Blumberg and Mellor, 1987[16]; Mellor, 2003[17]). This then gives extra vertical resolution in the shallow water. In deep water it suffers from the same numerical mixing problem as fixed layer models (Willebrand et al., 2001[18]). The sloping surfaces also introduce small errors in calculating the horizontal pressure gradient (Shchepetkin and McWilliams, 2003[19]). This does not matter in turbulent shallow water but in the deep ocean it can generate spurious currents.

Once the grid is chosen, grid variables are used to represent the standard momentum and tracer equations describing the ocean (Griffies, 2006[20]). They are written so that momentum, heat and salinity are conserved, but as they miss sub-gridscale processes, they can only provide approximate solutions to the full set of differential equations.

Most of the errors discussed can be reduced by using higher order numerical schemes or by using finer horizontal or vertical resolution. There is even a suspicion that we are starting to see a convergence in the results obtained from the different fine resolution models, so that in the end computational efficiency may be the only criteria left at this level of model choice.

Sub-grid scale and process models

Within the large scale model framework there are usually a number of process models representing the surface mixed layer, the bottom boundary layer, sea ice, floating ice shelves, small scale mixing and other processes – especially those that act at a scale smaller than the model resolution. Usually such process models can be adapted for use in all of the different large-scale models and, because a large variety of such process models have been developed, there is usually a selection available for each run of a large scale model (Griffies et al., 2004[3]). The skill of the final model may depend critically on the choices made at this stage.

Sea ice and the ice shelves

Sea ice in the Southern Ocean differs from that of the Arctic in two key ways. First, away from the coastline, the ice usually lasts for only a single year, so complex multi-year ice models are not essential. Secondly, the primarily westerly winds mean that the flow is generally divergent, so modelling of ice rheology is not so essential. As a result Southern Ocean models that use simple ice models (i.e. Semtner, 1976[21]; Hibler, 1979[22]), give good results away from the coastlines. Near the coast, especially in parts of the Weddell and Ross Seas, more complicated models are required. In such areas the Hunke and Dukowicz (1997[23]) elastic–viscous–plastic scheme is often chosen.

In the past the ocean under floating ice shelves has often been ignored in ocean models. Recently the situation has changed, and successful models of the flows under such ice shelves have been developed. These are discussed later. As confidence in these models develops they are likely to be more widely adopted.

Overflows and bottom boundary layer

The cooling and ice formation processes that occur on the continental shelf around Antarctica result in an important source of very dense water which eventually sinks down the continental slope to form some of the densest waters of the deep ocean. This 'overflow' occurs in a very thin layer, typically 30 m thick, which has proved almost impossible to represent correctly unless the model itself has a finer vertical resolution (Legg et al., 2006[24]).

Initially it was thought that analytic sub-grid scale models could be used (Baringer and Price, 1997[25]), but with realistic flows and topography these were found to be unstable. The lack of a realistic sub-grid model has its greatest effect on horizontal or z-layer models but schemes such as those of Döscher and Beckmann (2000[26]) can be used to reduce the error.

Isopycnal models can be more successful as long as one of the density layers corresponds to the thin descending plume (Willebrand et al., 2001[18]). Density is not fixed but depends non-linearly on both pressure and water properties, so the assumed existence of constant potential density layers produces small errors. Also the large range of densities found in the ocean, and the necessity of modelling small density differences in regions of weak stratification, means that the number of density layers in the model needs to be large.

Sigma coordinates have the advantage that extra resolution can be provided near the bottom. The bottom layer also follows the descending plume, its thickness increasing with depth. In principal, constant thickness lower layers can also be added to the z-layer models, the normal preference for global ocean models.

For long-term climate integrations, the realistic representation of overflows remains a major problem that still needs to be solved. Its importance arises because it affects the replenishment of bottom waters and thus the large-scale vertical structure of the ocean. However for the study of short-term and near surface processes, the effect of any such error is usually small.

Upwelling, subduction and the mixed layer

Offshore in the Southern Ocean, key processes include the upwelling of dense water in the south, the transport northwards of this water in the surface Ekman layer, and the mixing and sinking of intermediate waters in the north. If the wind stress is correct, then momentum conservation ensures that the total transport in the surface Ekman layer is also correct. The velocity of the Ekman layer, which affects sea ice, depends on near surface mixed layer processes, and these vary from model to model. Thus for any research involving sea ice, a good mixed-layer model is essential.

