Ice sheets with rapid basal sliding Ian Hewitt, University of Oxford - PowerPoint PPT Presentation

Ice sheets with rapid basal sliding Ian Hewitt, University of Oxford

Antarctic ice sheet Ice near the margins has accelerated substantially over the last decade Mouginot et al 2014 McMillan et al 2014 Marine Ice Sheet Collapse Potentially Under Way for the Thwaites Glacier Basin, West Antarctica Collapse of the West Antarctic Ice Sheet after local Ian Joughin, Benjamin E. Smith, Brooke Medley destabilization of the Amundsen Basin Johannes Feldmann a,b and Anders Levermann a,b,1

Antarctic ice sheet Ice speed Bed elevation 0°E C 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 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 Vertical exaggeration x80 Bentley Subglacial Trench

Sea level The glacial period is punctuated by several periods of rapid sea level rise (~1m/century) Time 0 50 100 150 200 250 300 350 400 450 500 0 RSL (m) 60 –40 –80 dRSL (m kyr –1 ) 40 –120 20 0 Red Sea Relative Sea Level Grant et al 2014

Observations show rapid changes in ice dynamics can occur What mechanisms cause massive ice loss, and how rapid ? The Greenland and Antarctic ice sheets contain ice equivalent to around 65m sea level

Basal sediments Most fast-moving ice is thought to be underlain by water-saturated sediments Laboratory experiments on till samples suggest very weak dependence of stress on strain rate Shear strength depends on effective pressure (i.e. on pore pressure) τ 0 = c + p e tan φ Iverson 2010 This talk: Explore the dynamics of an ice sheet with a perfectly plastic bed

Glacier flow ~10,000,000 x real time Extreme Ice Survey - Time-lapse camera Khumbu glacier, Nepal

A simple ice-sheet model + a z = s ( x, t ) u ( x, z, t ) ( b h = s − b Ice z = b ( x ) z Bedrock x Net accumulation - melting (climate forcing) ∂ h ∂ t + ∂ q Mass conservation ∂ x = a ∂ t ∂ x Z Z s q ( x, t ) = hu = u d z Ice flux b u ( x, z, t ) ( b Force balance and boundary conditions (Stokes flow)

Sliding at the bed � � The fastest motion occurs as a plug flow u ( x, z, t ) ≈ u ( x, t ) � z = s � + h z = b Two mechanisms for sliding - - a thin film of water between ice and bedrock | | τ b = Cu Nye 1969 - lubrication by a layer of underlying water-saturated sediments (viscous) τ b = η s u d s τ b ≈ − ρ i gh ∂ s From force balance ∂ x Combining with mass conservation gives a diffusion equation for ice thickness ✓ ◆ ∂ h ∂ t = ∂ h 2 ∂ h + a ∂ x ∂ x

Plastic bed model Consider the friction law τ b ≤ τ 0 u = 0 τ b = τ 0 u ≥ 0 ( Assuming deformation occurs, force balance becomes an equation for ice thickness − ∂ x ≤ ρ i g � ∂ h ∂ x = ∂ b τ 0 h = 0 at x = x m ∂ x + ρ i gh So ice thickness is (almost) determined without reference to mass conservation or velocity Z x m Z x m ˙ Ice volume V = h d x Global mass conservation V = a d x 0 0

Example Accumulation, a h 0 = τ 0 p 2 h 0 ( x m − x ) 1 / 2 Eg. Flat bed h = ρ i g Position-dependent accumulation a = λ ( x 0 − x ) ( Elevation, z = 2 x 0 ere x m Distance, x Global mass conservation ✓ x 0 − 1 ◆ p 2 h 0 x 1 / 2 x m = λ ˙ ere x m = 2 x 0 2 x m x m Stable steady state m ‘accumulation = ablation’

Example h 0 = τ 0 p 2 h 0 ( x m − x ) 1 / 2 h = Eg. Elevation-dependent accumulation ρ i g a = λ ( s − s 0 ) ( Elevation, z Distance, x Global mass conservation − ✓ 2 ◆ s 2 x m = 9 p p 2 h 0 x 1 / 2 2 h 0 x 3 / 2 0 x m = λ ˙ m − s 0 x m Unstable steady state m 3 8 h 0 Ice sheet shrinks to nothing, or fills continental shelf

Marine-terminating glaciers Extreme Ice Survey - Time-lapse camera Columbia Glacier, Alaska

Marine-terminating glaciers � z = b ( x ) + h f ( x ) − z z = 0 x q c h ( x, t ) = x m z = b ( x ) − = h f ( x ) = − ρ o b ( x ) Flotation thickness ρ i q m � h m ˙ x m = q c Mass conservation calving rate (includes ocean-driven melting) − z = b + h f z = 0 h x m

Approximate force balance [ h ] 0 = − ∂ p ∂ x + ∂τ xx ∂ x + ∂τ xz [ x ] ∼ 10 − 3 Full force balance Small aspect ratio ∂ z − ∂ x ∂ x ∂ z 0 = − ∂ p ∂ z + ∂τ xz ∂ x + ∂τ zz ∂ z − ρ i g p − τ zz = ρ i g ( s − z ) ( vertical balance approximately hydrostatic ∂ u depth integrate horizontal balance, with τ xx ≈ 2 η i ∂ x ✓ ◆ ∂ 4 η i h ∂ u � τ b � ρ i gh ∂ s ∂ x = 0 ∂ x ∂ x ✓ ◆ Z s − p + τ xx d x = − 1 2 ρ i gh 2 + 4 η i h ∂ u Note also, depth-integrated horizontal stress ∂ x b

