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wiley/fff2407c_b944_49e2_9739_985c13298e70.md | # Water Resources Research
Research Article
10.1029/2024 WR037068
A Fully Coupled Numerical Solution of Water, Vapor, Heat, and Water Stable Isotope Transport in Soil
[PERSON]
1 College of Hydraulic and Civil Engineering, Luding University, Yantai, Shandong Province, China, 1 Department of Soil Science, College of Agriculture and Bioresources, University of Saskatchewan, Saskatoon, Saskatchewan, Canada
[PERSON]
1 College of Hydraulic and Civil Engineering, Luding University, Yantai, Shandong Province, China, 1 Department of Soil Science, College of Agriculture and Bioresources, University of Saskatchewan, Saskatoon, Saskatchewan, Canada
[PERSON]
1 College of Hydraulic and Civil Engineering, Luding University, Yantai, Shandong Province, China, 1 Department of Soil Science, College of Agriculture and Bioresources, University of Saskatchewan, Saskatoon, Saskatchewan, Canada
[PERSON]
1 College of Hydraulic and Civil Engineering, Luding University, Yantai, Shandong Province, China, 1 Department of Soil Science, College of Agriculture and Bioresources, University of Saskatchewan, Saskatoon, Saskatchewan, Canada
###### Abstract
Modeling water stable isotope transport in soil is crucial to sharpen our understanding of water cycles in terrestrial ecosystems. Although several models for soil water isotope transport have been developed, many rely on a semi-coupled numerical approach, solving isotope transport only after obtaining solutions from water and heat transport equations. However, this approach may increase instability and errors of model. Here, we developed an algorithm that solves one-dimensional water, heat, and isotope transport equations with a fully coupled method (MOIST). Our results showed that MOIST is more stable under various spatial and temporal discretization than semi-coupled method and has good agreement with semi-analytical solutions of isotope transport. We also validated MOIST with long-term measurements from a lysimeter study under three scenarios with soil hydraulic parameters calibrated by HYDRUS-1D in the first two scenarios and by MOIST in the last scenario. In scenario 1, MOIST showed an overall _NSE_, _KGE_, and _MAE_ of simulated 8\({}^{18}\)O of 0.47, 0.58, and 0.92%, respectively, compared to the 0.31, 0.60, and 1.00% from HYDRUS-1D; In scenario 2, these indices of MOIST were 0.33, 0.52, and 1.04%, respectively, compared to the 0.19, 0.58, and 1.15% from HYDRUS-1D; In scenario 3, calibrated MOIST exhibited the highest _NSE_ (0.48) and _KGE_ (0.76), the smallest _MAE_ (0.90) among all scenarios. These findings indicate MOIST has better performance in simulating water flow and isotope transport in simplified ecosystems than HYDRUS-1D, suggesting the great potential of MOIST in furthering our understandings of ecohydrological processes in terrestrial ecosystems.
1
[PERSON], [PERSON], H.
theory to model isotopic signals in soil water. This simplification facilitates computations, making the models easily applicable at large scales, but may overlook the influence of physical mechanisms on isotope transport, such as phase changes and vapor diffusion.
Physical processes associated with isotope transport within soil receive greater attention at smaller scales, such as one-dimensional isotope movement. Based on [PERSON] et al. (1996), [PERSON] et al. (2005) developed the \"SiSPAT-Isotope\" (Simple Soil-Plant-Atmosphere Transfer) model, which incorporated the resistance to isotope transport between the soil surface and atmosphere ([PERSON], et al., 2009, 2009b). Subsequently, [PERSON] and [PERSON] (2010) developed \"Soil-Litter-Iso,\" which is also a one-dimensional model for the transport of heat, water, and stable isotopes in soil containing surface litter and actively transpiering vegetative cover. This model extended the linearization method of [PERSON] (2003) to vapor transport and emphasized the importance of isotope vapor transport, which is later supported by [PERSON] et al. (2018) using a SWIS model ([PERSON] et al., 2014). Compared to SiSPAT-Isotope, Soil-Litter-Iso is more efficient for thicker soil layers and larger time steps because of the implementation of the special linearization scheme in the model framework. However, Soil-Litter-Iso does not consider liquid and vapor heat capacity variation and the changes in vapor volume during heat transport, which may substantially bias the estimation of heat flux transport within the soil ([PERSON] et al., 2006), thereby resulting in biased isotope transport fluxes.
