Reformulating the full-Stokes ice sheet model for a more efficient computational solution
The first-order or Blatter-Pattyn ice sheet model, in spite of its approximate nature, is an attractive alternative to the full Stokes model in many applications because of its reduced computational demands. In contrast, the unapproximated Stokes ice sheet model is more difficult to solve and comput...
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ftdoajarticles:oai:doaj.org/article:1d4e1adb7d7b46e9ae1332f6a00e5cf1 2023-05-15T16:40:11+02:00 Reformulating the full-Stokes ice sheet model for a more efficient computational solution J. K. Dukowicz 2012-01-01T00:00:00Z https://doi.org/10.5194/tc-6-21-2012 https://doaj.org/article/1d4e1adb7d7b46e9ae1332f6a00e5cf1 EN eng Copernicus Publications http://www.the-cryosphere.net/6/21/2012/tc-6-21-2012.pdf https://doaj.org/toc/1994-0416 https://doaj.org/toc/1994-0424 doi:10.5194/tc-6-21-2012 1994-0416 1994-0424 https://doaj.org/article/1d4e1adb7d7b46e9ae1332f6a00e5cf1 The Cryosphere, Vol 6, Iss 1, Pp 21-34 (2012) Environmental sciences GE1-350 Geology QE1-996.5 article 2012 ftdoajarticles https://doi.org/10.5194/tc-6-21-2012 2022-12-31T02:28:10Z The first-order or Blatter-Pattyn ice sheet model, in spite of its approximate nature, is an attractive alternative to the full Stokes model in many applications because of its reduced computational demands. In contrast, the unapproximated Stokes ice sheet model is more difficult to solve and computationally more expensive. This is primarily due to the fact that the Stokes model is indefinite and involves all three velocity components, as well as the pressure, while the Blatter-Pattyn discrete model is positive-definite and involves just the horizontal velocity components. The Stokes model is indefinite because it arises from a constrained minimization principle where the pressure acts as a Lagrange multiplier to enforce incompressibility. To alleviate these problems we reformulate the full Stokes problem into an unconstrained, positive-definite minimization problem, similar to the Blatter-Pattyn model but without any of the approximations. This is accomplished by introducing a divergence-free velocity field that satisfies appropriate boundary conditions as a trial function in the variational formulation, thus dispensing with the need for a pressure. Such a velocity field is obtained by vertically integrating the continuity equation to give the vertical velocity as a function of the horizontal velocity components, as is in fact done in the Blatter-Pattyn model. This leads to a reduced system for just the horizontal velocity components, again just as in the Blatter-Pattyn model, but now without approximation. In the process we obtain a new, reformulated Stokes action principle as well as a novel set of Euler-Lagrange partial differential equations and boundary conditions. The model is also generalized from the common case of an ice sheet in contact with and sliding along the bed to other situations, such as to a floating ice shelf. These results are illustrated and validated using a simple but nontrivial Stokes flow problem involving a sliding ice sheet. Article in Journal/Newspaper Ice Sheet Ice Shelf The Cryosphere Directory of Open Access Journals: DOAJ Articles Lagrange ENVELOPE(-62.597,-62.597,-64.529,-64.529) The Cryosphere 6 1 21 34 |
institution |
Open Polar |
collection |
Directory of Open Access Journals: DOAJ Articles |
op_collection_id |
ftdoajarticles |
language |
English |
topic |
Environmental sciences GE1-350 Geology QE1-996.5 |
spellingShingle |
Environmental sciences GE1-350 Geology QE1-996.5 J. K. Dukowicz Reformulating the full-Stokes ice sheet model for a more efficient computational solution |
topic_facet |
Environmental sciences GE1-350 Geology QE1-996.5 |
description |
The first-order or Blatter-Pattyn ice sheet model, in spite of its approximate nature, is an attractive alternative to the full Stokes model in many applications because of its reduced computational demands. In contrast, the unapproximated Stokes ice sheet model is more difficult to solve and computationally more expensive. This is primarily due to the fact that the Stokes model is indefinite and involves all three velocity components, as well as the pressure, while the Blatter-Pattyn discrete model is positive-definite and involves just the horizontal velocity components. The Stokes model is indefinite because it arises from a constrained minimization principle where the pressure acts as a Lagrange multiplier to enforce incompressibility. To alleviate these problems we reformulate the full Stokes problem into an unconstrained, positive-definite minimization problem, similar to the Blatter-Pattyn model but without any of the approximations. This is accomplished by introducing a divergence-free velocity field that satisfies appropriate boundary conditions as a trial function in the variational formulation, thus dispensing with the need for a pressure. Such a velocity field is obtained by vertically integrating the continuity equation to give the vertical velocity as a function of the horizontal velocity components, as is in fact done in the Blatter-Pattyn model. This leads to a reduced system for just the horizontal velocity components, again just as in the Blatter-Pattyn model, but now without approximation. In the process we obtain a new, reformulated Stokes action principle as well as a novel set of Euler-Lagrange partial differential equations and boundary conditions. The model is also generalized from the common case of an ice sheet in contact with and sliding along the bed to other situations, such as to a floating ice shelf. These results are illustrated and validated using a simple but nontrivial Stokes flow problem involving a sliding ice sheet. |
format |
Article in Journal/Newspaper |
author |
J. K. Dukowicz |
author_facet |
J. K. Dukowicz |
author_sort |
J. K. Dukowicz |
title |
Reformulating the full-Stokes ice sheet model for a more efficient computational solution |
title_short |
Reformulating the full-Stokes ice sheet model for a more efficient computational solution |
title_full |
Reformulating the full-Stokes ice sheet model for a more efficient computational solution |
title_fullStr |
Reformulating the full-Stokes ice sheet model for a more efficient computational solution |
title_full_unstemmed |
Reformulating the full-Stokes ice sheet model for a more efficient computational solution |
title_sort |
reformulating the full-stokes ice sheet model for a more efficient computational solution |
publisher |
Copernicus Publications |
publishDate |
2012 |
url |
https://doi.org/10.5194/tc-6-21-2012 https://doaj.org/article/1d4e1adb7d7b46e9ae1332f6a00e5cf1 |
long_lat |
ENVELOPE(-62.597,-62.597,-64.529,-64.529) |
geographic |
Lagrange |
geographic_facet |
Lagrange |
genre |
Ice Sheet Ice Shelf The Cryosphere |
genre_facet |
Ice Sheet Ice Shelf The Cryosphere |
op_source |
The Cryosphere, Vol 6, Iss 1, Pp 21-34 (2012) |
op_relation |
http://www.the-cryosphere.net/6/21/2012/tc-6-21-2012.pdf https://doaj.org/toc/1994-0416 https://doaj.org/toc/1994-0424 doi:10.5194/tc-6-21-2012 1994-0416 1994-0424 https://doaj.org/article/1d4e1adb7d7b46e9ae1332f6a00e5cf1 |
op_doi |
https://doi.org/10.5194/tc-6-21-2012 |
container_title |
The Cryosphere |
container_volume |
6 |
container_issue |
1 |
container_start_page |
21 |
op_container_end_page |
34 |
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1766030559331483648 |