Editorial Type:
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Article Type:
Research Article

The Numerical Implementation of Land Models: Problem Formulation and Laugh Tests

Martyn P. Clark aCentre for Hydrology, University of Saskatchewan, Canmore, Alberta, Canada
bDepartment of Geography and Planning, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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https://orcid.org/0000-0002-2186-2625
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Reza Zolfaghari cDepartment of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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Kevin R. Green cDepartment of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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Sean Trim cDepartment of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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Wouter J. M. Knoben aCentre for Hydrology, University of Saskatchewan, Canmore, Alberta, Canada

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Andrew Bennett dDepartment of Civil and Environmental Engineering, University of Washington, Seattle, Washington

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Bart Nijssen dDepartment of Civil and Environmental Engineering, University of Washington, Seattle, Washington

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Andrew Ireson eGlobal Institute for Water Security, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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Raymond J. Spiteri cDepartment of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

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Online Publication:
07 Jun 2021
Print Publication:
01 Jun 2021
DOI:
https://doi.org/10.1175/JHM-D-20-0175.1
Page(s):
1627–1648
Open access
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Abstract

The intent of this paper is to encourage improved numerical implementation of land models. Our contributions in this paper are twofold. First, we present a unified framework to formulate and implement land model equations. We separate the representation of physical processes from their numerical solution, enabling the use of established robust numerical methods to solve the model equations. Second, we introduce a set of synthetic test cases (the laugh tests) to evaluate the numerical implementation of land models. The test cases include storage and transmission of water in soils, lateral subsurface flow, coupled hydrological and thermodynamic processes in snow, and cryosuction processes in soil. We consider synthetic test cases as “laugh tests” for land models because they provide the most rudimentary test of model capabilities. The laugh tests presented in this paper are all solved with the Structure for Unifying Multiple Modeling Alternatives (SUMMA) model implemented using the Suite of Nonlinear and Differential/Algebraic Equation Solvers (SUNDIALS). The numerical simulations from SUMMA/SUNDIALS are compared against 1) solutions to the synthetic test cases from other models documented in the peer-reviewed literature, 2) analytical solutions, and 3) observations made in laboratory experiments. In all cases, the numerical simulations are similar to the benchmarks, building confidence in the numerical model implementation. We posit that some land models may have difficulty in solving these benchmark problems. Dedicating more effort to solving synthetic test cases is critical in order to build confidence in the numerical implementation of land models.

Denotes content that is immediately available upon publication as open access.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Martyn P. Clark, martyn.clark@usask.ca

Abstract

The intent of this paper is to encourage improved numerical implementation of land models. Our contributions in this paper are twofold. First, we present a unified framework to formulate and implement land model equations. We separate the representation of physical processes from their numerical solution, enabling the use of established robust numerical methods to solve the model equations. Second, we introduce a set of synthetic test cases (the laugh tests) to evaluate the numerical implementation of land models. The test cases include storage and transmission of water in soils, lateral subsurface flow, coupled hydrological and thermodynamic processes in snow, and cryosuction processes in soil. We consider synthetic test cases as “laugh tests” for land models because they provide the most rudimentary test of model capabilities. The laugh tests presented in this paper are all solved with the Structure for Unifying Multiple Modeling Alternatives (SUMMA) model implemented using the Suite of Nonlinear and Differential/Algebraic Equation Solvers (SUNDIALS). The numerical simulations from SUMMA/SUNDIALS are compared against 1) solutions to the synthetic test cases from other models documented in the peer-reviewed literature, 2) analytical solutions, and 3) observations made in laboratory experiments. In all cases, the numerical simulations are similar to the benchmarks, building confidence in the numerical model implementation. We posit that some land models may have difficulty in solving these benchmark problems. Dedicating more effort to solving synthetic test cases is critical in order to build confidence in the numerical implementation of land models.

Denotes content that is immediately available upon publication as open access.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Martyn P. Clark, martyn.clark@usask.ca
Keywords: Diagnostics; Hydrologic models; Land surface model

1. Introduction

Land models encompass a broad class of models that are used to simulate biogeophysical processes (and sometimes biogeochemical processes) in terrestrial systems (e.g., Wigmosta et al. 1994; Baldocchi and Wilson 2001; Tague and Band 2004; Lawrence et al. 2019). Land models have varying degrees of spatial complexity and process complexity, with much focus on methods to represent spatial heterogeneity (e.g., Peters-Lidard et al. 2017; Fisher and Koven 2020) and parameterizations to represent the dominant biogeophysical and biogeochemical processes (e.g., Walker et al. 2015; Fatichi et al. 2016). Land models are used for a myriad of applications, including applications to simulate the terrestrial component of the Earth system (e.g., Lawrence et al. 2019), to provide a lower boundary condition for the atmosphere in numerical weather prediction models (e.g., Balsamo et al. 2011), to assess risks to water security (e.g., Wada et al. 2014), and to predict hydrological extremes (e.g., Wanders et al. 2019).

In the last decade, we have arguably seen a convergence across different land modeling communities. Land modeling is inherently an interdisciplinary endeavor, requiring expertise from hydrologists, biogeochemists, ecologists, geologists, atmospheric scientists, applied mathematicians, and many other disciplines (e.g., Fisher and Koven 2020). Although different land models have evolved in different ways, land models developed by different communities now include a similar set of processes. Such convergence is creating new opportunities to collaborate across disciplines, to learn from each other, and to share insights across different modeling communities (e.g., Coon et al. 2016; Bisht and Riley 2019).

A key challenge is that the decades-long development of land models means that they have become rather unwieldy. Many land models have evolved to include a menagerie of physical processes that are cobbled together in haphazard ways: New representations of individual processes are typically “bolted on” to previous model versions, with new parameterizations of physical processes (e.g., flux parameterizations) closely intertwined with their numerical solutions (e.g., Clark et al. 2015b; Fisher and Koven 2020). This causes two key problems. First, it can be difficult to add new modeling capabilities when the existing model structure is disorganized. For example, as pointed out by Bisht and Riley (2019), the monolithic structure of land models makes it cumbersome to systematically isolate and evaluate individual process parameterizations (see also Clark et al. 2011, 2015b). The second problem with land models is that their numerical implementation often relies on an ad hoc collection of simplistic solution methods that are tied to individual process representations (e.g., Bisht and Riley 2019). The underlying system of equations is rarely defined explicitly, and the details of the numerical implementation are often obscure [e.g., see the critique in Clark et al. (2015b)]. The failure to cleanly separate the model equations from their numerical solution makes it difficult for land modelers to benefit from more advanced numerical methods, such as the development of robust nonlinear equation solvers, adaptive time-stepping and temporal error control, and proper handling of discontinuities.

The intent of this paper is to encourage improved numerical implementation of land models. In our opinion, the implementation details of land models do not get the attention they deserve in the peer-reviewed literature. Such a lack of focus on model implementation can affect the credibility of land models. After all, the governing equations are the fundamental building blocks of land models—if the equations or the methods for their solution are not implemented correctly, then all predictions from such land models are suspect.

Our contributions in this paper are twofold. 1) We present a unified framework to formulate and implement land model equations [extending the work of Clark et al. (2015b,c)]. We separate the representation of physical processes from their numerical solution, enabling the use of established robust numerical methods to solve the model equations. 2) We introduce a set of synthetic test cases—the laugh tests—to evaluate the numerical implementation of land models. We make the inputs and outputs from the test cases publicly available so that other modeling groups can use the test cases in their own model development efforts. By presenting the equations, the numerical methods, and tractable test cases, our broader goal is to provide information in a form that is useful for both land modelers and applied mathematicians to improve the numerical implementation of land models.

The remainder of this paper is organized as follows. Section 2 specifies the general equations solved in land models, the spatial approximations of the model equations, and the solution of the resulting ordinary differential equations. Section 3 describes the synthetic test cases and their solutions, and section 4 describes checks for the conservation of energy and mass. Section 5 describes the availability of the test case solutions and the code repository used to reproduce them. Section 6 concludes the paper with a summary and discussion.

2. Land model equations: Description and numerical implementation

In this section, we present a unified framework to formulate and implement land models. Our framework is limited to the terrestrial water and energy fluxes shown in Fig. 1 (see also Clark et al. 2015b). We do not include terrain effects on the radiation balance, vapor transport within snow and soil, or horizontal transport of snow associated with avalanching or drifting (e.g., Marsh et al. 2018), and we do not explicitly represent glaciers, depression storage, wetlands and lakes, major aquifers, losses from the stream to the aquifer, and human impacts on the terrestrial water cycle such as reservoir operations and irrigation (e.g., Bowling and Lettenmaier 2010; Fan and Miguez-Macho 2011; Miguez-Macho and Fan 2012; Li et al. 2013; Yigzaw et al. 2019).

Fig. 1.
Fig. 1.

Conceptual depiction of the dominant thermodynamical and hydrological processes solved in land models. The red and blue font define thermodynamic and hydrological processes respectively; the flux qiceΩ in Eq. (2) is only defined in the vegetation and snow domains, and includes snowfall qsf, throughfall of snow through the vegetation canopy qtf,snowveg, and unloading of snow from the vegetation canopy qunloadveg. The model forcing variables are snowfall qsf (kg m−2 s−1), rainfall qrf (kg m−2 s−1), downwelling shortwave radiation Qswdatm (J m−2 s−1), downwelling longwave radiation Qlwdatm (J m−2 s−1), temperature, Tatm (K), wind speed uatm (m s−1), specific humidity hatm (kg kg−1), and atmospheric pressure patm (Pa). All other variables are defined in the text.

Citation: Journal of Hydrometeorology 22, 6; 10.1175/JHM-D-20-0175.1

From a land modeling perspective, our representation of vegetation is rather limited compared to modern land models (e.g., Lawrence et al. 2019). In particular, we do not include the storage and fluxes of carbon, nitrogen, and phosphorous (e.g., Zaehle and Friend 2010; Melton et al. 2020), and we do not explicitly simulate the pressure gradients throughout the soil–water–atmosphere continuum (e.g., Bonan et al. 2014; Mackay et al. 2015) that are necessary to simulate the cavitation of the xylem, delayed xylem infilling, and hence the memory in hydraulic dysfunction.

