Progress in Physical Geography 26,1 (2002) pp. 96–122

Calving glaciers C.J. van der Veen Byrd Polar Research Center, The Ohio State University, Columbus OH, USA

Abstract: Based on a review of observations on different types of calving glaciers, a simple calving model is proposed. Glaciers that exist in a sufficiently cold climate can form floating ice shelves and ice tongues that typically do not extend beyond confinements such as lateral fjord walls or mountains, and ice rises. If the local climate exceeds the thermal limit of ice shelf viability, as is the case for temperate glaciers, no floating tongue can be maintained and the position of the terminus is determined by the thickness in excess of flotation. If the snout is sufficiently thick, a stable terminus position at the mouth of the confining fjord – usually marked by a terminal shoal – can be maintained. Further advance is not possible because of increasing sea-floor depth and diverging flow resulting from lack of lateral constraints. If a mass balance deficiency causes the terminal region to thin, retreat is initiated with the calving front retreating to where the thickness is slightly in excess of flotation. In that case, the calving rate is determined by glacier speed and thickness change at the glacier snout. Advance or retreat of the calving front is not driven by changes in the calving rate, but by flow-induced changes in the geometry of the terminal region. This model is essentially different from prior suggestions in which some empirical relation – most commonly the water-depth model – is used to calculate calving rate and the rate of retreat or advance of the terminus. Key words: calving, Columbia Glacier, icebergs, ice shelves, lacustrine glaciers, tidewater glaciers.

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The calving debate

The large ice sheets of Antarctica and Greenland, together with numerous smaller ice caps, glaciers and ice fields, contain more than 87% of the Earth’s supply of fresh water. These frozen depots are in constant state of flux, nourished by surface precipitation while mass is lost through melting in the marginal regions and under floating ice shelves and ice tongues, and by iceberg calving at marine and lacustrine termini. For the cryosphere as a whole, the amount of annual mass exchange is impressive indeed. Surface accumulation adds about 2880 Gton water equivalent annually; surface melting and subsequent meltwater runoff removes about one-third of this (~980 Gton per year), while iceberg calving may account for as much as 2400 Gton per year lost from glaciers and ice sheets around the world (Van der Veen, 1999a: ch. 1). While the uncertainties in these numbers are large, they serve to illustrate the importance of iceberg calving as © Arnold 2002

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a mechanism for, at times rapid, mass transfer from the cryosphere to the world’s oceans. Calving permits much larger volumes of ice to be lost to the glacier than would be possible through surface ablation. The comparatively rapid disappearance of the Northern Hemisphere ice sheets at the end of the last glacial period cannot be explained by surface melting on the southern-most lobes. Instead, calving into pro-glacial lakes and embayments appears to have been the main cause for the rapid demise of these ice sheets (e.g., Pollard, 1984; c.f. Van der Veen, 1999a: section 9.8). Deep-sea sediment records suggest that iceberg production may have occurred in recurrent outbursts, associated with periodic surges of the former Laurentide Ice Sheet (Heinrich Events) during the last deglaciation (Heinrich, 1988; Bond and Lotti, 1995). These outbursts may have affected the large-scale oceanic thermohaline circulation that, in turn, may have caused climate to regress temporarily to full ice-age conditions (Broecker, 1992). On a smaller, but no less spectacular, scale, increased calving accompanies the ongoing collapse of the lower reach of Columbia Glacier, Alaska, USA. The front of this glacier became unstable in the early 1980s and has since retreated more than 13 km up its fjord at rates exceeding several kilometres per year in recent years (Krimmel, 1997; Pfeffer et al., 2000). In West Antarctica, finally, the break up of ice shelves in the Antarctic Peninsula has been attributed to increased calving rates possibly induced by a warming trend in the region (Vaughan and Doake, 1996). While the warming is, as yet, insufficient to threaten the major ice shelves surrounding the West Antarctic Ice Sheet, this situation could change if greenhouse warming becomes a reality. It has been argued that removal of the peripheral ice shelves could lead to the collapse of this ice sheet, with a consequent 7-m rise in global sea level (Weertman, 1974; Mercer, 1978; Oppenheimer, 1998). All these observations point to iceberg calving as a major factor in rapid ice sheet changes. However, the fundamental question that remains largely unanswered is whether increased iceberg production is a driver for collapse or merely the consequence of the several-fold increase in ice discharge and associated thinning usually observed during the phase of glacier collapse. Hughes (1986, 1998) proposed the Jakobshavn Effect, a powerful instability mechanism whereby small perturbations are amplified and may lead to dramatic retreat and possible collapse of calving outlet glaciers and ice streams. The trigger for this effect is surface melting and increased iceberg calving. Similarly, rapid retreat of tidewater glaciers has been attributed to greater calving rates once the terminus retreats from the terminal moraine into deeper water farther into the fjord (Meier and Post, 1987). In both models, retreat of the terminus, caused by increased calving, leads to larger stretching rates upglacier and, consequently, greater ice speeds, causing the glacier to thin which, together with enhanced stretching, facilitates full-thickness fracturing and thus iceberg calving. This view of calving as catalyst for collapse was challenged by Van der Veen (1996) who proposed that on grounded tidewater glaciers, the position of the calving front is controlled by local geometry such that, at the terminus, the ice thickness in excess of flotation cannot become less than a certain threshold value (~50 m for Columbia Glacier). According to this ‘height-above-buoyancy’ model, rapid retreat is caused by glacier thinning with the calving rate increasing in response to maintain a grounded terminus. To a large extent, the debate about the importance of calving on the dynamics of calving glaciers stems from a lack of understanding of the physical processes involved

