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 Tidewater glacier retreat is abrupt and typically irreversible. The mechanics of abrupt onset are evaluated by considering the behavior of a sliding law that depends inversely upon effective pressure (pice − pwater); for such a law, flux can increase with decreasing thickness if the loss of bed traction exceeds the loss of driving stress. Characteristic values of increased flux and glacier thinning are shown to propagate upstream if ice flux increases with decreasing thickness below an ice thickness threshold determined by the chosen sliding law. Upstream propagation leads to instability, described in earlier literature, in which increased flow leads to thinning, loss of bed traction, and continued acceleration. In simplest terms, the instability arises when alterations in glacier geometry act to reduce resistive stresses more than driving stresses. The effects of diffusion are also analyzed, and it is shown that for the form of sliding law considered here, diffusion will not necessarily eliminate upstream-propagating instabilities. Application of this theory shows that a transition from stable to unstable conditions occurred at Columbia Glacier, Alaska, near the time of onset of its retreat in the early 1980s. The theory also successfully predicts stable or rapidly retreating states for 12 other Alaska tidewater glaciers, and its applicability to Greenland outlet glaciers is discussed.
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 The process of retreat has been described [Post, 1975; Meier and Post, 1987] in terms of “drawdown,” in which accelerated flow at the glacier terminus causes thinning of nearby upstream ice, and thinning in turn increases flow to the terminus by reducing effective pressure at the bed. Retreat of the glacier terminus occurs when the rate of iceberg calving exceeds the delivery of ice to the terminus from upstream. On the basis of observations, the drawdown process seems to be self-sustaining and is known to propagate upglacier from the terminus, as has been demonstrated at Columbia Glacier, Alaska [Meier et al., 1985]. Retreat also appears to be irreversible in nearly all instances: once initiated, retreat continues until the terminus reaches shallow water.
 Observations of tidewater glacier retreat give rise to a number of important questions, including (1) To what degree does glacier mass balance control or influence tidewater glacier retreat? (2) Is the process of drawdown and retreat truly irreversible; that is, once started, must the terminus retreat until the glacier bed rises above sea level? and (3) What initiates the retreat, and how do tidewater glaciers occupy stable maximum (extended) positions for long periods of time, in spite of short-lived episodes of retreat from buttressing terminal moraines?
 Numerical modeling experiments can be used to investigate the interactions between flow processes and boundary conditions, but are complicated by the strongly three-dimensional character of tidewater glacier behavior (strong along-flow and cross-flow gradients and large geometry changes) and especially by the absence of a firmly constrained basal boundary condition to describe sliding. Substantial insights have been gained from explicit models, including those of Sikonia , Bindschadler and Rasmussen , Hooke et al. , Hanson and Hooke , and Vieli et al. . The analysis presented here is focused on discovering the nature of the interaction between basal sliding and kinematics, considering the governing equations but without using a full numerical model, and using an imperfect but widely accepted sliding rule.
 In addition to basal sliding, there is also considerable uncertainty as to what the role of iceberg calving is in near-terminus dynamics and about what processes control the rate of iceberg calving. No well-developed theoretical basis exists for iceberg calving, although many aspects have been investigated [Holdsworth, 1978; Brown et al., 1982; Hughes, 1992; van der Veen, 1996; O'Neel et al., 2003]. Calving processes at the glacier terminus are not considered here, however, and the focus is instead only on drawdown: the longitudinal (along-flow) dynamic interactions in the glacier channel above the terminus that result in accelerated flow, thinning, and rapid discharge of ice toward the glacier terminus. It should be possible to integrate the role of calving in tidewater retreat with the longitudinal dynamics of drawdown, since iceberg calving is either coupled with other parts of the glacier through longitudinal dynamics or hydrology, or calving is a strictly local effect, in which case the drawdown dynamics are unaffected by its action.
 The object of this work is to develop a simple theoretical framework for abrupt onset of accelerated flow and thinning, to provide an explanation for observations supporting the existence of upglacier propagation of thinning and increased flow velocity during tidewater glacier retreat, and to address the three questions outlined above. The model is set in the one-dimensional continuity equation, and is applied to observations of Columbia Glacier, Alaska, from the onset of its retreat in the early 1980s to 2001.
 The analysis applies to the portion of a tidewater glacier grounded below sea level and for which ice motion is predominantly by basal sliding. Consider the representation of a glacier as a one-dimensional time-varying system described by thickness h(x, t) and flux q(x, t) = u(x, t) h(x, t), where u(x, t) is the mean ice flow velocity, either depth-averaged or all basal sliding. Since the onset of retreat, the flow velocity at Columbia Glacier has been virtually all sliding [Rasmussen, 1989]. Given the very high discharge flux at Columbia Glacier (∼10 times the estimated mass balance flux [O'Neel et al., 2005]), surface mass balance is neglected and the continuity equation is written
 The following analysis follows Nye's  kinematic wave analysis in its first steps, but unlike Nye's kinematic wave formulation, equation (1) is not a first-order departure from a datum state; the mass balance term b that ordinarily appears on the right-hand side of the continuity equation is eliminated because it is negligible under conditions of rapid tidewater retreat. Equation (1) is thus the datum-state continuity equation and remains valid over large changes in geometry.