It is straightforward to extend these model equations to include these additional physical processes both through expanding model couplings (e.g., coupling with a groundwater model) or through expanding the current set of state equations. The limited implementation presented here is intended to provide an example of an extensible framework that can be adopted for multiple physical processes and process couplings.

a. State equations

Most land models solve a variant of the coupled conservation equations for thermodynamics and hydrology for the subdomains of the canopy air space (cas), vegetation (veg), snow, and soil (Fig. 1). The state equations for thermodynamics and hydrology can be written for a given model subdomain Ω as
∂HΩ∂t=−∂FΩ∂z+FsinkΩ, Ω=cas,veg,snow,soil,
∂ΘmΩ∂t=−∂qiceΩ∂z−∂qliqΩ∂z+MsinkΩ, Ω=veg,snow,soil,
where HΩ (J m−3) is the mixture enthalpy, ΘmΩ (—) is the volumetric fraction of total water, z (m) is the vertical coordinate (positive downward), and t (s) is time. The remaining terms on the RHS of Eqs. (1) and (2) are defined below.
In the development that follows, we link the temperature and enthalpy form of the heat equation in order to define conservative equations for thermodynamics. In land models the mixture enthalpy HΩ is equal to the internal energy of the system (Tubini et al. 2020). The mixture enthalpy HΩ is defined as (e.g., Voller et al. 1990; Nedjar 2002; Dall’Amico et al. 2011; Tubini et al. 2020)
HΩ=∑iHT, iΩ−LfusρiceΔθiceΩ,
where HT, iΩ is the temperature component of the enthalpy of constituent i (see Table 1 for the constituents of the model subdomains) and LfusρiceΔθiceΩ is the energy associated with melt/freeze, with Lfus (J kg−1) the latent heat of fusion and ρice (kg m−3) the intrinsic density of ice. The terms on the RHS of Eq. (3) are
HT, iΩ=ciρi∫TrefTΩθiΩ dT,
ΔθiceΩ=θiceΩ⁡(TΩ)−θiceΩ⁡(Tref),
with ci (J kg−1 K−1), ρi (kg m−3), and θiΩ (—) the specific heat, the intrinsic density, and the volumetric fraction of constituent i, respectively, TΩ (K) the temperature, with
∑iθiΩ=1.
The mixture enthalpy HΩ in Eq. (3) is defined such that it is negative when the temperature is below the reference temperature and positive when the temperature is above the reference temperature (the reference temperature is often set to the freezing point). The mixture enthalpy HΩ can be defined as a diagnostic variable computed from the state variables TΩ and ΘmΩ.
Table 1.

Constituents for model subdomains.

Table 1.
The volumetric fraction of total water ΘmΩ (—) is defined as
ΘmΩ=θliqΩ+ρiceρliq θiceΩ.
We note that no water is stored in the canopy air space, and hence the subdomains for Eq. (2) are restricted to Ω = veg, snow, soil.

The terms on the RHS of Eqs. (1) and (2) are defined as follows. In Eq. (1), FΩ⁡(TΩ,ΘmΩ) (J m−2 s−1) is the vertical energy flux and FsinkΩ⁡(TΩ,ΘmΩ) (J m−3 s−1) is the source–sink flux term for energy; in Eq. (2), qiceΩ⁡(TΩ,ΘmΩ) (m s−1) and qliqΩ⁡(TΩ,ΘmΩ) (m s−1) are the vertical fluxes of solid and liquid water, and MsinkΩ⁡(TΩ,ΘmΩ) (s−1) is the fractional source–sink flux term for hydrology. The source–sink flux terms are meant to capture physical processes that do not correspond to vertical fluxes, e.g., the transpiration sink term in soil hydrology associated with the latent heat flux. The vertical fluxes of solid water in Eq. (2), qiceΩ, include both snowfall (above the vegetation canopy) and throughfall and unloading of snow (within the vegetation canopy).

Many models separate the temperature change and phase change terms in Eq. (1) (see, e.g., Koren et al. 1999; Hansson et al. 2004; Clark et al. 2015b,c; Bao et al. 2016; Wang et al. 2017; Clow 2018). Using Eq. (3) in Eq. (1) results in
∂HTΩ∂t−Lfusρice∂θiceΩ∂t=−∂FΩ∂z+FsinkΩ,
where
HTΩ=∑iHT, iΩ.
Equation (1) can then be written as
CpΩ ∂TΩ∂t−Lfusρice∂θiceΩ∂t=−∂FΩ∂z+FsinkΩ,
where Cp (J m−3) is the volumetric heat capacity
CpΩ=∑iciρiθiΩ,
obtained by differentiating Eq. (4), i.e., ∂HT, iΩ/∂TΩ=ciρiθiΩ, and accounting for the volumetric fraction of each of the i constituents. The energy fluxes on the RHS of Eq. (10) are used to either change the temperature of the system (first term on the LHS) or for phase transition (second term on the LHS).
We note that Eq. (1) can also be formulated in terms of apparent heat capacity (e.g., Harlan 1973; Hansson et al. 2004):
CaΩ∂TΩ∂t=−∂FΩ∂z+FsinkΩ,
where CaΩ (J m−3 K−1) is the apparent heat capacity
CaΩ=∂HΩ∂TΩ=CpΩ−Lfusρice∂θiceΩ∂TΩ.
In Eqs. (10)–(13), θiceΩ is a diagnostic variable computed from TΩ and ΘmΩ.
The conservation equation for hydrology defined in Eq. (2) is a general equation, and variants of Eq. (2) are used in most land models. There are, however, two important deviations from Eq. (2). First, the fluxes of water vapor in the canopy air space must be defined. The vapor fluxes can be defined as (Flerchinger et al. 1998; Bonan et al. 2018)
ρair∂ϑcas∂t=−∂∂z⁡(ρairke∂ϑcas∂z)+FlatvegLvap,
where ϑcas (kg kg−1) is the specific humidity in the canopy air space, Lvap (J kg−1) is the latent heat of vaporization, and ke (m2 s−1) is the transfer coefficient for heat and water vapor within the canopy air space. Second, Eq. (2) cannot be used to simulate saturated flow in soils because the spatial gradients in Θmsoil are zero under saturated conditions. To address this issue, the hydrology state equation for the soil subdomain is modified to use the pressure-based form of Richards’ equation (Miller et al. 1998) with matric head ψ (m) as the state variable:
SsΘmsoilθs∂ψ0∂t+dΘmsoildψ0∂ψ0∂t=−∂qliqsoil∂z+Msinksoil,
where ψ0 is the matric potential that corresponds to the volumetric fraction of total water Θmsoil, Ss (m−1) is the specific storage coefficient, and θs (—) is the saturated water content (i.e., the soil porosity). Equation (15) is an extension of Eq. (2), where the first term on the LHS accounts for the compressibility of the fluid and the aquifer. The first term on the LHS of Eq. (15) is necessary to simulate saturated flow. Equation (15) excludes the vertical flux of frozen water (e.g., snowfall), ∂qicesoil/∂z, because qicesoil only occurs in the vegetation and snow domains. Solving Eq. (15) requires the closure relationship
Θmsoil⁡(ψ0)=S*⁡(ψ0),
where S*⁡(⋅) is a soil-water-retention function, e.g., the van Genuchten (1980) function, which is used to calculate dΘmsoil/dψ0 in Eq. (15). The state variable ψ0 in Eq. (15) is used to generalize the hydrology state equation for the soil subdomain to simulate the storage and transmission of water in partially frozen soils: ψ0 is the “unfrozen equivalent” matric potential, obtained from ψ0=ψ*⁡(θs−θair), where ψ*⁡(⋅) is the inverse of S*⁡(⋅).

b. Model fluxes

The vertical energy fluxes in Eq. (1) are defined for the vegetation domains as
FΩ={−Cpcaske∂Tcas∂z,Ω=cas,Qswdveg+Qlwdveg+Qswuveg+Qlwuveg+Qpveg,Ω=veg.
These fluxes for the snow and soil subdomains (Ω = ss = snow, soil) are defined as
Fss={Qswdsfc+Qlwdsfc+Qswusfc+Qlwusfc+Qsensfc+Qlatsfc+Qpsfc,z=zsfc,−λ∂Tss∂z,z>zsfc,
where zsfc is the vertical position of the surface–atmosphere interface—when the land is snow free, zsfc defines the top of the soil profile (i.e., zsfc = 0); when the land is snow covered, zsfc defines the top of the snowpack. The vertical coordinate z is defined as positive downward, so the vertical positions z > zsfc are below the surface–atmosphere interface.

In Eq. (17), ke (m2 s−1) is the transfer coefficient for heat and water vapor within the canopy air space; in Eq. (18), λ (J m−1 K−1 s−1) is the thermal conductivity within the snow–soil domain, and depends on the fraction of constituents at depth z. In Eqs. (17) and (18), Qswd (J m−2 s−1) and Qlwd (J m−2 s−1) are the downwelling shortwave and longwave radiation fluxes, Qswu (J m−2 s−1) and Qlwu (J m−2 s−1) are the upwelling shortwave and longwave radiation fluxes, Qp (J m−2 s−1) is the heat advected with precipitation, Qsen (J m−2 s−1) and Qlat (J m−2 s−1) are the sensible and latent heat fluxes.

The energy source–sink flux terms in Eq. (1) are
FsinkΩ={−Fsenveg,Ω=cas,Fsenveg+Flatveg,Ω=veg,0,Ω=snow,soil,
where Fsenveg (J m−3 s−1) and Flatveg (J m−3 s−1) are the volumetric sensible and latent heat fluxes, respectively, from the vegetation canopy to the canopy air space, and defined as positive toward the vegetation canopy. The sensible and latent heat fluxes are not part of the vertical fluxes in the vegetation subdomain, Eq. (17), because the source–sink flux term in Eq. (1), FsinkΩ, includes the sensible and latent heat fluxes between the vegetation canopy and the canopy air space.
The hydrology source–sink flux terms in Eq. (2) are
MsinkΩ={0,Ω=cas,Msub+Mevap,Ω=veg,snow,Msub+Mevap+Mtrans+Mlf,Ω=soil,
where Msub (s−1) defines the source–sink flux for frost/sublimation, Mevap (s−1) defines the source–sink flux for condensation/evaporation, Mtrans (s−1) defines the transpiration losses, and Mlf (s−1) is the net lateral flux of water. We note that no water is stored in the canopy air space, and hence Msinkcas=0.

c. Spatial discretization

The spatial discretization of land models is handled differently for the different model subdomains. For vegetation, the general model equations in section 2a can be discretized into a single canopy layer, i.e., the so-called “big-leaf” model (e.g., Dai et al. 2004), or multiple canopy layers (e.g., Bonan et al. 2018). For snow, the spatial (vertical) discretization is typically time varying. The snow subdomain is discretized into layers, and the depth of each snow layer is recomputed at the end of each time step. Snow layers are subdivided or combined when layer depths meet specific criteria; i.e., a snow layer is divided into two when its depth is larger than a prescribed threshold and combined with one of its neighbors when its depth is smaller than a prescribed threshold. For soil, the spatial discretization is typically time invariant. The soil subdomain can be discretized into any number of layers of variable thickness. Thinner soil layers are used near the top of the soil subdomain to better capture the effects of diurnal temperature forcing and large moisture gradients.

