Basal channels on ice shelves



[1] Recent surveys of floating ice shelves associated with Pine Island Glacier (Antarctica) and Petermann Glacier (Greenland) indicate that there are channels incised upward into their bottoms that may serve as the conduits of meltwater outflow from the sub-ice-shelf cavity. The formation of the channels, their evolution over time, and their impact on ice-shelf flow are investigated using a fully-coupled ice-shelf/sub-ice-shelf ocean model. The model simulations suggest that channels may form spontaneously in response to meltwater plume flow initiated at the grounding line if there are relatively high melt rates and if there is transverse to ice-flow variability in ice-shelf thickness. Typical channels formed in the simulations have a width of about 1–3 km and a vertical relief of about 100–200 m. Melt rates and sea-water transport in the channels are significantly higher than on the smooth flat ice bottom between the channels. The melt channels develop through melting, deformation, and advection with ice-shelf flow. Simulations suggest that both steady state and cyclic state solutions are possible depending on conditions along the lateral ice-shelf boundaries. This peculiar dynamics of the system has strong implications on the interpretation of observations. The richness of channel morphology and evolution seen in this study suggests that further observations and theoretical analysis are imperative for understanding ice-shelf behavior in warm oceanic conditions.

1 Introduction

[2] The ice shelf at the seaward margin of Pine Island Glacier (PIG) in the Amundsen Sea embayment (Antarctica) has melt rates on the order of tens of meters per year [Pritchard et al., 2012; Payne et al., 2007; Jacobs et al., 2011] due to interaction with warm Circumpolar Deep Water (CDW) [Jacobs et al., 1996]. The floating tongue of Peterman Glacier (Greenland) experiences melt rates of a similar order [Rignot and Steffen, 2008] due to interaction with warm modified Atlantic water [Johnson et al., 2011]. Inverted channels have been observed in the basal surface of both ice shelves [Rignot and Steffen, 2008; Bindschadler et al., 2011; Vaughan et al., 2012]. On Petermann Glacier, the channels are aligned with the direction of ice flow [Rignot and Steffen, 2008]; on PIG, they are both aligned with and transverse to the direction of ice flow [Bindschadler et al., 2011; Mankoff et al., 2012; Vaughan et al., 2012]. The presence of these channels at the base of ice shelves can influence their stability in two ways: First, ice thickness can be significantly reduced at the crests of the channels (where the ice-shelf bottom is shallower) due to concentrated melting. This creates zones of weakness and promotes fracture development. Rignot and Steffen [2008] report that the ice-shelf surface elevation of only 8 m at the crest of a melt channel. Second, spatial variations in ice thickness tend to promote fracturing and crevassing as a result of enhanced flexural stresses [Sergienko, 2010; Vaughan et al., 2012; MacAyeal and Sergienko, 2013].

[3] Gladish et al. [2012] provide insight into the mechanism that may govern the formation of channels observed on Petermann Glacier. They have demonstrated that channels can form in the presence of topographic features upstream of the grounding line that introduce grooves in the ice bottom that subsequently initiate the channels. In contrast to Petermann Glacier, topographic features that can be easily related to the melt channels were not observed upstream of the PIG grounding line [Vaughan et al., 2012]. This suggests that there might be more than one mechanism that can cause melt channels in ice shelves and ice tongues.

[4] The present study aims to explore processes that may lead to formation of melt channels at the ice-shelf base. To accomplish this, an ice-shelf flow model is coupled to a mixed-layer plume model simulating ocean circulation under the ice shelf. This coupled model is applied to an idealized geometry, in order to identify and explore fundamental aspects controlling melt channel dynamics.

[5] Exploratory numerical simulations presented here are intended as a first step that motivates further observational, numerical, and theoretical studies. This is because sub-ice-shelf ocean cavities remain largely the most remote and inaccessible parts of the world ocean. Traditional observational techniques in oceanography cannot be undertaken below hundreds of meters of ice to give a consistent, comprehensive picture of ocean circulation conditions. As a result of this inaccessibility, exploratory numerical modeling has become a necessary tool in the investigation of ice-shelf ocean interaction. Numerical modeling provides a relatively well-defined platform, albeit one not without flaws, for experiments that can delineate ranges of behavior and explore parameter spaces that guide and inspire more refined specific observations and focussed theoretical analysis. It is in this spirit that the coupled ice-shelf/sub-ice-shelf cavity model presented here will be used to explore possible interactions between processes that governs coevolution of ice shelf and ocean flows.

[6] The manuscript is organized into six sections and an Appendix: It starts with the coupled model description, followed by the investigation of the ice-shelf/sub-ice-shelf cavity system behavior in the absence of the Coriolis force under different glaciological and oceanic conditions. The effects of the Coriolis force are investigated next. The following section presents results of analysis of the effects of melt channels on the ice-shelf stress regime. The major results and their implication are summarized in the last two sections. Numerical aspects of this study are discussed in Appendix A.

2 Model Description

[7] The idealized model geometry is shown in Figure 1. The dimensions of the domain are 80 km long, 50 km wide, and chosen to be similar to those of the PIG floating ice shelf. All boundaries are fixed during simulations.

Figure 1.

Model geometry. Ice flows from the grounding line toward ice front (indicated by the red arrow).

