# A Model of Ice Wedge Polygon Drainage in Changing Arctic Terrain

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Conceptual Model

_{w}, is treated as constant.

_{w}) and the head in the polygon subsurface across from the trough at this elevation, $h\left(R,z,t\right)$. In general, the flux direction may vary.

_{,}responds rapidly to the center pool level at the surface (H(t)) due to the small vertical scale of the active layer. The polygon thickness L(t) may vary from zero to a maximum value of D and can be scaled as follows: L(t) = D$\xb7$f(t), where f(t) is a dimensionless function that varies from 0 to 1 by the end of the season. The thickness of the thawed layer can also be treated as constant for qualitative analyses or simulations of a short time duration.

## 3. Methods

#### 3.1. Problem Statement For Constant Thaw Depth

#### 3.2. Reduction to Dimensionless Steady-State Problem

_{drain}.

#### 3.3. Solutions

#### 3.3.1. Head h*(r*, z*)

#### 3.3.2. Fluxes and Stokes Stream Function ψ*(r*, z*)

#### 3.3.3. Center Pool Level H(t): Constant Thaw Depth

#### 3.3.4. Center Pool Level H(t): Dynamic Thaw Depth, Analytical Approach

#### 3.3.5. Center Pool Level H(t): Dynamic Thaw Depth, Numerical Approach

_{w}(t). An example using this approach to simulate the early-summer disappearance of a center pond near Utqiagvik, Alaska is presented in Section 4.3.

#### 3.4. Computation of Solutions

_{L}and pool depth dynamics, by Equations (10) or (27). Geometric and soil physical parameters are defined in the opening lines of the scripts and can easily be manipulated by the user. An example of this model is presented in Section 4.2.

#### 3.5. Model Parameters for Demonstration

#### 3.5.1. Constant Thaw-Layer Thickness

#### 3.5.2. Dynamic Thaw-Layer Thickness

_{r}and K

_{z}), the discharge conductance (κ), and the initial ponded water level (H(0)). We calibrated the model using a Levenberg–Marquardt approach implemented in the Julia LsqFit package (https://github.com/JuliaNLSolvers/LsqFit.jl). The objective function was the sum of the squared errors between the measured and simulated ponded water levels in the polygon center. Regularization terms were added to the objective function to penalize solutions where K

_{r}and κ deviated from physical values. Assuming preferential horizontal (radial) flow, the value of K

_{z}was required to be less than K

_{r}.

## 4. Results

#### 4.1. Flow Nets: Head and Flux Distributions

^{−1}(assuming k as a fraction of ${K}_{r}$ (e.g., 50%) and l on the order of 0.25 m). For a polygon $L=D=0.5\mathrm{m}$ thick, several horizontal polygon sizes were considered: R =0.5, 1.0, 1.5, 5.0, and 10.0 m.

#### 4.2. Water Level Dynamics and the Role of Evapotranspiration

_{w}remains approximately constant during the time period of interest. Figure 3a indicates that discharge increases linearly with the anisotropy-adjusted aspect ratio, ${R}^{*}$, for practically any value of $Bi$, the hydraulic conductance of the polygon drainage interface. This general graph can be utilized to evaluate of the polygon drainage dynamics, indicating that discharge is linearly dependent on the anisotropy-adjusted aspect ratio and is only dependent on the drainage interface when its conductance is significantly low.

_{r}= 1 m/day, K

_{z}= 0.2 m/day, and k = 0.5 m day

^{−1}(i.e., κ = 1.0 day

^{−1}). The initial center pool level was H(0) = 0.65 m (i.e., 0.25 m above the ground surface) and the water level in the trough was h

_{w}= 0.45 m. These parameters yielded dimensionless values R* = 16.8 and Bi = 0.89. The characteristic time of this problem was ${t}_{L}$ = 18 days. (Sensitivity of results to κ is moderate at these values, as discussed in Appendix B).

^{−3}m day

^{−1}. Therefore, we used E = $0.5\xb7{10}^{-3}$m day

^{−1}and E = $2\xb7{10}^{-3}$m day

^{−1}to explore the effect on t

_{drain}values.

_{w}= 0.35 m. Equation (27) shows that the pool will drain entirely at any evaporation rate (E = 0, $0.5\xb7{10}^{-3}$, and $2\xb7{10}^{-3}$ m day

^{−1}). Corresponding drainage times, calculated from Equation (30) are t

_{drain}= 33, 30, and 25 days, and are similar to typical conditions observed in the Arctic, either at the start of the summer or following a large precipitation event in mid-summer [5,8].

