# Ice Sheet Modeling

## Participants

^{Ice group on a glacier in Svalbard in 2015}

- Cheng Gong, PhD student, Uppsala University
- Per Lötstedt, professor in numerical analysis, Uppsala University
- Lina von Sydow, senior lecturer in numerical analysis, Uppsala University
- Nina Kirchner, senior Lecturer in numerical ice sheet modeling, Stockholm University

### Former members

- Josefin Ahlkrona
- Evan Gowan, Postdoc, Stockholm University
- James Lea, Postdoc, Stockholm University

### Former Master Students

- Hanna Holmgren
- Cecilia Håård
- Asako Fujisaki

## Research

Ice sheets, with connected ice shelves, are important components of the global climate system. If all the ice on Greenland and Antarctica melts then the sea level will rise by 7 m and 70 m, respectively. Numerical models help us both in understanding past ice configurations, and in predicting future ones. Investigating past ice sheets typically require simulating long time spans such as 100 000 years. These type of simulations are called paleosimulations and are computationally demanding. **The main goal of our research is to develop methods for paleosimulations of coupled ice sheet/ice stream/ice shelf systems**.

Schematic picture of a grounded ice sheet connected with a floating ice shelf. Image courtesy LIMA: Meet Antarctica. | A paleosimulation of the Cordilleran ice sheet in North America over 23000 years by A. Fujisaki [4]. Colors correspond to ice sheet thickness in meters. |

Ice can be regarded as a fluid with a very high viscosity, and governing its dynamics are the Stokes equations. Stress and strain-rate are related by a non-linear constitutive relation, the so-called Glen's flow law. An ice sheet (the grounded part) deforms due to its own weight and flows out onto the sea, forming a so-called ice shelf. The equations governing the ice flow are computationally heavy to solve and in order to run paleosimulations, approximations are required. The by far most common approximation is the Shallow Ice Approximation (SIA). This project has received supports from eSSENCE and FORMAS.

The methods in this research project are mainly implemented in the code Elmer/ICE that solves the full Stokes equation with a finite element method.

### Projects

#### Grounding Line Migration (Ongoing)

The grounding line is the transition zone between the grounded ice sheet and floating ice shelf. The location of the grounding line is important since it is sensitive to the change the climate. The mass loss from the ice sheet is strongly related to the position of the grounding line. It contains physical discontinuity which introduces difficulties in numerical simulations. The project is concerned with the development of efficient methods for numerical solution of the ice in the grounding line region.

Participants: Cheng Gong , Per Lötstedt , Lina von Sydow

#### Adaptive Time Stepping Method for Ice-sheet Modeling (Completed)

The efficiency, stability and accuracy are the three major issues in the numerical simulation of paleo ice sheet modeling. We propose an adaptive time stepping method to efficiently solve the coupled system which contains the velocity field equations and surface evolution equation (or the thickness equation). It automatically determines the maximum possible time step and adjusts it throughout a simulation. The time step is changed in an efficient way such that the velocity field equation is only solved once per time step. The adaptive time stepping method is based on a predictor-corrector scheme. The time step varies to control the errors estimated by the difference between the predictor and corrector, which guarantees the stability and accuracy of the method for best computational efficiency.

Participants: Cheng Gong , Per Lötstedt , Lina von Sydow

Adaptive time stepping for different tolerances with velocity fields solved by Shallow Ice Approximation (SIA) and Full Stokes (FS). The experiment is based on a flow line model from one of the EISMINT benchmark tests within a domain of 1000 km and a spatial resolution of 1.25 km. The three figures at the bottom are the height of the ice cap correspond to the certain time. |

#### Coupling of Approximation Levels (Ongoing)

The SIA is computationally cheap but models using the SIA typically fail in modeling ice streams (narrow bands of faster flowing ice) and the coupling between ice sheet and ice shelves, see left panel below. To remedy this we dynamically couple the exact equations with the SIA. The method is implemented in Elmer/ICE and is tested by application to the SeaRISE-setup and by various conceptual tests. It will also be applied to the Svalbard-Barents Sea ice-sheet.

Participants: Josefin Ahlkrona, Evan Gowan, Per Lötstedt, Nina Kirchner in collaboration with the Elmer team at Finnish IT Center for Science Ltd., CSC.

Computed surface velocities in the Greenland Ice Sheet, using the shallow ice approximation. In the north eastern part (at the red arrow) a large ice stream is missing. [6] |
Left part of an symmetric ice cap (a Vialov profile). In nodes for which the error is sufficiently low the SIA is computed (red), and elsewhere the exact equations are solved (blue). Elements containing both SIA-nodes and exact nodes couple the two approximations together (green). |

#### Iterative SOSIA (Ongoing)

In this project we incorporate the SOSIA in an iterative method.

Participants: Lina von Sydow, Josefin Ahlkrona and Per Lötstedt

#### Accuracy and Validity of the SIA and SOSIA (Completed)

The SIA is based on scaling arguments and perturbation expansion. Based on the same theory, the Second Order Shallow Ice Approximation (SOSIA) can be derived. In this project we have investigated the limits of validity of both the SIA and the SOSIA.

