Mathematical and numerical modelling of ice sheets and glaciers
Abstract
Tutorial held at the Workshop "Mathematical Approach to Climate Change Impacts", Istituto Nazionale di Alta Matematica Francesco Severi (INdAM), Rome, Italy, 2017.03.15/16. Summary. Ice sheets (with their attached ice shelves), ice caps and glaciers are active, dynamic components of the climate system of the Earth. As a common feature, these ice bodies show gravity-driven creep flow ("glacial flow"), sustained by the underlying land. This leads to thinning and horizontal spreading, which is essentially compensated by snow accumulation in the higher (interior) areas and melting and calving in the lower (marginal) areas. Any imbalance of this dynamic equilibrium leads to either growing or shrinking ice masses. In this tutorial, we will first review the underlying field theories of continuum mechanics and thermodynamics. Basic concepts are the tensorial measures for stress and strain, and the balance equations for mass, momentum and energy. We will then discuss the material (rheological) properties of polycrystalline ice, which are expressed by Glen's flow law (or variations thereof), Fourier's law of heat conduction and a caloric equation of state. Combining the balance and material equations leads to the field equations for the flow of ice sheets, ice shelves, ice caps and glaciers, which must be complemented by suitable boundary conditions. We will treat a hierarchy of simplifications of the field equations, from the most sophisticated full Stokes formulation to the most simplified shallow-ice and shallow-shelf approximations. Analytical solutions only exist for highly simplified problems, while in general numerical solution techniques must be applied. We will present some examples of numerical solutions for selected problems of ice sheet and glacier flow.
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Mathematical and numerical modelling of ice sheets and glaciers Ralf Greve Institute of Low Temperature Science, Hokkaido University, Sapporo, Japan Tutorial held at the Workshop “Mathematical Approach to Climate Change Impacts”, Istituto Nazionale di Alta Matematica Francesco Severi (INdAM), Rome, Italy, 2017.03.15/16
Ralf Greve: Modelling of ice sheets and glaciers 2/69 Contents 1. Introduction 2. Mechanisms of ice flow 3. Dynamics 4. Thermodynamics 5. Ice thickness equation 6. Sketch of the coupled initial–boundary value problem 7. Analytical solutions 8. Numerical solutions and models 9. Selected applications
1. Introduction
Ralf Greve: Modelling of ice sheets and glaciers 4/69 Terminology Vertical exaggeration factor ~200…500 Ice sheets → grounded ice masses of continental size, area > 50,000 km2 (Antarctica, Greenland). Ice shelves →floating ice masses, connected to an ice sheet (Antarctica).
Ralf Greve: Modelling of ice sheets and glaciers 5/69 Terminology Glaciers → small grounded ice masses in mountainous regions, constrained by topographical features. Ice caps → extended grounded ice masses, area < 50,000 km2 (Austfonna, Vatnajökull, North/South Patagonian Icefields...). Credit: Christoph Mayer Remark: “Glacier” is sometimes also used as an umbrella term for all grounded ice bodies (ice sheets, ice caps and glaciers as defined above).
Ralf Greve: Modelling of ice sheets and glaciers 6/69 Ice sheets Antarctic ice sheet (with ice shelves) Greenland ice sheet
Ralf Greve: Modelling of ice sheets and glaciers 7/69 Glaciers and ice caps Can be found on every continent (polar/sub-polar areas, mountains). Number: ~ 200,000 (~ 70 ice caps). Many different types: Valley glaciers, cirque glaciers, hanging glaciers, tidewater glaciers, rock glaciers… Photo credit: www.glaciers-online.net
Ralf Greve: Modelling of ice sheets and glaciers 8/69 Inventory Main source: Vaughan et al. (2013) [IPCC AR5 Ch. 4]. (*) Sum for all glaciers and ice caps. (**) Range of values for individual glaciers and ice caps. Glaciers and ice caps Greenland ice sheet Antarctic ice sheet Area (106km2)0.73* 1.80 12.3 Volume (metres of sea level equivalent) 0.41* 7.36 58.3 Turnover time (vol/accum, years) ~ 50 –1000** ~ 5000 ~ 12000
2. Mechanisms of ice flow
Ralf Greve: Modelling of ice sheets and glaciers 16/69 Isotropic, non-linear viscous fluid: Glen’s flow law Fluidity factors: TD = 2 ×fluidity = 1 / (2 ×viscosity)
Ralf Greve: Modelling of ice sheets and glaciers 17/69 Basal sliding Difficult to measure, not well understood! Often “Weertman-type” parameterization is used: Two different processes: sliding on hard rock vs. sliding on deformable sediment.