Another approach capitalizes on the capability of the well-known model, HYDRUS-1D, for simulating water flow and chemical transport. [PERSON] et al. (2012) simulated oxygen-18 movement in soil by modifying a solute transport module of HYDRUS-1D. However, [PERSON] et al. (2012) originally neglected fractionation and thus, their model was only applicable in situations where the slope of the isotopic evaporation line from soil water was close to the local meteoric water line ([PERSON] et al., 2012). The latest version of HYDRUS-1D incorporated isotopic fractionation by integrating with the SiSPAT-Isotope model ([PERSON] et al., 2021).
A common problem of these models related to isotope simulation is that each of these modeling approaches considers water and isotope transport separately, where the isotope transport is solved after solving the water and heat transport at each time step (semi-coupled method). This is reasonable because isotope transport does not affect soil water and heat transport. However, numerical errors and instabilities from soil, water and heat transport equations can be transferred into isotope transport equations at each time step (step 2 and three in Figure 1). Thus,
Figure 1: Demonstration of semi-coupled and fully coupled numerical solution method for water, heat, and isotope transport in soil at each time step. For the semi-coupled method, step 1 is to obtain the water and heat solutions at time step \(t+\Delta t\) from \(t\). Then, the soil water contents and temperature from \(t\) and \(t+\Delta t\) are used as inputs in the isotope transport equation (step 2). Finally, the isotope transport is solved in step 3. By contrast, the fully coupled method solves the three transport equations simultaneously.
either a tiny spatial discretization at the soil surface (\(10^{-6}\) m, [PERSON] et al., 2005) for extremely small mass balance errors (in the magnitude of \(10^{-16}\), [PERSON] et al., 2021), or complicated discretization schemes ([PERSON] & [PERSON], 2010) were required to protect the isotope solutions from numerical oscillation. We hypothesize that steps 2 and 3 from the semi-coupled method can be incorporated into step 1 in a fully coupled method (Figure 1) and thus, the fully coupled method reduces the impact of mass balance errors from soil water and heat transport equations on the isotope transport equation (step 2, Figure 1) compared to the semi-coupled method. Consequently, the fully coupled method should be more accurate and less affected by numerical instability than semi-coupled methods.
Therefore, the objectives of this study are: (a) to develop a one-dimensional model which can solve fully coupled soil water, heat, vapor, and isotope transport simultaneously and (b) to validate MOIST through analytical simulations with specific boundary conditions and field measurements. MOIST was written in MATLAB programming language and it is expected to be more efficient for simulating isotope transport within soil under various spatial and temporal scales than existing models by solving fully coupled soil water, heat, vapor, and isotope transport equations.
## 2 Materials and Methods
### Model Description
#### 2.1.1 Soil Water and Heat Transport
One-dimensional water and heat transport within soil can be described by mass and energy conservation equations with a downward-positive coordinate system ([PERSON], 2017; [PERSON] et al., 2006):
\[\frac{\partial(\theta+\theta_{v})}{\partial t}=-\frac{\partial(q_{t}+q_{v})}{ \partial z}-S_{p} \tag{1}\]
\[\frac{\partial C_{v\alpha\beta}T}{\partial t}+\lambda_{k}\frac{\partial\theta _{v}}{\partial t}=\frac{\partial\left(K_{H}\frac{\partial T}{\partial z} \right)}{\partial z}-C_{v}\frac{\partial(q_{t}T)}{\partial z}-C_{v\alpha} \frac{\partial(Tq_{v})}{\partial z}-\rho_{k}\frac{\partial q_{v}}{\partial z }-C_{v}S_{p}T \tag{2}\]
where \(\theta\) and \(\theta_{v}\) are the volumetric soil water (\(\mathrm{m^{3}\ m^{-3}}\)) and vapor content (\(\mathrm{m^{3}\ m^{-3}}\)); \(t\) is the time (s); \(q_{t}\) is the liquid flux (\(\mathrm{m\ s^{-1}}\)) and \(q_{v}\) is the vapor flux (\(\mathrm{m\ s^{-1}}\)); \(z\) is the spatial distance (m); \(S_{p}\) is the sink term (\(\mathrm{s^{-1}}\)), which is zero when there is no sink during simulation (e.g., root water uptake); \(C_{v\alpha\beta}\) is the soil volumetric heat capacity (J \(\mathrm{m^{-3}\ K^{-1}}\)); T is the soil temperature (K); \(\lambda_{k}\) is the latent heat of vapourization (J \(\mathrm{kg^{-1}}\)); \(K_{H}\) is the soil thermal conductivity (\(\mathrm{W\ m^{-1}\ K^{-1}}\)); \(C_{v\alpha}\) and \(C_{v\alpha}\) are the heat capacities of liquid water (J \(\mathrm{m^{-3}\ K^{-1}}\)) and vapor (\(\mathrm{J\ m^{-3}\ K^{-1}}\)); and \(\rho\) is the water density (\(\mathrm{kg\ m^{-3}}\)). Detailed explanations of these variables can be found in the Supporting Information S1.
