Röthlisberger channel model accounting for antiplaneshear loading and undeforming bed

Matheus C. Fernandes∗1 Colin R. Meyer1Thibaut Perol1 James R. Rice1,2

1John A. Paulson School of Engineering and Applied SciencesHarvard University - Cambridge MA, USA

2Department of Earth and Planetary SciencesHarvard University - Cambridge MA, USA

International Symposium on theHydrology of Glaciers and Ice Sheets

June 23, 2015

Antiplane and undeforming bed channel model Introduction

Questions of interest

For a conduit along thebed, how does antiplaneshear and locking a�ectthe channel closure andstresses at the bed?What implications doesthis have on the diameterof a Röthlisbergerchannel?Create Finite ElementMethod (FEM) models for 2separate cases.Compliment and verifyWeertman (1972) analysis.

Antiplane

N τAPτAP

σoy

z

x

Locked Bed

N u=0

σo N=σo−pfluid

u=0

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 2

Antiplane and undeforming bed channel model Numerical model

Numerical model description

Ice rheology is modeled as a shear thinning �uid with apower law (Glen’s law, n = 3) relationship between stressand strain rate given as:

ε̇E = AτnE , where ε̇E =

√1

2ε̇ij ε̇ij and τE =

√1

2sijsij.

The model assumes incompressibility and plane straindescribed by:

ε̇RR + ε̇θθ = 0 and ε̇zz = 0.

Over the span of it’s 1km domain, the numerical model hasan average error of 0.42% when benchmarking the Nyesolution.

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 3

Antiplane and undeforming bed channel model Antiplane shear stress

Antiplane shear stress

Constant shear stress τAPis applied along the bed andacts in the x-direction.

A pressure di�erence ofN = 1 is applied alongchannel boundary and iceoverburden is σo = 1.

Non-dimensionalizestresses by σo and lengthscales by channel diametera.

Channel radius a = 1 anddomain radius b = 1000.

z

y

x

N

Channel

Boundary

ÜAP

θ

R

a

b−a

uz=0

SAP=τAP

N

N=σo−pfluid

σo = 1

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 4

Antiplane and undeforming bed channel model Antiplane shear stress

Weertman (1972) antiplane model

Weertman sees that the presence of large antiplane basalshear stress makes in-plane �ow law e�ectively Newtonianas regards to tunnel closure.

Claims that Nye solution hoop stresses are tensile if creeprheology exponent n > 2 or compressive if n < 2.

Describes radial velocity for τAP/N < 1 by matching twoasymptotic solutions divided by a critical radius obtainedcomparing magnitudes of antiplane to in-plane stresses as:

Rcr = a(N/τAP )n/2 = aS

−n/2AP

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 5

Antiplane and undeforming bed channel model Antiplane shear stress

Antiplane shear - hoop stress along bed

Weertman sees that largeantiplane shear makesin-plane �ow Newtonian.

We see that antiplane shearincreases compressivestress up to ∼3 timesoverburden near channel.

Intermediate values for SAPshow the growth of a humpin the hoop stress next tothe channel indicatingpossible channel migration. distance from center of channel ’R’

100

101

102

σθθ,hoop

stress

0.5

1

1.5

2

2.5

3 Nye Sol’n n=3

Nye Sol’n n=1

SAP=1e-04

SAP=5e-02

SAP=2e-01

SAP=5e-01

SAP=2e+01

uz=0N ÜAP

z

yx

R

Ò

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 6

Antiplane and undeforming bed channel model Antiplane shear stress

Weertman antiplane shear model

100

102

103

ur

10-5

100

103

SAP=1e-02

100

102

103

10-5

100

103

SAP=5e-02

R10

010

210

3

ur

10-5

100

103

SAP=3e-01

R10

010

210

310

-5

100

103

SAP=9e-01

Numerical Sol’n

Nye Sol’n

Weertman Sol’n

Rcr=S−3/2AP

For small SAP Weertman’s modelfollows the Nye solution as do thenumerical results.Weertman’s solution follows thenumerical results for SAP ∼0.3.Weertman assumes that there tobe a transition in the domainbetween the two dominantregimes.Numerical results suggest thatthe magnitude of the entiredomain shifts, as expected giventhat the shear stress is applieduniformly along the bed.

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 7

Antiplane and undeforming bed channel model Antiplane shear stress

Antiplane shear - channel opening

SAP

10-2

10-1

100

101

102

ucreep/uN

ye

100

101

102

103

104

105

Numerical Sol’n

Weertman Approx.

uz=0N ÜAP

z

yx

R

Ò

SAP = τAP

N

∼S2AP

For SAP ∼ 10−2 thechannel opening isdescribed by the Nyesolution.

For SAP > 1 (antiplaneshear greater than channelpressure) and constant N ,the channel closure ratescales with τ 2

AP .

Weertman’s solutionsuggests that for large SAP ,ucreep

uNye∼ nn(τAP/N)n−1.

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 8

Antiplane and undeforming bed channel model Antiplane shear stress

Röthlisberger Channel Implications

The Röthlisbergerchannel diameter isdescribed by:

D=4

(ρiceL

ρwg

ucreepnm

sin3/2(α)

)3/5

Ice stream shearmargins can have SAPup to O(1).

Mountain glaciers canhave SAP up toO(10−1). SAP

10-2

10-1

100

101

DAP/D

Nye

100

101

102

SAP = τAP

N

Ice Stream Shear Margins

Mountain Glaciers

∼S6/5AP

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 9

Antiplane and undeforming bed channel model Ice locking along undeforming bed

In-plane shear stress and locking bed

Consider the extreme casein which no slippage occursat the bed, namelyu(θ = 0, π) = 0.