 The flux q is assumed to be a function of ice thickness h and surface slope α, as q = f(h, a) and the total derivative of flux q is written as
 Substituting equation (2) into equation (1) yields the following partial differential equation for thickness changes dh written as a function of x and t:
 Primes denote differentiation with respect to x. The coefficients D = (∂q/∂α) and c = (∂q/∂h) = u + h(∂u/∂h). To achieve constraints on the behavior of this system, the method of characteristics [e.g., Farlow, 1993] is used to identify how values of dependant variables h and q vary over time and space. The coefficients of equation (3) describe advection and diffusion of h in x-t space. The coefficient (c − D′) is the time- and space-varying velocity at which characteristic values of thickness h are propagated through the glacier system, and for a nondiffusive (D = 0) system, characteristic values of h propagate at speed c. The formulation up to equation (3) is parallel to that of Nye  with the exception of the fact that here q and h are 0-order datum variables and not first-order perturbations. The development departs from Nye at this point. In the following, the independent variables x and t will not be written but will be assumed to apply.
 Ice velocity u must be specified in terms of measurable local quantities, and for this a commonly used form is chosen:
where u is the sliding speed, k is a dimensional scaling parameter, τd = ρighsin(α) ≈ ρighα (for small α) is the local driving stress given ice thickness h, surface slope α, ice density ρi (917 kg m−3) and gravitational acceleration g. Locally determined expressions of this general type for glacier sliding have been investigated and used in a variety of settings [Budd et al., 1979, 1984; Bindschadler, 1983; Kamb et al., 1985; Iken and Truffer, 1997; Vieli et al., 2001] and are chosen here for Columbia Glacier principally on account of the strong correspondence observed between increased velocity and decreased ice thickness on multiyear timescales during retreat [Krimmel, 2001; O'Neel et al., 2005] and on account of the more complex but compelling relationship between borehole water levels and velocity on diurnal to weekly timescales [Meier et al., 1994; Kamb et al., 1994]. It will be seen that the character of the theoretical results derived do not depend sensitively on specific magnitudes of the exponents n and m. The actual relationship between velocity at Columbia Glacier and spatially determined independent variables is more complicated than this, as is born out by force balance calculations during the progress of the retreat [O'Neel et al., 2005]. It is not possible, at Columbia Glacier, or elsewhere where the thin-ice approximation does not apply, to accurately reproduce velocity distributions in space and time using a three-parameter expression as simple as equation (4) [Raymond and Harrison, 1987]. This sliding law is not intended to reconstruct all details of the evolving velocity and geometry at Columbia Glacier during its retreat but to explore the relative roles of driving stress and effective pressure in an environment undergoing large changes in geometry. The results derived below will be expressed as factors multiplying the velocity u and attention will be focused on the sign of the results and to a lesser extent on their relative value. Since the velocity u is known to be positive and the actual magnitudes of the results are not needed, this approach allows useful information to be extracted from the analysis without explicit knowledge of values of u.
 The effective pressure at the bed, peff, is defined as the difference between the ice overburden and the basal water pressure. Near the terminus,
where pw is water pressure at the bed, water depth hw is the depth of the glacier bed below sea level, ρw is the density of ocean water (1025 kg m−3) and Δp is an additional head above sea level pressure required to drive the drainage of water from the subglacial system. The pressure term ρwghw varies in x and the thickness h varies in x and t. While some nonzero Δp exists, it is difficult to quantify and for simplicity it is assumed to be zero; it will be seen that this simplification is conservative: the conclusions are still valid in the presence of a positive nonzero Δp. The basal water pressure pw is fixed at the water pressure at depth hw, and as the ice thickness h diminishes, Peff becomes small and flotation occurs at Peff = 0.
where κ = k(ρig)n−m is a dimensional constant and w = hwρw/ρi is an ice thickness exerting a pressure equivalent to the local water pressure pw. The velocity of Columbia Glacier was determined by Rasmussen  to be more than 90% sliding for the time period 1977–1982, and accordingly the velocity can be adequately represented as being entirely due to sliding, especially in subsequent years when the near-terminus velocity increased by nearly an order of magnitude over the 1977–1982 values. The flux q is thus written as
 Appropriate values for the exponents m and n are not precisely known; values used include n = m = 1 [Budd et al., 1984] and n = 3, m = 1 [Bindschadler, 1983]. The range of appropriate values for n and m can be constrained by considering the behavior of equation (6) as w → 0 (equivalent to pw → 0). In order to achieve better compatibility with conventional power law sliding (u ∝ τdm [Weertman, 1957]) at zero basal water pressure, consider that for w = 0, equation (6) reduces to
so the closest match to Weertman-type sliding is obtained for 0 < m ≪ n. Of the previously published values, Bindschadler's  choice of n = 3 and m = 1 comes the closest to meeting this criterion.
