Characteristics Of Surging Glaciers

In their review of the characteristics of surging glaciers Reference Meier and PostMeier and Post (1969) include the following observations:

1. “All surging glaciers surge repeatedly” with an apparently periodic surge cycle which is relatively constant for any given glacier.Footnote *

2. The active surge phase commonly lasts 2-3 years followed by a quiescent phase of approximately 15 to over 100 years with a common range of 20-30 years.

3. During the active phase, the flow velocity increases to at least ten times its quiescent value.

4. “Glacier surges occur in almost all climatic environments. . . . Marine, temperate glaciers in areas of high accumulation and ablation activity as well as subpolar glaciers in continental, cold areas can surge.”

5. The geographic distribution of surging glaciers is not random. In North America, all but a few surging glaciers are located in a relatively small area near the Alaska–Yukon border and none are found in the Brooks Range, Kenai Mountains, west and central Chugach Mountains, west and central Wrangell Mountains, Coast Mountains, Rocky Mountains, Cascade Range, Olympic Mountains, or Sierra Nevada (Reference PostPost, 1969). (A good example of the non-random distribution of surge-type glaciers is the remarkable concentration in the Steele Creek basin, Yukon Territory (Fig. 1).)

Fig. 1.

Surge-type glaciers in the Steele Creek basin, rukon Territory, Canada (after Reference Ragle, Ragle, Collins and TolmanRagle and others, unpublished). The shading indicates the probability that a glacier is of surge type. Darkest shading denotes a glacier which definitely surges; lightest shading denotes a glacier which probably surges. Unshaded glaciers show no sign of surge behavior.

An important corollary of the hypothesis that surging is thermally controlled is that temperate glaciers cannot surge by this mechanism. The statement “temperate glaciers can surge” therefore demands special attention, for if temperate glaciers do surge one of the following statements must be true:

1. No glaciers surge by thermal instability.

2. There are several surge mechanisms one of which could be thermal instability.

When Meier and Post asserted that temperate glaciers surge, their claim was based, not on direct temperature measurements, but on the fact that some surge-type glaciers are located in temperate marine environments. In recent years more geophysical data have become available and these results are summarized in Table I. Because so few temperature measurements have been taken in surge-type glaciers, seismic and radar sounding results are also useful in distinguishing between temperate and cold ice, although these indirect determinations should be viewed with a critical eye. It is a well-known fact that seismic P-wave velocity decreases as the melting point of ice is approached (Reference Robin deRobin, 1958) and velocities of 3 650 m s^-1 or less are characteristic of temperate ice. Seismic velocity measurements therefore give bulk temperature information. Attempts to sound temperate ice masses using pulsed radar systems operating in the frequency range 35-620 MHz have not met with notable success and we take the successful completion of a radar-echo survey in this frequency range as evidence that a glacier is predominantly cold. A major drawback of using shallow ice temperature measurements, seismic refraction, and radar sounding to detect cold ice is that they cannot rule out the presence of an extensive layer of temperate ice at the bed, and it is the bottom regime which is critical to surging. Finally, the thermal regimes of surge-type glaciers may be sufficiently complex that they cannot be satisfactorily determined by one or two measurements.

At least three surging glaciers listed in Table I, the Kolka, Black Rapids and Variegated, are probably temperate so it appears that all surges cannot be thermally controlled. These determinations are based on a small number of shallow ice or firn temperature measurements, so are not definitive. The thermal regimes of surging outlet glaciers from Vatnajökull have not been measured; the characterization of Vatnajökull as temperate is based on measurements in the accumulation zone and these may be misleading. Possibly parts of these glaciers are not temperate in which case a thermal mechanism is still admissible.

Even if it is true that no glaciers surge by thermal instability there are many cold glaciers which surge and for these glaciers thermal processes must influence the timing of the surge cycle—a surging glacier cannot be frozen to a large fraction of its bed during a surge advance. We shall therefore continue our discussion of thermal control along two separate lines:

1. To examine whether thermal instability can account for surge behavior in cold glaciers.

2. To examine the thermal effects of surging on cold glaciers. A necessary condition for a glacier to surge is that much of its bed be at the melting point. If the deep temperature of a cold glacier alternates between periods when the bed is mainly cold and when it is mainly temperate, surges are likely to be thermally regulated.

Table I.

Apparent thermal regimes of some surge-type glaciers

* May not surge.

Cold Surge-Type Glaciers

Only two cold surge-type glaciers, the Rusty and Trapridge in (he Yukon Territory, have a reasonably well-known thermal structure. Both glaciers are small but 15 other surge-type glaciers, spanning a wide range of sizes, lie in the immediate vicinity (Fig. 1) so it is reasonable to suppose that all these glaciers surge by the same mechanism.

