The pathways and timing of drainage from the inundated centers of ice-wedge polygons in a warming climate have important implications for carbon flushing, advective heat transport, and transitions from methane to carbon dioxide dominated emissions. Here, we expand on previous research using a recently developed analytical model of drainage from a low-centered polygon. Specifically, we perform (1) a calibration to field data identifying necessary model refinements and (2) a rigorous model sensitivity analysis that expands on previously published indications of polygon drainage characteristics. This research provides intuition on inundated polygon drainage by presenting the first in-depth analysis of drainage within a polygon based on hydrogeological first principles. We verify a recently developed analytical solution of polygon drainage through a calibration to a season of field measurements. Due to the parsimony of the model, providing the potential that it could fail, we identify the minimum necessary refinements that allow the model to match water levels measured in a low-centered polygon. We find that (1) the measured precipitation must be increased by a factor of around 2.2, and (2) the vertical soil hydraulic conductivity must decrease with increasing thaw depth. Model refinement (1) accounts for runoff from rims into the ice-wedge polygon pond during precipitation events and possible rain gauge undercatch, while refinement (2) accounts for the decreasing permeability of deeper soil layers. The calibration to field measurements supports the validity of the model, indicating that it is able to represent ice-wedge polygon drainage dynamics. We then use the analytical solution in non-dimensional form to provide a baseline for the effects of polygon aspect ratios (radius to thaw depth) and coefficient of hydraulic conductivity anisotropy (horizontal to vertical hydraulic conductivity) on drainage pathways and temporal depletion of ponded water from inundated ice-wedge polygon centers. By varying the polygon aspect ratio, we evaluate the relative effect of polygon size (width), inter-annual increases in active-layer thickness, and seasonal increases in thaw depth on drainage. The results of our sensitivity analysis rigorously confirm a previous analysis indicating that most drainage through the active layer occurs along an annular region of the polygon center near the rims. This has important implications for transport of nutrients (such as dissolved organic carbon) and advection of heat towards ice-wedge tops. We also provide a comprehensive investigation of the effect of polygon aspect ratio and anisotropy on drainage timing and patterns, expanding on previously published research. Our results indicate that polygons with large aspect ratios and high anisotropy will have the most distributed drainage, while polygons with large aspect ratios and low anisotropy will have their drainage most focused near their periphery and will drain most slowly. Polygons with small aspect ratios and high anisotropy will drain most quickly. These results, based on parametric investigation of idealized scenarios, provide a baseline for further research considering the geometric and hydraulic complexities of ice-wedge polygons.

Polygonal tundra occurs in continuous-permafrost landscapes lacking exposed bedrock or active sedimentation

Polygonal tundra forms in cold environments by the cyclic process of vertical cracking of frozen ground due to thermal contraction, water infiltration into these cracks, freezing of this infiltrated water, and subsequent re-cracking.
Over many cycles, this process leads to the growth of subsurface ice wedges connected in polygonal patterns known as ice-wedge polygons

Pie wedge schematic diagram of 3D axisymmetric analytical model of inundated low-centered polygon drainage. The diagram represents an idealized

Recent observational studies have shaped our current conceptualization of low-centered polygonal tundra hydrology

The microtopographic features of polygonal tundra result in pronounced fine-scale spatial gradients in thermal and hydrologic conditions

Although there have been numerous field observations of inundated low-centered polygon drainage to troughs

The geometric shape of low-centered polygons along with soil hydraulic properties (for example, hydraulic conductivity) affects the distribution of hydraulic heads that control the pathways and timing of inundated ice-wedge polygon drainage.
In this paper, we use a recently developed model

This model idealizes the thawed subsurface of an inundated low-centered polygon as a thin cylinder overlain with an initial height of ponded water that drains through the cylindrical thawed soil layer to the surrounding trough (outer vertical boundary; an annular ring defined by a line from (