Available mixed layer models include those of Pacanowski and Philander (1981[27]), Mellor and Yamada (1982[28]), Price et al. (1986[29]), Large et al. (1994[30]) and Gaspar et al. (1990[31]), and extend from simple bulk models to detailed turbulent closure models. Their effectiveness has not been seriously tested with the range of conditions (ice, stratification and surface forcing) found in the Southern Ocean, so at present all should be used with caution.

Both upwelling and subduction involve advection of water along sloping density layers. This is handled best by isopycnal models (Willebrand et al., 2001[18]). However subduction also involves interaction with the mixed layer and capping at the end of winter and here the isopycnal layer models have problems (Large and Nurser, 1998[32]).

Mixing – tides, topography, currents

The representation of mixing in the ocean is a huge subject, which can only be briefly discussed here. Near the surface the effect of wind and breaking surface waves is included in the mixed layer models. On continental shelves the extra effect of bottom turbulence due to the currents may fully mix the water column, and as tides produce their own currents, their influence also needs to be included.

Away from the boundaries, vertical mixing occurs primarily due to breaking internal waves. The energy for these waves may come from the wind acting via the surface mixed layer, from the propagation of internal tides and from the interaction of currents with bottom topography. However the processes are still only poorly understood. Most mixing models represent such effects by simple Laplacian diffusion, possibly with larger values near topography. Recent research indicates that in the Southern Ocean mixing is largest in areas of strong bottom currents so there is a case for increasing the values in these regions as well. However while numerical mixing remains a problem it is likely that, except in isopycnal models, the effective vertical mixing will be too large.

Horizontal mixing is also important in the ocean, especially in frontal regions where gradients are large. In low-resolution models the main sub-gridscale process that needs to included is baroclinic instability. The Gent and McWilliams scheme (Gent and McWilliams, 1990[33]; Gent et al. 1995[34]; Griffies, 1998[35]) has been used to represent such processes, but there are concerns about how realistic it is, especially near the ocean surface or bottom topography. As a result, if frontal regions are important then it is best to use a model with a resolution of less than the Rossby radius.

Finally, because the Southern Ocean is only weakly stratified, bottom topography effectively steers the currents throughout the whole water column. It is therefore essential that topography is accurately represented. If smoothing is carried out, as it is usually done in sigma coordinate models to reduce errors in the pressure term, then the errors produced by the smoothing need to be addressed.

Available models of the Southern Ocean

For large-scale studies of the Southern Ocean it is probably best to start with the fine resolution global models for which model data is readily available. These are OCCAM and the Parallel Ocean Model POP (Maltrud and McClean, 2005[36]; Collins et al., 2006[37]) with a resolution of 0.1 degrees or less. POP is available in both the original and the NCAR community versions.

OCCAM and POP are related to the original Bryan-Cox-Semtner code. If you want to run or develop your own version of this code, the best supported version is MOM (Griffies et al., 2004[3], 2005[38]) but both POP and OCCAM have made code available, for example, for developing biological models (Popova et al., 2006[39]). The NEMO (Nucleus for European Modelling of the Ocean), ORCA025 and ORCA12 (Madec et al., 1998[40]) models are primitive equation models adapted for regional and global ocean circulation problems. With a resolution of ¼° or 1/12° (Madec, 2008[41]) NEMO is intended to be a flexible tool for studying the ocean and its interactions with the others components of the Earth’s climate system (atmosphere, sea-ice, biogeochemical tracers etc) over a wide range of space and time scales.

Other global studies which have been carried out at lower resolution include the HYCOM (Chassignet et al., 2006[42], 2007[12]) and POM (Mellor, 2003[17]) models. HYCOM is an isopycnal model adapted to use level coordinates in the near surface layer. POM is a widely used sigma coordinate model. At low resolution we are also starting to see more operational models. These combine one of the regular models with data from satellites and other sources to provide a more accurate view of the ocean and its circulation (Chassignet and Verron, 2006[43]).

Regional Models

A major problem that arises when developing regional models is how best to deal with the open boundary. Flow through the boundary usually dominates the large-scale circulation within the region under study, so any errors seriously affect the results. In such cases the best solution is to specify the boundary conditions using data taken from one of the global models. A less satisfactory solution is to specify the boundary conditions using climatology.

At the largest scale there have been three major models that cover just the Southern Ocean. The earliest, FRAM, was a rigid-lid level model without sea ice. It relaxed to climatology at the surface and at the open boundary (FRAM Group, 1991[44]).

