19 May 2022
19 May 2022
Status: this preprint is currently under review for the journal TC.

On the periodicity of free oscillations for a finite ice column

Daniel Moreno1,2, Alexander Robinson1,2, Marisa Montoya1,2, and Jorge Alvarez-Solas1,2 Daniel Moreno et al.
  • 1Departamento de Física de la Tierra y Astrofísica, Universidad Complutense de Madrid, Facultad de Ciencias Físicas, 28040 Madrid, Spain
  • 2Instituto de Geociencias, Consejo Superior de Investigaciones Cientifícas-Universidad Complutense de Madrid, 28040 Madrid, Spain

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)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • 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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Short summary
Our study tries to understand how the ice temperature evolves in a large mass as in the case of Antarctica. We found a relation that tells us the ice temperature at any point. These results are important because they also determine how the ice moves. In general, ice moves due to slow deformation (as if pouring honey from a jar). Nevertheless, in some regions the ice base warms enough and melts. The liquid water then serves as lubricant and the ice slides and its velocity increases rapidly.