## Elementary kinematic wave theory

Let us now develop these ideas analytically. In this development, following an analysis by Nye (1960), we consider a slab of ice on a slope, 3(x), with thickness, h(x, t), and surface slope, a(x, t) (Figure 14.2). We assume that dh/dx is small and that the slab is of infinite extent in the horizontal direction normal to the x-axis. The surface slope is related to the bed slope by:

Figure 14.2. Relation among surface slope, a, bed slope, p, and thickness, h.

Figure 14.3. Contributions to change in mass in an element of a glacier of length dx.

Note that if h decreases downglacier, d h/dx is negative so a > p.

Consider conservation of mass in an element of a glacier of length dx (Figure 14.3). For convenience, we will express mass fluxes in terms of the equivalent volumes of ice, based on a standard density. Ice flows into the element at a rate, q (x, h, a, t), and out of it at a rate, q + (d q/dx) dx. Here, q is the flux per unit of glacier width, and thus has the dimensions m3a-1m-1. In addition, there is accumulation at a rate bdx. If more ice flows into, or accumulates in, the element than leaves it, the glacier increases in thickness at a rate (dh/d t), so the increase in volume of ice in the element is (dh/dt) dx. Thus:

or, simplifying:

Because q is a function of h and x, the functional dependence expressed by Equation (14.2) leads to a general class of motions in flow systems known as kinematic waves (Lighthill and Whitham, 1955). Our objective next is to gain some appreciation for the nature of such waves on glaciers.

Let us begin by considering the wave speed. Suppose we multiply both sides of Equation (14.2) by (dq/dh)x = c, where c is the change in flux resulting from a change in thickness at point x, thus:

d q d q d h d q d q c_I + = bc or cā + ā = bc (14.3)

Figure 14.4. Numerical interpretation of the terms in Equation (14.3).

1050

1050

100 m

1000

100 m

Equation (14.3) is known as the kinematic wave equation; c has the dimensions m3a-1m-1/m = ma-1. Thus, it is a speed. In fact, it is the celerity (or speed) of the wave. (Because q = uh, where u is the mean (depth-averaged) speed, dq/dh = u + h(du/dh).)

To gain some appreciation for the implications of Equation (14.3), consider the situation in Figure 14.4. The ice flux into the element of the glacier is 1050 m3a-1m-1, while that out is 1000 m3a-1m-1. The element is in the ablation area and the ablation rate is -0.4 m a-1, or -40 m3a-1m-1 over the length of the element. As a result of this positive balance, with more ice entering the element than leaving it, the glacier is increasing in thickness. With the use ofEquation (14.3) we can calculate the rate of increase in mass flux, dq/d t, resulting from this increase in thickness as follows. The flux gradient, dq/dx, over the 100 m long element is - 50 m3 a-1 m-1 / 100mor -0.5 m a-1. Suppose c = 200 ma-1. Then, dq/dt is 20 m3a-1m-1. In other words, owing to the increase in thickness and the resulting increase in speed, the mass flux increases by 20 m3a-1m-1.

The relationship among q, h, u, and c is illustrated in Figure 14.5, in which q is plotted against h. Because of the nonlinearity of the flow law, we expect q to increase nonlinearly with h as shown. The mean speed, u, of a glacier with a thickness and ice flux given by the values of q and h at point P in the figure is u = q/h. This is represented by the slope of the dashed line connecting P with the origin. However, the speed, c, of a kinematic wave is (d q/d h)P, which is the slope of a line drawn tangent to the q-h curve at point P. In other words, as mentioned earlier, the speed of the kinematic wave is appreciably larger than the mean speed of the glacier.

To get a sense of how much faster the kinematic wave moves, consider the case of a glacier moving entirely by internal deformation such that (see Equation 5.19):

Figure 14.5. Relation between mean speed of a fdq\ glacier and speed of a \dk /d kinematic wave.

Based on flow law

Figure 14.5. Relation between mean speed of a fdq\ glacier and speed of a \dk /d kinematic wave.

Based on flow law q

In other words, the kinematic wave moves with a speed that is roughly five times the depth-averaged velocity of the glacier. If there is basal sliding and the sliding speed varies as t2 (Equation (7.10)), the ratio is likely to be slightly less than 5. This relation applies, rigorously, only to infinitesimal waves. Waves of finite amplitude may have higher speeds.

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