On the periodicity of free oscillations for a finite ice column
Abstract. The temperature distribution in ice sheets is worthy of attention given the strong relation with ice dynamics and the intrinsic information about past surface temperature variations. Here we refine the classical analysis of free oscillations in an ice sheet by analytically solving the thermal evolution of an ice column. In so doing, we provide analytical solutions to the one-dimensional Fourier heat equation over a finite motionless ice column for a general boundary condition problem. The time evolution of the temperature profiles appears to be strongly dependent on the column thickness L and largely differs from previous studies that assumed an infinite column thickness. Consequently, the time required for the column base to thaw depends on several factors: the ice column thickness L, the initial temperature profile and the boundary conditions. This timescale is classically considered to be the period of a binge-purge oscillator, a potential mechanism behind the Heinrich Events. Our analytical solutions show a broad range of periods for medium-size column thicknesses. In the limit of L → ∞, the particular values of the prescribed temperature at the top of the column become irrelevant and the reference value of ~7000 years, previously estimated for an idealised infinite domain, is retrieved. More generally, we prove that solutions with different upper boundary conditions, yet covered by our formulation, converge to the same result in such a limit. These results ultimately manifest a subtle connection between internal free and externally-driven (in the sense of a time-dependent boundary condition at the top) mechanisms caused by the finitude of the domain. Thermomechanical instabilities, inherent to internal free oscillations, are in fact sensitive to the particular climatic forcing imposed as a boundary condition at the top of the ice column. Lastly, analytical solutions herein presented are applicable in any context where our general boundary problem is satisfied.
Daniel Moreno et al.
Status: final response (author comments only)
RC1: 'Comment on tc-2022-97', Anonymous Referee #1, 13 Jun 2022
- AC1: 'Reply on RC1', Daniel Moreno, 20 Sep 2022
RC2: 'Comment on tc-2022-97', Anonymous Referee #2, 20 Jul 2022
- AC2: 'Reply on RC2', Daniel Moreno, 20 Sep 2022
RC3: 'Comment on tc-2022-97', Ed Bueler, 23 Jul 2022
- AC3: 'Reply on RC3', Daniel Moreno, 20 Sep 2022
Daniel Moreno et al.
Daniel Moreno et al.
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This manuscript highlights the computational efficiency of using the method of separation of variables to solve the diffusion equation rather than direct numerical methods. While this is well-known, the application to the basal temperature evolution in an ice column had previously only been done for infinitely deep ice; this manuscript considers the case of a shallow ice sheet and performs some example calculations for different initial conditions and boundary conditions.
Overall, I was somewhat disappointed that the authors did not go into more depth in analysing and describing their results, in particular exploring the wealth of curious trends shown in figure 4 - most of the paper is instead given over to a routine description of the method of separation of variables. In particular, given the stated threshold of 2km for the solution to approach the infinite depth limit, it would be nice to explore what factors set this threshold. Looking at figure 5 there seems to be a rather narrow band of depth values for which T is finite but larger than the MacAyeal solution. I think figure 4b also shows this rather sudden regime change.
If Equation (6) were given as cot(L*sqrt(λ)) = βλ, there would be no need to treat β=0 as a special case.
Figure 4 - the values of the parameters held fixed are not given.
Figure 4d - interesting that T is non-montonic with L at -14°C. Why is this?
Figure 4c - this figure shows the most interesting trends, but is barely discussed in the text. Perhaps using θL/L as the primary variable instead would clarify the impact of the temperature gradient on the basal evolution.
Line 162 - where T saturates to above 25kyr, are we in fact in a limit where T is infinite?
Convergence towards no dependence on the detailed surface boundary conditions as L→∞ could be moved to an appendix for better flow of the manuscript.
Figure 4 colorbar caption could be oriented to match the axis label.