2.1. Toy model of glacier calving: discrete asymmetric random walk

In this model, the ‘state’ of the glacier is determined solely by the position of the ice front/terminus, assumed to be confined to discrete, equally spaced nodes (Fig. 1). The notation used throughout is that P_{i ,j} means the probability of transition from the jth state to the ith state. The random walk proceeds according to the following rules: (1) If the glacier terminus is located at the ith node, the probability itwill ‘hop’ forward or transition to the (i + 1)th node during a (short) time interval, Δt, is P_{i +1}, _i = α_i Δt. (2) The probability the terminus will hop backwards (by calving) to the (i − 1)th node is P_{i −1}, _i = β_i Δt. (3) The probability that the terminus position neither transitions forward nor backward is P_i , _i = 1 − (α_i + β_i )Δt, since probability mass distribution must sum to unity. The parameters α_i and β_i are called transition rates and have units of inverse time.

Fig. 1.

Illustration showing the discrete geometry used in the asymmetric random walk. The length of the glacier is assumed to be discrete and confined to equally spaced nodes, x = 0, Δx, 2Δx,. . . , NΔx. Transitions are permitted only to adjacent nodes.

The probability of two hops in a single time-step will be of order , three hops and so on. Taking Δt sufficiently small, these higher-order terms can be neglected. A realization of the model corresponds to a set of N steps, Δx, n = 1,…, N, where Δx_n takes on values of +Δx, 0, −Δx, chosen at random, based on the transition rates α_i and β_i . The α_i and β_i may be functions of position that depend on, for example, ice thickness, stress or strain rate or terminus velocity, but are assumed to be otherwise independent of time.

The simplest case corresponds to constant α_i and β_i . Figure 2 shows realizations of the random walk for three different choices of α and β. Units are chosen such that Δx = 1 and the time-step interval Δt = 0.1. In Figure 2a, the probability of advance is equal to the probability of retreat (α = β) and there is no preference for advance or retreat. Nonetheless, it is evident that even though the mean position of the terminus does not change, there are fluctuations in the magnitude of displacement that increase with time. In Figure 2b, α > β and, on average, the glacier advances, although there are again fluctuations that cause the occasional retreat. Similarly, Figure 2c shows that the glacier tends to retreat when α < β, also with occasional fluctuations superimposed on the mean trend.

Fig. 2.

Three realizations of the random walk for different transition rates. (a) Equal probability of advance and retreat for each time-step. Even though there is no preferred direction, the magnitudes of fluctuations away from x = 0 increase over time. (b) An example of glacier advance where α > β. (c) An example of glacier retreat where α < β.

For this simple example, the rate of terminus advance (negative implies retreat) is undefined during each hop and is zero between hops. This relationship does not particularly illuminate the longer-timescale trends in terminus advance. It is, however, emblematic of the observational problem of determining a calving rate or rate of terminus advance from a time series of individual calving events. Macroscopic variables, such as the rate of terminus advance and calving rate, are only meaningful when considering averages over time, lest the discrete nature of calving events manifests itself in the signal. Moreover, when observations are constrained to a small interval of time it is difficult to distinguish trends from fluctuations. For instance, Figure 2c shows that, although the overall trend is decreasing terminus position, fluctuations about this trend lead to periods of time with a net rate of advance.

2.2. Ensemble averages

It is often useful to calculate the time-averaged rate of terminus advance dL/dt:

where τ is the averaging interval, N is the number of steps and angle brackets are used to denote averages. Defined this way, the average rate of terminus advance is implicitly a function of the averaging interval, τ; different averaging intervals will yield different rates of terminus advance. Although rarely explicitly stated, this definition is typically implied whenever the terms ‘rate of terminus advance’ or ‘calving rate’ are used. Because of the dependence on averaging interval, τ, Equation (1) is of limited utility, unless, for sufficiently large τ, the average value becomes insensitive to the precise value of τ.