The liquid flux, \(q_{p}\) is calculated by [PERSON]'s law (positive downwards) ([PERSON], 2010):
\[q_{t}=-K(h)\,\frac{dh}{dz}+K(h) \tag{3}\]
where \(K(h)\) is the unsaturated hydraulic conductivity (\(\mathrm{m\ s^{-1}}\)) as a function of \(h\). The vapor flux, \(q_{v}\), is calculated by the [PERSON]'s law ([PERSON] et al., 2009):
\[q_{v}=-D_{v\alpha}\frac{\partial c_{v}}{\partial z} \tag{4}\]
where \(D_{v\alpha}\) is the diffusivity of vapor within soil (\(\mathrm{m^{2}\ s^{-1}}\)). Note that \(c_{v}\) is the function of \(h\) and \(T\). Thus, the thermal vapor conduction is implicitly considered. Detailed derivations can be found in the Supporting Information S1.
Note that Equation 1 is the mixed form of [PERSON] equation. However, the current study uses the head-based Richards' equation:\[(1-c_{v,sat}R_{H})\frac{\partial\theta}{\partial h}\frac{\partial h}{\partial t}+( \theta_{sat}-\theta)\frac{\partial(c_{v,sat}R_{H})}{\partial t}=-\frac{\partial (q_{t}+q_{t})}{\partial z}-S_{p} \tag{5}\]
where \(c_{v,sat}\) is the saturated vapor concentration within soil air (m\({}^{3}_{\text{liquid water}}\) m\({}^{-3}_{\text{air}}\)); and \(R_{H}\) is the relative humidity in soil. Equations for \(c_{v}\), \(D_{\text{ex}}\), \(c_{v,sat}\), and \(R_{H}\) can be found in the Supporting Information S1.
Both the head-based and the mixed forms of [PERSON] equation are commonly used for modeling water flow in porous media. The current study utilizes the head-based form of [PERSON] equation because the head is continuous across the soil interface under typical natural conditions, especially for the layered soils ([PERSON] et al., 2019). However, \(\frac{\partial\theta}{\partial h}\frac{\partial h}{\partial t}\) may not be numerically equal to \(\frac{\partial\theta}{\partial t}\)([PERSON] et al., 1990; [PERSON] et al., 2021) because \(\frac{\partial\theta}{\partial t}\) could introduce mass balance errors during linearization. Nevertheless, the mass balance of head-based Richards' equation can be significantly improved by a second-order approximation to the time derivative ([PERSON] et al., 1990) and effectively controlled by adaptive time-stepping schemes ([PERSON] et al., 2023). MATLAB provides well-tested ordinary differential equations solvers, which implement adaptive time stepping schemes. As such, these solvers effectively meet the criteria.
#### 2.1.2 Soil Water Isotope Transport
The abundance of oxygen and hydrogen stable isotopes is conventionally expressed as \(\delta\) values, in units of per mil (%). However, for convenience, the abundances in MOIST were presented as concentrations (kg m\({}^{-3}\)), \(c_{i}\), where the relationship between isotopic ratio (\(R_{i}\)) and concentration (\(c_{i}\)) can be expressed as ([PERSON] et al., 2005; [PERSON] et al., 1996):
\[c_{i}=\frac{M_{i}}{M_{w}}R_{i}\rho \tag{6}\]
where \(M_{i}\) and \(R_{i}\) are the molar mass (kg mol\({}^{-1}\)) and the isotopic ratio of isotopic species \(i\), respectively, for a given isotope species; \(R_{i}\) can be calculated by:
\[R_{i}=\delta\frac{R_{ref}}{1000}+R_{ref} \tag{7}\]
where \(R_{ref}\) is the reference value for the isotopic ratio, which is 155.76 \(\times\) 10\({}^{-6}\) and 2,005.2 \(\times\) 10\({}^{-6}\) for HDO and H\({}_{2}\)\({}^{18}\)O, respectively ([PERSON], 1978).
