Analysis of permafrost evolution under a thermokarst lake based on the Euler-Lagrange equation

Thermokarst lakes, which cover substantial portion of the Arctic coastal plains (ACP), can cause permafrost thaw underneath due to the larger heat supply from lake surface than tundra. This study obtained a three-dimensional Stefan equation to characterize the thaw bulb or talik geometry under an idealized thaw lake in the continuous permafrost region. Using the Euler-Lagrange equation in the calculus of variations, the talik boundary was determined as an extremal of the functional describing the total talik volume and phase boundary area. This analysis showed that the semi-ellipsoid geometry of a talik minimized the lake-integrated phase boundary area that represents the total depth of permafrost thaw under the lake for a given summertime heat supply. Thus, the ellipsoidal shape of talik can be mathematically shown to be optimum under the quasi-steady thermal conditions. The result shows that the talik width-depth relationship is a direct indicator of the vertical ground temperature slope since the top-down permafrost thawing is much less than the horizontal direction thaw due to the vertical heat conduction to the permafrost (the energy flux anisotropy). Additionally, this theory is useful for partitioning the factors affecting the orientation of thermokarst lakes in certain ice-rich permafrost regions. This theory suggests that the thermally optimum ellipsoidal talik geometry influences the thermokarst lake shape while wind induced currents in the lake water are likely responsible for elongation and orientation of shallow thermokarst lakes. Although this presentation mainly focuses on the theoretical development, comparisons to the field evidence by ground-based observations, geophysical methods, and remote sensing measurements will be discussed.