We demonstrate that the model is able to accurately simulate ice-wedge polygon drainage by performing a calibration to field measurements over an entire thaw season. We calibrate the model parameters to fit water levels measured in the center of a low-centered polygon within the Barrow Environmental Observatory near Utqiaġvik, Alaska. We force the model with measured trough water levels, precipitation, thaw depth, and evapotranspiration. The goodness of fit of initial calibration efforts indicated that the model was unable to capture the polygon-center water level trends (i.e., the model was falsified). However, by considering (1) increased precipitation and (2) a thaw-depth-dependent (and therefore time-variable) vertical hydraulic conductivity, the model produces a good fit to the water levels. Model refinement (1) accounts for pond accumulation due to runoff from surrounding microtopographic highs (e.g., rims), while model refinement (2) accounts for the decreasing hydraulic conductivity of deeper tundra soil layers. The calibration indicates the minimum refinements necessary to a hydrology model to capture ice-wedge polygon drainage dynamics and indicates the important factors in this process. This calibration, based on transient boundary conditions, provides confidence in the physical meaningfulness of the steady-state snapshots presented in the sensitivity analysis.

We then use the model to investigate the pathways and timing of inundated polygon drainage through a sensitivity analysis of polygon geometry and anisotropy.
The analysis of the model enhances our intuition into the pathways and timing of inundated ice-wedge polygon drainage for polygons with different geometries and degrees of anisotropy.
Varying the geometry of the cylinder allows us to capture not only relative differences in drainage between different-sized polygons (in other words, polygons with different radii) but also seasonal and inter-annual variations in polygon thaw-layer depths within a single polygon (in other words, as the thawed soil layer increases during a single thaw season or as the active layer increases from year to year).
By allowing for different effective hydraulic conductivities between the horizontal and vertical directions, we can evaluate the effects of preferential flow directions on drainage.
In addition to geologic layering, processes such as frost heave and horizontal ice lens thaw can result in preferential horizontal flow in cold climates

The calibration to field data indicates that the model is able to capture ice-wedge polygon drainage dynamics. Our sensitivity analysis provides a new perspective on inundated polygon hydrogeology, indicating the effects of polygon geometry and hydraulic conductivity anisotropy on drainage pathways and timing. Although the simplifications of the model may limit its applicability to some scenarios, they allow general intuitive insights to be drawn which would be obfuscated without them. The findings here provide a basis to quantify and understand deviations from our idealized scenarios.

We use recently developed 3D axisymmetric analytical solutions, derived and validated in

Non-dimensional hydraulic head lines are drawn from 0.05 to 0.95 with increments of 0.1.
Dimensional heads (

We use a Robin boundary condition (a third-type boundary condition allowing both head and flux to be specified) for the outer vertical boundary.
This allows the model to represent the resistance of drainage under the rim due to elevated frozen ground and due to the accumulation of fine soil particles at the soil–water interface of the trough

We normalize the stream function

The change in non-dimensional ponded water height in the polygon center due solely to drainage (the depletion curve) over time can be expressed by a simple exponential decay function as

As described in

We performed nonlinear least-squares calibrations using a Levenberg–Marquardt approach

In calibration case 1,

In calibration case 2, we add a precipitation multiplier

In calibration case 3, we add vertical hydraulic conductivity as a super-elliptical function of thaw depth to calibration case 2 as

The measured data used in the calibration were collected during the thaw season of 2013 from 16 June to 18 September from a low-centered polygon within the Barrow Environmental Observatory near Utqiaġvik, Alaska (Fig.

We selected aspect ratio and anisotropy scenarios based on existing literature and observations.
While a pan-Arctic survey of ice-wedge polygon diameters does not to our knowledge currently exist, researchers have provided general characteristics based on extensive observations.