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in iceberg production, as well as factors controlling calving rate. While some attempts have been made to develop theoretical models for calving (e.g., Reeh, 1968; Holdsworth, 1978; Hughes, 1992; Hughes and Fastook, 1997) there is, at present, no theoretical model available that can explain the observations. At the same time, deriving empirical models or ‘calving laws’ based on interpretation of observations has also proven to be ambiguous, with different authors arriving at seemingly opposite conclusions despite using essentially the same – Columbia Glacier – dataset (Meier, 1994, 1997; Van der Veen, 1996). Calving occurs when a piece of ice breaks off from the main glacier and this process involves the propagation of a fracture. Several modeling studies have been conducted to simulate the calving process on grounded termini (Iken, 1977; Fastook and Schmidt, 1982; Hanson and Hooke, 2000) and results indicate the importance of oversteepening of the ice cliff owing to differential ice flow (greater ice speed at the surface than near the glacier base). This oversteepening may result in subaerial launch of seracs (Figure 1a) but may also lead to bending moments that allow crevasses upglacier from the calving cliff to penetrate the full ice thickness, thereby forming relatively large icebergs (Figure 1b). Other factors that may affect the calving process include thermal erosion at the waterline (Kirkbride and Warren, 1997) (Figure 1c) and subaqueous calving of submarine platforms or ‘toes’ (Motyka, 1997) (Figure 1d). In addition, the presence of surface and bottom crevasses, as well as bands of structural weakness along which fractures may preferentially propagate, are apt to control the rate of iceberg production to some extent. Given the complexity of fracture propagation in an actively deforming material such as glacier ice, one might be skeptical about any mathematical relationship describing the calving process. Indeed, it is probably illusory to expect a universally applicable ‘calving law’ to exist. However, at the present time it appears that even a conceptual model for calving is lacking – or at least under debate. On the one hand, the waterdepth model has gained widespread acceptance for describing the calving process on grounded tidewater glaciers (e.g., Brown et al., 1982; Meier, 1994, 1997; Clarke et al., 1999; Hanson and Hooke, 2000). According to this model, the calving rate is linearly related to water depth at the calving terminus. On the other hand, Van der Veen (1996) argues that the position of the terminus is controlled by geometric factors, including the rate of thickness change. Consequently, in this height-above-buoyancy model, calving rate must be considered the slave, and ice speed and change in terminus position – dictated by local thinning – as masters. In light of the unresolved nature of the calving process and its obvious importance to the dynamical behavior of calving glaciers, a re-evaluation of the water-depth model is warranted. This is done here using updated measurements from the rapidly retreating Columbia Glacier, Alaska, shown in Figure 2 (Pfeffer et al., 2000; Krimmel, 2001). Terminus position and ice speed derive from repeat aerial photogrammetry that continues to the present. However, starting around 1994, the combination of the length of time between successive flights and the great speed of the glacier resulted in few trackable features near the glacier terminus and thus a lack of accurate terminus speeds (R.M. Krimmel, personal communication, 2001; an exception is the period March–May, 1997, when several flights were conducted at close intervals and for which extensive velocity coverage is available). For that reason, only the data up to 1994 are considered here. The water depth shown in the lower panel is more accurate than used in previous

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Figure 1 Four mechanisms for production of icebergs from a grounded terminus

studies (Meier, 1994; Van der Veen, 1996) as it is based on ship-borne fathometry of the forebay conducted on three occasions in 1995–97 when the bay became mostly free of ice (Krimmel, 2001). While documentation of the retreat of Columbia Glacier constitutes the best available and most detailed description of a retreating tidewater glacier, further insight into the calving process can be gained by considering observations on all types of calving glaciers. There are two main factors that determine the type of calving, namely whether the glacier is cold or temperate, and whether the terminus is grounded or floating. This gives four possible combinations of which three are known to exist. Temperate glaciers do not form floating ice tongues and the combination warm/floating does not occur. Within the category of temperate and grounded glaciers, a further distinction can be made between glaciers ending in fresh-water lakes versus those terminating in sea water. Admittedly, there are, at present, no convincing arguments for assuming that the processes controlling calving from grounded and floating termini are the same and it may well be that various types of calving should be described by different calving relations. Further, it remains possible that the water-depth relation for grounded glaciers is not valid at times of rapid retreat as the terminus approaches flotation (Meier and Post, 1987) and that different calving laws should be applied depending on the dynamical regime of the glacier under consideration. On the other hand, a conceptual framework that does not invoke seemingly ad hoc differences between various glaciers or stages of glacier evolution seems meritorious, if only because of its simplicity. It is argued here that observations on all types of glaciers can be explained within the context of the height-above-buoyancy model proposed by Van der Veen (1996).

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Figure 2 Columbia Glacier data, 1976–94. The upper panel shows the rate of change in terminus position and the second panel the ice speed at or near the terminus, both measured from repeat photogrammetry. The third panel shows the calving rate, obtained from the difference between terminus change and speed. The fourth panel shows the water depth at the terminus, based on fathometry of the forebay. Smooth curves represent Gaussian averages Source: based on data from Krimmel (2001)

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Questioning the water-depth model

The water-depth relation for calving rate from grounded tidewater glaciers was proposed by Brown et al. (1982) based on statistical analysis of data on change in terminus position, ice speed, ice thickness and water depth, from 12 Alaskan tidewater glaciers. Since then, similar relations have been found for other glaciers (e.g., Pelto and Warren, 1991; Kennett et al., 1997). Yet a number of questions remain that continue to cast doubt on the water-depth relation as a viable explanatory and predictive model for iceberg calving. First, the water-depth model applies to annual-averaged calving rates only (Brown et al., 1982). Calving is, of course, a discrete process with icebergs detaching from the main body of glacier periodically. Thus, to accommodate the stochastic nature of the calving process, calving rates must be determined over time intervals that are significantly longer than the typical time span separating calving events. On a tidewater glacier such as Columbia Glacier, icebergs are produced on a daily basis and an averaging time of, say, a few weeks should be sufficient for obtaining representative rates of calving. When seasonal rates determined over time spans of one or more months are considered, the water-depth model breaks down (Sikonia, 1982; Krimmel, 1997; Meier, 1997). On Columbia Glacier, calving rates are lowest in winter to late spring and are highest during the second half of the year (Krimmel, 1997). This is illustrated in Figure 3, which shows the difference between the calving rate for each flight interval and the long-term trend obtained from Gaussian smoothing (solid curve in Figure 2), plotted as a function of the time of year. While the scatter is large, when values are grouped and averaged per month, the seasonal trend becomes clear. Similar seasonal variations have been observed on other glaciers, including Hans Glacier, a tidewater glacier in Spitsbergen whose terminus has remained grounded in water of nearly constant depth (Jania and

Figure 3 Seasonal variation in calving rate, defined as the difference between the actual calving rate and the Gaussian smoothed trend shown in Figure 2. The block curve represents monthly variations, obtained by averaging all values within one month Source: based on data from Krimmel (2001)

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Kaczmarska, 1997). Such seasonal fluctuations in calving rate cannot be explained by the water-depth model. A second observation that casts doubt on the water-depth model is that, during its rapid retreat, the speed of Columbia Glacier increased almost as much as did the calving rate. In fact, an excellent correlation exists between these two quantities, not only for Columbia Glacier during its retreat, but also for the other Alaskan glaciers considered by Brown et al. (1982) (Van der Veen, 1995, 1996). This relation between ice speed and calving rate not only applies to smoothed averages (Figure 4, right panel), but to a lesser extent also to seasonal fluctuations (Figure 4, left panel). Similarly, on Hans Glacier, the calving rate starts to increase in June as the ice speed increases, with both reaching a maximum in late July (Jania and Kaczmarska, 1997). Moreover, the correlation between calving rate and glacier speed also applies to glaciers calving into proglacial fresh-water lakes, as illustrated in Figure 5. The correlation between ice speed and calving rate is a surprising result if the waterdepth model indeed describes the calving process. Glacier speed is primarily controlled by processes acting at the glacier bed, such as effective basal pressure or amount of water present beneath the glacier, as well as by the drag at the glacier base. There is no a priori reason why these processes would affect the rate of iceberg production in a similar way as glacier speed is affected. It might well be that larger speeds are associated with increased along-flow stretching, or that larger basal water pressure – resulting in faster sliding – could facilitate upward propagation of bottom crevasses. Both effects may enhance the possibility of full-thickness fracturing (Van der Veen, 1998a,b; 1999b) and thus lead to increased calving. However, the enigma that remains is why this increase would be linearly proportional to the increase in glacier speed. A third objection against the water-depth model is that it clearly does not explain all of the variations in calving rate, even if annual-mean values are considered. From late 1984 to early 1989, the calving rate on Columbia Glacier remained more or less constant, yet the terminus continued retreating into deeper water, with the water depth