1) Thermodynamics

The spatial discretization of the conservation equation for thermodynamics in the vegetation domain is obtained from Eqs. (1), (17), and (19). Using a single canopy layer, the volumetric sensible and latent heat fluxes in Eq. (19) can be defined as
Fsenveg=QsenvegDcan,
Flatveg=Qsubveg+Qevapveg+QtransvegDcan,
where Qsenveg (J m−2 s−1) is the sensible heat flux from the vegetation canopy to the canopy air space, Qsubveg (J m−2 s−1), Qevapveg (J m−2 s−1), and Qtransveg (J m−2 s−1) are respectively the heat fluxes associated with sublimation, evaporation, and transpiration, and Dcan=zbotveg−ztopveg (m) is the depth of the vegetation canopy. The spatially discretized thermodynamic state equation for the vegetation canopy can then be given as
Cpveg dTvegdt−Lfusρicedθicevegdt=Qsw,netveg+Qlw,netvegDcan+QsenvegDcan+Qsubveg+Qevapveg+QtransvegDcan+QpvegDcan,
where Qsw,netveg and Qlw,netveg are the net shortwave and longwave fluxes absorbed by the vegetation canopy. The first term on the RHS of Eq. (23) is the net radiation balance for the vegetation canopy, the second term is the sensible heat flux from the vegetation canopy to the canopy air space, the third term is the latent heat flux from the vegetation canopy to the canopy air space, and the fourth term is the heat advected with precipitation.
The spatial discretization in the canopy air space can be obtained similarly. With one vegetation layer, it is only necessary to compute the vertical fluxes at the bottom and top of the canopy air space subdomain. The latent heat fluxes in the canopy air space are defined in Eq. (14); i.e., Qlatcas⁡(z)=−ρairke∂ϑcas/∂z. Given Eq. (14), the latent heat fluxes can be defined as
Qlatsfc=−ρairkeϑsfc−ϑcaszsfc−zcas,
Qlattotal=−ρairkeϑcas−ϑatmzcas−zatm,
where Qlatsfc (J m−2 s−1) and Qlattotal (J m−2 s−1) define the latent heat flux at the bottom and the top of the canopy air space. The sensible heat flux within the canopy air space is defined in Eq. (17); i.e., Qsencas⁡(z)=−Cpcaske∂Tcas/∂z. Given Eq. (17), the sensible heat fluxes at the boundaries of the canopy air space subdomain are computed from the gradients in temperature from the surface to the canopy air space and from the canopy air space to the upper boundary:
Qsensfc=−CpcaskeTsfc−Tcaszsfc−zcas,
Qsentotal=−CpcaskeTcas−Tatmzcas−zatm,
where Qsensfc (J m−2 s−1) and Qsentotal (J m−2 s−1) define the sensible heat flux at the bottom and the top of the canopy air space.
The conservation equations for temperature and water vapor in the canopy air space are treated differently. The spatially discretized conservation equation for temperature in the canopy air space is formulated directly in order to simplify the stability corrections. The conservation equation for temperature in the canopy air space is
CpcasdTcasdt=−Qsensfc−QsentotalDcan−QsenvegDcan.
We note that no water is stored in the canopy air space; hence, θicecas≡0, and the phase change term in Eq. (3) is neglected. We do not formulate a conservation equation for water vapor in the canopy air space. Instead, we make the simplification that
Qlattotal=Qevapveg+Qtransveg+Qlatsfc.
In this formulation, ϑcas in Eqs. (14), (24), and (25) is a diagnostic variable.
The snow and soil subdomains are discretized into multiple layers. Suppose that Δzj is the depth of layer j in the snow or soil subdomain; the spatial discretization of the conservation equation for thermodynamics can then be written for layer j as
CpssdTjssdt−Lfusρicedθicejssdt=−Fj+1/2ss−Fj−1/2ssΔzj,j=1,2,…,Jss,
where the vertical energy fluxes are as defined in Eq. (18).

2) Hydrology

The spatial discretization of the conservation equation for the vegetation canopy hydrology can be obtained from Eqs. (2) and (20). The net solid and liquid precipitation fluxes are
qice,netveg=(qtf,snowveg+qunloadveg)−qsf,
qliq,netveg=(qtf,rainveg+qdripveg)−qrf,
where the first and second terms on the RHS define the fluxes at the bottom and top of the vegetation canopy, respectively. Here, qsf (m s−1) and qrf (m s−1) are respectively snowfall and rainfall, qtf,snowveg (m s−1) and qtf,rainveg (m s−1) are the throughfall of snow and rain, qunloadveg (m s−1) is the unloading of snow from the vegetation canopy, and qdripveg (m s−1) is the drip of liquid water from the vegetation canopy. The loss of water from evaporation and sublimation, i.e., contributing to the source–sink flux terms Sevap and Ssub in Eq. (20), are related to the energy used for evaporation and sublimation, i.e.,
Msubveg=QsubvegLsubρliqDcan,
Mevapveg=QevapvegLvapρliqDcan,
where Lvap (J kg−1) and Lsub (J kg−1) are respectively the latent heat of evaporation and sublimation. The spatially discretized conservation equation for the vegetation canopy hydrology is then
dΘmvegdt=−qice,netvegDcan−qliq,netvegDcan+Msubveg+Mevapveg,
where all other terms are as defined previously.
The spatial discretization of the conservation equations for snow and soil hydrology is obtained in an analogous manner to vegetation. The net vertical flux of solid water is
qice,netjss={−(qtf,snowveg+qunloadveg),j=1,0,j>1,
where j = 1 defines the top layer in the snow+soil domain. We note that if there is no vegetation or if vegetation is buried, then qtf,snowveg=qsf and qunloadveg=0. Similarly, the sublimation and evaporation fluxes are apportioned to the uppermost layer in the snow + soil domain
Msubjss+Mevapjss={QsubsfcLsubρliqΔzjΩ+QevapsfcLvapρliqΔzjΩ,j=1,0,j>1.
The transpiration fluxes are apportioned to the soil layers via
Mtransjss={0,Ω=snow,fj QtranssfcLvapρliqΔzjsoil,Ω=soil,
where fj is the fraction of the transpiration flux apportioned to layer j and depends on soil water content and root density.
The spatially discretized conservation equation for soil and snow hydrology is then
dΘmjsnowdt=−qice,netjsnowΔzj−qliqj+1/2snow−qliqj−1/2snowΔzj+Msinkjsnow,j=1,2,…,Jsnow,
where Msinkjsnow=Msubjsnow+Mevapjsnow and
SsΘmjsoilθsjdψ0jdt+dΘmjsoildψ0jdψ0jdt=−qliqj+1/2soil−qliqj−1/2soilΔzj+Msinkjsoil,j=1,2,…,Jsoil,
where
Msinkjsoil=Msubjsoil+Mevapjsoil+Mtransjsoil+Mlfjsoil,
with Mlfjsoil the source–sink flux term for lateral flow.

d. Temporal discretization

The temporal discretization of the state equations requires special attention in order to conserve energy and mass. The following subsections describe methods for conservative temporal approximations of Eqs. (1) and (2). To show application in existing land models, appendix A provides details on the temporal discretization and solution of the model equations using the first-order implicit Euler method.

1) Thermodynamics

We write a conservative discretization of the enthalpy form of the heat equation, Eq. (1), as
ΔHj,n+1ΩΔtn=FnetjΩΔzjΩ,
where ΔHj,n+1Ω=[HjΩ⁡(tn+1)−HjΩ⁡(tn)] is the change in enthalpy (J m−3) over the time interval [tn, tn+1], with tn+1 = tn + Δtn, and ΔzjΩ=Dcan=zbot−ztop for Ω = cas, veg and ΔzjΩ=zj+1/2−zj−1/2 for Ω = snow, soil. Making use of Eqs. (23), (28), and (30),
FnetjΩ={−(Qsensfc−Qsentotal)−(Qsenveg),Ω=cas,⁡(Qsw,netveg+Qlw,netveg)+(Qsenveg)+(Qsubveg+Qevapveg+Qtransveg)+(Qpveg),Ω=veg,−(Fj+1/2ss−Fj−1/2ss),Ω=snow,soil.
Although it is valid to discretize the enthalpy form of the heat equation, Eq. (1), we are interested in the temperature form of the heat equation, Eqs. (10) and (12), because the net fluxes FnetjΩ are parameterized as a function of temperature. Formulating the heat equation in terms of temperature is also useful in order to directly control the numerical error of the temperature state variables. The temperature form of the heat equation in Eq. (12) can be discretized as
C˜ajΩΔTj,n+1ΩΔtn=FnetjΩΔzjΩ,
where C˜a (J m−3 K−1) is the apparent heat capacity and ΔTj,n+1Ω=[TjΩ⁡(tn+1)−TjΩ⁡(tn)] is the change in temperature (K) over the time interval [tn, tn+1].
Comparing Eqs. (41) and (43), it is apparent that Eq. (43) conserves energy if C˜ajΩΔTj,n+1Ω=ΔHj,n+1Ω (e.g., see Tubini et al. 2020). Indeed, estimates of Ca at an arbitrary point in the time step (e.g., tn or tn+1) do not typically conserve energy. A conservative method for Eq. (43) can be obtained using
C˜ajΩ=ΔHj,n+1ΩΔTj,n+1Ω.
We can interpret C˜a as the value of the apparent heat capacity that is necessary to conserve energy. Because the approximation in Eq. (44) only requires modifying the estimate C˜a, this approach can easily be implemented in existing land models.
Separating out the temperature and phase change terms on the LHS of Eq. (43), i.e., as in Eqs. (10), (23), (28), and (30), and making use of Eq. (44), in the limit as Δt → 0, we have
ΔHTjΩΔTjΩdTjΩdt−LfusρiceΔθicejΩΔtn=FnetjΩΔzjΩ,j=1,2,…,JΩ,
where JΩ is the number of layers in subdomain Ω. In Eq. (45), the enthalpy HTΩ and volumetric ice content θiceΩ are both diagnostic variables. The methods used to calculate enthalpy are detailed in appendix B. The quantity θicesnow⁡(Tsnow,Θmsnow) is calculated using Eqs. (7) and (62); θicesoil⁡(Tsoil,ψ0) is calculated using Eqs. (67)–(73).

