Basal Processes (i.e. boundary conditions)

Transcription

1 Basal Processes (i.e. boundary conditions) Alan Rempel, University of Oregon Motivation: a) the sliding law b) weathering & erosion Outline: - temperate sliding - drainage elements - intemperate effects - further complications

2 Conventional Sliding Theory (e.g. Weertman, Nye, Kamb) Big bumps -> deformation Little bumps -> regelation Net result: u s!! = A" n! i L T m " positive roughness ( T m " T ) = p i " p l 0 # $ % ( ) + ( p l " p m ) 1"! i! l & ' ( Problem: difficult to reconcile with inferred effects of liquid water

3 Water Issues (a) u s Q b Q f Q g Supply: a) external (seasonal and/or episodic) b) local (geothermal/frictional) - ubiquitous, but slow ( )! Q b!m = Q g + Q f " i L ( )! Q b = Q g + #u s " i L Problem 1. Ignoring Q f and Q b., what is a typical value for m?

4 Water Issues (b) Small bumps progressively drowned with addition of water. Frictional resistance depends on effective stress N (note: typically N smaller than for Byerlee s law). Till deformation (if any) depends on N. Summary: plenty to implicate N, but which N? Spatial averaging and hydrological modelling are needed.

5 Drainage Elements a) Channels: R-channels, N-channels, Fowler & Walder s canals b) Films: Walder showed Weertman s were too simple to be stable, - extensions by Creyts and Schoof (JGR in press) are promising c) Porous flow: in combination with a) and/or b) - locally important Constraints on drainage networks: - must get rid of or supply enough water for heat/mass balance - must yield N distribution that leads to basal balance with! - not likely to be unique -> history dependence/energy minimisation - observations of rapid surface uplift must be explained

6 Basal Equilibrium - How Temperate? high K film thickness d Equilibrium between ice and liquid need not be temperate (i.e. ) Generalized Clapeyron equation:! i L T m Curvature and substrate interactions: Problem 2. Derive the generalized Clapeyron equation. Note, changes in chemical potential satisfy: ( T m " T ) = ( p i " p l ) + p l " p m p i! p l = " il K + #(d) p i! p l # $ % ( ) 1"! i µ(t, p) = µ(t m, p m ) + 1! (p " p m ) " s(t " T m )! l & ' (

7 Basal Equilibrium - How Temperate? ice-liquid interface has: p i! p l = " il K + #(d) Implications: - high N corresponds to large p i - p l and freeze-on d! - temperate theory appropriate where molecular distances (evaluate d using water flow requirements and force balance) - partitioning of! between bed bumps and granular friction is key

8 Towards a composite model side view Sliding over and through bumps produces resistance! d (u s )+! r (u s ). front view!m(! f,u s ) Driving stress partitioned so that: Darcy flow through sediments evacuates melt water to conduits and resulting distribution of N controls! f = µn.! =! d +! r +! f.

9 A Simple Composite Model: #! d +! r " a l - deformation + regelation -> $ % - $ % 2aA' ( 0,- / 0 - seepage transport to conduit - conduits flow down steepest potential gradient (~ surface gradient) - seepage driven by gradients in effective stress - average effective stress N = s u 0 " N C + Q g! Q b % where D u s # $ µu s & ' tanh " D u s % $ s u ' # 0 &! Q g! Q b u 0 =! lk 0 L µu µ"s s -! f = µ " =!#! d #! r - can achieve needed! f by adjusting D and/or N C Problem 3. a) From mass balance considerations, with the simplest reasonable assumptions, what equation must N satisfy? b) Justify the expression above for N (see notes at end). 2 & ' ( + ) i L k* au s + # u s & 1/n.

10 Additional Considerations - some transients likely to drive and reflect drainage reorganizations - observed surface uplift likely requires films/extremely linked cavities - bedform generation and till constitutive behavior poorly constrained - stick-slip suggests granular friction; regelation/deformation -> healing?

11 thanks for your time funding from NSF-OPP reprints (freeze-on/seepage flow/dirty friction) Problem 4. If you had a large budget geared towards acquiring field data to constrain more appropriate basal boundary conditions for your ice sheet model, what measurements would you request be made?

12 A simple model for the effective stress distribution produced by seepage flows (see Rempel, JGlac, 2009) Darcy s Law: Q b <Q g +Q f ( ) u =! k " # p + $ l gz Continuity:!"!t + #iu = 0 drainage divide Q b >Q g +Q f drainage divide Integrate over depth (e.g. D>>l-b=s); replace p with $ n -N u xy =! k " # xy ( ) &' $ n! N + % l g l! h ()! "( l # b # h)!t $% &' + ( xy i$( % l # b # h)u xy & ' = # ) i V ) l $ n : glacier weight (less bridging stresses) N: effective stress at l-h (sliding surface) V: local freezing rate V=(Q b -Q g -Q f )/(% i L)