2.1. Threshold Thickness h* for Increased Discharge Given Decreased Thickness (∂q/∂h < 0)
 For a sliding law of the form of equation (6), a reduction in thickness h reduces the magnitude of both the numerator (driving stress) and the denominator (effective pressure); whether the ice velocity (or flux) increases or decreases as a result depends upon the relative magnitude of h and w. When water depths are small in comparison to ice thickness, a reduction in ice thickness forces a decrease in peff which is small compared to the reduction in driving stress τd, causing the velocity (and flux) to decrease. In contrast, as ice thickness approaches flotation, a reduction in thickness h results in an increase in velocity and flux because the basal decoupling effect of reduced peff dominates the reduction in driving stress. Figure 1 shows how flux varies in response to the ratio of ice thickness to water depth for a variety of values of sliding law parameters n and m. Values of flux increase to the right (∂q/∂h > 0) for large values of the dimensionless thickness h/hw as increasing driving stress dominates increasing effective pressure, and increase to the left (∂q/∂h < 0) for thicknesses approaching flotation (h/hw →1) as loss of effective pressure dominates loss of driving stress. The minimum flux located between these increases occurs at the threshold thickness h* where ∂q/∂h = 0. The value of h* can be determined by evaluating ∂q/∂h with flux q determined from the sliding law (equation (6)).
2.2. Nondiffusive Wave Propagation
 If the effect of perturbations in slope on flux (∂q/∂α) is neglected, then D = 0 and equation (3) becomes purely advective, with characteristic values of h propagating at c = (∂q/∂h). Using the chosen form for the sliding velocity (equation (6)), the propagation speed c is then
Note that c is composed of a factor (in square brackets) multiplying the sliding velocity u; this allows the propagation speed c to be expressed in terms of the ratio c/u and it is not necessary to explicitly calculate the sliding speed u. The algebraic form of this expression shows the potential for the speed c to reverse sign. Positive values of c correspond to downglacier propagation of characteristic values of thickness h, while negative values correspond to upglacier propagation and occur when the term ((n − m) h − nw)/(h − w) < −1, or equivalently, for the condition
For grounded tidewater glaciers h/w is always greater than 1 (Peff ≥ 0), so equation (10) places the additional restriction that n ≥ n − m. This merely requires that n ≥ m ≥ 0.
 Values of normalized characteristic speed c/u are shown in Figure 2 for normalized effective pressures ranging from (h − w)/h = 1 (no water pressure present) to (h − w)/h = 0 (flotation), and for a variety of values for exponents n and m. The particular choice of n and m shifts the value of effective pressure at which the kinematic wave speed becomes negative, but the transition to negative wave speed occurs between values of (h − w)/h falling between 0.20 and 0.65 in all cases. For example, for n = 3 and m = 1, if the ratio h/w ≈ 1.33 or less (equivalent to h/hw(ρi/ρw) = h/hw(0.895) ≈ 1.33 or h/hw ≈ 1.49), ∂q/∂h < 0 and thinning will propagate upstream.
 Note here that including a positive increment of water pressure Δp required to drive water out of the glacier system reduces peff and shifts but does not diminish or eliminate the effect being considered.
2.3. Diffusive Wave Propagation
 Including the influence of slope perturbations on flux (∂q/∂α) gives rise to the full advection-diffusion form of equation (3). The diffusion D is then
and the full diffusive advection speed becomes
The diffusive advection speed (c − D′) differs from the nondiffusive advection speed c in the presence of the last bracketed term on the right hand side of equation (12). Unlike the nondiffusive wave speed c (equation (9)), the velocity u now cannot be factored out of (c − D′), so the full diffusive wave speed must be calculated explicitly. Higher-order terms (especially α′) introduce noise which makes the diffusive advection speed very difficult to evaluate from data. Nevertheless, several conclusions can be drawn from the algebraic form of the equations (11) and (12). The diffusion D becomes negligible in comparison to c for small slopes (α → 0) and for water depths approaching flotation ((h − ) → 0). The effect of approaching flotation can be seen by calculating the ratio D/c = (∂q/∂α)/(∂q/∂h). From equations (9) and (11),
 As flotation is approached ((h − w) → 0), this ratio goes to zero, indicating that diffusion (∂q/∂α) becomes negligible in comparison to advection (∂q/∂h). This is a limiting case, however, given that temperate marine-ending glaciers do not normally reach full flotation. Some additional characterization of diffusive effects must be made for intermediate thickness/water depth ratios.
 Further insight into the behavior of the diffusive advection speed at intermediate thicknesses can be illustrated by examining the special case of a uniform inclined slab (h′ = 0; α′ = 0), in which case equation (12) reduces to
As in equation (9), the propagation speed can still be expressed as its ratio with the sliding speed, and the reversibility of the sign of the speed is apparent in the term in brackets on the right-hand side. Choosing n = 3 and m = 1, the bracketed term becomes negative, and the condition for upglacier propagation is met for water depths such that h/w ≈ 1.66 (equivalent to (h − w)/h ≈ 0.4 or h/hw ≈ 1.85), or less. Upglacier propagation thus happens for thicker ice (relative to water depth) than in the nondiffusive case, because diffusion spreads out the local thinning (directly due to −∂q/∂x = ∂h/∂t), resulting in further thinning. Thus, in this restricted case, while diffusion may act to eliminate a thickness perturbation over time, it does not block the transition to upstream-propagating thickness values, but actually reduces the threshold for upstream propagation. The diffusivity D can also be calculated directly from observations using equation (11); this is done for Columbia Glacier in section 3.2.
