Ice-sheet dynamics: the influence of glacier sliding on ice loss and sea level Ian Hewitt, Mathematical Institute, University of Oxford
Greenland
How does meltwater penetrating to the bed affect ice-sheet motion? What implications does this have for ice loss (sea level)?
Sea level history 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 Global sea level has been at least 6m higher in previous interglacials.
Antarctic Ice Sheet Current volume ~27x10 6 km 3 (~58m sea level equivalent) Net mass loss currently ~100 Gt/yr (~0.3 mm/yr sea level rise)
Greenland Ice Sheet Current volume ~2.7x10 6 km 3 (~7m sea level equivalent) Net mass loss currently ~200 Gt/yr (around 0.6 mm/yr sea level rise) Timescale ~10,000 years
Laura Stevens
Greenland ice sheet mass balance 800 D 600 Discharge calving 400 SMB Surface balance -1 ) 200 Mass flux (Gt yr accumulation - runoff 0 0 Eq. SLR (mm yr 0.5 MB -200 1.0 -400 -1 ) 1.5 -600 1960 1970 1980 1990 2000 2010 Year van den Broeke et al 2016 Greenland is losing mass due to decreased SMB and increased discharge
Satellite-derived ice surface speeds
Greenland ice sheet velocities Summer drainage of surface meltwater to the bed causes large fluctuations in ice speed. suggests potential for significant changes in ice velocity heat for are rates, stud- the distinguish small Ice speed (GPS) Runoff Time van de Wal et al 2015 obtained The
Greenland ice sheet velocities Longer term measurements show a slight decreasing trend in average velocity, while runoff shows an increasing trend. suggests possible negative relationship between runoff and average velocity? km 4 Melt (w.e. m yr − 1 ) 50 a 0 10 20 Greenland 3 40 2 c 30 120 Area (km 2 ) 2,000 68.6° N 1 80 C N 20 1,000 120 40 b B A 0 0 10 110 400 600 800 1,000 1 1 , Change (%) , 0 Elevation (m.a.s.l.) 2 0 –0.1 m yr − 2 , P = 0.80 0 0 0 100 0 8 6 0 Velocity (m yr − 1 ) 0 0 0 90 –10 80 1,200 a –20 Area (km 2 ) 800 67.9° N 70 400 0 –30 –30 0 30 Change (%) 60 10 b Change (%) 0 –40 –1.5 m yr − 2 , P < 0.01 R 2 = 0.79 –10 50 –20 –30 400 600 800 1,000 –50 40 Elevation (m.a.s.l.) 51° W 50° W 49° W 1985 1990 1995 2000 2005 2010 2015 Tedstone et al 2015 Year
Evolution of the subglacial drainage system Increased efficiency of drainage Isolated water pockets High average water pressure Large melt-enlarged channels Lower average water pressure
Subglacial discharge Ice speed (areal m 2 /s) Time Hewitt 2013, EPSL
Mathematical model = a z = s ( x, t ) u ( x, z, t ) ( b h = s � b Ice z = b ( x ) z Bedrock x Vertically-integrated mass conservation ∂ t ∂ x Z s @ h @ t + @ q @ x = a = a = net accumulation - melting q ( x, t ) = hu = u d z b � 0 = r · σ + ρ i g Force balance � Z ✓ ◆ ⌧ b = � ⇢ i gh @ s @ x + @ @ u 4 h ⌘ i ⌧ b = f ( u, N ) ( N = p i � p w p i = ρ i g ( s � z ) @ x @ x
Theoretical framework for modelling sliding ~1000 km Ice ~4 km u b Bedrock f τ b τ b p i N = p i � p w ~1 m � p w u b ~10 m Theory and some measurements suggest a friction / slip law of the form ⌧ b = f ( u b , N ) ( to be applied to the large-scale ice flow
Mathematical model = a z = s ( x, t ) u ( x, z, t ) ( b h = s � b Ice z = b ( x ) z Bedrock x ✓ ◆ Boundary conditions land-terminating h = 0 , q = 0 at x = x m @ x = 1 @ u ⇢ i gh 2 � ⇢ o gb 2 � � h ˙ x m = q � q c , 4 h ⌘ i at x = x m marine-terminating 2 + calving condition h = fh f at x = x m h f = � ρ o flotation thickness b ρ i
Glacier flow ~10,000,000 x real time Extreme Ice Survey - Time-lapse camera Khumbu glacier, Nepal
Glacier flow ~10,000,000 x real time Extreme Ice Survey - Time-lapse camera Khumbu glacier, Nepal
Marine-terminating / tidewater glaciers Extreme Ice Survey - Time-lapse camera Columbia Glacier, Alaska
Marine-terminating / tidewater glaciers Extreme Ice Survey - Time-lapse camera Columbia Glacier, Alaska
Mathematical model = a z = s ( x, t ) u = u b h = s � b Ice z = b ( x ) z Bedrock x If friction law is invertible, , and bed topography relatively flat, u = F ( ⌧ b , N ) ( the problem is seemingly diffusive ✓ ◆ ✓ ◆ @ h @ t = @ K @ h + a K K = ⇢ i ghF τ a @ x @ x However, the diffusion coefficient may be highly non-linear, and accumulation rate varies with ice thickness. a e.g. a = � ( s � s e ) s e equilibrium line altitude (ELA) λ ( s s e generic behaviour is ‘blow-up’ (cf. reaction-diffusion problems)
