C43C-0627 S CHANGES IN ICE SHEET MOTION DUE TO MELT … · 2015. 9. 9. · SEASONAL CHANGES IN ICE...
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SEASONAL CHANGES IN ICE SHEET MOTION DUE TO MELT WATER LUBRICATION IAN J. HEWITT ([email protected] ), UNIVERSITY OF BRITISH COLUMBIA, CANADA C43C-0627 AGU 2012 Motion of the Greenland Ice Sheet is influenced by routing of surface meltwater to its bed on seasonal and diurnal timescale. Numerical model used to estimate how mean annual velocity is affected by surface melting rate for an ‘ideal’ ice sheet. Meltwater lubrication is not accounted for in current ice sheet models. SUMMARY WHY? More melting generally results in faster flowing ice, but with complex pattern of acceleration and deceleration. METHODS Fixed ice geometry. Surface melting decreases with elevation. Model of subglacial drainage system Model of ice force balance Runoff into moulins Sliding law Cavity opening Ice Velocity Input : Output : Sliding law depends on effective pressure : Vertically-integrated model for ice flow; incorporates membrane stresses and internal shearing. Drainage model incorporates flow in cavities and channels. τ b = f (U b ,N ) u b U b 0 50 1 1.5 2 1 1.5 2 A 0 50 1 1.5 2 1 1.5 2 B 0 50 1 1.5 2 1 1.5 2 C 0 50 121 152 182 213 244 274 1 1.5 2 1 1.5 2 D 0 1 2 3 4 5 r [ m ] 0 10 20 30 40 50 0.9 1 1.1 1.2 1.3 A B C D 0.9 1 1.1 1.2 1.3 Channels z b z s Creep closure Cavity opening m = G + τ b · u b ρ w L ∂ h ∂ t = ρ w ρ i m + U b (h r − h)/l r − 2A n n hN n Porous sheet (cavities) Melting h Channels ∂ S ∂ t = ρ w ρ i M − 2A n n SN n ρ 2 Dissipation in channel ... and underlying cavities Creep closure M = |Q∂φ/∂ s| + λ c |q · ∇φ| ρ w L Melting S −∇φ = Ψ 0 + ∇N Surface slope N = p i − p w Effective pressure 0 40 r [ mm / d ] 1 91 182 274 365 q = − Kh 3 ρ w g ∇φ Q = −K c S 5/4 ∂φ ∂ s −1/2 ∂φ ∂ s ∂ S ∂ t = ρ w ρ i M − 2A n n S |N | n−1 N ρ 2A ∂ h ∂ t = ρ w ρ i m + U b (h r − h)/l r − 2A n n h|N | n−1 N γ ∂ h ∂ t + ∇ · q + ∂ S ∂ t + ∂ Q ∂ s δ (x c )+ ∂ Σ ∂ t = m + M δ (x c )+ Rδ (x m ) Mass conservation Water flow pa- rameterizations Evolution parame- terizations m = G + τ b · u b ρ w L M = |Q∂φ/∂ s| + λ c |q · ∇φ| ρ w L Englacial storage Energy conservation f (U b ,N ) U b u b = −ρ i gH ∂ s ∂ x + ∂ ∂ x [H (2 τ xx + τ yy )] + ∂ ∂ y [H τ xy ] f (U b ,N ) U b v b = −ρ i gH ∂ s ∂ y + ∂ ∂ y [H ( τ xx +2 τ yy )] + ∂ ∂ x [H τ xy ] τ xx = 2˜ η ∂ u b ∂ x , τ yy = 2˜ η ∂ v b ∂ y , τ xy =˜ η ∂ u b ∂ y + ∂ v b ∂ x τ xz = η ∂ u ∂ z , τ yz = η ∂ v ∂ z τ xz = −ρ i g (s − z ) ∂ s ∂ x + ∂ ∂ x s z (2τ xx + τ yy )dz + ∂ ∂ y s z τ xy dz τ yz = −ρ i g (s − z ) ∂ s ∂ y + ∂ ∂ y s z (τ xx +2τ yy )dz + ∂ ∂ x s z τ xy dz ˜ η = 1 2 A −1 ˜ τ 2 xz +˜ τ 2 yz + τ 2 xy + τ 2 xx + τ xx τ yy + τ 2 yy (1−n)/2 η = 1 2 A −1 τ 2 xz + τ 