Of cold ice, warm ice and water: thermodynamics of ice sheets and glaciers

Of cold ice, warm ice and water: thermodynamics of ice sheets and glaciers IGS Global Seminar, 2021.10.13, 20:00 UTC Ralf Greve Institute of Low Temperature Science, Hokkaido University, Sapporo, Japan Ralf Greve: Of cold ice, warm ice and water 2/29 Thermodynamics From Greek θέρμη (therme), meaning “heat”, and δύναμις (dynamis), meaning “power”. Branch of physics that deals with heat, work and temperature, and their relation to energy, radiation and physical properties of matter. (https://en.wikipedia.org/wiki/Thermodynamics) Ralf Greve: Of cold ice, warm ice and water 3/29 Why bother? Two mechanisms contribute to ice flow Internal deformation (ice = viscous fluid) Basal sliding (on hard rock or soft sediment) Ralf Greve: Of cold ice, warm ice and water 4/29 Why bother? Both internal deformation and basal sliding depend strongly on temperature (and water content) Viscosity of polycrystalline ice (Greve and Blatter 2009) Basal sliding → Flow of ice sheets and glaciers: Thermo-mechanically coupled problem! Ralf Greve: Of cold ice, warm ice and water 5/29 Why me? (https://doi.org/10.5281/zenodo.3815324) Ralf Greve: Of cold ice, warm ice and water 6/29 (https://doi.org/10.5281/zenodo.3815324) Why me? Based on… Ralf Greve: Of cold ice, warm ice and water 7/29 Temperature computation Temperature equation: Material time derivative (local derivative + 3D advection) Heat conduction (diffusion) Strain heating (dissipation) Boundary conditions:Surface temperature Ts Geothermal heat flux qgeo →𝜕𝜕𝑇𝑇 𝜕𝜕𝐧𝐧 d𝑇𝑇 d𝑡𝑡=1 𝜌𝜌𝜌𝜌div 𝜅𝜅grad 𝑇𝑇+𝛷𝛷 𝜌𝜌𝜌𝜌 Ralf Greve: Of cold ice, warm ice and water 8/29 Some MATLAB tests for an ice column… H= 100 m, α= 10°, Ts= –10°C, qgeo = 50 mW m−2, Tinit = –10°C, t= 0…1000 a Good! Ts qgeo z H α Ralf Greve: Of cold ice, warm ice and water 9/29 Some MATLAB tests for an ice column… Let’s make it a bit thicker: H= 120 m Not good! Ralf Greve: Of cold ice, warm ice and water 16/29 Polythermal method Temperature equation as before, but only solved in cold ice. Water-content equation in temperate ice: Melting conditions: d𝑊𝑊 d𝑡𝑡=1 𝜌𝜌div 𝜈𝜈grad 𝑊𝑊+𝛷𝛷 𝜌𝜌𝜌𝜌 𝑎𝑎m ⟂> 0 𝑎𝑎m ⟂< 0 Freezing conditions: Ice flow from cold to temperate → ∂T/∂nand Wcontinuous across the CTS. Ice flow from temperate to cold → ∂T/∂nand Wjump across the CTS. Energy jump condition at the CTS: (Gusmeroli et al., 2010) Ralf Greve: Of cold ice, warm ice and water 17/29 Steady-state solution for an ice column Melting conditions, am┴= +0.2 m a−1 Freezing conditions, am┴= –0.2 m a−1 H= 200 m, α= 4°, Ts= –3°C / –10°C, am┴= +0.2 m a−1 / –0.2 m a−1 𝑊𝑊+=𝑊𝑊−= 0 𝜕𝜕𝑇𝑇 𝜕𝜕𝜕𝜕 + =𝜕𝜕𝑇𝑇 𝜕𝜕𝜕𝜕 − = 0 𝑊𝑊+= 0 𝑊𝑊−> 0 𝜕𝜕𝑇𝑇 𝜕𝜕𝜕𝜕 + < 0 𝜕𝜕𝑇𝑇 𝜕𝜕𝜕𝜕 − = 0 Ralf Greve: Of cold ice, warm ice and water 18/29 Steady-state solution for the Greenland ice sheet (Greve, 1995, 1997) Ice-sheet model 𝑇𝑇b ′ Areas with cold base temperate base temperate layer occur. At 40 km resolution, freezing conditions only detected for a single grid point ( ) → not that important. Ralf Greve: Of cold ice, warm ice and water 19/29 with 𝑘𝑘= 𝜅𝜅 𝜌𝜌𝜌𝜌 for cold ice 𝜈𝜈 𝜌𝜌for temperate ice Enthalpy method One common thermodynamic field for cold and temperate ice:(Aschwanden et al., 2012) Enthalpy equation for cold and temperate ice: dℎ d𝑡𝑡=div 𝑘𝑘grad ℎ+𝛷𝛷 𝜌𝜌 ℎ 𝑇𝑇,𝑊𝑊=� 𝑇𝑇0 𝑇𝑇𝜌𝜌 𝑇𝑇′d𝑇𝑇𝑇+𝜌𝜌𝑊𝑊 Enthalpy ℎ=fct(Temperature 𝑇𝑇,water content 𝑊𝑊) Ralf Greve: Of cold ice, warm ice and water 20/29 Thermodynamics solvers in SICOPOLIS Cold-ice method (COLD) Polythermal method Terrain-following coordinates (sigma transformation), one common domain ζc= 0…1 for cold and temperate ice. Two separate domains ζc= 0…1, ζt= 0…1. Enforcement of the energy jump condition at the CTS: Melting and freezing conditions →POLY1. Only melting conditions →POLY2. 𝜕𝜕𝜁𝜁c 1 0 Base CTS Surface 𝜕𝜕𝜁𝜁c 1 0 Base CTS Surface 𝜁𝜁t 1 0 Ralf Greve: Of cold ice, warm ice and water 21/29 Thermodynamics solvers in SICOPOLIS Enthalpy method One common domain ζc= 0…1 for cold and temperate ice. Enforcement of the continuity of the temperature gradient at the CTS: No →conventional enthalpy scheme (ENTC). Yes →melting-CTS enthalpy scheme (ENTM). 𝜕𝜕𝜁𝜁c 1 0 Base CTS Surface Ralf Greve: Of cold ice, warm ice and water 22/29 EISMINT Phase 2 SGE experiment A1 produces a Greenland-like ice sheet (Payne et al., 2000; Greve and Blatter, 2016) POLY2, Δx = 10 km, Δt = 2 a Vtot = 2.1 × 106 km3 Ralf Greve: Of cold ice, warm ice and water 23/29 Exp. A1: Evolution of the temperate ice volume COLD ENTC ENTM POLY2 Δx = 10 km Δt = 20 a Δx = 10 km Δt = 2 a Δx = 10 km Δt = 2 a HVR (Greve and Blatter, 2016) COLD: Much too thick. ENTC: Somewhat too thick. ENTM: Converges to POLY2. HVR: high vertical resolution (5 ×standard) Ralf Greve: Of cold ice, warm ice and water 24/29 Exp. A1: Thickness of temperate ice layer Δx = 10 km Δt = 20 a (Greve and Blatter, 2016) POLY2: A bit wavy (instability). COLD: Much too thick. ENTC & ENTM: Somewhat noisy. Ralf Greve: Of cold ice, warm ice and water 25/29 Exp. A1: Thickness of temperate ice layer Δx = 10 km Δt = 2 a POLY2: Now fine (stable). COLD: Still much too thick. ENTC & ENTM: Still somewhat noisy. (Greve and Blatter, 2016)