Maximum thaw depths are increasing throughout the Arctic

Based on these considerations, we evaluate aspect ratios from 2.5 to 20. For example, given a thaw depth of 1 m, an aspect ratio of 2.5 would represent a 2.5 m radius polygon center, while an aspect ratio of 20 would represent a 20 m radius polygon center, more than covering the observed range. To consider cases with thinner thawed soil layers, larger aspect ratios can be used. For example, given a thaw depth of 0.5 m, an aspect ratio of 20 would represent a polygon with a 10 m radius center.

Comprehensive measurements of hydraulic conductivity anisotropy are lacking from ice-wedge polygons.
Based on tracer arrival times,

A physical interpretation of our selected values of anisotropy coefficient can be obtained by considering that ice-wedge polygon soils are typically layered and that the horizontal and vertical hydraulic conductivities can therefore be considered to be

The hydraulic conductivity of the interface between soil and open water can be half that of the rest of the soil due to the accumulation of fines

In practice, as in our calibration, the water level in troughs (

Although included in the calibration, we have neglected the effects of evaporation and precipitation in the sensitivity analysis as they will not affect the drainage patterns we present (based on non-dimensional heads), and their effect on drainage timing (based on non-dimensional depletion curves) is straightforward, shifting the non-dimensional exponential drainage curve upwards or downwards. In other words, using non-dimensional variables is a powerful approach to gain intuition into the fundamentals of inundated ice-wedge polygon drainage irrespective of variable magnitude.

We verify the model through calibration to water level measurements, identifying refinements necessary for hydrologic models to match field observations of polygon drainage.
We present drainage flow nets for various aspect ratios and anisotropy coefficient values, polygon-center ponded height depletion curves, and maps of the percent of the thawed soil accessed by 95 % of the drainage flow and depletion characteristic times as a function of aspect ratio and anisotropy coefficient.
We compare the effect of aspect ratio on drainage pathways in an isotropic and a highly anisotropic (

We present the best-fit polygon-center water levels for the calibration cases along with the measured values in Fig.

Considering that precipitation will run off from rims and collect in the polygon-center pond and that rain gauges may have undercatch issues (e.g.,

The compromised fit in calibration case 2 indicated that the effective vertical hydraulic conductivity likely decreases as the thaw depth increases, which would be consistent with observations of reduced hydraulic conductivities at depth.
In calibration case 3, we implemented vertical hydraulic conductivity as a function of thaw depth.
This model refinement allowed the calibration to achieve a good overall fit to the data (RMSE

Calibrated parameters values, root mean square error (RMSE), coefficient of determination (

The standard errors in the calibrated parameters for calibration case 3 listed in Table

The calibration verifies that the model is able to capture ice-wedge polygon drainage characteristics. In the next sections, we perform sensitivity analyses using non-dimensional forms of this verified analytical solution to gain insights into ice-wedge polygon drainage characteristics. The use of non-dimensional solution snapshots eliminates the need to consider the precipitation multiplier and thaw-depth-dependent vertical hydraulic conductivity explicitly. Instead, their effects are implicit in the relative differences between snapshots.

Effect of polygon aspect ratio on polygon drainage with isotropic hydraulic conductivity. Plots along the left contain polygon radial transect head contours (colored lines) and filled stream tubes (gray regions) for several polygon aspect ratios (radius/thickness). The gray shaded region denotes the portion of the transect accessed by 95 % of the flow. The plots along the right contain corresponding gray rings indicating the surface area where 95 % of the polygon flow infiltrates. Each plot along the left contains a rectangle drawn to the actual proportions for the given polygon aspect ratio. In all cases, the anisotropy coefficient (horizontal conductivity/vertical conductivity) is fixed at unity.

The relative effect of aspect ratio on drainage pathways when the hydraulic conductivities are isotropic (

Effect of hydraulic conductivity anisotropy coefficient on polygon drainage. Plots along the left contain polygon radial transect head contours (colored lines) and filled stream tubes (gray regions) for several anisotropies (horizontal conductivity/vertical conductivity). The gray shaded region denotes the portion of the transect accessed by 95 % of the flow. The plots along the right contain corresponding gray rings indicating the surface area where 95 % of the polygon flow infiltrates. Each plot along the left contains a rectangle drawn to the actual proportions for the given polygon aspect ratio. In all cases, the polygon aspect ratio (radius/thickness) is fixed at 10.

