Ice-sheet model

In climate modelling, '''ice-sheet models''' use numerical methods to simulate the evolution, dynamics and thermodynamics of ice sheets, such as the Antarctic ice sheet, the Greenland ice sheet or the large ice sheets on the Northern Hemisphere during the Last Glacial Period. They are used for a variety of purposes, from studies of the glaciation of Earth over glacial–interglacial cycles in the past to projections of ice-sheet decay under future global warming conditions.

== History == Beginning in the mid-18th Century, investigation into ice sheet behavior began.<ref name="Blatter-2010">{{Cite journal|last1=Blatter|first1=Heinz|last2=Greve|first2=Ralf|last3=Abe-Ouchi|first3=Ayako|date=2010|title=A short history of the thermomechanical theory and modeling of glaciers and ice sheets|journal=Journal of Glaciology|language=en|volume=56|issue=200|pages=1087–1094|doi=10.3189/002214311796406059|bibcode=2010JGlac..56.1087B |issn=0022-1430|doi-access=free|hdl=2115/46879|hdl-access=free}}</ref> Since the Journal of Glaciology's founding, physicists have been publishing glacial mechanics.<ref name="Blatter-2010"/> thumb|upright=1.2|Barnes Ice Cap The first 3-D model was applied to the Barnes Ice Cap.<ref name="Blatter-2010"/> In 1988, the first thermodynamically coupled model incorporating ice-shelves, sheet/shelf transition, membrane stress gradients, isostatic bed adjustment and basal sliding using more advanced numerical techniques was developed and applied to the Antarctic ice sheet.<ref name="Blatter-2010"/> This model had a resolution of 40 km and 10 vertical layers.<ref name="Blatter-2010"/>

When the first IPCC assessment report came out in 1990, ice sheets were not an active part of the climate system model, their evolution was based on a correlation between global temperature and surface mass balance.<ref name="Shepherd-2017">{{Cite journal|last1=Shepherd|first1=Andrew|last2=Nowicki|first2=Sophie|date=October 2017|title=Improvements in ice-sheet sea-level projections|url=http://www.nature.com/articles/nclimate3400|journal=Nature Climate Change|language=en|volume=7|issue=10|pages=672–674|doi=10.1038/nclimate3400|bibcode=2017NatCC...7..672S |issn=1758-678X|url-access=subscription}}</ref> When the second IPCC assessment report came out in 1996, the beginning of both 2D and 3D modelling was shown with ice sheets.<ref name="Shepherd-2017" /> The 1990s heralded several more computational models, bringing with it the European Ice Sheet Modelling Initiative (EISMINT).<ref name="Blatter-2010"/><ref name="Philippe-1997">{{Cite book|last=Philippe|first=Huybrechts|url=https://epic.awi.de/id/eprint/6776/1/Huy1998b.pdf|title=Report of the Third EISMINT Workshop on Model Intercomparison|year=1997}}</ref> The EISMINT produced several workshops throughout the 1990s of an international collaboration, comparing most models of Greenland, Antarctic, ice-shelf, thermomechanical and grounding-line.<ref name="Philippe-1997" />

The 2000s included integrating first-order approximation of full Stokes Dynamics into an ice-sheet model.<ref name="Blatter-2010" /> The fourth IPCC assessment report showed ice-sheet models with projections of rapid dynamical responses in the ice, which led to evidence of significant ice loss.<ref name="Shepherd-2017" />

In 2016, part of the Coupled Model Intercomparison Project Phase 6 (CMIP Phase 6) was the Ice Sheet Model Intercomparison Project, which defined a protocol for all variables related to ice sheet modelling.<ref>{{Cite journal|last1=Nowicki|first1=Sophie M. J.|last2=Payne|first2=Anthony|last3=Larour|first3=Eric|last4=Seroussi|first4=Helene|last5=Goelzer|first5=Heiko|last6=Lipscomb|first6=William|last7=Gregory|first7=Jonathan|last8=Abe-Ouchi|first8=Ayako|last9=Shepherd|first9=Andrew|date=2016-12-21|title=Ice Sheet Model Intercomparison Project (ISMIP6) contribution to CMIP6|journal=Geoscientific Model Development|language=en|volume=9|issue=12|pages=4521–4545|doi=10.5194/gmd-9-4521-2016|issn=1991-9603|pmc=5911933|pmid=29697697 |bibcode=2016GMD.....9.4521N |doi-access=free }}</ref> The project allowed for both improvement in numerical and physical approaches to ice sheets.<ref>{{Cite journal |last=Pattyn |first=Frank |author-link=Frank Pattyn |date=December 2018 |title=The paradigm shift in Antarctic ice sheet modelling |journal=Nature Communications |language=en |volume=9 |issue=1 |pages=2728 |doi=10.1038/s41467-018-05003-z |issn=2041-1723 |pmc=6048022 |pmid=30013142|bibcode=2018NatCo...9.2728P }}</ref>