Alternatively, instead of averaging over time, many realizations (such as those in Fig. 2) can be computed and the rate of terminus advance determined by the average of these realizations:

where Δx_k are the individual realizations of terminus position (as a function of time) and M is the number of realizations. The system is said to be ergodic if time averaging over a suitably long time is equivalent to ensemble averaging over a large number of realizations. Although many systems are assumed to be ergodic (and this is the fundamental ansatz of statistical mechanics), there is no reason to suppose that glacier retreat/advance is ergodic. Nonetheless, the ensemble averages are often more convenient to perform and it is this expectation value that is used throughout the remainder of this paper, with the understanding that this ensemble-averaged expectation value is only equivalent to the time-averaged expectation value in the special circumstance that the system is ergodic (which it is for the random walk).

The advantage of the ensemble average is illustrated in Figure 3, which shows the probability distribution computed from a normalized histogram of 1000 realizations of the random walk at three different points in time for the same values of a and β as in Figure 2. In all three cases the probability distribution broadens over time, implying a larger probability that the glacier will be found a greater distance from the mean position. Figure 3b and c show that the centroid of the probability distribution drifts over time when a ≠ β with drift velocity determined by a − β. In all examples, fluctuations can constructively add up to drive the glacier terminus away from the expected position. Even when a = β and 〈dL/dt〉 = 0, the expected value of the total change in length of the glacier increases with increasing time, i.e. 〈L ²〉 > 0. This hints that it may be important to determine not only the peak of the distribution, but also its width.

Fig. 3.

Normalized probability distribution of terminus position at three points in time. The probability distribution is computed using two different methods. The first method is an ensemble average of 1000 realizations (shaded regions). The second method obtains the PDFs directly by solving the master equation (smooth black curves). The three panels correspond to the three different choices of transition probabilities used in Figure 2.

The advantage of the ensemble method is that individual trajectories can be tracked through time and compared with observations. However, the brute force method of computing the probability distribution by performing thousands of iterations is neither feasible nor desirable for large-scale ice-sheet models, where the computational cost of a single model run may be prohibitive. If individual trajectories are not needed, it is possible to formulate an equation – the so-called master equation – that describes the time evolution of probability density directly.

2.3. The master equation

The master equation (sometimes less prosaically referred to as the equation) is a differential equation describing the rate of change of probability, obtained in the limit Δt → 0 (e.g. Reference Van KampenVan Kampen, 1992). In the discrete case, it is usually written in the form:

where P_n is the probability that the terminus is located at the nth node at time t and the elements W_nn ′ are transition rates from state n′ to state n (e.g. Reference Van KampenVan Kampen, 1992, p. 96–97). Written this way, it is clear that the first term on the right-hand side of the master equation is a source term, representing the increased probability that the nth state is occupied by transitions from other states into the nth state. The second term on the right-hand side is a loss term, representing the decrease in probability due to transitions out of the nth state into other states. Note that in the master equation formulation, it is not necessary to know W_nn since the change in probability distribution is unaffected by self-transitions as these are neither gains nor losses. Equation (3) represents a linear system of first-order differential equations for the mass probability distribution that the glacier occupies any of the n sites.

For the discrete asymmetric random walk, excluding self-transitions (which do not change the probability), the only nonzero transition rates are:

Substituting these transition rates into Equation (3) provides the master equation for the asymmetric random walk:

The solution of Equation (5) for all n provides the evolution of the probability of finding the terminus at any position as a function of time.

The solid black curve in Figure 3 shows the evolution of the probability distribution, computed by solving Equation (5) numerically using a forward Euler time-step with a delta function initial condition. The initially narrow probability distribution broadens over time and drifts with rate α − β, analogous to what was observed using the brute force method of section 2.2. This time, the probability distribution is a smooth Gaussian, lacking the spikiness of the ensemble method. So long as individual realizations of the process are not what is of interest, solving the master equation is much more accurate and efficient than averaging over realizations. However, even when the mean terminus position is zero (α = β), the PDF does not have a steady-state solution. Instead, it broadens over time, and uncertainty in the location of the terminus position increases until all prognostic information is lost. Crucially, without a steady-state probability distribution, there is no macroscopic calving law that can be extracted from this model. Although simplistic, this model highlights the fundamental difference between the stochastic approach advocated here and fully deterministic approaches. In the absence of a sharply peaked probability distribution, the mean of the distribution may be meaningless. To obtain a useful mean rate of terminus advance, it is also necessary to know something about the width of the distribution and how this changes over time.