2) Soil hydrology

Care is also needed to conserve mass in soil. The discretized form of Eq. (15) can be written as
SsΘmjsoilθsjΔψ0j,n+1Δt+C˜ψjΔψ0j,n+1Δt=qnetjsoilΔzjsoil+Msinkjsoil,j=1,2,…,Jsoil,
with
qnetjsoil=−[⁡(qliqsoil)j+1/2−⁡(qliqsoil)j−1/2].
In Eq. (46), Δψ0j,n+1=[ψ0j⁡(tn+1)−ψ0j⁡(tn)] and ΔΘmj,n+1soil=[Θmjsoil⁡(tn+1)−Θmjsoil⁡(tn)] are respectively the change in matric head (m) and total volumetric water content (—) over the time interval [tn, tn+1], and C˜ψjsoil=dΘmjsoil/dψ0j (m−1) is the slope of the water retention curve. Similar to Eq. (43), conservation of mass can be satisfied using
C˜ψjsoil=ΔΘmj, n+1soilΔψ0j, n+1,
which corresponds to the standard chord slope approximation examined by Rathfelder and Abriola (1994). Further discussions of mass conservation issues are in Milly (1984) and Celia et al. (1990).
Given Eqs. (15) and (48), in the limit as Δt → 0, we have
SsΘmjsoilθsjdψ0jdt+ΔΘmjsoilΔψ0jdψ0jdt=qnetjsoilΔzjsoil+Msinkjsoil,j=1,2,…,Jsoil.
In Eq. (49), Θmsoil is a diagnostic variable computed from ψ0.

e. Numerical solutions

Up to this point, the model development has been largely general. Most existing land models are some variant of Eqs. (1) and (2). Although land models apply to 3D space (x, y, z), as is common, we formulate the conservation laws for fixed x and y and solve each 1D column in the vertical dimension z independently. The lateral flux across soil columns is handled as a source–sink flux term in the 1D equations. The differences among land models depend on aspects such as what physical processes are included, how the fluxes are parameterized, and how the model domain is discretized.

In this study, we solve this system of equations using IDA, which is part of the Suite of Nonlinear and Differential/Algebraic Equation Solvers (SUNDIALS) suite (Hindmarsh et al. 2005), as implemented in the Structure for Unifying Multiple Modeling Alternatives (SUMMA) model (Clark et al. 2015b). IDA is a package for the solution of implicit ordinary differential equation (ODE) and differential-algebraic equation (DAE) systems in the general form f(t, y, y′) = 0. The integration method in IDA uses variable-length time steps and methods from the backward differentiation formula (BDF) family of orders 1–5. IDA is written in C, but for use with FORTRAN applications like SUMMA, a set of FORTRAN/C interface routines, called FIDA, is also supplied in the SUNDIALS suite.

The simulations take place over a time interval described in terms of a sequence of “data windows” (i.e., data time steps) over which certain inputs or other quantities are fixed and then updated from one window to the next. Fluxes such as precipitation and shortwave radiation are often taken to be averages over a data window, e.g., downwelling shortwave radiation changes during daylight hours and can change rapidly during otherwise innocuous events such as the sun going behind a cloud; similarly, precipitation can start and stop abruptly. Because of the potential discontinuities arising in the terms of the differential equations, the integration in SUMMA/SUNDIALS is restarted on each data window.

3. Synthetic test cases

Synthetic test cases—the laugh tests—are used to evaluate the implementation of the model equations, including impacts of the numerical approximations. As discussed by Clark et al. (2015a), synthetic test problems have been used to evaluate the numerical implementation of Richards’ equation (Celia et al. 1990; Boone and Wetzel 1996; Tocci et al. 1997; Lee and Abriola 1999; Vanderborght et al. 2005; Miller et al. 2006), model simulations of cryosuction processes during soil freezing (Hansson et al. 2004; Painter 2011; Noh et al. 2012; Kurylyk et al. 2014; Grenier et al. 2018), model simulations of water movement through snow during rain-on-snow events (Colbeck 1976; Clark et al. 2017), model simulations of lateral subsurface flow (Wigmosta and Lettenmaier 1999; Rupp and Selker 2005, 2006; Bogaart et al. 2013), and overland flow (Parlange et al. 1981; Govindaraju et al. 1990; Mizumura 2006; Mizumura and Ito 2011a,b). Synthetic experiments are also used to simulate interlinked physical processes. For example, Maxwell et al. (2014) examined cases of runoff and return flow on an inclined hillslope, runoff over a tilted V-shaped catchment, and runoff generation on a heterogeneous 1D slab.

Synthetic test cases are important because they provide evidence that the land model equations are formulated and implemented correctly. We consider synthetic test cases as “laugh tests” for land models—if a land model fails a laugh test, then it is difficult to seriously consider the use of the land model for general applications. Synthetic test cases provide the most rudimentary test of model capabilities, yet they are not widely reported in land model evaluation efforts. Such test cases are important not just in the initial stage of model development; the tests are also important as part of continuous model testing to check that new modeling enhancements do not introduce unintended errors.

The laugh tests are designed to test the numerical solution of the coupled energy and mass equations in different model subdomains. To the extent possible, the test cases are designed to analyze model behavior across different model subdomains (e.g., the soil subdomain, the snow subdomain) and directly evaluate the coupling processes (e.g., the coupling between vertical processes and lateral flow, the coupling between thermodynamic and hydrological processes). The test cases are hence less focused on the parameterization of individual processes; instead, the test cases focus on how individual processes combine to affect the temporal evolution of model state variables for different model subdomains. Critically, the laugh tests do not evaluate whether the land model equations are appropriate; the laugh tests simply check that the model equations are implemented correctly and are able to reproduce known results.

Although the laugh tests introduced here are by no means comprehensive, they do provide a useful starting point to assess the numerical implementation of land models. The laugh tests include three test cases for storage and transmission of water in soils (section 3a), one test case for lateral subsurface flow (section 3b), one test case for coupled hydrological and thermodynamic processes in snow (section 3c), and one test case for cryosuction processes in soil (section 3d). All of the laugh tests are solved using the same model (SUMMA) using different model parameters and different boundary conditions. To ensure that the spatial discretization does not dominate the overall error compared to the time stepping, we impose fine vertical discretizations for all test cases. The laugh tests use an accuracy tolerance of 1 × 10−6 for all model state variables.

a. Storage and transmission of water in soils

Richards’ equation is a common approach used in land models to simulate the storage and transmission of water in soils (Clark et al. 2015a). As discussed earlier and defined in Eq. (15), variably saturated flow problems require formulating Richards’ equation with matric head ψ (m) as the state variable, i.e., the pressure-based form of Richards’ equation. In this formulation, θliqsoil is a diagnostic variable computed from ψ using the soil-water-retention function defined in Eq. (16). The conservation equation for soil hydrology, Eq. (15), can then be written for unfrozen conditions as
Ssθliqsoilθs∂ψ∂t+dθliqsoildψ∂ψ∂t=−∂qliqsoil∂z+Msinksoil,
with
qliqsoil⁡(ψ)=−k⁡(ψ)∂ψ∂z+k⁡(ψ),
where k(ψ) is the hydraulic conductivity (m s−1). We note that Eq. (50) is formulated for unfrozen conditions, so Θmsoil=θliqsoil and ψ0 = ψ.
The test cases examined here evaluate infiltration into initially dry soil, as formulated by Celia et al. (1990) and Miller et al. (1998). These test cases use the closure relations from van Genuchten (1980),
θliqsoil⁡(ψ)={θs−θr[1+⁡(−αψ)n]m+θr,ψ≤0,θs,ψ>0,
k⁡(ψ)={ks{1−⁡(−αψ)n−1[1+⁡(−αψ)n]−m}2[1+⁡(−αψ)n]m/2,ψ≤0,ks,ψ>0,
where θr is the residual water content (—), α is the capillary length scale (m−1), the parameters m (—) and n (—) are related to the pore size distribution [typically m = 1− (1/n)], and ks is the saturated hydraulic conductivity (m s−1). The parameters for the test cases of Celia et al. and Miller et al. are fully specified in Table 2.
Table 2.

Parameters for the test cases of Celia et al. (1990) and Miller et al. (1998).

Table 2.
The test case formulated by Celia et al. (1990) considers the propagation of a sharp infiltration front in a 0.6-m soil column. For this test case, we solve Eqs. (50)–(53) subject to the following initial and boundary conditions
ψ⁡(z,0)={−0.75−9.250.006z,0≤z≤0.006,−10,z>0.006,
ψ⁡(0,t)=−0.75,
ψ⁡(0.6,t)=−10.
The soil domain is discretized into 100 uniform layers, and the simulation time consists of 120 data windows of length 1800 s for a total simulation time of 60 h.

Although the test case of Celia et al. has strong spatial and temporal gradients in pressure, its boundary conditions restrict simulations to unsaturated flow. Figure 2 illustrates the vertical profiles of pressure at various times throughout the simulation. The simulations shown in Fig. 2 closely match the independent numerical solutions shown in Celia et al. (1990) and Kavetski et al. (2001), building confidence in the numerical solutions.

Fig. 2.
Fig. 2.

Solution of the test problem from Celia et al. (1990), showing pressure profiles at various times during the model simulation.

Citation: Journal of Hydrometeorology 22, 6; 10.1175/JHM-D-20-0175.1

The test cases formulated by Miller et al. (1998) evaluate infiltration into an initially hydrostatic soil profile, with a fixed water table at the lower boundary, subject to ponding at the surface. The test cases of Miller et al. hence evaluate model simulations of variably saturated flow. For these cases, we solve Eqs. (50)–(53) subject to the following initial and boundary conditions:
ψ⁡(z,0)=z−zmax,
ψ⁡(0,t)=0.1,
ψ⁡(zmax,t)=0.
The domains are discretized into 800, 400, and 320 uniform layers for sand, loam, and clay, respectively, and the simulation time in each case consists of 240 data windows of length 900 s for a total simulation time of 60 h.

