Sticky spots

The term 'sticky spots' has been proposed to describe localized sources of high basal resistance within an otherwise weak till bed (Alley, 1993). Although they represent a pretty logical theoretical construction and most likely do exist, the physical nature of sticky spots and their exact role in controlling soft-bedded ice stream motion are not constrained by observations. The best constraints on spatial distribution and physical nature of sticky spots come from passive seismic experiments described by Anandakrishnan & Bentley (1993) and Anandakrishnan & Alley (1993). Whillans et al. (1993) examined the strain rate field on parts of Whillans Ice Stream but failed to find clear evidence of sticky spots. Engelhardt and Kamb (Kamb, 2001) drilled an array of four boreholes in a location where geophysical data suggested the presence of a sticky spot (Rooney et al., 1987) but found that subglacial conditions in these boreholes were not distinguishably different from those revealed by boreholes drilled in other parts of the same ice stream.

Notwithstanding the limited success with direct investigations of sticky spots, analyses of ice-stream velocity data indicate that bed resistance has some level of spatial non-uniformity, at least over spatial scales longer than an ice thickness (MacAyeal et al., 1995; Price et al., 2002; Joughin et al., 2004). However, because ice acts as a low-pass filter with respect to basal inhomogeneities, inversions constrained by ice- surface velocity data do not provide direct constraints on sticky spots, the dimensions of which are much smaller than the ice thickness. In other words, a till bed that is peppered with such small sticky spots is indistinguishable in these inversions from a uniform till bed.

The most geographically extensive inversion of basal shear stress in the Siple Coast region is that found by Joughin et al. (2004). It is based on the vertically integrated stress-balance equations for ice streams proposed by MacAyeal (1989a,b). Hence, it yields more realistic quantitative estimates of basal stress when basal shear stress is a small fraction of the gravitational driving stress, Td > Tb, and internal ice deformation is negligible. However, it provides less reliable results when that condition is not met. The inversion of Joughin et al. (2004) is constrained by satellite-derived ice-surface velocity data from autumn of 1997 available at a horizontal resolution of 500 m. Results of this work suggest that over most of the bed beneath ice-stream trunks basal shear stress is weak (ca. 10kPa or less) but that smaller areas (tens and hundreds of square kilometres) of elevated resistance (up to ca. 100kPa) do exist. Out of the Siple Coast ice streams, the Whillans Ice Stream has the most uniform and the weakest bed, whereas the MacAyeal Ice Stream has abundant and strong sticky spots. In addition, the work of Joughin et al. (2004) and Price et al. (2002) shows that the basal resistance beneath ice-stream tributaries tends to be more spatially variable than beneath ice-stream trunks.

70.5 Temporal variability of ice-stream flow

Perhaps the most important aspect of ice-stream flow is the ability of ice streams to change their velocity, and thus alter their contribution to ice-sheet mass balance, on different time-scales (e.g. Bindschadler & Vornberger, 1998; Joughin & Tulaczyk, 2002). This is dramatically illustrated by the recent discovery of stickslip behaviour on the ice plain of Whillans Ice Stream, which is stationary over most of the time and moves in short, tidally triggered spurts during which ice-flow velocity is equivalent to about 10kmyr-1 (Bindschadler et al., 2003). The same ice stream also experienced a continuous slow down in average velocities, at least in the period between 1974 and 1997, with deceleration rates reaching ca. 5 myr-1 on the ice plain (Joughin et al., 2002). If this process continues, the ice stream may shut down in ca. 80 yr (Bougamont et al., 2003b). Another key example of ice-stream flow variability is the shutdown of Kamb Ice Stream, which occurred ca. 150yr ago (Retzlaff & Bentley, 1993; Smith et al.,

2002). This event has widespread implications because it pushed the Siple Coast sector of the ice sheet into a positive mass balance (Joughin & Tulaczyk, 2002), caused the growth of an ice bulge at the confluence of the two tributaries of this ice stream (Joughin et al., 1999; Price et al., 2001), and is already forcing rearrangements in the regional ice flow pattern (Conway et al., 2002).

Observations suggest that the past few hundred years were not overly unusual in terms of ice-stream flow variability when compared with all of the past millennium (Jacobel et al., 1996, 2000; Nereson & Raymond, 1998; Fahnestock et al., 2000; Gades et al., 2000; Nereson et al., 2001). Thus, internal ice-stream processes are able to cause significant changes in ice-flow patterns and rates during time periods when climate forcing is relatively stable. The most important question, however, is whether the documented changes in ice-stream flow are consistent with the idea of a continuing Holocene collapse of the West Antarctic Ice Sheet or they are indicative of an end to the post-glacial retreat of this ice mass (Bindschadler, 1998; Bentley, 1999; Conway et al., 2002).