3. Dynamics
Ralf Greve: Modelling of ice sheets and glaciers 19/69 Geometry
Ralf Greve: Modelling of ice sheets and glaciers 20/69 Grounded vs. floating ice Plug flow, dominated by p, txx , tyy , tzz, txy D D D Shear flow, dominated by p, txz, tyz Ice sheet Ice shelf Ice stream
Ralf Greve: Modelling of ice sheets and glaciers 21/69 Full Stokes (FS) flow problem → Fr = [U]2 / (g[H]) ~ 10–15 3-d momentum balance on a flat Earth → Fr/Ro = 2Ω[U][L]/ (g[H]) ~ 5 x 10–8 >>> FS
Ralf Greve: Modelling of ice sheets and glaciers 22/69 Grounded ice: Hydrostatic and shallow ice approximations Full Stokes → Hydrostatic approximation → SIA
Ralf Greve: Modelling of ice sheets and glaciers 23/69 SIA force balance Hydrostatic pressure: Vertical shear stresses: At the ice base (z= b):
Ralf Greve: Modelling of ice sheets and glaciers 24/69 Floating ice: Hydrostatic and shallow shelf approximations Full Stokes → Hydrostatic approximation → SSA
Ralf Greve: Modelling of ice sheets and glaciers 25/69 SSA force balance Hydrostatic vertical normal stress: Vertically integrated horizontal force balance:
Ralf Greve: Modelling of ice sheets and glaciers 32/69 Cold-ice method Temperature equation: Secondary condition: Water content:
Ralf Greve: Modelling of ice sheets and glaciers 33/69 Polythermal method Temperature equation as before, but only solved in cold ice. Water-content equation in temperate ice: Transition conditions at the CTS: (1) melting conditions (am┴> 0) (2) freezing conditions (am┴< 0)
Ralf Greve: Modelling of ice sheets and glaciers 34/69 Polythermal method Geometry Melting conditions, am┴= +0.2 m/a Freezing conditions, am┴= –0.2 m/a Analytical solution for the parallel-sided slab
Ralf Greve: Modelling of ice sheets and glaciers 35/69 Enthalpy method One common thermodynamic field Enthalpy h= fct(temperature T, water content W) for cold and temperate ice:(Aschwanden et al. 2012) Enthalpy equation for cold and temperate ice:
5. Ice thickness equation
Ralf Greve: Modelling of ice sheets and glaciers 37/69 Ice thickness equation Geometry, processes:
Ralf Greve: Modelling of ice sheets and glaciers 38/69 Ice thickness equation Volume balance (due to incompressibility):
6. Sketch of the coupled initial–boundary value problem
Ralf Greve: Modelling of ice sheets and glaciers 40/69 Sketch of the coupled initial–boundary value problem Rectangular boxes: prognostic model components. Ovals: model input.
7. Analytical solutions
8. Numerical solutions and models
Ralf Greve: Modelling of ice sheets and glaciers 49/69 Numerical solutions Finite difference methods (FDM). Finite elements methods (FEM). Finite volume methods (FVM). Others... Non-linear, thermo-mechanically coupled, free-surface flow problem → in general, numerical solution techniques are required:
Ralf Greve: Modelling of ice sheets and glaciers 50/69 Model SICOPOLIS “SImulation COde for POLythermal Ice Sheets” Open-source model, mainly delevoped at ILTS (www.sicopolis.net). Coded in Fortran. Shallow ice + shallow shelf approximations. Finite difference method.
Ralf Greve: Modelling of ice sheets and glaciers 51/69 SICOPOLIS –Sigma transformation Vertical ice columns mapped on [0,1] intervals. Separate mappings for cold-ice layer, temperate-ice layer [polythermal method only], lithosphere (rock) layer →vertical coordinates ζc, ζt, ζr. Cold-ice layer: Densification of grid points close to the base →parameter a.
Ralf Greve: Modelling of ice sheets and glaciers 52/69 SICOPOLIS –Numerical solution technique Finite difference method. Staggered grid (Arakawa-C grid): –Velocities (vx, vy, vz) and volume fluxes (Qx, Qy) are defined in between grid points. –Other field quantities (Ψ) are defined on grid points.
Ralf Greve: Modelling of ice sheets and glaciers 53/69 SICOPOLIS –Numerical solution technique 2nd-order central differences for diffusive terms. 1st-order upstreaming for advective terms. Time-stepping (ice thickness equation): –Time-step Δt (same for velocity and isostasy). –Over-implicit in the linear part, explicit in the non-linear part. Time-stepping (temperature, water content and age): –Time-step Δt (integer multiple of Δt). –Implicit in the vertical, explicit in the horizontal derivatives. ~
Ralf Greve: Modelling of ice sheets and glaciers 54/69 Model Elmer/Ice elmerice.elmerfem.org Add-on package to Elmer (multi-physics FEM suite mainly developed by CSC –IT Center for Science, Espoo, Finland). Open-source model. Solves the full Stokes (FS) equations. Applicable to ice sheets, ice shelves, ice caps and glaciers.
9. Selected applications
Ralf Greve: Modelling of ice sheets and glaciers 56/69 Application of SICOPOLIS to the Austfonna ice cap, Svalbard Objective: To reproduce the observed surge-recovery cycles of several drainage basins of Austfonna. Austfonna (Dunse et al. 2011)
Ralf Greve: Modelling of ice sheets and glaciers 57/69 Simulated surface velocity field over 1000 years of present-day climate conditions Animation → supplementary material
Ralf Greve: Modelling of ice sheets and glaciers 64/69 Application of Elmer/Ice to Bowdoin Glacier, Greenland Bowdoin Glacier: Marine-terminating outlet glacier located in NW Greenland. Field surveys (2013–2016), satellite data analysis. (Sugiyama et al. 2015)
Ralf Greve: Modelling of ice sheets and glaciers 65/69 Control run –Set-up Diagnostic simulation, resolution ~ 70 m. Temperature field: Steady state w/o basal sliding. Control inverse method: Minimize cost function Jtot = J0+ λJreg (J0:misfit between modelled and observed surface velocities, Jreg:regularization) → distribution of the basal drag β (simplified sliding law τb= βvb)Observed surface velocities: Sugiyama et al. (2015)
Ralf Greve: Modelling of ice sheets and glaciers 66/69 Control run –Results Simulated surface velocity Observed surface velocity Slip ratioBasal drag coefficient Seddik et al. (in preparation)
Acknowledgements Many, many colleagues (development of the theory, the models SICOPOLIS and Elmer/Ice, applications…). Funding by the Japan Society for the Promotion of Science (JSPS) (KAKENHI grants, fellowships). INdAM for the kind invitation.
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