The utility of the master equation goes beyond merely being an efficient means of computing the probability distribution. If only the expected value of the probability distribution is sought, it is possible to solve for this (and higher-order moments) directly instead of solving for the entire distribution. This provides the potential for further gains in efficiency and foreshadows the techniques used to obtain calving laws that will be developed later.

The expected value of terminus position can be computed from Equation (5) by multiplying by terminus position, n, and summing over all states:

where 〈L〉 denotes the expected value of the terminus position and 〈α〉 and 〈β〉 denote the expected value of the transition rates. (In this case the expected values of α_i and β_i are trivial since both are constant.)

Equation (6) is valid for unit spacing between discrete nodes. If the nodes are instead spaced a distance Δx apart, the expected rate of change of terminus position is obtained by multiplying Equation (5) by nΔx and summing over all states to find

Equation (7) is a macroscopic equation describing the evolution of the terminus as a function of expected values of forward and backward transition rates and node spacing. Physically, the left-hand side of Equation (7) describes the expected rate of change of terminus position where, unlike previous calving laws, the expected value is an ensemble average instead of a time average. The term 〈αΔx〉 corresponds to the expected value of terminus velocity, u; the term 〈βΔx〉 corresponds to the magnitude of the expected calving rate, V _c. With these definitions, Equation (7) can be rewritten in the more familiar form:

The so-called calving law for the random walk is thus

For the random walk, the calving rate has an obvious physical interpretation: it is the expected value of the product of iceberg size and iceberg detachment frequency. With this in mind, the expected value of terminus advance is zero when α ≠ β, as previously demonstrated. When α − β, the expected rate of terminus advance (or retreat) is α − β.

The random walk model can be made more physically relevant by, for example, choosing spatially variable α_i and β_i that correspond to functions of glacier geometry/dynamics. For instance, the α_i must be related to the terminus velocity, u(x), while the probability of calving, β_i , may be chosen to be suitable functions of terminus ice thickness, water depth, etc. However, the model will still suffer from the defect that all icebergs have the same size and, in the absence of calving events, the terminus position hops forward discretely rather than continuously advancing. In section 3, a more realistic continuous model is proposed, that permits calving events of arbitrary size while allowing continuous advection of the terminus position (by ice flow).

3.1. Continuous advection of the terminus

In the discrete model, during each time-step there are three options: the terminus position either remains fixed, or hops forward a distance Δx or hops back a distance Δx. A more realistic model may assume that while calving is a random process, the terminus still advances every time-step based on the terminus velocity (determined from an ice-sheet model). Replacing the previously random transition rate, α, with a deterministic forward transition rate, u/Δx, where u is the velocity at the terminus, the master equation in the absence of calving is

where P_n = P(x_n ), P_{n −1} = P(x_n − Δx) and so on. In anticipation of taking the limit Δx → 0, the probabilities have been written explicitly as functions of the positions separated by a distance Δx. Writing them this way makes clear the connection with the upwind finite-difference version of the continuity equation.

Taking the limit Δx → 0, while keeping the product u(x) = α(x_n )Δx constant, yields an advection equation for probability:

where the lower-case symbol p has been substituted for P to indicate that the discrete probability distribution is now a probability density distribution (probability per unit length). Equation (10) is a probability continuity statement and shows that probability advects forward over time with velocity u. If the initial condition is a very narrow delta-like function, the delta function advects forward over time and the deterministic relationship with ice-front advance in the absence of calving is recovered. Numerical diffusion, associated with discretization of Equation (10) will cause an initially narrow distribution to broaden (or diffuse) over time, in a manner that is analogous to the broadening due to assuming a forward transition rate. Because of this, in some applications it may be convenient to exploit the equivalence of the discretized continuity equation with the discrete stochastic process with transition rates determined by the terminus velocity.

3.2. Icebergs of arbitrary size

To include calving of events of arbitrary size, the master equation can be generalized from discrete to continuous states by replacing the sum in Equation (3) by an integral over all states:

where the notation W(x′|x) indicates the transition rate from state x to x′ (i.e. from terminus position x to x′). In the continuous limit, the transition rates, W(x|x′), have the dimension of inverse time per unit length.