Figure 3 illustrates the vertical profiles of pressure for the three soil types considered by Miller et al. (1998). The simulations in Fig. 3 have more numerical diffusion than the simulations in Miller et al. (1998). As with the test case from Celia et al. (1990), the test cases from Miller et al. (1998) present a didactic set of problems because of the sharp vertical gradients at the toe of the infiltration front.

Fig. 3.
Fig. 3.

Solution of the test problem from Miller et al. (1998), showing pressure profiles for sand at t = 0.18 days, loam at t = 2.25 days, and clay loam at t = 1 day.

Citation: Journal of Hydrometeorology 22, 6; 10.1175/JHM-D-20-0175.1

The next test case was defined by Vanderborght et al. (2005) to evaluate the steady-state flux in layered soil profiles. This test case is useful because an analytical solution can be derived to yield depth as a function of pressure head (Vanderborght et al. 2005),
Δz=z1−z2=∫ψ⁡(z2)ψ⁡(z1)[qliqsoil⁡(0,t)k⁡(ψ)−1]−1dψ.
The Vanderborght case is defined as one soil type for depth 0–0.5 m overlying another soil type for depth 0.5–2 m for loam over sand, sand over loam, and clay over sand. The soil parameters are defined in Table 3. We solve Eqs. (50)–(53) subject to the initial and boundary conditions
ψ⁡(z,0)=−20,
qliqsoil⁡(0,t)=5.79×10−8,
∂ψ∂z⁡ (2,t)=0.
In each case, the domain is discretized into 200 uniform layers, and the simulation time consists of data windows of length 3600 s for a total simulation time of 24 000 h. We note that the Vanderborght test cases are for steady-state conditions, so the simulation time is more than is required to bring the model to steady state.
Table 3.

Parameters for the soil types used in the Vanderborght et al. (2005) test cases.

Table 3.

Figure 4 illustrates the vertical profiles of pressure for the layered soil profiles studied by Vanderborght et al. (2005). Similar to the results presented by Vanderborght et al. (2005), the simulations in Fig. 4 provide close approximations to the analytical solutions.

Fig. 4.
Fig. 4.

Solution of the three test problems from Vanderborght et al. (2005), showing the vertical profiles of pressure head at steady state. The solid red lines are the model simulations, and the dashed black lines are the analytical solutions.

Citation: Journal of Hydrometeorology 22, 6; 10.1175/JHM-D-20-0175.1

b. Lateral subsurface flow

Lateral subsurface flow is a dominant hydrological process that has received limited attention in land models (Clark et al. 2015a; Fan et al. 2019). Recent approaches represent lateral flow as subgrid or “meta” hillslopes (Chaney et al. 2018; Swenson et al. 2019). This approach defines hillslopes with realistic length and height dimensions and assumes that a single hillslope or an ensemble of hillslopes are representative of the model grid cell. It is hence prudent to use simple synthetic test cases to evaluate the implementation of subsurface lateral flow.

The continuity equation for subsurface storage (below the water table) can be given in the x dimension as (Wigmosta et al. 1994)
ϕ ∂zwt∂t=−∂qx∂x1xlen−qr,
where ϕ (—) is the drainable porosity, zwt (m) is the water table depth, qx (m3 s−1) is the rate of saturated subsurface flow, xlen (m) is the flow width, and qr (m s−1) is the recharge flux.
Assuming an unconfined horizontal aquifer, the rate of saturated subsurface flow can be given as (Wigmosta and Lettenmaier 1999)
qx⁡(zwt)=xlen tanβ Taq⁡(zwt),
where β (—) is the water table slope and Taq⁡(zwt) (m2 s−1) is the aquifer transmissivity at water table depth zwt. The transmissivity can be obtained from the vertical profile of saturated hydraulic conductivity, ks (m s−1), e.g., using the power-law profile described by Rupp and Woods (2008), as
ks⁡(z)=ks0⁡(1−zzmax)nsf−1,
where ks0 (m s−1) is the saturated hydraulic conductivity at the soil surface, zmax is the depth of the soil profile, and nsf is the shape of the transmissivity profile. The transmissivity is hence
Taq⁡(zwt)=∫zwtzmaxks⁡(w) dw,
yielding
Taq⁡(zwt)=ks0zmaxnsf⁡(1−zwtzmax)nsf,nsf≥1,
as used in some applications of TOPMODEL (Duan and Miller 1997). The subsurface lateral flow is included as part of the source–sink flux term in Eq. (2).

Here, we use the synthetic experiments described by Wigmosta and Lettenmaier (1999) to evaluate model representations of hillslope hydraulics under steady-state and transient conditions. The Wigmosta and Lettenmaier experiments provide a test for the conditions where topography dominates subsurface flow, i.e., steep hillslopes with thin, permeable soils. Under these conditions, the slope of the water table is assumed to be parallel to the ground surface, and saturated subsurface flow can be modeled using a kinematic wave equation. Such simplifications are useful because the kinematic wave equation can be solved analytically using the method of characteristics.

The Wigmosta and Lettenmaier test case applies a constant rainfall rate of 5.56 kg m−2 s−1 (0.002 m h−1) for 550 h to an example hillslope of length 50 m and height 15 m (i.e., slope tanβ = 0.3), with soil depth zmax = 1.5 m and drainable porosity ϕ = 0.25. For this test case, we discretize the hillslope into 50 independent soil columns; the lateral flow across soil columns is represented using the lateral flow source–sink flux term, i.e., Slf defined in Eq. (20).

For this test case, we solve Eqs. (50)–(53), (55), (56), and (59) subject to the initial and boundary conditions
zwt⁡(z,0)=1.5 m,
qliqsoil⁡(0,t)={0.002 m h–1,t≤550 h,0,t>550 h,
qliqsoil⁡(zmax,t)=0.
The spatial domain spans a single hillslope consisting of 50 soil columns. Each soil column is vertically discretized into 8 soil layers. The simulation time consists of 1000 data windows of length 3600 s for a total simulation time of 1000 h.

Figure 5 illustrates the simulated water table profiles and hillslope outflow for the following two experiments:

  1. nsf = 1 and ks0=8.33×10−5 m s−1 (0.3 m h−1) and

  2. nsf = 3 and ks0=8.33×10−4 m s−1 (3.0 m h−1).

As in Wigmosta and Lettenmaier (1999), the model simulations have close correspondence to the analytical solutions.
Fig. 5.
Fig. 5.

Simulated (left) water table profiles and (right) hillslope outflow for the test problems from Wigmosta and Lettenmaier (1999). The figure shows both the linear experiment (nsf = 1, blue) and the nonlinear experiment (nsf = 3, red).

Citation: Journal of Hydrometeorology 22, 6; 10.1175/JHM-D-20-0175.1

c. Coupled thermodynamic and hydrological processes in snow

Snow modeling can be challenging because of the strong coupling between the thermodynamic and hydrological processes. For example, the propagation of liquid water in snow requires satisfying both thermal and liquid deficits. It requires both freezing a sufficient volume of water onto the snow grains to raise the snow temperature to 0°C as well as filling the storage of water that is retained as an adsorbed film and the water held immobile in the menisci between the snow grains (Colbeck 1976). It is useful to define test cases that evaluate these coupled processes.

As in Eqs. (8) and (2), the coupled conservation equations for energy and mass in snow can be defined by the state variables of temperature T (K) and volumetric fraction of total water Θmsnow,
Cp∂Tsnow∂t−Lfusρice∂θicesnow∂t=−∂Fsnow∂z,
∂Θmsnow∂t=−∂qliqsnow∂z.
The volumetric fraction of liquid water is
θliqsnow⁡(Tsnow,Θmsnow)={Θmsnow1+[ϖ⁡(Tfrz−Tsnow)]2,Tsnow<Tfrz,Θmsnow,Tsnow≥Tfrz,
with Tfrz = 273.16 K the triple point of pure water and ϖ (K−1) is a scaling parameter. Equation (62) defines a smoothed step function over a small temperature interval below Tfrz, ranging from zero to Θmsnow at Tfrz. The parameter ϖ defines the degree of smoothing (e.g., ϖ = 50 K−1 defines a function that is visually indistinguishable from a step function over a typical diurnal temperature range).
The vertical liquid flux in snow is related to the volumetric fraction of liquid water; i.e.,
qliqsnow={0,θliqsnow≤θr,kssnow⁡(θliqsnow−θrθs−θr)ns,θliqsnow>θr,
where kssnow (m s−1) is the saturated hydraulic conductivity, θr is the residual volumetric liquid water content, θs=1−θicesnow is the porosity, and ns defines the shape of the hydraulic conductivity function. The saturated hydraulic conductivity is given as
kssnow=κ ρliqgμ,
where κ (m2) is the permeability, g (m s−2) is the gravitational acceleration, and μ (kg m−1 s−1) is the dynamic viscosity. The values of these parameters are given in Table 4.
Table 4.

Physical constants and parameter values for the water flow examples presented by Colbeck (1976).

Table 4.

Here, we use the synthetic experiments of water movement through snow that were introduced by Colbeck (1976) and further developed by Clark et al. (2017). Colbeck (1976) considered a constant rain rate of 0.1 kg m−2 s−1 applied at the top of a 1-m snowpack for a time period of 3 h for different initial snow conditions: ripe snow, refrozen snow, and fresh snow (see Table 5). The colder initial temperatures for refrozen and fresh snow require that both the heat deficit and the residual saturation deficit be satisfied before flow occurs; the smaller grain size for fresh snow substantially reduces the permeability of the snowpack and slows the flow of water. Colbeck’s examples provide a useful test case for snow models because they evaluate the coupling of hydrology and thermodynamics (i.e., the freezing of water onto the snow grains necessary to supply sufficient latent heat to raise the snow temperature to 0°C), as well as providing a means to evaluate the unsaturated flow of water through porous media.

Table 5.

Initial snow conditions and parameter values for the water flow examples presented by Colbeck (1976). See Clark et al. (2017) for more details.

Table 5.