 Finally, while diffusion influences the propagation of characteristic values of thickness and flux, at intermediate thickness values it remains to be determined whether diffusion of perturbations of h through x-t space has the capacity to restore stability to an initially unstable system (h < h*) by leading to an increase in thickness above the threshold value h*. This is analogous to the elimination of kinematic waves by diffusion in Nye's theory, but with an important difference: in downstream propagation of thickness disturbances (addressed in Nye's theory) a spatially confined perturbation (a topographic bulge) is eliminated by locally elevated velocity resulting from a combination of increased slope and increased thickness. In the case considered here, instability (i.e., increasing flux at decreasing thickness) is determined by the condition h < h*, and diffusion will not restore stability unless the result of the diffusion is an increase in thickness above the threshold h*. Whether this will occur can be evaluated simply by calculating the flux gradient ∂q/∂x at a point where instability(h < h*) exists. By continuity, only if ∂q/∂x < 0 will ∂h/∂t = −∂q/∂x > 0, resulting in thickening of the ice above the threshold h*.
 To illustrate this condition, the relationship between diffusion and changes in thickness is shown schematically in Figure 3. An initial thickness perturbation Δh (Figure 3a) reduces h below the threshold h*; the perturbation then diffuses (Figures 3b and 3c), but locally the ice thickness h thins or thickens depending upon how the center of diffusion is shifted upstream or downstream. If diffusion does not result in thickening (e.g., above the inflection point xi in Figure 3b or anywhere in Figure 3c), the instability is not eliminated: h is still less than h*. An upstream shift as depicted in Figure 3c is consistent with a regionally extensile strain environment accompanied by upstream propagation of elevated velocities (e.g., the conditions at Columbia Glacier during its retreat). This is an essential difference between the situation considered here and the classical Nye kinematic wave problem: continued propagation of a disturbance in the classical setting requires maintenance of a locally confined geometric perturbation, while in the tidewater glacier case, continued propagation depends only on the condition that h < h*; diffusion can (and does) act to eliminate any locally confined geometric perturbation, but it cannot eliminate the velocity instability unless it also causes thickening so that h > h*.
 The potential for diffusion to restore stability can be determined by evaluating ∂q/∂x, and seeking conditions for which either ∂q/∂x < 0, indicating local thickening and potential restoration of stability, or ∂q/∂x > 0, indicating local thinning and continued instability despite the action of diffusion. From equation (4),
As with equation (9) for the nondiffusive propagation speed, the flux gradient ∂q/∂x can be expressed as a coefficient multiplying the ice velocity u and thus stability or instability can be interpreted from the sign of the result alone without the need for explicit values of u. Alternatively, the flux gradient ∂q/∂x can be calculated directly from observations of velocity and geometry. In section 3.2, the flux gradient for Columbia Glacier is evaluated by direct calculation.
3. Application to Columbia Glacier
 The retreat of Columbia Glacier from its onset in the early 1980s through ca. 2000 has been described extensively elsewhere [Meier et al., 1985, 1994; Meier and Post, 1987; Kamb et al., 1994; Pfeffer et al., 2000]. The photogrammetric elevation and velocity data derived from the program of aerial photography at Columbia Glacier are cataloged and described by Krimmel . Force balance on Columbia Glacier in the region approaching the terminus has been calculated by van der Veen and Whillans  for preretreat conditions in 1977. Force balance from the onset of retreat to 2001 is presented by O'Neel et al. , as well as an overview of the glacier geometry, including newly inferred and measured basal topography. The magnitude of changes since the onset of retreat are dramatic: since 1982, the glacier terminus has retreated 13 km and at the 2004 terminus has thinned 320 m (ca. 32% of total thickness) in the central portion of the glacier channel ca. 14 km upstream from the preretreat terminus. During 2001, seasonal terminus velocities exceeded 10 km yr−1, and discharge flux of ice into the ocean reached 7 km3 yr−1.
 For this analysis, velocity and geometry data were averaged over a 1.5-km-wide swath taken along the flow centerline of the main glacier channel along the lowermost ca. 30 km, between the preretreat terminus and (depending upon availability of data for a given year) approximately the location at which the bed of Columbia Glacier rises above sea level. Surface slopes were averaged over 1 km in the alongflow direction, or approximately twice the ice thickness on average. Basal topography, however, has been calculated [O'Neel et al., 2005] only for positions up to ca. 17 km above the preretreat terminus, and the calculations discussed below are made only up to this point. The centerline coordinate system defined by Meier et al.  is used to refer to glacier locations. This is a curvilinear coordinate system which follows the flow centerline of the main glacier channel with coordinate values increasing in the downflow direction from km 0 at the head of the glacier to km 66 at the position of the preretreat terminus.
