Introduction
An exciting and still unsolved phenomenon in glaciology is the occurrence of surges. A glacier stagnant for some tens of years suddenly starts to move with exceptionally high velocity. Great masses of ice are transported from the upper to the lower part of the glacier in one or two years. As a result of this movement, the surface level in the upper part is lowered and the lower part is raised, but the ice does not move beyond the limits of the previously stagnant ice. As a result of the surge, medial moraines are deformed into loops. From this characteristic it can be deduced that a surge is a repeating phenomenon. No external cause can be found for a surge (Reference PostPost, 1960) and it seems very probable that it is caused by a variation of the sliding conditions at the glacier bed (Reference WeertmanWeertman, 1962).
In this paper we try to find a theoretical model which explains the sequence of motion during a surge, assuming a sudden improvement of the sliding conditions. As a prototype the surge of Muldrow Glacier described by Reference PostPost (1960) will be used.
General Equations
We consider a two-dimensional model of a glacier. As in the theory of advance and retreat of glaciers, we need two equations. First the continuity equation and secondly a relation between the discharge q(x,t) and the ice thickness h(x,t) and its derivative ∂h(x,t)/∂x. This relation can be deduced from a general flow law which implies the flow law of the ice and the sliding conditions at the glacier bed. As in the present problem we are concerned with changes of the sliding conditions, we have to consider q as a function, of the parameters α i of the general flow law too. We make the assumption, that the mathematical form of the general flow law always remains the same.
If we regard only perturbations (suffix 1) from a steady state (suffix 0) we may write the two equations:
where a 1(x,t) is the deviation from a mean annual balance a 0. x is measured positive from the upper end of the glacier along the surface and h is perpendicular to the x-direction.
Eliminating h 1 and q 1 respectively we obtain from Equations (1) and (2) the equations:
The third term on the right side of Equation (3) represents the influence of a change in the sliding conditions on q 1. It is equivalent to a variation of the annual balance a 1′ which can be found from the first-order differential equation
In the same manner we can replace the fourth term on the right side of Equation (4) by an equivalent variation of the annual balance a 1″ given by
With Equations (5) and (6) we have reduced our problem to the response of glaciers to climatic changes. We are therefore able to apply the theory of advance and retreat of glaciers (Reference NyeNye, 1963) with only small modifications.
Special Model
For the description of the glacier flow we adopt a power law (Reference NyeNye, 1959).
Where τ is the shear stress at the glacier bed and as a first approximation is given by
β is the inclination of the glacier bed, ρ the density of the ice, and g the gravity acceleration.
At any datum state of the glacier we are able to choose the coefficients U and T in such a manner that q becomes independent of n. If we do this, U is consequently the mean velocity of the ice and T the basal shear stress at this datum state.
For (∂q/∂h)0 which represents the velocity of the kinematic waves, and [∂q/∂(∂h/∂x)]0 which is the diffusion coefficient, we adopt the same polynomials as discussed and used by Reference NyeNye (1963).
where l(1—δ) is the length L of the glacier and σ is a natural time unit. For a glacier of L ≈ 10 km a rough estimate gives σ ≈ 10 year and δ ≈ 0.01. This estimate of σ is only valid if the exponent n is about 3 or 4. If we vary n → n 1, we have to multiply σ by the factor (n+1)/(n 1+1).
If we choose U and T in Equation (7) in such a manner that (∂q/∂n)0 = 0, Σ(∂q/∂α i )0 reduces using Equations (7) and (9) to
We adopt for h 0/T 0 the constant value (h 0/T 0) x = L/2. Following from Equation (8), this simplification means a uniform surface inclination of the whole glacier. Therefore T 0 tends to zero for x → 0 and (1−δ)l in the same manner as h 0. From Equations (10) and (11) we see that [∂q/∂(∂h/∂x)]0 and (∂q/∂T)0 should have the same analytical form. But as we need only the approximate form of the coefficients, the difference between the adopted polynomials will not falsify the results.
Solution
After having determined the coefficients of Equation (3) we look for a solution which satisfies the following conditions: a 1 = 0 for any time; for t < 0 we assume a steady state with h 1, ∂h 1/∂x, ∂T 1/∂t = 0 and an arbitrary n; for the small time interval 0 ⩽ t < ∆t we take a constant ∂T 1/∂t = T 1/∆t. For t ⩾ ∆t we take ∂T 1/∂t = 0.
In our model T 0 tends to zero for x → 0 or (1–δ) l. As we shall assume T 1 constant and negative, this will lead to a basal shear stress in the direction of the glacier flow, which is physically meaningless. But as this discrepancy is restricted to the highest and lowest part of the glacier it can hardly affect the solution of Equation (3) for the main part of the glacier.
The variation of the annual balance equivalent to the adopted variation of the sliding conditions is found by Equation (5) to be
With a 1 and the coefficients given by Equations (9), (10) and (11), the solution of Equation (3) for the time interval 0 ≤ t < ∆t is given by (Reference NyeNye, 1963)
We expand exp (−t/σ) and exp (−2δt/σ) into series and, taking ∆t → 0, we find for q 1 at t = 0
A sudden change of T 0 to T 0 + T 1, yields therefore a q 1 given by Equation (12).
For t ≥ ∆t we have to solve the homogenous differential equation (3) with the initial condition (12). The solution is
The solution for h 1 is found from Equations (13) and (1) :
Discussion
From Equation (13) we see that a sudden change of T 0 to T 0 + T 1 (T 1 negative) causes an additional discharge q 1 which can be separated into two phases. The first phase, represented by the first term in Equation (13), describes a discharge which reaches its maximum in the middle of the glacier and vanishes completely at the head and nearly at the snout. The motion caused by this discharge yields a transport of ice from the upper part of the glacier to the lower part and decays from a maximum at the beginning with a time constant σ. No essential transport of ice beyond the previous limits of the glacier takes place.
The second phase of motion, represented by the second term in Equation (13), becomes important after the first has decayed. It increases quadratically toward the end of the glacier and therefore causes an advance. Its magnitude is smaller by the factor 2δ, but, decaying with the time constant σ/2δ, it continues correspondingly longer. σ/2δ is known as the long time constant from the theory of advance and retreat of glaciers.
We find a similar behaviour in h 1. During the first phase of rapid motion the surface level lowers in the upper part of the glacier and rises in the lower part. During the second phase, decaying with the long time constant σ/2δ, the surface level in the lower part drops too, till a constant decrease of ice thickness h 1 = nh 0 T 1/(n+1) T 0 over the whole glacier is reached. The resulting diminution of the length or the glacier is of no physical significance for |T 1| > T 0 at the extreme ends of the glacier as mentioned in the previous section.
Qualitatively the first phase of motion is very similar to a surge. To fit our model quantitatively we assume 
to get the right magnitude of the surface level variation. σ, normally about 10 years, must be more than 10-times smaller during a surge to yield the very fast motion observed. This corresponds to a very high value of the exponent n. Normally no advance is produced by a surge. The first phase of motion of our theoretical model meets this observation. The second phase, transporting ice beyond the steady-state end of the glacier, is weaker by the factor 2δ than the first phase. One could imagine that this second phase is suppressed if a consolidation of the sliding conditions takes place after the first phase of strong motion is over.
