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**Model description paper**
08 Jan 2018

**Model description paper** | 08 Jan 2018

Impacts of microtopographic snow redistribution and lateral subsurface processes on hydrologic and thermal states in an Arctic polygonal ground ecosystem: a case study using ELM-3D v1.0

Impacts of microtopographic snow redistribution and lateral subsurface processes on hydrologic and thermal states in an Arctic polygonal ground ecosystem: a case study using ELM-3D v1.0
Impacts of microtopographic snow redistribution and lateral subsurface processes on hydrologic...
Gautam Bisht et al.

^{1}Climate & Ecosystem Sciences Division, Lawrence Berkeley National Laboratory,1 Cyclotron Road, Berkeley, CA 94720, USA^{2}Environmental Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6301, USA^{3}Geophysical Institute, University of Alaska Fairbanks, Fairbanks, AK 99775, USA

^{1}Climate & Ecosystem Sciences Division, Lawrence Berkeley National Laboratory,1 Cyclotron Road, Berkeley, CA 94720, USA^{2}Environmental Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6301, USA^{3}Geophysical Institute, University of Alaska Fairbanks, Fairbanks, AK 99775, USA

**Correspondence**: Gautam Bisht (gbisht@lbl.gov)

**Correspondence**: Gautam Bisht (gbisht@lbl.gov)

Abstract

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Microtopographic features, such as polygonal ground, are characteristic
sources of landscape heterogeneity in the Alaskan Arctic coastal plain. Here,
we analyze the effects of snow redistribution (SR) and lateral subsurface
processes on hydrologic and thermal states at a polygonal tundra site near
Barrow, Alaska. We extended the land model integrated in the E3SM to
redistribute incoming snow by accounting for microtopography and incorporated
subsurface lateral transport of water and energy (ELM-3D v1.0). Multiple
10-year-long simulations were performed for a transect across a polygonal
tundra landscape at the Barrow Environmental Observatory in Alaska to isolate
the impact of SR and subsurface process representation. When SR was included,
model predictions better agreed (higher *R*^{2}, lower bias and RMSE) with
observed differences in snow depth between polygonal rims and centers. The
model was also able to accurately reproduce observed soil temperature
vertical profiles in the polygon rims and centers (overall bias, RMSE, and
*R*^{2} of 0.59 ^{∘}C, 1.82 ^{∘}C, and 0.99, respectively). The
spatial heterogeneity of snow depth during the winter due to SR generated
surface soil temperature heterogeneity that propagated in depth and time and
led to ∼ 10 cm shallower and ∼ 5 cm deeper maximum annual thaw
depths under the polygon rims and centers, respectively. Additionally, SR led
to spatial heterogeneity in surface energy fluxes and soil moisture during
the summer. Excluding lateral subsurface hydrologic and thermal processes led
to small effects on mean states but an overestimation of spatial variability
in soil moisture and soil temperature as subsurface liquid pressure and
thermal gradients were artificially prevented from spatially dissipating over
time. The effect of lateral subsurface processes on maximum thaw depths was
modest, with mean absolute differences of ∼ 3 cm. Our integration of
three-dimensional subsurface hydrologic and thermal subsurface dynamics in
the E3SM land model will facilitate a wide range of analyses heretofore
impossible in an ESM context.

How to cite

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How to cite.

Bisht, G., Riley, W. J., Wainwright, H. M., Dafflon, B., Yuan, F., and Romanovsky, V. E.: Impacts of microtopographic snow redistribution and lateral subsurface processes on hydrologic and thermal states in an Arctic polygonal ground ecosystem: a case study using ELM-3D v1.0, Geosci. Model Dev., 11, 61–76, https://doi.org/10.5194/gmd-11-61-2018, 2018.

1 Introduction

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The northern circumpolar permafrost region, which contains ∼ 1700 Pg
of organic carbon down to 3 m (Tarnocai et al., 2009), is predicted to
experience disproportionately larger future warming compared to the tropics
and temperate latitudes (Holland and Bitz, 2003). Recent warming in the
Arctic has led to changes in lake area (Smith et al., 2005), snow cover
duration and extent (Callaghan et al., 2011a), vegetation cover (Sturm et
al., 2005), growing season length (Smith et al., 2004), thaw depth (Schuur et
al., 2008), permafrost stability (Jorgenson et al., 2006), and
land–atmosphere feedbacks (Euskirchen et al., 2009). Future predictions of
Arctic warming include northward expansion of shrub cover in tundra (Sturm et
al., 2001; Tape et al., 2006), decreases in snow cover duration (Callaghan et
al., 2011a), and emissions of CO_{2} and CH_{4} from decomposition of
belowground soil organic matter (Koven et al., 2011; Schaefer et al., 2011;
Schuur and Abbott, 2011; Xu et al., 2016).

Several recent modeling studies have predicted a positive global carbon–climate feedback at the global scale (Cox et al., 2000; Dufresne et al., 2002; Friedlingstein et al., 2001; Fung et al., 2005; Govindasamy et al., 2011; Jiang et al., 2011; Jones et al., 2003; Koven et al., 2015; Matthews et al., 2005, 2007; Sitch et al., 2008; Thompson et al., 2004; Zeng et al., 2004), although the strength of this predicted feedback at the year 2100 was shown to have a large variability across models (Friedlingstein et al., 2006). In contrast to the ocean carbon cycle, the terrestrial carbon cycle is expected to be a more dominant factor in the global carbon–climate feedback over the next century (Matthews et al., 2007; Randerson et al., 2015).

Snow, which covers the Arctic ecosystem for 8–10 months each year (Callaghan
et al., 2011b), is a critical factor influencing hydrologic and ecologic
interactions (Jones, 1999). Snowpack modifies surface energy balances (via
high reflectivity), soil thermal regimes (due to low thermal conductivity),
and hydrologic cycles (because of meltwater). Several studies have shown
that warm soil temperatures under snowpack support the emission of greenhouse
gases from belowground respiration (Grogan and Chapin III, 1999; Sullivan, 2010) and nitrogen mineralization (Borner et
al., 2008; Schimel et al., 2004) during winter. Additionally, decreases in
snow cover duration have been shown to increase net ecosystem CO_{2} uptake
(Galen and Stanton, 1995; Groendahl et al., 2007). Recent snow manipulation
experiments in the Arctic have provided evidence of the importance of snow in
the expected responses of Arctic ecosystems under future climate change
(Morgner et al., 2010; Nobrega and Grogan, 2007; Rogers et al., 2011; Schimel
et al., 2004; Wahren et al., 2005; Welker et al., 2000).

Apart from the spatial extent and duration of snowpack, the spatial heterogeneity of snow depth is an important factor in various terrestrial processes (Clark et al., 2011; Lundquist and Dettinger, 2005). As synthesized by López-Moreno et al. (2015), the following processes are responsible for snow depth heterogeneity at three distinct spatial scales: microtopography at 1–10 m (López-Moreno et al., 2011); wind-induced lateral transport processes at 100–1000 m (Liston et al., 2007); and precipitation variability at catchment scales of 10–1000 km (Sexstone and Fassnacht, 2014). The spatial distribution of snow not only affects the quantity of snowmelt discharge (Hartman et al., 1999; Luce et al., 1998), but also the water chemistry (Rohrbough et al., 2003; Wadham et al., 2006; Williams et al., 2001). Lawrence and Swenson (2011) demonstrated the importance of snow depth heterogeneity in predicting responses of the Arctic ecosystem to future climate change by performing idealized numerical simulations of shrub expansion across the pan-Arctic region using the Community Land Model (CLM4). Their results showed that an increase in active layer thickness, which is the maximum annual thaw depth, under shrubs was negated when spatial heterogeneity in snow cover due to wind-driven snow redistribution was accounted for, resulting in an unchanged grid cell mean active layer thickness.