Combining Equations (10) and (11), the master equation can be expressed as an integro-differential equation

Because calving always decreases the length of the glacier, transitions x′ → x are prohibited if x > x′, thus W(x|x′) = 0 for x > x′. Choosing a coordinate system where x measures the length of the glacier so that x ≥ 0, the lower limit of the integrals is zero and the master equation can be more intuitively written

where the first term on the right-hand side represents the probability that a glacier longer than x calves back to x and the second term represents the probability that a glacier with initial length x calves back to x′, a length that is less than x.

It remains to specify the form for the calving transition rates, W(x′|x). Rather than attempting a complete specification of transition rates here, the flexibility of the theory is illustrated using two glaciologically relevant limiting cases to show the theory is able to reproduce (qualitatively at least) both the infrequent, sporadic detachment of large tabular bergs from ice tongues and ice shelves and the more frequent, smaller-sized calving events from tidewater glaciers. These two limiting cases are used in section 4.3 to illustrate how approximate calving laws can be deduced from the master equation, providing the motivation for the more general, systematic expansion of the master equation in section 5, from which universal calving laws are determined.

4.1. Example 1: calving from ice tongues and ice shelves

In this first example, it is assumed that calving from any where along the ice tongue length is permitted and is equally likely everywhere. At first sight this appears to be a severe and unrealistic assumption. However, for ice tongues and ice shelves that protrude beyond their embayments/pinning points it may be justified by noting that, away from the grounding line, ice thickness, strain rate, stress, etc. have small gradients and are nearly constant over length scales that are large compared to the ice thickness. If the transition rates are assumed to be functions of these large-scale internal variables, it is reasonable to suppose that transition rates are also nearly constant. This line of reasoning neglects the small additional bending stresses that occur near the icetongue/ice-shelf terminus that might favor the detachment of ice-thickness-sized bergs (Reference ReehReeh, 1968). However, since mass lost by calving from ice shelves and ice tongues is dominated by large tabular bergs, neglecting these stresses may be a reasonable first approximation. In principle, it is possible to include bending stresses along with structural features such as pre-existing rifts, suture zones that might hinder or favor detachment of icebergs from some places, by allowing for spatially variable transition rates.

The assumption of constant transition rates also corresponds to a modest generalization of a calving law proposed by Reference Benn, Warren and MottramBenn and others (2007). They argued that an iceberg will detach from a glacier anywhere the glaciological stress is sufficient to cause a fracture to either propagate all the way to the bed or, if the glacier terminates in a body of water, to sea or lake level. The latter condition was invoked under the assumption that a well-developed drainage network exists and this causes crevasses that penetrate to water level to immediately fill with water and hydraulically fracture to the bed. This condition is also assumed to be true irrespective of the type of glacier (e.g. freely floating, tidewater, confined or unconfined) and irrespective of whether the body of water is a lake or an ocean. Cold Antarctic ice shelves and ice tongues are unlikely to have a well-developed drainage system. However, one may argue that this condition may be approximately satisfied if basal crevasses intersect with surface crevasses.

In the absence of meltwater, the ratio of the depth to which a surface crevasse will propagate, d, to the ice thickness, H, can be calculated based on the depth when those forces opening a crevasse are exactly balanced by the hydrostatic weight of the ice closing the crevasse (Reference Benn, Warren and MottramBenn and others, 2007):

where is the depth-averaged longitudinal strain rate, ρ _i is the density of ice, g is the acceleration due to gravity, A is the rheological rate parameter and n is the flow-law exponent. A straightforward, heuristic generalization of the crevasse penetration depth criterion supposes that an iceberg detaches when the ratio of surface crevasse depth to ice thickness exceeds a predefined threshold that is ≤1. However, for a freely floating ice tongue, unencumbered by lateral drag or ice rises, the predicted ratio, d/H, in the long-wavelength shallow-shelf approximation is constant. This model then leads to the unsatisfyingly unstable prediction that ice tongues either advance to infinity or retreat to the grounding line, depending on the value chosen for the calving threshold.

In keeping with the spirit of the present work, one might instead assume that the transition rates are functions of the penetration ratio, i.e. λ = f(d/H) = constant. Conceptually, the transition rates would govern the probability that a surface crevasse intersects with a basal crevasse and this probability is controlled by the penetration ratio, d/H. Unlike in the purely deterministic viewpoint, random perturbations to the stress field caused by ocean waves, collisions with passing icebergs, large local variations in basal melt/refreeze rates, etc., may result in basal or surface crevasses that penetrate to a greater (or lesser) depth than one would otherwise expect.