The effect of anisotropy coefficient on drainage pathways when the aspect ratio is held constant (

Effect of polygon aspect ratio on polygon drainage with high-hydraulic-conductivity anisotropy coefficient (

The effect of aspect ratio on drainage pathways when the hydraulic conductivities are highly anisotropic (

Contour map of the percent of the polygon access by 95 % of the flow as a function of polygon aspect ratio and anisotropy coefficient. Locations associated with other figures are indicated by labeled points. By choosing a constant value for

To gain a global perspective on the trends in drainage pathways with aspect ratio and anisotropy coefficient, Fig.

Depletion of ponded water height over time (

Depletion curves for the non-dimensional ponded water height in the polygon center for various aspect ratios (at fixed high and low anisotropy coefficient) and anisotropies are shown in Fig.

Characteristic time as a function of

It is apparent from these plots that small, deeply thawed (lower aspect ratio) polygons will drain faster than wide, shallowly thawed (high aspect ratio) polygons, while polygons with a higher anisotropy coefficient will drain faster than those with a lower anisotropy coefficient.
It is also apparent that the increase in drainage time with aspect ratio is nearly linear, while for anisotropy coefficient, it is exponential.
This is further illustrated in Fig.

Contour map of the characteristic time of polygon drainage as a function of polygon aspect ratio and anisotropy coefficient. Locations associated with other figures are indicated by labeled points. By choosing a constant value for

A global perspective on drainage timing trends, where characteristic time is mapped as a function of aspect ratio and anisotropy coefficient, is shown in Fig.

Attaining the same mathematical solution by modifying polygon aspect ratio (

It is important to note that aspect ratio and anisotropy have similar effects on drainage.
Within the model used here, in a similar fashion to aspect ratio, the anisotropy coefficient stretches the domain by multiplying the non-dimensional radius by

In the bottom plot in Fig.

Our analysis provides new insights into the relative effects of geometry and anisotropy on the manner in which inundated ice-wedge polygons retain and slowly release water from their centers to their troughs, which form the drainage network of polygonal tundra landscapes.
Using a mathematical representation of inundated ice-wedge polygon drainage

The calibration identifies factors which need to be considered by any hydrologic model to simulate drainage from an inundated polygon center. Using a parsimonious model, we were able to identify the refinements required in a polygon drainage model to capture center water levels. Using more complex models would likely obfuscate the identification of these refinements. The final model formulation provides a fast model for predicting the manner and timing of polygon drainage driven by environmental factors. The calibration of the model, driven by transient boundary conditions, provides confidence in the real-world applicability of the sensitivity analysis based on comparisons of steady-state snapshots of the model.

The first refinement is a precipitation multiplier and is based on a simple mass balance indicating that the measured precipitation cannot account for the total increase in ponded water levels after precipitation events.
The precipitation multiplier accounts for the fact that precipitation will run off from the rims into the center pond, resulting in a larger increase in ponded water than precipitation measured by a rain gauge.
The precipitation multiplier also accounts for any potential rain gauge undercatch.
The second refinement is a thaw-depth-dependent vertical hydraulic conductivity that accounts for the decreased hydraulic conductivity of deeper soil layers.
This refinement accounts for the decrease in vertical hydraulic conductivities as the thaw depth increases and includes less hydraulically conductive soils.
The final calibration provides a good match to measurements, with a sub-centimeter RMSE of 0.37 cm and an

Due to covariance in the effects of the hydraulic conductivity parameters on water levels, the hydraulic conductivity parameters are loosely constrained by the calibration (based on local sensitivities). This indicates the importance of constraining the hydraulic conductivity parameters with field measurements if possible (field measurements are not available in our case). However, despite relatively large standard errors in the hydraulic conductivity parameter estimates, the calibration identifies physically realistic values. It should also be noted that despite vertical hydraulic conductivity being loosely constrained in the final calibration (calibration case 3), the model was unable to match water levels with a constant vertical hydraulic conductivity (calibration case 2). This indicates the importance of the thaw-depth-dependent vertical hydraulic conductivity in the model. The other parameters (discharge conductance, initial polygon-center water level, and precipitation multiplier) are all well constrained by the analysis.