2 yz + τ 2 xy + τ 2 xx + τ xx τ yy + τ 2 yy (1−n)/2 ˜ τ xz = −ρ i g (s − z ) ∂ s ∂ x ˜ τ yz = −ρ i g (s − z ) ∂ s ∂ y Basal force balance In-plane force balance Approximate constitutive law MODEL Discharge Ice surface velocity Runoff at A Velocity at stakes A-D Annual mean velocity MODEL OBSERVATIONS DIURNAL VARIABILITY Highest velocities occur during early summer near margin and progressively later fur- ther inland, due to evolution of drainage system. SEASONAL MELT CYCLE Steady state never realized in practice STEADY STATE More melting provokes larger initial acceleration, followed by compensating deceleration near margin. Channelization reduces ice velocities, but it is hard to gen- erate significant slow down when averaged over the year. Some parameters are very uncertain : need to fit to observations. Sensitivity to annual melt rate is generally higher further from the margin. Channels shrink entirely over winter : rejuvenation in early summer takes too long (?) Quasi-steady sliding law used here : need to develop transient sliding law Model is overly sensitive to extreme water pressures : need improvements to cope with rapid drainage events. FUTURE ISSUES Normalized velocity Runoff [mm/d] Time [d] Distance [m] Normalized velocity Annual melt [m] Time [d] Effective pressure Ice surface velocity Discharge [mm/d] 0 40 182 183 Time [d] [mm/d] Runoff at A Discharge Ice surface velocity Velocity at stakes A-D 0 50 1 1.5 2 2.5 A 0 20 40 0 50 1 1.5 2 2.5 h [ cm ] B 0 20 40 0 50 1 1.5 2 2.5 C 0 20 40 0 50 121 152 182 213 244 1 1.5 2 2.5 D 0 20 40 Runoff [mm/d] Sheet depth [cm] Normalized velocity Time [d] 1 1.5 16 17 1 1.5 h [ cm ] 32 33 1 1.5 23 24 1 1.5 10 11 Sheet depth [cm] Uplift [cm] Normalized velocity f (U b ,N )= μ I NU b f (U b ,N )= μ II N 1/3 U 1/3 b f (U b ,N )= μ III N U b U b + λ b AN n 1/n u Sliding law I II III 0 2 4 1 1.05 1.1 1.15 1.2 1.25 A B C D I II III r [ m ] Annual melt [m] ‘Speed up’ Σ = σ p w ρ w g Low melt Medium melt High melt
Transcript of C43C-0627 S CHANGES IN ICE SHEET MOTION DUE TO MELT … · 2015. 9. 9. · SEASONAL CHANGES IN ICE...
SEASONAL CHANGES IN ICE SHEET MOTION DUE TO MELT WATER LUBRICATIONIAN J. HEWITT ([email protected]), UNIVERSITY OF BRITISH COLUMBIA, CANADA
C43C-0627AGU 2012
Motion of the Greenland Ice Sheet is influenced by routing of surface meltwater to its bed on seasonal and diurnal timescale.
Numerical model used to estimate how mean annual velocity is affected by surface melting rate for an ‘ideal’ ice sheet.
Meltwater lubrication is not accounted for in current ice sheet models.
SUMMARY
WHY?
More melting generally results in faster flowing ice, but with complex pattern of acceleration and deceleration.
METHODS
Fixed ice geometry. Surface melting decreases with elevation.
