Röthlisberger channel model accounting for antiplane shear loading and undeforming bed

Transcription

1 öthlisberger channel model accounting for antiplane shear loading and undeforming bed Matheus C. Fernandes Colin. Meyer Thibaut Perol James. ice,2 John A. Paulson School of Engineering and Applied Sciences Harvard University - Cambridge MA, USA 2 Department of Earth and Planetary Sciences Harvard University - Cambridge MA, USA International Symposium on the Hydrology of Glaciers and Ice Sheets June 23, 205

2 Questions of interest Introduction For a conduit along the bed, how does antiplane shear and locking affect the channel closure and stresses at the bed? What implications does this have on the diameter of a öthlisberger channel? Create Finite Element Method (FEM) models for 2 separate cases. Compliment and verify Weertman (972) analysis. τ AP u = 0 σ o Antiplane z Locked Bed σ o x τ AP = σ o p fluid u = 0 y M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 2

3 umerical model umerical model description Ice rheology is modeled as a shear thinning fluid with a power law (Glen s law, n = 3) relationship between stress and strain rate given as: ɛ E = AτE n, where ɛ E = 2 ɛ ij ɛ ij and τ E = 2 s ijs ij. The model assumes incompressibility and plane strain described by: ɛ + ɛ θθ = 0 and ɛ zz = 0. Over the span of it s km domain, the numerical model has an average error of 0.42% when benchmarking the ye solution. M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 3

4 Antiplane shear stress Antiplane shear stress Constant shear stress τ AP is applied along the bed and acts in the x-direction. z A pressure difference of = is applied along channel boundary and ice overburden is σ o =. on-dimensionalize stresses by σ o and length scales by channel diameter a. b a a σ o = S AP = τ AP = σ o p fluid Channel Boundary θ u z = 0 ÜAP x y Channel radius a = and domain radius b = 000. M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 4

5 Antiplane shear stress Weertman (972) antiplane model Weertman sees that the presence of large antiplane basal shear stress makes in-plane flow law effectively ewtonian as regards to tunnel closure. Claims that ye solution hoop stresses are tensile if creep rheology exponent n > 2 or compressive if n < 2. Describes radial velocity for τ AP / < by matching two asymptotic solutions divided by a critical radius obtained comparing magnitudes of antiplane to in-plane stresses as: cr = a(/τ AP ) n/2 = as n/2 AP M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 5

6 Antiplane shear stress Antiplane shear - hoop stress along bed Weertman sees that large antiplane shear makes in-plane flow ewtonian. We see that antiplane shear increases compressive stress up to 3 times overburden near channel. Intermediate values for S AP show the growth of a hump in the hoop stress next to the channel indicating possible channel migration. σθθ, hoop stress 3 ye Sol n n=3 ye Sol n n= z S AP =e-04 Ò 2.5 y S AP =5e-02 x S AP =2e-0 uz=0 S AP =5e-0 ÜAP 2 S AP =2e distance from center of channel M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 6

7 Antiplane shear stress Weertman antiplane shear model ur ur 0 3 S AP =e S AP =5e umerical Sol n ye Sol n Weertman Sol n 0 3 S AP =3e S AP =9e cr =S 3/2 AP For small S AP Weertman s model follows the ye solution as do the numerical results. Weertman s solution follows the numerical results for S AP 0.3. Weertman assumes that there to be a transition in the domain between the two dominant regimes. umerical results suggest that the magnitude of the entire domain shifts, as expected given that the shear stress is applied uniformly along the bed. M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 7

8 Antiplane shear stress Antiplane shear - channel opening ucreep/uye x z Ò y uz=0 ÜAP S AP = τ AP S 2 AP umerical Sol n Weertman Approx S AP For S AP 0 2 the channel opening is described by the ye solution. For S AP > (antiplane shear greater than channel pressure) and constant, the channel closure rate scales with τ 2 AP. Weertman s solution suggests that for large S AP, u creep u ye n n (τ AP /) n. M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 8

9 Antiplane shear stress öthlisberger Channel Implications The öthlisberger channel diameter is described by: ( ρice L u creep n m D = 4 ρ w g sin 3/2 (α) Ice stream shear margins can have S AP up to O(). Mountain glaciers can have S AP up to O(0 ). ) 3/5 DAP/Dye 0 2 Mountain Glaciers 0 S AP = τ AP Ice Stream Shear Margins S 6/5 AP S AP M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 9