Apply an overburden stressσo = 1.

Let pressure di�erence inthe channel be N = 1.

Weertman’s 1972 attemptused in-plane shear.

z

y

x

N

Channel

Boundary

Locked ! ! Bed !!

θ

R

a

b−a

u=0

N=σo−pfluidσo = 1

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 10

Antiplane and undeforming bed channel model Ice locking along undeforming bed

In-plane shear - displacement along bed

For large SIP , ur ∼ R1/2

not ur ∼ R−1 as claimed byWeertman.

Radial creep rate does notdecay to 0 away from thechannel, as the shearbecomes prevalent.

Furthermore, themagnitude of radial creeprate also scales as S3

IP .

100

101

102

103

ur

0

0.2

0.4

0.6

0.8

1

SIP=1e-03

Numerical Sol’n

Nye Sol’n

100

101

102

103

-1

-0.5

0

0.5

1

SIP=3e-02

R10

010

110

210

3

ur

-30

-20

-10

0

10

SIP=1e-01

R10

010

110

210

3

×104

-3

-2

-1

0

1

SIP=1e+00

SIP = τIPN

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 11

Antiplane and undeforming bed channel model Ice locking along undeforming bed

In-plane shear - hoop stress along bed

Weertman predicts a hoopstress along the bed asσθθ ∼ R−2.

Hoop stress does not scaleas Nye solution and isheavily in�uenced by theshear along bed.

When shear along the bedis the same magnitude aspressure di�erence, thehoop stress can increase upto 10 times overburdenpressure near channel.

100

101

102

σθθ,hoop

stress

0.6

0.8

1

1.2

SIP =1e-03

Numerical Sol’n

Nye Sol’n

100

101

102

0.6

0.8

1

1.2

SIP =1e-02

R10

010

110

2

σθθ,hoop

stress

0.5

1

1.5

2

SIP =1e-01

R10

010

110

20

5

10

SIP =1e+00

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 12

Antiplane and undeforming bed channel model Ice locking along undeforming bed

Locked bed - hoop stress along bed

Distance from center of channel10

010

110

210

3

σθθ,hoop

stress

-1.5

-1

-0.5

0

0.5

1

1.5

Numerical Sol’n

Nye Sol’n

u=0N

Locked !! Bed !!

z

yx

R

Ò

If the channel is locked thehoop stress is nearlyoverburden across theentire length of the bed.Results are very di�erentthan applying a constantshear stress.Singular tensile stressoccurs near the channel asa result of constraining thedisplacement and imposinga closing stress boundarycondition.

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 13

Antiplane and undeforming bed channel model Ice locking along undeforming bed

Locked bed - channel opening

Highest value of creepclosure is equivalent to∼0.73 times the Nyeclosure rate.Total channel closing rate is∼0.67 the Nye solution.Independent in magnitudeof N .Also very di�erent thanapplying a constant shearstress along the bed.Locking the bed, changesR-channel diameter by afactor of ∼0.79. θ(π)

0 1/4 1/2

ucreep/uN

ye

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

-1 -0.5 0 0.5 1

y

0

0.5

1

u=0N

Locked !! Bed !!

z

yx

R

Ò

umean≈0.67∼0.73

M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 14

Antiplane and undeforming bed channel model Conclusion

ConclusionsRöthlisberger channels in ice stream shear margins, whereantiplane stresses are signi�cantly contribute to thein-plane viscosity, may see up to a factor of 6 change inR-channel diameter.

Channels in mountain glaciers are not expected to bea�ected by antiplane shear.

Although Weertman’s scaling for channel opening matchesthe numerical results, we do not see a transition betweenasymptotic solutions over the displacement along the bed.

Applying an in-plane shear stress at the bed is notequivalent to locking the bed.

A locked bed can change R-channel diameter by a factor of∼0.79.M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 15

Antiplane and undeforming bed channel model References

ReferencesBartholomous, T. C., Anderson, R. S., and Anderson, S. P. (2011). Growth andcollapse of the distributed subglacial hydrologic system of kennicott glacier,alaska, usa, and its e�ects on basal motion. J. Glaciol., 57(206):985–1002.

Joughin, I., Tulaczyk, S., Bindschadler, R., and Price, S. F. (2002). Changes inwest antarctic ice stream velocities: Observation and analysis. Journal ofGeophysical Research: Solid Earth, 107(B11):EPM 3–1–EPM 3–22. 2289.

Meyer, C., Fernandes, M., and Rice, J. (2015). Röthlisberger channels underantiplane shear. Journal of Fluid Mechanics, submitted.

Nye, J. F. (1953). The �ow law of ice from measurements in glacier tunnels,laboratory experiments and the jungfrau�rn borehole experiment.Proceedings of the Royal Society of London. Series A. Mathematical and PhysicalSciences, 219(1139):477–489.

Perol, T. and Rice, J. R. (2011). Control of the width of west antarctic icestreams by internal melting in the ice sheet near the margins. AGU FallMeeting Abstracts, 1:0677.

Röthlisberger, H. (1972). Water pressure in intra- and subglacial channels.11(62):177–203.

Weertman, J. (1972). General theory of water �ow at the base of a glacier orice sheet. Reviews of Geophysics, 10(1):287–333.M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 2015 16

Matheus C. Fernandes http://fer.me

Thank You!

Mendenhall Glacier Ice Cave in Alaska - Photograph by Greg Newkirk

Acknowledgments:Harvard SEAS Blue Hill HydrologyEndowment (MCF Masters program).

Harvard University