The isotope mass conservation equation for both liquid and vapor phases is:
\[\frac{\partial(c_{i}\theta+c_{i}^{\prime}(\theta_{sat}-\theta))}{\partial t} =-\frac{\partial q_{t}}{\partial z}-c_{i}^{\prime}S_{p} \tag{8}\]
where \(c_{i}^{\prime}\) and \(c_{i}^{\prime}\) are the concentrations of isotope species \(i\) in liquid (kg m\({}^{-3}\)) and vapor phases (kg m\({}^{-3}\)). Assuming an instantaneous equilibrium between liquid and vapor phases ([PERSON] et al., 2017), a relationship between liquid and vapor isotopic concentration can be expressed as ([PERSON] et al., 2005; [PERSON] and [PERSON], 2010):
\[c_{i}^{\prime}=c_{v}\:a_{i}^{*}c_{i}^{\prime} \tag{9}\]
where \(a_{i}^{*}\) is the equilibrium fractionation coefficient of isotope species \(i\), which can be written as ([PERSON] et al., 2005):
\[a_{i}^{*}=e^{\left(\frac{c_{i}^{\prime}}{(\theta+\lambda_{i})^{2}}\frac{c_{i}^ {\prime}}{\tau^{2}\lambda_{i}-c_{i}^{\prime}}\right)} \tag{10}\]where \(a^{*}\), \(b^{*}\), \(c^{*}\) have values of 24,844, \(-\)76.248, and 0.052612 for \({}^{2}\)H, and 1,137, \(-\)0.4156, and \(-\)0.0020667 for \({}^{18}\)O.
The total isotopic flux, \(q_{i}\) (positive downwards), consists of liquid isotopic flux, \(q_{i}^{I}\), and vapor isotopic flux, \(q_{i}^{I}\):
\[q_{i}^{I}=c_{i}^{I}q_{i}-D_{i,a}^{I}\frac{\partial c_{i}^{I}}{\partial z} \tag{11}\]
\[q_{i}^{r}=-D_{i,a}^{r}\frac{\partial c_{i}^{r}}{\partial z} \tag{12}\]
where \(D_{i,a}^{I}\) and \(D_{i,a}^{r}\) are the liquid (m\({}^{2}\) s\({}^{-1}\)) and vapor (m\({}^{2}\) s\({}^{-1}\)) diffusivity in soil for isotope species \(i\), respectively. Detailed equations of \(D_{i,a}^{I}\) and \(D_{i,a}^{r}\) can be found in the Supporting Information S1.
Combing Equations 4 and 9 (Equation S3-10 in Supporting Information S1), and (Equation S3-11 in Supporting Information S1) with Equation 12 leads to ([PERSON] & [PERSON], 2010):
\[q_{i}^{r}=-(\alpha_{diff})^{D}\ a_{i}^{*}c_{i}^{I}q_{i}-D_{i,a}^{r}c_{i}\left( a_{i}^{r}\frac{\partial c_{i}^{I}}{\partial z}+c_{i}^{I}\frac{\partial a_{i}^{r}}{ \partial z}\right) \tag{13}\]
Equation 13 illustrates that the isotope transport in vapor phase consists of convection and diffusion terms, where the latter can be expressed as a concentration gradient in the liquid phase (first term within the brackets), and an equilibrium fractionation coefficient gradient (second term within the brackets). Detailed derivations can be found in Supporting Information S1.
#### 2.1.3 Root Water Uptake
Root water uptake is modeled as a sink term in Equations 1 and 8 ([PERSON] et al., 2001):
\[S_{p}=\frac{\Omega F_{i}P_{i}}{\Delta z} \tag{14}\]
where \(\Omega\) is the efficiency coefficient (between 0 and 1), which can be obtained from a prescribed stress function; \(\Delta z\) is the thickness of the considered layer (\(m\)) and \(P_{i}\) is the potential transpiration rate (m s\({}^{-1}\)), which can be obtained from the Penman-Monteith equation ([PERSON] et al., 1999) and the leaf area index. The current version of MOIST does not account for isotope fractionation during root water uptake. Root water uptake in MOIST is described by the [PERSON] model ([PERSON] et al., 1978). Essentially, potential evapotranspiration (\(pET\)) is calculated using the Penman-Monteith equation ([PERSON] et al., 1999). Following this, \(pET\) partitioned into potential evaporation and potential transpiration based on leaf area index ([PERSON] et al., 2013). The potential transpiration is then distributed across individual soil layers by multiplying root proportion (\(F_{i}\)) of each layer. Ultimately, the actual root water uptake flux from each layer is determined by multiplying the allocated potential root water uptake flux by the stress factor (Equations 14 and 15, Figure 2).