Adopting the hypothesis that transition rates are (nearly) constant implies

where λ is a constant transition rate. The master equation is then written

where x = 0 represents the grounding line, a point beyond which the tongue is (for illustrative purposes) prohibited from retreating. For large x, the right-hand side of Equation (15) is negative, implying very long ice tongues are more likely to calve back than to advance. In contrast, for small x, the second term on the right-hand side will be negligible and the right-hand side will be positive. This means short ice tongues will be more likely to advance than retreat. The probability of calving increases (decreases) with increasing (decreasing) ice tongue length. Because of this, a steady-state distribution of ice tongue lengths is possible.

To provide a more concrete visualization of the calving behavior of the ice tongue model, Figure 4a shows an example of a realization of this model, computed using an analytic solution for the ice tongue velocity (e.g. Reference Van der VeenVan der Veen, 1999, p.163, equation (6.6.8)) with ice thickness and ice flow velocity at the grounding line prescribed to be 1000 m and 250 m a⁻¹, softness of the ice B = 10⁸ Pa s^1/3 and an initial condition of zero length. For this example, the terminus exhibits a sawtooth pattern of advance and sporadic retreat, reminiscent of the detachment of large, tabular bergs from ice tongues and ice shelves. Although calving events of all sizes are equally likely, large calving events remove more of the ice tongue and dominate the calving portion of the signal.

Fig. 4.

(a) The blue line shows a realization of the ice-tongue calving model with terminus velocity computed using an analytic solution for terminus velocity (e.g. Reference Van der VeenVan der Veen, 1999, p.163, equation (6.6.8)). For this example, the thickness at the grounding line is 1000 m, the ice-flow velocity at the grounding line is 250 m a⁻¹, the rate factor is B = 10⁸ Pa s^1/3 and the (constant) transition rate λ = 0.01 km⁻¹ s⁻¹. The red bars show the size of icebergs that detached. The solid black curve shows the terminus position computed using the macroscopic calving law (Equation (30) with corrections due to fluctuations omitted). The shaded gray area traces out the variance in terminus position, computed using Equation (44). (b) The steady-state probability density distribution, computed from the master equation using the same parameters as in (a) (shaded blue region) and for an increased ice-flow velocity at the grounding line of 800 m a⁻¹ (shaded red region). Here, unlike in the random walk, the PDF shown corresponds to a steady-state solution of the master equation that is reached after a few decades.

The recurrence interval between calving events is determined by the magnitude of the transition rates. Larger transition rates lead to more frequent calving events and a shorter ice tongue. Conversely, a larger terminus velocity will result in faster terminus advance between calving events and hence a greater mean ice tongue length. However, since calving is equally likely from anywhere within the ice tongue, a longer ice tongue permits larger calving events. This will result in a longer-tailed probability distribution. Hence, longer ice tongues have a larger variance and skewness in the probability distribution than smaller tongues. This is illustrated in Figure 4b, which shows the probability distribution computed by solving Equation (15) for two different inflow velocities. In this example, in direct contrast with the random walk, a steady-state probability distribution is reached after a few decades. Furthermore, the shape of the probability density distribution is clearly non-Gaussian, with a long tail that increases with increasing terminus velocity. To reiterate, the prediction of sporadic calving of large tabular bergs emerges in this example solely because transition rates are nearly constant and is independent of any assumptions about the actual physics that governs the transition rates.

4.2. Example 2: calving from tidewater and lacustrine glaciers

In the previous example, stress, strain rate, ice thickness, etc., were all assumed to have relatively small gradients along the entire ice tongue length. In contrast, many calving glaciers (defined here to be any glacier with a grounded terminus, including tidewater and lacustrine) exhibit large variability in strain rate, stress and ice thickness in the near-terminus region, especially as the glacier terminus approaches flotation (Reference O’Neel, Echelmeyer and MotykaO’Neel and others, 2001). In keeping with the previous discussion, it may still be appropriate to assume that transition rates are a function of the penetration ratio, λ = f(d/H). However, it is now assumed that f(d/H) is strongly peaked in a near-terminus region of characteristic size Δx. In this case one might suppose that the characteristic size of this region ought to be approximately proportional to the ice thickness, H. This appears to be reasonable, in that icebergs that calve from calving glaciers tend to be ice-thickness and/or sub-ice-thickness sized rather than tabular.

