Ice-sheet dynamics: the influence of glacier sliding on ice loss and - - PowerPoint PPT Presentation

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)?

20 40 60 –120 –80 –40 RSL (m) 50 100 150 200 250 300 350 400 450 500 dRSL (m kyr–1)

Grant et al 2014

Red Sea Relative Sea Level The glacial period is punctuated by several periods of rapid sea level rise (~1m/century) Time Sea level history Global sea level has been at least 6m higher in previous interglacials.

Antarctic Ice Sheet Net mass loss currently ~100 Gt/yr (~0.3 mm/yr sea level rise) Current volume ~27x106 km3 (~58m sea level equivalent)

Net mass loss currently ~200 Gt/yr (around 0.6 mm/yr sea level rise) Greenland Ice Sheet Current volume ~2.7x106 km3 (~7m sea level equivalent) Timescale ~10,000 years

Laura Stevens

• 600
• 400
• 200

200 400 600 800 1960 1970 1980 1990 2000 2010 Mass flux (Gt yr

• 1)

Year D SMB MB 0.5 1.0 1.5

• Eq. SLR (mm yr
• 1)

van den Broeke et al 2016

Surface balance Discharge

accumulation - runoff

Greenland ice sheet mass balance

calving

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.

heat for are rates, stud- the distinguish small

• btained

The

van de Wal et al 2015

Ice speed (GPS) Runoff

Time

suggests potential for significant changes in ice velocity

67.9° N 68.6° N 51° W 50° W 49° W 6 8 1 , 1 , 2 C B A 10 20 km –50 –40 –30 –20 –10 10 20 30 40 50 Change (%) –30 30 400 800 1,200 a 400 600 800 1,000 Elevation (m.a.s.l.) –30 –20 –10 10 Change (%) Change (%) Area (km2)

b

Greenland

1 2 3 4

Melt (w.e. m yr−1)

a

1985 1990 1995 2000

Year

2005 2010 2015 40 50 60 70 80 90 100 110 120

Velocity (m yr−1)

–0.1 m yr−2, P = 0.80 –1.5 m yr−2, P < 0.01 R2 = 0.79

b

400 600 800 1,000

Elevation (m.a.s.l.)

1,000 2,000

N

c

40 80 120

Area (km2)

Tedstone et al 2015

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?

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

Time Ice speed Subglacial discharge (areal m2/s)

Hewitt 2013, EPSL

Ice Bedrock

x z

z = s(x, t) z = b(x) Vertically-integrated mass conservation = net accumulation - melting = a Force balance

• 0 = r · σ + ρig

Z pi = ρig(s z) = a

• ⌧b = f(u, N)

(

N = pi pw

h = s b @h @t + @q @x = a ⌧b = ⇢igh@s @x + @ @x ✓ 4h⌘i @u @x ◆ Mathematical model b u(x, z, t) ( ∂t ∂x q(x, t) = hu = Z s

b

u dz

Ice Bedrock

Theoretical framework for modelling sliding

~10 m ~1000 km ~4 km ~1 m

pw ub f τb pi Theory and some measurements suggest a friction / slip law of the form ⌧b = f(ub, N) (

N = pi pw

to be applied to the large-scale ice flow τb ub

Boundary conditions ✓ ◆ h = 0, q = 0 at x = xm h ˙ xm = q qc, 4h⌘i @u @x = 1 2

• ⇢igh2 ⇢ogb2

at x = xm land-terminating marine-terminating Mathematical model

Ice Bedrock

x z

z = s(x, t) z = b(x) = a h = s b b u(x, z, t) (

hf = ρo ρi b

flotation thickness + calving condition h = fhf at x = xm

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

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

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

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

@h @t = @ @x ✓ K @h @x ◆ + a K a K = ⇢ighFτ If friction law is invertible, , and bed topography relatively flat, the problem is seemingly diffusive u = F(⌧b, N) ( ✓ ◆ However, the diffusion coefficient may be highly non-linear, and accumulation rate varies with ice thickness.