Model of subglacial drainage system
Model of ice force balance
Runoff into moulins
Sliding law
Cavity opening
Ice VelocityInput : Output :
Sliding law depends on effective pressure :
Vertically-integrated model for ice flow; incorporates membrane stresses and internal shearing.
Drainage model incorporates flow in cavities and channels.
β > 0
n = 3
N = Nc
Sk
β(x, y) = β0 [1 + sin(2πx/L) sin(2πy/L)]
τ b = fub
Ub
f(Ub, N) = µaNpU q
b
Ub = (u2b + v2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
τ b = µNub
τ b = µNpU qub
τ b = µbN
�Ub
Ub + λbANn
�1/n ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1N
∂h
∂t=
ρwρi
m+RUb −2A
nnh|N |n−1N
∂S
∂t=
ρwρi
M − 2A
nnSNn
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnhNn
h = hc
�pwpi
�γ
7
0
50
1
1.5
2
1
1.5
2A
0
50
r [ m
m /
d ]
1
1.5
2
Velo
city
[ no
rmal
ized
]
1
1.5
2B
0
50
1
1.5
2
1
1.5
2C
0
50
121 152 182 213 244 274
1
1.5
2
Time [ d ]
1
1.5
2D
012345
r tota
l [ m
]
0 10 20 30 40 500.9
11.11.21.3
Velo
city
[ no
rmal
ized
]
x [ km ]ABCD
35 mm / d25 mm / d15 mm / d
0.911.11.21.3
Channels
q = −khα|∇φ|β−1∇φ
∂S
∂t+
∂Q
∂s= κ
∂S
∂t=
Ξ
ρiL− K̂S|N |n−1N
Ξ = kcSα|Ψ|β+1 + lrkh
α|Ψ|β+1
∂S
∂t=
kcSα|Ψ|β+1
ρiL+ lr
khα|Ψ|β+1
ρiL− K̂S|N |n−1N
Q = −kcSα|Ψ|β−1Ψ
Q = −kcSα|∇sφ|β−1∇sφ
∂S
∂t=
kcSα|∇sφ|β+1
ρiL+ lr
khα|∇φ|β+1
ρiL− K̂S|N |n−1N
zs
zb
ε̇ = Aτn
A(T ) n
0 = −∇p+∇ · τ
ε̇ = A(T )τn−1τ
∇ · u = 0
ub = f(x, τb, N)τ b
∂T
∂t+ u ·∇T = κ∇2T
pi ≈ ρig(zs − zb)
3
q = −khα|∇φ|β−1∇φ
∂S
∂t+
∂Q
∂s= κ
∂S
∂t=
Ξ
ρiL− K̂S|N |n−1N
Ξ = kcSα|Ψ|β+1 + lrkh
α|Ψ|β+1
∂S
∂t=
kcSα|Ψ|β+1
ρiL+ lr
khα|Ψ|β+1
ρiL− K̂S|N |n−1N
Q = −kcSα|Ψ|β−1Ψ
Q = −kcSα|∇sφ|β−1∇sφ
∂S
∂t=
kcSα|∇sφ|β+1
ρiL+ lr
khα|∇φ|β+1
ρiL− K̂S|N |n−1N
zs
zb
ε̇ = Aτn
A(T ) n
0 = −∇p+∇ · τ
ε̇ = A(T )τn−1τ
∇ · u = 0
ub = f(x, τb, N)τ b
∂T
∂t+ u ·∇T = κ∇2T
pi ≈ ρig(zs − zb)
3
Creep closure
Cavity opening
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂V
∂tδ(xm) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z,
7
N = Nc
Sk
β(x, y) = β0 [1 + sin(2πx/L) sin(2πy/L)]
τ b = fub
Ub
f(Ub, N) = µaNpU q
b
Ub = (u2b + v2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
τ b = µNub
τ b = µNpU qub
τ b = µbN
�Ub
Ub + λbANn
�1/n ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1N
∂h
∂t=
ρwρi
m+RUb −2A
nnh|N |n−1N
∂S
∂t=
ρwρi
M − 2A
nnSNn
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnhNn
h = hc
�pwpi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