As demonstrated by Figure 2, \(\Omega\) is a function of pressure head, \(h\), and can be modeled with four critical pressure heads (\(h_{I}\), \(h_{2}\), \(h_{y}\), \(h_{\Delta}\)) that may vary among different plant species:
\[\Omega=\begin{cases}0,h<h_{4}\ \text{or}\ h>h_{1}\\ \frac{h-h_{4}}{h_{3}-h_{4}},h_{4}\leq h\leq h_{3}\\ \frac{h-h_{1}}{h_{2}-h_{1}},h_{2}\leq h\leq h_{1}\\ 1,h_{3}\leq h\leq h_{2}\end{cases} \tag{15}\]
\(F_{i}\) is the fraction of root length in the depth interval of \(z\) to \(z+\Delta z\), defined as:\[F_{i}=\frac{\int_{z}^{z+\Delta z}R(x)\,dx}{\int_{0}^{z}R(x)\,dx} \tag{16}\]
where \(z\) is the depth (\(m\)) of the upper boundary of layer \(i\); \(z+\Delta\)\(z\) is the depth of the lower boundary (m); \(Z_{i}\) is the maximum depth of the soil column (m); and \(R(x)\) is a predefined root length density distribution function which can be obtained by fitting to fine root length density measurements. For convenience, here we define \(R(z)\) as ([PERSON] et al., 1988):
\[R(z)=RD_{0}\left[1-\frac{1}{1+e^{-\Delta z}}+\frac{e^{-\Delta z}}{2}\right] \tag{17}\]
where \(RD_{0}\) is root length density at \(z=0\) and it is canceled out when calculating \(F_{i}\); \(b\) is an empirical root distribution parameter (m\({}^{-1}\)), and the equation was given by [PERSON] et al. (2001).
Note that \(R(z)\) can be static or treated as dynamic with \(Z_{r}\) being described as the classical Verhulst-Pearl logistic growth function ([PERSON], 2001):
\[Z_{r}(t)=L_{m}\frac{L_{0}}{L_{0}+\left(L_{m}-L_{0}\right)e^{-\alpha t}} \tag{18}\]
where \(L_{0}\) is the initial value of the rooting depth at the beginning of the growing season (m); \(L_{m}\) is the maximum rooting depth during the growing season (m); \(r\) is the growth rate (day\({}^{-1}\)) ([PERSON] et al., 2013).
#### 2.1.4 Boundary Conditions
#### 2.1.4.1 Boundary Conditions for Soil Water Flow
According to the Soil-Litter-Iso model ([PERSON], 2010), the coupled energy and water mass conservation equations were solved at the soil-air interface:
\[\frac{1}{r_{bw}}[c_{av}(T_{s},R_{hi})-c_{un}]=-D_{v}\frac{\partial c_{v,att}R _{hi}}{\partial z}-K\frac{\partial h}{\partial z}+K \tag{19}\]
\[R_{net}= \frac{C_{p}}{r_{hh}}(T_{s}-T_{a})+\frac{\rho\lambda_{E}}{r_{bw}}[c_{ av}(T_{s},R_{hi})-c_{un}]-K_{H}\frac{\partial T}{\partial z} \tag{20}\]
Figure 2: Illustration of a stress function, based on [PERSON] et al. (1978).
where \(r_{bw}\) and \(r_{hb}\) are the resistance to vapor (s m\({}^{-1}\)) and heat (s m\({}^{-1}\)) transfer at the soil-air interface, respectively; \(c_{vis}\) and \(c_{uu}\) are the vapor concentrations (m\({}^{3}\)\({}_{\text{water}}\) m\({}^{-3}\)\({}_{\text{air}}\)) at the soil surface and in the atmosphere, respectively; \(R_{ant}\) is the net radiation (W m\({}^{-2}\) K\({}^{-1}\)); \(C_{p}\) is the volumetric heat capacity of air at constant pressure (J m\({}^{-3}\) K\({}^{-1}\)); and \(T_{a}\) is the air temperature (K).
Equations 19 and 20 were used only in the top half of the soil surface layer for obtaining the unknown soil surface temperature (\(T_{s}\)) and surface relative humidity (\(R_{th}\)) by using a Jacobian iteration method at the beginning of each time step. \(T_{s}\) and \(R_{th}\) are required to calculate surface evaporation flux, surface liquid and vapor fluxes, surface sensible heat flux, surface latent heat flux and heat flux into the soil ([PERSON] & [PERSON], 2010).