== Modelling ==

=== Ice-Flow ===

==== Shallow Ice Approximation ==== Shallow Ice Approximation (SIA) is a simple method to model ice flow without having to solve full-Stokes equations.<ref name="Oerlemans-1982">{{Cite journal|last=Oerlemans|first=J.|date=December 1982|title=Glacial cycles and ice-sheet modelling|url=http://link.springer.com/10.1007/BF02423468|journal=Climatic Change|language=en|volume=4|issue=4|pages=353–374|doi=10.1007/BF02423468|bibcode=1982ClCh....4..353O |hdl=1874/21024 |s2cid=189889177 |issn=0165-0009|hdl-access=free|url-access=subscription}}</ref> The approximation is best applied to ice sheet with a small depth-to-width ratio, without many sliding dynamics and a simple bed topography.<ref name="Davies-2020">{{Cite web|last=Davies|first=Bethan|title=A hierarchy of ice-sheet models|url=http://www.antarcticglaciers.org/glaciers-and-climate/numerical-ice-sheet-models/hierarchy-ice-sheet-models-introduction/|access-date=2021-10-18|website=AntarcticGlaciers.org|date=22 June 2020 |language=en-US}}</ref> SIA does not include many forces on an ice sheet, and can be considered a 'zero-order' model.<ref>{{Cite web|last=Davies|first=Bethan|title=A hierarchy of ice-sheet models|url=http://www.antarcticglaciers.org/glaciers-and-climate/numerical-ice-sheet-models/hierarchy-ice-sheet-models-introduction/|access-date=2021-10-18|website=AntarcticGlaciers.org|date=22 June 2020 |language=en-US}}</ref> The model assumes that ice sheets are mostly split up by basal sheer stress, and it is not necessary to consider the other forces.<ref name="Van Den Berg-2006">{{Cite journal |last1=Van Den Berg |first1=J. |last2=Van De Wal |first2=R. S. W. |last3=Oerlemans |first3=J. |date=2006 |title=Effects of spatial discretization in ice-sheet modelling using the shallow-ice approximation |journal=Journal of Glaciology |language=en |volume=52 |issue=176 |pages=89–98 |bibcode=2006JGlac..52...89V |doi=10.3189/172756506781828935 |issn=0022-1430 |doi-access=free}}</ref> It also assumes that the basal shear stress and the gravitational driving stress of the grounded ice balance one another out.<ref>{{Cite web|last=Davies|first=Bethan|title=A hierarchy of ice-sheet models|url=http://www.antarcticglaciers.org/glaciers-and-climate/numerical-ice-sheet-models/hierarchy-ice-sheet-models-introduction/|access-date=2021-10-18|website=AntarcticGlaciers.org|date=22 June 2020 |language=en-US}}</ref> The method is computationally inexpensive.<ref name="Van Den Berg-2006" />

==== Shallow Shelf Approximation ==== Shallow Shelf Approximation is another method to model ice flow, in particular a membrane-type flow of floating ice, or of sliding grounded ice over a base.<ref>{{Cite web|title=Two stress balance models: SIA and SSA – PISM, a Parallel Ice Sheet Model 1.2.1 documentation|url=https://pism-docs.org/sphinx/manual/highlevelview/stress-balance-models.html|access-date=2021-10-19|website=pism-docs.org}}</ref> Also known as a membrane model, they are similar to free-film models in fluid dynamics.<ref>{{Cite journal|last1=Schoof|first1=Christian|last2=Hewitt|first2=Ian|date=2013-01-03|title=Ice-Sheet Dynamics|url=https://www.annualreviews.org/doi/10.1146/annurev-fluid-011212-140632|journal=Annual Review of Fluid Mechanics|language=en|volume=45|issue=1|pages=217–239|doi=10.1146/annurev-fluid-011212-140632|bibcode=2013AnRFM..45..217S |issn=0066-4189|url-access=subscription}}</ref> As opposed to Shallow Ice Approximation, Shallow Shelf Approximation models ice flow when longitudinal forces are strong; sliding and vertical forces.<ref>{{Cite web|last=Davies|first=Bethan|title=A hierarchy of ice-sheet models|url=http://www.antarcticglaciers.org/glaciers-and-climate/numerical-ice-sheet-models/hierarchy-ice-sheet-models-introduction/|access-date=2021-10-18|website=AntarcticGlaciers.org|date=22 June 2020 |language=en-US}}</ref> SSA can also be considered a 'zero-order' model.<ref>{{Cite web|last=Davies|first=Bethan|title=A hierarchy of ice-sheet models|url=http://www.antarcticglaciers.org/glaciers-and-climate/numerical-ice-sheet-models/hierarchy-ice-sheet-models-introduction/|access-date=2021-10-18|website=AntarcticGlaciers.org|date=22 June 2020 |language=en-US}}</ref>