2.1 Ice-Flow Model

[8] The ice-shelf flow model is a widely used Shallow Shelf Approximation model [e.g., MacAyeal, 1989]. Ice velocity, inline image, with components u along and v across the flow, respectively, is independent of the vertical coordinate z and obey the vertically integrated momentum-balance equations

display math(1a)
display math(1b)

where H is ice thickness, ν is ice viscosity, ρi is ice density, ρw is sea-water density (assumed to be constant in the ice model), g is acceleration due to gravity, and subscripts x and y denote partial derivatives with respect to x and y. The ice viscosity ν is assumed to be strain-rate dependent in accordance with Glen's flow law

display math(2)

where inline image is the vertically-averaged ice stiffness (assumed to be constant 1.68·108 Pa inline image, that corresponds to −15°C, MacAyeal [1989]), inline image is the second invariant of the strain-rate tensor, and n is Glen's flow law exponent (assumed to be 3).

[9] Ice thickness evolution is governed by the mass-balance equation

display math(3)

where inline image is melting/freezing rate at the base of the ice shelf (positive for melting). In this study, we disregard ablation/accumulation at the upper ice-shelf surface due to the fact that its characteristic magnitude is much less than that of basal melting.

2.1.1 Boundary Conditions

[10] As shown in Figure 1, ice flows from left to right. At the upstream (left) boundary, a uniform ice thickness H0=1400 m is prescribed (unless specified otherwise). The velocity at this boundary is prescribed as well, with components given by

display math(4a)
display math(4b)

where u0 is the velocity magnitude at the corners of the grounding line, uc is the velocity magnitude at the center of the grounding line, taken to be 1000 m yr−1, and Ly is the half width of the domain (25 km). The shape of this profile is chosen to be similar to the velocity distribution in confined ice streams [Raymond, 1996]. The constant u0 specifies slip conditions at the lateral boundaries: it is zero for no-slip conditions, 610 m yr−1 for partial-slip conditions with the lateral shear, τs≠0. In the case of free slip (described below), τs=0 and u=u0=1000 m yr−1.

[11] At the ice-front boundary, stress conditions are prescribed. The vertically integrated normal stress balances vertically integrated sea-water hydrostatic pressure, and the shear stress is absent, i.e.,

display math(5a)
display math(5b)

[12] Along the lateral boundaries, two types of conditions are considered. First is the no-slip condition with u and v being zero. Second is a slip condition fashioned after a Weertman-type law

display math(6a)
display math(6b)

The lateral shear τs is prescribed, where C is the sliding law constant and m is the sliding law exponent, taken to be m=3 [Weertman, 1957; Fowler, 1981]. The constant C is either 9·106Pa inline image or 0. Simulations with these sliding boundary conditions result either in partial or free slip of the ice shelf along its lateral boundaries.

2.2 Plume Model

[13] The plume model for sub-ice-shelf ocean circulation was originally developed by Holland and Feltham [2006]. Here it is used in a modified form (excludes formation of the frazil ice) and is identical to the configuration used by Payne et al. [2007]. The model represents the ocean as a two-layer system with an immobile, deep ambient layer with horizontally uniform stratification beneath a buoyancy-driven plume layer which evolves according to the momentum, mass, energy, and salt balance equations

display math(7a)
display math(7b)
display math(7c)
display math(7d)
display math(7e)

Here D is the plume thickness, inline image is the velocity of the plume layer, ρ is density, T is temperature, and S is salinity of the plume. Entrainment is given by inline image. A is the level of the interface between the plume and the ambient ocean layer. Kh is a horizontal eddy viscosity (taken as 10 m2s−1). ρ0 is the reference density. inline image is the reduced gravity; ρw is plume density; ρa, Ta, and Sa are the density, temperature, and salinity of the ambient ocean representing the lower layer of the ocean, respectively. γt is the heat transfer coefficient. f is the Coriolis parameter, which is either set to zero or is f−1.415· 10−4s−1, corresponding to 76°S—the location of the PIG floating tongue. The entrainment rate inline image is parametrized as in Holland et al. [2007] and Payne et al. [2007], and the melt rate inline image is computed using the three-equation formulation describing conservation of energy and salt at the ice-ocean interface, along with a linearized freezing temperature relation [Holland and Jenkins, 1999]:

display math(8a)
display math(8b)
display math(8c)

where cw and ci are the specific heat capacities of sea water and ice, respectively; T* and S* are equilibrium temperature and salinity at ice/water interface, L is the latent heat of ice, Ti is interior ice temperature (−15°C), p is the pressure at the ice/water interface, and a, b, and c are empirical constants. The expressions for the heat and salt transfer coefficients are

display math(9a)
display math(9b)

where inline image is the friction velocity, cd is a drag coefficient, νw is the sea-water viscosity, and Pr and Sc are the Prandtl and Schmidt numbers, respectively. All constant values are the same as in Holland and Jenkins [1999].

[14] Gladish et al. [2012] use the same plume model with a different formulation of the entrainment rate inline image. Similar to their study, a minimum value of 10 cm for the plume thickness is imposed in this study. Reduction of this minimum value to 1 cm does not alter the results presented below.

2.2.1 Boundary Conditions

[15] At the grounding line and at the lateral boundaries, the plume flow is prescribed to be free slip as follows:

display math(10)

At the ice front, an open boundary condition is approximated by imposing

display math(11)

where F is a corresponding field of the plume model, F={D,U,V,T,S}.

[16] The behavior of this plume model has been explored in a number of uncoupled studies investigating sub-ice-shelf cavity circulation [e.g., Holland et al., 2007, 2009]. Payne et al. [2007] have performed a sensitivity analysis of the melt rates calculated with this model to various parameters. In this study, the plume model parameters are the same as in the control run of Payne et al. [2007].