Large portions of the Arctic are characterized by polygonal ground features,
which are formed in permafrost soil when frozen ground cracks due to thermal
contraction during winter and ice wedges form within the upper several meters
(Hinkel et al., 2005). Polygons can be classified as “low-centered” or
“high-centered” based on the relationship between their central and mean
elevations. Polygonal ground features are dynamic components of the Arctic
landscape in which the upper part of ice-wedge thaw under low-centered
polygon troughs leads to subsidence, eventually (∼ o(centuries))
converting the low-centered polygon into a high-centered polygon (Seppala et
al., 1991). Microtopography of polygonal ground influences soil hydrologic
and thermal conditions (Engstrom et al., 2005). In addition to controlling
CO_{2} and CH_{4} emissions, soil moisture affects (1) partitioning of
incoming radiation into latent, sensible, and ground heat fluxes (Hinzman and
Kane, 1992; McFadden et al., 1998); (2) photosynthesis rates (McGuire et al.,
2000; Oberbauer et al., 1991; Oechel et al., 1993; Zona et al., 2011); and
(3) vegetation distributions (Wiggins, 1951).

Our goals in this study include (1) analyzing the effects of spatially heterogeneous snow in polygonal ground on soil temperature and moisture and surface processes (e.g., surface energy budgets); (2) analyzing how model predictions are affected by the inclusion of lateral subsurface hydrologic and thermal processes; and (3) developing and testing a three-dimensional version of the E3SM Land Model (ELM; Tang and Riley, 2016; Zhu and Riley, 2015), called ELM-3D v1.0 (hereafter ELM-3D). We then applied ELM-3D to a transect across a polygonal tundra landscape at the Barrow Environmental Observatory in Alaska. After defining our study site, the model improvements, model tests against observations, and analyses, we apply the model to examine the effects of snow redistribution and lateral subsurface processes on snow microtopographical heterogeneity, soil temperature, and the surface energy budget.

2 Methodology

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Our analysis focuses on sites located near Barrow, Alaska (71.3^{∘} N,
156.5^{∘} W) from the long-term Department of Energy (DOE)
Next-Generation Ecosystem Experiment (NGEE-Arctic) project. The four primary
NGEE-Arctic study sites (A, B, C, D) are located within the Barrow
Environmental Observatory (BEO), which is situated on the Alaskan Coastal
Plain. The annual mean air temperature for our study sites is approximately
−13 ^{∘}C (Walker et al., 2005) and mean annual precipitation is
106 mm, with the majority of precipitation occurring during the summer season
(Wu et al., 2013). The study site is underlain with continuous permafrost
(Sellmann et al., 1975) and the annual
maximum thaw depth (active layer depth) ranges between 30 and 90 cm (Hinkel et
al., 2003). Although the overall topographic relief for the BEO is low, the
four NGEE study sites have distinct microtopographic features:
low-centered (A), high-centered (B), and transitional polygons (C, D).
Contrasting polygon types are indicative of different stages of permafrost
degradation and were the primary motivation behind the choice of study sites
for the NGEE-Arctic project. Lidar digital elevation model (DEM) data were
available at 0.25 m resolution for the region encompassing all four NGEE
sites. In this work, we perform simulations along a two-dimensional transect
in low-centered polygon Site-A as shown by the dotted line in Fig. 1.

The original version of ELM is equivalent to CLM4.5 (Ghimire et al., 2016; Koven et al., 2013; Oleson et al., 2013) and represents vertical energy and water dynamics, including phase change. We developed ELM-3D by expanding on that model to explicitly represent soil lateral energy and hydrological exchanges and fine-resolution snow redistribution. We run ELM-3D here with prescribed plant phenology (called the “satellite phenology” (SP) mode) since our focus is on thermal dynamics of the system, rather than C cycle dynamics.

The flow of water in the unsaturated zone is given by the *θ*-based
Richards equation as

$$\begin{array}{}\text{(1)}& {\displaystyle \frac{\partial \mathit{\theta}}{\partial t}}=-\mathrm{\nabla}\cdot \mathit{q}-Q,\end{array}$$

where *θ* (m^{3} m^{−3}) is the volumetric soil water content, *t*
(s) is time, ** q** (ms

$$\begin{array}{}\text{(2)}& \mathit{q}=-k\mathrm{\nabla}(\mathit{\psi}+z),\end{array}$$

where *k* (ms^{−1}) is the hydraulic conductivity, *ψ* (m) is the soil
matric potential, and *z* (m) is the height above a reference datum. The
hydraulic conductivity and soil matric potential are nonlinear functions of
volumetric soil moisture. ELMv0 uses the modified form of the Richards equation
of Zeng and Decker (2009) that computes the Darcy flux as

$$\begin{array}{}\text{(3)}& \mathit{q}=-k\mathrm{\nabla}(\mathit{\psi}+z-C),\end{array}$$

where *C* is a constant hydraulic potential above the water table,
*z*_{∇}, given as

$$\begin{array}{}\text{(4)}& C={\mathit{\psi}}_{\mathrm{E}}+z={\mathit{\psi}}_{\mathrm{sat}}{\left[{\displaystyle \frac{{\mathit{\theta}}_{\mathrm{E}}\left(z\right)}{{\mathit{\theta}}_{\mathrm{sat}}}}\right]}^{-B}+z={\mathit{\psi}}_{\mathrm{sat}}+{z}_{\mathrm{\nabla}},\end{array}$$

where *ψ*_{E} (m) is the equilibrium soil matric potential, *ψ*_{sat} (m) is the saturated soil matric potential,
*θ*_{E} (m^{3} m^{−3}) is the volumetric soil water content at
equilibrium soil matric potential, *θ*_{sat} (m^{3} m^{−3})
is the volumetric soil water content at saturation, *z*_{∇} (m) is the height
of water table above the reference datum, and *B* (−) is a fitting parameter
for soil water characteristic curves. Substituting Eqs. (3) and (4) into
Eq. (1) yields the equation for the vertical transport of water in ELMv0:

$$\begin{array}{}\text{(5)}& {\displaystyle \frac{\partial \mathit{\theta}}{\partial t}}={\displaystyle \frac{\partial}{\partial z}}\left[k\left({\displaystyle \frac{\partial \left(\mathit{\psi}-{\mathit{\psi}}_{\mathrm{E}}\right)}{\partial z}}\right)\right]-Q.\end{array}$$

A finite volume spatial discretization and implicit temporal discretization with Taylor series expansion leads to a tridiagonal system of equations. We extended this 1-D (one-dimensional) Richards equation to a 3-D (three-dimensional) representation integrated in ELM-3D, which is presented next.