==== Full Stokes Equations ==== It is considered advantageous to model ice using Navier-Stokes equations as ice is a viscous fluid and these capture all forces exerted on the ice.<ref name="Oerlemans-1982"/> As these equations are computationally expensive, it is important to include many approximations to reduce running time.<ref name="Oerlemans-1982"/> Because of their computational expense, they are not easily used at a large scale and can be used in specific sections or scenarios, such as at grounding lines.<ref name="Davies-2020"/> thumb|upright=2|A diagram of some of the aspects of an ice-sheet model

=== Interactions with other climatic components === Ice sheets interact with the surrounding atmosphere, ocean and sub-glacial earth.<ref name="Goelzer-2017">{{Cite journal |last1=Goelzer |first1=Heiko |last2=Robinson |first2=Alexander |last3=Seroussi |first3=Helene |last4=van de Wal |first4=Roderik S. W. |date=December 2017 |title=Recent Progress in Greenland Ice Sheet Modelling |journal=Current Climate Change Reports |language=en |volume=3 |issue=4 |pages=291–302 |bibcode=2017CCCR....3..291G |doi=10.1007/s40641-017-0073-y |issn=2198-6061 |pmc=6959375 |pmid=32010550}}</ref> All of these interactive components need to be included to be able to have a comprehensive ice-sheet model.<ref name="Goelzer-2017" />

'''Basal Conditions''' play an important role in determining the behavior of ice sheets. The basal thermal state (if the ice is thawed or frozen) and the basal topography are difficult to map.<ref name="Goelzer-2017" /> The most favored method is to apply mass conservation constraints.<ref name="Goelzer-2017" /> For long-term projections, it is important to project the topography onto the continental shelf or into the fjords, and this can be difficult when the sub-glacial topography is not well-known.<ref name="Goelzer-2017" />

'''Summer Insolation''' drive temperature responses that have an effect on the rate of melting and mass balance of the ice sheet.<ref name="Ruddiman-2014">{{Cite book|last=Ruddiman|first=William|title=Earth's Climate: Past and Future|publisher=W.H. Freeman and Company|year=2014|isbn=978-1-4292-5525-7|location=New York}}</ref> For example, the dependence of ice volume on summer insolation can be represented with <math>{d(I) \over d(t)} = {1 \over T} (S - I)</math>, where I is ice volume, <math>{d(I) \over d(t)}</math> is the rate of change of ice volume per unit of time, T is the response time of the ice sheet and S is the insolation signal.<ref name="Ruddiman-2014" />

'''Air Temperature''' is needed in a model as it informs surface melt and runoff rates.<ref name="Albrecht-2020">{{Cite journal|last1=Albrecht|first1=Torsten|last2=Winkelmann|first2=Ricarda|author2-link=Ricarda Winkelmann|last3=Levermann|first3=Anders|date=2020-02-14|title=Glacial-cycle simulations of the Antarctic Ice Sheet with the Parallel Ice Sheet Model (PISM) – Part 1: Boundary conditions and climatic forcing|url=https://tc.copernicus.org/articles/14/599/2020/|journal=The Cryosphere|language=en|volume=14|issue=2|pages=599–632|doi=10.5194/tc-14-599-2020|bibcode=2020TCry...14..599A |issn=1994-0424|doi-access=free}}</ref> For example, surface air temperature can be expressed with latitude 'lat', surface elevation ''h'' and mean temperature to provide an estimate of annual mean temperatures:<ref name="Albrecht-2020" /> <math>T_{aml} = temperature - c_1 * max(h, 1000) - c_2 * lat </math>. This example assumes the ice shelf 's surface is as cold as at 1000m altitude.<ref name="Albrecht-2020" />

'''Precipitation''' is directly tied to air temperature, and also depends on moisture above and around the ice sheet.<ref name="Albrecht-2020" /> Precipitation plays an important part in ice-sheet melting and accumulation.<ref name="Albrecht-2020" />