2.2.2 Ambient Stratification

[17] The ambient ocean stratification is based on the stratification observed in front of the PIG ice shelf [Jacobs et al., 2011]. It is described by the following approximation

display math(12a)
display math(12b)

The subscript sw refers to the surface water with temperature Tsw=−1.9°C and salinity Ssw = 33.5‰, and the subscript bw refers to the bottom water salinity Sbw=34.69‰ and temperature Tbw=1.2°, unless indicated otherwise. Finally, Dsw=−300m is the depth of the bottom of the surface layer and Dtc=−800m is the bottom of the thermocline. All numerical aspects of the models and simulations are discussed in Appendix A.

3 Channel Formation

[18] In order to focus on processes that lead to channel formation, we begin with simulations that disregard the effect of the Coriolis force, and set the Coriolis parameter f to 0. In this circumstance, the behavior of the coupled ice-shelf/sub-ice-shelf cavity system is symmetric with respect to the centerline of the domain. All results are shown at steady states (see Appendix A for details) unless indicated otherwise.

[19] Several points about the ice-shelf/sub-ice-shelf cavity system behavior need to be emphasized before proceeding with an exploration of the conditions that lead to melt channel formation. First, the ice shelf and plume mutually affect each other. The morphology of the ice-shelf base determines how fast the plume flows: the steeper the slope, the faster the plume flows (equations (7b) and (7c)) [e.g., Holland and Feltham, 2006; Gladish et al., 2012; Goldberg et al., 2012a]. As the plume velocity increases, its ability to turbulently entrain underlying warm seawater increases, and thus the basal melting rate increases (equations (8a) and (9a)) [Holland and Jenkins, 1999; Holland et al., 2007, 2009]. This increased entrainment of warm underlying water increases melting and gives rise to even greater plume buoyancy, hence initiates a positive feedback between steep slopes, fast plume flow and enhanced melt rates. Second, any morphological feature presented at the ice-shelf base is advected and deformed with ice-shelf flow, and therefore does not preserve its original shape.

[20] Formation of the melt channels is prohibited when all characteristics of the ice-shelf/sub-ice-shelf cavity system (e.g., ice velocity and thickness, plume velocity, thickness, temperature, and salinity) are transversely uniform. This is due to the fact that melt channels are formed as a result of spatial heterogeneity of melt rates. However, melt rates are transversely uniform when all other characteristics are uniform. Such transversely homogeneous flows occur in circumstances where the ice shelf experiences free slip on its lateral boundaries. Simulations with free-slip boundary conditions for ice-shelf flow confirm this assessment. As Figure 2 shows, all characteristics of the ice-shelf/sub-ice-shelf cavity system are transversely homogeneous.

Figure 2.

Configuration of the ice shelf experiencing free slip at the lateral boundaries. Color shows melt rates (m yr−1). Contour lines show ice thickness with 100 m intervals. Arrows show plume velocity vectors.

3.1 Channel Formation Due to Preexisting Topographic Features

[21] The transverse heterogeneity of ice-shelf thickness can originate from ice flow over undulated topography upstream of the grounding line. Gladish et al. [2012] have explored melt channel formation under these circumstances. Figure 3a shows the results of simulations with free-slip conditions at the lateral boundaries and with periodic perturbations in ice thickness prescribed at the grounding line. The ice thickness perturbation, δH is

display math(13)

where b=50 m is the perturbation amplitude, inline image and inline image are wave numbers corresponding to four and eight undulations across the span of the grounding line. The shape and amplitude of the perturbations are arbitrary and similar to those used by Gladish et al. [2012] with the only difference being the number of smaller undulations (8 instead of 12). There are several things to notice in the results shown in Figure 3a. The first, an obvious one, is the presence of the melt channels. The mechanism of their formation is described by Gladish et al. [2012]—plume flow accelerates in the locations where local basal slopes are steep, hence causes locally enhanced melting that makes preexisting undulations even deeper. The deepened channels are advected downstream with the ice-shelf flow, which is essentially a plug flow downstream of the grounding zone (~10 km near the grounding line), as a result of the free-slip conditions at the lateral boundaries. The second feature to notice is that the shape of the channels is not identical to the shape of the perturbations. Figure 3b shows the transverse ice-draft profile along the transect at inline image km. The channels (blue curve) have amplitudes that are fairly similar in all channels, and their wavenumber is close to k8, in contrast to the perturbation shape (green curve) in which undulations have significantly different amplitudes. The smaller undulations are deepened while the larger undulations are shallowed. Thus, in circumstances where the melt channels are caused by the preexisting topographic features upstream of the grounding line, they do not preserve the shape and amplitudes of these topographic features.

Figure 3.

Configuration of the ice shelf with ice-draft perturbation at the grounding line (equation (13)) experiencing free slip at the lateral boundaries. (a) Color shows melt rates (m yr−1). Contour lines show ice thickness with 50 m intervals. Arrows show plume velocity vectors. (b) Blue line shows the transverse profile of the ice draft at x=40 km; green line shows the shape of the ice-draft perturbation at the grounding line (equation (13)) shifted to the level of the ice draft at x=40 km.

3.2 Channel Formation Due to Shear at the Ice-Shelf Lateral Boundaries

[22] In circumstances where shear at ice-shelf lateral boundaries is nonnegligible, its flow, hence flux, and as a result ice thickness are transversely nonuniform. The greater the lateral shear at the sides, the larger the difference between the horizontal velocity at the centerline and at the edge. As the magnitude of lateral shear increases, the basal draft difference also becomes larger between the centerline of the ice shelf and the edge. This, ice dynamics related, transverse variability is present on ice shelves regardless of whether melting or freezing occurs at their bases. Figure 4 shows the transverse ice-draft profiles along the transect at inline image km in the absence of basal melting or refreezing for ice shelves with different amounts of lateral shear. The shape of the ice-shelf draft progressively steepens as the lateral shear at the boundaries increases, and is steepest when the ice shelf experiences no-slip conditions at its lateral boundaries.