We use a cell-centered finite volume discretization to decompose the spatial
domain into *N* non-overlapping control volumes, Ω_{n}, such that
$\mathrm{\Omega}={\bigcup}_{n=\mathrm{1}}^{N}{\mathrm{\Omega}}_{n}$, and Γ_{n} represents the boundary of the *n*th control volume. Applying a finite
volume integral to Eq. (1) and the divergence theorem yields

$$\begin{array}{}\text{(6)}& {\displaystyle \frac{\partial}{\partial t}}\underset{{\mathrm{\Omega}}_{n}}{\int}\mathit{\theta}\mathrm{d}V=-\underset{{\mathrm{\Gamma}}_{n}}{\int}\left(\mathit{q}\cdot \mathrm{d}\mathit{A}\right)-\underset{{\mathrm{\Omega}}_{\mathrm{n}}}{\int}Q\mathrm{d}V.\end{array}$$

The spatially discretized equation for the *n*th grid cell that has *V*_{n}
volume and *n*^{′} neighbors is given by

$$\begin{array}{}\text{(7)}& {\displaystyle \frac{\mathrm{d}{\mathit{\theta}}_{n}}{\mathrm{d}t}}{V}_{n}=-\sum _{{n}^{\prime}}\left({\mathit{q}}_{n{n}^{\prime}}\cdot {\mathit{A}}_{n{n}^{\prime}}\right)-Q{V}_{n}.\end{array}$$

For the sake of simplicity in presenting the discretized equation, we assume
the 3-D grid is a Cartesian grid, with each grid cell having a thickness of
Δ*x*, Δ*y*, and Δ*z* in the *x*, *y*, and *z* directions,
respectively. Using an implicit time integral, the 3-D discretized equation
at time *t*+1 for a ($i,j,k)$ control volume is given as

$$\begin{array}{ll}\text{(8)}& {\displaystyle}\left({\displaystyle \frac{\mathrm{\Delta}{\mathit{\theta}}_{i,j,k}^{t+\mathrm{1}}}{\mathrm{\Delta}t}}\right){V}_{i,j,k}& {\displaystyle}=\left({q}_{{x}_{i-\mathrm{1}/\mathrm{2},j,k}}^{t+\mathrm{1}}-{q}_{{x}_{i+\mathrm{1}/\mathrm{2},j,k}}^{t+\mathrm{1}}\right)\mathrm{\Delta}y\mathrm{\Delta}z{\displaystyle}& {\displaystyle}+\left({q}_{{y}_{i,j-\mathrm{1}/\mathrm{2},k}}^{t+\mathrm{1}}-{q}_{{y}_{i,j+\mathrm{1}/\mathrm{2},k}}^{t+\mathrm{1}}\right)\mathrm{\Delta}x\mathrm{\Delta}z\\ {\displaystyle}& {\displaystyle}+\left({q}_{{z}_{i,j,k-\mathrm{1}/\mathrm{2}}}^{t+\mathrm{1}}-{q}_{{z}_{i,j,k+\mathrm{1}/\mathrm{2}}}^{t+\mathrm{1}}\right)\mathrm{\Delta}x\mathrm{\Delta}y-Q{V}_{i,j,k},\end{array}$$

where *q*_{x}, *q*_{y}, and *q*_{z} are the Darcy flux in the *x*, *y*, and
*z* directions, respectively, and $\mathrm{\Delta}{\mathit{\theta}}_{i,j,k}^{t+\mathrm{1}}$ is the
change in volumetric soil liquid water in time, Δ*t*. Using the same
approach as Oleson et al. (2013), the Darcy flux in all three directions is
linearized about *θ* using Taylor series expansion. The linearized
Darcy flux in the *x* direction at the ($i-\mathrm{1}/\mathrm{2},j,k)$ interface is a function
of ${\mathit{\theta}}_{i-\mathrm{1},j,k}$ and ${\mathit{\theta}}_{i,j,k}$:

$$\begin{array}{ll}\text{(9)}& {\displaystyle}{q}_{{x}_{i-\mathrm{1}/\mathrm{2},j,k}}^{t+\mathrm{1}}& {\displaystyle}={q}_{{x}_{i-\mathrm{1}/\mathrm{2},j,k}}^{t}+{\displaystyle \frac{\partial {q}_{{x}_{i-\mathrm{1}/\mathrm{2},j,k}}^{t}}{\partial {\mathit{\theta}}_{i-\mathrm{1},j,k}}}\mathrm{\Delta}{\mathit{\theta}}_{i-\mathrm{1},j,k}^{t+\mathrm{1}}{\displaystyle}& {\displaystyle}+{\displaystyle \frac{\partial {q}_{{x}_{i-\mathrm{1}/\mathrm{2},j,k}}^{t}}{\partial {\mathit{\theta}}_{i,j,k}}}\mathrm{\Delta}{\mathit{\theta}}_{i+\mathrm{1},j,k}^{t+\mathrm{1}}.\end{array}$$

The linearized Darcy fluxes in the *y* and *z* directions are computed
similarly. Substituting Eq. (9) in Eq. (8) results in a banded matrix of the
form

$$\begin{array}{ll}\text{(10)}& {\displaystyle}\mathit{\alpha}\mathrm{\Delta}{\mathit{\theta}}_{i-\mathrm{1},j,k}^{t+\mathrm{1}}& {\displaystyle}+\mathit{\beta}\mathrm{\Delta}{\mathit{\theta}}_{i,j-\mathrm{1},k}^{t+\mathrm{1}}+\mathit{\gamma}\mathrm{\Delta}{\mathit{\theta}}_{i,j,k-\mathrm{1}}^{t+\mathrm{1}}+\mathit{\eta}\mathrm{\Delta}{\mathit{\theta}}_{i+\mathrm{1},j,k}^{t+\mathrm{1}}{\displaystyle}& {\displaystyle}+\mathit{\mu}\mathrm{\Delta}{\mathit{\theta}}_{i,j+\mathrm{1},k}^{t+\mathrm{1}}+\mathit{\varphi}\mathrm{\Delta}{\mathit{\theta}}_{i,j,k+\mathrm{1}}^{t+\mathrm{1}}+\mathit{\zeta}\mathrm{\Delta}{\mathit{\theta}}_{i,j,k}^{t+\mathrm{1}}=\mathit{\phi},\end{array}$$

where *α*, *β*, and *γ* are subdiagonal entries; *η*,
*μ*, and *ϕ* are superdiagonal entries; *ζ* is a diagonal entry
of the banded matrix and is given by