Figure 4 Relation between calving rate and glacier speed on Columbia Glacier, using values determined for each flight interval (left) and Gaussian smoothed values (right) Source: based on data from Krimmel (2001)

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Figure 5 Relation between calving rate and glacier speed for 12 lacustrine calving glaciers Source: based on data from Funk and Röthlisberger (1989) and Warren et al. (1995b)

increasing by about 100 m (Figures 2 and 6). Since then, the calving rate almost doubled despite the terminus remaining grounded at about the same depth. Clearly, other factors must be important in controlling the rate of iceberg production. Apart from these specific objections, a concern with the water-depth model is that this can, obviously, apply to grounded ice fronts only. Thus, calving from floating ice shelves and ice tongues must be assumed to obey a different relation. The difficulty with this concept is that a transition from calving from a grounded terminus to calving from a floating front is not included or allowed for. While there is no a priori reason to presume that calving from an Alaskan tidewater glacier is similar in nature to calving from peripheral Antarctic ice shelves, the distinction between grounded and floating termini becomes less clear if collapsing tidewater glaciers are considered. The terminus of Columbia Glacier, for example, while mostly grounded during its rapid retreat, has experienced short-lived periods of flotation (Krimmel, 1997). Under conditions in which the terminus is effectively – almost – floating, one would expect the calving mechanism to resemble that of floating ice shelves or, at the very least, exhibit characteristics of both types of calving. Yet, according to the water-depth model, during its collapse, the calving rate on Columbia Glacier obeyed the same relation as does the calving rate on a stable tidewater glacier grounded firmly on its terminal moraine. It could be countered that a transition from grounded to floating calving can be achieved if, instead of water depth, the ice depth below water level is used in the calving relation, since on grounded glaciers both quantities are the same. In that case, calving from floating termini would be related to the ice thickness below the water line or, equivalently (since the terminus is floating), to the thickness itself. There is, however, no convincing evidence supporting this model – quite the opposite. The left panel in Figure 7 shows calving rate as a function of thickness at the terminus for 12 floating tidal glaciers in north Greenland. Using aerial photographs taken between 1947 and 1978, Higgins (1990) determined the long-term average glacier speed and change in terminus position. Measurements of surface altitude were made using 1978 vertical aerial photographs and from these altitudes the frontal thickness was calculated

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Figure 6 Relation between calving rate and water depth at the terminus for Columbia Glacier, using values determined for each flight interval (left) and Gaussian smoothed values (right) Source: based on data from Krimmel (2001)

assuming flotation. For some of the larger glaciers, such as Petermann Gletscher, different calving rates for the central portion and the two marginal regions were estimated so the number of data points in Figure 7 exceeds the number of studied glaciers. Even admitting potentially large errors in the data, there is no evidence to suggest or support that calving rate is in any way related to the terminal thickness. Instead, a statistically significant (R2 = 0.94) linear correlation exists between calving rate and glacier speed (Figure 7, right panel).

Figure 7 Relation between calving rate and ice thickness (left) and glacier speed (right) for 12 floating tidal glaciers in northern Greenland Source: based on data from Higgins (1990)

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Spurious correlations

One could argue that, despite the poor performance of the water-depth model in explaining prominent features of the calving process, this empirical relation not only remains useful for predicting calving rates from tidewater glaciers but, in fact, represents an important physical control. Indeed, Meier (1997: 112) argues that ‘an empirical relation between iceberg calving speed (averaged over a seasonal cycle) and water depth appears to be fairly robust, and therefore a physical model should produce a relation that is not inconsistent with this’. Hanson and Hooke (2000: 190) express a similar opinion as they state that ‘when a strong correlation exists for some data sets or some parts of a data set, we should seek a physical cause for this correlation’. These views are questioned by Pelto and Warren (1991) who propose that the water-depth relation cannot be a causal one, and who pose the question ‘is it a spurious result, due to limited data?’. Hanson and Hooke (2000) consider calving as a multivariate problem with water depth, longitudinal strain rate (or accumulated strain), and temperature as the three key factors controlling the rate of calving. At different times, and on different glaciers, one factor may dominate the calving process while at other times, another variable may be most important. Using a finite-element model to calculate the distribution of stress and strain rate near the calving front, these authors seek a physical explanation for the dependency of calving rate on water depth. One pervasive feature of the model results is a region of high speed just below the waterline at the calving front that becomes more pronounced as the ice thickness increases. Hanson and Hooke (2000: 192) argue that the velocity gradient will tend to develop an overhang that would promote failure of the upper part of the calving front (as in Figure 1a), particularly if the glacier surface is heavily crevassed. Moreover, the tendency to develop an overhang is greater on thicker glaciers. This, then, could be a physical mechanism making the calving rate increase with water depth. A second suggested physical reason is the zone of high extending deviatoric stresses along the bed near the calving front, which may arise to balance the bending moment originating from the unequal distribution of hydrostatic and cryostatic pressure on either side of the calving face (Hanson and Hooke, 2000: 193). This tensile stress, which increases with water depth, may initiate bottom crevasses that would promote submarine calving. Hence, this mechanism could be another explanation for the depth-dependency of calving rate. Together then, Hanson and Hooke (2000) argue that these processes may provide a possible physical explanation for the observed water-depth calving relation. The interpretation of Hanson and Hooke (2000) may be challenged. First, it is not evident that an overhang develops sufficiently rapidly to significantly affect the calving rate. For a 300-m thick terminus, with a 60-m freeboard, the velocity difference between the base and the waterline maximum is of the order of 40 m yr–1 (Hanson and Hooke, 2000: figure 4), suggesting an associated calving rate of perhaps several tens of meters per year, or about 1–10% of observed rates. In other words, this contribution to calving, and its possible depth-dependency, is generally too small to explain the water-depth relation. Second, the longitudinal deviatoric stress at the bed calculated by Hanson and Hooke (2000: figure 8) may not be sufficient to initiate bottom fracturing. For a glacier 300-m thick, with a freeboard of 60 m, the inferred deviatoric stress is about 155 kPa, which gives a net longitudinal stress of near zero at the base. However, accounting for