10 Antiplane and undeforming bed channel model Ice locking along undeforming bed In-plane shear stress and locking bed Consider the extreme case in which no slippage occurs at the bed, namely u(θ = 0, π) = 0. x z y Apply an overburden stress σ o =. Let pressure difference in the channel be =. Weertman s 972 attempt used in-plane shear. b a a σ o = = σ o p fluid Channel Boundary θ u= 0 L o c k e d Bed M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 0

11 Ice locking along undeforming bed In-plane shear - displacement along bed For large S IP, u r /2 not u r as claimed by Weertman. adial creep rate does not decay to 0 away from the channel, as the shear becomes prevalent. ur S IP=e-03 umerical Sol n ye Sol n S IP=e S IP=3e S IP=e+00 0 Furthermore, the magnitude of radial creep rate also scales as S 3 IP. ur S IP = τip M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205

12 Ice locking along undeforming bed In-plane shear - hoop stress along bed Weertman predicts a hoop stress along the bed as σ θθ 2. Hoop stress does not scale as ye solution and is heavily influenced by the shear along bed. σθθ, hoop stress.2 S IP =e umerical Sol n ye Sol n S IP =e S IP =e S IP =e+00 When shear along the bed is the same magnitude as pressure difference, the hoop stress can increase up to 0 times overburden pressure near channel. σθθ, hoop stress M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 2

13 Antiplane and undeforming bed channel model Ice locking along undeforming bed Locked bed - hoop stress along bed σθθ, hoop stress x u=0 Locked Bed umerical Sol n ye Sol n Distance from center of channel z Ò y If the channel is locked the hoop stress is nearly overburden across the entire length of the bed. esults are very different than applying a constant shear stress. Singular tensile stress occurs near the channel as a result of constraining the displacement and imposing a closing stress boundary condition. M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 3

14 Locked bed - channel opening Ice locking along undeforming bed Highest value of creep closure is equivalent to 0.73 times the ye closure rate. Total channel closing rate is 0.67 the ye solution. Independent in magnitude of. Also very different than applying a constant shear stress along the bed. Locking the bed, changes -channel diameter by a factor of ucreep/uye u mean 0.67 y x 0 0 /4 /2 θ(π) x z 0.73 Ò y u=0 Locked Bed M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 4

15 Conclusions Conclusion öthlisberger channels in ice stream shear margins, where antiplane stresses are significantly contribute to the in-plane viscosity, may see up to a factor of 6 change in -channel diameter. Channels in mountain glaciers are not expected to be affected by antiplane shear. Although Weertman s scaling for channel opening matches the numerical results, we do not see a transition between asymptotic solutions over the displacement along the bed. Applying an in-plane shear stress at the bed is not equivalent to locking the bed. A locked bed can change -channel diameter by a factor of M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 5

16 eferences eferences Bartholomous, T. C., Anderson,. S., and Anderson, S. P. (20). Growth and collapse of the distributed subglacial hydrologic system of kennicott glacier, alaska, usa, and its effects on basal motion. J. Glaciol., 57(206): Joughin, I., Tulaczyk, S., Bindschadler,., and Price, S. F. (2002). Changes in west antarctic ice stream velocities: Observation and analysis. Journal of Geophysical esearch: Solid Earth, 07(B):EPM 3 EPM Meyer, C., Fernandes, M., and ice, J. (205). öthlisberger channels under antiplane shear. Journal of Fluid Mechanics, submitted. ye, J. F. (953). The flow law of ice from measurements in glacier tunnels, laboratory experiments and the jungfraufirn borehole experiment. Proceedings of the oyal Society of London. Series A. Mathematical and Physical Sciences, 29(39): Perol, T. and ice, J.. (20). Control of the width of west antarctic ice streams by internal melting in the ice sheet near the margins. AGU Fall Meeting Abstracts, :0677. öthlisberger, H. (972). Water pressure in intra- and subglacial channels. (62): Weertman, J. (972). General theory of water flow at the base of a glacier or ice sheet. eviews of Geophysics, 0(): M.C.Fernandes et al. - Harvard University IGS Symposium, Höfn - June 23, 205 6

17 Matheus C. Fernandes Thank You! Mendenhall Glacier Ice Cave in Alaska - Photograph by Greg ewkirk Acknowledgments: Harvard SEAS Blue Hill Hydrology Endowment (MCF Masters program). Harvard University