$$\begin{array}{ll}\text{(11)}& {\displaystyle}& {\displaystyle}\mathit{\alpha}={\displaystyle \frac{\partial {q}_{{x}_{i-\mathrm{1}/\mathrm{2},j,k}}^{t}}{\partial {\mathit{\theta}}_{i-\mathrm{1},j,k}}}\mathrm{\Delta}y\mathrm{\Delta}z\text{(12)}& {\displaystyle}& {\displaystyle}\mathit{\beta}={\displaystyle \frac{\partial {q}_{{y}_{i,j-\mathrm{1}/\mathrm{2},k}}^{t}}{\partial {\mathit{\theta}}_{i,j-\mathrm{1},k}}}\mathrm{\Delta}x\mathrm{\Delta}z\text{(13)}& {\displaystyle}& {\displaystyle}\mathit{\gamma}={\displaystyle \frac{\partial {q}_{{z}_{i,j,k-\mathrm{1}/\mathrm{2}}}^{t}}{\partial {\mathit{\theta}}_{i,j,k-\mathrm{1}}}}\mathrm{\Delta}x\mathrm{\Delta}y\text{(14)}& {\displaystyle}& {\displaystyle}\mathit{\eta}={\displaystyle \frac{\partial {q}_{{x}_{i+\mathrm{1}/\mathrm{2},j,k}}^{t}}{\partial {\mathit{\theta}}_{i+\mathrm{1},j,k}}}\mathrm{\Delta}y\mathrm{\Delta}z\text{(15)}& {\displaystyle}& {\displaystyle}\mathit{\mu}={\displaystyle \frac{\partial {q}_{{y}_{i,j+\mathrm{1}/\mathrm{2},k}}^{t}}{\partial {\mathit{\theta}}_{i,j+\mathrm{1},k}}}\mathrm{\Delta}x\mathrm{\Delta}z\text{(16)}& {\displaystyle}& {\displaystyle}\mathit{\varphi}={\displaystyle \frac{\partial {q}_{{z}_{i,j,k+\mathrm{1}/\mathrm{2}}}^{t}}{\partial {\mathit{\theta}}_{i,j,k+\mathrm{1}}}}\mathrm{\Delta}x\mathrm{\Delta}y\text{(17)}& {\displaystyle}& {\displaystyle}\mathit{\zeta}=\left({\displaystyle \frac{\partial {q}_{{x}_{i-\mathrm{1}/\mathrm{2},j,k}}^{t}}{\partial {\mathit{\theta}}_{i,j,k}}}-{\displaystyle \frac{\partial {q}_{{x}_{i+\mathrm{1}/\mathrm{2},j,k}^{t}}}{\partial {\mathit{\theta}}_{i,j,k}}}\right)\mathrm{\Delta}y\mathrm{\Delta}z{\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+\left({\displaystyle \frac{\partial {q}_{{y}_{i,j-\mathrm{1}/\mathrm{2},k}}^{t}}{\partial {\mathit{\theta}}_{i,j,k}}}-{\displaystyle \frac{\partial {q}_{{y}_{i,j+\mathrm{1}/\mathrm{2},k}^{t}}}{\partial {\mathit{\theta}}_{i,j,k}}}\right)\mathrm{\Delta}x\mathrm{\Delta}z\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+\left({\displaystyle \frac{\partial {q}_{{z}_{i,j,k-\mathrm{1}/\mathrm{2}}}^{t}}{\partial {\mathit{\theta}}_{i,j,k}}}-{\displaystyle \frac{\partial {q}_{{z}_{i,j,k+\mathrm{1}/\mathrm{2}}}^{t}}{\partial {\mathit{\theta}}_{i,j,k}}}\right)\mathrm{\Delta}x\mathrm{\Delta}y-{\displaystyle \frac{\mathrm{\Delta}x\mathrm{\Delta}x\mathrm{\Delta}z}{\mathrm{\Delta}t}}.\end{array}$$

The column vector *φ* is given by

$$\begin{array}{ll}\text{(18)}& {\displaystyle}\mathit{\phi}=& {\displaystyle}-\left({q}_{{x}_{i-\frac{\mathrm{1}}{\mathrm{2}},j,k}}^{t}-{q}_{{x}_{i+\frac{\mathrm{1}}{\mathrm{2}},j,k}}^{t}\right)\mathrm{\Delta}y\mathrm{\Delta}z{\displaystyle}& {\displaystyle}-\left({q}_{{y}_{i,j-\frac{\mathrm{1}}{\mathrm{2}},k}}^{t}-{q}_{{y}_{i,j+\frac{\mathrm{1}}{\mathrm{2}},k}^{t}}\right)\mathrm{\Delta}x\mathrm{\Delta}z\\ {\displaystyle}& {\displaystyle}-\left({q}_{{z}_{i,j,k-\frac{\mathrm{1}}{\mathrm{2}}}}^{t}-{q}_{{z}_{i,j,k+\frac{\mathrm{1}}{\mathrm{2}}}}^{t}\right)\mathrm{\Delta}x\mathrm{\Delta}y+{Q}_{i,j,k}^{t+\mathrm{1}}\mathrm{\Delta}x\mathrm{\Delta}x\mathrm{\Delta}z.\end{array}$$

The coefficients of Eq. (10) described in Eqs. (11)–(18) are for an internal grid cell with six neighbors. The coefficients for the top and bottom grid cells are modified for infiltration and interaction with the unconfined aquifer in the same manner as Oleson et al. (2013). Similarly, the coefficients for the grid cells on the lateral boundary are modified for a no-flux boundary condition. See Oleson et al. (2013) for details about the computation of hydraulic properties and derivative of the Darcy flux with respect to soil liquid water content.

ELMv0 solves a tightly coupled system of equations for soil, snow, and standing water temperature (Oleson et al., 2013). The model solves the transient conservation of energy:

$$\begin{array}{}\text{(19)}& c{\displaystyle \frac{\partial T}{\partial t}}=-\mathrm{\nabla}\cdot \mathbf{F},\end{array}$$

where *c* is the volumetric heat capacity (J m^{−3} K^{−1}),
**F** is the heat flux (W m^{−2}), and *t* is time (s). The heat
conduction flux is given by

$$\begin{array}{}\text{(20)}& \mathbf{F}=-\mathit{\lambda}\mathrm{\nabla}T,\end{array}$$

where *λ* is thermal conductivity (W m^{−1} K^{−1}) and
*T* is temperature (K). Applying a finite volume integral to
Eq. (20) and divergence theorem yields

$$\begin{array}{}\text{(21)}& c{\displaystyle \frac{\partial}{\partial t}}\underset{{\mathrm{\Omega}}_{n}}{\int}T=-\underset{{\mathrm{\Gamma}}_{\mathrm{n}}}{\int}\mathit{F}\cdot \mathrm{d}\mathit{A}.\end{array}$$

The spatially discretized equation for a *n*th grid cell that has *V*_{n}
volume and *n*^{′} neighbors is given by

$$\begin{array}{}\text{(22)}& {c}_{n}{\displaystyle \frac{\mathrm{d}{T}_{n}}{\mathrm{d}t}}{V}_{n}=-\sum _{{n}^{\prime}}\left({\mathit{F}}_{n{n}^{\prime}}\cdot {\mathit{A}}_{n{n}^{\prime}}\right).\end{array}$$

Similar to the approach taken in Sect. 2.3.1,
ELM-3D assumes a 3-D Cartesian grid, with each grid cell having a thickness
of Δ*x*, Δ*y*, and Δ*z* in the *x*, *y*, and *z*
directions, respectively. Temporal integration of Eq. (22) is carried
out using the Crank–Nicholson method that uses a linear combination of
fluxes evaluated at time *t* and *t*+1:

$$\begin{array}{ll}\text{(23)}& {\displaystyle}& {\displaystyle}{c}_{{n}_{i,j,k}}{\displaystyle \frac{\left({T}_{i,j,k}^{t+\mathrm{1}}-{T}_{i,j,k}^{t}\right)}{\mathrm{\Delta}t}}\mathrm{\Delta}x\mathrm{\Delta}y\mathrm{\Delta}z{\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}=\mathit{\omega}\left\{\left({F}_{{x}_{i-\frac{\mathrm{1}}{\mathrm{2}},j,k}}^{t}-{F}_{{x}_{i+\frac{\mathrm{1}}{\mathrm{2}},j,k}}^{t}\right)\mathrm{\Delta}y\mathrm{\Delta}z\right.\\ {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}+\left({F}_{{y}_{i,j-\frac{\mathrm{1}}{\mathrm{2}},k}}^{t}-{F}_{{y}_{i,j+\frac{\mathrm{1}}{\mathrm{2}},k}}^{t}\right)\mathrm{\Delta}x\mathrm{\Delta}z\\ {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\left.+\left({F}_{{z}_{i,j,k-\frac{\mathrm{1}}{\mathrm{2}}}}^{t}-{F}_{{z}_{i,j,k+\frac{\mathrm{1}}{\mathrm{2}}}}^{t}\right)\mathrm{\Delta}x\mathrm{\Delta}y\right\}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}+\left(\mathrm{1}-\mathit{\omega}\right)\left\{\left({F}_{{x}_{i-\frac{\mathrm{1}}{\mathrm{2}},j,k}}^{t+\mathrm{1}}-{F}_{{x}_{i+\frac{\mathrm{1}}{\mathrm{2}},j,k}}^{t+\mathrm{1}}\right)\mathrm{\Delta}y\mathrm{\Delta}z\right.\\ {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}+\left({F}_{{y}_{i,j-\frac{\mathrm{1}}{\mathrm{2}},k}}^{t+\mathrm{1}}-{F}_{{y}_{i,j+\frac{\mathrm{1}}{\mathrm{2}},k}}^{t+\mathrm{1}}\right)\mathrm{\Delta}x\mathrm{\Delta}z\\ {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}\left.+\left({F}_{{z}_{i,j,k-\frac{\mathrm{1}}{\mathrm{2}}}}^{t+\mathrm{1}}-{F}_{{z}_{i,j,k+\frac{\mathrm{1}}{\mathrm{2}}}}^{t}+\mathrm{1}\right)\mathrm{\Delta}x\mathrm{\Delta}y\right\},\end{array}$$