The lower boundary condition for soil water flow can be set to free drainage (zero gradient of soil water pressure head at the lower boundary), seepage surface (constant soil water pressure head at the lower boundary), or zero water flux. The lower boundary condition for heat transport can be set to zero temperature gradient.
#### 2.1.4.2 Boundary Conditions for Isotope Transport
Isotopic evaporation fluxes through the soil surface to the atmosphere were calculated using the [PERSON] (1965) model:
\[E_{i}=\frac{\alpha_{k}}{r_{bw}}[c_{v}c_{i,u}^{l}\sigma_{i}^{*}(T_{s})-c_{i}^{ uu}] \tag{21}\]
where \(\alpha_{k}\) is the kinetic fractionation coefficient; \(c_{i}^{uu}\) is the isotopic concentration of the atmospheric water vapor (kg m\({}^{-3}\)); \(c_{i,u}^{l}\) is the isotopic concentration at the soil surface (kg m\({}^{-3}\)); and \(E_{i}\) is the surface soil isotopic flux (m s\({}^{-1}\)) and can be specified as:
\[E_{i}=-p_{i}^{c}\frac{\partial(c_{v}c_{i}^{l}\alpha_{i}^{*}T)}{\partial z}-D_{ i}^{c}\frac{\partial c_{v}^{l}}{\partial z}+q_{h}c_{v,u}^{l} \tag{22}\]
where \(q_{h}\) is the liquid flux at the soil surface (m s\({}^{-1}\)).
The kinetic fractionation coefficient, \(\alpha_{k}\), can be written as ([PERSON], 2010; [PERSON], 1996):
\[\alpha_{k}=(\alpha_{diff})^{nk} \tag{23}\]
with
\[nk=\frac{(\theta_{s}-\theta_{res})\,n_{a}+(\theta_{sat}-\theta_{s})\,n_{s}}{ \theta_{sat}-\theta_{res}} \tag{24}\]
where \(n_{a}\) and \(n_{s}\) are coefficients with values of 0.5 and 1.0, respectively, and \(\theta_{s}\) is the soil surface water content (m\({}^{3}\) m\({}^{-5}\)). There are several equations available to calculate \(\alpha_{k}\). In the current study, a sensitivity analysis was conducted on these equations (Supporting Information S1), which revealed that MOIST was not sensitive to available formulations. Thus, we chose Equation 21 for its relatively simple formulation, and it considers the influence of surface soil water content on isotope fractionation.
Combining Equations 21 and 22 leads to the final expression for surface soil isotopic concentration, \(c_{i,u}^{l}\) (kg m\({}^{-3}\)):
\[\frac{\alpha_{k}}{r_{bw}}[c_{i,u}^{l}\alpha_{k}(T_{s})-c_{i}^{uu}]=-D_{i}^{c }\frac{\partial(c_{v}c_{i}^{l}\alpha_{i}^{*}T)}{\partial z}-D_{i}^{l}\frac{ \partial c_{v}^{l}}{\partial z}+q_{h}c_{v,u}^{l} \tag{25}\]
Using this relationship, \(c_{i,u}^{l}\) can be readily solved as it is the only unknown quantity.
It is important to note that energy, water, and isotope mass conservation equations (Equations 19, 20 and 25) at the air-soil interface are identical to that of [PERSON] and [PERSON] (2010), except that the influence of litter layer is not considered. This means that the fluxes from the center of the top-soil layer to the soil-air interface are balanced by the fluxes from the soil-air interface to the atmosphere (Equations 19, 20 and 25). The lower boundary conditions for isotope transport are determined by soil water flux or can be customized. Normally, the lower boundary condition for isotope transport is zero gradient or zero flux.
#### 2.1.5 Numerical Implementations
MOIST adopts a cell-centered spatial discretization scheme along with the finite volume method (FVM). While the finite element method (FEM) is another commonly used approach for solving partial differential equations, it is particularly advantageous for complex problems such as two- and three-dimensional water transport in intricate geometries with complex boundary conditions. However, this study focuses on one-dimensional soil water movement, where the simplicity of the scenario reduces performance differences between FVM and FEM, making the choice of method less impactful on the performance of the fully coupled scheme. Furthermore, compared to FEM, FVM is inherently suited to conservation law problems, as it conserves fluxes both locally and globally, and it also offers simpler mathematical implementation. Therefore, FVM is used in MOIST.
