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