Glacier Hydrology II: Theory and Modeling McCarthy Summer School 2018 Matt Hoffman Gwenn Flowers, Simon Fraser Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA
Observations Theory Modeling 6/13/18 2
Governing equations for subglacial drainage Continuity Source/sinks Fluid potential Water flux Evolution equation for element(s) Gwenn Flowers 6/13/18 3
Conservation of mass (continuity) in a 1-D water sheet or film Gwenn Flowers 6/13/18 4
Water sources, sinks, and fluxes 6/13/18 5
Governing equations for subglacial drainage Continuity Source/sinks Fluid potential Water flux Evolution equation for element(s) Gwenn Flowers 6/13/18 6
Laminar and turbulent flow Navier-Stokes equations from Conservation of Momentum for motion of fluids (Newton s Second Law) Inertial forces Pressure forces Viscous forces External forces Reynolds number: Used to predict laminar and turbulent flow regimes Low Re Inertial forces Viscous forces High Re 6/13/18 7
Laminar flow in sheet or film hh Navier-Stokes equation from Conservation of Momentum (Newton s Second Law) Here assuming: Incompressible Laminar Parallel shear flow Inertial forces Pressure forces p w Viscous forces External forces Φ = hydraulic potential v = water velocity ρ w = Density of water μ = Dynamic viscosity of water Poiseuille Flow Assumptions: Steady-state Horizontal 1-d Viscous forces Pressure forces Solve for velocity Integrate twice Apply appropriate boundary conditions Solve for flux Integrate velocity with depth Apply appropriate boundary conditions Solve for depth-averaged velocity Laminar flow: q 6/13/18 8
Laminar flow in pipe or channel Navier-Stokes equation Radial coordinates Solve for velocity Integrate twice Apply appropriate boundary conditions 6/13/18 9
Porous media flow: Darcy s Law Porous media flow Creeping, laminar flow Tortuous paths Quantity of interest is not fluid velocity but effective velocity Relevant to: Uneven water films Bulk behavior of linked cavity system ( microporous sheet ) Till canals Nye channels Drainage through till Flux Hydraulic conductivity layer thickness Hydropotential gradient Ian Hewitt 6/13/18 10
Turbulent flow in pipe or channel Can be derived from Navier-Stokes equations More intuitive: balance of driving pressure force and resistive viscous force Potential gradient Area Perimeter Wall stress Wall stress: Empirical from engineering hydraulics Darcy-Weisbach Gauckler-Manning- Strickler Friction factor Combine to solve for velocity: Turbulent flow: q 1/2 6/13/18 11
Governing equations for subglacial drainage Continuity Source/sinks Fluid potential Water flux Evolution equation for element(s) Gwenn Flowers 6/13/18 12
Evolution of conduit (channel or cavity) volume ice Modified from Anderson, et al. 2004 v bed v Conduit Evolution: Melt opening Sliding over bumps Creep closure of ice Energy balance for melt: Geothermal heat flux Frictional heating Viscous dissipation (laminar or turbulent) Model output: sheet thickness, water pressure
Melt opening of a channel Viscous dissipation power per unit length of conduit release of potential energy (gravitational + pressure) Pressure-melt dependence when pressure varies some power required to keep water at melt temperature Pressure melting coefficent ~0.3 Specific heat capacity of water Net power As a melt rate Instantaneous heat transfer Temperate ice More complex treatment using full energy balance 6/13/18 14
Creep closure for a channel Glen s flow law Radial coordinates Rewrite as area change Complexities Conduit shape channels can be low and broad, non-channel shapes Addition of shape factor Effective pressure typically substituted for normal stress 6/13/18 15
Classic conduit evolution mechanisms Image: Ian Hewitt Distributed Drainage Image: Tim Creyts Channelized Drainage Opening Sliding Melting Passive sources Viscous dissipation Closing Creep YES (but other mechanisms also possible) Maybe No (can lead to channelization) Yes (but other mechanisms also possible) No/not important (but maybe can destroy) No/not important YES (can also cause closing) Yes 6/13/18 16
Classic conduit evolution mechanisms Image: Ian Hewitt Distributed Drainage Image: Tim Creyts Channelized Drainage Conduit Evolution: Melt Sliding opening over bumps Creep closure of ice Melt Sliding opening over bumps Creep closure of ice Energy balance for melt: Geothermal heat flux Frictional heating Viscous dissipation (laminar or turbulent) Geothermal heat flux Frictional heating Viscous dissipation (laminar or turbulent) 6/13/18 17