#### 4.3. Calibration of Center Pond Drainage Scenario with Dynamic Thaw-Layer Thickness at Utqiagvik

_{r}, K

_{z}, and κ fit squarely within the ranges defined in Section 3.5.1, with an anisotropy ratio K

_{r}/K

_{z}of ~7.6. The close match between observed and simulated head and the reasonable calibration of hydraulic parameters affirms the physical realism of our model in a scenario characterized by rapid changes in thaw-layer thickness and trough pool water level. A video illustrating these results is also available online in the Supplementary Material.

## 5. Conclusions

- Polygons are flushed most intensively at the edges for practically all existing physical and geometric parameters. The streamline patterns in this zone change little when the aspect ratio (radius-to-thickness of active layer) exceeds a value of three.
- Anisotropy in hydraulic conductivity (horizontal-to-vertical hydraulic conductivity ratio) has a secondary influence on the intensity of flushing. Increases of anisotropy values counteract the effects of increased geometrical aspect ratio increases and vice versa (as discussed in Appendix B).
- Hydraulic resistance of the drainage interface between the polygon and trough also has some limited, but not overriding influence within the typical range of Arctic tundra conditions.
- Drainage time scales are consistent with observed duration of the center pool drainage within low-centered polygons. The parameter ${t}_{L}$ can be used to characterize drainage times from an inundated polygon center.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of the Head Distribution and Stokes Stream Function

_{n}uses identities:

_{n}, resulting in Equation (18).

## Appendix B. Role of Hydraulic Resistance of the Polygon-trough Interface and Anisotropy

**Figure A1.**Three sets of streamlines corresponding to values of Bi = 0.01, 1.0, and 10. Each stream tube carries 10% of the flux, so the area (volume) above the lower streamline in each set contains 90% of the water flux.

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**Figure 1.**Schematic diagram of our three-dimensional axisymmetric analytical model of inundated low-centered polygon drainage. The diagram represents an idealized “pie wedge” section of a low-centered polygon, including pools in the center and trough.

**Figure 2.**Effects of variable R* on the spatial distribution of fluxes across the polygon. The color bar applies to the head ${h}^{*}\left({r}^{*},{z}^{*}\right)$. All streamlines (in black) are contour lines of the stream function ${\Psi}^{*}\left({r}^{*},{z}^{*}\right)$, defining 10 stream tubes in total. The area (volume), conducting 90% of the water flux to the trough shifts toward the polygon edge with increasing R*.

**Figure 3.**Drainage dynamics through the polygon: (

**a**) dimensionless function ${Q}^{*}\left({R}^{*},B{i}^{*}\right)$, (

**b**) example of center pool depth dynamics for specific parameter values with characteristic time scale ${t}_{L}=18$ days and different evaporation rates and water levels in the trough. Continuous lines show actual water level dynamics in the pool before complete draining; dashed lines are auxiliary and are mathematical, non-physical parts of the solution after drainage is complete. Times of draining t

_{drain}are indicated by stem plots on the time axis.

**Figure 4.**Observed and simulated center pool water level, from a piecewise-linear simulation of early summer drainage at a low-centered polygon near Utqiagvik, Alaska, accounting for temporal variability in thaw-layer thickness beneath the polygon center and water level in the trough pool. A video of this simulation is available online in the Supplementary Material.

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**MDPI and ACS Style**

Zlotnik, V.A.; Harp, D.R.; Jafarov, E.E.; Abolt, C.J.
A Model of Ice Wedge Polygon Drainage in Changing Arctic Terrain. *Water* **2020**, *12*, 3376.
https://doi.org/10.3390/w12123376

**AMA Style**

Zlotnik VA, Harp DR, Jafarov EE, Abolt CJ.
A Model of Ice Wedge Polygon Drainage in Changing Arctic Terrain. *Water*. 2020; 12(12):3376.
https://doi.org/10.3390/w12123376

**Chicago/Turabian Style**

Zlotnik, Vitaly A., Dylan R. Harp, Elchin E. Jafarov, and Charles J. Abolt.
2020. "A Model of Ice Wedge Polygon Drainage in Changing Arctic Terrain" *Water* 12, no. 12: 3376.
https://doi.org/10.3390/w12123376