where *ω* is the weight in the Crank–Nicholson method and is set to 0.5
in this study. Substituting a discretized form of heat flux using Eq. (20)
in Eq. (23) results in a banded matrix of the form

$$\begin{array}{ll}\text{(24)}& {\displaystyle}\mathit{\alpha}{T}_{i-\mathrm{1},j,k}^{t+\mathrm{1}}& {\displaystyle}+\mathit{\beta}{T}_{i,j-\mathrm{1},k}^{t+\mathrm{1}}+\mathit{\gamma}{T}_{i,j,k-\mathrm{1}}^{t+\mathrm{1}}+\mathit{\eta}{T}_{i+\mathrm{1},j,k}^{t+\mathrm{1}}{\displaystyle}& {\displaystyle}+\mathit{\mu}{T}_{i,j+\mathrm{1},k}^{t+\mathrm{1}}++\mathit{\varphi}{T}_{i,j,k+\mathrm{1}}^{t+\mathrm{1}}+\mathit{\zeta}\mathrm{\Delta}{T}_{i,j,k}^{t+\mathrm{1}}=\mathit{\phi},\end{array}$$

where *α*, *β*, and *γ* are subdiagonal entries; *η*,
*μ*, and *ϕ* are superdiagonal entries; *ζ* is a diagonal entry of
the banded matrix and is given by

$$\begin{array}{ll}\text{(25)}& {\displaystyle}& {\displaystyle}\mathit{\alpha}=\left({\displaystyle \frac{-\left(\mathrm{1}-\mathit{\omega}\right)\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}x}}\right)\left({\displaystyle \frac{{\mathit{\lambda}}_{i-\mathrm{1}/\mathrm{2},j,k}}{{x}_{i,j,k}-{x}_{i-\mathrm{1},j,k}}}\right)\text{(26)}& {\displaystyle}& {\displaystyle}\mathit{\beta}=\left({\displaystyle \frac{-\left(\mathrm{1}-\mathit{\omega}\right)\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}y}}\right)\left({\displaystyle \frac{{\mathit{\lambda}}_{i,j-\mathrm{1}/\mathrm{2},k}}{{y}_{i,j,k}-{y}_{i-\mathrm{1},j,k}}}\right)\text{(27)}& {\displaystyle}& {\displaystyle}\mathit{\gamma}=\left({\displaystyle \frac{-\left(\mathrm{1}-\mathit{\omega}\right)\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}z}}\right)\left({\displaystyle \frac{{\mathit{\lambda}}_{i,j,k-\mathrm{1}/\mathrm{2}}}{{z}_{i,j,k}-{z}_{i,j,k-\mathrm{1}}}}\right)\text{(28)}& {\displaystyle}& {\displaystyle}\mathit{\eta}=\left({\displaystyle \frac{-\left(\mathrm{1}-\mathit{\omega}\right)\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}x}}\right)\left({\displaystyle \frac{{\mathit{\lambda}}_{i+\mathrm{1}/\mathrm{2},j,k}}{{x}_{i+\mathrm{1},j,k}-{x}_{i,j,k}}}\right)\text{(29)}& {\displaystyle}& {\displaystyle}\mathit{\mu}=\left({\displaystyle \frac{-\left(\mathrm{1}-\mathit{\omega}\right)\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}y}}\right)\left({\displaystyle \frac{{\mathit{\lambda}}_{i-\mathrm{1}/\mathrm{2},j,k}}{{y}_{i+\mathrm{1},j,k}-{y}_{i,j,k}}}\right)\text{(30)}& {\displaystyle}& {\displaystyle}\mathit{\varphi}=\left({\displaystyle \frac{-\left(\mathrm{1}-\mathit{\omega}\right)\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}z}}\right)\left({\displaystyle \frac{{\mathit{\lambda}}_{i-\mathrm{1}/\mathrm{2},j,k}}{{z}_{i+\mathrm{1},j,k}-{z}_{i,j,k}}}\right)\text{(31)}& {\displaystyle}& {\displaystyle}\mathit{\zeta}=\mathrm{1}+\left({\displaystyle \frac{\left(\mathrm{1}-\mathit{\omega}\right)\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}x}}\right)\left[{\displaystyle \frac{{\mathit{\lambda}}_{i-\mathrm{1}/\mathrm{2},j,k}}{{x}_{i,j,k}-{x}_{i-\mathrm{1},j,k}}}\right.{\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\left.+{\displaystyle \frac{{\mathit{\lambda}}_{i+\mathrm{1}/\mathrm{2},j,k}}{{x}_{i+\mathrm{1},j,k}-{x}_{i,j,k}}}\right]+\left({\displaystyle \frac{\left(\mathrm{1}-\mathit{\omega}\right)\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}y}}\right)\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\left[{\displaystyle \frac{{\mathit{\lambda}}_{i,j-\mathrm{1}/\mathrm{2},k}}{{y}_{i,j,k}-{y}_{i-\mathrm{1},j,k}}}+{\displaystyle \frac{{\mathit{\lambda}}_{i-\mathrm{1}/\mathrm{2},j,k}}{{y}_{i+\mathrm{1},j,k}-{y}_{i,j,k}}}\right]\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}+\left({\displaystyle \frac{\left(\mathrm{1}-\mathit{\omega}\right)\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}z}}\right)\left[{\displaystyle \frac{{\mathit{\lambda}}_{i,j,k-\mathrm{1}/\mathrm{2}}}{{z}_{i,j,k}-{z}_{i,j,k-\mathrm{1}}}}+{\displaystyle \frac{{\mathit{\lambda}}_{i-\mathrm{1}/\mathrm{2},j,k}}{{z}_{i+\mathrm{1},j,k}-{z}_{i,j,k}}}\right].\end{array}$$

The column vector *φ* is given by

$$\begin{array}{ll}\text{(32)}& {\displaystyle}\mathit{\phi}& {\displaystyle}={T}_{i,j,k}^{t}+\left({\displaystyle \frac{\mathit{\omega}\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}x}}\right)\left({F}_{{x}_{i-\mathrm{1}/\mathrm{2},j,k}}^{t}-{F}_{{x}_{i+\mathrm{1}/\mathrm{2},j,k}}^{t}\right){\displaystyle}& {\displaystyle}+\left({\displaystyle \frac{\mathit{\omega}\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}y}}\right)\left({F}_{{y}_{i,j-\mathrm{1}/\mathrm{2},k}}^{t}-{F}_{{y}_{i,j+\mathrm{1}/\mathrm{2},k}}^{t}\right)+\left({\displaystyle \frac{\mathit{\omega}\mathrm{\Delta}t}{{c}_{{n}_{i,j,k}}\mathrm{\Delta}z}}\right)\\ {\displaystyle}& {\displaystyle}\left({F}_{{z}_{i,j,k-\mathrm{1}/\mathrm{2}}}^{t}-{F}_{{z}_{i,j,k+\mathrm{1}/\mathrm{2}}}^{t}\right).\end{array}$$

The coefficients of Eq. (24) described in Eqs. (25)–(32) are for an internal grid cell with six neighbors. The coefficients for the top grid cells are modified for the presence of snow and/or standing water. A no-flux boundary condition was applied on the bottom grid cells; thus no geothermal flux was accounted for in this study. The coefficients for the grid cells on the lateral boundary are modified for a no-flux boundary condition. ELM handles ice–liquid phase transitions by first predicting temperatures at the end of a time step and then updating temperatures after accounting for deficits or excesses of energy during melting or freezing. See Oleson et al. (2013) for details about the computation of thermal properties and phase transition.