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the fracture toughness of ice, a net basal tensile stress of ~50 kPa is required for a basal fracture to form (Van der Veen, 1998b). Indeed, the study of Van der Veen (1998b) indicates that, on grounded glaciers, basal crevasses can only occur if the basal water pressure is close to the ice overburden or, in other words, if the glacier is very near flotation. It appears therefore that neither of the features revealed by the modeling study of Hanson and Hooke (2000) can adequately account for any significant dependency of calving rate on water depth. The lack of any reasonable physical explanation for why calving rate should depend on water depth raises the possibility that the correlation – which only applies at some times during the life cycle of a tidewater glacier – is a spurious result and does not signify causality. In his Presidential Address delivered to the Royal Statistical Society on 17 November 1925, G. Udny Yule presented what has become one of the celebrated examples of a spurious statistical correlation. Over the period 1866 to 1911 the proportion of Church of England marriages to all marriages declined from 780 per 1000 in 1866 to 609 per 1000 in 1911. At the same time, the standardized mortality rate decreased from 22.1 to 14.3 per 1000 persons. A very high correlation (R2 = 0.95) exists between the two quantities (Yule, 1926; Larsen and Marx, 1986: 490) yet few would interpret this correlation as implying that marrying in the Church of England could be lethal. Glaciology is no more or less immune to spurious correlations occurring than any other branch of science. As an example, consider glacier speed along a flowline extending from the ice divide to the glacier terminus. If the glacier is close to steady state and the rate of thickness change is small compared with the average accumulation rate, the ice flux at any point equals approximately the integrated upglacier net accumulation. Thus, a linear correlation between ice flux and upglacier accumulation may be expected. However, this correlation does not signify a direct physical control on glacier speed that can be used to calculate or estimate the ice velocity or changes therein should accumulation change. Rather, the correlation reflects the tendency of glaciers to adjust to external forcing – surface accumulation and ablation – in order to achieve equilibrium. Similarly then, the correlation between calving rate and water depth may reflect some inherent property of grounded calving glaciers, but it does not necessarily express a direct causal mechanism. This means that, while any proposed calving model should be able to explain why, generally, calving rates are larger if the terminus is grounded in deeper water, the emphasis on water depth in the calving process may well be a red herring, diverting attention from the real physical controls. To summarize the discussion so far, observations on calving glaciers suggest there may be, at times, an increase in calving rate as the terminus retreats into deeper water. Observations on the rapidly retreating Columbia Glacier indicate the water-depth model for calving breaks down during rapid retreat, possibly, as suggested by Meier and Post (1987), because the terminus approached flotation. Further, there is no obvious physical control that can explain any depth dependency of the calving rate. So, at best, the water-depth model represents an empirical relation that applies sometimes – this does not appear to be a strong foundation for incorporation into models that simulate evolution of calving glaciers. Instead, what is desirable is a heuristic boundary condition for the calving terminus that can be applied at all stages of glacier evolution, and to all types of calving glaciers. Such a model is developed below based on observations from a number of different calving glaciers.

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Observations on calving glaciers

The glaciological literature abounds with observations made on calving glaciers, ranging from accounts of single and perhaps unusual calving events (Epprecht, 1987; Harrison, 1992), to detailed descriptions of the nature of iceberg formation observed over a short time period (Kirkbride and Warren, 1997), to the decades-long monitoring program on Columbia Glacier (Brown et al., 1982; Meier and Post, 1987; Krimmel, 1997, 2001; Meier, 1997; Pfeffer et al., 2000). The objective here is not to provide a complete and detailed summary of all observations reported in the literature but, rather, to highlight what appear to be the most important features. The following review serves to set the stage for the conceptual calving model described in the next section. 1

Grounded glaciers

The best documented tidewater glacier is without doubt Columbia Glacier, Alaska, USA, which has been surveyed by aerial photography at regular intervals since mid1976. Prior to 1980 the terminus position remained stable at the terminal moraine, but in the early 1980s, contact was lost with this morainal shoal and the terminus began to retreat (Krimmel, 1997). The onset of retreat followed a sustained period of generally negative annual surface balances from 1949 to 1974 (Tangborn, 1997; Tangborn and Post, 1998) suggesting that the onset of retreat may have been the consequence of several decades of gradual thinning of the lower reach of the glacier. During the period of general retreat, short periods of terminus advance occurred when the lower reach was thickening, while maximum retreat rates coincide with periods of large thinning (Van der Veen, 1996: figure 7). Venteris et al. (1997) use extension rate as a proxy for thinning because this is believed to be more precise than thinning rates inferred from the direct measurements of surface altitude. Again, a close and in-phase correlation between thinning rate – or rather, extension rate – and terminus position is found. Observations on other calving glaciers confirm the importance of thinning on terminus stability. For example, Ventisquero Marinelli in the Darwin Cordillera, southern Chile, started retreating rapidly during the 1960s, following decades of glacier thinning that led to the terminus losing contact with the terminal moraine (Holmlund and Fuenzalida, 1995). An inventory of calving glaciers in southern Patagonia indicates that large thinning rates (>10 m yr–1) occur on glaciers that are retreating rapidly (>300 m yr–1), while thinning rates are generally small on slowly retreating glaciers (Naruse et al., 1995; Aniya et al., 1997). One of the glaciers for which thickness changes could be determined is undergoing thickening and the terminus is advancing (Aniya et al., 1997). On Tasman Glacier, New Zealand, the terminus region shows signs of recent changes in dynamic behavior that may be precursors of imminent onset of lacustrine calving (Kirkbride and Warren, 1999). These changes follow a prolonged period of sustained thinning and generally increasing ice velocities. Kirkbride and Warren (1999) suggest that the increase in glacier speed near the terminus that occurred over the previous five years may be a prerequisite for faster calving through the formation of many crevasses just upglacier of the calving front that may facilitate fracture propagation and production of icebergs. The important observation here is that changes in terminus dynamics and glacier thinning occurred before increased calving was observed.