To solve Equations 1, 2 and 8 simultaneously, we expand them into:
\[\left[\left(1-c_{s,sat}R_{H}\right)\frac{\partial\theta}{\partial t }+\left(\theta_{sat}-\theta\right)c_{s,sat}\frac{\partial R_{H}}{\partial t} \right]\frac{\partial t}{\partial
\[B=\ C_{s,sat}+\rho\lambda_{E}\left[\left(\theta_{sat}-\theta\right)c_{s,sat}\frac{ \partial h_{r}}{\partial T}+\left(\theta_{sat}-\theta\right)R_{H}\frac{\partial c _{s,sat}}{\partial T}\right] \tag{33}\]
\[\mathrm{C}=\left(1-c_{s,sat}R_{H}\right)\frac{\partial\theta}{\partial h}+ \left(\theta_{sat}-\theta\right)c_{s,sat}\frac{\partial R_{H}}{\partial h} \tag{34}\]
\[\mathrm{D}=\left(\theta_{sat}-\theta\right)c_{s,sat}\frac{\partial R_{H}}{ \partial T}+\left(\theta_{sat}-\theta\right)R_{H}\frac{\partial c_{s,sat}}{ \partial T} \tag{35}\]
\[\mathrm{E}=\theta+\alpha_{c}^{*}c_{s,sat}R_{H}(\theta_{sat}-\theta) \tag{36}\]
\[\mathrm{F}=-\frac{\partial q_{i}}{\partial z}-c_{i}^{*}S_{p}-c_{i}^{ \dagger}\frac{\partial\theta}{\partial h}\frac{\partial h}{\partial h}-c_{i}^ {\dagger}\theta_{sat}\left[c_{s,sat}R_{H}\frac{\partial\sigma}{\partial T} \frac{\partial T}{\partial t}+\alpha_{i}^{*}R_{H}\frac{\partial\sigma_{s,sat} }{\partial T}\frac{\partial T}{\partial t}+\alpha_{i}^{*}c_{s,sat}\left(\frac {\partial R_{H}}{\partial h}\frac{\partial h}{\partial t}+\frac{\partial R_{H}} {\partial T}\frac{\partial T}{\partial t}\right)\right]\] \[\quad\quad+c_{i}^{*}\left[c_{s,sat}R_{H}\theta\frac{\partial \alpha_{i}^{*}}{\partial T}\frac{\partial T}{\partial t}+\alpha_{i}^{*}R_{H} \theta\frac{\partial c_{s,sat}}{\partial T}\frac{\partial\sigma}{\partial t}+ \alpha_{i}^{*}c_{s,sat}\theta\left(\frac{\partial R_{H}}{\partial h}\frac{ \partial h}{\partial t}+\frac{\partial R_{H}}{\partial T}\frac{\partial T}{ \partial t}\right)+\alpha_{i}^{*}c_{s,sat}R_{H}\frac{\partial\theta}{\partial h }\frac{\partial h}{\partial t}\right] \tag{37}\]
Equations 26-28 are expressed as a system of coupled ordinary differential equations of Equations 29-37. This system of equations (\(\frac{\partial T}{\partial t}\), \(\frac{\partial T}{\partial t}\), and \(\frac{\partial^{*}}{\partial t^{*}}\)) are solved at the center of each layer (cell-centered discretization) by MATLAB solver ode113 (or ode23 tb). Furthermore, the semi-coupled scheme (Figure 1) is also incorporated in MOIST, where the derivative vector of the soil water and heat transport equations (Equations 29 and 30) are constructed by Equations 32-35 and passed to the solver first. Then, Equation 31 is rewritten as:
\[\frac{\partial C_{i}^{\dagger}}{\partial t}=\frac{-\frac{\partial q_{i}}{ \partial z}-H}{I} \tag{38}\]
with
\[H=\ c_{i}^{\dagger}\left[\frac{\partial\theta}{\partial t}-c_{s,sat}R_{H}\ \alpha_{i}^{*}\frac{\partial\theta}{\partial t}+\frac{\partial\left(c_{s,sat}R _{H}\ \alpha_{i}^{*}\ \theta_{sat}\right)}{\partial t}-\theta\frac{\partial\left(c_{s,sat}R_{H}\ \alpha_{i}^{*}\ \right)}{ \partial t}\right] \tag{39}\]
\[I=\theta+c_{s,sat}R_{H}\alpha_{i}^{*}\left(\theta_{sat}-\theta\right) \tag{40}\]
The solutions from the coupled soil water and heat transport equation (\(h\) and \(T\)) at new time point are used to update parameter values (\(c_{s,sat}\), \(R_{H}\), \(\alpha_{i}^{*}\), \(\theta\)) in Equations 39 and 40. Subsequently, the derivative vector of isotope transport can be constructed and solved (Equations 38-40).