Σ = σρwgpw
7
Porous sheet (cavities)
Melting
q = −khαe |∇φ|β−1∇φ
h�e
∂N
∂t+∇ · q = m
h = he(N)
n = 3
α = 5/4
β = 1/2
pi ≈ ρigH
τ b ≈ −ρigH∇s
C = C0N
φ = ρwgb+ pw
φ = ρigs+∆ρgb−N
z = s
z = b
H = s− b
S
Sw
Sw = S
Sw < S
h = hw
h
hw
0 < pw < pi
0 ≤ pw ≤ pi
φm < φ < φ0
φm ≤ φ ≤ φ0
pw = pi
pw = 0
φ = φm
φ = φ0
1
Channels
N = Nc
Sk
β(x, y) = β0 [1 + sin(2πx/L) sin(2πy/L)]
τ b = fub
Ub
f(Ub, N) = µaNpU q
b
Ub = (u2b + v2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
τ b = µNub
τ b = µNpU qub
τ b = µbN
�Ub
Ub + λbANn
�1/n ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1N
∂h
∂t=
ρwρi
m+RUb −2A
nnh|N |n−1N
∂S
∂t=
ρwρi
M − 2A
nnSNn
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnhNn
h = hc
�pwpi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
Σ = σρwgpw
7
Dissipation in channel... and underlying cavities
Creep closure
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂V
∂tδ(xm) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z,
7
Melting
q = −khαe |∇φ|β−1∇φ
h�e
∂N
∂t+∇ · q = m
h = he(N)
n = 3
α = 5/4
β = 1/2
pi ≈ ρigH
τ b ≈ −ρigH∇s
C = C0N
φ = ρwgb+ pw
φ = ρigs+∆ρgb−N
z = s
z = b
H = s− b
S
Sw
Sw = S
Sw < S
h = hw
h
hw
0 < pw < pi
0 ≤ pw ≤ pi
φm < φ < φ0
φm ≤ φ ≤ φ0
pw = pi
pw = 0
φ = φm
φ = φ0
1
∂h
∂t= Rub −Kh|N |n−1N +
khα|Ψ|β+1
ρiL
∂h
∂t= Rub −Kh|N |n−1N+
khα|∇φ|β+1
ρiL
∂h
∂t= Rub −Kh|N |n−1N +
Ξ
ρiL
Ξ = khα|∇φ|β+1
−∇φ = Ψ0 +∇N
R =hr
lr
φ = ρwgzb + pw (15)
= ρigzs +∆ρgzb −N (16)
N = pi − pw
Ψ = −∇φ
∂h
∂t+∇ ·
�khα|Ψ0 +∇N |β−1(Ψ0 +∇N)
�= m
∂h
∂t+∇ · q = m
∂h
∂t−∇ ·
�khα|∇φ|β−1∇φ
�= m
q = −khα|∇φ|β−1∇φ
∂S
∂t+
∂Q
∂s= M
∂S
∂t− ∂
∂s
�kcS
α
����∂φ
∂s
����β−1 ∂φ
∂s
�= M
κ = [q · n]+−
4
Surface slope
∂h
∂t= Rub −Kh|N |n−1N +
khα|Ψ|β+1
ρiL
∂h
∂t= Rub −Kh|N |n−1N+
khα|∇φ|β+1
ρiL
∂h
∂t= Rub −Kh|N |n−1N+
khα|∇φ|β+1
ρiL
∂h
∂t= Rub −Kh|N |n−1N +
M
ρiL
Ξ = khα|∇φ|β+1
−∇φ = Ψ0 +∇N
R =hr
lr
φ = ρwgzb + pw (15)
= ρigzs +∆ρgzb −N (16)
= ρigzs + (ρw − ρi)gzb −N
= ρwgzb + pi −N
≈ ρwgzb + pi
N = pi − pw
N = pi − pw
Ψ = −∇φ
∂h
∂t+∇ ·
�khα|Ψ0 +∇N |β−1(Ψ0 +∇N)
�= m
∂h
∂t+∇ · q = m
∂h
∂t−∇ ·
�khα|∇φ|β−1∇φ
�= m
4
Effective pressure
0
40
r [ m
m /
d ]
Time [ d ]1 91 182 274 365
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂V
∂tδ(xm) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
7
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂V
∂tδ(xm) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
7
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂V
