(In)consistent modelling of suglacial drainage and sliding Ian Hewitt, University of Oxford Greenland, NASA
Ice acceleration and deceleration due to surface melting Greenland surface velocities (GPS) Ice speed Zwally et al. 2002
Ice speed varies diurnally Ice speed Hoffman et al 2011
Ice-dynamic influences on ice-sheet mass alance Accumulation Radiation Speed up and surface lowering Accumulation Radiation Accumulation Radiation Speed up and increased calving Calving
): 15 Sliding and water pressure 10 June 10 - - - - ve lo c ity fall 1982 ad 30 [mm/h] 1 Horizontal Velocity i I I I 60 Depth of wate r leve l [mm Ih] r30 Speed Horizontal Velocity 140 180 I[m] I I I gla c ie r el ow 20 surfac 25 25! 25 201-h-..-t=1 [mm / h] I 15 1 T/. Water pressure Speed 20...0. /11 or I I Fig. 2. a. Velocity records for poles near the centre line of the glacier. Broken lilies indicate mean velocities over periods when no short-interval measurements were made.. Top: velocity 0/ two poles 0/ profile c. Broken lines indicate mean velocities over periods whell no short-interval measurements were made. Upper middle: depth of the water level elow surface in four ore holes. Lower middle: discharge of the terminal stream ( y courtesy of Grande Dixence. S.A.). Broken line: a dam had roken alld discharge data refer to only part of the outlet stream. Bol/om: thick line: air temperature near the glacier terminus ( y courtesy of Grande Dixence. S.A. ). Thin line: temperature of the free atmosphere at the 700 mar level near Payerne ( y court esy 0/ S chwei;;erische Meteorologische Anstalt). x... x 15 Iken & Bindschadler 1986 C JjC. t 60 Water pressure C I 20 [m] 15-30 May 1982 30 May - 4 June 1982 Iken & Bindschadler 1986 0 4-20 June 1982 x 1980 Fig. 6. pressure Velocity of pole C3 as a function of the Correlation etween surface velocity and orehole water water pressure (shown as depth of water that is within the water level was very roughly, m approximately 10 m, at the same depth elow surface in all of them. This applies also to holes 8 and 9 during the rief period when they had connections. In the two semi-marginal holes, I and 5, where the ice 2700 is only 110 and 85 m thick, respectively, the water levels2004, Harper et Sugiyama & Gudmundson tended to e somewhat deeper. That is, the piezometric sur2600 face, which approximately paralleled the ice surface in the surface). The water pressure. equal to the ice pressure 00 at the centre line. corresponds to water level of 18 m elow the surface. Differe refer to different periods: o large open symols indicate that the scatter o 2008, Fudge et levels al 2009 water in different ore holes was small. the scatter of velocity values at profiles B to D A3 BUT this relationship is not always consistent '""! D2 al 2007, Howat et al 7 10
Suglacial drainage system Lots of evidence suggests suglacial drainage systems are constantly evolving. Inefficient drainage through sediments and cavities. Efficient drainage through tunnels.