In [2] we solved the exact equations using Elmer/Ice for different aspect ratios of an ice sheet, and thereby made conclusions about the underlying assumptions in the theory of the SIA and the SOSIA. In [6] the SOSIA was implemented. In [1] we derived analytical solutions to the SIA and made numerical experiments where we investigated the practical order of accuracy of the SIA and the SOSIA. The assumptions in the derivation of the SOSIA are not correct and thus the accuracy is much lower than second order.

Participants: Josefin Ahlkrona, Nina Kirchner and Per Lötstedt

#### SIA Temperature and Thickness Evolution (Completed)

The SIA temperature equation and ice thickness evolution are implemented into a MATLAB-code and compared with the exact solution by using Elmer/Ice [3].

Participants: Cecilia Håård, Lina von Sydow and Per Lötstedt

#### Hydraulics in Paleosimulations (Completed)

A reconnaissance analysis of the glacio-hydraulic algorithms in the ice sheet model ARCTIC-TARAH is performed [5]. ARCTIC-TARAH is a Bolin Center for Climate Research spin-off from the Pennsylvania State University Ice sheet model (PSUI).

Participants: Hanna Holmgren and Nina Kirchner

#### Simulating the Cordilleran Ice Sheet (Completed)

ARCTIC-TARAH is adjusted for the simulation of the Cordilleran Ice Sheet (CIS), which periodically appears in the northwestern corner over North America during glacial periods, and a 30 000 year long period is simulated [4].

Participants: Asako Fujisaki, Nina Kirchner and Per Lötstedt

## Publications

Publications related to the project.

#### PhD thesis

- Ahlkrona, J., Computational Ice Sheet Dynamics - Error control and efficiency, PhD thesis, Uppsala University, 2016.

#### Publications

- Cheng, G., Shcherbakov, V., Anisotropic radial basis function methods for continental size ice sheet simulations, Journal of Computational Physics, 372, 161-177, 2018.
- van Dongen, E.C.H., Kirchner, N., van Gijzen, M.B., van de Wal, R.S.W., Zwinger, T., Cheng, G., Lötstedt, P., and von Sydow, L., Dynamically coupling Full Stokes and Shallow Shelf Approximation for marine ice sheet flow using Elmer/Ice (v8.3), Geosci. Model Dev., 11, 4563-4576, 2018.
- Ahlkrona, J., Shcherbakov, V., A meshfree approach to non-Newtonian free surface ice flow: Application to the Haut Glacier d'Arolla, Journal of Computational Physics, 330, 633-649, 2017.
- Cheng, G., Lötstedt, P., von Sydow, L., Accurate and stable time stepping in ice sheet modeling, Journal of Computational Physics, 329, 29-47, 2017.
- Ahlkrona, J., Lötstedt, P., Kirchner, N., Zwinger, T., ''Dynamically coupling the non-linear Stokes equations with the Shallow Ice Approximation in glaciology: Description and first applications of the ISCAL method'', Journal of Computational Physics, 308, 1-19, 2016.
- Kirchner, N., Ahlkrona, J., Gowan, E., Lötstedt, P., Lea, J., Noormets, R., von Sydow, L., Dowdeswell J., Benham T., ''Shallow Ice Approximation, Second Order Shallow Ice Approximation and Full Stokes models: A discussion of their roles in palaeo-ice sheet modelling and development, Quaternary Science Review, 135, 103-114, 2016.
- Ahlkrona, J., Kirchner, N., Lötstedt, P., ''Accuracy of the zeroth and second order shallow ice approximation - numerical and theoretical results'', Geoscientific Model Development, 6, 2135-2152, 2013
- Ahlkrona, J., Kirchner, N., Lötstedt, P., ''A Numerical Study of Scaling Relations for Non-Newtonian Thin-film Flows with Applications in Ice Sheet Modelling'', Quarterly Journal of Mechanics and Applied Mathematics, 66, 417-435, 2013
- Håård, C., ''Implementation of temperature variations and free surface evolution in the Shallow Ice Approximation (SIA)'', Master's Thesis with Lina von Sydow (supervisor) and Per Lötstedt, 2013.
- Fujisaki, A., ''Numerical simulations of the Cordilleran Ice Sheet (CIS): Implementing a new module to the ice code ARCTIC-TARAH'', Master's Thesis with Nina Kirchner and Per Lötstedt, 2013.
- Holmgren, H., ''Computational Ice Dynamics and Hydraulics: Towards a Coupling in the Ice Sheet Code ARCTIC-TARAH'', Master's Thesis with Nina Kirchner (supervisor) and Per Lötstedt (reviewer), 2012.
- Ahlkrona, J., ''Implementing Higher Order Dynamics into the Ice Sheet Model SICOPOLIS'', Master's Thesis with Nina Kirchner (supervisor) and Per Lötstedt (reviewer), 2011.

## Events

8th of February, 2012: Workshop in Ice Modeling and Simulation