3.1. Nondiffusive Propagation
 Given the difficulty in calculating values of (c − D′) accurately, the unstable upstream propagation of perturbations of h are calculated using the nondiffusive characteristic propagation speed c. The rationale for using the nondiffusive speed c was discussed in section 2.3, and the possible effects of the neglected diffusion will be considered in section 3.2. The speed c was calculated for particular times as a function of position x and time t from ice thickness hi(x, t), water depth hw(x), surface slope α(x, t), and the densities of glacier ice ρi and ocean water ρw. Values of ice velocity are not needed, as described in section 2. Surface elevation and velocity data for Columbia Glacier was taken from Krimmel . Water depth is known from bathymetry measurements in the forebay downglacier from near the 1997 terminus and by continuity calculations above the 1997 terminus [O'Neel et al., 2005]. Photogrammetrically determined surface elevation profiles and velocity profiles are shown in Figures 4a and 4b. The retreat of the terminus position can be seen in Figure 4a, although the end of these lines do not correspond precisely to the terminus position for cases where positions could not be determined photogrammetrically in the immediate area of the terminus. Discrepancies between the actual terminus position and the downstream end of photogrammetric positions are typically less than a few hundred meters. Only one elevation profile (1957) is available prior to the mid-1970s. During the 1970s the lower ca. 20 km of the glacier was calving at rates in excess of its balance flux, and thinning at rates approaching 10 m/yr [Meier et al., 1978]. While significant terminus retreat did not occur until the early 1980s, the earlier observed thinning and increased flow suggests that instability was either imminent in the late 1970s or possibly underway but controlled by continued contact with the terminal moraine. The 1957 topography, however, is in all likelihood a good representative of the stable preretreat geometry.
 Since the propagation speed is locally determined (c is calculated for a specific x), the speed changes along its path as well as through time when the glacier geometry is changing. Figure 5 shows distributions of normalized speed c/u along the glacier centerline for representative years spanning 1957 (prior to the onset of retreat) through 2001. The preretreat glacier geometry for 1957 produces positive values of c/u over the entire length, while all times following 1982 show progressively more negative values of c/u. Positive values of c/u in the preretreat (1957) geometry indicate conventionally propagating (downstream) waves, while negative values indicate upstream propagation and instability. Because the glacier terminus retreats over time, the profiles of c/u recede to the left in Figure 5; as with the profiles of velocity (Figure 4b), values are not carried all the way to the end of the glacier in cases where photogrammetric data are not available close to the terminus. Along-flow variations in the magnitude of c/u result from changes in the thickness/water depth ratio, with the negative magnitude of c/u generally declining in the upstream direction where thinning was less relative to downstream locations, and with negative magnitudes of c/u generally increasing over time at a point as overall thinning progressed. The reduction of negative magnitude of c/u between km 62 and 64 in 1983 are caused by the adverse (thickening downstream) surface profile observed in 1983 around km 64.
3.2. Diffusive Propagation
 The diffusion coefficient D can be calculated from equation (11), either theoretically in terms of the sliding law expression for u (equation (6)) or directly using observed geometry and thickness. Since the theoretical values are expressed as a ratio with the sliding speed (as the propagation speed was), the numbers are difficult to compare to other calculations of diffusion and thus a direct computation is preferable. The average diffusion coefficient over the lowest 2.5 km of Columbia Glacier (approximately 4–5 ice thicknesses at the terminus) was calculated from observations over the period 1957–2002, and expressed as D/W has an average value of 8.33 × 108 m2 yr−1, comparable to Bindschadler's  calculation for typical ice stream values. The preretreat value for Columbia Glacier (D/W = 9.3 × 107 m2 yr−1) was lower than the overall average, but not outside the range of variability following the onset of retreat.
 The computed diffusive wave speed (c − D′) is exceptionally noisy, for reasons discussed above, and meaningful values could not be extracted from the Columbia Glacier observations. As an alternative, the potential for diffusion to restore stability is addressed by calculating the flux gradient ∂q/∂x as given in equation (15). The flux gradient was evaluated directly from observations over the period 1957–2002, and averaged over the lowest 2.5 km of Columbia Glacier. The results are shown in Table 1.
Table 1. Flux Gradient Averaged Over Lowermost 2.5 km of Columbia Glacier at Various Times Before and During Retreat
∂q/∂x Averaged Over Lowest 2.5 km
Positive values, which indicate conditions where diffusion cannot result in thickening and stabilization.