The water level

A key result of this study is that the geometry and anisotropy of the polygon subsurface have a significant effect on the region of the polygon subsurface predominantly accessed by drainage flow and the time to transition from inundated to drained.
Due to the geometry of inundated polygons, the primary drainage pathway is restricted to an annular, radially peripheral region of the ice-wedge polygon center near the rim.
As a result, the middle and lower portions of the polygon center are excluded from the majority of the drainage to varying degrees depending on the polygon geometry and anisotropy.
Also, the majority of the ponded water will flow towards the rim of the polygon before infiltrating into the subsurface.
Field observations have indicated not only the existence of intra-polygon biogeochemical diversity

The potential that the annular, radially peripheral region near the rims will be well flushed of nutrients, while the middle may not, indicates the need for additional field studies designed to measure the effects of anisotropy and preferential flow paths on thermal hydrology and biogeochemistry.
For isotropic cases, it should also be considered that the drainage will spread out further towards the middle of the polygon center as the thaw season progresses, and the thawed soil layer thickens (in other words, the aspect ratio decreases; Figs.

For a given thaw depth, advective heat transport will be more focused near the rim for larger polygons and may result in enhanced ice-wedge degradation

Small polygons with deeply thawed soil layers (low aspect ratios) and high horizontal preferential flow (high anisotropy coefficient) have the potential to drain most quickly. Therefore, all other factors being equal, regions of polygonal tundra characterized by small, deeply thawed, anisotropic polygons will drain more quickly and consequently will have a greater potential for nutrient flushing, transition from methane to carbon dioxide atmospheric emissions, and biological succession than regions with large, shallowly thawed, isotropic polygons. Concerning temporal changes during the thaw season, polygon pond depletion will slow down as the thaw depth increases, and this reduction will become more dramatic over the thaw season as inter-annual active-layer thickness increases. This implies that a thickening active layer due to warming trends may result in slower pond depletion. For a given location, factors such as regional flow patterns, large-scale topography, etc. will influence the region's overall drainage timing. However, along with these other factors, our analysis indicates that aspect ratio will have a nearly linear positive relationship, while the anisotropy coefficient will have an exponential negative relationship with drainage timing.

The drainage pathways and timing presented here are based on hydrogeological first principles

In our sensitivity analysis, we use idealized conceptualizations and dimensionless variables, which allow hydrologic characteristics of drainage to be exposed in their most fundamental form. To clearly and concisely expose these characteristics, we neglect factors such as evaporation, precipitation, and non-idealized polygon geometry (evaporation and precipitation are included in the calibration). As a result, our analysis is not intended to provide predictive capability across all polygonal tundra scenarios but to provide hydrologic intuition into the relative effects of geometry and anisotropy on inundated ice-wedge polygon drainage and timing.

As the analytical solution applies to ponded conditions, it does not apply to freeze-up at the end of the thaw season, after the ponded water in the polygon center freezes.
At this time, the active layer freezes simultaneously from the top and bottom, and cryosuction draws water towards the freezing fronts.
This redistribution of water affects ice-wedge polygon drainage, and

Since the model is based on the saturated groundwater flow equation, in its current form it cannot be applied to non-inundated low-centered polygons. Since it is based on having a ponded center, the model is also not applicable to high-centered polygons. However, a similar approach to that presented here could be taken with the unsaturated groundwater flow equation to capture these other polygon scenarios and types.