The main features of the thermal regimes of Rusty and Trapridge Glaciers are that substantial portions of the beds are at or near the melting point and the glacier snouts appear to be frozen to the beds (Fig. 2a). This situation is similar to the “cold ring” structure which Reference SchyttSchytt (1969) noted in the surge-type ice caps of Svalbard, although in that case temperate ice was held in by a ring of cold ice (Fig. 2b).

The model of thermal instability in cold glaciers which we shall now consider in mathematical detail closely resembles that originally proposed by Reference Robin deRobin (1955). The surge starts when the basal ice temperature reaches the melting point. Upon the onset of basal melting the thermal boundary condition at the ice rock interface changes from continuity of flux to a fixed temperature at the interface and discontinuous flux. Two competing processes control the duration of the sliding phase: frictional heat melts water at the ice-rock interface and sustains the surge, but the extension of the glacier as it advances causes the basal ice temperature gradient to increase. Eventually thermal flux away from the ice-rock interface exceeds

Fig. 2.

a. The thermal regime of Trapridge Glacier Drilling sites and maximum drilling depths are indicated by vertical lines. Depth to bedrock was found by radar sounding, b. The “cold ring” thermal regime of ice caps in Nordaustlandet (after Reference SchyttSchytt, 1969).

the frictional heat generation and the glacier refreezes to its bed. A quiescent period follows until the combined effects of surface accumulation and compressive glacier flow cause the bed temperature to return to the melting point. When certain conditions on the ice thickness, the surface temperature, and the geothermal flux are satisfied, a periodic cycle of advance and quiescence results.

There is disagreement over whether a finite thickness of temperate basal ice must form before sliding can occur (Reference WeertmanWeertman, 1967; Reference LliboutryLliboutry, 1967). For simplicity we assume that a temperate layer is not required, but it would be quite possible to develop an analogous model of thermal instability based on changes in thickness of a temperate basal layer.

Heat conduction equations

The equations of heat conduction in bedrock and glacier ice for two-dimensional flow are

(1)

(2)

(see Reference Carslaw and JaegerCarslaw and Jaeger, 1959, ch. 1). For a material satisfying (lien’s flow law

(3)

where B ₀ = B(T ₀) exp (E/RT ₀) and T ₀ = 273 K. The justification for neglecting all thermal effects due to water flow through the glacier in Equation (2) is that the glacier is assumed to be cold so that an extensive internal drainage system is unlikely to form. If this assumption is untrue, prior knowledge of the nature and extent of the internal drainage system would be required to correct for the transport of heat by water flow.

The boundary condition at the ice-rock interface takes one of two forms. If the temperature at the interface is below the melting point, no melting occurs at the interface and the thermal flux from the bed is exactly equal to the flux into the bottom ice; thus

(4)

at the boundary. If the bed is at the melting point the boundary condition changes to T ₁ = T ₂ = T _m. When the bed is at the melting point the flux may become discontinuous at the boundary and any imbalance is accounted for by melting basal ice or freezing basal water. This change in boundary condition plays an important role in the thermal control of surging.

Equations of motion

A completely rigorous model of glacier surging would require a solution of equations for creep motion together with a realistic sliding law. For several reasons this is not a practical course to follow. At the present time it is probably not computationally feasible to solve the full equations of motion for a glacier, so one must settle for approximate flow solutions. The sliding law, and especially the lubricating effect of basal water, is still a matter of great uncertainty and until some general agreement is reached there seems little point in adding unnecessary complexity to surge models. But until the shortcomings of sliding theory are removed no complete theory of surging can be constructed.

The use of specific approximations to the flow and sliding equations will be described in greater detail in the section on numerical modelling.

Water generation

Sliding at the bed creates frictional heat at a rate proportional to U _bη. The rate of melting from the glacier bottom is

(5)

where the temperature derivatives are taken at the ice-rock interface. This expression is a straightforward generalization of an equation given by Reference WeertmanWeertman (1962), and is true regardless of the sliding law. It is assumed that because the glacier is cold, water from the surface does not reach the bed.

Basal water flow

The rate of sliding is dependent on the amount of water at the glacier bed which in turn depends on the rate of water production and the rate at which water flows from the bed. The principle of conservation of water volume gives the continuity equation

(6)

(Reference WeertmanWeertman, 1962) where

(7)

u(x,y, t) is the water flow velocity at the bed averaged across the width of the glacier.

