Empirically derived ‘calving rate functions’

Initial attempts to quantify calving rates have seen authors attempt to correlate them with measured independent external variables, the most noteworthy being the relationship between water–depth and calving rate. Brown et al. (1982) first quantified the correlation between water depth at the calving margin and calving rate for Alaskan tidewater glaciers, with water depth governing calving rate (Funk and Rothlisberger, 1989; Hanson and Hooke, 2000). Following Brown et al. (1982) a number of authors (e.g., Funk and Rothlisberger, 1989; Pelto and Warren, 1991; Warren et al., 1995a; Warren and Kirkbride, 2003) have elaborated on this idea by identifying a variety of empirically based linear relationships with the form:

ݑ_ୡ=݄ܿ_୵ (2.7) where, uc is width–averaged calving rate , hw is water depth, and c is a constant.

Linear regression applied to a number of datasets has produced several different relationships between uc and hw for tidewater and freshwater calving–glaciers (Table 2.1). The most recent

attempt for freshwater–terminating glaciers by Warren and Kirkbride (2003) incorporated a total of 21 glaciers, from Patagonia (Warrenet al., 1995b), Svartisheibreen, Norway (Kennett et al., 1997), Glacier Ameghino, Argentina (Warren, 1999), and six New Zealand glaciers (Warren and Kirkbride, 2003), producing Equation (2.14). However, the relationship described by Warren and Kirkbride (2003) is questionable, as shallower, less realistic, proglacial water depths are reported in their study (Dykes and Brook, 2010). Considering that subaqueous processes act on the entire submerged portion of the glacier terminus, the measurement of ݄୵ should take into account the

average total depth of the glacier terminus not just to a subaqueous foot, which is apparently the case in this study. Furthermore, studies have also indicated that this linear relationship between calving rate and water depth does not hold true over long–term datasets (e.g., Robertson et al., 2013). Nevertheless, the difference between Warren and Kirkbride’s (2003) coefficients and Pelto and Warren’s (1991), for freshwater and tidewater glaciers respectively, does highlight the difference in rates of retreat between tidewater and freshwater glaciers. However, it has recently been shown that there could be just as much variation between glacial regions and even between glaciers within the same mountain zone (Haresign, 2004).

Table 2.1: Relationship between calving rate (uc) and water depth (hw). uc is width averaged calving speed (ma-1), hw and hmax are the average and maximum water depth at the calving front

(m), and c is a coefficient.

Equation Coefficent (c) Glacier type Source Equation

ݑୡ=݄ܿ୵ 27.1 a-1 12 tidewater

glaciers Brown et al. (1982) (2.8)

ݑୡ=݄ܿ୵+ 70 8.33 a-1 22 tidewater _glaciers Pelto and Warren (1991) (2.9)

ݑୡ=݄ܿ୵+ 12 1.9 a-1

5 freshwater glaciers

Funk and Rothlisberger

(1989) (2.10)

ݑୡ=݄ܿ୵ 2.5 ± 0.5 a-1 14 freshwater

glaciers Warren et al. (1995b) (2.11)

ݑୡ=݄ܿ୫ୟ୶ 1.8 ± 0.5 a-1

14 freshwater

glaciers Warren et al. (1995b) (2.12)

ݑୡ=݄ܿ୵ 3.62 a-1

19 freshwater

glaciers Skvarca et al. (2002) (2.13)

ݑୡ=݄ܿ୵+ 17.4 2.3 a-1 21 freshwater

glaciers

Warren and Kirkbride

The water depth model has been criticised from a number of sources, which have identified the

ݑ_ୡ/݄_୵ correlation as being spurious (e.g., van der Veen, 2002). Calving rates are typically given as annually–averaged data (m a-1), which appears to predict calving rates at this timescale quite well. However, this is not always the case, with calving rate actually accelerating at some glaciers (e.g., Columbia Glacier, Alaska) as the glacier retreated into shallower waters (Hanson and Hooke, 2000). The water depth model also breaks down at shorter timescales (e.g., seasonally) as fluctuations in calving rate occur at a near constant water depth (Sikonia, 1982; Krimmel, 1997; Meier, 1997; van der Veen, 2002; Bennet al., 2007b). For example, during the initial retreat of Columbia Glacier, Alaska, seasonal variations in calving rate occurred (with the lowest calving rates occurring in winter and early spring (Krimmel, 1997)), whilst water depth remained relatively constant throughout (Sikonia, 1982; Krimmel, 1997; Meier, 1997; van der Veen, 2002).

An equally important criticism of the ݑୡ/݄୵ correlation is that as οܮ/οݐ is typically smaller

than ݑ୧, a correlation between ice velocity and calving rate exists (van der Veen, 2002). This is,

however, implicit as ice velocity is used as a proxy for ice flux into the calving front, and therefore coeval fluctuations would be expected. The example of the Columbia Glacier has again been used to describe the relationship between glacier velocity and calving rate, with van der Veen (2002) finding a strong correlation (r2 = 0.94) between the two variables. This correlation between glacier velocity and calving rate was identified during the retreat of the Columbia Glacier, with similar relationships having been identified for other Alaskan glaciers in the study by Brown et al. (1982). The implication of this is that as ice velocity is dependent on a large number of factors (such as basal water pressure, valley/glacier geometry, and longitudinal strain rates) there must be underlying factors affecting calving.

As a result of this analysis, a number of authors (e.g., Pelto and Warren, 1991; van der Veen, 1996; Hanson and Hooke, 2000; van der Veen, 2002) have speculated that the strong correlation that exists between ݑୡ/݄୵ actually represents a dynamically linked third (unspecified) variable

that has not yet been conclusively identified (Hanson and Hooke, 2000). This is based on the assumption that as no physical mechanism has been found to explain the correlation between the two, there must be a third hidden variable which would explain the relationship. As yet this has not occurred. However, the water–depth model has provided the basis from which other studies have built on.