As our analysis is the first detailed examination of inundated ice-wedge polygon drainage patterns, our results provide a new perspective to existing mathematical analyses.
Much of the existing mathematical analyses investigate drainage from polygonal landscapes.
For example,

Our results affirm that existing conceptualizations of polygon drainage need revisiting, as was implied by a preliminary analysis conducted by

A simple analytical solution based on hydrological first principles is able to capture ice-wedge polygon drainage dynamics over an entire thaw season. Consideration within the model of runoff from topographic highs (rims) during rain events and decreasing effective vertical hydraulic conductivity with increasing thaw depth is required to match field measurements. We were able to identify these necessary model components by using a parsimonious model that has the ability to fail.

We provide rigorous confirmation that the majority of drainage from inundated ice-wedge polygon centers occurs along an annular region along their radial periphery; however, polygon geometry and hydraulic conductivity anisotropy significantly impact the drainage pathways, as originally postulated by

A combination of high aspect ratio (wide, shallow polygons) and high anisotropy coefficient (preferential horizontal flow) results in the greatest spreading of drainage flow towards the middle of the polygon center and the largest fraction of the polygon volume being accessed by drainage flow. In these cases, the nutrient flushing will be more uniform than other cases, and advective heat transport towards the ice-wedge top will be less focused and therefore less able to thaw the ice-wedge top.

A combination of high aspect ratio (wide, shallow polygons) and low anisotropy coefficient (preferential vertical flow) results in the greatest focusing of drainage flow and the smallest fraction of the polygon volume being accessed by drainage flow. In these cases, nutrient flushing will be more localized along the outer edge of the polygon (non-uniform) and the advective heat transport towards the ice-wedge tops most focused, possibly resulting in greater degradation of ice wedges than in other cases.

Combinations of aspect ratio and anisotropy coefficient have counteracting effects of radial versus vertical extension and contraction of drainage pathways, producing non-monotonic relationships between aspect ratio or anisotropy coefficient and accessed volume (curved feature in accessed volume response surface in Fig.

Polygon drainage time has an approximately positive linear relationship with aspect ratio when anisotropy is held constant; in other words, wide, shallow polygons drain slowly, while small, deep polygons drain quickly. This implies that polygonal tundra with larger polygons may drain more slowly than tundra composed of smaller polygons.

Polygon drainage time has a negative exponential relationship with anisotropy coefficient when aspect ratio is held constant. In other words, increases in preferential horizontal flow lead to exponentially faster drainage. Therefore, polygonal tundra with greater preferential horizontal flow, due to more pronounced horizontal stratigraphy or ice lens development, will drain faster, while less horizontally stratified tundra, due to cryoturbation, for example, will drain more slowly.

We idealize the subsurface below the center region of a low-centered polygon as a thin cylinder with its radius in the horizontal direction and length in the vertical direction (refer to Fig.

Using dimensionless coordinates and parameters defined as

The solutions can be verified by direct substitution of Eqs. (

The ponded height depletion curve can be defined as

MATLAB code of the analytical solution is available via

All data sets are freely available to the public at the provided references in the paper.

A video illustrating the validation of the analytical solution to an inundated ice-wedge polygon drainage event in 2012 near Utqiaġvik is available via

DRH and VZ developed the conceptual model of inundated polygonal tundra hydrology. VZ derived the analytical solutions. VZ and DRH encoded the analytical solutions. CJA created Fig.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The Next Generation Ecosystem Experiments Arctic (NGEE-Arctic) project (DOE ERKP757), funded by the Office of Biological and Environmental Research within the US Department of Energy's Office of Science, supported this research. Sofia Avendaño and Bulbul Ahmmed provided reviews during the development of this article, providing technical and editorial improvements.

This research has been supported by the US Department of Energy, Office of Science (grant no. DOE ERKP757).

This paper was edited by Moritz Langer and reviewed by Jan Nitzbon and one anonymous referee.