∂tδ(xm) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
7
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂V
∂tδ(xm) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
7
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
7
Mass conservation
Water flow pa-rameterizations
Evolution parame-terizations
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
7
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
7
Englacial storage
Energy conservation
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
τ b = µNub
τ b = µNpU
qub
τ b = µbN
�Ub
Ub + λbANn
�1/n ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
Σ = σρwgpw
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
τxz = −ρig(s− z)∂s
∂x+
∂
∂x
�� s
z
(2τxx + τyy) dz
�+
∂
∂y
�� s
z
τxy dz
�
τyz = −ρig(s− z)∂s
∂y+
∂
∂y
�� s
z
(τxx + 2τyy) dz
�+
∂
∂x
�� s
z
τxy dz
�
7
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
τ b = µNub
τ b = µNpU
qub
τ b = µbN
�Ub
Ub + λbANn
�1/n ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
Σ = σρwgpw
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
τxz = −ρig(s− z)∂s
∂x+
∂
∂x
�� s
z
(2τxx + τyy) dz
�+
∂
∂y
�� s
z
τxy dz
�
τyz = −ρig(s− z)∂s
∂y+
∂
∂y
�� s
z
(τxx + 2τyy) dz
�+
∂
∂x
�� s
z
τxy dz
�
7
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
τ b = µNub
τ b = µNpU
qub
τ b = µbN
�Ub
Ub + λbANn
�1/n ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
Σ = σρwgpw
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
τxz = −ρig(s− z)∂s
∂x+
∂
∂x
�� s
z
(2τxx + τyy) dz
�+
∂
∂y
�� s
z
τxy dz
�
τyz = −ρig(s− z)∂s
∂y+
∂
∂y
�� s
z
(τxx + 2τyy) dz
�+
∂
∂x
�� s
z
τxy dz
�
7
Ub = (u2b + v
2b )
1/2
f(Ub, N) = µbN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
τ b = µNub
τ b = µNpU
qub
τ b = µbN
�Ub
Ub + λbANn
�1/n ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1
N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1
N
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
Σ = σρwgpw
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂Σ
∂t= m+Mδ(xc) +Rδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
τxz = −ρig(s− z)∂s
∂x+
∂
∂x
�� s
z
(2τxx + τyy) dz
�+
∂
∂y
�� s
z
τxy dz
�
τyz = −ρig(s− z)∂s
∂y+
∂
∂y
�� s
z
(τxx + 2τyy) dz
�+
∂
∂x
�� s
z
τxy dz
�
7
η̃ = 12A
−1 �τ̃ 2xz + τ̃ 2yz + τ 2xy + τ 2xx + τxxτyy + τ 2yy
�(1−n)/2
τ̃xz = −ρig(s− z)∂s
∂xand τ̃yz = −ρig(s− z)
∂s
∂y
η = 12A