Governing equations for subglacial drainage Continuity Source/sinks Fluid potential Water flux Evolution equation for element(s) Gwenn Flowers 6/13/18 18
Putting it together: implications Gwenn Flowers 6/13/18 19
Implications: channel vs. distributed drainage efficiency distributed channel 6/13/18 20
Observations Theory Modeling 6/13/18 21
Literature: models of subglacial drainage Gwenn Flowers 6/13/18 22
Overview: models of subglacial drainage Gwenn Flowers 6/13/18 23
Elements of the subglacial drainage system 6/13/18 24
Elements of the subglacial drainage system 6/13/18 25
Overview: Models of subglacial drainage Flowers (2015) Proc. Royal Soc. A 6/13/18 26
Recipe for a model of subglacial drainage Gwenn Flowers 6/13/18 27
Early models from groundwater hydrology (1970s 1990s) Gwenn Flowers 6/13/18 28
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Motivated by an interest in basal till deformation, fast flow, fate of basal melt, impact of glaciation on groundwater Hydrogeologic units beneath ice sheets rarely capable of evacuating even basal melt from ice sheets Some interfacial drainage system required, though rarely formalized Gwenn Flowers 6/13/18 31
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Next-generation glaciological models (1990s ±) Focus on drainage at the ice-bed interface Models employ more of the theoretical drainage elements Gwenn Flowers 6/13/18 33
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Integrating multiple drainage elements (2000s present) Gwenn Flowers 6/13/18 38
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Subglacial drainage model zoo Subglacial Hydrology Model Intercomparison Project (SHMIP) de Fleurian et al. in review, J. Glac. 6/13/18 46
Features of current models Correct (?) physics applied to fast and slow drainage systems Numerical formulation (2D distributed system, network of 1D channel elements) Dynamic evolution of drainage system Two-way coupling to sliding/friction laws Modelling challenges and limitations Resolution of individual channel elements Assumption of fully connected drainage system Temperate bed assumption Dearth of calibration/validation data, resolution of input data 6/13/18 47
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Too simple? Field observations more complex and richer than current models can explain Out of phase borehole pressure variations Water pressure above floatation pressure Large pressure gradients between neighboring boreholes Switching between connected and disconnected behavior Very high winter water pressure despite negligible water input Rada and Schoof (2018) Subglacial drainage characterization from eight years of continuous borehole data on a small glacier in the Yukon Territory, Canada. The Cryosphere Discussions. 6/13/18 49
Summer Winter Summer Winter Downs, J. Z., et al. (2018) Dynamic hydraulic conductivity reconciles mismatch between modeled and observed winter subglacial water pressure, J. Geophys. Res. Earth Surf., 123, 818 836. 6/13/18 50
Hoffman, et al. (2016) Greenland subglacial drainage evolution regulated by weakly-connected regions of the bed, Nat. Commun., 7, 13903. 6/13/18 51
Ice dynamics responds to integrated basal traction over entire bed. Spring Iken, A., and M. Truffer (1997), The relationship between subglacial water pressure and velocity of Findelengletscher, Switzerland, during its advance and retreat, J. Glaciol., 43(144), 328 338. Summer Fall Hoffman, et al. (2016) Greenland subglacial drainage evolution regulated by weakly-connected regions of the bed, Nat. Commun., 7, 13903. 6/13/18 52
Rada and Schoof (2018) Subglacial drainage characterization from eight years of continuous borehole data on a small glacier in the Yukon Territory, Canada. The Cryosphere Discussions. 6/13/18 53
Recommendations for ongoing and future work Efficient representation of drainage networks in largescale models Continuum approached for efficient drainage that converge with grid resolution Alternative approaches to continuum models or explicit treatment of connected and unconnected bed areas Unified treatment of hard and soft beds? More attention to polythermal conditions (c.f. e.g. Bueler and Brown, 2009; Creyts and Clarke, 2010) Surface, englacial, groundwater drainage Develop methods to measure basal water pressure at scales commensurate with ice dynamics 6/13/18 54