SLIDE 1
Tidewater glaciers currently discharge around half of Greenland’s ice loss to the ocean, and can change rapidly. Certain positions of the ice front act as ‘pinning points’, at which the glacier achieves a roughly steady balance (accumulation≈discharge). Changes in forcing can cause such points to disappear or lose stability. Central question: what controls ice discharge? and, related to that, what determines the location of the ice front? The highly non-linear response is inherent to the dynamics of tidewater glaciers, whether forcing is from the ocean or from upstream. Summary
Upstream Forcing of Tidewater Glacier Retreat
Ian Hewitt, University of Oxford I’ll describe a simplified model that helps elucidate this.
SLIDE 2 Motivation: impact of subglacial lubrication
Drainage of surface meltwater to the bed affects ice speed (due to influence on water pressure).
Tedstone et al 2015
c d
Zwally et al 2002
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
Possibility that increased surface melt could cause increased ice speeds, and consequently (perhaps?) ice loss. Some recent studies suggest increased surface melt may have caused a slight decrease in average ice speeds (due to more efficient subglacial drainage). What impact do we expect such changes to have
SLIDE 3 Ice front evolution
xm
calving + frontal melting margin ice flux
= qm qc hm
Primary control on discharge is ice depth at the margin.
dt
dxm dt = qm qc
Kinematic condition It is typically observed that qm ⇡ qc
Z V ⇡ r a m
ice volume
SLIDE 4 calving + frontal melting margin ice flux
= qm qc hm dt
dxm dt = qm qc
Kinematic condition
xm
V ⇡ r a m
ice volume
hm
Primary control on discharge is ice depth at the margin. It is typically observed that qm ⇡ qc
Z
Ice front evolution
SLIDE 5 calving + frontal melting margin ice flux
= qm qc hm dt
dxm dt = qm qc
Kinematic condition
xm
V ⇡ r a m Z dV dt = qb qc Z Z qb = Z xm (a m) dx ✓ ◆
balance flux Global mass balance ice volume
hm
Primary control on discharge is ice depth at the margin. It is typically observed that qm ⇡ qc
Z
Ice front evolution
SLIDE 6
Extreme Ice Survey - Time-lapse camera Columbia Glacier, Alaska
Time-lapse movie
SLIDE 7
Extreme Ice Survey - Time-lapse camera Columbia Glacier, Alaska
Time-lapse movie
SLIDE 8 qc = a
m
xm
x z
A simplified model of a tidewater glacier
Mass conservation Force balance Two-dimensional. Ice motion dominated by basal sliding.
✓Z ◆ ∂ ∂x ✓ 4ηih∂u ∂x ◆ − ρigh ∂ ∂x(b + h) − τb = 0 ✓Z ◆ ∂h ∂t + ∂q ∂x = a − m q = hu ◆
At ice front
z = b ✓Z h at x = xm ◆
(stress balance) (calving criteria) A plastic (rate-independent) friction law.
✓ ◆ 4ηih∂u ∂x = 1 2
h = f ✓ −ρo ρi b ◆
flotation factor flotation depth
z = se
◆ τb = µN ✓ ◆
SLIDE 9 qc = a
m
xm
x z
A simplified model of a tidewater glacier
Mass conservation Force balance Two-dimensional. Ice motion dominated by basal sliding.
✓Z ◆ ∂ ∂x ✓ 4ηih∂u ∂x ◆ − ρigh ∂ ∂x(b + h) − τb = 0 ✓Z ◆ ∂h ∂t + ∂q ∂x = a − m q = hu ◆
At ice front
z = b ✓Z h at x = xm ◆
(stress balance) (calving criteria) A plastic (rate-independent) friction law. ice thickness and volume determined purely by margin position and basal friction
- cf. Nye 1951, Weertman 1961, Ultee & Bassis 2016
Model reduction Away from the front, force balance −ρigh ∂
∂x(b + h) ≈ τb ✓ ◆ 4ηih∂u ∂x = 1 2
h = f ✓ −ρo ρi b ◆
flotation factor flotation depth
z = se
◆ τb = µN ✓ ◆
SLIDE 10 qc = a
m
xm
x z
A simplified model of a tidewater glacier
Mass conservation Force balance Two-dimensional. Ice motion dominated by basal sliding.
