glacier hydrology ii: theory and modeling · glacier hydrology ii: theory and modeling matt hoffman...
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Glacier Hydrology II: Theory and Modeling
Matt HoffmanGwenn Flowers, Simon Fraser
McCarthy Summer School 2018
Observations
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Theory
Modeling
Governing equations for subglacial drainage
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Continuity
Source/sinks
Fluid potential
Water flux
Evolution equation for element(s)
Gwenn Flowers
Conservation of mass (continuity) in a 1-D water sheet or film
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Gwenn Flowers
Water sources, sinks, and fluxes
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Governing equations for subglacial drainage
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Continuity
Source/sinks
Fluid potential
Water flux
Evolution equation for element(s)
Gwenn Flowers
Laminar and turbulent flow
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Navier-Stokes equationsfrom Conservation of Momentum for motion of fluids(Newton’s Second Law)
Inertialforces
Pressureforces
Viscousforces
Externalforces
Reynolds number:Used to predict laminar and turbulent flow regimes
InertialforcesViscousforces
Low Re
High Re
Laminar flow in sheet or film
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Navier-Stokes equationfrom Conservation of Momentum
(Newton’s Second Law)
Here assuming:
• Incompressible
• Laminar
• Parallel shear flow
μ = Dynamic
viscosity of water
ρw = Density of water
v = water velocity
Φ = hydraulic potential
Inertial
forces
Pressure
forces
Viscous
forces
External
forces
Poiseuille FlowAssumptions:
• 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∝
pw
hh
Laminar flow in pipe or channel
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Navier-Stokes equationRadial coordinates
Solve for velocity• Integrate twice• Apply appropriate boundary
conditions
Porous media flow: Darcy’s Law
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Porous media flow• Creeping, laminar flow• Tortuous paths• Quantity of interest is not fluid
velocity but effective velocity
Hydraulic conductivityFlux Hydropotential
gradient
layerthickness
Relevant to:• Uneven water films • Bulk behavior of
linked cavity system (“microporous sheet”)
• Till canals• Nye channels• Drainage through till
Ian Hewitt
Turbulent flow in pipe or channel
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• 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
Frictionfactor
Combine to solve for velocity: Turbulent flow: q∝
1/2
Governing equations for subglacial drainage
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Continuity
Source/sinks
Fluid potential
Water flux
Evolution equation for element(s)
Gwenn Flowers
Energy balance for melt:
Model output: sheet thickness, water pressure
Meltopening
Slidingover
bumps
Creep closure of ice
v
v
ice
bed
Modified from Anderson, et al. 2004
Evolution of conduit (channel or cavity) volume
ConduitEvolution:
Geothermalheat flux
Frictionalheating
Viscousdissipation
(laminar or turbulent)
Melt opening of a channel
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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
Pressuremeltingcoefficent
Specificheat capacityof water
~0.3
Net power
As a melt rate• Instantaneous heat transfer• Temperate ice• More complex treatment
using full energy balance
Creep closure for a channel
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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
Classic conduit evolution mechanisms
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Channelized DrainageImage: Tim CreytsImage: Ian Hewitt
Distributed Drainage
Opening
Closing
Sliding
Melting
Creep
YES(but other mechanisms also possible)
Passive sources
Viscous dissipation
Yes(but other mechanisms also possible)
Yes
No/not important(but maybe can destroy)
Maybe No/not important
No(can lead to channelization)
YES(can also cause closing)
Classic conduit evolution mechanisms
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Channelized DrainageImage: Tim CreytsImage: Ian Hewitt
Distributed Drainage
Energy balance for melt:
Meltopening
Slidingover
bumps
Creep closure of
ice
ConduitEvolution:
Geothermalheat flux
Frictionalheating
Viscousdissipation
(laminar or turbulent)
Meltopening
Slidingover
bumps
Creep closure of
ice
Geothermalheat flux
Frictionalheating
Viscousdissipation
(laminar or turbulent)
Governing equations for subglacial drainage
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Continuity
Source/sinks
Fluid potential
Water flux
Evolution equation for element(s)
Gwenn Flowers
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Putting it together: implications
Gwenn Flowers
Implications: channel vs. distributed drainage efficiency
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channel
distributed
Observations
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Theory
Modeling
Literature: models of subglacial drainage
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Gwenn Flowers
Overview: models of subglacial drainage
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Gwenn Flowers
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Elements of the subglacial drainage system
Elements of the subglacial drainage system
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Overview: Models of subglacial drainage
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Flowers (2015) Proc. Royal Soc. A
Recipe for a model of subglacial drainage
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Gwenn Flowers
Early models from groundwater hydrology (1970s – 1990s)
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Gwenn Flowers
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Gwenn Flowers
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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
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Gwenn Flowers
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Focus on drainage at the ice-bed interfaceModels employ more of the theoretical drainage elements
“Next-generation” glaciological models (1990s ±)
Gwenn Flowers
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Gwenn Flowers
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Gwenn Flowers
Integrating multiple drainage elements (2000s – present)
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Gwenn Flowers
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Gwenn Flowers
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Gwenn Flowers
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Subglacial drainage model zoo
Subglacial Hydrology Model Intercomparison Project (SHMIP)de Fleurian et al. in review, J. Glac.
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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
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Gwenn Flowers
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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• …
Too simple?
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.
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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.
Summer Winter Summer Winter
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Hoffman, et al. (2016) Greenland subglacial drainage evolution regulated by weakly-connected
regions of the bed, Nat. Commun., 7, 13903.
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Spring
Summer
Fall
Hoffman, et al. (2016) Greenland subglacial drainage evolution regulated by weakly-connected
regions of the bed, Nat. Commun., 7, 13903.
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.
Ice dynamics responds to integrated
basal traction over entire bed.
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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.
Recommendations for ongoing and future work
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• Efficient representation of drainage networks in large-
scale 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