Earth and Planetary Science Letters 395 (2014) 184–193 Contents lists available at ScienceDirect Earth and Planetary Science Letters www.elsevier.com/locate/epsl Ice-shelf flexure and tidal forcing of Bindschadler Ice Stream, West Antarctica , ,∗ , Ryan T. Walker a e , Byron R. Parizek b, Richard B. Alley c, Kelly M. Brunt d e, Sridhar Anandakrishnan c a Earth System Science Interdisciplinary Center, University of Maryland, College Park, MD 20742, USA b Mathematics and Geoscience, 181 Smeal Building, Penn State DuBois, College Place, DuBois, PA 15801, USA c Department of Geosciences, 517 Deike Building, Pennsylvania State University, University Park, PA 16802, USA d Goddard Earth Sciences Technology and Research (GESTAR), USA e Cryospheric Sciences Laboratory, Mail Code 615, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA article info abstract Article history: Viscoelastic models of ice-shelf flexure and ice-stream velocity perturbations are combined into a single Received 25 November 2013 efficient flowline model to study tidal forcing of grounded ice. The magnitude and timing of ice- Received in revised form 28 February 2014 stream response to tidally driven changes in hydrostatic pressure and/or basal drag are found to depend Accepted 22 March 2014 significantly on bed rheology, with only a perfectly plastic bed allowing instantaneous velocity response Available online xxxx at the grounding line. The model can reasonably reproduce GPS observations near the grounding zone of Editor: J. Lynch-Stieglitz Bindschadler Ice Stream (formerly Ice Stream D) on semidiurnal time scales; however, other forcings such Keywords: as tidally driven ice-shelf slope transverse to the flowline and flexurally driven till deformation must also ice shelf be considered if diurnal motion is to be matched. ice stream © 2014 Elsevier B.V. All rights reserved. ocean tides viscoelastic 1. Introduction suggest that the ice-stream beds can have markedly different char- acteristics, with Gudmundsson (2011) finding a basal flow law The influence of ocean tides on ice shelves and their tribu- for Rutford Ice Stream consistent with hard-bed sliding or low- tary ice streams provides opportunities for investigation of fun- stress-exponent till deformation while Walker et al. (2012) find damental processes controlling flow at ice-sheet margins. Early an effectively plastic basal flow law for Bindschadler Ice Stream observational (Robin, 1958) and theoretical (Holdsworth, 1969, (on the Siple Coast) consistent with sliding over weakly velocity- 1977, 1981) studies of vertical ice-shelf flexure led to more com- strengthening till (Tulaczyk, 2006). plex modeling studies (Schmeltz et al., 2002; Reeh et al., 2003; The precise mechanisms by which tidal motion causes modu- Sergienko, 2010; Sayag and Worster, 2011, 2013), and ultimately lation of ice-stream flow are not yet well known, but it is likely to the use of remote sensing to detect the grounding zone over that improved understanding of these mechanisms will provide which ice transitions between grounded and freely floating states insight to grounding-zone processes that affect ice-stream dynam- (Scambos et al., 2007; Fricker et al., 2009; Brunt et al., 2010, ics. The two most obvious processes are the change in hydro- 2011; Bindschadler et al., 2011; Rignot et al., 2011). Meanwhile, static pressure with tidal height and transient changes in ground- GPS observations demonstrated that the flow of several Siple Coast ing due to flexure, although modeling of the latter in an elas- ice streams into the Ross Ice Shelf (Anandakrishnan et al., 2003; tic, non-hydrostatic manner remains an active area of research Bindschadler et al., 2003; Winberry et al., 2009, 2011) and the (e.g., Sayag and Worster, 2011, 2013). Other potentially signif- flow of Rutford Ice Stream into the Ronne Ice Shelf (Gudmundsson, icant processes include changes in ice-shelf slope due to spa- 2006, 2007; Murray et al., 2007; King et al., 2010) are modu- tially nonuniform tides (Brunt, 2008), flexurally induced variation lated by the tides. Modeling studies of these velocity perturbations in subglacial pressure affecting water flow (Walker et al., 2013), and formation of “sticky spots” due to repeated vertical com- paction of till (Christianson et al., 2013). Anandakrishnan and Al- Corresponding author at: Cryospheric Sciences Laboratory, Mail Code 615, NASA * ley (1997) were able to explain tidally forced basal seismicity of Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA. Tel.: +1 301 614 6933. ice stream C (now Kamb Ice Stream) using a simplified model E-mail address: [email protected] (R.T. Walker). of an elastic ice stream with no explicit shelf overlying a viscous