The later BRIOS model is a sigma co-ordinate model based on Haidvogel's SPEM code (Haidvogel et al., 1991[14]), which uses a Hibler type ice model. The Southern Ocean versions use 24 layers in the vertical and have a horizontal resolution of 1.5 degrees or less, with finer resolution in areas such as the Weddell Sea (Beckmann et al., 1999[45]). It is important because it is the first of the large-scale models to include the ocean under the ice-shelves.

Another important large-scale model is the isopycnal model which Hallberg and Gnanadesikan (2006[46]) used to investigate the effect of horizontal resolution in the Southern Ocean. The model uses 20 density layers but has no sea ice. Of the three model types this is probably the best suited to studies of the transport of mid-depth water masses through the ocean.

The BRIOS Model

In the past, most large-scale models ignored the regions of ocean under the ice shelves. This was a serious omission but it arose because no suitable computer codes had been developed for handling the revised upper boundary condition.

Such codes have now been developed, one of the first to be widely used being part of the BRIOS model, discussed above. It was originally used to study flows in the Weddell Sea region (Beckmann et al., 1999[45]; Timmermann et al., 2002[47]) but has also been used elsewhere around Antarctica (Assmann et al., 2003[48]). The model extends the sigma coordinate scheme under the ice shelves with the 'ocean surface' coordinate following the bottom contour of the ice shelf. As a result, vertical resolution is good under the ice shelf. The fact that the model can be run in circumpolar mode also means that problems with the open boundary condition have little effect on the shelf circulation. As with other sigma coordinate models the main problems are due to numerical mixing and the necessity to smooth topography to reduce pressure gradient errors.

Isopycnal model

An alternative approach is that of Holland (Holland et al., 2003[49]; Jenkins et al., 2004[50]), who has modified the MICOM isopycnal model to include ice shelves. As with other pure isopycnal schemes, vertical resolution is obtained by making a judicious choice of model density levels for the area under study. The model typically uses ten density layers, but the weak stratification of some regions of the ice cavity means that only a few layers are involved, so the effective vertical resolution can be very coarse. However the advantage of the method is that it does not suffer from numerical mixing in the same way as the sigma coordinate model, and there is no pressure gradient error. The model is thus a useful independent check on the circulation.

The model of Dinniman et al. (2007[51])

A second sigma coordinate model for use under the ice shelves has been developed by Dinniman et al. (2007[51]), based on the ROMS model (Shchepetkin and McWilliams, 2005[52]). Both this and the BRIOS model use 24 layers in the vertical with a concentration of layers towards the top and bottom, so the main differences are at the process model level. Thus where the BRIOS model uses a Hibler ice model, Dinniman et al. preferred to impose an ice climatology based on satellite observations. They did this because during the period of study (2001-2003) the sea ice was affected by large ice islands and behaved in a complex way, which was unlikely to be reproduced by a standard ice model. Such parallel developments need be encouraged because of the insights they give into the strengths and weaknesses of different approaches.

Under Ice Shelf Models

It is important to have realistic models of the ocean flow under ice shelves because of the crucial role that the shelves play in the climate system of the Antarctic, and their sensitivity to changes in water masses. At present global and regional climate models do not include sub-ice shelf cavities; these must be included in the future and the current generation of ocean/shelf models provides a step in this direction.

At the moment high resolution ocean models are run across limited areas of the Southern Ocean with atmospheric forcing being provided by the reanalysis data sets or global or regional atmospheric models. These allow the investigation of the changes of water masses under the ice shelves and the interaction with the broader scale ocean environment (e.g. Hellmer, 2004[53]).

Concluding comments

In future there are likely to be two areas where specialised models of the Southern Ocean need development. The first is in the study of the biology of the Southern Ocean, and especially the communities that develop under the immense areas of sea ice. The second is in the study of land ice and its response to climate change. Here the processes occurring under the ice shelves may have a significant impact.

For the biological studies, the main weaknesses of the physical models is likely to be in the representation of the surface mixed layer and the detailed properties of the surface ice field. It is easy to suggest possible improvements to the process models, but what are lacking are sets of good year-round data from the Southern Ocean that can be used to test them. Data from the Argo floats is helping to fill the gaps but there is still very little data from the large areas of open ocean covered by sea ice.