A reduced ‘plastic bed’ model = a z = s ( x, t ) u = u b h = s � b Ice z = b ( x ) z Bedrock x Vertically-integrated mass conservation @ h @ t + @ q q = hu a = � ( s � s e ) @ x = a net accumulation - melting s e equilibrium line altitude (ELA) Force balance + friction parameterisation ✓ ◆ ⌧ b = � ⇢ i gh @ s @ x + @ @ u 4 h ⌘ i τ b = µ h N i = τ 0 bed ‘strength’ @ x @ x s e Goal: Consider effect of a long-term changes in and = τ 0
Land-terminating glacier = a z = s ( x, t ) Ice h ( x, t ) = u Z z = τ 0 ✓ ◆ Bedrock x m x Boundary conditions h = 0 , q = 0 at x = x m ( t ) ( ✓ ◆ Z x m τ 0 = − ρ i gh ∂ s Force balance ice thickness & volume V = h d x ∂ x 0 r 2 τ 0 ✓ ρ i g ( x m − x ) 1 / 2 h = e.g. for a flat bed profile Z √ Z x m d V d t = a d x Mass conservation an ODE for the evolution of ice volume 0 " # ◆ 1 / 3 ✓ 9 ρ i g d V V − V 2 / 3 e.g. d t = λ s e 8 τ 0 cf. Nye 1951, Weertman 1961, 1976
Land terminating glacier A gradual decrease in bed strength results in increased velocities and mass loss (an increase in bed strength causes the opposite) 120 0 y 100 τ b [kPa] Velocity [m/y] Elevation [m] 80 10 y Time 190 x m [km] 50 y 180 170 10 2000 200 100 y 0 ∆ V [km 2 ] 0 0 -10 0 100 200 0 100 200 Distance [km] Time [y] Rate of ice loss controlled by SMB
Marine-terminating glacier = a z = s ( x, t ) Ice � q c h ( x, t ) = u z x h f Z = τ 0 z = b ( x ) Bedrock x m ✓ ◆ ∂ u ∂ x = ρ i g h − ρ i h 2 Boundary conditions h = fh f , h ˙ x m = q − q c , 4 h η i at x = x m ( t ) f 2 ρ o Z x m τ 0 = − ρ i gh ∂ s Z thickness + volume V = h d x Force balance ∂ x Z 0 ✓ Z x m d V q c = ρ i g ˆ Q ( f ) h 3 d t = a d x � q c Calving flux Mass conservation f η i µ 0 Z an ODE for the evolution of ice volume
Marine terminating glacier A gradual decrease in bed strength results in increased velocities and mass loss (as for land terminating glaciers) 120 0 y 100 τ b [kPa] Velocity [m/y] Elevation [m] 80 50 y Time 190 x m [km] 180 100 y 170 20 1000 500 200 y 0 0 0 ∆ V [km 2 ] -20 100 150 200 0 100 200 Distance [km] Time [y]
Marine terminating glacier An increase in bed strength results in initially decreased velocities … but this initiates terminus retreat and acceleration . 120 0 y 100 τ b [kPa] Velocity [m/y] Elevation [m] 80 50 y Time 180 x m [km] 100 y 160 0 1000 500 200 y -20 -40 0 0 ∆ V [km 2 ] -60 100 150 200 0 100 200 Distance [km] Time [y] Rate of ice loss controlled by ice mechanics (& topography)
Marine terminating glacier � � Z x m = a d V d t = a d x � q c Z s e 0 ρ g q c = ρ i g � q c ˆ Q ( f ) h 3 z V f η i µ x Z fh f = τ 0 x m d V d t = F ( V ; t ) V
Marine terminating glacier � � Z x m = a d V d t = a d x � q c Z s e 0 ρ g q c = ρ i g � q c ˆ Q ( f ) h 3 z V f η i µ x Z fh f = τ 0 x m d V d t = F ( V ; t ) V
Marine terminating glacier � � Z x m = a d V d t = a d x � q c Z s e 0 ρ g q c = ρ i g � q c ˆ Q ( f ) h 3 z V f η i µ x Z fh f = τ 0 x m d V d t = F ( V ; t ) V
Marine terminating glacier � � Z x m = a d V d t = a d x � q c Z s e 0 ρ g q c = ρ i g � q c ˆ Q ( f ) h 3 z V f η i µ x Z fh f = τ 0 x m d V d t = F ( V ; t ) V
Marine terminating glacier An essentially indistinguishable response occurs to an increase in calving or an increase in ELA Increased calving Increased ELA 0 y 0 y Velocity [m/y] Velocity [m/y] Elevation [m] Elevation [m] 50 y 50 y Time 100 y 100 y 1000 500 1000 500 200 y 200 y -20 -20 -40 -40 0 0 0 0 -60 -60 100 150 200 100 150 200 Distance [km] Distance [km]
Summary Subglacial meltwater can both increase and decrease average ice speeds. Changes in either direction have potential to influence ice loss. A simplified model suggests ice-sheet slow down can help induce tidewater- glacier retreat , and hence may facilitate rapid ice loss. Recent retreat and speed-up of tidewater glaciers in Greenland may be as much a response to inland forcing as ocean forcing.
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