For these test cases, we solve Eqs. (60)–(64) subject to the boundary conditions
qliqsnow⁡(0,t)={0.1,0≤t<10 800 s,0,t>10 800 s,
F⁡(0,t)=0,
F⁡(1,t)=0,
with initial conditions given in Table 5.

The snow domain is discretized into 100 uniform layers, and the simulation time consists of 600 data windows of length 60 s for a total simulation time of 10 h.

The solutions for the test cases for water movement through snow are presented in Figs. 6 and 7 along with analytical solutions derived using the method of characteristics [see Clark et al. (2017) for full development of the analytical solutions]. As presented earlier by Clark et al. (2017), the results in Fig. 6 demonstrate close correspondence between the numerical simulations and the analytical solutions, building confidence in the numerical model implementation. The results in Fig. 7 also show similar results in the numerical simulations and the analytical solutions, with numerical diffusion causing some water to flow from the bottom of the snowpack ahead of the wetting front.

Fig. 6.
Fig. 6.

Contour plots of the volumetric fraction of liquid water for the problem of Colbeck (1976) from (left) the analytical solution and (right) the numerical simulations for (top) the ripe snow test case, (middle) the refrozen snow test case, and (bottom) the fresh snow test case.

Citation: Journal of Hydrometeorology 22, 6; 10.1175/JHM-D-20-0175.1

Fig. 7.
Fig. 7.

The outflow hydrograph for the problem of Colbeck (1976) from numerical simulations (solid lines) and analytical solutions (dashed lines) for (top) the ripe snow test case, (middle) the refrozen snow test case, and (bottom) the fresh snow test case.

Citation: Journal of Hydrometeorology 22, 6; 10.1175/JHM-D-20-0175.1

d. Cryosuction processes in soil

The coupled thermodynamic and hydrological processes in soil cause the migration of water toward the freezing front, known as cryosuction (Hansson et al. 2004; Dall’Amico et al. 2011). Specifically, freezing of water in soil pores results in a low-pressure head, creating a gradient in pressure in the vicinity of the freezing front. Such cryosuction processes provide an interesting test case for land models because their solution requires accurate coupling of thermodynamic and hydrological processes as well as generating negative pressures that induce hydraulic gradients toward the freezing front.

Cryosuction processes can be simulated in a similar way to the coupled thermodynamic and hydrological processes in snow. Making use of Eqs. (1) and (15), the conservation equations for partially frozen soils can be written as
Cpsoil∂Tsoil∂t−Lfusρice∂θicesoil∂t=−∂Fsoil∂z,
dΘmsoildψ0∂ψ0∂t=−∂qliqsoil∂z,
where, as in Eq. (15), ψ0 (m) is the matric potential that corresponds to the volumetric fraction of total water Θmsoil. Equation (66) does not include the compressibility term in Eq. (15) because we restrict attention to unsaturated flow. The total water content in Eq. (66) is computed using the closure relation Θmsoil⁡(ψ0)=S*⁡(ψ0), i.e., Eq. (16), where S*⁡(⋅) is a soil-water-retention function, e.g., the van Genuchten function Eq. (52). The term dΘmsoil/dψ0 in Eq. (66) is the derivative of the closure relation.
We use the generalized Clapeyron equation to separate the volumetric fraction of total water into the volumetric fractions of liquid water and ice. The Clapeyron equation can be defined for the state of water as [Dall’Amico et al. 2011, their Eq. (11)]
dPwdT=LfusρliqT,T<T*,
where Pw (Pa) is the water pressure, T* (K) is the soil temperature where all water is unfrozen (note T*<Tfrz), and all other variables are as defined previously. The liquid water matric potential from the Clapeyron equation can then be computed during subfreezing conditions. Given Pw = ρliqgψ and noting that T*=Tfrz at ψ = 0; i.e., ψ(Tfrz)= 0, Eq. (67) can be approximated as
ψ⁡(T)−ψ⁡(Tfrz)T−Tfrz=Lfusg Tfrz,T<T*.
The liquid water matric potential from the Clapeyron equation, ψc, is then
ψc⁡(T)=LfusgT−TfrzTfrz,T<T*.
We note that all water is unfrozen when ψ0 < ψc. Derivations of Eq. (69) are provided in Dall’Amico et al. (2011) and Painter and Karra et al. (2014).
Using the “freezing = drying” assumption (Spaans and Baker 1996), i.e., the same closure relations for soil-water-retention and hydraulic conductivity, Eqs. (52) and (53) can be used under both freezing and drying conditions, ψc can be used to separate Θmsoil into the volumetric fractions of liquid water and ice. The soil freezing temperature, T*, used to delineate unfrozen and partially frozen conditions, can be obtained by setting ψc = ψ0 in Eq. (69) and solving for T* as a function of ψ0:
T*⁡(ψ0)={Tfrz,ψ0>0,Tfrz+gTfrzLfusψ0,ψ0≤0,
and the volumetric fraction of liquid water computed using a soil-water-retention function, S*⁡(⋅),
θliqsoil⁡(ψ0, T)={S*⁡(ψc),T<T*,S*⁡(ψ0),T≥T*.
The volumetric fraction of ice can then be defined based on the volume constraints
θairsoil⁡(ψ0)=θs−S*⁡(ψ0)and
θicesoil⁡(ψ0, T)=θs−θairsoil⁡(ψ0)−θliqsoil⁡(ψ0, T).
In this approach, it is assumed that the intrinsic density of ice in soil is equal to the intrinsic density of liquid water, i.e., ρice = 1000 kg m−3, and hence Θmsoil=θliqsoil+θicesoil. This assumption avoids the need for a mechanical model to handle the deformation of the soil matrix caused by volume expansion during freezing [this assumption was also made by Dall’Amico et al. (2011)].

The vertical fluxes of water in soil, qliqsoil used in Eq. (66), can (in principle) be computed from Eqs. (51) and (69), i.e., qliqsoil⁡(ψc). In practice, however, this procedure creates two practical issues: first, the presence of ice may substantially reduce the liquid water flux due to the apparent pore blockage effect exerted by ice (Dall’Amico et al. 2011); second, ψc can have large negative values (substantial suction) as the volumetric fraction of air approaches zero, and this behavior can complicate the numerical solution. A common approach to address these issues is to apply a multiplicative ice impedance factor to the hydraulic conductivity function, e.g., Eq. (53). Specifically, to solve Eq. (66), previous studies have used a modified hydraulic conductivity function k˜⁡(ψc)=10−ζνk⁡(ψc) with parameter ζ (e.g., ζ = 7) and ν=θicesoil/θs−θr (e.g., Hansson et al. 2004; Dall’Amico et al. 2011).

Here, we implement an alternative modification to the vertical flux parameterizations. We follow the approach of Zhao et al. (1997) and only use the Clapeyron equation to separate the phases of liquid water and ice within soil (i.e., ψc ≠ ψ). The liquid water matric potential ψ (m), is computed by assuming that ice forms part of the solid matrix (Zhao et al. 1997). The effective saturation of soils Se (—) is
Se⁡(ψ0,T)=θliqsoil−θrθs−θicesoil−θr,
and using the soil-water-retention function, e.g., Eq. (52),
ψ⁡(ψ0, T)=1α ⁡(Se−1/m−1)1/n,
and qliqsoil⁡(ψ) can be computed using Eq. (51). Including ice as part of the solid matrix both reduces suction when ice is present (i.e., it impedes flow) and also simplifies the numerical solution because ψ → 0 as θairsoil→0. Although more complex approaches are available (e.g., Karra et al. 2014), the simple modification to the flux parameterization described here is straightforward to implement in most land models.
We use the laboratory experiments of Mizoguchi (1990) to evaluate cryosuction processes. These experiments are described in detail (and in English) by Hansson et al. (2004). Briefly, Mizoguchi packed four identical cylinders (8-cm diameter, 20-cm length) with sandy loam and brought them to the same initial state of uniform temperature of 6.7°C and close to uniform volumetric liquid water content of 0.33. One of the cylinders was excluded from the freezing tests; the other cylinders were insulated on the sides and bottom and subjected to a circulating fluid with temperature −6°C at the top. The cylinders were removed from the freezing apparatus after 12, 24, and 50 h, divided into 1-cm slices, and dried in an oven to obtain volumetric fraction of total water (liquid plus ice). The experiments demonstrated redistribution of water to the top of the soil column. Soil hydraulic parameters for the Mizoguchi case are ks = 3.2 × 10−6 m s−1, θr = 0.05, θs = 0.535, α = 1.11 (m−1), and n = 1.48. We solve Eqs. (65), (66), and (69)–(75) subject to the initial and boundary conditions
Θmsoil⁡(z,0)=0.33,T⁡(z,0)=6.7°C,qliqsoil⁡(0,t)=0,T⁡(0,t)=−6.0°C,qliqsoil⁡(0.2,t)=0,F⁡(0.2,t)=0.
The soil domain is discretized into 200 uniform layers, and the simulation time consists of 3600 data windows of length 60 s for a total simulation time of 60 h.

Figure 8 compares model simulations to the observations from the Mizoguchi experiment. Similar to the simulations presented by Hansson et al. (2004) and Dall’Amico et al. (2011), the model simulations and laboratory measurements both show the redistribution of water toward the top of the soil profile.

Fig. 8.
Fig. 8.

Simulated and measured values of the total volumetric water content for the Mizoguchi experiment for (left) 12 h, (center) 24 h, and (right) 50 h after the start of the experiment.

Citation: Journal of Hydrometeorology 22, 6; 10.1175/JHM-D-20-0175.1

4. Energy and water balance

All of the test problems described in this paper enable evaluating the coupled processes in the different model subdomains. A fundamental metric that is relevant for all of the numerical experiments is whether the numerical methods conserve energy and mass. To this end, we evaluate the energy and mass balance for all of the test problems.

a. Energy balance

The energy balance error εnrgj,nΩ for Eq. (1) is
εnrgj,nΩ=ΔHTj,n+1ΩΔzjΩ−FnetjΩ¯Δtn, Ω=snow,soil,
where, as in Eqs. (41) and (43), ΔHTj,n+1Ω=HTjΩ⁡(tn+1)−HTjΩ⁡(tn) and
FnetjΩ¯Δtn=∫tntn+1Fnetj,tΩ dt.
The net fluxes FnetjΩ in Eq. (77) are defined in Eq. (42). The SUNDIALS solver provides variable-order approximations of FnetjΩ¯; we note that the first-order implicit Euler method corresponds to the approximation of FnetjΩ¯ by Fnetj,n+1Ω (i.e., the right-hand rectangle rule). The energy source–sink flux term FsinkΩ in Eq. (1) is omitted because FsinkΩ=0 in all test problems presented. Formulating Eq. (76) with ΔzjΩ included in the first term on the RHS accounts for the compaction of snow layers. The total energy balance error for the snow or soil domain for time step n is then
εnrgnΩ=∑j=1JΩεnrgj,nΩ.
The statistics of the energy balance errors are quantified in Table 6 using the maximum absolute energy balance error ‖εnrgΩ‖∞=maxn |εnrgnΩ| and the mean absolute energy balance error ‖εnrgΩ‖1/n=∑n|εnrgnΩ|/n.
Table 6.

