Glacier Sliding Ian Hewitt, University of Oxford hewitt@maths.ox.ac.uk
Sliding / friction laws - Numerical models Classical sliding theory ( hard bed ) - Regelation - Viscous deformation - Cavitation Soft bed sliding - Till strength - Bed deformation
Satellite-derived ice surface speeds Rignot et al 2010 Rignot Morlighem 2012
Sliding law / Friction law z = s ( x, t ) h Stokes flow 0 = �⇥ p + ⇥ · τ + ρ i g τ b ⇤ · u = 0 z = b ( x ) U b To calculate ice flow, we need a basal boundary condition. τ b U b Friction law relates basal shear stress and basal speed τ b = | τ b | U b = | u b | ⇤ b = f ( U b , . . . ) This is a parameterization of unresolved processes close to the bed. Historically thought of as ‘sliding’ law U b = F ( ⇤ b , . . . ) τ b ⇡ � ρ i gh ∂ s May be multi-valued Shallow ice approximation ∂ x Modern view point ⇤ b = f ( U b , . . . )
Numerical ice-sheet models u b Most numerical models use a friction law of the form τ b = CU m or in vector form τ b = CU m b b U b The ‘ slipperiness ’ is usually C = C ( x, y ) treated as a fitting parameter(s), chosen to achieve a good fit with observations of surface velocities. Isaac et al 2015 The slipperiness reflects unresolved properties of the bed that may vary with time. We want to understand what physical processes govern variations of the slipperiness .
Numerical ice-sheet models Models show different results depending on the form of the basal friction law. 0.05 Plastic a Weertman + Plastic Viscous + Weertman + Plastic 0.04 0.03 Probability Density density 0.02 0.01 0.00 0 10 20 30 40 50 60 Sea level equivalent (cm) Sea-level-equivalent mass loss by 2100 (cm) Ritz et al 2016 0.020
Example friction laws τ b τ b τ b N N u b /N n u b u b Power law Sliding with cavitation Coulomb plastic Weertman 1957, Budd et al 1979 Lliboutry 1968, Schoof 2005, Kamb 1991, Clarke 2005, Gagliardini et al 2007, Budd et al Iverson 2010 1979, Zoet & Iverson 2015
Hard-bedded sliding
Hard-bedded sliding Weertman 1957 τ b β u b A film of water exists between ice and the underlying bedrock (a few microns thick). Microscopically , free slip is allowed (i.e. ). τ b micro = 0 Macroscopic resistance comes from the roughness of the bedrock ( ). τ b macro = f ( U b ) Flow over roughness occurs via regelation and viscous (plastic) deformation .
Viscous flow and regelation The ice deforms viscously around obstacles in the bed � ⇥ Dimensional analysis, using Glen’s flow law ⇥ ⇧ n � aA b ice a U V � 2 n ⇤ 2 n ⇤ rock ⇤ = a ‘roughness’ � ⇧ � Regelation: pressure difference across obstacles causes a temperature difference - results in upstream melting and downstream freezing low p high p water flow high T low T Balance of conductive / latent heat flow melting refreezing ✓ k Γ ◆ τ b ice U R = ν 2 ρ i La heat flow rock
Viscous flow and regelation � ⇥ Combining these two mechanisms: ⇥ ⇧ n � aA b effective for LARGE bumps U V � 2 n ⇤ 2 n ⇤ ice a ✓ k Γ ◆ τ b effective for SMALL bumps U R = ν 2 ρ i La rock ⇤ = a � ⇧ ⇤ � There is a ‘ controlling obstacle size ’ for which stress / speed cross over: a ∝ U − ( n − 1) / ( n +1) b ⇥ − ✓ ρ i L ◆ 1 / ( n +1) ⇧ b = ⇤ 2 R U 2 / ( n +1) Weertman sliding law R = b 2 k Γ A
SLIDING MOTION OF GLACIERS 679' melting. This process of melting and refreezing is here called 'regelation,' and its contribution to the sliding process is called 'regelation sliding' (the use of the term 'regelation' in this context is discussed by Kamb and LaChapelle [1964, p. 160]). (2) The ice responds plastically to the increased normal pres- sure on the upstream faces, arching upward at these points and thereby permit- ting the ice to move forward; correspondingly, it closes down behind the irreg- Nye-Kamb theory Nye 1969, Kamb 1970 ularities, in response to the reduced normal