The Column model

Budd et al. (1971) solved Equation (6.13) in a more general form than those we have considered so far. Calculations using their model, which they refer to as the Column model, can be done by hand.

The coordinate system they use is shown in Figure 6.10. The temperature profile is to be calculated at a point a distance x from the divide. Starting again with Equation (6.13), we restrict the model to two dimensions, thus eliminating derivatives in the y-direction; we assume that temperature gradients in the x-direction are sufficiently small that

Figure 6.10. Coordinate system and parameters involved in Column model calculations. their derivative is negligible; and we assume a steady state. With these assumptions, Equation (6.13) becomes:

d20 do do q

Let us now consider the strain heating term, Q/pC. From Equation (6.9) it will be seenthatthis term increases approximately as d 4, where d is the depth below the surface. In other words, because the strain rate increases rapidly near the bed, most of the strain heating occurs in the basal few meters of ice. (The student may find it interesting to study this effect by solving Equation (6.13) with the assumption that all advection terms and the horizontal conduction terms are negligible, and that a steady state exists. Equation (6.9) is used for Q/pC. The problem is most easily tackled by using a coordinate system in which the z-axis points vertically downward.) Recognizing that significant strain heating occurs only near the bed, Budd (1969) assumed that it occurs only at the bed, and that this heat could thus be added to the geothermal flux. The basal boundary condition, ftb, thus becomes:

K K m2 Q K — — m . a = — 1 Nm = 1J m m J m maK

and Q/pC is set to zero. Here, as before, ftG is the gradient required to conduct the geothermal flux upward into the ice; tb is the basal drag, approximated by pgda; and u is the mean horizontal velocity. For u at X we use the balance velocity:

(Equation (5.1)). In calculations, care must be taken to ensure that the sign of the (Tbu) / K term is the same as that of ftG; this sign is determined by the choice of coordinate axes.

We turn now to the term u ■ dQ/dx in Equation (6.36). In Equation (6.34) we set dQ/dx = aX, as this is the rate at which the atmospheric temperature increases as one moves to lower elevations along the ice surface. This is, therefore, the rate at whichnear-surface ice must warm as a result of horizontal advection. If the glacier surface slope is sufficiently low, the deeper ice will warm at the same rate, with negligible lag. This led Budd (1969) to suggest that, to a reasonable first approximation, u ■ dQ/dx can be replaced with uaX in Equation (6.36). The consequences of this assumption are discussed below.

With the additional substitution of (ws - wb)z/H for w, Equation (6.36) becomes:

H dz

which is to be solved by using the boundary condition of Equation (6.37). Here, ws and wb are the vertical velocities at the surface and bed respectively; ws can be calculated from the submergence velocity (Equation (5.26)) when bn, us, and a are known. If the velocity atthe bed is assumed to be parallel to the bed, wb can be estimated from knowledge of ub and the bed slope. Assuming that ub = u = us is probably a reasonable approximation in this calculation, but knowing u, one could also calculate us and ub from Equations (5.18) and (5.19).

The solution to Equation (6.39) is (Budd, 1969; Budd et al., 1971):

where:

et2 dt dy

Note that this solution uses Budd's definition of the error function, erf(x). E(x) is the integral of a function known as Dawson's integral (the quantity in square brackets), and it too has been tabulated. A plot of it for a reasonable range of Z is shown in Figure 6.5.

Temperature profiles can be calculated readily by using Equation (6.40) and Figure 6.5. A typical one is shown in Figure 6.11 (profile (a)).

Figure 6.11. Temperature profiles calculated from

(b) Equation (6.41). The following values of the parameters, approximately appropriate for Camp Century, Greenland, were used in the calculations:

a = -0.01, u = 15 m a-1, k = -0.01 K m-1, k = 37.2 m2a-1, H = 1368 m, eH = -24 °C, and fib = -0.0508 K m-1.

Figure 6.11. Temperature profiles calculated from

(b) Equation (6.41). The following values of the parameters, approximately appropriate for Camp Century, Greenland, were used in the calculations:

a = -0.01, u = 15 m a-1, k = -0.01 K m-1, k = 37.2 m2a-1, H = 1368 m, eH = -24 °C, and fib = -0.0508 K m-1. 1000

Temperature, oC

1000

The minimum temperature occurs at some depth below the glacier surface. This represents, as in Figure 6.9, cold ice that is advected downward and laterally from some point further upglacier where the surface is at a higher elevation and hence colder. However, the Column model does not include this longitudinal advection rigorously, but simply specifies a warming rate. Thus the temperature at depth is only an approximation that becomes better as the warming rate decreases. This approximation is best, therefore, where surface slopes (a) and lapse rates (k) are lowest.

Both the magnitude and the curvature ofthe positive temperature gradient near the surface are adjusted so that heat conducted downward from the surface, in combination with heat advected downward, is sufficient to warm the ice everywhere above the point of minimum temperature at the rate uak. Ice below this point is warmed at this rate by heat from the bed -both geothermal and frictional. When the warming rate at depth that is specified (by representing it by uak) is larger than that in the natural situation being modeled, the positive temperature gradient at the surface becomes too high, and the temperatures at depth, thus, too cold.

Further insight into the mathematical properties of the solution can be gained by comparing the profile calculated from Equation (6.40) with one calculated from:

together, again, with the boundary condition of Equation (6.37). (The integration of Equation (6.41) is left as an exercise for the reader.) This is profile (b) in Figure 6.11. Equations (6.39) and (6.41) differ in that the vertical advection term is omitted in Equation (6.41). By analogy with our discussion of Equation (6.24) and Figure 6.7, one might think that because vertical advection moves cold ice downward from the surface, omission of this term would make profile (b) warmer than profile (a) at depth. However, in this case the ice at depth is colder than that at the surface, and because the vertical advection term operates on ice at the surface at the point where the profile is being calculated, not at some point upglacier therefrom, the ice advected downward is warmer than the ice at depth. As a result of this downward advection of heat, included in Equation (6.40), the uaX warming rate does not need to be satisfied entirely by conduction from the surface in profile (a).

0 0