Assuming the transition rates beyond a distance H are negligibly small, the master equation for near-terminus calving is:

It is further assumed that the probability that an iceberg detaches at point x depends only locally on properties at x (and not the terminus position, x′) and hence the transition rates can be written more simply in the form W(x|x′) = λ(x), where λ(x) is a spatially variable transition rate. The integrals on the right-hand side of Equation (16) can be approximated using the midpoint rule

followed by a Taylor series expansion about x ± H/2,

to transform the master equation from an integro-differential equation into a partial differential equation:

The iceberg size vanishes as H → 0, as does the calving rate. This shows that the calving rate is finite for finite ice thickness, but decreases (increases) as the ice thickness decreases (increases) and is analogous to the dependence of calving rate on node spacing, Δx, determined for the random walk (section 2.3).

Finally, the transition rate per unit ice thickness is approximately constant, so defining λ(x) = 2β(x)/H(x), where the factor of two has been introduced for convenience, and making use of the chain rule followed by the product rule, the master equation now becomes

Note that β(x) has units of inverse time, consistent with the discrete transition rates for the discrete random walk discussed previously.

Taking the ice thickness to be approximately constant in the near-terminus region (i.e. averaged over a one-icethickness spatial scale), the last term on the right-hand side of Equation (18) can be dropped and the master equation further approximated as

This last simplification corresponds to neglecting the variability in the characteristic size of icebergs (and hence calving rate) induced by gradients in the near-terminus ice thickness.

If the spatial derivatives are approximated using upwind differences and defining the discrete transition rates

the discretized version of Equation (19) has the same form a the asymmetric random walk with spatially varying transition probabilities:

where P_n = p_n Δx. Unlike the random walk, where the terminus can advance, retreat or stay in the same position, here the terminus continually advances, by a distance uΔt each time-step (where u is the terminus velocity).

Figure 5a, shows one realization for this calving model, computed with constant terminus velocity (u = 1 in dimensionless units) and transition rates linearly increasing with distance, α = x/40. This choice is arbitrary, but if the transition rates increase with water depth (increased probability that a surface crevasse intersects with a basal crevasse or plumbing system), this example would correspond to a calving glacier advancing into deeper water. Inspection of Figure 5 shows that there is an initial period of slow terminus advance with superimposed small fluctuations until t ≈ 100, after which the terminus position fluctuates about a steady position. The probability distribution, determined by solving the master equation at three points in time with a delta function initial condition at x = 0, is shown in Figure 5b. The peak of the PDF advances and broadens over time until a steady-state mean and width are reached. Unlike the ice shelf/tongue, the distribution quickly evolves to a (nearly) Gaussian shape with little obvious asymmetry.

Fig. 5.

(a) A single realization of the calving model with constant terminus velocity (u = 1 in dimensionless units) and transition rates linearly increasing with distance, β = x/40. The solid black curve shows the terminus position computed using the macroscopic law (Equation (22)). The shaded gray area traces out the variance in the terminus position, computed using Equation (46). (b) The probability distribution computed from the master equation at three points in time. The initial condition is a delta function centered at x = 0. As in (a), the peak of the PDF advances and broadens over time until a steady-state mean and width are reached. Unlike the ice-shelf/icetongue example, the distribution appears nearly Gaussian at all but the earliest stages of evolution.

4.3. Approximate calving laws

The previous examples showed how the probability distribution for terminus position could be computed directly from the master equation using different limiting assumptions about the form of the underlying transition rates. In this subsection, approximate macroscopic calving laws for both glaciological regimes previously considered are deduced by computing expected values of terminus position directly from the master equations determined for each of the regimes. Calving laws for the two limiting cases are considered separately below. However, the reader should keep in mind that, despite the more involved mathematical manipulations, the procedure used is identical to that used for the random walk discussed previously.