pressure on the downstream faces. m• • ..... * '•"*:^• ^f this process is called 'plastic-flow sliding.' The plastic- flow response of the ice is assumed to be that of a generalized Newtonian fluid, rather than plasticity in the strict sense. A more sophisticated approach to (Newtonian) viscous flow and regelation The sliding process defined by these specifications is illustrated schematically in Figure 1. We seek a relationship between the sliding velocity v and the drag � Stokes flow � 4 ψ = 0 ∼ ⇤ Z b ( x ) Heat equation � 2 T = 0 Surface z = Zo- FX, y) '"•,•,,,•//' Regelation BEDROCK layer Kamb 1970 Fig. la. Schematic representation of ice sliding with velocity v over an arbitrary bedrock topographic surface z -- Zo (x, y). Coordinate system � is as shown, with the x axis in the sliding direction. A cross section of the bed in the y direction (perpendicular to the direction of sliding) would ⇤ ∞ ˆ k 2 Z b ( k ) k 3 be ge{erally similar to the x cross section shown. The regelation layer τ b = η i U b d k via Fourier transform ∗ consists of ice formed by the refreezing of v•ater that has migrated along k 2 + k 2 π the ice-rock contact from areas of high normal pressure. Warping of the 0 ∗ ✓ ◆ ice flow vectors near the bedrock surface is an indication of plastic de- formation taking place in the ice. � ⇥ stress or basal shear stress r, as a function of bed roughness, appropriately η i ∼ 1 /A τ n − 1 effective ice viscosity defined. For this purpose the bedrock topography zo(x, y) is Fourier ana- b lyzed, and the regelation and plastic-flow contributions to the sliding motion regelation important ✓ ρ i L ◆ 1 / 2 are calculated separately for the individual Fourier components. This analysis for short wavelengths k ∗ = ∼ 2 π / 50 cm transitional wavenumber can be carried out rigorously wheil the amplitudes of the Fourier components 4 k Γ η i � are small compared to their wavelengths, that is, when the roughness is low ⌅ M enough that the heat and plastic-flow problems can be treated as problems in � � 1 Z b ( x ) e ikx d x ˆ a half-space. For the plastic-flow problem, a rigorous analysis is possible in the � � power spectrum of bed profile Z b ( k ) = lim � � M case of linear rheology (Newtonian viscosity), and a practical approximation M →∞ � � − M to nonlinearity can'be d•v•loped frbn•.• this,: as a starting point. •T'he results can then be appropriately combined to obtain r as a function of v for sliding under the simultaneous operation of regelation and plastic flow. In making this com- bination, there emerges a eharaeteris•tie length .X•, here called the transition
Sliding with cavitation Lliboutry 1968 τ b β u b ≈ Cavitation occurs when pressure on downstream face of bumps reduces to critical level p c Sliding law becomes dependent on effective pressure = p i (macroscopic) ice N = p i � p c normal stress ⇥ b = f ( U b , N ) ≈ ≈ ≈ Ice p c p c p c Bed ≈ ≈ Increased p c p c Decreased High Low N N
Sliding with cavitation Lliboutry 1968, Iken 1981,1983 Lliboutry suggested the sliding relationship was non-monotonic - a ‘multivalued’ sliding law ⇧ b U b Iken suggested there should be a maximum shear stress ⇧ b associated with cavities ‘drowning’ the bed roughness. = µN U b
Sliding with cavitation Fowler 1986, Schoof 2005, Gagliardini et al 2007 Fowler suggests cavities never really ‘drown’ bed - stress is just transferred to larger bumps ⇥ ⇥ ⇤ τ b /N ⇥ U b ⇤ τ b = Nf N n ‘Generalized’ Weertman law τ b = CU p b N q 0 < p, q < 1 U b /N n Some experimental support for this law with p = q = 1 (Budd et al 1979) 3 ⇥ τ b /N Schoof suggests an alternative with a maximum shear stress = µN ⇥ 1 /n � ⌅ b U b Regularised ‘Coulomb’ law N = µ U b + ⇥ AN n U b /N n
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