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The studies of Patagonian glaciers have confirmed the contrasting behavior between tidewater glaciers and lacustrine glaciers whose termini are grounded in proglacial fresh water lakes (Warren and Aniya, 1999). For comparable geometries – in particular water depth – calving rates on lacustrine glaciers are an order of magnitude smaller than those on tidewater glaciers (Funk and Röthlisberger, 1989; Warren et al., 1995a; Warren and Aniya, 1999). The cause for this difference has remained unexplained. The formation of a proglacial lake appears to initiate increased calving that may cause pronounced retreat of the terminus or accelerate retreat already underway (Funk and Röthlisberger, 1989; Warren and Aniya, 1999). Funk and Röthlisberger (1989) propose that retreat may in part be associated with melting of the terminus, where ice is in contact with lake water. Kirkbride and Warren (1999) on the other hand argue that a heavily crevassed glacier surface may be necessary for rapid calving. In the absence of crevasses along which fractures can propagate, the calving rate is controlled by the rate of waterline melting and formation of subhorizontal notches in the calving front that act as initiators of calving (Kirkbride and Warren, 1997). The rate of notch melting is about 1 m day–1 and therefore cannot explain rapid calving rates of the order of several kilometres per year as observed on Upsala West Glacier (Warren and Aniya, 1999). Calving glaciers that have fewer crevasses generally have more stable terminus positions (Warren et al., 1995a; Aniya et al., 1997). Observations on Glaciar San Rafael, Chile, which terminates in a low-salinity lake, indicate that locations at which fracturing occurs are often related to crevasses immediately behind the calving front. These crevasses, which are formed upglacier and carried to the glacier front by ice flow, constitute lines of weakness along which mechanical failure or fracturing occurs (Warren et al., 1995b). While calving from this glacier is a continuous process, with smaller bergs breaking off frequently, there is no correlation between the total calving flux and the frequency of calving events, implying that the flux is dominated by a small number of larger calving events. The time spacing between subsequent large events suggests that about one week is needed for stresses to reach the critical level needed for breaking off a large piece of the terminus (Warren et al., 1995b). Earlier observations on this glacier indicate that the calving dynamics constitute a modified response driven by the history of precipitation, although the response appears to be both damped and amplified at times (Warren, 1993). Generally then, it appears that retreat of grounded calving glaciers may be linked to changes in climate and consequent thinning of, in particular, the lower glacier reaches. In Alaska, Columbia Glacier is the last of the major tidewater glaciers to undergo rapid retreat following the regional warming of the last few centuries (Pfeffer et al., 2000). Similarly, in southern South America, a majority of calving glaciers has retreated as climate gradually warmed (Warren and Aniya, 1999) and it appears probable that climate is the dominant control on calving glaciers (Holmlund and Fuenzalida, 1995; Aniya et al., 1997; Aniya, 1999; Warren and Aniya, 1999). Local topographic, sedimentary and micro-climate conditions may result in different and sometimes contrasting behavior of neighboring glaciers but this is no different from noncalving mountain glaciers (Letréguilly and Reynaud, 1989). All observations indicate that a prerequisite for rapid retreat is a prolonged period of mass deficiency and glacier thinning. Perhaps the most intriguing characteristic of temperate grounded calving glaciers, whether tidewater or terminating in freshwater lakes, is that, irrespective of the nature of the calving process, the thickness at the terminus of retreating glaciers always

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remains near the flotation thickness. During its retreat, the height-above-buoyancy on Columbia Glacier remained more or less constant at about 50 m (Van der Veen, 1996), although short-lived periods of flotation did occur (Krimmel, 1997; Pfeffer et al., 2000). On lacustrine glaciers, the terminus may become temporarily buoyant, that is, remain grounded while the thickness is somewhat less than that needed for flotation. Icebergs that calve rise above the terminus freeboard (‘pop up’) as the terminus is prevented from floating through the tensile strength of the ice (Lingle et al., 1993; Warren et al., 2001). As the terminus thins further, the length of the buoyant zone increases until the moment generated by the upward force of the water becomes sufficiently large to break off a large tabular section. This mechanism has been suggested for a number of lacustrine calving glaciers including Bering Glacier in Alaska (Lingle et al., 1993), Austerdalsisen, Svartisen, Norway (Theakstone and Knudsen, 1986; Theakstone, 1989), Baffin Island, Canada (Holdsworth, 1973), Breidamerkurjökull, Iceland (Howarth and Price, 1969) and Glaciar Nef, Chilean Patagonia (Warren et al., 2001). Buoyancy-driven lacustrine calving appears to be restricted to glaciers whose surface is mostly free from crevasses, which may well inhibit regular calving that would otherwise keep the terminus thickness at or slightly above flotation. Thus, with the possible exception of short-lived events, the terminus of grounded temperate calving glaciers that are undergoing retreat remains close to the flotation thickness. If iceberg calving is a separate physical control on terminus position, the fact that retreating termini are always close to flotation is surprising. Whether the water-depth relation is adopted for calving rate, or perhaps some other calving law, there is no a priori reason why the combination of calving rate and glacier speed would cooperate to keep the terminus near flotation, rather than, for example, the terminus retreating much more rapidly and thus maintaining a larger frontal thickness. Instead, it appears more plausible that the position of the calving front is determined by the flotation criterion with calving rate a secondary parameter, determined by the difference between the glacier speed and the rate of retreat required to maintain the calving front at or near flotation. The nature of the glacier may determine the minimum thickness above buoyancy that can be maintained during retreat. On heavily crevassed glaciers such as Columbia Glacier, on which bottom crevasses may also occur (Venteris, 1997, 1999) a greater thickness may be necessary than on slower-moving glaciers mostly devoid of crevasses and whose termini may, at times, become buoyant. 2

Floating ice shelves and glacier tongues

Several recent events have greatly increased the interest of the glaciological community in calving from floating ice shelves. In March 2000, what appears to be the largest iceberg ever to be observed – B-15 – broke off the Ross Ice Shelf near Roosevelt Island (Lazzara et al., 1999) following a steady advance of the ice front after the ‘B-9’ calving event on the eastern part of the ice front between Edward VII Peninsula and Roosevelt Island in October 1987 (Keys et al., 1998). In the Antarctic Peninsula, a number of ice shelves has recently disintegrated, including the Wordie Ice Shelf (Doake and Vaughan, 1991), and the northern part of the Larsen Ice Shelf (Rott et al., 1996). More recently, other ice shelves in this region have started to retreat and the total area of ice shelves lost since the mid-1960s is about 10 000 km2 (Skvarca et al., 1999).