Joughin & Tulaczyk (2002) speculated that the latter may be the case because the last two major events in the Siple Coast region involved the stoppage of the Kamb Ice Stream and the slowdown of the Whillans Ice Stream. Nonetheless, they acknowledged that these individual events may simply be a part of a century-scale oscillation, which will ultimately return the Siple Coast region to a negative mass balance and enable further retreat of the WAIS. The physical argument is that the Holocene retreat of the ice sheet may have thinned it sufficiently for ice streams to start freezing to their beds, particularly near their grounding lines (Bougamont et al., 2003a,b; Christoffersen & Tulaczyk, 2003a,b). This proposition has been questioned by Parizek et al. (2002,

2003) who used a flowline model to infer that there is enough subglacial water beneath Siple Coast ice streams to keep their beds lubricated. Although ultimately the contention of Parizek et al. (2002, 2003) may turn out to be correct, their flowline model neglects the dissipation of gravitational energy in lateral shear margins of ice streams, thereby overestimating the basal shear heating and water generation.

The existing thermodynamic models of ice-stream flow evolution suggest that major oscillations in ice-stream flow ('on/off cycles') are longer than hundreds of years and should last a few thousands of years or more (MacAyeal, 1993a; Marshall & Clarke, 1998a,b; Payne, 1998, 1999; Tulaczyk et al., 2000b; Calov et al., 2002; Bougamont et al., 2003a,b; Christoffersen & Tulaczyk, 2003a,b). However, these models make no, or limited, provision for the control of subglacial water dynamics on ice dynamics and they do not incorporate behaviour of ice bulges, such as the one growing on the stopped Kamb Ice Stream. In addition, models with more realistic ice-stream physics tend to consider only single ice streams, neglecting potential interactions between neighbouring ice streams (e.g. Bougamont et al., 2003a; Christoffersen & Tulaczyk, 2003a,b). On the other hand, ice-sheet models represent ice streams in a rudimentary fashion (Marshall & Clarke,

1998a,b; Payne, 1998, 1999; Calov et al., 2002). For these reasons, it is too early to reject the hypothesis that the observed recent changes in ice-stream velocities are just a part of some relatively short, century-scale fluctuation, which will not amount to cessation of the Holocene collapse of WAIS.

Whereas understanding of the dynamics of a system of interacting ice streams is still difficult to achieve, significant advances have been made in the analysis of velocity evolution for a single ice stream (MacAyeal, 1993a; Tulaczyk et al., 2000b; Joughin et al., 2002; Bougamont et al., 2003a,b; Christoffersen & Tulaczyk, 2003a,b). Of greatest interest is the variability in ice-stream velocity over socially relevant time-scales of dozens to a few hundred years. Susceptibility of an ice stream to different perturbations can be illustrated by taking a time derivative of Raymond's (1996, equation 33) analytical equation for centre-line velocity of an ice stream flowing in a rectangular channel. When expressed as a percentage of the original ice-stream velocity, the linear accelerations in response to independent and small perturbations in ice stream width, W, basal shear stress, Tb, and gravitational driving stress, Td, can be expressed through relatively simple equations:

where W, Tb, Td denote the magnitude of the small perturbations and n is the stress exponent in the flow-law of ice (here assumed equal to 3). As equations (1b) and (1c) differ only in a sign, Fig. 70.2 contains just two panels because panel (B) illustrates the magnitude of ice-stream velocity variations in response to changes in either driving stress or basal shear stress. The clear message of Fig. 70.2 is that stress perturbations, such as weakening or strengthening of basal resistance, have much greater impact on ice-stream velocity than changes in ice-stream width. Even relatively large ice-stream widening/narrowing rates of 100 myr-1 (Clarke et al., 2000; Bindschadler & Vornberger, 1998) will result in velocity changes of the order of 0.1-1.0% per year, with only weak dependence on the initial width. However, reasonable perturbations in stresses (order of ca. 0.1kPayr-1) can cause ice stream velocity to change by tens, hundreds, or even thousands of per cent per year, particularly where the initial difference between driving stress and basal stress (Td-Tb) is small (of the order of 1kPa or less). Such small stress differences are characteristic for ice plains, where driving stresses are just a few to several kPa. Hence, the recent observations of high flow-rate variability near the grounding line of Whillans Ice Stream (Bindschadler et al., 2003) are consistent with this simple model. The high sensitivity of ice-stream velocity to changes in stress difference is quite unfortunate because of the difficulty with constraining basal shear stress and quantifying its likely temporal evolution in response to changes in subglacial water flow/storage and other forcings.

