A new Stefan equation to characterize the evolution of thermokarst lake and talik geometry

Abstract

Thermokarst lake dynamics, which play an essential role in carbon release due to permafrost thaw, are affected by various geomorphological processes. In this study, we derive a three-dimensional (3D) Stefan equation to characterize talik geometry under a hypothetical thermokarst lake in the continuous permafrost region. Using the Euler equation in the calculus of variations, the lower bounds of the talik were determined as an extremum of the functional describing the phase boundary area with a fixed total talik volume. We demonstrate that the semi-ellipsoid geometry of the talik is optimal for minimizing the total permafrost thaw under the lake for a given annual heat supply. The model predicting ellipsoidal talik geometry was compared to talik thickness observations using transient electromagnetic (TEM) soundings in Peatball Lake on the Arctic Coastal Plain (ACP) of northern Alaska. The depth : width ratio of the elliptical sub-lake talik can characterize the energy flux anisotropy in the permafrost, although the lake bathymetry cross section may not be elliptic due to the presence of near-surface ice-rich permafrost. This theory suggests that talik development deepens lakes and results in more uniform horizontal lake expansion around the perimeter of the lakes, while wind-induced waves and currents are likely responsible for the elongation and orientation of shallow thermokarst lakes without taliks in certain regions such as the ACP of northern Alaska.