Statistics of the energy balance errors (J m−2): ‖εnrgΩ‖∞=maxn|εnrgnΩ| is the maximum absolute energy balance error and ‖εnrgΩ‖1/n=∑n|εnrgnΩ|/n is the mean absolute energy balance error.

Table 6.

b. Water balance

The water balance error in Eqs. (2) and (15) for the snow and soil domains is
εmassj,nsnow=ΔΘmj,n+1snowΔzjsnow−qnetjsnow¯Δtn, Ω=snow,
εmassj,nsoil=(ΔΘmj,n+1soil+SsΘmj,n+1soilθsjΔψ0j,n+1)Δzj−qnetjsoil¯Δtn, Ω=soil,
where Δψ0j,n+1=ψ0j⁡(tn+1)−ψ0j⁡(tn), ΔΘmj,n+1Ω=ΘmjΩ⁡(tn+1)−ΘmjΩ⁡(tn), and
qnetjΩ¯Δtn=∫tntn+1qnetjΩ dt, Ω=snow, soil,
where qnetjΩ−[⁡(qliqΩ)j+1/2−⁡(qliqΩ)j−1/2]. As in Eq. (77), we note that the SUNDIALS solver provides variable-order approximations of qnetjΩ¯ and that the first-order implicit Euler method corresponds to the approximation of qnetjΩ¯ by qnetj,n+1Ω. In Eq. (81), the source–sink flux term MsinkΩ is omitted because MsinkΩ=0 in all test problems. Moreover, Eq. (81) ignores the vertical flux of ice (i.e., snowfall and sublimation) because these fluxes are also not included in the test problems. As in Eq. (76), the total mass balance error for the snow and soil domains for time step n is
εmassnΩ=∑j=1JΩεmassj,nΩ.
The statistics of the mass balance errors are quantified in Table 7 using the maximum absolute mass balance error ‖εmassΩ‖∞=maxn|εmassnΩ| and the mean absolute mass balance error ‖εmassΩ‖1/n=∑n|εmassnΩ|/n.
Table 7.

Statistics of the mass balance errors (kg m−2): ‖εmassΩ‖∞=maxn|εmassnΩ| is the maximum absolute mass balance error and ‖εmassΩ‖1/n=∑n|εmassnΩ|/n is the mean absolute mass balance error.

Table 7.

5. Data availability

Results for all six synthetic test cases are available on https://github.com/CH-Earth/laughTests. The repository includes both simulations of the test cases from SUMMA/SUNDIALS as well as the Python code (in the form of Jupyter notebooks) that was used to read the SUMMA/SUNDIALS output and generate Figs. 2–8 in this paper. The analytical solutions (for the Vanderborght, Wigmosta, and Colbeck test cases) as well as the laboratory data (Mizoguchi test case) needed for comparison with the model outputs are included in these notebooks.

We expect that the SUMMA/SUNDIALS solutions will be useful for other model development groups because we ensure that the SUMMA/SUNDIALS solutions satisfy prescribed error tolerances (i.e., we have quantitative evidence that our numerical solutions are correct). The availability of the SUMMA/SUNDIALS solutions enables different modeling groups to use these solutions in their own model development work and publications. For example, it is possible for authors to compare their numerical solutions against the numerical solutions from SUMMA/SUNDIALS. Land model users and developers are invited to extend laugh test applications beyond the collection of tests presented in this paper.

6. Summary and discussion

The intent of this paper is to encourage improved numerical implementation of land models. Our contributions in this paper are twofold. First, we present a unified framework to formulate and implement land model equations. We separate the representation of physical processes from their numerical solution, enabling the use of established robust numerical methods to solve the model equations. Second, we introduce a set of synthetic test cases (the laugh tests) to evaluate the numerical implementation of land models. The test cases include storage and transmission of water in soils, lateral subsurface flow, coupled hydrological and thermodynamic processes in snow, and cryosuction processes in soil. We consider synthetic test cases as laugh tests for land models because they provide the most rudimentary test of model capabilities.

The results presented in this paper give solutions to all of the test cases using the SUNDIALS solver (Hindmarsh et al. 2005) implemented in the SUMMA model (Clark et al. 2015b). The numerical simulations from SUMMA/SUNDIALS are compared against 1) solutions to the synthetic test cases from other models documented in the peer-reviewed literature (for infiltration into initially dry soil); 2) analytical solutions (for the steady-state flux in layered soil, lateral flow across a hillslope, and coupling of thermodynamic and hydrological processes in snow); and 3) observations made in laboratory experiments (for cryosuction processes in soil). In all cases, the numerical simulations are similar to the benchmarks, building confidence in the numerical model implementation.

We posit that some land models may have difficulty in solving these benchmark problems. For example, land models often implement the moisture-based form of Richards’ equation and hence do not have the capabilities to simulate variably saturated flow, making it impossible to solve the Miller test cases presented. Moreover, the simplistic implementation of Richards’ equation in land models, e.g., noniterative solutions of the moisture-based form of Richards’ equation that require ad hoc numerical “fixes” (Clark et al. 2015a), can also create challenges to effectively solve the test case of Celia et al. of infiltration into initially dry soil and the Vanderborght test case for steady-state flux in layered soil. Land models do not typically include the lateral flow across soil columns and hence may not have the capability to solve the Wigmosta test case. More generally, land models may not adequately couple the thermodynamic and hydrological processes, creating challenges to simulate the Colbeck rain-on-snow test case and the Mizoguchi cryosuction experiments. Dedicating more effort to solving synthetic test cases is critical in order to determine when land models are suitable for specific applications.

The synthetic test cases presented here should become an integral part of broader efforts to evaluate the numerical implementation of land models across large geographical domains (i.e., a river basin, a continent, the globe). First, the laugh tests should be used with a very fine spatial discretization that limits numerical approximation errors (as shown in this paper) to confirm that model equations were implemented and solved correctly. Next, it is critical to carefully analyze accuracy–efficiency tradeoffs in order to optimize land model configurations. Large-domain applications of land models typically require accepting some numerical simplifications, for example, accepting numerical errors that arise from a spatial (vertical) discretization that is too coarse and accepting numerical errors that arise from time steps that are too long. The analysis of accuracy–efficiency tradeoffs enable us to complete large-domain model simulations with full awareness of any numerical errors that we introduce.

More work is also needed to extend the laugh tests to include a more extensive set of biophysical and biogeochemical processes. The test cases presented here are limited in that they only evaluate a small subset of processes in land models. More test cases are needed that evaluate some of the processes that are neglected in this study (e.g., biophysical processes such as evapotranspiration). We encourage the community to continue to collect and define laugh tests such as these so that laugh tests become a critical plank in comprehensive benchmarking platforms.

Acknowledgments

This work was supported by the Global Water Futures programme in Canada. We greatly appreciate the constructive comments from the three reviewers.

APPENDIX A

Implicit Euler Solutions

The numerical solvers in land models are generally much simpler than those in IDA; most land models use some variant of a first-order implicit Euler approximation over long time steps. Specifically, the nonlinear equations are typically solved using Newton–Raphson iteration, where every iteration consists of solving the linear system in Eq. (A1) followed by the update in Eq. (A1):
Jn+1⁡(ℓ)δζn+1⁡(ℓ+1)=−rn+1⁡(ℓ),
ζn+1⁡(ℓ+1)=ζn+1⁡(ℓ)+δζn+1⁡(ℓ+1),
where the superscript ℓ indexes the nonlinear solver iterations, the subscript n indexes the time step, ζ is the model state vector, r is the error (in this case, the residual in the energy and mass balance over the time step), and J is the Jacobian matrix.

A straightforward solution to conserve energy and mass in land models is to use finite differences for the residual in Eq. (A1) and analytical derivatives for the Jacobian matrix. Note that the solution at convergence does not depend on the analytical derivatives because Jn+1⁡(ℓ)δζn+1⁡(ℓ+1)→0 at convergence [see, e.g., Rathfelder and Abriola (1994) for a discussion of mass-conservative solutions of Richards’ equation].

a. Implicit Euler solutions for thermodynamics

An energy-conserving solution of Eq. (1) can be derived starting with the implicit Euler discretization of Eq. (45):
C˜pjΩΔTj,n+1ΩΔtn−LfusρiceΔθicej,n+1ΩΔtn=Fnetj,n+1Ω,
where ΔTj,n+1Ω=[TjΩ⁡(tn+1)−TjΩ⁡(tn)] and Δθicej,n+1Ω=[θicejΩ⁡(tn+1)−θicejΩ⁡(tn)] are respectively the change in temperature (K) and volumetric ice content (—) over the time interval [tn, tn+1]. In Eq. (A3), FnetjΩ is computed using Eq. (42).
Given the energy conservation condition C˜pjΔTj,n+1Ω=ΔHj,n+1Ω and making use of Eq. (8), the time differences for the heat and phase change terms on the LHS of Eq. (A3) can be expanded as
ΔHTj,n+1Ω⁡(ℓ+1)=ΔHTj,n+1Ω⁡(ℓ)+⁡(dHTj,n+1ΩdTj,n+1Ω)⁡(ℓ)δTj,n+1Ω⁡(ℓ+1),
Δθicej,n+1Ω⁡(ℓ+1)=Δθicej,n+1Ω⁡(ℓ)+⁡(dθicej,n+1ΩdTj,n+1Ω)⁡(ℓ)δTj,n+1Ω⁡(ℓ+1),
where HTΩ is defined in Eqs. (3), (4), and (9) and
ΔHTj,n+1Ω⁡(ℓ)=HTj,n+1Ω⁡(ℓ)−HTj,nΩ,
Δθicej,n+1Ω⁡(ℓ)=θicej,n+1Ω⁡(ℓ)−θicej,nΩ,
δTj,n+1Ω⁡(ℓ+1)=Tj,n+1Ω⁡(ℓ+1)−Tj,n+1Ω⁡(ℓ).
In Eqs. (A4) and (A5), dHTΩ/dTΩ and dθiceΩ/dTΩ are computed analytically using the state variables at iteration ℓ. We recall from Eq. (11) that dHTΩ/dTΩ=CpΩ.
Given the definition of CaΩ in Eq. (13) and noting that CaΩ=dHΩ/dTΩ, an implicit finite difference discretization of the heat equation in Eqs. (1), (8), (10), and (12) can be defined as
[Caj,n+1Ω⁡(ℓ)−Δtn⁡(dFnetj,n+1ΩdTj,n+1Ω)⁡(ℓ)]δTj,n+1Ω⁡(ℓ+1)=ΔtnFnetj,n+1Ω⁡(ℓ)−ΔHTj,n+1Ω⁡(ℓ)+LfusρiceΔθicej,n+1Ω⁡(ℓ).
In Eq. (A6), the enthalpy HTΩ and volumetric ice content θiceΩ are both diagnostic variables. See appendix B for the methods used to calculate enthalpy, Eqs. (7) and (62) for methods to calculate θicesnow⁡(Tsnow,Θmsnow), and Eqs. (67)–(73) to calculate θicesoil⁡(Tsoil,ψ0).