Before one can solve Equation (6) it is necessary to introduce an equation of state q = q(h) relating q to the mean thickness of basal water. Unfortunately the form of this equation is a matter of current disagreement. If basal water flows as if between two parallel plates separated by a distance h and inclined at an angle δ

(8)

This form for the equation of slate was proposed by Reference WeertmanWeertman (1962) and later, modified to allow for a generalized pressure gradient, defended by him on theoretical grounds (Reference WeertmanWeertman, 1972), If water does not flow as a sheet some other form of the equation of state must replace Equation (8).

Finite-difference scheme

Because longitudinal temperature gradients in glaciers are usually small in comparison to those measured perpendicular to the bed the longitudinal heat conduction can be neglected so that the ice temperature equation becomes

(9)

(Reference PatersonPaterson, 1969, p. 180). To write Equation (9) as a finite-difference equation we first replace the variables x, y, and t by the discrete variables iδx, jδy and nδt where the Indices take integer values; T_{i ,j} ⁿ therefore represents the temperature at the mesh point (i, j)evaluated at the nth time step. Following a partially implicit numerical scheme, we write

(10)

where λ = kδt/(δy)² is a dimensionless quantity. For U_{i, j} ⁿ = V_{i, j} ⁿ = A_{I, j} ⁿ = 0 we obtain the form of the difference equations in rock. At the nth time step all quantities on the right-hand side are known and the unknowns are the temperatures at the (n+1)th time step. These can be computed by solving a system of tridiagonal equations obtained from Equation (10) and the boundary conditions. Reference Carnahan, Carnahan, Luther and WilkesCarnahan and others (1969) give a clear account of the solution of tridiagonal systems; a comprehensive discussion of finite-difference methods can be found in Reference Forsythe and WasowForsythe and Wasow (1960).

Stretched slab model

The aim of the approximations which lead to the stretched slab model is to eliminate x dependence from the temperature equations and to obtain a very simple velocity distribution. An earlier version of this model (Reference Hoffmann and ClarkeHoffmann and Clarke, 1973) suffered from lack of generality and a computational error.

Fig. 4.

The "stretched slab" model of surging.

Let us represent a surge-type glacier by an inclined slab truncated at x = X and attached to immobile ice or rock at x = 0; thus o ≼ x ≼ X represents the “active zone” (Fig. 4) The upper surface of the slab y = γ (t) is held at a constant temperature T_s and the thickness of the glacier changes in response to accumulation at the surface and motion of the slab When the snout advances during a surge, the glacier thins: during the quiescent phase it thickens. Because the creep contribution to the flow is small compared to the sliding contribution, we neglect it entirely during the surge and assume that the forward ice velocity is given by the sliding rate which is taken to vary linearly with distance from the snout so that

(11)

where U _o is the snout velocity. During an advance the active zone is consequently under uniform tension, and the vertical component of velocity is

(12)

where V ₀ — γU ₀/X is the rate at which the glacier changes thickness due to extension and W is the rate at which basal melting removes ice from the bed. During quiescence the glacier is assumed to thicken due to a uniform compressive strain-rate VQ/γ. The shearing contribution to the flow is neglected though not its effect on internal heat generation.

If the water production does not vary rapidly on the time-scale of hours the basal water flow will be steady and

(13)

Giving

(14)

By assuming a relationship between q and h such as Equation (8) the thickness of the basal water film could be determined and from that the sliding rate found.

We have done this for Weertman’s sliding laws

(15)

(16)

(Weertman, Reference Weertman1962,1969) and water-flow equation (8) and find that unreasonably smooth beds must be assumed before satisfactory surge velocities are achieved. It is, in part, this difficulty that led Reference Robin de and WeertmanRobin and Weertman (1973) to propose stress-dammed basal water flow as a cause of surges. Apart from their lack of plausibility, very smooth beds make the friction term overwhelm the conduction term in Equation (5) so the model surges do not stop.

Owing to the uncertainties in the sliding and water-flow equations a simplified approach is followed. The two equations are combined into a single equation to which no physical interpretation is ascribed. We assume that for lubricated sliding

(17)

and that a general equation of state for water flow has the form

(18)

where D ₃, D ₄, λ and ξ are constants. For Weertman’s theories λ = 1 and ξ = 3. Thus Equations (17) and (18) combine to give

(19)

Certain choices of the parameters D ₀ and v were found to give acceptable results. The sliding velocity computed in this way was taken as the velocity at x = X/2 and the velocity was taken to vary linearly between x = o and the snout at x = X according to Equation (11).

Table II.