−1 �τ 2xz + τ 2yz + τ 2xy + τ 2xx + τxxτyy + τ 2yy
�(1−n)/2
8
η̃ = 12A
−1 �τ̃ 2xz + τ̃ 2yz + τ 2xy + τ 2xx + τxxτyy + τ 2yy
�(1−n)/2
τ̃xz = −ρig(s− z)∂s
∂xand τ̃yz = −ρig(s− z)
∂s
∂y
η = 12A
−1 �τ 2xz + τ 2yz + τ 2xy + τ 2xx + τxxτyy + τ 2yy
�(1−n)/2
8
η̃ = 12A
−1 �τ̃ 2xz + τ̃ 2yz + τ 2xy + τ 2xx + τxxτyy + τ 2yy
�(1−n)/2
τ̃xz = −ρig(s− z)∂s
∂xand τ̃yz = −ρig(s− z)
∂s
∂y
η = 12A
−1 �τ 2xz + τ 2yz + τ 2xy + τ 2xx + τxxτyy + τ 2yy
�(1−n)/2
8
η̃ = 12A
−1 �τ̃ 2xz + τ̃ 2yz + τ 2xy + τ 2xx + τxxτyy + τ 2yy
�(1−n)/2
τ̃xz = −ρig(s− z)∂s
∂xand τ̃yz = −ρig(s− z)
∂s
∂y
η = 12A
−1 �τ 2xz + τ 2yz + τ 2xy + τ 2xx + τxxτyy + τ 2yy
�(1−n)/2
8
Basal force balance
In-plane force balance
Approximate constitutive
law
MODEL
Discharge Ice surface velocity
Runoff at AVelocity at stakes A-D
Annual mean velocity
MODEL OBSERVATIONS
DIURNAL VARIABILITY
Highest velocities occur during early summer near margin and progressively later fur-ther inland, due to evolution of drainage system.
SEASONAL MELT CYCLE
Steady state never realized in practice
STEADY STATE
More melting provokes larger initial acceleration, followed by compensating deceleration near margin.
Channelization reduces ice velocities, but it is hard to gen-erate significant slow down when averaged over the year.
Some parameters are very uncertain : need to fit to observations.
Sensitivity to annual melt rate is generally higher further from the margin.
Channels shrink entirely over winter : rejuvenation in early summer takes too long (?)
Quasi-steady sliding law used here : need to develop transient sliding law
Model is overly sensitive to extreme water pressures : need improvements to cope with rapid drainage events.
FUTURE ISSUES
Nor
mal
ized
vel
ocity
Run
off [
mm
/d]
Time [d]
Distance [m]
Nor
mal
ized
vel
ocity
Ann
ual m
elt
[m]
Time [d]
Effective pressure
Ice surface velocity
Discharge
[mm
/d]
0
40
r [ m
m /
d ]
Time [ d ]182 183
Time [d]
[mm
/d]
Runoff at A
Discharge Ice surface velocity
Velocity at stakes A-D
0
50
11.5
22.5 A
0
20
40
0
50
r [ m
m /
d ]
11.5
22.5
Velo
city
[ no
rmal
ized
]
h [ c
m ]
B
0
20
40
0
50
11.5
22.5 C
0
20
40
0
50
121 152 182 213 2441
1.52
2.5
Time [ d ]
D
0
20
40
Run
off [
mm
/d]
She
et d
epth
[cm
]
Nor
mal
ized
vel
ocity
Time [d]
0
50
1
1.5
16
17
0
50
r [ m
m /
d ]
Velo
city
[ no
rmal
ized
]
1
1.5
h [ c
m ]32
33
0
50
1
1.5
23
24
0
50
1
1.5
10
11
She
et d
epth
[cm
]U
plif
t [c