ELMv0, which considers flow only in the vertical direction, solves a tridiagonal and banded tridiagonal system of equations for water and energy transport, respectively. In ELM-3D, accounting for lateral flow in the subsurface results in a sparse linear system, Eqs. (10) and (24), where the sparsity pattern of the linear system depends on grid cell connectivity. In this work, we use the PETSc (Portable, Extensible Toolkit for Scientific Computing) library (Balay et al., 2016) developed at the Argonne National Laboratory to solve the sparse linear systems. PETSc provides object-oriented data structures and solvers for scalable scientific computation on parallel supercomputers. A description of the numerical tests that were conducted to ensure that the lateral coupling of hydrologic and thermal processes was correctly implemented is presented in the Supplement (Figs. S1 and S2)

The snow model in ELM-3D is the same as that in the default ELMv0 and CLM4.5 (Anderson, 1976; Dai and Zeng, 1997; Jordan, 1991), except for the inclusion of snow redistribution (SR). The snow model allows for a dynamic snow depth and up to five snow layers, and explicitly solves the vertically resolved mass and energy budgets. Snow aging, compaction, and phase change are all represented in the snow model formulation. Additionally, the snow model accounts for the influence of aerosols (including black and organic carbon and mineral dust) on snow radiative transfer (Oleson et al., 2013). ELMv0 uses the methodology of Swenson and Lawrence (2012) to compute fractional snow cover area, which is appropriate for ESM-scale grid cells (∼ 100 km × 100 km). Since the grid cell resolution in this work is sub-meter, we modified the fractional cover to be either 1 (when snow was present) or 0 (when snow was absent).

Two main drivers of SR include topography and surface wind (Warscher et al., 2013); previous SR models include mechanistically based (Bartelt and Lehning, 2002; Liston and Elder, 2006) and empirically based (Frey and Holzmann, 2015; Helfricht et al., 2012) approaches. To mimic the effects of wind, we used a conceptual model to simulate SR over the fine-resolution topography of our site by instantaneously re-distributing the incoming snow flux such that lower elevation areas (polygon center) receive snow before higher elevation areas (polygon rims). This relatively simple and parsimonious approach is reasonable given the observed snow depth heterogeneity, as described below, and small spatial extent of our domain.

Hydrologic and thermal properties differ by depth and landscape type. We
used the horizontal distribution of organic matter content from
Wainwright et al. (2015) to infer soil hydrologic and thermal
properties following the default representations in ELM. Vegetation cover
was classified as arctic shrubs in polygon centers and arctic grasses in
polygon rims. The default representation of the plant wilting factor assigns
a value of zero for a given soil layer when its temperature falls below a
threshold (*T*_{threshold}) of −2 ^{∘}C. This default value leads to
overly large predicted latent and sensible heat fluxes during winter,
compared to nearby eddy covariance measurements. We modified *T*_{threshold}
to be 0 ^{∘}C in this study, resulting in improved predicted wintertime
latent heat fluxes compared to the default version of the model (Fig. S3).
Although biases compared to the observations remain, particularly for
sensible heat fluxes in the spring, the improvement is substantial and,
given the observational uncertainties, we believe sufficient to justify our
use of the model for investigations of the role of snow heterogeneity in
this polygonal tundra system.

Because of computational constraints, we investigated the role of snow redistribution and physics representation using a two-dimensional transect through site A (Fig. 1). The transect was 104 m long and 45 m deep and was discretized horizontally with a grid spacing of 0.25 m and an exponentially varying layer thickness in the vertical with 30 soil layers. The transect does not align with the sensor locations because our objective was not to validate the model for a few grid cells, but to focus on relative differences between predictions for rims and centers of a polygon field. No flow conditions for mass and energy were imposed on the east, west, and bottom boundaries of the domain. Temporal discretization of 30 min was used in the simulations. All simulations were performed in the satellite phenology (SP) mode; i.e., Leaf Area Index (LAI) was prescribed from MODIS observations.

Simulations were run for 10 years using long-term climate data gathered at the Barrow Alaska Observatory site (https://www.esrl.noaa.gov/gmd/obop/brw/) managed by the Global Monitoring Division of NOAA's Earth System Research Laboratory (Mefford et al., 1996). The missing precipitation time series was gap-filled using daily precipitation at the Barrow Regional Airport available from the Global Historical Climatology Network (http://www1.ncdc.noaa.gov/pub/data/ghcn/daily). We tested the model by comparing predictions to high-frequency observations of snow depth and vertically resolved soil temperature for September 2012–September 2013. Temperature observations were taken at discrete locations in a polygon center and rim (Fig. 1), and they were combined to analyze comparable landscape positions in the simulations (Fig. 2).

After testing, the model was used to investigate the effects of snow redistribution and 2-D subsurface hydrologic and thermal physics by analyzing three scenarios: (1) no snow redistribution and 1-D physics; (2) snow redistribution and 1-D physics; and (3) snow redistribution and 2-D physics. Between these scenarios, we compared vertically resolved soil temperature and liquid saturation, active layer depth, and mean and spatial variation of latent and sensible heat fluxes across the 10 years of simulations. For each soil column, the simulated soil temperature was interpolated vertically, and the active layer depth was estimated as the maximum depth that had above-freezing soil temperature.

3 Results and discussion

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In the absence of SR, predicted snow depth exactly follows the topography. With SR, a much smaller dependence of winter average snow depth on topography is predicted (Fig. 2). Further, for the winter average, there are very small differences in snow depth between simulations with SR and 1-D or 2-D subsurface physics representations. Compared to observations, considering SR led to (1) a factor of ∼ 2 improvement in snow depth bias for the polygon center; (2) modest increase and decrease in average bias on the rims for September through February and March through June, respectively; and (3) a dramatic improvement in bias of the difference in snow depth between the polygon centers and rims (Fig. 3). There was no discernible difference in snow depth bias between the 1-D and 2-D physics (Table 1), although the predicted subsurface temperature fields were different, as shown below.

The temporal variation of the mean snow depth (Fig. 4a) and its spatial standard deviation (Fig. 4b) also differed based on whether SR was considered, but it was not affected by considering 2-D thermal or hydrologic physics. With SR, the snow depth coefficient of variation (Fig. 4c) was about 0.5 from December through the beginning of the snowmelt period, indicating relatively large spatial heterogeneity. Simulated snow depths for the three simulation scenarios are included in the Supplement (4).

Broadly, ELM-3D accurately predicted the polygon center soil temperature at
depth intervals corresponding to the temperature probes (0–20, 20–50,
50–75, and 75–100 cm; Fig. 5a). Recall that the observed temperatures
for the polygon center and rims were taken at single points in site A
(Fig. 1), while the predicted temperatures were calculated as averages across the
transect for each of the two landscape position types. The model was able to
simulate early freeze-up of the soil column under the rims as compared to
centers in November 2012 because of differences in accumulated snowpack. The
transition to thawed soil in the 0–20 cm depth interval in early June 2013
and the subsequent temperature dynamics over the summer were very well
captured by ELM-3D. Minimum temperatures during the winter were also
accurately predicted, although the temperatures in the deepest layer (75–100 cm)
were overestimated by ∼ 3 ^{∘}C in March. For figure clarity
we did not indicate the standard deviation of the observations but provide
that information in the Supplement (Figs. S5–S8).