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Major calving events from Antarctic ice shelves are associated with large rifts extending through the entire thickness. With the advent of satellite imagery it has become clear that rifts are common features, some reaching several hundred kilometers in length and several kilometers in width (e.g., Lazzara et al., 1999: figure 1). Precursor rifts that delineate the outline of future icebergs can be identified sometimes decades prior to the actual calving event (e.g., Keys et al., 1990; Lazzara et al., 1999). For example, Neuburg et al. (1959) describe the Grand Chasms on the Filchner Ice Shelf as a ‘gigantic rupture’ extending some 100 km from Berkner Island on the west to the coast of Coats Land on the east, with a width ranging from 400 m to 5 km. When the feature was surveyed in 1957, the depth of the chasm was about 53 m with a bottom consisting of ice blocks floating on the sea. Since the initial survey, the chasm continued to widen to 11 km at its widest point in 1973 and 19 km in 1985 (Swithinbank, 1988). In the Austral winter of 1986, some 11 500 km2 detached from the Filchner Ice Shelf along the Grand Chasms (Ferrigno and Gould, 1987). Similarly, between 1965 and 1971, the rift along which iceberg B-9 calved in October 1987 extended about 100 km east-northeast from a crevassed area between the Bay of Whales and Roosevelt Island, outlining much of the perimeter of the future iceberg (Keys et al., 1990). The recent calving of B-15 from the Ross Ice Shelf may have been somewhat unusual in that the precursor rifts are believed to have formed sometime in 1988, some 12 years prior to the calving event (Lazzara et al., 1999) whereas the lifespan of rifts associated with other prominent calving events appears to have been greater. Rifts extend through the entire thickness and are filled with sea water, sea ice and fragments from the ice shelf and the occasional seal (Rignot and MacAyeal, 1998), and could either be formed by a bottom crevasse propagating all the way to the surface, or a surface fracture opening to the bottom (Shabtaie and Bentley, 1982). Rignot and MacAyeal (1998) suggest that rifts develop where an ice rise is present or in the vicinity of coastal roughness. Inspection of the distribution of rifts on the Ross Ice Shelf indicates no clear physical control and Lazzara et al. (1999) speculate that rifts develop laterally across the ice flow as a result of extensional stresses in the ice and crack nucleation initiated by some ice-weakening event upstream. Analysis of SAR interferograms covering several large rifts on the Filchner-Ronne Ice Shelf suggests that the riftfilling melange possesses a mechanical integrity and is capable of supporting a spatially continuous velocity field. Moreover, the melange appears to acts as a bonding agent, holding together large tabular fragments of the ice shelf, thus preventing premature detachment (Rignot and MacAyeal, 1998). The extent to which major calving events impact the stability of the calving front may be limited. The ice front of the Ross Ice Shelf appears to have been fairly stable over the last century or so (Keys et al., 1998). Generally, calving rates are low and insufficient to maintain a fixed terminus position. Thus, infrequent large events such as the calving of B-9 and B-15 are necessary to maintain the position of the ice front between Ross and Roosevelt islands. Also, there is no clear observational evidence supporting the suggestion made by Hughes (1986, 1998) that the formation of a large berg leads to increased stretching near the calving front, which would facilitate further fracturing and iceberg production. While detailed measurements of ice speed and stretching rates prior and subsequent to a large calving event are lacking, the front of the Ross Ice Shelf seems to have accelerated in the decade prior to the calving of B-9, with the velocity decreasing subsequently. It may be, however, that the acceleration reflected the

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widening of existing transverse rifts and the formation of an additional rift over the period 1983–87, rather than an actual increase in glacier speed (Keys et al., 1998). In either case, there is no indication of increased ice flow following the calving event. On the northern Larsen Ice Shelf, significant increases in ice velocity occurred from 1975 to 1989 (Bindschadler et al., 1994), concurrent with significant retreat of the ice shelf front (Skvarca, 1993). However, the pattern of velocity changes suggests that these were not driven by increased calving (Bindschadler et al., 1994). The calving of large tabular icebergs is controlled by the occurrence and spacing of rifts in the ice shelf. In contrast, the breakup of ice shelves in the Antarctic Peninsula can be attributed to a local warming trend (Doake and Vaughan, 1991; Rott et al., 1996, 1998; Vaughan and Doake, 1996; Doake et al., 1998; Skvarca et al., 1999). The first shelf observed to disintegrate was the Wordie Ice Shelf whose size decreased from about 2000 km2 in 1966 to little more than a few disconnected and retreating ice tongues in the late 1980s (Doake and Vaughan, 1991; Vaughan, 1993). Retreat of the ice front appears to have started in the early 1970s (Doake, 1982) but was interrupted at times by shorter periods of apparent stability during which the ice front rested on ice rises that may have compressed the ice upstream, thus providing temporary restraint preventing further calving and frontal retreat. However, during retreat of the ice front, the velocity and thickness distribution on the shelf changed such that crevasses appeared upstream of the ice rises. Vaughan (1993) suggests that the increased stresses induced by the separation from the ice rises as the velocity increased, may have resulted in increased fracturing. At this stage of collapse, the ice rises may have behaved more like wedges indenting the ice shelf, contributing to its weakening, rather than acting as stabilizing pinning points (Vaughan, 1993). Perhaps the most spectacular changes occurred on the northern Larsen Ice Shelf, the largest of the Peninsula ice shelves. Between 1975 and 1989, the calving front between the Sobral Peninsula and Lindenburg Island retreated at a rate of about 1 km yr–1 (Skvarca, 1993) but during the Austral summer of 1992–93 this rate accelerated because of the loss of 209 km2 within a ten-week period (Rott et al., 1996). Subsequently, the retreat rate slowed but several rifts running parallel to the surface formed, separating sections of the shelf with surface elevations differing by several meters, suggesting that these rifts extended through the entire ice thickness. In the northern-most part, downstream of D-B-E glaciers, the ice surface was heavily crevassed in 1994. Rott et al. (1996) suggest that, while at this time most of the shelf was heavily fractured, breakup was prevented by a combination of cold temperatures and dense sea ice cover. This situation changed in January 1995, however, during a period of intense northwesterly winds and unusually high temperatures and, in early March 1995, most of the northern Larsen Ice Shelf had disappeared, dispersing thousands of smaller icebergs into the Weddell Sea (Rott et al., 1996). Two styles of calving contributed to the rapid breakup of the Larsen Ice Shelf. The first style involved the calving of a single large tabular iceberg from Larsen B between Jason Peninsula and Robertson Island, while the second style involved the disintegration of Larsen A north of the Seal Nunataks into numerous smaller bergs. The rapid collapse of ice shelves in the Antarctic Peninsula has been interpreted as confirming the suggestion by Mercer (1978) that floating ice shelves cannot be maintained if the annual mean air temperature rises above a threshold value, the socalled thermal limit of ice shelf viability (Vaughan and Doake, 1996). Analysis of

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atmospheric observations show a warming trend on both sides of the Antarctic Peninsula (Vaughan and Doake, 1996; Skvarca et al., 1999) and a gradual southward migration of annual-mean isotherms. Vaughan and Doake (1996) show that ice shelves south of the –5°C mean annual isotherm have shown little change in terminus position, while those shelves north of this isotherm have undergone dramatic retreat or complete collapse. It appears, then, that an abrupt climatic limit exists that, if exceeded, causes floating ice shelves to rapidly disintegrate. The existence of extensive floating ice tongues in Greenland also appears to be related to the annual mean temperature. The –5°C isotherm is to the north of Jakobshavn on the west coast and south of Scoresbysund on the east coast (Putnins, 1970). South of this isotherm, no significant floating tongues exist, although some of the outlet glaciers such as Jakobshavn Isbræ on the west and Kangerdlugssuaq Glacier on the east have termini that are floating over several kilometers upstream of the calving front. In north and northeast Greenland, on the other hand, extensive floating tongues exist, such as Petermann Glacier, whose floating part extends over almost 60 km to the end of the confining fjord (Weidick, 1995). Summarizing observations on ice shelves, under favorable climate conditions, the calving front of floating ice shelves and ice tongues is relatively stable with its position determined by lateral confinements such as ice rises or fjord walls. The stable position is maintained through calving which may involve sparse large events such as those observed recently on the Ross Ice Shelf. When the ambient climate warms above a certain threshold, ice shelves cannot maintain their integrity and disintegrate rapidly. This disintegration may involve wholesale breakup into a few large tabular icebergs, or the splintering into numerous smaller bergs. There is no evidence supporting the model that large calving events, or enhanced production of smaller bergs, significantly impact the upglacier flow. V