m]
Nor
mal
ized
vel
ocity
β > 0
n = 3
N = Nc
Sk
β(x, y) = β0 [1 + sin(2πx/L) sin(2πy/L)]
τ b = fub
Ub
f(Ub, N) = µaNpU q
b
Ub = (u2b + v2b )
1/2
f(Ub, N) = µINUb
f(Ub, N) = µIIN1/3U1/3
b
f(Ub, N) = µIIIN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
τ b = µNub
τ b = µNpU qub
τ b = µbN
�Ub
Ub + λbANn
�1/n ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1N
∂h
∂t=
ρwρi
m+RUb −2A
nnh|N |n−1N
∂S
∂t=
ρwρi
M − 2A
nnSNn
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnhNn
7
β > 0
n = 3
N = Nc
Sk
β(x, y) = β0 [1 + sin(2πx/L) sin(2πy/L)]
τ b = fub
Ub
f(Ub, N) = µaNpU q
b
Ub = (u2b + v2b )
1/2
f(Ub, N) = µINUb
f(Ub, N) = µIIN1/3U1/3
b
f(Ub, N) = µIIIN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
τ b = µNub
τ b = µNpU qub
τ b = µbN
�Ub
Ub + λbANn
�1/n ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1N
∂h
∂t=
ρwρi
m+RUb −2A
nnh|N |n−1N
∂S
∂t=
ρwρi
M − 2A
nnSNn
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnhNn
7
β > 0
n = 3
N = Nc
Sk
β(x, y) = β0 [1 + sin(2πx/L) sin(2πy/L)]
τ b = fub
Ub
f(Ub, N) = µaNpU q
b
Ub = (u2b + v2b )
1/2
f(Ub, N) = µINUb
f(Ub, N) = µIIN1/3U1/3
b
f(Ub, N) = µIIIN
�Ub
Ub + λbANn
�1/n
τ b = f(Ub, N)ub
Ub
τ b = µNub
τ b = µNpU qub
τ b = µbN
�Ub
Ub + λbANn
�1/n ub
Ub
q = −Kh3
ρwg∇φ and Q = −KcS
5/4
����∂φ
∂s
����−1/2 ∂φ
∂s
∂S
∂t=
ρwρi
M − 2A
nnS|N |n−1N
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnh|N |n−1N
∂h
∂t=
ρwρi
m+RUb −2A
nnh|N |n−1N
∂S
∂t=
ρwρi
M − 2A
nnSNn
∂h
∂t=
ρwρi
m+ Ub(hr − h)/lr −2A
nnhNn
7
Sliding law I II III
0 2 41
1.05
1.1
1.15
1.2
1.25
A
BC
D
IIIIII
Mea
n
rtotal [ m ]0 2 4
1
1.2
1.4
1.6
1.8
2
A
BC
D
Max
imum
rtotal [ m ]Annual melt [m]
‘Spe
ed u
p’
h = hc
�pw
pi
�γ
m =G+ τ b · ub
ρwLand M =
|Q∂φ/∂s|+ λc|q ·∇φ|ρwL
Σ = σpw
ρwg
∂h
∂t+∇ · q+
�∂S
∂t+
∂Q
∂s
�δ(xc) +
∂Σ
∂t= m+Mδ(xc) + Iδ(xm)
f(Ub, N)
Ubub = −ρigH
∂s
∂x+
∂
∂x[H (2τxx + τ yy)] +
∂
∂y[Hτxy]
f(Ub, N)
Ubvb = −ρigH
∂s
∂y+
∂
∂y[H (τxx + 2τ yy)] +
∂
∂x[Hτxy]
τxx = 2η̃∂ub
∂x, τyy = 2η̃
∂vb∂y
, τxy = η̃
�∂ub
∂y+
∂vb∂x
�
τxz = η∂u
∂z, τyz = η
∂v
∂z
τxz = −ρig(s− z)∂s
∂x+
∂
∂x
�� s
z
(2τxx + τyy) dz
�+
∂
∂y
�� s
z
τxy dz
�
τyz = −ρig(s− z)∂s
∂y+
∂
∂y
�� s
z
(τxx + 2τyy) dz
�+
∂
∂x
�� s
z
τxy dz
�
η̃ = 12A
−1 �τ̃ 2xz + τ̃ 2yz + τ 2xy + τ 2xx + τxxτyy + τ 2yy
�(1−n)/2
τ̃xz = −ρig(s− z)∂s
∂xand τ̃yz = −ρig(s− z)
∂s
∂y
η = 12A
−1 �τ 2xz + τ 2yz + τ 2xy + τ 2xx + τxxτyy + τ 2yy
�(1−n)/2
8
Low melt Medium melt High melt
