Supplement of The current state and 125 kyr history of permafrost on the Kara Sea shelf: modeling constraints

Abstract. The evolution of permafrost on the Kara shelf is reconstructed for
the past 125 kyr. The work includes zoning of the shelf according to
geological history; compiling sea level and ground temperature scenarios
within the distinguished zones; and modeling to evaluate the thickness of
permafrost and the distribution of frozen, cooled and thawed deposits.
Special attention is given to the scenarios of the evolution of ground
temperature in key stages of history that determined the current state of
the Kara shelf permafrost zone: characterization of the extensiveness and
duration of the existence of the sea during stage 3 of the marine oxygen isotope
stratigraphy (MIS-3),
the spread of glaciation and dammed basins in MIS-2. The present shelf is
divided into areas of continuous, discontinuous-to-sporadic and sporadic
permafrost. Cooled deposits occur at the western and northwestern water
zones and correspond to areas of MIS-2 glaciation. Permafrost occurs in the
periglacial domain that is within a zone of modern sea depth from 0 to 100 m, adjacent to the continent. The distribution of permafrost is mostly
sporadic in the southwest of this zone, while it is mostly continuous in the
northeast. The thickness of permafrost does not exceed 100 m in the
southeast and ranges from 100 to 300 m in the northeast. Thawed deposits
are confined to the estuaries of large rivers and the deepwater part of the
St. Anna trench. The modeling results are correlated to the available field
data and are presented as a geocryological map. The formation of frozen,
cooled and thawed deposits of the region is inferred to depend on the spread
of ice sheets, sea level, and duration of shelf freezing and thawing
periods.


Supplement. Extrapolation of the modeling results of permafrost thickness to real geological sections.
Сarrying out such a procedure is necessary to reduce the number of simulated thermal tasks to a reasonable limit. There are several approaches.
A. In the first case, we consider the uniform alternation of homogeneous layers with a relatively low thickness (relative to the total thickness of the permafrost). The linear interpolation is simply performed in accordance with the percentage of the thickness of frozen ground obtained during modeling for two "pure" soils at a given moment.
There are 2 options here: 1. We have the thicknesses of frozen layers in sand (T s ) and loam (T l ) (clay) sediments from the simulation in similar conditions (scenario, heat flux). Then the formula will be: where n s is the relative content of sand layers in the section.
The same formula can also be used to evaluate the thawing of permafrost from above, if this occurs in accordance with a specific scenario.
2. More complicated is the very common case when, for example, relict permafrost is preserved in sandy sediments within the water area, and as for the loam it thaws completely long before modern times.
In this case, we apply the following approximate approach.
For a reference soil in which permafrost is not preserved up to the present (and this is always loam (clay) in the considered pair of reference soils), the last point of the existence of permafrost at the  deg (time of degradation) is found at the corresponding graph. This point, in the absence of thawing from above, is located on the surface, otherwise, at a certain depth from the bottom z deg , where the upper and lower fronts of thawing of permafrost in the loams meet.
At this point, according to the slope of curve representing the movement of the base of permafrost, the rate of thawing  th from below is determined right before the complete disappearance of the frozen layer. Based on this rate of the upward movement of the lower boundary of the permafrost, the value of the potential thawing of loams from the moment  deg is calculated: and the estimated (fictitious) position of the modern lower permafrost boundary in the loamy rocks relative to the bottom surface will be equal to The value can be either positive (below the bottom) or negative (above the bottom).
Putting in (1) instead of Tc allows to find the position of the lower boundary of frozen soils in a layered sandy-loamy section with a given relative content of sand layers ( ). A negative value clearly indicates the complete degradation of permafrost so far even in the absence of thawing from above.
To obtain the thawing from above for the alternating layers of sediments, the following approximate approach is used. At the moment of the complete degradation of frozen loams  deg , according to the graph of permafrost dynamics, the thawing value from above ( deg ) is found for sandy soil and the ratio = z / ( ) is calculated. It is assumed that the indicated ratio of thawing depths from above for soils of different compositions at the time of permafrost degradation in loamy sediments is preserved up to the present. Then the potential thawing on top of the loams at the moment will be = × (4), where and are the calculated depth of thawing from the top for the loams and the model depth of thawing from the top for the sand accordingly, to the current time The current position of the permafrost top under the sea floor for a layered section of bottom sediments with a given relative content of sand ( ) is found from a dependence of type (1), which in this case has the next form: Next, the residual thickness of the layered permafrost under the sea bottom should be found.
If the value ℎ < 0 then the relict permafrost do not persist at a specific point in the water area.
B. Extrapolation of simulation results of submarine permafrost in homogeneous soils into a two-layer geological section.
The case is considered when the thickness of the sediments is relatively small and bedrock lies from a shallow depth under the bottom. An approximate approach is known to estimate the freezing depth of a two-layer strata with known freezing depths of the rocks of the upper and lower layers separately, all other things being equal.
This approach is based on obvious points. At zero thickness of the upper layer of sediments (th 1 =0), freezing of a two-layer system is equal to freezing in the lower layer = , and at a thickness of the upper layer equal to the depth of freezing of the upper layer (th = ) freezing of the system is equal to freezing of the upper layer = .
Intermediate freezing depths of this two-layered system for random values of the upper layer thickness are considered varying linearly between these extreme values, where the following relationship is obtained.
It is clear that the limits of variation in the thickness of the upper layer of ground are limited -when th > the system becomes single-layer. Equation (7) is valid as well for thawing of the permafrost for two-layer structure section.
There are also two cases of interpolation.
1. The modern thickness of permafrost in sediments ( ℎ ) and bedrocks ( ℎ ), is obtained under the same other conditions (scenario, heat flow) from the simulation results. Then the formula for determining the thickness of permafrost in a two-layer system will obviously have the next form: where ℎ is the thickness of the upper layer of sediments or the depth of the top of the bedrocks.
We consider the sediments as the strata of alternating sand and loam with a given ratio in the section. The extrapolation of modeling results of homogeneous reference soils to a layered stratum was considered in the previous section.
The same dependence can also be used to evaluate thawing of two-layer permafrost from above.

Often there is a situation when the relict permafrost is preserved in sediments but in rocky deposits it is
completely degraded to the current moment. In this situation, the same approach is used as previously considered (see section A). Using data on the depth and period of complete thawing of permafrost in bedrocks as well as on the dynamics of the movement of thawing fronts immediately before the disappearance of frozen rocks , the calculated (fictitious) position of the boundaries of frozen rocks at the modern time is determined using the dependencies (2-4), and the prediction of thawing of frozen rock from above is carried out according to (4) using the dynamics of sandy permafrost.
Using the same methodology, the fictitious positions of the upper and lower boundaries of the permafrost in sediments are calculated -i.e. layered strata with a given relative sand content. Further, substituting the obtained fictitious positions of the permafrost boundaries in sediments and rocks in (8)