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APPENDIX Transformation of the Governing Equation
The one-dimensional energy balance considered is:
where z is positive upward with its origin at the rock/ice interface, w is the vertical velocity and both the heat capacity, ρc, and the thermal conductivity, k, are, in general, functions of temparature. The heat flux at the base of the domain, located at some large depth z = –h
0, remains fixed at q
The upper boundary, or the ice/air interface, is located at z = h, where h is an arbitrarily specified function of time. The surface tempaerature, T
s, is also an arbitraarily specified function of time:
In the present case, the time-dependence of T
s is given indirectly through its dependence on h(t).
The vertical velocity in the varies linearly through the upper three-quarters of the ice sheet and quadratically below that (Dansgaard and Johnsen, 1969):
m is the elevation above the bed at which the upper, linear velocity profile intersects the lower, quadratic profile. (In all calculations shown in this paper, we take z
m/h = 0.25) In the limit that z
m → 0, this recovers the linear profile of Nye (1951, 1957, 1963). W is the magnitude of the advective velocity at the surface. Here, W is the volume flux of ice across the upper surface of the ice sheet, i.e. it represents the accumulation rate over and above that which goes into growth of the ice sheet. Of course, a divergent, lateral flow must balance the downward flow of ice. An assumption implicit in Equation (A4) is that the velocity field in the ice is able to adjust rapidly to increasing h in order to maintain W in the form given throughout the growth history.
The “front-fixing” method entails coordinate transformation ζ = z/h, in terms of which Equation (A1) can be written for the domain within the ice (0 ≤ ζ ≤ h) in the form:
where w (Equation (A4)) is now written in terms of ζ.
Note that, even in the absence of downward advection in the fixed reference frame (w = 0), the coordinate transformation has the effect of introducing into Equation (A5) an advection-like term that accounts for the moving boundary, h(t). The effect of the boundary moving upward at velocity dh/dt and that of advection cold ice across the surface at velocity −W are additive. The boundary condition at the top surface (Equation (A3)) is now applied at the fixed point ζ = 1, which is convenient for numerical implementation. In the scheme adopted here, we solve the conduction problem in the underlying rock domain (–h
0 ≤ z ≤ 0) in the untransformed coordinate, z, while solving Equation (A5) for the ice. The temperature and heat flux are matched at the rock/ice interface, z = ζ = 0.
We consider two particular forms of the growth function, h(t):
x is the final ice thickness and tr
the rise time. In the text, we refer to Equation (A6) as an exponential growth modal and Equation (A7) as a linear modal.