Glacier Hydrology II: Theory and Modeling · Glacier Hydrology II: Theory and Modeling Matt Hoffman Gwenn Flowers, Simon Fraser McCarthy Summer School 2018. Observations Los Alamos

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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