B

Figure 70.2 Sensitivity of ice-stream velocity to changes in ice-stream width (A) and basal or driving stress (B) based on Equation (1a-c). The range of ice-stream widths in (A) covers tributaries (ca. 20km), ice plains (ca. 100km) and ice-stream trunks in between (Joughin et al., 1999; Raymond, 2000; Tulaczyk et al., 2000b). The stress difference in (B) is defined as the difference between driving stress and basal resistance. The stress change rate represents variations in either driving stress or basal resistance. Ice accelerations result from an increase in driving stress or a decrease in basal resistance. Velocity changes are plotted for the case of acceleration (positive per cent per year) but equivalent decelerations can be obtained by simply switching signs. The velocity change isolines are plotted on a linear scale in (A) and on a logarithmic scale in (B).

Figure 70.2 Sensitivity of ice-stream velocity to changes in ice-stream width (A) and basal or driving stress (B) based on Equation (1a-c). The range of ice-stream widths in (A) covers tributaries (ca. 20km), ice plains (ca. 100km) and ice-stream trunks in between (Joughin et al., 1999; Raymond, 2000; Tulaczyk et al., 2000b). The stress difference in (B) is defined as the difference between driving stress and basal resistance. The stress change rate represents variations in either driving stress or basal resistance. Ice accelerations result from an increase in driving stress or a decrease in basal resistance. Velocity changes are plotted for the case of acceleration (positive per cent per year) but equivalent decelerations can be obtained by simply switching signs. The velocity change isolines are plotted on a linear scale in (A) and on a logarithmic scale in (B).

70.6 Representation of ice streams in ice-sheet models

If, as it is assumed here, the ultimate goal of glaciological research on ice streams is to produce a quantitative model of ice-stream behaviour that can be used to predict future ice-sheet evolution and to reconstruct past ice-sheet flow, then considerable future efforts are needed to achieve this objective. At the present time, treatment of ice streams in numerical ice-sheet models lags behind our understanding of the complexity of ice-stream physics. This is because some of the key physical processes determining ice-stream velocity and its changes occur on relatively short spatial scales and cannot be directly resolved in ice-sheet models, which are mostly vertically integrated and have horizontal resolution of tens of kilometres.

Numerical models based on the shallow-ice approximation are particularly unsuitable for capturing the complex processes taking place in and around lateral ice-stream margins because their force balance equation neglects the very terms which in reality give rise to marginal shear stress and to stress transfer from the margins to the bed (Whillans & van der Veen, 2001). To properly capture these phenomena in a numerical ice-sheet model, a higher order, three-dimensional treatment (e.g. Pattyn, 2003) is necessary with horizontal and vertical resolution of the order of 100m, at least in and around lateral margins. A considerable increase in computational power would be necessary to achieve such a resolution over the whole ice sheet given that modern ice-sheet models have ca. 100,000 elements and the number of elements in a suitable three-dimensional model would be ca. 1000-100,000 greater. Although these numbers may seem daunting, it is important to note that further advancements in computer processor technology are likely to increase computational power of a single processor by a factor of 1000 in the next decade (Moore, 1965). Additional technological advances, e.g. grid computing, may increase the availability of inexpensive computational time for advanced ice-sheet modelling. Moreover, shallow-ice approximation works well over much of an ice-sheet area (Pattyn, 2003) and development of 'smart' grids, which would apply the three-dimensional treatment only where needed (e.g. lateral shear margins), can further cut down the number of elements needed to achieve realistic representation of ice streams.

In addition to increasing spatial resolution and deploying three-dimensional force balance, ice-sheet models need to take into account that the mass balance of subglacial water is the key factor controlling the distribution and efficiency of lubrication enabling rapid basal motion. Subglacial water is a conserved quantity and its flow and storage can be treated in ice-sheet models (Johnson & Fastook, 2002). Sliding coefficients used to simulate rapid basal motion in areas of melted bed cannot be arbitrary constants but have to be coupled to subglacial water dynamics, even if the exact nature of the relationships between ice velocity, basal shear stress and subglacial water flow/storage is not yet determined. Without such parametrizations ice-sheet models will not be able to predict the future mass balance of modern ice sheets and reconstruct the behaviour of past ice sheets because subglacial water is one of the key factors controlling fast ice-flow rates and their changes (e.g. Zwally et al., 2002a). In spite of the remaining challenges, the rapid progress in ice-stream research observed over the past few decades provides solid foundations for future advances in this important area of glaciologi-cal research.

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