Conditions at the marine margin q m = q c ✓ ◆ Continuity of longitudinal stress at the margin − 1 2 ρ i gh 2 + 4 η i h ∂ u ∂ x = − 1 2 ρ o gb 2 at x = x m x m ρ Z q c = q m − h ˙ x m Some models for calving prescribe a ‘rate’ Others prescribe an equilibrium-like ‘criteria’ - e.g. flotation condition h = h f ≡ − ρ o b ρ i calving rate (includes ocean-driven melting)

Full model ◆ Non-dimensional equations r = ρ o ⇡ 1 . 1 ( ρ i ∂ h ∂ t + ∂ q ∂ x = a ε = η i [ a ] [ τ ][ x ] ⌧ 1 ( ✓ ◆ 4 h ∂ u − τ ∗ − h ∂ ( b + h ) ε ∂ = 0 τ ∗ = 1 ∂ x ∂ x ∂ x Boundary conditions q = 0 at x = 0 ε 4 h ∂ u ∂ x = 1 2( h 2 − h 2 h = h f f /r ) at x = x m ( t ) ( flotation condition � z = b ( x ) + h f ( x ) z x h ( x, t ) = x m z = b ( x )

Numerical solutions Steady states for constant accumulation 2 Elevation, z a = 0 . 1 0 -2 0 0.5 1 1.5 2 Distance, x 4 Velocity, u 2 Elevation, z a = 1 0 -2 0 0.5 1 1.5 2 Distance, x 4 Velocity, u 2 Elevation, z a = 2 0 -2 0 0.5 1 1.5 2 Distance, x 4 Velocity, u

Margin boundary layer Longitudinal stress most important near margin h = ε 1 / 4 H u = ε − 1 / 4 U x m ( t ) � x = ε 1 / 2 X H f = ε − 1 / 4 rb ( x m ) rescale ( Equations become ∂ ∂ X ( HU ) = 0 HU = Q ✓ ◆ 4 H ∂ U � τ ∗ + H ∂ H ∂ H f , 1 − 1 /r ( Q ) ∂ X = 0 8 ∂ X ∂ X V U X = � V � 4 U 2 V � V 2 V X = � τ ⇤ U 4 Q + Q U Boundary / matching conditions U U = Q V = 1 8(1 � 1 /r ) H f at X = 0 H f Q = Q m ( H f ) ⇡ (1 � 1 /r ) H 4 U ! 0 V ! 0 as X ! 1 f 8 τ ∗ Only one value of allows the required trajectory Q

Reduced model Away from the margin, ignore longitudinal stress ∂ h ∂ t + ∂ q ∂ x = a at b ( x ) = O ( ε 1 / 4 ), � τ ∗ � h ∂ h h f = ε 1 / 4 H f ∂ x = 0 h = 0 at x = x m √ 2 τ ∗ ( x m − x ) 1 / 2 h = 2 Q m ( H f ( x )) Integrating mass conservation 1.5 Z x m √ 2 τ ∗ x 1 / 2 m ˙ x m = a d x − Q m ( H f ( x m )) Flux, q 0 1 Z x 0.5 a d x 0 Multiple steady states depending on bed slope 0 (analogous to grounding lines, Schoof 2007) 0 0.5 1 1.5 2 Distance, x

More about calving The role of calving in this model was to evacuate ice delivered to the margin If the processes responsible for calving cannot keep up, a floating ice shelf will form But if calving is more efficient, it may result in margin thickness above flotation � h m ˙ x m = q m � q c Replace flotation condition with The boundary layer analysis can be generalised to find q m = Q m ( h m , h f ) ( h m ˙ x m = Q m ( h m , h f ) � q c Q m m m f

Reduced model II Away from the margin, ignore longitudinal stress ∂ h ∂ t + ∂ q ∂ x = a at b ( x ) = O ( ε 1 / 4 ), � τ ∗ � h ∂ h h f = ε 1 / 4 H f ∂ x = 0 h = 0 at x = x m √ 2 τ ∗ ( x m − x ) 1 / 2 h = Integrating mass conservation Z x m √ Z 2 τ ∗ x 1 / 2 m ˙ x m = a d x − q c 0 Q m ( H m , H f ) = q c Height above flotation adjusts to balance calving rate (Hindmarsh 2012)

Could meltwater-induced ice-sheet slow-down be a greater problem than acceleration? Recent observations suggest surface meltwater Melt (w.e. m yr − 1 ) 4 a penetrating to the bed may slightly reduce, rather 3 than increase, ice velocities. 2 c 120 Area (km 2 ) 2,000 80 1 N 1,000 120 40 b 0 0 110 400 600 800 1,000 Elevation (m.a.s.l.) –0.1 m yr − 2 , P = 0.80 100 Velocity (m yr − 1 ) 90 80 70 60 –1.5 m yr − 2 , P < 0.01 R 2 = 0.79 50 40 Zwally et al 2002 1985 1990 1995 2000 2005 2010 2015 Year Tedstone et al 2015 Changes in accumulation are the primary driver of the marine ice-sheet instability in this model. For outlet glaciers, ‘accumulation’ includes inflow of ice from catchment basin - slow-down of surrounding ice will reduce ice supply, with potential rapid retreat.

Summary Ice-sheet beds can be modelled using plastic rheology (though should more realistically couple with water pressure) - enables simple studies of stability. Calving processes act to keep the glacier tongue near flotation - but small differences in depth have large effect on ice flux. Marine-terminating glaciers have potential for rapid retreat - slow-down of ice in the feeding catchment basin may initiate retreat.