Similarly, the soil temperatures were accurately predicted in the polygon
rims (Fig. 5b). The largest discrepancies between measured and predicted
soil temperatures were in the shallowest layer (0–25 cm), where the
predictions were up to a few ^{∘}C cooler than some of the observations
between December 2012 and March 2013. In the polygon center, a thicker snowpack acts as a heat insulator and keeps soil temperature higher in winter as
compared to the polygon rims.

Three recent studies have used other mechanistic models to simulate soil temperature fields at this site and achieved comparably good comparisons with observations (Kumar et al., 2016 applied a 3-D version of PFLOTRAN; Atchley et al., 2015 and Harp et al., 2016 applied a 1-D version of the Arctic Terrestrial Simulator – ATS). However, those models used measured soil temperatures near the surface as the top boundary condition. In contrast, the top boundary condition in this work is the climate forcing (air temperature, wind, solar radiation, humidity, precipitation), and the ground heat flux is predicted based on ELM's vegetation and surface energy dynamics. We note that no parameter calibration was done in this work or that of Kumar et al. (2016), while the ATS parameterizations were calibrated to match the soil temperature profile.

Snow redistribution impacts spatial variability of soil temperature throughout the soil column. Absence of SR results in no significant spatial variability of soil temperature (Fig. 6a). Inclusion of SR on the surface modifies the amount of energy exchanged between the snow and the top soil layer, thereby creating spatial variability in the temperature of the top soil, which propagates down into the soil column (Fig. 6b). With SR, energy dissipation in the lateral direction reduces the penetration depth of the soil temperature spatial variance (compare Fig. 6c and b).

With 1-D physics, the average spatial and temporal difference of the active
layer depth (ALD) between simulations with and without SR was 1.7 cm (Fig. 7a),
and the absolute difference was 6.5 cm. As described above, we
diagnosed the ALD to be the maximum soil depth during the summer at which
vertically interpolated soil temperature is 0 ^{∘}C. On average, the
rims had ∼ 10 cm shallower ALD with (blue line) than without (green
line) SR, consistent with the loss of insulation from SR on the rims during
the winter. In the centers (e.g., at location 42–55 m), the thaw depth was
deeper by ∼ 5 cm with SR because of the higher snow depth there from
SR. The effect of SR on the ALD was largest on the rims because, compared to
centers, they (1) on average lost more snow with SR and (2) are more
thermally conductive. Since rims are therefore colder at the time of snowmelt
with SR, the ground heat flux during the subsequent summer was unable to thaw
the soil column as deeply as when SR is ignored. For comparison, Atchley et
al. (2015) found in their sensitivity analysis using the 1-D version of ATS
that SR resulted in deeper thaw depths in both polygon centers (by
∼ 3 cm) and rims (∼ 0.3 cm). Thus, their results for polygon
centers are consistent in sign but lower in magnitude than ours, but opposite
in sign for the rims.

Across 10 years of simulation, the interannual variability (IAV) in ALD varied substantially between the three scenarios (Fig. 7b). As expected, for the 1-D physics without SR scenario (green line), the IAV in ALD was determined by a landscape position because of differences in soil and vegetation parameters. With SR and 1-D physics, the model shows largest differences over the rims, again highlighting the relatively larger effects of SR on the rim soil temperatures.

The effect of 1-D versus 2-D physics on the ALD across the transect was modest (mean absolute difference ∼ 3 cm). Generally, because 2-D physics allows for lateral energy diffusion, the horizontal variation of ALD was slightly lower (i.e., the red line is smoother than the blue line; Fig. 7a) than with 1-D physics. This difference was also reflected in the thaw depth IAV across the transect, where 2-D physics led to a smoother lateral profile of interannual variability than with 1-D physics.

The impact of physics formulation (i.e., 1-D or 2-D) alone was investigated by
analyzing differences between soil temperature profiles over time for polygon
rims and centers in simulations with snow redistribution. Inclusion of 2-D
subsurface physics resulted in soil temperatures with depth and time that
were lower in the polygon rims (Fig. 8a) and higher in polygon centers
(Fig. 8b). Using the simulations from the scenario with SR and 2-D physics,
we evaluated the extent to which soils under rims and centers can be
separately considered as relatively homogeneous single column systems by
evaluating the soil temperature standard deviation as a function of depth and
time (Fig. 9). During winter, both polygon rims and centers were predicted
to have soil temperature spatial variability > 1 ^{∘}C up to
a depth of ∼ 2 m. The soil temperature spatial variability in winter
due to snow redistribution was dissipated over the summer. During the summer,
polygon centers were relatively more homogeneous vertically compared to
polygon rims.

Predicted monthly mean and spatial mean (*μ*) surface latent heat fluxes
across the transect were very similar between the three scenarios (Fig. 10a),
with a growing seasonal mean difference of < 1.0 W m^{−2}.
However, the spatial variability (SV =*σ*; Fig. 10b) and
coefficient of variation (CV $=\mathit{\sigma}/\mathit{\mu}$; Fig. 10c) of latent
heat fluxes were different between the scenarios with SR (1-D and 2-D physics)
and without SR. With SR, the latent heat flux spatial standard deviation
peaked after snowmelt and declined until the fall when snow began, from about
∼ 100 to 10 % of the mean. This relatively larger spatial
variation in latent heat flux occurred because of large spatial heterogeneity
in near-surface soil moisture in the beginning of summer, indicating a
residual effect of SR from the previous winter.

The predicted temporal monthly mean and spatial mean surface sensible heat
fluxes across the transect were also similar between the three scenarios
(Fig. 11a), with a growing season mean absolute difference of < 3.5 W m^{−2}.
Also, the sensible heat flux spatial variability differences
occurred earlier than snowmelt, in contrast to the latent heat flux. Both the
standard deviation and CV of the sensible heat fluxes were larger than those
of the latent heat fluxes, with early season standard deviations of
∼ 50 W m^{−2} (Fig. 11b) and CVs of ∼ 1.5 (Fig. 11c). As
for the latent heat fluxes, the differences in standard deviation and CV of
sensible heat fluxes were small between the 1-D and 2-D scenarios with SR,
arguing that the subsurface lateral energy exchanges associated with the 2-D
physics did not propagate to the mean surface heat fluxes. However, as for
the latent heat flux, there was a relatively large difference in spatial
variation between the scenarios with and without SR (e.g., of about 25 W m^{−2}
in May; Fig. 10b).

Neither SR nor 2-D lateral physics affected the spatial mean moisture across time (not shown). However, spatial heterogeneity of predicted soil moisture content differed substantially between scenarios during the snow-free period (Fig. 12). For the 1-D simulations, the effect of SR was to increase growing season soil moisture spatial heterogeneity by factors of 5.2 and 1.6 for 0–10 and 10–65 cm depth intervals, respectively (compare Fig. 12a and b). Compared to 1-D physics, simulating 2-D thermal and hydrologic physics led to an overall reduction in soil moisture spatial heterogeneity by factors of 0.8 and 0.7 for 0–10 and 10–65 cm depth intervals, respectively (compare Fig. 12b and c). Thus, with respect to dynamic spatial mean soil moisture, SR effects dominated those associated with lateral subsurface water movement.