5.1. Expansion of the master equation

The previous two examples illustrate that two seemingly different calving laws can be derived from the same underlying master equation, depending on the form of the transition rates. Moreover, these examples hint at one of the limitations of so-called calving laws; they appear to be restricted to cases where fluctuations are small enough that they can be neglected. This fact is exploited in this subsection, where a systematic expansion of the master equation, based on the size of the fluctuations measured relative to the system size, is developed. This systematic expansion reduces the effort of determining the calving law to that of evaluating an integral by quadrature. An additional benefit of the expansion is that it provides a method of determining the magnitude of the fluctuations. The magnitude of fluctuations can then be used to determine when the expansion is expected to fail, indicating when the expansion must either be abandoned or higher-order terms in the expansion retained.

The expansion presented is formidable and readers may wish to skip ahead to subsections 5.2 and 5.3, where the calving law along with an evolution equation for the variance are finally deduced. The expansion closely follows the standard Ω expansion posed by Reference Van KampenVan Kampen (1992), and readers are encouraged to consult this well-written textbook for supplementary descriptions of the arcane sequence of variable transformations and Taylor expansions that follow. The reader should keep in mind that the overarching purpose of the expansion is to (1) formulate an expression for the expected value of terminus position that is independent of the variance or higher-order moments of the distribution and (2) provide a separate evolution equation for the variance in the system.

The expansion proceeds as follows. Let Ω denote the characteristic size of the system. The size may be the characteristic lateral extent of an ice tongue/shelf, glacier or ice sheet or some subsection thereof. Since calving occurs from the ice-sheet margin, the ratio of terminus position (current glacier length) to system size (characteristic glacier length) is a number of order one, irrespective of system size. The size of icebergs (and hence fluctuations in terminus position) will usually be much less than the system size. The smallness of the ratio of fluctuations to system size provides the basis of the expansion method. With this in mind, a coordinate transformation is made to the intensive variable X = x/Ω. Time is simultaneously rescaled by setting τ = t/Ω. Making a transformation to the new variables, the transition rates may be written as a function of starting position and jump size (i.e. initial terminus position and iceberg size)

where the dependence on system size has been explicitly retained and r = x′ − x represents the iceberg size. The Ω subscript has been appended to W to emphasize the dependence on system size, and Φ is used to denote transition rates that depend on the intensive variable, X′, and iceberg size, r. Insisting that calving only results in transitions that decrease glacier length implies Φ(X′; r) = 0 unless r > 0 (i.e. negative iceberg size is prohibited).

Noting that W _Ω (X′|X) = Φ(X′; −r), because iceberg size in this case is x − x’ = −r, and making use of the fact that X′ = X + r/Ω, the master equation can be written

where the symbols U and are introduced to denote the probability density, p(x, t), and terminus velocity, u(x, t), as functions of the intensive variables, X and τ:

With this rescaling and change of variables, the master equation is now dependent on the ratio of terminus position to system size and on iceberg size. Furthermore, the transition rates and probability densities now have arguments shifted by r/Ω and it is this ratio of iceberg size to system size that can serve as the small parameter in the expansion.

First, however, it is necessary to adopt the ansatz that the non-dimensional terminus position, X, can be represented by

where φ(τ) will be chosen to follow the peak (or some approximation thereof) of the probability density distribution and ξ is a random variable that accounts for fluctuations about the mean terminus position (see, e.g., Van Kampen, Reference Van Kampen1992, for a more detailed motivation of the ansatz). More physically, the change of variables can be viewed as introducing a variable, φ, that follows the ratio of terminus position to system size and a variable, Ω^−1/2 ξ, that describes how the width of the distribution fluctuates. These two variables together provide something like a Lagrangian description of the terminus position probability distribution. Note that as Ω → ∞, the width of the distribution (or size of the fluctuations) vanishes and the distribution becomes sharply peaked. This limit, in which icebergs are infinitesimally small compared to the size of the glacier, corresponds to a deterministic calving process. The ansatz is the key step in the expansion.

Denoting the new probability density distribution, Π,

now a function of ξ and τ, and substituting the ansatz into Equation (32) shows that the first term in the integral on the right-hand side can be written in the form

whereas the second term in the integral becomes

The argument in the first integral relative to the second is shifted by Ω^−1/2 r. This shift can be accounted for with a Taylor expansion, whence the right-hand side of the master equation becomes

where the first term in the expansion cancelled with the second integral. The integrals can be expressed more concisely by introducing the iceberg jump moments

for ν = 1,2, etc. The iceberg jump moments correspond to integrals of iceberg size, raised to an integer power weighted by the transition rates.