 Conditions of diffusive thinning are indicated by the keyed footnote in Table 1, and mark times (1992, 1994, 1998, 1999, and 2000) when diffusion could not restore stability by increasing thickness, indicated by ∂q/∂x > 0. Diffusive thinning did not occur consistently throughout the retreat, but stability could not be accomplished even while diffusive thickening was indicated (1983, 1988, 1998, and 2001) if the thickening did not raise h above the threshold value h*. By the time retreat was established, thinning had brought the terminus region so far below the threshold for instability that recovery by this means would have been extremely unlikely. For example, in 1988, when ∂q/∂x had its most negative value following the onset of retreat, the average thickness h over the last 2 km approaching the terminus was 353 m; the water depth hw over the same average was 315 m, giving a ratio h/hw = 1.12. For stability to be restored by thickening (i.e., h/hw > 1.49), the terminus ice thickness would have to be increased by 116 m to 469 m.
4. Application to Other Alaskan Glaciers
 The condition for upglacier propagation of thinning can be tested against other tidewater glaciers where water depth and thickness is available. Meier et al.  present appropriate data for 13 Alaska tidewater glaciers, including Columbia Glacier and two others (McCarty Glacier and Muir Glacier) at different times during and after their retreat (similar but less complete data are available in the work of Brown et al. ). These data are shown in Table 2. The fifth column of Table 2 shows the terminus advance rate for each case, with rapid retreat (retreat rates in excess of 200 m yr−1) indicated by the keyed footnote “b” The sixth column shows values of ρih/ρwhw calculated from the values given in the second and third columns. Entries for which ρih/ρwhw < 1.33 (also indicated by the keyed footnote “b”) correspond to upglacier propagation, using the nondiffusive criterion established in section 2.3. Note that the nondiffusive criterion for upglacier propagation (based on equation (9)) is more restrictive than the estimate for diffusive propagation (based on equation (14)). Note also that Tyndall Glacier (retreating at −206 m yr−1) was essentially at the threshold of ρih/ρwhw < 1.33, McCarty Glacier (for which ρih/ρwhw = 1.23), was retreating at 160 m yr−1, only slightly less than the rather arbitrary cutoff of 200 m yr−1 and Columbia Glacier, which in 1977–1978 was at ρih/ρwhw = 1.47 and not yet quite past the threshold, was at that time on the verge of rapid retreat.
Table 2. Alaskan Tidewater Glacier Geometry, Speed, and Advance Rate, Ordered by Declining Advance Ratea
Glacier Name, Period of Observation
Water Depth, m
Surface Height, m
Calving Speed, m/yr
Rate of Advance, m/yr
Negative values indicate retreat. Values in parentheses are standard error.
 In their own discussion of conditions dictating tidewater advance or retreat, Meier et al.  propose that terminus water depth greater than 80 m is a sufficient condition for rapid retreat. All of the stable or advancing glaciers listed in Table 2 have termini in shallow water, and support this simpler criterion. Hubbard Glacier, however, is a revealing member of the group: in 1977–1978, it was retreating at a rate of 32 m yr−1, sliding rapidly (2630 m yr−1) and had a terminus water depth of 80 m, deep enough to place it at the threshold of rapid retreat by the Meier et al.  criterion. In contrast, the value of ρih/ρwhw = 1.96, and by the upstream-propagation criterion proposed here, Hubbard should have been stable at that time. Over the longer term, of course, the Hubbard Glacier advanced, and continues to advance today. This is an example of a glacier terminus making a short-term retreat into deeper water, loosing contact with (part or all) of its terminal moraine but nevertheless re-advancing and re-establishing contact with its moraine. The mechanism of recovery is contained in the upstream-propagation theory: unless the ratio ρih/ρwhw lies below the threshold defined by equation (10), thinning (by any process, including calving) will cause a reduction in flux at the terminus and consequently, by equation (1), thickening, and by equation (6), readvance. Only when the thickness relative to water depth is diminished and the threshold of equation (10) is crossed, will thinning result in further thinning, acceleration, and further retreat.
5. Application to Greenland
 The criterion for irreversible retreat as presented depends critically on the choice of the sliding law (equation (6)) and its dependence on effective pressure. Equation (6) works well for Columbia Glacier in part because the coastal Alaskan tidewater glaciers do not float, and the singularity at Peff = 0 is never encountered. Some other formulation for sliding must be found for Greenland, where floating termini exist, probably involving explicit inclusion of lateral stresses. Such a formulation has been proposed by Benn et al. . Nevertheless, preliminary calculations were made using channel depth and ice thickness data for the Helheim Glacier in east Greenland, and the Upernavik Glacier in West Greenland (I. Howat, personal communication, 2006) using the same criterion as for Columbia Glacier. Helheim Glacier is a rapidly flowing glacier with maximum speeds of ca. 10 km yr−1 [Rignot and Kanagaratnam, 2006], and would be expected to have a thickness/water depth ratio below the threshold for instability ((n − m)h − nw)/(h − w) < −1). Upernavik is a slowly flowing glacier which has a negative mass balance and a record of long-term retreat [Weidick, 1999], and would be expected to have a thickness above the threshold for instability. For Helheim, ice thickness was found to be well below the threshold value, consistent with its rapid flow. Upernavik was found to have thicknesses above the threshold for rapid flow, except in the lowermost ca. 10 km, where the ice is very close to floatation. This result may reflect the need for a term in the sliding law that includes lateral shear stress. Alternatively, Upernavik may be unstable and may soon accelerate and retreat. Successful prediction of outlet glacier stability depends on the formulation of a sliding law applicable at both basal and lateral boundaries that includes effective pressure but produces finite velocities at floatation.
