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Water 2019, 11, 706
3. Model Description
3.1. Lake Radius Expanding Model
A simple lake radius expanding model was employed to simulate the thermal regimen of
permafrost and talik development under thermokarst lakes in the Beiluhe Basin on the QTP in this
study. This finite element model was modified from a well-tuned two-dimensional heat transfer model
with phase change in a cylindrical coordinate system [2] by forcing the intermittent thermokarst lake
radius expansion as an external influence [40]. Neglecting water migration in the system, the governing
equations describing the thermal regimen of talik and permafrost underneath an expanding thermokarst
lake can be written in the following form:
C& = -(-1k + k dT +a-(kaT )
at=ar\ar r ar+dz\ (z)' (1)
(0 < t < D, 0 < z < Z,0 < r < R).
Cf, T <Te - AT,
C=Cf +Lw pb-" , Te - AT < T <Te .(2)
Cu, T > Te.
kf, T <Te - AT,
k= kf+k4T-[Ti(Te-AT)] , Te - AT<T<Te . (3)
ku, T > Te.
r(t) = r0 + vhet (4)
where T is temperature in C; t is time in seconds; z is the depth from the ground surface downward in
m; Z and R are the total depth and radius of the analysis domain in m; D is the total simulation period
in seconds; C and k are the volumetric heat capacity in J m-3 OC-1 and the thermal conductivity in
W m-1 C-1; the subscriptsf and u are the frozen and unfrozen phases, respectively; Te is the thawing
or freezing temperature and is set at 0 C; AT is the temperature interval in which the phase change
occurs in C and is assumed to be 1.0 C; [Te - AT, Te] is the temperature range in which the phase
change occurs; Lw is the mass specific latent heat of freezing in J kg-1; Pb is the dry buck density of
soil in kg m-3; W is the total water content percent of soil by mass; r and r0 are the radius from the
lake centerline at time t and the initial radius of the lake in m; and vhe is the mean lake radius lateral
expansion rate in m/year.
3.2. Thermokarst Lake Expansion
Natural thermokarst lakes expand their basin margins by thawing ground ice and redistributing
the sediment. The lake radius increases gradually by lake shoreline retreat via complex mechanical
and thermal erosion processes [41,42]. Several studies have shown that the long-term mean lake lateral
expansion rate can be simply described as a linear relation between the lake radius and time [18,26,43].
Based on the field investigations and research results in the Beiluhe Basin [26,30,43], the initial radius
of the lake formed over the ground surface was set to be r0 = 6.0 m, and the mean lake radius lateral
expansion rate which increased linearly in Equation (4) was set to be vhe = 0.25 m/year. Because lake
radius expansion caused by lake bank slumping and retreat is a transient and intermittent process,
rather than a gradual and continuous process, the mean lake radius lateral expansion rate of 0.25 m/year
was treated as a stepwise change, i.e., the lakeshore collapsed and the ground surface boundary
transformed into the lake bottom boundary instantaneously on 30 September every four years, with a
lake radius increment of 1.0 m after each collapse.
This study focuses on the sensitivity of talik development and permafrost degradation under
thermokarst lakes to variations in the MALBT. The lake depth was set to be a constant determined
by the depth of the ice-rich permafrost thickness and permafrost thaw settlement coefficient. The
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