=== Calving === Calving is still an active area of investigation in ice-sheet modelling.<ref name="Goelzer-2017" /> A total picture of calving will include many different aspects, including but not limited to tides, basal crevasses, collisions with ice bergs, thickness and temperature.<ref>{{Cite journal|last1=Alley|first1=Richard B.|last2=Horgan|first2=Huw J.|last3=Joughin|first3=Ian|last4=Cuffey|first4=Kurt M.|last5=Dupont|first5=Todd K.|last6=Parizek|first6=Byron R.|last7=Anandakrishnan|first7=Sridhar|last8=Bassis|first8=Jeremy|date=2008-11-28|title=A Simple Law for Ice-Shelf Calving|url=https://www.science.org/doi/full/10.1126/science.1162543|journal=Science|volume=322|issue=5906|pages=1344|doi=10.1126/science.1162543|pmid=19039129 |bibcode=2008Sci...322.1344A |s2cid=206514828 |url-access=subscription}}</ref> The recent development of the concepts of Marine Ice Sheet Instability and Marine Ice Cliff Instability have contributed to more accurate results of ice-sheet calving processes.<ref>{{Cite journal|last1=Pattyn|first1=Frank|last2=Favier|first2=Lionel|last3=Sun|first3=Sainan|last4=Durand|first4=Gaël|date=2017-09-01|title=Progress in Numerical Modeling of Antarctic Ice-Sheet Dynamics|url=https://doi.org/10.1007/s40641-017-0069-7|journal=Current Climate Change Reports|language=en|volume=3|issue=3|pages=174–184|doi=10.1007/s40641-017-0069-7|bibcode=2017CCCR....3..174P |s2cid=134517464 |issn=2198-6061|hdl=2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/262952|hdl-access=free}}</ref>

== Examples ==

=== CISM === The Community Ice Sheet Model is part of the Community Earth Systems Model funded by the National Science Foundation and models ice dynamics.<ref name="cism.github.io">{{Cite web|title=CISM / The Community Ice Sheet Model|url=https://cism.github.io/|access-date=2021-10-14|website=cism.github.io}}</ref><ref name="websrv.cs.umt.edu">{{Cite web|title=Development of a Community Ice Sheet Model - Interactive System for Ice sheet Simulation|url=http://websrv.cs.umt.edu/isis/index.php/Main_Page|access-date=2021-10-28|website=websrv.cs.umt.edu|archive-date=2 December 2021|archive-url=https://web.archive.org/web/20211202152754/http://websrv.cs.umt.edu/isis/index.php/Main_Page|url-status=dead}}</ref> It is written in Fortran 90 and is open-source.<ref name="cism.github.io" /> The US Department of Energy has begun to contribute to CISM.<ref name="websrv.cs.umt.edu" /> The CISM project works on other adjacent projects in developing a cirriculum to expand knowledge on ice sheets, and engaging a broader community in ice-sheet modelling.<ref name="websrv.cs.umt.edu" /> Many ice-sheet modelling softwares have influenced CISM, including the [https://pism-docs.org/wiki/doku.php Parallel Ice Sheet Model] (PSIM) and Glimmer.<ref>{{Cite web |title=Software – Interactive System for Ice sheet Simulation |url=http://websrv.cs.umt.edu/isis/index.php/Software |access-date=2021-10-28 |website=websrv.cs.umt.edu |archive-date=29 October 2021 |archive-url=https://web.archive.org/web/20211029022628/http://websrv.cs.umt.edu/isis/index.php/Software |url-status=dead }}</ref><ref>{{Cite web |title=Documentation for PISM, the Parallel Ice Sheet Model |url=https://pism-docs.org/wiki/doku.php |access-date=2021-10-28 |website=pism-docs.org |language=en}}</ref>

=== seaRISE === Sea-level Response to Ice Sheet Evolution (SeaRISE) is a subcommunity of CISM that sets out to estimate the upper limit of sea level rise from ice sheets.<ref name="websrv.cs.umt.edu-2">{{Cite web |title=SeaRISE Assessment – Interactive System for Ice sheet Simulation |url=http://websrv.cs.umt.edu/isis/index.php/SeaRISE_Assessment |access-date=2021-10-28 |website=websrv.cs.umt.edu |archive-date=29 October 2021 |archive-url=https://web.archive.org/web/20211029022629/http://websrv.cs.umt.edu/isis/index.php/SeaRISE_Assessment |url-status=dead }}</ref> The project sets out to develop a set of experiments and assessments for ice sheet and sea level rise modelling, as well as make a unified input dataset for ice sheet models.<ref name="websrv.cs.umt.edu-2" />