✓Z ◆ ∂ ∂x ✓ 4ηih∂u ∂x ◆ − ρigh ∂ ∂x(b + h) − τb = 0 ✓Z ◆ ∂h ∂t + ∂q ∂x = a − m q = hu ◆
At ice front
z = b ✓Z h at x = xm ◆
(stress balance) (calving criteria) A plastic (rate-independent) friction law. ice thickness and volume determined purely by margin position and basal friction
- cf. Nye 1951, Weertman 1961, Ultee & Bassis 2016
Model reduction Away from the front, force balance −ρigh ∂
∂x(b + h) ≈ τb ✓ ◆ 4ηih∂u ∂x = 1 2
h = f ✓ −ρo ρi b ◆
flotation factor flotation depth Near the front, a boundary-layer analysis relates calving flux to local water depth
- cf. Schoof 2007, Tsai et al 2015
✓ ◆ qc = A(2ρig)n µ ˆ Q(f) ✓ − ρi ρo b ◆n+2
f
1 1.05 1.1 1.15 1.2
ˆ Q
0.2 0.4 0.6 0.8 1 104 × ˜ Q 103 × ˆ Q
)n ˆ Q(f)
z = se
◆ τb = µN ✓ ◆
SLIDE 11
qc xm
z = se
Stable equilibria
= f ✓ −ρo ρi b ◆ ◆ τb = µN ✓ ◆
Global mass balance
A simplified model of a tidewater glacier
dxm dt = F( xm ; N, se, f) Z dV dt = qb − qc dxm dt = F(xm) = xm
SLIDE 12
qc xm
z = se
Stable equilibria Bed strength Forcing parameters: µN
◆
Equilibrium line altitude (ELA) Calving parameter
= f N, se = f ✓ −ρo ρi b ◆ ◆ τb = µN ✓ ◆
Global mass balance
A simplified model of a tidewater glacier
dxm dt = F( xm ; N, se, f) Z dV dt = qb − qc dxm dt = F(xm) = xm
SLIDE 13
Decreasing basal friction Increasing basal friction Speed up Time Increased mass loss Slow down Retreat Advance Greater mass loss Speed up Velocity Steady state
Motivation: impact of subglacial lubrication
SLIDE 14
qc xm
z = se
Stable equilibria Bed strength Forcing parameters: µN
◆
Equilibrium line altitude (ELA) Calving parameter
= f N, se = f ✓ −ρo ρi b ◆ ◆ τb = µN ✓ ◆
Global mass balance
A simplified model of a tidewater glacier
dxm dt = F( xm ; N, se, f) Z dV dt = qb − qc dxm dt = F(xm) = xm
SLIDE 15
qc xm
z = se
Stable equilibria Bed strength Forcing parameters: µN
◆
Equilibrium line altitude (ELA) Calving parameter
= f N, se = f ✓ −ρo ρi b ◆ ◆ τb = µN ✓ ◆
Global mass balance Decrease basal friction
A simplified model of a tidewater glacier
dxm dt = F( xm ; N, se, f) Z dV dt = qb − qc dxm dt = F(xm) = xm
SLIDE 16
qc xm
z = se
Stable equilibria Bed strength Forcing parameters: µN
◆
Equilibrium line altitude (ELA) Calving parameter
= f N, se = f ✓ −ρo ρi b ◆ ◆ τb = µN ✓ ◆
Global mass balance
A simplified model of a tidewater glacier
dxm dt = F( xm ; N, se, f) Z dV dt = qb − qc dxm dt = F(xm) = xm
SLIDE 17
qc xm
z = se
Stable equilibria Bed strength Forcing parameters: µN
◆
Equilibrium line altitude (ELA) Calving parameter
= f N, se = f ✓ −ρo ρi b ◆ ◆ τb = µN ✓ ◆
Global mass balance
A simplified model of a tidewater glacier
dxm dt = F( xm ; N, se, f) Z dV dt = qb − qc dxm dt = F(xm) = xm
Increase basal friction
SLIDE 18
qc xm
z = se
Stable equilibria Bed strength Forcing parameters: µN
◆
Equilibrium line altitude (ELA) Calving parameter
= f N, se = f ✓ −ρo ρi b ◆ ◆ τb = µN ✓ ◆