Figure 4.

Transverse profiles of ice-shelf drafts experiencing different slip regimes at the lateral boundaries and no melting/refreezing at their bases. The transect is taken at inline image.

[23] The transversely nonuniform ice-draft results in a speedup of plume flow where basal slopes are high. This means that there is the possibility of enabling a positive feedback between steep slopes, plume speed, and melt rates, as described above. The following simulation explores this feedback. Figure 5 and Movies 1–3 of the supporting information illustrate the initiation and time evolution of melt channels arising spontaneously where there is only a lateral ice thickness gradient generated by lateral shear in the ice-shelf velocity profile. A melt channel starts to form at a location where the thinning rate, Ht, is substantial and is of the same order of magnitude as other terms in the ice-shelf mass balance, equation (3) (Figure 5d). This initial undulation continues to deepen as a result of the positive feedback between the ice-shelf slope, fast plume flow, and stronger melting. Simultaneously, the undulation is elongated due to advection with ice-shelf flow. The presence of this undulation causes additional ice-shelf deformation, hence, changes the shape of the ice-shelf base, and promotes initiation of other undulations that develop into channels, leading to spontaneous channelization at the ice-shelf base. Melt rates and thinning rates are not equivalent to each other (Figures 5a–5f), and their patterns evolve differently through the course of the channel development. At the beginning of the simulations (Figures 5a, 5b, 5d, and 5e), when the ice-shelf shape and its flow are still close to their initial steady state configurations, the thinning rates reflect melt rate patterns. However, at later times, when the ice shelf adjusts to its evolving state, the thinning rate patterns reflect ice-shelf advection patterns (Figures 5c and 5f).

Figure 5.

Melt channel formation on the ice shelf with no slip at the lateral boundaries. Ice flows from the grounding line toward ice front. (a–c) The color shows melt rates (m yr−1); (d–f) the color shows thinning rates, Ht (m yr−1). Contour lines show ice thickness with 50 m intervals. Arrows show plume velocity vectors.

[24] The channel patterns have a structure that features several distinct deep channels (~200 m) aligned in the direction of ice-shelf flow (x axis) and a number of smaller channels (~30 m) that initiate near the ice plain oriented with ~30–40° angle to x axis. The latter channels appear to be similar to ones described by Bindschadler et al. [2011] and Vaughan et al. [2012]. In the present study, however, they are not related to seasonality of the sea-water heat content, as was suggested by Bindschadler et al. [2011] but which is not supported by the recent observations [Walker et al., 2013]. The mechanism for formation and evolution of these channels is the same as for the along-flow oriented channels. They form between the along-flow oriented deep channels as a result of enhanced local slopes. In the regions where transverse channels form, the across-flow (y) velocity component of ice-shelf flow attains its extreme values (Figure 6a) and has magnitudes comparable to the x velocity component. Therefore, these channels while advecting with ice-shelf flow propagate at an angle to x axis.

Figure 6.

The transverse velocity component of ice-shelf flow, v (m yr−1), for different slip conditions at the lateral boundaries. (a) No slip at t=700 years; (b) partial slip at steady state.

[25] It is important to note that the channel morphology evolves into a equilibrium cyclic state where the ice-shelf/sub-ice-shelf cavity system experiences similar states with periodicity of ~30–35 years. The periodicity is close to the time required for an ice column to travel from the ice plain (Figures 5c and 5f) to locations where melting is suppressed as a result of the shallow ice-shelf base (~20–30 km from the ice front). The temporal evolution of these melt channels is driven by internal dynamics of the ice-shelf/sub-ice-shelf cavity system. This temporal variability happens in the absence of variations in external forcing, and so represent an internal mode of an ice-shelf/ocean system.

[26] The channels act as pathways for the plume flow from the grounding line toward the ice front. The plume water is a mixture of melt water and ambient ocean water, and is thus colder and fresher in the channels than outside them. This cold and fresh water is efficiently delivered through channels from the cavity interior to the ice front as observed at the PIG front [Mankoff et al., 2012]. As Figure 7 shows, the plume speed and thickness in the channels are significantly larger than outside them. The strongest flow takes place near the lateral boundaries where the ice-shelf base slopes are the largest (Figure 7a). As a result, the plume flux there is almost 2 orders of magnitude larger than the area average (Figure 8). Such a channelized outflow of the plume water may impact ocean circulation and water mass properties in the open ocean adjacent to the ice-shelf cavity and may provide an important observational constraint on what is happening below the ice shelf where direct observations are logistically difficult.

Figure 7.

Plume flow at t=700 years: (a) plume speed, inline image (m s−1); (b) plume thickness, D, (m).

Figure 8.

Melt channel configurations for different slip conditions at the lateral boundaries (a and c) no slip, at t=700 years; (b and d) partial slip with velocity 60% of the centerline velocity, steady state. (a and b) the color shows melt rates (m yr−1), contour lines show ice thickness with 50 m intervals. Arrows show plume velocity vectors. (c and d) color shows plume flux (m2 s−1) contour lines show ice thickness with 50 m intervals.