The good agreement between ELM-3D predictions and soil temperature observations demonstrates the model's capabilities to represent this very spatially heterogeneous and complex system. However, several caveats to our conclusions remain due to uncertainties in model parameterizations, model structure, and climate forcing data.

ELMv0, a one-dimensional model, is embarrassingly parallel with no cross-processor communication. The current implementation of the three-dimensional solver in ELM-3D only supports serial computing. Support of parallel computing will be included in a future version of the model. Because of computational constraints, we applied a 2-D transect domain to the site, instead of a full 3-D domain. We are working to improve the computational efficiency of the model, which will facilitate a thorough analysis of the effects of 3-D subsurface energy and water fluxes. A related issue is our simplified treatment of surface water flows. A thorough analysis of the effects of surface water redistribution would require integration of a 2-D surface thermal flow model in a 3-D domain, which is another goal for our future work. However, we note that the good agreement using the 2-D model domain supports the idea that a two-dimensional simplification may be appropriate for this system. The expected geomorphological changes in these systems over the coming decades (e.g., Liljedahl et al., 2016), which will certainly affect soil temperature and moisture, are not currently represented in ELM, although incorporation of these processes is a long-term development goal.

The current representation of vegetation in ELM-3D for these polygonal tundra systems is oversimplified. For example, non-vascular plants (mosses and lichens) are not explicitly represented in the model, but can be responsible for a majority of evaporative losses (Miller et al., 1976) and are strongly influenced by near-surface hydrologic conditions (Williams and Flanagan, 1996). Our use of the satellite phenology mode, which imposes transient LAI profiles on each plant functional type in the domain, ignores the likely influence of nutrient constraints (Zhu et al., 2016) on photosynthesis and therefore the surface energy budget. Other model simplifications, e.g., the simplified treatment of radiation competition, may also be important, especially as simulations are extended over periods where vegetation change may occur (e.g., Grant et al., 2015).

Development of subgrid parameterizations to parsimoniously capture fine-scale processes will be pursued in the future. For example, a two-tile approach to represent hydrologic and thermal processes in coupled polygon rims and centers with snow redistribution should be evaluated. Inclusion of lateral subsurface processes has a greater impact on predicted subgrid variability than on spatially averaged states. Thus, one possible extension of the current model would be to explicitly include an equation for the temporal evolution of subgrid variability using the approach of Montaldo and Albertson (2003). The use of reduced-order models (e.g., Pau et al., 2014; Liu et al., 2017) is an alternate approach to estimate fine-scale hydrologic and thermal states from a coarse resolution representation. Additionally, lateral subsurface processes can be included in the land surface model via a range of numerical discretization approaches of varying complexity, e.g., adding lateral water and energy fluxes as source–sink terms in the existing 1-D model, implementing an operator split approach to solve vertical and lateral processes in a non-iterative approach, or solving a fully coupled 3-D model. Trade-offs between approaches that represent lateral processes and computational costs need to be carefully studied before developing quasi or fully three-dimensional land surface models. While the present study focused on application and validation of ELM-3D at fine-scale, future work will focus on regional-scale applications using comprehensive datasets and the Distributed Model Intercomparison Project Phase 2 modeling protocol (Smith et al., 2012). Although we found no significant effect of topography and SR on the 100 m × 100 m grid-averaged exchanges with the atmosphere, future work needs to analyze intermediate-scale (e.g., 100 m–10 km) topographical variation and the potential effects on biogeochemical and plant processes and surface exchanges.

4 Summary and Conclusions

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In a polygonal tundra landscape, we analyzed effects of microtopographical surface heterogeneity and lateral subsurface transport on soil temperature, soil moisture, and surface energy exchanges. Starting from the climate-scale land model ELMv0, we incorporated in ELM-3D numerical representations of subsurface water and energy lateral transport that are solved using PETSc. A simple method for redistributing incoming snow along the microtopographic transect was also integrated in the model.

Over the observational record, ELM-3D with snow redistribution and lateral
heat and hydrological fluxes accurately predicted snow depth and soil
temperature vertical profiles in the polygon rims and centers (overall bias,
RMSE, and *R*^{2} of 0.59 ^{∘}C, 1.82 ^{∘}C, and 0.99,
respectively). In the rims, the transition to thawed soil in spring, summer
temperature dynamics, and minimum temperatures during the winter were all
accurately predicted. In the centers, a ∼ 2 ^{∘}C warm bias in
April in the 75–100 cm soil layer was predicted, although this bias
disappeared during snowmelt.

The spatial heterogeneity of snow depth during the winter due to snow
redistribution generated surface soil temperature heterogeneity that
propagated into the soil over time. The temporal and spatial variation of
snow depth was affected by snow redistribution, but not by lateral thermal
and hydrologic transport. Both snow redistribution and lateral thermal fluxes
affected spatial variability of soil temperatures. Energy dissipation in the
lateral direction reduced the depth to which soil temperature variance
penetrated. Snow redistribution led to ∼ 10 cm shallower active layer
depths under the polygon rims because of the residual effect of reduced
insulation during the winter. In contrast, snow redistribution led to
∼ 5 cm deeper maximum thaw depth under the polygon centers. The effect
of lateral energy fluxes on active layer depths was ∼ 3 cm. Compared
to 1-D physics, the 2-D subsurface physics led to lower (higher) soil
temperatures with depth and time in the polygon rims (centers). The larger
than 1 ^{∘}C wintertime spatial temperature variability down to
∼ 2 m depth in rims and centers indicates the uncertainty associated
with considering rims and centers as separate 1-D columns. During the summer,
polygon center temperatures were relatively more vertically homogeneous than
temperatures in the rims.

The monthly mean and spatial mean predicted latent and sensible heat fluxes were unaffected by snow redistribution and lateral heat and hydrological fluxes. However, snow redistribution led to spatial heterogeneity in surface energy fluxes and soil moisture during the summer. Excluding lateral subsurface hydrologic and thermal processes led to an overprediction of spatial variability in soil moisture and soil temperature because subsurface gradients were artificially prevented from laterally dissipating over time. Snow redistribution effects on soil moisture heterogeneity were larger than those associated with lateral thermal fluxes.

Overall, our analysis demonstrates the potential and value of explicitly representing snow redistribution and lateral subsurface hydrologic and thermal dynamics in polygonal ground systems and quantifies the effects of these processes on the resulting system states and surface energy exchanges with the atmosphere. The integration of a 3-D subsurface model in the E3SM Land Model also allows for a wide range of analyses heretofore impossible in an Earth System Model context.

Code availability

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Code availability.

The ELM-3D v1.0 code and data used in study are publicly available at https://bitbucket.org/gbisht/lateral-subsurface-model (commit hash 2a31e42d911c1d7539230848d3479533d7605912) and https://bitbucket.org/gbisht/notes-for-gmd-2017-71.

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/gmd-11-61-2018-supplement.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This research was supported by the Director, Office of Science, Office of
Biological and Environmental Research of the US Department of Energy under
contract no. DE-AC02-05CH11231 as part of the NGEE-Arctic and Energy
Exascale Earth System Model (E3SM) programs.

Edited by: Philippe Peylin

Reviewed by: three anonymous referees

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Short summary

The land model integrated into the Energy Exascale Earth System Model was extended to include snow redistribution (SR) and lateral subsurface hydrologic and thermal processes. Simulation results at a polygonal tundra site near Barrow, Alaska, showed that inclusion of SR resulted in a better agreement with observations. Excluding lateral subsurface processes had a small impact on mean states but caused a large overestimation of spatial variability in soil moisture and temperature.

The land model integrated into the Energy Exascale Earth System Model was extended to include...

Geoscientific Model Development

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