For the flows under the ice shelves data are becoming available from boreholes and by other means. Here a model intercomparison experiment, along the lines of the DYNAMO project (Willebrand et al., 2001[18]), would be useful. This should compare the results of a sigma coordinate model with isopycnal and z-layer models of comparable vertical and horizontal resolution, and be designed initially to investigate the size and effect of the error terms. Once these are quantified then people would have a lot more confidence in using the models to predict future changes.

References

  1. ↑ Iskandarani, M., Haidvogel, D.B. and Levin, J.C. 2003. A three-dimensional spectral element model for the solution of the hydrostatic primitive equations, Journal of Computational Physics, 186, 397-425.
  2. ↑ Arakawa, A. 1966. Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow, Part 1, Journal of Computational Physics, 1, 119-143.
  3. ↑ 3.0 3.1 3.2 Griffies, S.M., Harrison, M.J., Pacanowski, R.C. and Rosati, A. 2004. A technical guide to MOM4. GFDL ocean group technical report No. 5. NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, USA.
  4. ↑ Coward, A. and De Cuevas, B. 2005. The OCCAM 66 level model: model description, physics, initial conditions and external forcing, Tech. rep., Southampton Oceanography Centre, Internal Document No. 99, 83 pp. (http://eprints.soton.ac.uk/18792).
  5. ↑ Webb, D.J. and De Cuevas, B.A. 2007. On the fast response of the Southern Ocean to changes in the zonal wind, Ocean Science, 3, 417-427.
  6. ↑ Penduff, T., Le Sommer, J., Barnier, B., Treguier, A.-M., Molines, J.-M., and Madec, G., 2007: Influence of numerical schemes on current-topography interactions in 1/4° global ocean simulations, Ocean Science, 3, 509-524.
  7. ↑ Bryan, K. and Cox, M.D. 1968. A non-linear model of an ocean drivem by wind and differential heating: Part 1. Description of the Three-dimensional velocity and density fields, Journal of the Atmospheric Sciences, 25 (6), 945-967.
  8. ↑ Semtner, A.J. 1974. An oceanic general circulation model with bottom topography, Technical Report No. 9, Dept. of Meteorology, UCLA, Los Angeles, CA 90095, USA.
  9. ↑ Cox, M.D. 1984. A primitive equation 3-dimensional model of the ocean. Tech. Rep. 1, Geophys. Fluid Dyn. Lab, Natl. Oceanic and Atmos. Admin., Princeton Uni. Press, Princeton, N.J.
  10. ↑ Bleck, R., Rooth, C., Hu, D. and Smith, L.T. 1992. Ventilation patterns and mode water formation in a wind- and thermodynamically driven isopycnic coordinate model of the North Atlantic, Journal of Physical Oceanography, 22, 1486-1505.
  11. ↑ Hallberg, R.W. 1997. Stable split time stepping schemes for large-scale ocean modelling, Journal of Computational Physics, 135, 54-6.
  12. ↑ 12.0 12.1 Chassignet, E.P., Hurlburt, H.E., Smedstad, O.M., Halliwell, G.R., Hogan, P.J., Wallcraft, A.J., Baraille, R. and Bleck, R. 2007. The HYCOM (HYbrid Coordinate Ocean Model) data assimilative system, Journal of Marine Systems, 65, 60-83.
  13. ↑ Gnanadesikan, A., Griffies, S.M. and Samuels, B.L. 2007. Effects in a climate model of slope tapering in neutral physics schemes, Ocean Modelling, 16, 1-16.
  14. ↑ 14.0 14.1 Haidvogel, D.B., Wilkin, J.L. and Young, R.E. 1991. A semi-spectral primitive equation ocean circulation model using vertical sigma and orthogonal curvilinear horizontal coordinates, Journal of Computational Physics, 94, 151-185.
  15. ↑ Song, Y.T. and Haidvogel, D. 1994. A semi-implicit ocean circulation model using a generalized topography following coordinate system, Journal of Computational Physics, 115, 228-248.
  16. ↑ Blumberg, A.F. and Mellor, G.L. 1987. A description of a three-dimensional coastal ocean circulation model, In Three-Dimensional Coastal Ocean Models, Heaps, N.S. (Ed.), 1-16, American Geophysical Union, Washington, DC.
