Two-phase flow dynamics in ice sheets Ian Hewitt, University of Oxford Thanks to: Christian Schoof, Richard Katz, ERC
1. Subglacial drainage systems as a compacting/reacting porous medium 2. Two-phase mechanics of temperate ice
Scene setting B B ' West Antarctica 4,000 Ellsworth Ice Elevation (m) Mountains Land 2,000 Ocean Ronne Ice Shelf Ross Ice Shelf (MSL) 0 -2,000 0°E C Vertical exaggeration x80 Bentley Subglacial Trench 30°E 30°W B Ice velocity 60°W 60°W 60°W 60°E 60°E 60°E (m year –1 ) 1,000 100 90°W 90°E 90°E 90°E 10 S ° 0 8 S <1.5 ° 0 7 120°E 120°E 120°E 120°W B ' 150°E 150°E 150°E 150°W C ' 180°E East Antarctica C C ' 4,000 Elevation (m) Gamburtsev Subglacial Vostok Subglacial Mountains 2,000 Highlands Gunnerus Bank (MSL) 0 -2,000 Vertical exaggeration x80 Aurora Vincennes Astrolabe Subglacial Basin Subglacial Basin Subglacial Basin
Introduction Ice is a polycrystalline material. It deforms through a combination of diffusion and dislocation creep. Models usually treat ice sheets as a shear-thinning viscous fluid, with effective viscosity dependent on temperature. τ ij = 2 η ˙ ε ij η = η ( T, φ ) Basal boundary condition is crucial Cold Ice τ b = β ( x , U b , . . . ) u b Temperate Ice Bedrock Ice temperature controlled by: radiation, advection, conduction, shear heating, basal friction and geothermal heat flux.
Inferred basal slipperiness Subglacial effective pressure Evidence suggests that basal slipperiness depends on subglacial water pressure τ b = β ( x , U b , N ) u b Effective pressure N = p i � p w Isaac et al 2015 τ b β u b Ice surface Water-filled cavities z = s � τ b β u b p i ⇡ ρ i g ( s � b ) β u b Deformation Basal water z = b of wet till
Subglacial drainage systems h h S
Mount Robson, Canada 10 m
Cordillera Blanca, Peru
A distributed drainage model ∂ h Mass conservation ∂ t + r · q = m + r Flux law q = � K ( h ) r φ Hydraulic potential φ = ρ w gb + p w φ = ρ i gs + ( ρ w � ρ i ) gb � N �r r � ∂ h Evolution law ∂ t = . . . ◆ Ice surface h ‘Water table’ � φ z = s ρ w g Basal water z = b
A distributed drainage model Sliding Melting Creep = h h r l r G @ h @ t = ⇢ w m + U b chN Evolution law ( h r � h ) � ˜ ⇢ i ` r ⌘ Creep closure Sliding m = G + kT z + τ b · u b + | q · r � | Melting ⇢ w L ∂ h Mass conservation ∂ t � ⌅ · [ K ( h ) ⌅ φ ] = m + r ∂ s S Hydraulic potential �r φ = Φ + r N Walder 1986, Fowler 1986, Kamb 1987, Schoof et al 2012
A distributed drainage model ∂ h ∂ t � h as specific heat
A conduit drainage model ⇤ ⇤ ∂ S ∂ t + ∂ Q ⇤ ⇤ Mass conservation ∂ s = M + q in ⇤ ⇤ ⇤ ⇤ − 1 / 2 ∂φ ⇤ ⇤ ∂φ Flux law Q = − K c S α ⇤ ⇤ ⇤ ⇤ ∂ s ∂ s ⇤ ⇤ Hydraulic potential φ = ρ w gb + p w Viscous creep φ = ρ i gs + ( ρ w � ρ i ) gb � N Melting Cross-sectional area evolution w @ S @ t = ⇢ w C SN M � ˜ ⇢ i ⌘ ⇤ ⇤ 1 ⇤ Q ∂φ ⇤ ⇤ Melting M = ⇤ ⇤ ρ w L ∂ s ⇤ ⇤ ⇤ � φ z = s ρ w g z = b Röthlisberger 1972, Nye 1976
Subglacial conduits ∂ S ∂ t C 2 S Q Mass conservation prevents unbounded growth ... but neighbouring channels compete with one other
A hybrid sheet-conduit drainage model h S Numerical method combines 1d and 2d finite elements Werder et al 2013
A hybrid sheet-conduit drainage model Time Werder et al 2013
1. Subglacial drainage systems as a compacting/reacting porous medium 2. Two-phase mechanics of temperate ice
Temperature of glaciers and ice sheets Many mid-latitude alpine glaciers are entirely ‘temperate’. a) ‘Canadian type’ b) In high latitudes many glaciers are ‘polythermal’. ‘Scandinavian type’ temperate cold Aschwanden et al (2012) Most of the large ice-sheets (Antarctica & Greenland) are ‘cold’, but may be locally temperate near base Temperate = in thermodynamic equilibrium at T = T m ( p ) Lüthi et al (2009)
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