Ice Bedrock

x z

z = s(x, t) z = b(x) = a u = ub h = s b Mathematical model a = (s se) se equilibrium line altitude (ELA) e.g. generic behaviour is ‘blow-up’ (cf. reaction-diffusion problems)

a λ(s

se

Ice Bedrock

x z

z = s(x, t) z = b(x) Vertically-integrated mass conservation net accumulation - melting = a u = ub Force balance + friction parameterisation h = s b @h @t + @q @x = a q = hu ⌧b = ⇢igh@s @x + @ @x ✓ 4h⌘i @u @x ◆ A reduced ‘plastic bed’ model τb = µhNi = τ0 a = (s se) bed ‘strength’ se equilibrium line altitude (ELA) Goal: Consider effect of a long-term changes in and = τ0 se

Mass conservation Force balance

x z

h(x, t) =

Ice Bedrock

ice thickness & volume xm Land-terminating glacier u Z = τ0 = a Boundary conditions τ0 = −ρigh∂s ∂x ✓ z = s(x, t) e.g. for a flat bed profile ✓ ◆ h = r2τ0 ρig(xm − x)1/2 √ ✓ ◆ h = 0, q = 0 at x = xm(t) (

• cf. Nye 1951, Weertman 1961, 1976

V = Z xm h dx Z dV dt = Z xm a dx an ODE for the evolution of ice volume dV dt = λ " V − V 2/3 ✓9ρig 8τ0 ◆1/3 se # e.g.

Land terminating glacier A gradual decrease in bed strength results in increased velocities and mass loss Time

50 y

Distance [km]

100 200 2000

100 y

200

Elevation [m]

10 y

Velocity [m/y]

0 y

80 100 120 τb [kPa]

Time [y]

100 200

• 10

10 ∆V [km2] 170 180 190 xm [km]

(an increase in bed strength causes the opposite) Rate of ice loss controlled by SMB

Marine-terminating glacier Boundary conditions Mass conservation Force balance thickness + volume τ0 = −ρigh∂s ∂x ✓ V = Z xm h dx Z dV dt = Z xm a dx qc

x z

h(x, t) =

Ice Bedrock

xm u Z = τ0 z = s(x, t) = a qc z = b(x) hf

Z qc = ρig ηiµ ˆ Q(f)h3

f

Z

Calving flux an ODE for the evolution of ice volume

h = fhf, h ˙ xm = q − qc, 4hηi ∂u ∂x = ρig 2 ✓ h − ρi ρo h2

f

◆ at x = xm(t)

Marine terminating glacier

100 y

Distance [km]

100 150 200 1000

200 y

500

Elevation [m]

50 y

Velocity [m/y]

0 y

80 100 120 τb [kPa]

Time [y]

100 200

• 20

20 ∆V [km2] 170 180 190 xm [km]

Time A gradual decrease in bed strength results in increased velocities and mass loss (as for land terminating glaciers)

Marine terminating glacier An increase in bed strength results in initially decreased velocities … but this initiates terminus retreat and acceleration.

100 y

Distance [km]

100 150 200 1000

200 y

500

Elevation [m]

50 y

Velocity [m/y]

0 y

80 100 120 τb [kPa]

Time [y]

100 200

• 60
• 40
• 20

∆V [km2] 160 180 xm [km]

Time Rate of ice loss controlled by ice mechanics (& topography)

• dV

dt = Z xm a dx qc ρ g

x z

= a qc

V

xm = τ0

Z qc = ρig ηiµ ˆ Q(f)h3

f

Z fhf V dV dt = F(V ; t)

se Marine terminating glacier

• dV

dt = Z xm a dx qc ρ g

x z

= a qc

V

xm = τ0

Z qc = ρig ηiµ ˆ Q(f)h3

f

Z fhf V dV dt = F(V ; t)

se Marine terminating glacier

• dV

dt = Z xm a dx qc ρ g

x z

= a qc

V

xm = τ0

Z qc = ρig ηiµ ˆ Q(f)h3

f

Z fhf V dV dt = F(V ; t)

se Marine terminating glacier

• dV

dt = Z xm a dx qc ρ g

x z

= a qc

V

xm = τ0

Z qc = ρig ηiµ ˆ Q(f)h3

f

Z fhf V dV dt = F(V ; t)

se Marine terminating glacier

100 y

Distance [km]

100 150 200 1000

200 y

500

Elevation [m]

50 y

Velocity [m/y]

0 y

• 60
• 40
• 20

Marine terminating glacier An essentially indistinguishable response occurs to an increase in calving

• r an increase in ELA

100 y

Distance [km]

100 150 200 1000

200 y

500

Elevation [m]

50 y

Velocity [m/y]

0 y

• 60
• 40
• 20

Time Increased calving Increased ELA

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.