3.2.1 Sensitivity to Lateral Shear

[27] Two cases of the ice-shelf lateral boundary conditions considered so far—no-slip and free-slip conditions—represent the two end-members of a range of possible slip conditions. As Figure 4 shows, the amount of slip at the lateral boundaries determines the curvature of the ice-shelf draft, and this, in turn, determines the channel formation. Sensitivity experiments show that the melt channels do not form spontaneously if the ice-shelf slip velocity at its lateral boundaries exceeds 70% of its centerline value. Figure 8 shows the results of simulations with the ice-shelf slip velocity ~60% of its centerline velocity. The number of distinct channels is significantly reduced, and the amplitudes of these channels are smaller compared to the no-slip simulations (Figure 5). In the simulations with slip at the boundaries, the ice-shelf/sub-ice-shelf cavity system achieves steady state as opposed to a cyclic state achieved when there is reduced or no slip at the lateral boundaries. This is due to the fact that the transverse variability in the ice shelf caused by partial slip at the lateral boundaries is not as strong as in the no-slip case, and the ice-shelf/sub-ice-shelf cavity system is capable of achieving its time-independent steady state after approximately 100 years (see Appendix A for details). In the case of partial slip at the lateral boundaries, melt channels transverse to the direction of ice-shelf flow do not form. In this case, magnitudes of the transverse velocity of ice-shelf flow are at most half of those in the no-slip case (Figure 6).

3.2.2 Sensitivity to Ocean-Bottom Temperature

[28] In order to test whether melt channels can form when the bottom ocean layer is colder, two additional simulations have been done with Tbw=−1.3°C and Tbw=−1.7°C. Figures 9a–9c show that melt channels still form in the Tbw=−1.3°C simulations, with average melt rates of ~5–6 m yr−1; but do not form in the Tbw=−1.7°C case, where melt rates are significantly less, ~2–3 m yr−1. Therefore, in addition to the transverse nonuniformity of the ice-shelf draft, a necessary condition for the channel formation is “sufficiently strong” basal melting, or sufficiently warm ocean temperatures that allow to produce such melt rates. In order to quantify the level of “sufficiently strong,” we estimate magnitudes of different terms in the ice-shelf mass balance, equation (3). Figures 9d–9i show the magnitudes of the advection term, uHx+vHy, and the deformation term, H(ux+vy). In the Tbw=−1.7°C simulation, these two terms are of the same order of magnitude and balance each other, while melt rates are significantly smaller compared to them. In contrast, in the Tbw=1.2°C simulation, the deformation contribution to the mass balance is negligible compared to other terms, and the leading order balance is between the advection term and melt rate in the vicinity of the ice plain, and between the advection term thinning rate Ht over the rest of the ice shelf. Thus, “sufficiently strong” melting means that melt rates have to be of the same order as other terms of the ice-shelf mass balance.

Figure 9.

Various terms of the mass balance of ice shelves (equation (3)) with no slip at the lateral boundaries in different ocean environments: (a, d, g) at t=700 years, (b, c, e, f, h, i) steady state. (a–c) The color shows melt rates, (m yr−1), contour lines show ice thickness with 50 m intervals. Arrows show plume velocity vectors. (d–f) the ice-shelf advection, uHx+vHy (m yr−1). (g–i) The ice-shelf deformation, H(ux+vy) (m yr−1). Notice different color scales on different panels.

[29] The configurations of the ice-shelf/sub-ice-shelf cavity obtained in simulations with colder ocean temperatures differ from the states obtained in simulations with warmer ocean temperature in many ways. The number of channels and their depth are significantly reduced in simulations with colder ocean temperatures. Also, in the simulations with colder ocean temperatures, the ice-shelf/sub-ice-shelf cavity system attains the time-invariant steady states. The reason for this is that, in colder oceans, melt channel undulations have smaller amplitudes than in a warmer ocean. This makes the ice-shelf configuration less sensitive to the presence of channels, and the ice-shelf/sub-ice-shelf cavity system is able to reach a time-invariant steady state.

[30] It should be pointed out that the same considerations as used above to quantify the strength of melting determine a degree of coupling between an ice-shelf and sub-ice-shelf cavity models required to investigate ice-shelf/ocean interaction processes. As Sergienko et al. [2013] show, such investigations can be done using uncoupled models if the dominant ice-shelf mass balance is between ice advection and deformation, as in the case of Tbw=−1.7°C in this study. However, in all other circumstances coupling between ice-shelf and sub-ice-shelf cavity models are required.

4 The Effect of the Coriolis Force

[31] The effect of the Coriolis force on channel formation was investigated using a coupled model simulations with the Tbw=1.2°C and both the no-slip and partial-slip boundary conditions. The Coriolis parameter f in equations (7b)(7c) was taken to be −1.415· 10−4s−1, the value corresponding to 76°S, an approximate location of the PIG ice shelf. Figures 10a and 10c and Movies 4–5 of the supporting information show the results of these simulations. The Coriolis force has a strong effect on the alignment of the channels. In contrast to simulations described above, where the along-the-flow channels are parallel to the lateral boundaries, (Figures 8c and 8d) the Coriolis force diverts the channels to the left, and the strongest plume flow takes place along the left boundary of the domain. The Coriolis force also imposes an asymmetric structure to the channels. Their left sides are steeper than their right sides. The orientation of the transverse channels is affected by the Coriolis force as well, they are not symmetric with respect to the centerline (as in Figures 5c and 5f). A reason for such asymmetry is enhanced melting caused by acceleration of plume on the Coriolis favorite side.

Figure 10.

The effects of the Coriolis force. (a, c) no slip at the later boundaries at t=1000 years; (b, d) partial slip at the lateral boundaries (60% of the centerline velocity) at steady state. (a) and (b) The color shows melt rates (m yr−1), contour lines show ice thickness with 50 m intervals; (c) and (d) the color shows plume flux (m2 s−1), contour lines show ice thickness with 50 m intervals.