=== Glimmer === Glimmer ([https://www.ercim.eu/publication/Ercim_News/enw61/price.html GENIE] Land Ice Model with Multiply-Enabled Regions) is an ice-sheet model initially made to contribute to a more comprehensive earth system model, GENIE.<ref>{{Cite journal|last1=Rutt|first1=I. C.|last2=Hagdorn|first2=M.|last3=Hulton|first3=N. R. J.|last4=Payne|first4=A. J.|date=2009|title=The Glimmer community ice sheet model|journal=Journal of Geophysical Research: Earth Surface|language=en|volume=114|issue=F2|doi=10.1029/2008JF001015|bibcode=2009JGRF..114.2004R |issn=2156-2202|doi-access=free|hdl=20.500.11820/fd14bca6-3e08-4c40-a099-6cb4ced05157|hdl-access=free}}</ref>

=== PISM === The Parallel Ice Sheet Model is an open-sourced 3D ice sheet model capable of high resolution.<ref name="pism-docs.org">{{Cite web |title=Documentation for PISM, the Parallel Ice Sheet Model |url=https://pism-docs.org/wiki/doku.php?id=home |access-date=2021-10-17 |website=pism-docs.org |language=en |archive-date=19 October 2021 |archive-url=https://web.archive.org/web/20211019171028/https://pism-docs.org/wiki/doku.php?id=home |url-status=dead }}</ref> PISM is written in C++ and Python, and takes NetCDF files as input for the model.<ref>{{Citation|title=PISM, a Parallel Ice Sheet Model|date=2021-10-11|url=https://github.com/pism/pism|publisher=Parallel Ice Sheet Model|access-date=2021-10-17}}</ref> PISM uses a "SIA+SSA hybrid" model, using both the shallow shelf approximation and shallow ice approximation models as stress balance models and does not solve full Stokes equations.<ref name="pism-docs.org" /> The model gets climatic information from an external General Circulation Model, and needs information like boundary temperature, mass flux into the ice, precipitation and air temperature.<ref>{{Cite web |title=Climate inputs, and their interface with ice dynamics – PISM, a Parallel Ice Sheet Model 1.2.1 documentation |url=https://pism-docs.org/sphinx/manual/highlevelview/climate-inputs.html |access-date=2021-10-17 |website=pism-docs.org}}</ref>

A horizontal grid of equal distance is used, with a variable vertical axis, and runs on a year timescale.<ref>{{Cite web |title=Spatial grid – PISM, a Parallel Ice Sheet Model 1.2.1 documentation |url=https://pism-docs.org/sphinx/manual/modeling-choices/computational/grid.html |access-date=2021-10-18 |website=pism-docs.org}}</ref><ref>{{Cite web |title=Model time – PISM, a Parallel Ice Sheet Model 1.2.1 documentation |url=https://pism-docs.org/sphinx/manual/modeling-choices/computational/time.html |access-date=2021-10-18 |website=pism-docs.org}}</ref>

== Ice-sheet models on the web == * CISM<ref>{{Cite web|url=https://www.lanl.gov/errors/system-notification.php|title=System Unavailable|first=Los Alamos National Laboratory, Operated by Los Alamos National Security, LLC, for the U. S. Department of|last=Energy|website=www.lanl.gov|accessdate=24 March 2024}}</ref> – Community Ice Sheet Model, under development as a land-ice component of the [http://www2.cesm.ucar.edu Community Earth System Model (CESM)] * Elmer/Ice<ref>{{Cite web|url=http://elmerice.elmerfem.org|title=Elmer/Ice|website=elmerice.elmerfem.org|accessdate=24 March 2024}}</ref> – a multi-physics finite element code with special modules for full-stress ice dynamics analysis * ISSM<ref>{{cite web|url=https://github.com/ISSMteam/ISSM|title=ISSM|website=github.com/ISSMteam/ISSM}}</ref> – Ice-Sheet and Sea-level System Model, a large-scale thermo-mechanical 2D/3D parallelized multi-purpose finite-element software dedicated to ice sheet and sea-level modeling. * PISM<ref>{{Cite web|url=https://pism-docs.org/|title=Banger Casino Bangladesh: Where Winners Are Made|date=28 November 2023|accessdate=24 March 2024}}</ref> – Parallel Ice Sheet Model, which includes ice shelves and ice streams * SICOPOLIS<ref>{{Cite web|url=https://www.sicopolis.net/|title=Ice-sheet model SICOPOLIS|accessdate=24 March 2024}}</ref> – SImulation COde for POLythermal Ice Sheets, a 3D ice-sheet model which accounts for polythermal conditions (coexistence of ice at and below the melting point in different parts of an ice sheet)

==References== {{Reflist}}

Category:Glaciology * Category:Climate modeling