 Profiles of the nondiffusive propagation velocity c in the Columbia Glacier channel indicate the presence of stabilizing downglacier propagation of characteristic values of thickness (i.e., positive speed c) before the onset of retreat in the early 1980s, and persistent and growing instability thereafter (negative speed c). This result is obtained through a highly simplified theory, but a number of robust conclusions can be drawn.
6.1. Role of Mass Balance
 Changes in mass balance were neglected in the continuity equation used here (equation (1)). At Columbia Glacier, ample justification can be found for this by considering the post-1982 changes in flux: the 2001 discharge flux was ca. 7 km3 yr−1 while the balance flux (the terminus discharge required to balance mass input by net accumulation) is poorly known but was roughly +0.4 km3 yr−1 in 1978 [Mayo et al., 1979]. To balance the 7 km3 yr−1 discharge in 2001, a glacier-wide average positive balance of +7 m yr−1 water equivalent would be required. The drastic changes that have occurred in glacier elevation and volume since 1982 were not accompanied by any equally strong negative climatic mass balance shift at that time. The reduction in glacier volume was the result of a change in dynamic mass balance, that is, a change in glacier volume caused not by varying accumulation and ablation but by a dynamic deviation of the velocity field of the glacier from that needed to redistribute mass in a steady state fashion. In terms of the continuity equation (1), the flux gradient term ∂q/∂x changed not in response to thickness changes (∂h/∂t) arising from variations in mass balance (), but because of direct interaction between flow speed and thickness (through reduction in Peff).
 Dynamic changes of this type clearly play a critical role in overall mass balance of a glacier system, not only for the tidewater glaciers typical of the Alaska coast, but also for Greenland as well, where similar abrupt transitions to high flow rates and thinning are now occurring [e.g., Rignot and Kanagaratnam, 2006].
 Nevertheless, climatic mass balance has a role to play. For marine-based ice that is thick relative to water depth, that is, for large (h − w)/h, perturbations in thickness propagate downglacier and are stable. The glacier may suffer episodes of negative mass balance and consequent thinning, but thinning will lead to a stabilizing reduction in velocity, along-flow compression, and thickening. This explains why tidewater glaciers persist so stably in their extended positions: they arrive at their maximum extent with thickness well above the threshold value h* for upstream propagation. Long-term thinning, however, can lead to the situation which occurred for Columbia Glacier around 1982, when the threshold of stability was crossed and thinning led to acceleration and further thinning. Long-term climatic mass balance thus controls the transition from stable to unstable glacier geometry, but once the threshold of instability is crossed, climatic mass balance is effectively decoupled from the glacier response, as proposed by Meier and Post  and Clarke , among others. This is similar to the dynamics of glacier surges, where threshold values of basal shear stress are crossed (in part as a consequence of mass balance–induced thickening and increase in surface slope), after which significant changes in flux and geometry occur with insignificant modulation by mass balance [Raymond, 1987].
 The cycle of thinning, acceleration, longitudinal extension, and further thinning suggests that once the threshold of instability is crossed, the retreat is more or less irreversible. However, all aspects of the theory as applied to Columbia Glacier depend upon the validity of the chosen form of the sliding law (equation (4)). If this sliding law is valid, then for ice thickness h less than the threshold value for stability, thinning will always result in a further increase in speed and the process is indeed irreversible, short of a nondynamic thickening such as a drastic increase in climatic mass balance. The shortcomings of equation (4) are discussed further below. At the simplest level, though, the theory offers a concrete picture of the unstable behavior of glacier sliding when basal drag reduced by buoyancy forces overcomes driving stress reduced by thinning. Qualitatively, this idea is supported by the increasing velocity of Columbia Glacier under conditions of diminishing thickness and effective pressure. Note that in this discussion “stability” and “instability” refer to the behavior of glacier flux and thickness changes, and not directly to terminus position. The terminus retreat of Columbia Glacier slowed substantially during its passage through the Kadin-Great Nunatak constriction (km 53–54) between ca. 2001 and 2005 (the glacier cleared this gap in 2005 and the retreat rate is now increasing); ice velocity and flux have been highly variable during this period, but the glacier did not “stabilize” at the constriction in any real sense. True stabilization would only occur through a cessation of terminus retreat (i.e., downstream flux balances calving speed) in a geometry in which mass balance can resupply discharge flux without reducing glacier volume. This is highly unlikely in the present state of Columbia Glacier, with the terminus grounded in approximately 500 m of water and thinned by nearly 500 m from an initial geometry that had an only slightly positive glacier-wide mass balance in the late 1970s.