  17. ↑ 17.0 17.1 Mellor, G.L. 2003. Users guide for a three-dimensional, primitive equation, numerical ocean model (June 2003 version), 53 pp., Prog. in Atmos. and Ocean. Sci, Princeton University.
  18. ↑ 18.0 18.1 18.2 18.3 Willebrand, J., Barnier, B., Boening, C., Dieterich, C., Killworth, P., Le Provost, C., Jia, Y., Molines, J. and New, A.L. 2001. Circulation characteristics in three eddy-permitting models of the North Atlantic, Progress in Oceanography, 48, 123-161.
  19. ↑ Shchepetkin, A.F. and McWilliams, J.C. 2003. A method for computing horizontal pressure-gradient force in an oceanic model with a nonaligned vertical coordinate, Journal of Geophysical Research, 108 (c3), 3090, doi:10.1029/2001JC001047.
  20. ↑ Griffies, S.M. 2006. Some ocean model fundamentals. in: Ocean Weather Forecasting: an integrated view of Oceanography. E. P. Chassignet and J. Verron, eds., Berlin, Germany: Springer, 19-74.
  21. ↑ Semtner, A. 1976. A model for the thermodynamic growth of sea ice in numerical investigations of climate, Journal of Physical Oceanography, 6 (3), 379-389.
  22. ↑ Hibler III, W.D. 1979. A dynamic-thermodynamic sea ice model, Journal of Physical Oceanography, 9, 815-846.
  23. ↑ Hunke, E.C. and Dukowicz, J.K. 1997. An Elastic–Viscous–Plastic Model for Sea Ice Dynamics, Journal of Physical Oceanography, 27 (9), 1849-1867.
  24. ↑ Legg, S., Hallberg, R.R. and Girton, J.B. 2006. Comparison of entrainment in overflows simulated by z-coordinate, isopycnal and non-hydrostatic models, Ocean Modelling, 11 (1-2), 69-79.
  25. ↑ Baringer, M.O. and Price, J.F. 1997. Momentum and Energy Balance of the Mediterranean Outflow, Journal of Physical Oceanography, 27 (8), 1678-1692.
  26. ↑ Döscher, R. and Beckmann, A. 2000. Effects of a Bottom Boundary Layer Parameterization in a Coarse-Resolution Model of the North Atlantic Ocean, Journal of Atmospheric and Oceanic Technology, 17(5), 698-707.
  27. ↑ Pacanowski, R.C. and Philander, S.G.H. 1981. Paramerization of vertical mixing in numerical models of tropical oceans, Journal of Physical Oceanography, 11, 1443-1451.
  28. ↑ Mellor, G.L. and Yamada, T. 1982. Development of a turbulent closure model for geophysical fluid problems, Geophysics and Space Physics, 20, 851-875.
  29. ↑ Price, J.F., Weller, R.A. and Oinkel, R. 1986. Duirnal cycling: Observations and models of the upper ocean response to diurnal heating, cooling and wind mixing, Journal of Geophysical Research, 91, 8411-8427.
  30. ↑ Large, W.G., Williams, J.C. and Doney, S.C. 1994. Oceanic vertical mixing: a review and a model with nonlocal boundary layer parameterization, Reviews of Geophysics, 32, 363-403.
  31. ↑ Gaspar, P., Gregoris, Y., and Lefevre, J.-M. 1990. A simple eddy kinetic energy model for stimulations of the oceanic vertical mixing: tests at station Papa and Long-Term Upper Ocean Study Site, Journal of Geophysical Research, 95, 16179-16193.
  32. ↑ Large, W.G. and Nurser, A.J.G. 1998. Ocean Surface Water Mass Transformations, 317-336 in: Eds: Siedler, G., Church, J. and Gould, J., Ocean Circulation and Climate. Academic Press, San Diego, 715 pp.
  33. ↑ Gent, P.R. and McWilliams, J.C. 1990. Isopycnal mixing in ocean circulation models, Journal of Physical Oceanography, 20, 150-155.
  34. ↑ Gent, P.R, Willebrand, J., McDougall, T.J. and McWilliams, J.C. 1995. Parameterizing eddy-induced tracer transports in ocean circulation models, Journal of Physical Oceanography, 25, 463-474.
  35. ↑ Griffies, S.M. 1998. The Gent–McWilliams Skew Flux, Journal of Physical Oceanography, 28, 831-841.
  36. ↑ Maltrud, M.E. and McClean, J.L. 2005. An eddy resolving global 1/10 degrees ocean simulation, Ocean Modelling, 8(1-2), 31-54.