A conceptual calving model

The observations discussed in the preceding suggest a simple – albeit heuristic – model for calving glaciers. The important observation on which the conceptual model proposed here is based is the existence of a thermal limit of viability for floating ice shelves. That is, ice shelves cannot exist if the annual-mean air temperature rises above a certain threshold. If the mean air temperature is below this threshold, the calving front advances to the end of the fjord or embayment, or to the most advanced obstruction such as ice rises or islands (Sanderson, 1979). Further advance of the calving front is limited because diverging flow would lead to large stretching rates and presumably easier full-thickness fracturing and iceberg production. Because most tidewater glaciers are located in temperate climates well above the thermal limit, these glaciers cannot form floating ice tongues. Consequently, if the terminus becomes sufficiently thin, the snout breaks off to maintain a thickness close to the flotation thickness. In colder climates, such as central and northern Greenland, tidal glaciers can form sometimes extensive ice tongues occupying coastal fjords, provided air temperatures remain below the thermal limit. According to the height-above-buoyancy model, if ambient climate is above the thermal limit of ice-shelf viability, the thickness at the calving front cannot become

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less that the critical thickness, Hc, given by Hc =

rw ri

D + Ho.

(1)

In this expression, rw represents the density of fresh water in the case of lacustrine calving, or that of sea water in the case of tidewater glaciers, ri represents the density of glacier ice, D the water depth, and Ho the thickness in excess of flotation that can be maintained by the glacier. On Columbia Glacier, this thickness is ~50 m (Van der Veen, 1996) but it may be less for smaller glaciers grounded in shallow water, and for glaciers that are not so heavily crevassed. Vieli et al. (2001) propose a modified flotation criterion with the fixed height above flotation, Ho, replaced by a small fraction, q, of the flotation thickness at the terminus. In that case, the minimum terminus thickness is Hc =

rw ri

(1 + q) D.

(2)

Perhaps the most important aspect of the model advanced here is that the focus is diverted away from trying to establish a calving law that relates calving rate to geometric parameters and possibly other factors. Indeed, even without full understanding of the physical processes involved in iceberg calving, the model provides a simple boundary condition for application in numerical models simulating the evolution of calving glaciers. Moreover, a number of the observations discussed above can be readily explained based on the conceptual model. Important to note is that these observations are not restricted to any one particular type of calving glacier or to the stage of advance or retreat. 1

Ice speed and calving rate

An intriguing observation is the close linear correlation between ice speed at the terminus and calving rate. This correlation holds for grounded tidewater glaciers, whether grounded on their terminal moraine or undergoing rapid retreat, for lacustrine glaciers, as well as for floating ice tongues, except perhaps during the short period of wholesale ice-shelf collapse. If the calving front is at its most advanced position – the end of the fjord or embayment – ice speed and calving rate must balance to maintain a stable terminus position. For a glacier undergoing slow retreat or advance, the correlation is also maintained because the rate of advance or retreat forced by glacier thickening or thinning is generally much smaller than the glacier speed. On a rapidly retreating grounded glacier, the calving rate must exceed the frontal ice speed, yet the correlation between speed and calving rate is maintained but with a regression slope slightly greater than unity. Many such collapsing glaciers experience increases in glacier speed (which may, in fact, be the main cause for terminus thinning and consequent retreat) that are significantly greater than the rate at which the terminus must retreat to remain grounded (Figure 2). Thus, calving rates must increase slightly more than the ice speed for the thinning terminus to retreat.

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2

Calving glaciers

Calving rate and water depth

Vieli et al. (2001) apply a numerical ice flow model to investigate the dynamics of tidewater glaciers and to simulate the cycle of slow advance and rapid retreat. As boundary condition at the calving front the modified flotation criterion is prescribed and at each time step the terminus is moved to the position where the ice thickness equals the critical thickness given by eq. (2). The calving rate is thus controlled by ice speed and thickness change near the snout, and is a predicted result of the model. Retreat and advance across a submarine depression is simulated starting with a steady state glacier and uniformly decreasing or increasing the surface mass balance. The model results indicate that if the terminus is grounded on a subglacial shoal, with a depression upglacier, a small change in mass balance can lead to drastic retreat of the terminus and that the thinning resulting from a sustained negative mass balance is the trigger for this retreat. During periods of slow advance or retreat, calculated calving rate correlates with water depth, but during times of rapid retreat, the linear dependency breaks down. Moreover, the model experiments show that whether or not retreat or advance is rapid or slow depends not only on water depth, but also on the basal slope. Rapid changes in terminus position occur where the bed slopes upward in the direction of flow, while the rate of terminus change is small where the bed slopes down in the flow direction (Vieli et al., 2001). The significance of the model study of Vieli et al. (2001) is, of course, that it demonstrates that the apparent correlation between calving rate and water depth is a consequence of the imposed boundary condition that restricts the minimum thickness at the terminus, rather than signifying a causal physical mechanism. As, generally, speeds are greater on glaciers grounded in deeper water because of lower effective basal pressure, calving rates may be expected to be larger for termini grounded in deep water. However, this does not imply that water depth can be used to infer the rate of iceberg production. 3

Seasonal variations in calving rate

On the seasonal scale, the calving rate from grounded glaciers increases as the seasonal speed increases. Meltwater input to the subglacial drainage system increases water pressures leading to greater speed and, in order to remain a grounded terminus, the calving rate must increase accordingly. Indeed, this interaction between speed and calving may explain the correlation between seasonal calving rates from Columbia Glacier and discharge from the nearby Knik River, considered a proxy for subglacial discharge (Sikonia, 1982). Not all of the seasonal variations in calving rate need be associated with seasonal speed fluctuations. On Columbia Glacier, the annual amplitude of the seasonal cycle in calving rate – 1 km yr–1 deviation from the long-term average, as shown in Figure 3 – is greater than the seasonal variations in speed. In addition to changes in terminus thickness associated with ice flow, direct mass balance forcing also affects the rate of thickening or thinning and thus, indirectly, the calving rate. In winter and early spring, snow accumulation tends to thicken the terminal region, while during summer and fall, surface ablation leads to thinning additional to flow-induced thinning. This additional effect amplifies the seasonal cycle in calving rate.