6.3. Sliding Rule and Actual Glacier Response
 The simplicity of the expression for glacier sliding (equation (4)) has already been mentioned. While observations support a sliding law of this general form, it is extremely difficult to choose a single set of parameters that will match a set of observations. For the Columbia Glacier velocity data, there is no single choice of the parameters n, m and κ which allows an acceptable match between equation (4) and observations over all times and positions. This is hardly surprising, since influences not included in equation (4), such as longitudinal stress coupling, would be expected to play a role in determining sliding speed, and this is amply borne out by force-balance calculations [O'Neel et al., 2005]. Other effects of the approximate nature of the sliding law are apparent. Most important of these is the observation that the glacier has not, in fact, responded exactly as Figure 5 suggests. A value of c/u = −10, for example, would indicate that characteristic values of thickness h should be propagating upglacier at 10 times the ice velocity, or ca. 200 m d−1. Actual values of upstream propagation of characteristic values of h are on the order of 500 m yr−1, as can be seen in the upglacier translation of elevation profiles in Figure 4a. The difference lies in the fact that the sliding is not locally determined, in this case where “local” values are averages over length scales of approximately 2–4h. Longitudinal coupling and basal hydrological effects evidently exert influences over much larger length scales. For example, the onset of retreat was accompanied by significant increases in velocity over the entire length of Columbia Glacier, including the upper 35 km of the main glacier trunk that is grounded above sea level. The values of c/u shown in Figure 5 should be taken not as predictors or representations of actual flow history but as measures of instability at different locations on the glacier at different times. Far-field effects may modify the sliding response from what a sliding law dependent only upon local ice thickness, surface slope, and effective pressure indicates, but the basic source of instability, dominance of buoyancy over driving stress, initiates the thinning process, and c/u is a measure of that effect.
 Diffusion is well-known to quickly eliminate thickness perturbations in glacier geometries dictated by conventional sliding and deformation, but for an effective-pressure dependant sliding law of the form of equation (4), it is shown that the influence of diffusion declines with thickness and vanishes at floatation (equation (13)). Furthermore, since elevated velocities can be maintained by ice thickness h below the threshold h* established in equation (10), a locally confined topographic disturbance is not needed to sustain fast flow, and diffusion can act at intermediate thicknesses without eliminating the instability as long as it does not result in increased thickness. The tendency for diffusion to restore stability was investigated at Columbia Glacier by calculating the flux gradient ∂q/∂x from observations, and it was found that for 5 of the 9 years investigated after the onset of retreat, diffusion was accompanied by local thinning, and thus diffusion could not restore stability. For the remaining 4 years, diffusion could have resulted in stabilization, but only if the thickness increase accompanying ∂q/∂x < 0 caused h to exceed the threshold h*.
 A simple analysis based on the method of characteristics yields the result that for ice thickness sufficiently close to floatation, thinning disturbances propagate upstream. While there is no thoroughly validated specific form for a sliding law that depends on effective pressure, the results developed here apply to a reasonable range of parameter values in a generalized sliding law. Upstream propagation causes thinning to penetrate upstream from the near-terminus region, and leads to a thinning-accelerating instability compatible with the concept of tidewater instability expressed by Post  and Meier and Post , as well as by many writers subsequently. Bed and surface elevation data for Columbia Glacier are used to show that the theory predicts stability for the geometry of Columbia Glacier prior to the onset of its retreat in the early 1980s, a transition to instability around the time of onset of retreat, and continued instability for the subsequently evolving geometry up through the present day. The theory was applied to 13 other Alaskan glaciers for which bed and surface geometry and velocity and retreat rate are known, and the predicted stability or rapid retreat matched observations in all cases. The limited role of mass balance in tidewater retreat is also shown, reinforcing the idea expressed by Meier and Post , Clarke , and others that tidewater glacier retreat is not directly driven by climate. Tidewater glacier retreat has been widely described as being completely independent of mass balance, but in fact previous writers [Meier et al., 1980; Meier and Post, 1987; Clarke, 1987; Alley, 1991; Vieli et al., 2001] have suggested only that retreat is insensitive to, or indirectly related to, changing climate. Meier and Post  suggested that a succession of negative mass balance years may trigger retreat, and the model results of Vieli et al.  show such a response. It is proposed here that climatically induced long-term thinning triggers retreat through alterations of geometry that reduce resistive stresses more than driving stresses. Retreat is manifested in the reversal of the propagation speed c, and this reversal occurs at different times for different glaciers depending upon the details of glacier geometry and fjord depth.
 I thank Shad O'Neel for help with assembling information for this analysis and for very useful discussions, Robert Anderson for valuable comments and continuing encouragement, and Kees van der Veen, Andreas Vieli, and Faezeh Nick for careful reviews. Ian Howat kindly provided data for Helheim and Upernavik Glaciers in Greenland. Finally, I thank Robert Krimmel, Mark Meier, Austin Post, and the many other participants in the USGS Columbia Glacier Project, to whom the credit is due for assembling such a rich data set. This work was supported by National Science Foundation grants OPP-9614493, OPP-0121399, OPP-0228223, OPP-0240972, and OPP-0327345, awarded through the OPP Arctic Natural Sciences Program.