5 The Effect of Melt Channels on the Ice-Shelf Stress Regime

[32] The effect of melt channels on the ice-shelf stress regime has been assessed by comparing the effective stress (the second invariant of the deviatoric stress-tensor) in ice shelves with and without melt channels in the absence of the Coriolis force with Tbw=1.2°C. Ice-shelf configurations without melt channels have been computed using the imposed width-averaged melt rates (insets in Figures 11b and 12b). Figure 11a shows the effective stress in the ice shelf with no slip at its lateral boundaries computed in the coupled simulation at t=700 years. As a result of no slip at the lateral boundaries, the effective stress has large magnitudes (~200 kPa) along these boundaries. The largest stresses ~300 kPa are attained at locations where ice-shelf slopes are the largest. These stresses of the same magnitudes or smaller than those inferred from observations on the PIG and Petermann glaciers (Figure S1), where the effective stresses near the lateral boundaries are ~350–400 kPa. The general pattern of effective stress in the ice shelf with melt channels is similar to that of the ice shelf without melt channels (Figure 11b), where effective stress has large magnitudes at the lateral boundaries and gradually reduces toward the ice-shelf center line. However, there are substantial differences in the effective stress patterns caused by melt channels. As Figure 11c shows, effective stress is elevated ~10–20% in parts of the ice shelf that are immediately over melt channels. The largest difference in effective stresses (~50%) is observed near the grounding line and at the onset of longitudinal melt channels. Stress magnitudes over the longitudinal channels close to the ice-shelf lateral boundaries are sufficiently large to cause ice to fracture and develop crevasses. The strong spatial variations in ice-shelf stresses in the vicinity of melt channels will also contribute to fracture initiation and development. Similar enhanced stresses (~150–200 kPa) are observed over the channels on the PIG and Petermann glaciers (Figure S1) where surface crevasses are abundant.

Figure 11.

The effects of melt channels on the ice-shelf stress-state in simulations where there is no slip at the lateral boundaries. (a) Effective stress (kPa) at t=700 year; (b) effective stress (kPa) in the ice shelf with imposed width-averaged melt-rate distribution shown in the inset; (c) ratio of effective stresses shown in Figures 11a and 11b.

Figure 12.

The effects of melt channels on the ice-shelf stress-state in simulations where there is partial slip at the lateral boundaries. (a) Steady state effective stress (kPa); (b) effective stress (kPa) in the ice shelf with imposed width-averaged melt-rate distribution shown in the inset; (c) ratio of effective stresses shown in Figures 12a and 12b.

[33] In the case of partial slip at the ice-shelf lateral boundaries, the effects of melt channels on the ice-shelf effective stress are of comparable magnitudes to the no-slip case: ~10–20% increase compared to the ice-shelf configuration with no melt channels. However, effective stress over melt channels has lower magnitudes than the stress in the ice shelf without melt channels in the ~20–30 km zone near the ice front (Figure 12c). This is in contrast to the no-slip case, where effective stress over melt channels is larger than effective stress with no melt channels through the whole length of the ice shelf (Figure 11c). Overall, as expected, the magnitudes of effective stress are lower in the partial-slip case compared to the no-slip case, regardless of the presence or absence of the melt channels.

[34] The above stress analysis should be considered as an “order of magnitude” assessment. The ice-shelf configurations computed with the imposed width-averaged melt rates have different ice thickness and velocity distributions than those obtained in the coupled simulations. Therefore, the direct comparison of the stresses at a specific location in the ice shelf can be misleading.

6 Potential Implications for Interpretation of Ice-Shelf Observations

[35] As this study shows, melt channels form in response to the presence of the transverse variability in the ice-shelf configuration. One possible origin of such transverse variability is the presence of topographic features upstream of the grounding line. Gladish et al. [2012] attribute formation of the melt channels observed on Petermann Glacier to such features.

[36] Another origin of the ice-shelf transverse variability demonstrated by this study is strong shear at an ice shelf's lateral boundaries. In this circumstance, melt channels can form spontaneously without preexisting conditions in the grounding-line topography. The absence of pronounced features in observations of the basal topography upstream of the PIG grounding line [Vaughan et al., 2012] and the presence of the strong lateral shear on its lateral boundaries (Figure S1) suggest that melt channels observed on the PIG floating tongue are possibly formed spontaneously by the mechanism involving lateral shear.

[37] The channels produced in this study are a product of dynamical coupling of the ice-shelf flow and the sub-shelf ocean circulation and serve at the very least to indicate how this coupling could create complexities (i.e., periodic states and migration) that are yet to be accounted for in designing field observation strategies. The plume flow that causes melting of the ice shelf is controlled by the ice-shelf basal morphology which depends, in turn, on the ice-shelf's lateral thickness gradient that is determined by boundary conditions. Dynamic interaction of the ice-shelf and plume flow results in a highly variable spatial distribution of melting that manifests itself as self-organized melt channels. In some circumstances, as in the simulations with no-slip at the ice-shelf lateral boundaries, the ice-shelf/plume system settles into a cyclic state. This is notable, because it suggests that temporal variability in the basal melting rates and ocean circulation can be independent of external forcing.