http://dx.doi.org/10.1016/j.epsl.2014.03.049 0012-821X/© 2014 Elsevier B.V. All rights reserved. R.T. Walker et al. / Earth and Planetary Science Letters 395 (2014) 184–193 185 bed. More recently, Gudmundsson (2011) used a viscoelastic, full- In Walker et al. (2012), which was concerned only with grounded Stokes model to show that the fortnightly modulation of Rutford ice, it was assumed that neither the basal pressure Pb ∂xb nor the Ice Stream can be explained by interaction between semidiurnal basal drag coefficient β2 varied in time; however, this assumption tidal constituents provided that the basal sliding law is nonlinear. is no longer valid when vertical flexure of the ice shelf and/or In this study, we combine and expand earlier models of ice- tidally-driven changes in grounding are considered. For the basal shelf flexure (Walker et al., 2013) and ice-stream response to stress drag term, we now have perturbations (Walker et al., 2012) to analyze how tidal forcing β2 leads to modulation of ice-stream flow, with an emphasis on ve- 2 ˜ 1 ˜ 1 −1 ˜ 2 ˜ 1 ∂t τb = ∂t β (u + u) m = (u + u) m ∂t u + ∂t β (u + u) m , locity perturbations in the grounding zone. Beginning with exper- m iments on idealized domains, we demonstrate the importance of (7) basal rheology in determining the magnitude and timing of the which adds a term to the earlier model. As for the basal pressure, response to a given forcing. The model is then applied to Bind- the elevation of the base of an ice shelf subject to tidal flexure is schadler Ice Stream, resulting in broad agreement with the obser- z = b(x,t) = z (x)+ w(x,t), and its depth is η(x,t)−b = η− z − w, vations of Anandakrishnan et al. (2003). b b where η is the tidal height, w is the vertical flexure of the shelf (calculated from (1)), and z is the initial basal elevation when η = 2. Model derivation b w = 0. We define z = 0 at sea level (i.e., the sea surface elevation when η = 0). Assuming hydrostatic pressure at the ice-shelf base The vertical displacement w of the ice shelf/stream (at time and using the preceding expressions for basal depth and elevation t (s) at a horizontal position x (m) along a chosen flowline) is cal- leads to culated from the model of Walker et al. (2013), Ek E Pb ∂xb = ρw g(η − zb − w)∂x(zb + w), (8) ∂ kw + ∂2 D ∂2 w + w = ∂ q + q, (1) t x x ν( − λ2) t ν( − λ2) 2 1 2 1 so that which represents the ice as a viscoelastic beam with Young’s mod- ∂ ( ∂ ) = ρ ∂ (η − )∂ ( + ) + (η − − )∂ ∂ . ulus E, viscosity ν, and Poisson’s ratio λ resting (where grounded) t Pb xb w g t w x zb w zb w x t w on an elastic foundation with spring constant k. The flexural rigid- (9) ity of the ice is given by As in the earlier model, this term remains zero for grounded ice. 3 Eh We continue to take ice to be a Maxwell material, the simplest D ≡ , (2) 12(1 − λ2) viscoelastic rheology that displays both instantaneous elastic re- sponse and long-time viscous behavior. This can be thought of as where the thickness h can vary along flow. Hydrostatic pressure a spring and dashpot in series, so that the same stress acts across provides an applied load, each element and the total strain is the sum of the elastic and viscous strains, leading to the differential equation q = ρw g(η − w), (3) σ˙ = ˙ − E σ = ∂ − E σ , wherever the vertical displacement of the shelf differs from the E ν E xu ν (10) tidal height η. The vertical displacement calculated by the flex- 2 2 ural model comprises part of the forcing of the flowline model (e.g., Turcotte and Schubert, 2002) where E is Young’s modulus described below. and the viscosity ν, given by Glen’s flow law, is assumed to change To derive the perturbed flowline model (following Walker et al. sufficiently slowly that the viscosity of the background state can be (2012)), we first remove the assumption of viscous rheology from used in calculating the perturbations. The final perturbed momen- the Dupont and Alley (2005) momentum equation to obtain tum equation is obtained by substituting (7), (9), and (10) into (6): σ˜ 1 2 ∂x 2hσ − ρ gh = Pb ∂xb + τb(u), (4) ∂ 2hE ∂ u˜ − 2 x x ν 2 where u is velocity and σ is along-flow deviatoric stress. The = ρ g ∂ (η − w)∂ (η − z − w) w t x b along-flow component of basal pressure Pb ∂xb (which results from + (η − zb − w)∂x∂t (η − w) depth-integration) is the product of the pressure Pb (cryostatic or β2 hydrostatic) and slope at the bottom surface of the ice, while the ˜ 1 −1 ˜ 2 ˜ 1 + (u + u) m ∂t u + ∂t β (u + u) m , (11) basal drag under grounded ice is given by the power-law bed rhe- m τ ( ) = β2 1 = ology b u u m , which can range from linear viscous (m 1) where advective terms from the total time derivative in (10) are →∞ β2( , ) to perfectly plastic (m ) with the drag coefficient x t typ- negligible.