Global mass balance
A simplified model of a tidewater glacier
dxm dt = F( xm ; N, se, f) Z dV dt = qb − qc dxm dt = F(xm) = xm
SLIDE 19
qc xm
z = se
Stable equilibria Bed strength Forcing parameters: µN
◆
Equilibrium line altitude (ELA) Calving parameter
= f N, se = f ✓ −ρo ρi b ◆ ◆ τb = µN ✓ ◆
Global mass balance Increase calving or increase ELA
A simplified model of a tidewater glacier
dxm dt = F( xm ; N, se, f) Z dV dt = qb − qc dxm dt = F(xm) = xm
SLIDE 20
qc xm
z = se
Stable equilibria Bed strength Forcing parameters: µN
◆
Equilibrium line altitude (ELA) Calving parameter
= f N, se = f ✓ −ρo ρi b ◆ ◆ τb = µN ✓ ◆
Global mass balance
A simplified model of a tidewater glacier
dxm dt = F( xm ; N, se, f) Z dV dt = qb − qc dxm dt = F(xm) = xm
SLIDE 21 100 y
Distance [km]
100 150 200 1000
200 y
500
Elevation [m]
50 y
Velocity [m/y]
0 y
100 y
Distance [km]
100 150 200 1000
200 y
500
Elevation [m]
50 y
Velocity [m/y]
0 y
An essentially indistinguishable response can occur to very distinct forcing mechanisms.
100 y
Distance [km]
100 150 200 1000
200 y
500
Elevation [m]
50 y
Velocity [m/y]
0 y
Increased calving Increased ELA Increased bed strength
Response to different forcing
SLIDE 22
Summary
Episodic acceleration and deceleration of a tidewater glacier is generic, in response to changes in both oceanic and upstream forcing. Tidewater glaciers can be described as a forced dynamical system.
dxm dt = F(xm) = xm
SLIDE 23
Summary
dxm dt = F(xm) = xm
Episodic acceleration and deceleration of a tidewater glacier is generic, in response to changes in both oceanic and upstream forcing. Tidewater glaciers can be described as a forced dynamical system.
SLIDE 24
Summary
dxm dt = F(xm) = xm
Episodic acceleration and deceleration of a tidewater glacier is generic, in response to changes in both oceanic and upstream forcing. Tidewater glaciers can be described as a forced dynamical system.
SLIDE 25
Summary
dxm dt = F(xm) = xm
Episodic acceleration and deceleration of a tidewater glacier is generic, in response to changes in both oceanic and upstream forcing. Tidewater glaciers can be described as a forced dynamical system.
SLIDE 26
Summary
dxm dt = F(xm) = xm
Episodic acceleration and deceleration of a tidewater glacier is generic, in response to changes in both oceanic and upstream forcing. Tidewater glaciers can be described as a forced dynamical system.
SLIDE 27
Summary
dxm dt = F(xm) = xm
Episodic acceleration and deceleration of a tidewater glacier is generic, in response to changes in both oceanic and upstream forcing. Tidewater glaciers can be described as a forced dynamical system.
SLIDE 28
Summary
dxm dt = F(xm) = xm
Episodic acceleration and deceleration of a tidewater glacier is generic, in response to changes in both oceanic and upstream forcing. Tidewater glaciers can be described as a forced dynamical system.