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Fresh-water calving

The generally smaller calving rates observed on glaciers terminating in fresh water compared with similar tidewater glaciers is the consequence of the greater sensitivity of speed on water depth on tidewater glaciers. This is illustrated in Figure 8, which shows ice speed as a function of water depth for the 12 Alaskan glaciers given in Brown et al. (1982) and for 14 lacustrine glaciers based on data from Funk and Röthlisberger (1989) and Warren et al. (1995b). For glaciers terminating in sea water, the slope of the regression line between ice speed and water depth is about four times the corresponding slope applicable to freshwater glaciers. This difference in sensitivity to water depth is the main reason for generally smaller calving rates on lacustrine glaciers and is due to differences in densities of fresh lake water and salt sea water. While the precise nature of the sliding relation remains a topic of discussion, it is generally accepted that the sliding speed depends on the drag at the glacier bed, tb, and the effective basal pressure, N, according to U = As

t qb Np

,

(3)

(e.g., Budd et al., 1979; Bindschadler, 1983; Vieli et al., 2001). Near the terminus, the effective basal pressure may be calculated from the difference between the ice overburden and the hydrostatic pressure: N = ri gH - rwgD.

Figure 8 Relation between ice speed at the terminus and water depth for 12 Alaskan glaciers (black dots; data from Brown et al., 1982) and for 14 freshwater calving glaciers (open circles; data from Funk and Röthlisberger, 1989, and Warren et al., 1995b)

(4)

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Calving glaciers

Considering two identical glaciers with freeboard height Hf, but terminating in sea water and fresh water, respectively, the ratio of the speeds is given by Us Uf

=

[

(ri - rf)D + riH f (ri - rs)D + riH f

]

p

,

(5)

where Us and Uf represent the velocity on the tidewater and freshwater glaciers, and rs and rf the density of sea water and fresh water, respectively. This ratio is shown in Figure 9 for two values of the freeboard height and for p = 1 and p = 3. As the terminus approaches flotation – greater water depth – this ratio strongly increases. Because, as argued above, the calving rate closely follows the ice speed, this effect explains why calving rates on lacustrine glaciers are smaller than on comparable tidewater glaciers. VI

Processes controlling ice-shelf viability

The important, yet unanswered question is what determines the thermal limit above which floating ice shelves cannot maintain their integrity? The initial model of Mercer (1978) was based on the suggestion that downward percolation and subsequent refreezing of meltwater at depth could eliminate the cold thermal wave associated with the preceding winter and allow a sub-surface magnification of a small temperature rise at the surface. As a result, the temperature of the ice shelf could be raised to the pressure-melting temperature over the entire ice thickness. However, as noted by Vaughan and Doake (1996), if percolation of surface meltwater is restricted to the upper 10 m or so, it is unlikely that the entire thickness would become temperate. Thus, only

Figure 9 Ratio of the velocity on a tidewater and freshwater glacier as a function of water depth for two values of the freeboard height and for p = 1 (full curves) and p = 3 (dashed curves) for the exponent in the denominator of the sliding relation (3)

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a near-surface layer would experience warming. This heat would be conducted downward but this is a slow process and, further, the amplitude of the warming would decrease rapidly with depth. It could be, however, that downward percolation of meltwater reduces the fracture strength of ice. The experiments of Liu and Miller (1979) indicate that the presence of liquid water reduces the fracture strength of ice, so provided that surface meltwater penetrates deep enough into the ice, this process could weaken the ice and lead to the break up of the ice shelf. It is not entirely clear how water at the surface can reach the deeper layers without refreezing. Perhaps an englacial drainage network connected to moulins and surface crevasses exists that transports water to well within the ice shelf (Doake and Vaughan, 1991; Vaughan and Doake, 1996). A more direct effect of meltwater ponding on the surface may be easier full-thickness fracturing. Scambos et al. (2000) suggest that if surface melting is sufficiently extensive, meltwater ponds will form and the firn becomes saturated with water. Surface crevasses would then be filled with water and, provided the water level remains sufficiently close to the surface (~10 to 20 m below the surface; Van der Veen, 1998a), propagate the full thickness of the ice shelf. Scambos et al. (2000) propose this mechanism to explain how comparatively small changes in the surface climate can lead to the rapid disintegration of ice shelves. Rignot and MacAyeal (1998) speculate that enhanced melting and weakening of the melange filling rifts could be responsible for more widespread and rapid release of tabular icebergs as well as for the sudden breakup of Larsen A into small icebergs. These authors suggest that such a rapid disintegration is difficult to explain without the pre-existence of a network of fractures, which may have been held together by a melange of snow and ice. Following the unusually warm Austral summer of 1994/95, this melange may have weakened sufficiently to fail during a large storm, causing the shelf to crumble into small pieces. From observations on Jakobshavn Isbræ, West Greenland, it is evident that the onset of surface melting affects the calving rate, possibly through one of the mechanisms outlined above. This fast-moving glacier underwent a 26 km retreat over the period 1850 to 1950, from an extended position at the end of the fjord to the current position almost at the head (Weidick, 1992). Presently, the lower 10 km or so is floating with a periodic seasonal advance and retreat of the position of the calving front. Since 1950, the terminus has annually fluctuated about 2.5 km around a stable, average position (Sohn et al., 1998). Using histograms of brightness, surface melting can be detected on SAR imagery and Sohn et al. (1998) find that surface melting starts in the middle of April at which time the calving rate starts to increase rapidly. Surface melting reaches its maximum near the end of June, at about the same time as the calving rate reaches its maximum. From then on, the calving rate decreases gradually until the following April. The gradual decrease in calving rate over the winter suggests that, in addition to surface meltwater, confining fjord ice also affects the rate at which icebergs are produced. If surface melting on the glacier were the only control, one would expect a sudden decrease in calving rate at the end of the summer as available meltwater drains or refreezes. Instead, the calving rate decreases gradually as the sea ice and icebergs in the fjord gradually consolidate, suggesting that the fjord ice in part impedes iceberg calving. A similar observation was made by Higgins (1990) who notes that in northern Greenland, general ice-free summers are accompanied by break-up and dispersal of

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floating ice tongues. These events are exceptional, however, taking place at intervals of up to several decades (Higgins, 1990: 3). VII

Concluding remarks

The main point made here is that the rate of iceberg calving is not described by some diagnostic relation such as the water-depth model but, instead, is controlled by glacier speed and geometric changes in the terminal region. As noted by Vieli et al. (2001) this means that the rate of calving is the result of glacier dynamics and flow induced thinning, rather than the other way around. Many questions remain regarding the thermal limit of ice shelf viability. Formation of surface meltwater and the break up of confining fjord or sea ice appear to play a role in maintaining the integrity of a floating tongue, but the precise physical nature of these interactions is not entirely clear. While this is unsatisfactory, the model proposed here provides a simple boundary condition that can be applied to numerical models. And, of course, while the fact that the underlying physics may not be fully understood may be considered a shortcoming of the height-above-buoyancy calving model, this is no different from the empirical water-depth relation for which no convincing explanation has been put forward. Acknowledgements This work was supported by grants from the National Science Foundation (OPP9807521) and from NASA (NAG5-8632). This is Byrd Polar Research Center contribution no. C-1228.

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