[38] These results allow for speculation that some of the dramatic changes reported for Pine Island Glacier and its ice shelf [e.g., Shepherd et al., 2004] might have internal triggers (e.g., interaction of one of the melt channels with ocean bed topography [Jenkins et al., 2010]) in lieu of, or in addition to, external causes usually attributed to warming of the ocean waters and consequent enhanced melting of the ice shelf. This speculation has indirect support by observations reported by Jacobs et al. [2011] of a relatively small, 6% increase of the CDW temperature between 1994 and 2009. A warming of this small level may be insufficient to cause the significant changes observed on the PIG ice shelf. A possible chain of events could be the following: Interaction of one of the melt channels with the ridge discovered by Jenkins et al. [2010], on which PIG was grounded would lead to ungrounding of the ice shelf that would set off changes in the ice-shelf configuration, that would result in the grounding line retreat and further ice shelf thinning and changes in the ice-shelf cavity.

[39] The model results have important implications to be mindful of when interpreting available observations. Short-term and spatially sparse observations of ice-shelf thinning (e.g., surface elevation changes along satellite tracks [e.g., Pritchard et al., 2012]) could be caused by advection of melt channels that is accompanied by changes in ice-shelf thickness. Figure 13a shows a snapshot of the ice-shelf thinning/thickening rate, Ht, the first term on the left-hand side of equation (3). The magnitude of the thinning/thickening rate achieves tens of meters per year and reflects ice-shelf thinning due to both melting and ice-shelf advection and deformation. Snapshot and short-term observations of ice-shelf surface elevation taken at fixed sparse locations, e.g., along satellite tracks or flight lines (imitated by black lines in Figures 13a and 13c), thus, provide neither comprehensive temporal nor spatial patterns of ice-shelf thinning rates. As examples in Figure 13 show, snapshots taken 20 years apart along the black lines and consequently smoothed and interpolated on the 10 km grid (similar to a procedure performed by Pritchard et al. [2012]) have significantly different spatial patterns, none of which reflect the long-term evolution of the thinning rates (Figures 13b and 13d).

Figure 13.

Rate of ice thickness change Ht (m yr−1). (a and b) At t=500 years, (c and d) at t=520 years in the presence of the Coriolis force and with no slip at the ice-shelf lateral boundaries. Figures 13a and 13c show thinning rates over the whole ice shelf, black lines emulate repeated satellite tracks or flight lines. Inset shows values of Ht along the black lines. Figures 13b and 13d show thinning rates spatially smoothed and averaged.

[40] These examples, as well as results of the presented simulations, illuminate the following issues of the long-term monitoring of ice shelves experiencing strong melting. First is the spatial resolution of airborne and/or in situ observations. As the presented simulations show, ice shelves with melt channels experience spatial variability on small length scales. Similarly, the ice-penetrating radar survey carried out by Vaughan et al. [2012] revealing detailed structure of melt channels on the PIG was done on a dense spatial grid, ~500 m between the survey lines. In order to establish evolution of melt channels, either ground based or airborne surveys need to be continued with the same spatial resolution for the next 10–20 years (an approximate residence time of an ice column in the PIG ice shelf). Second, such surveys need to be accomplished in a fairly short time (e.g., during one field season). This is due to the fact that PIG flows ~4 km yr−1, hence every morphological feature is advected (and deformed) with ice flow at the same rate. Therefore, in order to estimate thinning rates in the Eulerian (fixed) coordinate frame (e.g., Dutrieux et al, under review in The Cryosphere Discussions) ice elevation or thickness needs to be observed at the same time on the whole ice shelf. A similar analysis in the Lagrangian (moving) coordinate frame requires precise knowledge of the ice velocities at the same time when ice elevation is observed.

7 Conclusions

[41] The presented results demonstrate that melt channel formation is a natural consequence of transverse variability of ice-shelf thickness at the grounding line. This variability can be a product of basal topography upstream of the grounding line or equally a product of lateral velocity variations caused by shear at the ice-shelf lateral boundaries. When generated by topography upstream of the grounding line, melt channels do not preserve the shape of these features. Where lateral shear at the ice-shelf boundaries generates the melt channels, they can form spontaneously, and the morphology of the melt channels is controlled by magnitudes of the lateral shear. Another necessary condition of the melt channel formation is relatively strong basal melting with magnitudes comparable to other components of the ice-shelf mass balance. None of these two conditions, however, is sufficient on their own to initiate and maintain melt channels.

[42] The significant result of this study is the demonstration that melt channels signify strongly-coupled coevolution of the ice shelf and ocean circulation underneath. Internal dynamics of this system thus controls the evolution of the channels in a manner that is unrelated to variations in the far-field ocean and grounded ice. The presented simulations provide a first glimpse of the characteristics of this internal dynamics that motivates more detailed and focused studies. One of the first steps should be establishing spatial and temporal scales that govern evolution of the coupled system.

[43] With regards to observations, the findings of this study call for reevaluation of the traditional view that ice-shelf changes are entirely caused by changes in the ambient ocean. Despite the simplicity of individual ice-shelf flow and sub-ice-shelf cavity circulation, their coupled interaction allows for complexity and internally driven temporal variability on decadal scales that heretofore may not have been fully appreciated.

Appendix A: Numerical Aspects

[44] All simulations are done with a finite element solver COMSOL™. Both models run on the same finite element mesh with resolution ~400 m increasing to 40 m near the lateral boundaries. All boundaries are fixed during all simulations.

A1 Spin-Up

[45] Initial configurations for the coupled simulations are computed in the following way. The ice-shelf momentum-balance equations (1) and the steady state mass-balance equation (3), with zero first term on the left-hand side, are solved to obtain an initial ice-shelf configuration with a prescribed melt rate distribution (uniform in the transverse direction)

display math(A1)

where inline image m yr−1 for Tbw={−1.7,−1.3,1.2}°C, respectively. Both, the values of inline image and the shape of the melt rate distribution are arbitrary and do not affect the achieved cyclic and steady states.

