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

6/13/18 | 2Los Alamos National Laboratory

Theory

Modeling

Governing equations for subglacial drainage


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


Gwenn Flowers

Water sources, sinks, and fluxes




Continuity

Source/sinks

Fluid potential

Water flux


Gwenn Flowers

Laminar and turbulent flow


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


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


Navier-Stokes equationRadial coordinates

Solve for velocity• Integrate twice• Apply appropriate boundary

conditions

Porous media flow: Darcy’s Law


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


• 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



Continuity

Source/sinks

Fluid potential

Water flux


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


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


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


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


Channelized DrainageImage: Tim CreytsImage: Ian Hewitt

Distributed Drainage

Energy balance for melt:

Meltopening

Slidingover

bumps

Creep closure of

ice

ConduitEvolution:

Geothermalheat flux

Frictionalheating

Viscousdissipation


Meltopening

Slidingover

bumps

Creep closure of

ice

Geothermalheat flux

Frictionalheating

Viscousdissipation




Continuity

Source/sinks

Fluid potential

Water flux


Gwenn Flowers


Putting it together: implications

Gwenn Flowers

Implications: channel vs. distributed drainage efficiency


channel

distributed

Observations


Theory

Modeling

Literature: models of subglacial drainage


Gwenn Flowers

Overview: models of subglacial drainage


Gwenn Flowers


Elements of the subglacial drainage system

Overview: Models of subglacial drainage


Flowers (2015) Proc. Royal Soc. A

Recipe for a model of subglacial drainage


Gwenn Flowers

Early models from groundwater hydrology (1970s – 1990s)


Gwenn Flowers


Gwenn Flowers


• 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


Gwenn Flowers


Focus on drainage at the ice-bed interfaceModels employ more of the theoretical drainage elements

“Next-generation” glaciological models (1990s ±)

Gwenn Flowers


Gwenn Flowers

Integrating multiple drainage elements (2000s – present)


Gwenn Flowers


Gwenn Flowers


Subglacial drainage model zoo

Subglacial Hydrology Model Intercomparison Project (SHMIP)de Fleurian et al. in review, J. Glac.


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


Gwenn Flowers


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.


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


Hoffman, et al. (2016) Greenland subglacial drainage evolution regulated by weakly-connected

regions of the bed, Nat. Commun., 7, 13903.


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.


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


• 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

glacier hydrology ii: theory and modeling · glacier hydrology ii: theory and modeling matt hoffman...

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