Armed with these simplifications and after making heavy use of the chain rule, Equation (32) can be written in the form

The dependence of ξ apparent in both jump moments and terminus velocity can be removed by yet another Taylor expansion

where primes denote differentiation with respect to ξ. Assembling all components, the expansion for the master equation takes the final form

In this form of the master equation an infinite hierarchy of closed equations can be constructed by equating terms of equal size and the series can now be truncated whenever additional terms become small.

5.2. The macroscopic calving law

Examining Equation (36) more closely shows that the expansion will fail unless both of the terms of order Ω^1/2 can be arranged so they exactly cancel for arbitrary Π. This can only be arranged if φ is chosen to obey the ordinary differential equation

where the first jump moment (or calving rate) is an integral over transition rates over all iceberg sizes

This, however, is precisely the macroscopic calving law sought in the first place. Recalling that φ is the ratio of terminus position to characteristic system size and U is terminus velocity scaled by characteristic system size, the macroscopic equation can be re-expressed in terms of extensive variables and rescaled back to the original variables:

where the connection between calving rate and the first jump moment has been made explicit and the rescaled version of Equation (38) provides the calving law:

This law, unlike previous laws, is valid for any glacier (floating, grounded, temperate or cold), provided the transition rates can be specified. Furthermore, since the calving rate and terminus velocity now depend on the current terminus position, L, and not at all on the variance of terminus position, the macroscopic equation is an ordinary differential equation that can easily be incorporated into numerical ice-sheet models.

In the extreme case that transition rates are constant, one must integrate over the entire length of the ice shelf/tongue/glacier. Under this assumption, the integral in Equation (38) reduces to the calving rate determined in Equation (27):

In contrast, if the transition rates are strongly peaked near the terminus, evaluating the quadrate under the assumption that the transition rate per unit ice thickness is constant over the ice thickness yields the tidewater calving rate:

where, β is defined as in section 4.2. Equation (40) provides a universally valid calving law. However, the calving law may take on widely disparate forms, including the two examples considered in section 4, depending on the processes that determine the transition rates. Given the wide variety of glaciological conditions in which vigorous calving is observed, it is no wonder that an all-encompassing calving law has not previously been discovered empirically. Despite the fact that Equation (38) provides a convenient recipe to compute the calving law, the accuracy of the calving law hinges on the supposition that the remaining terms in the series are small. To determine if this is true one must proceed to higher-order terms in the expansion.

5.3. Fluctuations and the white-noise approximation

Having determined the macroscopic behavior, it is now possible to proceed to the next order in the expansion. Equating terms of yields

where the reader is reminded that a ₂ is the second jump moment of the distribution

and primes denote differentiation with respect to ξ. Equation (41) is a linear Fokker–Planck equation. The equilibrium probability distribution can be shown to be Gaussian and the first and second moments can be found by multiplying Equation (41) by ξ and ξ ² and integrating over all states (e.g. Reference Van KampenVan Kampen, 1992):

Although not necessary, it is convenient to choose ξ = 0, such that it coincides with φ. With this choice, the first equation can be ignored and the second equation provides the evolution of the width (or variance) about φ. The implication of these equations is most concretely illustrated by returning to the two examples discussed previously.

5.4. Fluctuations, the Langevin equation and steady states

Combining the macroscopic equation with the white-noise approximation provides an equation for the evolution of terminus position:

where η is a source of random, uncorrelated noise with amplitude that varies with L and is determined by solving for ξ ². If fluctuations are negligible, this equation is entirely deterministic. However, in general, fluctuations do not vanish. To illustrate the potential effect of fluctuations, assume there are two or more steady-state terminus positions, denoted L ₀, L ₁ and so on. The multiple steady states might be due to bedrock topography, but are otherwise unspecified. The presence of fluctuations means that a glacier with steady-state terminus position L ₁ can fluctuate out of state L ₁ into L ₀ and vice versa. This type of behavior may have important consequences for the episodic retreat and stability of tidewater and outlet glaciers.