b. Implicit Euler solutions for soil hydrology

A mass-conserving solution for soil hydrology is the mixed formulation of Richards’ equation described by Celia et al. (1990). Starting with the implicit Euler discretization of Eqs. (15), (40), and (49),
SsΘmj,n+1soilθsjΔψ0j,n+1Δtn+C˜ψj,n+1soilΔψ0j,n+1Δtn=qnetj,n+1soil+Msinkj,n+1soil,j=1,2,…,Jsoil,
where C˜ψsoil=dΘmsoil/dψ0 and qnetjsoil is computed in Eq. (47). In Eq. (A7), Δψ0j,n+1=[ψ0j⁡(tn+1)−ψ0j⁡(tn)] can be expanded as
Δψ0j,n+1⁡(ℓ+1)=ΔΘmj,n+1soil⁡(ℓ)+⁡(dΘmj,n+1soildψ0j,n+1)⁡(ℓ)δψ0j,n+1⁡(ℓ+1),
with
ΔΘmj,n+1soil⁡(ℓ)=Θmj,n+1soil⁡(ℓ)−Θmj,nsoil,
δψ0j,n+1⁡(ℓ+1)=ψ0j,n+1⁡(ℓ+1)−ψ0j,n+1⁡(ℓ).
Combining Eqs. (15), (40), and (A7),
[SsΘmj,n+1soil⁡(ℓ)θsj+⁡(dΘmj,n+1soildψ0j,n+1)⁡(ℓ)−Δtn⁡(dqnetj,n+1soildψ0j,n+1)⁡(ℓ)]δψ0j,n+1⁡(ℓ+1)=Δtn⁡(qnetj,n+1soil+Ssinkj,n+1soil)⁡(ℓ)−SsΘmj,n+1soil⁡(ℓ)θsjΔψ0j,n+1⁡(ℓ)−ΔΘmj,n+1soil⁡(ℓ).
In Eq. (A9), Θmsoil is a diagnostic variable computed from ψ0.

APPENDIX B

Enthalpy Calculations

a. The snow domain

The mixture enthalpy for snow, Hsnow, is given from Eqs. (3) and (4):
Hsnow=HT, liqsnow+HT, icesnow+HT, airsnow−Lfusρice[θicesnow⁡(Tsnow)−θicesnow⁡(Tref)].
Given Eq. (4) and Tref = Tfrz,
HT, liqsnow+HT, icesnow=cliq∫TfrzTsnowρliqθliqsnow dT+cice∫TfrzTsnowρiceθicesnow dT
=ρliq∫TfrzTsnow[cice⁡(Θmsnow−θicesnow)+cliqθliqsnow] dT,
and making use of Eq. (62) to compute θliqsnow,
HT, liqsnow+HT, icesnow=ρliqΘmsnow∫TfrzTsnowcice+cliq−cice1+[ϖ⁡(Tfrz−T)]2 dT
=ρliqΘmsnow[atan⁡(ϖδT)cliq−ciceϖ+ciceδT],
where δT = Tsnow − Tfrz. We note that these equations are only developed for T < Tfrz because snow cannot exist at temperatures above freezing—i.e., in Eq. (62), all water is liquid when T > Tfrz.

b. The soil domain

The mixture enthalpy for soil, Hsoil, is also given from Eqs. (3) and (4). From Eq. (3),
Hsoil=HT, soilsoil+HT, liqsoil+HT, icesoil+HT, airsoil−Lfusρice[θicesoil⁡(Tsoil)−θicesoil⁡(Tref)],
where enthalpies of the constituents are computed using Eqs. (4) and (71)–(73). Starting by assuming saturated conditions, i.e., ψ0 ≥ 0, and setting Tref = Tfrz, Eq. (4) can be written as
H⁡(Tsoil)=∫TfrzTsoil[cliqρliqθliqsoil⁡(T)+ciceρiceθicesoil⁡(T)] dT,
noting that under saturated conditions
θliqsoil⁡(Tsoil)=S*⁡(ψc)=S*⁡(LfusgTsoil−TfrzTfrz),
θicesoil⁡(Tsoil)=θs−θliqsoil⁡(Tsoil).
The enthalpy for variably saturated soil can then be computed by making use of Eqs. (71)–(73),
HT, liqsoil+HT, icesoil={[H⁡(Tsoil)−H⁡(T*)]+cliqρliqS*⁡(ψ0)⁡(Tsoil−T*),Tsoil<T*,cliqρliqS*⁡(ψ0)⁡(Tsoil−T*),Tsoil≥T*,
noting from Eq. (70) that T* is the temperature where all water is unfrozen. In the test cases examined here, H⁡(⋅) is computed by fitting a cubic spline to a lookup table of Tsoil and Hsoil values.

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Conceptual depiction of the dominant thermodynamical and hydrological processes solved in land models. The red and blue font define thermodynamic and hydrological processes respectively; the flux qiceΩ in Eq. (2) is only defined in the vegetation and snow domains, and includes snowfall qsf, throughfall of snow through the vegetation canopy qtf,snowveg, and unloading of snow from the vegetation canopy qunloadveg. The model forcing variables are snowfall qsf (kg m−2 s−1), rainfall qrf (kg m−2 s−1), downwelling shortwave radiation Qswdatm (J m−2 s−1), downwelling longwave radiation Qlwdatm (J m−2 s−1), temperature, Tatm (K), wind speed uatm (m s−1), specific humidity hatm (kg kg−1), and atmospheric pressure patm (Pa). All other variables are defined in the text.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Solution of the test problem from Celia et al. (1990), showing pressure profiles at various times during the model simulation.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Solution of the test problem from Miller et al. (1998), showing pressure profiles for sand at t = 0.18 days, loam at t = 2.25 days, and clay loam at t = 1 day.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Solution of the three test problems from Vanderborght et al. (2005), showing the vertical profiles of pressure head at steady state. The solid red lines are the model simulations, and the dashed black lines are the analytical solutions.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Simulated (left) water table profiles and (right) hillslope outflow for the test problems from Wigmosta and Lettenmaier (1999). The figure shows both the linear experiment (nsf = 1, blue) and the nonlinear experiment (nsf = 3, red).

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Contour plots of the volumetric fraction of liquid water for the problem of Colbeck (1976) from (left) the analytical solution and (right) the numerical simulations for (top) the ripe snow test case, (middle) the refrozen snow test case, and (bottom) the fresh snow test case.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    The outflow hydrograph for the problem of Colbeck (1976) from numerical simulations (solid lines) and analytical solutions (dashed lines) for (top) the ripe snow test case, (middle) the refrozen snow test case, and (bottom) the fresh snow test case.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Simulated and measured values of the total volumetric water content for the Mizoguchi experiment for (left) 12 h, (center) 24 h, and (right) 50 h after the start of the experiment.

Fig. 1.

Conceptual depiction of the dominant thermodynamical and hydrological processes solved in land models. The red and blue font define thermodynamic and hydrological processes respectively; the flux qiceΩ in Eq. (2) is only defined in the vegetation and snow domains, and includes snowfall qsf, throughfall of snow through the vegetation canopy qtf,snowveg, and unloading of snow from the vegetation canopy qunloadveg. The model forcing variables are snowfall qsf (kg m−2 s−1), rainfall qrf (kg m−2 s−1), downwelling shortwave radiation Qswdatm (J m−2 s−1), downwelling longwave radiation Qlwdatm (J m−2 s−1), temperature, Tatm (K), wind speed uatm (m s−1), specific humidity hatm (kg kg−1), and atmospheric pressure patm (Pa). All other variables are defined in the text.
Fig. 2.

Solution of the test problem from Celia et al. (1990), showing pressure profiles at various times during the model simulation.
Fig. 3.

Solution of the test problem from Miller et al. (1998), showing pressure profiles for sand at t = 0.18 days, loam at t = 2.25 days, and clay loam at t = 1 day.
Fig. 4.

Solution of the three test problems from Vanderborght et al. (2005), showing the vertical profiles of pressure head at steady state. The solid red lines are the model simulations, and the dashed black lines are the analytical solutions.
Fig. 5.

Simulated (left) water table profiles and (right) hillslope outflow for the test problems from Wigmosta and Lettenmaier (1999). The figure shows both the linear experiment (nsf = 1, blue) and the nonlinear experiment (nsf = 3, red).
Fig. 6.

Contour plots of the volumetric fraction of liquid water for the problem of Colbeck (1976) from (left) the analytical solution and (right) the numerical simulations for (top) the ripe snow test case, (middle) the refrozen snow test case, and (bottom) the fresh snow test case.
Fig. 7.

The outflow hydrograph for the problem of Colbeck (1976) from numerical simulations (solid lines) and analytical solutions (dashed lines) for (top) the ripe snow test case, (middle) the refrozen snow test case, and (bottom) the fresh snow test case.
Fig. 8.

Simulated and measured values of the total volumetric water content for the Mizoguchi experiment for (left) 12 h, (center) 24 h, and (right) 50 h after the start of the experiment.
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