Modelling coupled hydrology and sliding Basal friction law Suglacial drainage system Force alance Surface runoff Cavitation Basal melting Ice speed Isothermal ice-sheet model (2d vertically integrated approximation incorporating longitudinal stresses and internal shear). Fixed ice-sheet geometry. Hewitt 2013, EPSL
= M + qq n= q n in t s τ = f Basal friction law + φ= ε = Aτ ε n = Aτ n N nρi gs + (ρw ρi )g +hn S Q u = fuτ = f τ S t + s = M + q n 1/m A(T ) A(T n τ = ) n τ = f (U,over N ) some repre N RU = p i pw. pi and pw are averaged Assume sliding depends oneffective effective pressure pressure h τ = fτu= f u sentative area. 1/n τ = RU 0 = p + τ 0 = p + τ Surface z = s S 1/3 Bed z = τ = CU N 1/3 τ = βτu= β u n 1 τ ε = A(T )τ τ τ p q ε = A(T )τ n 1 τ τ = CU N 11/24 1/12 T N φ Q τ U 2 + u T T = κ T U 2 += u 0 T = κ T t τ = µn u 1 t u=0 N Llioutry 1979, Budd et al 1979, Fowler 1986 U τ t 1/n 11/8 1/4 N φ Q U pi ρipg(z s z ) z ) ρ g(z U N i τ, N i )τ s τ = µn u = f (x, τ n u = f (x, τ, N )τ U + λn h 1/n 17 U N U + q = m + rµn 5 U τ = f 5 t Cavities τ = µn τ ρi gh s n n u = f u τ = f τ N U + λan τ N 1/3 1/9 1/n 1/m 17 U φ Schoof 2005, 2007 τ =Gagliardini RU3 N et al u τ = µn U q = Kh φ τ = f τu = f u 1/9 µn n Q Uτ+ λan 1/n u h τ = RU N + q = m + r 1/n τ = f τ = β τu = β u U U tτ = CU 1/3 N 1/3 n NµN µn τ = n h ρw hr p q U + λan N n τ T = m + U τ = CU 2 N γahn1 T + u T 1/ t ρi lr + u= κ T T= κ 2 T t µn U t U τ = µn τ = µn n U + λan h = h (N ) τ pi ρi g(z z ) e 1/n s ρ g(z N 1/n pi i s z ) U U τ =Kam µn1991, Tulaczyk 2000 n U τ = µn U + λn µn n 5 S N φ φ0 h s H U + λan Soft sediments 5 τ ρi gh s N 1/n z At ed elevation z =, z u = fu (x,=τ f, N )τ (x, τ, N )τ U +
Suglacial drainage model Saturated sediments Distriuted systems Channel systems h S
Suglacial drainage model h S Conduits Sliding Melting Creep Creep h r Melting l r h t = ρ w h r h m + U ρ i l r h t + q = m + r m = G + τ u ρ w L 2A n n hn n S t = ρ w M 2A ρ i n n S N n 1 N S t + Q s = M + q n + M = Q φ/ s + λ c q φ ρ w L
Annual cycle for a synthetic ice-sheet margin J A J O J Time Runoff [ mm / d ] Surface runoff (rain + melt - retention) decreases linearly with elevation. Runoff is collected from catchments and input to randomly distriuted moulins. Friction law τ = µu N
Time
Surface velocity response Velocity [ m / y ] Velocity [ m / y ] 150 50 150 50 A 0 J A J O J Time C 0 J A J O J Time Runoff [ mm / d ] Runoff [ mm / d ]
Increased surface melt Velocity [ m / y ] Velocity [ m / y ] 150 50 150 50 A 0 J A J O J Time C 0 J A J O J Time Runoff [ mm / d ] Runoff [ mm / d ]
Cavity friction law τ = µn U U + λan n 1/n Velocity [ m / y ] Velocity [ m / y ] 150 50 150 50 A 0 J A J O J Time C 0 J A J O J Time Runoff [ mm / d ] Runoff [ mm / d ]
Diurnal modulation of runoff Velocity [ m / y ] Velocity [ m / y ] 150 50 150 50 A 0 J A J O120 A J Time C Velocity [ m / y ] 0 120 C J A J O J Time Velocity [ m / y ] 80 Runoff [ mm / d ] 60 0 182 183 184 185 Time [ d ] 80 Runoff [ mm / d ] 60 0 182 183 184 185 Time [ d ] 30 20 10 30 20 10 Sheet depth [ cm ] Sheet depth [ cm ] Runoff [ mm / d ] Runoff [ mm / d ]
Cavitation τ U Normal stress Increase water pressure
Cavitation τ U Normal stress Growing cavities Increase water pressure Iken 1981
Cavitation τ U Normal stress Growing cavities Increase water pressure Iken 1981 τ = f(u,n,χ) χ fractional cavitated area
Connectedness of the drainage system Water pressure in the hydrology model is really an average over a connected drainage system. It cannot e expected to e the same as pressure in unconnected regions. Increase pressure? Comparison with orehole measurements is very difficult!
Summary Model comines an effective pressure-dependent friction law with calculation of water pressure in the suglacial drainage system. It can reproduce a range of oserved ice ehaviour. Quantitative evaluation is still needed. Important aspects of the model - evolution of the suglacial drainage system - feedack of sliding speed on the drainage system The assumption of local connectivity of the drainage system is almost certainly wrong. Oservations suggest we should consider - transient cavitation - unconnected regions of the ed Understanding the coupling of meltwater and ice dynamics is also important for erosion and long term landscape evolution.
τ U