  37. ↑ Collins, W.D. and coauthors. 2006. The Community Climate System Model Version 3 (CCSM3), Journal of Climate, 19, 2122-2143.
  38. ↑ Griffies, S.M., Gnanadesikan, A., Dixon, K.W., Dunne, J.P., Gerdes, R., Harrison, M.J., Rosati, A., Russell, J.L., Samuels, B. L., Spelman, M.J., Winton, M. and Zhang, R. 2005. Formulation of an ocean model for global climate simulations, Ocean Science, 1, 45-79.
  39. ↑ Popova, E.E., Coward, A.C., Nurser, G.A., De Cuevas, B., Fasham, M.J.R. and Anderson, T.R. 2006. Mechanisms controlling primary and new production in a global ecosystem model – Part I: Validation of the biological simulation, Ocean Science, 2, 249-266.
  40. ↑ Madec, G., Delecluse, P., Imbard, M. and Lévy, C. 1998. OPA8.1 Ocean general circulation model reference manual. Notes de l'IPSL, Université P. et M. Curie, B102 T15-E5, 4 place Jussieu, Paris cedex 5, N°11, 91p.
  41. ↑ Madec, G. 2008. "NEMO ocean engine". Note du Pole de modélisation, Institut Pierre-Simon Laplace (IPSL), France, No 27 ISSN No 1288-1619.
  42. ↑ Chassignet, E.P., Hurlburt, H.E., Smedstad, O.M., Halliwell, G.R., Wallcraft, A.J., Metzger, E.J., Blanton, B.O., Lozano, C., Rao, D.B., Hogan, P.J. and Srinivasan, A. 2006. Generalized vertical coordinates for eddy-resolving global and coastal ocean forecasts. Oceanography, 19, 20-31.
  43. ↑ Chassignet, E.P. and Verron, J. 2006. Ocean Weather Forecasting: An Integrated View of Oceanography. (Eds.). Springer, 577pp.
  44. ↑ FRAM Group (Webb, D.J. et al). 1991. An Eddy-Resolving Model of the Southern Ocean. EOS, Transactions, American Geophysical Union, 72(15), 169-174.
  45. ↑ 45.0 45.1 Beckmann, A., Hellmer, H.H. and Timmermann, R. 1999. A numerical model of the Weddell Sea: large-scale circulation and water mass distribution, J. Geophys. Res. 104(C10), 23375-23,391.
  46. ↑ Hallberg, R. and Gnanadesikan, A. 2006. The role of eddies in determining the structure and response of the wind-driven Southern Hemisphere overturning: Initial results from the Modelling Eddies in the Southern Ocean project, J. Phys. Oceanogr., 36, 3312-3330.
  47. ↑ Timmermann, R., Hellmer, H.H. and Beckmann, A. 2002. Simulations of ice-ocean dynamics in the Weddell Sea 2. Interannual variability 1985-1993, J. Geophys. Res., 107(C3), 3025, doi:10.1029/2000JC000742.
  48. ↑ Assmann, K., Hellmer, H.H. and Beckmann, A. 2003. Seasonal variation in circulation and water mass distribution on the Ross Sea continental shelf, Antarctic Science, 15 (1), 3-11 (doi: 10.1017/S0954102003001007).
  49. ↑ Holland, D.M., Jacobs, S.S. and Jenkins, A. 2003. Modelling the ocean circulation beneath the Ross Ice Shelf, Antarctic Science, 15 (1), 13-23 (doi: 10.1017/S0954102003001019).
  50. ↑ Jenkins, A., Holland, D.M., Nicholls, K.W., Schröder, M. and Østerhus, S. 2004. Seasonal ventilation of the cavity beneath Filchner-Ronne Ice Shelf simulated with an isopycnic coordinate ocean model, J. Geophys. Res., 109, C01024, doi:10.1029/2001JC001086.
  51. ↑ 51.0 51.1 Dinniman, M.S., Klinck, J.M. and Smith Jr., W.O. 2007. Influence of sea ice cover and icebergs on circulation and water mass formation in a numerical circulation model of the Ross Sea, Antarctica, Journal of Geophysical Research, 112, C11013, (doi:10.1029/2006JC004036).
  52. ↑ Shchepetkin, A. F. and McWilliams, J. C. 2005. The regional oceanic modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model. Ocean Modelling, 9(4), 347-404.
  53. ↑ Hellmer, H.H. 2004. Impact of Antarctic shelf basal melting on sea ice and deep ocean properties, Geophys. Res. Lett., 31, doi:10.1029/2004GL019506.
Retrieved from ‘http://acce.scar.org/w/index.php?title=Ocean_models_of_the_Southern_Ocean&oldid=257’