[46] Using this ice-shelf configuration, the plume model is run for 10 days in a stand-alone mode. This run starts with a time step of 1 s and increased to 1 h by the end of 10 day run. After that, the resulting plume configuration is used as an initial guess to compute a steady state plume flow by solving equation (7) with zero first terms on the left-hand side (time-derivative terms). These steady state ice-shelf and plume configurations are used as the initial configurations for coupled transient simulations.

A2 Coupled Simulations

[47] In contrast to Gladish et al. [2012] and Goldberg et al. [2012a, 2012b] who use a segregation method, i.e., circulation in the ice-shelf cavity and ice-shelf flow are computed sequentially using their respective values for the ocean and ice models' time steps; in the coupled transient simulations performed here, all equations are solved simultaneously at each time step using a modified form of the damped Newton iteration method paired with a direct solver Multifrontal Massively Parallel Sparse direct Solver (MUMPS) to solve a linearized system of equations (COMSOLUser Guide). At each time step, the solution (computed variables) satisfy all equations simultaneously within a convergence tolerance (10−6). Time steps are automatically adjusted to satisfy the chosen convergence tolerance. A typical time step is ~10 days at the beginning of simulations and gradually increases to ~3 months in simulations that do not achieve a steady state (high melting and no-slip conditions at the ice-shelf lateral boundaries), and ~1 year when the solution is getting close to a steady state in all other simulations. Results of a 100 year simulation with a fixed time step (1 day) were identical to those with the adaptive time step. As expected, this simulation took significantly more time to complete.

[48] It should be realized that in the chosen model configuration (constant ocean stratification), the sub-ice-shelf cavity circulation changes only as a result of changes in the ice-shelf cavity shape. These changes occur on ice-flow time scales (a few months to a year) that are significantly longer than the time scales of adjustment of the sub-ice-shelf cavity circulation (5–7 days). Thus, at every time step, the cavity circulation is in steady state with the ice-shelf configuration. This has been confirmed by running a test simulation for 100 years, in which the plume model equations were solved in their steady state form, i.e., with the time-derivative terms (first terms on the left-hand side of equations (7)) set to zero.

[49] After each 100 years of simulation, the solution is tested on its closeness to steady state. To do this, an attempt is made to directly calculate a steady state solution (time derivatives dropped from all model equations) using the current solution as an initial guess. If this attempt fails, i.e., the solution does not converge within a tolerance (10−6), the time evolution is continued. In all simulations, apart from those involving no-slip conditions at lateral boundaries, 100 years of time-dependent simulations were enough to bring the ice-shelf/sub-ice-shelf cavity system close to steady state. In simulations with no slip and Tbw=1.2°C, steady state was not achieved after 1000 years of simulation. Instead, it exhibited a periodic behavior, with similar states occurring with a period of ~35 years.

A3 The Effects of Mesh Resolution

[50] In order to investigate robustness of the results, several coupled simulations have been done using meshes with different resolutions. These simulations are performed for the ice shelf with no slip and partial slip at the lateral boundaries. Simulations with no-slip boundary conditions run for 100 years with the mesh sizes 200 m and 800 m, and simulations with the partial-slip boundary conditions are run to their steady states (described above) with the mesh sizes 200 m, 800 m, and 1600 m (all previous simulations have been done with 400 m mesh).

[51] Figures A1a–A1c, and A2a–A2c show the spatial patterns of the channels that are similar in all simulations. They form in the same locations, however, they appear more diffuse (i.e., wider and shallower) in simulations with coarser mesh resolutions (Figures A1d and A2d). Despite dissimilarities in simulations with no slip at the ice-shelf lateral boundaries with coarser, 800 m, resolution, the general pattern is very similar to results of simulations with finer resolution (Figures A1a–A1c). It is notable that the smaller transverse channels form in all simulations with no slip at the lateral boundaries, at approximately the same locations and of the same size. These results indicate several points. First, the melt channels obtained in this study are physical, and not caused by numerical artifacts, because their spatial patterns are robust with respect to mesh refinement. Second, the chosen mesh resolution (400 m) is sufficient enough to adequately resolve melt channels and their dynamics, however, coarser resolution may not be sufficient. Third, simulations of sub-ice-shelf cavity circulation with stand-alone ocean models (a) need to be performed with fine spatial resolutions (less than 500 m) and (b) require knowledge of ice-shelf draft on the same spatial scales—500 m and less. Thus, the recent field campaign that yielded high resolution observations of the PIG floating tongue [Vaughan et al., 2012] has to become a routine survey in order to monitor evolution of ice shelves floating in warm oceans.

Figure A1.

The effects of mesh resolution in simulations where there is no slip at the ice-shelf lateral boundaries. (a–c) Color shows plume flux (m2 s−1); contour lines show ice thickness with 50 m interval. (d) Transverse profiles of ice-shelf draft (m) at inline image km. All fields are shown at t=100 years.

Figure A2.

The effects of mesh resolution in simulations where there is partial slip at the ice-shelf lateral boundaries. (a–c) Color shows plume flux (m2 s−1); contour lines show ice thickness with 50 m interval. (d) Transverse profiles of ice-shelf draft (m) at inline image km. All fields are shown in steady states.


[52] I thank two anonymous referees for their constructive criticisms and thoughtful suggestions, Carl Gladish for fruitful discussions, and Doug MacAyeal for help with the manuscript. This research is supported by NSF grants ANT-0838811 and ARC-0934534.