Trapridge glacier model A

Model computations are carried out in the following manner: The surface temperature, geothermal flux, initial thickness, initial length, accumulation rate, and rate at which compressive flow raises the ice surface during the quiescent phase, and a sliding rule of the form of Equation (19) are chosen, and physical constants such as the flow-law coefficient and exponent (used in computing the deformational self-heating but not the actual flow) and the thermal properties of ice and rock assigned. If the bed is at the pressure-melting temperature sliding begins and the water flow rate at x = X/2 is calculated from Equation (14). Sliding continues at a rate determined by Equations (14) and (19) and the slab stretches and thins until conduction losses into the bottom ice cause the water layer to disappear and sliding to stop. Throughout the quiescent phase the glacier thickens due to the combined effects of surface accumulation and compressive flow until the bed returns to the melting point and a new surge cycle begins. The ice thickness, intensity, and duration of surges, and the length of quiescence, are allowed to adjust freely to the steady surface accumulation rate and the compressive thickening during quiescence. After 25 or more cycles the temperature distribution, thickness, and rate of advance become perfectly periodic.

Discussion of stretched slab modelling results

A model of cyclic surges for Trapridge Glacier was constructed using data extracted from Reference SharpSharp (1947,1951), Reference CollinsCollins (1972), Reference Goodman, Goodman, Clarke, Jarvis, Collins and MetcalfeGoodman and others (1975), and Reference Jarvis and ClarkeJarvis and Clarke (1975). The principal model inputs and outputs are given in Table II and the computed pre-surge and post-surge temperature distributions in ice and rock are presented in Figure 5.

Fig. 5.

Pre-surge and post-surge temperature distribution computed for Trapridge Glacier Model A. For this calculation the thermal conductivities of ice and rock have been taken as equal.

As the duration and intensity of the surge is largely determined by the non-physical parameters D ₀ and v while the duration of quiescence depends on the ill-determined upward rate of surface movement during the post-surge recovery, the 40 year surge cycle found for our Model A is not a reliable predictor of the actual periodicity for Trapridge surges.

Two-dimensional model

The purpose of the foregoing stretched-slab model was to demonstrate the possibility that thermal processes control the existence and thickness of the basal water layer in cold surge- type glaciers and may in this way determine the onset and termination of a surge advance. We shall now investigate the effects of cyclic surging on the temperature field within a cold glacier. In the following two-dimensional model we are not concerned with what actually drives the surge, but with the thermal consequences of surging. Instead of allowing the timing of the surge cycle and the ice displacement to be determined by processes at the glacier bed, we assume a fixed surge cycle with a constant snout velocity during each surge. The surface profile and temperature field are allowed to adjust to this forced cycle of surging and eventually become cyclic with time as well, losing all memory of the assumed initial conditions.

The conservation equation

(20)

governs the evolution of the surface profile where Q the ice flux per unit width is given by

(21)

(Reference Lighthill and WhithamLighthill and Whitham, 1955; Reference WeertmanWeertman, 1958; Reference NyeNye, 1958 ,1960; Reference WhithamWhitham, 1974). Rather than solve the equations of motion to calculate U and V we make a simplifying approximation. A bulk flow law of the form

(22)

wilt be assumed to govern the creep How contribution to the total flux where τ = pg γ sin α. Equation (22) is an integrated Form of Glen’s power law and has been used extensively in modelling glacier flow (Reference NyeNye, 1960,1963; Rudd, 1969, 1975; Reference Rasmussen and CampbellRasmussen and Campbell, 1973; Reference Budd and McInnesBudd and Mclnnes, 1974). If we assume that ice flows in a columnar fashion so that U varies with x but not with y, we can calculate U directly from Q and avoid solving the complete equations of motion for U and V. The velocity normal to the bed follows from the incompressibility of ice

(23)

where we neglect the small velocity contribution due to bottom melting.

During the surge phase the glacier is assumed to flow by both sliding and internal deformation. We shall take the snout velocity U ₀ as constant throughout the surge and assume that the sliding velocity varies over the active zone according to

(24)

This expression is probably a fairly good approximation to the actual distribution averaged over the surge. The derivative ∂U _b/∂x vanishes at x = 0 to prevent a discontinuity in the strain-rate, and at x = X so that the snout slides as a rigid block. Finally the choice of the form for U _b (x) greatly influences the pre-surge and post-surge profiles of a cyclically surging glacier. Equation (24) has proven to give acceptable results where other functions have yielded bizarre ones. If the bed near the snout is below the melting point just prior to a surge, Equation (24) requires that sliding can occur over a frozen bed. What actually happens in real surging glaciers with a cold snout is doubtless very complex. Ice may, perhaps, be thrust over a cold bed by shear, or basal water may be injected into the frozen zone to initiate sliding.

For the purposes of computing the velocity components U and V it is assumed that U is independent of depth, but for the computation of the internal heat generation the thermal effects of the creep flow contribution are distributed over the thickness of the glacier according to the following procedure. The shear strain is taken as ∂U/∂y = Bσ _xy ⁿ and the second invariant of the strain-rate tensor constructed from the strain-rate components ½ ∂U/∂y,∂ U/∂x and ∂V/∂y. From this invariant the second invariant of the stress deviator

is formed and Equation (3) used to calculate the internal heating.

Apart from assumptions of physical constants such as gravity, density, thermal conductivity, diffusivity, etc., the basic inputs are the surface temperature distribution T _s(x), the mass-balance function b(x), the geothermal flux G (measured normal to the bed), and the duration of the surge and quiescent phases. Together with the assumed flow and sliding relations these determine the surface profile and temperature field. A constant ice flux Q ₀ is assumed to feed ice into the active zone at x = o and this together with the assumed initial profile give the required boundary and initial conditions for Equation (20). More complicated functional forms for T _s, b and G involving dependence on time and the y coordinate could easily be introduced.

To start the computations, an initial surface profile and temperature field are assumed and the surface profile, temperature, and velocity fields are then computed as the glacier model is forced through a number of surge cycles. The durations of the surge and quiescent phases

Table III.

Nume rical inputs for traprldge glacier model b

and the snout velocity U ₀ remain the same for each cycle but the glacier profile and temperature field evolve until, after 25 or more complete cycles, they become perfectly cyclic. At this point the pre-surge and post-surge states can be examined free from any influence of the initial conditions.

Discussion of two-dimensional modelling results

Table III summarizes the inputs to our two-dimensional model. There is no information concerning the periodicity of surging for the Trapridge Glacier but the most recent surge was around 1940 so if it surges periodically our assumed 37 year cycle is a lower limit. A rather high geothermal flux is required to bring a substantial fraction of the bed to the melting

Fig. 6.

Evolution of temperature field for a two-dimensional model of Trapridge Glacier (Model B). Note that the relative amounts of temperate and cold bed change throughout the surge cycle and that immediately following a surge isotherms near the snout are overturned.

point. No geothermal data are available to support this assumption but it does seem reasonable in view of the Tertiary vulcanism in the region.

Figure 6 shows the glacier surface profile and temperature field at three stages of the surge cycle. In the upper region of the active zone the o°C isotherm follows the bed, but near the snout the proximity of cold surface ice forces it below the bed so (hat a “cold ring” regime is established. At the onset of a surge the warm basal ice zone is relatively large but after a surge, advection causes it to shrink.

One of the most interesting features in Figure 6 is the overturning of isotherms near the snout at the end of a surge advance (Fig. 6c). The temperature-depth profile in such a region would have a pronounced kink in it; similar features have actually been observed in the Trapridge and Steele Glaciers (Reference Jarvis and ClarkeJarvis and Clarke, 1975; Reference Clarke and JarvisClarke and Jarvis, 1976). This overturning is a simple consequence of the fact that deep ice temperature near the snout is colder than that of ice farther up-glacier so that a surge displaces warm ice over a cold bed. An unwelcome consequence of the overturned isotherms is that the longitudinal heat conduction term ∂² T/∂x²; in Equation (2) is not negligible so that Equation (10) may not be a satisfactory approximation in the snout region. Averaged over the entire surge cycle, however, the approximation remains acceptable and it is doubtful whether the slight error in computations immediately following a surge greatly changes the temperature distribution near the snout.

Close inspection of Figure 6c reveals that a thick layer of temperate basal ice did not form as a consequence of the surge. Evidently conduction losses due to advection more than compensated for the heat generated by internal friction during the surge. This is in keeping with Robin’s original concept of thermal instability (Reference Robin deRobin, 1955) but not with his later ideas (Reference Robin deRobin, 1969). As the bed-normal space increment y was taken as 5 m a thin temperate layer could pass undetected, but it is very unlikely this was the case.

To this point we have avoided the question of what might cause cyclic surges in a glacier similar to our Model B, It is interesting to compare the relative amounts of temperate and cold basal ice before and after a surge. Shortly after the completion of a surge the temperate zone recedes from the snout region leaving an extended tongue frozen to the bed (Eig. 6a). As the quiescent phase progresses, compressive flow increases the zone of melting at the expense of the frozen zone and ablation of the snout further diminishes the frozen zone (Fig. 6b). If this were allowed to continue indefinitely almost the entire bed would become temperate. Clearly the situation is unstable and at some point the glacier must begin to advance by sliding (Fig. 6c). At the end of a surge the temperate zone again shrinks and the cycle is completed.