#### 2.1.5.1 MATLAB Solver ode113
The ode113 solver uses an adaptive, variable-order, variable-step-size (VOVS) method ([PERSON], 2002). This is implemented with a variable order Adams-Bashforth-Moulton (ABM) method, which is a combination of an explicit type of the Adams-Bashforth (AB) and an implicit type of Adams-Moulton (AM) methods. Specifically, the AB method is used to obtain the solution at the new time step by taking multiple previous time steps into account, while the AM method is used to make corrections.
The ode113 selects automatically between the first and twelfth order approximation during the computation based on the estimation errors. This is helpful for minimizing the estimated errors and for achieving high efficiency in time. Moreover, the time step size is adjusted according to the estimation error. Therefore, ode113 can handle a wide range of problems with high accuracy and efficiency. The ode113 is well suited to transport in relatively uniform media but is susceptible to numerical oscillation when hydraulic conductivities between layers differ greatly.
#### 2.1.5.2 MATLAB Solver ode23 tb
Ode23 tb is a solver specifically designed for solving ordinary differential equations with highly oscillatory solutions, such as those arising from heterogeneity in hydraulic conductivities between soil layers. The solver combines a trapezoidal rule (sometimes referred as the second order AM method) with a second order backward differentiation formula (BDF), hence the \"tb\" suffix. As an implicit solver, ode23 tb is generally more computationally expensive than other solvers that adopt explicit numerical schemes. However, because it adopts the trapezoidal BDF method, it is more efficient than other types of implicit methods, such as the fully implicit Euler method or the backward Euler method. Therefore, ode23 tb is better suited than ode113 when the soil physical properties differed greatly between layers. Like ode113, ode23 tb can adjust the step size automatically based on the oscillatory behavior of the solution. Thus, ode23 tb is an efficient and accurate solver for stiff systems, making it less likely to have numerical instability.
#### 2.1.6 Modeling Efficiency
To evaluate the model performance quantitatively, the Nash-Sutcliffe efficiency (_NSE_) (Nash & Sutcliffe, 1970) is used:
\[NSE=1-\frac{\sum_{t=1}^{T_{cl}}\left[M_{0}(t)-M_{m}\right]^{2}}{\sum_{t=1}^{T _{cl}}\left[M_{0}(t)-\overline{M_{0}}\right]^{2}} \tag{41}\]
where \(M_{0}\) and \(M_{m}\) are observations (measurements) and modeling values (simulations) respectively. \(\overline{M_{0}}\) is the average of the observations over time, \(t\) is the temporal point, and \(T_{cl}\) is the total number of temporal steps. _NSE_ ranges between negative infinity to 1. _NSE_ of one is indicative of excellent performance of the model in predicting the temporal variations of variables, while an _NSE_ of 0 suggests the model can only reflect average values. A negative _NSE_ implies poor performance of the model in regenerating the temporal variations of variables.
Moreover, Kling-Gupta efficiency (_KGE_, [PERSON] et al., 2009) has become popular in recent years to evaluate the model performance ([PERSON] et al., 2019; [PERSON] et al., 2022). _KGE_ considers correlation, bias, and variability between simulations and measurements, providing a more comprehensive assessment of the model performance and can be written as:
\[KGE=1-\sqrt{(\text{cor}-1)^{2}+\left(\frac{\sigma_{m}}{\sigma_{o}}-1\right)^{ 2}+\left(\frac{\mu_{m}}{\mu_{o}}-1\right)^{2}} \tag{42}\]
where _cor_ is the linear correlation between observations and simulations, \(\sigma_{m}\) and \(\sigma_{o}\) are standard deviations of simulations and observations, respectively; \(\mu_{m}\) and \(\mu_{o}\) are mean of simulations and observations, respectively. Like _NSE_, _KGE_ = 1 indicates perfect agreement between simulations and observations. A model had a negative _KGE_ is generally considered as \"not satisfactory\" ([PERSON] et al., 2017).
Sometimes, _NSE_ and _KGE_ may disagree with each other during model comparisons ([PERSON] et al., 2019). A third index, the absolute error (_MAE_), is used to evaluate the model performance. Here, we decided to use _MAE_ rather than the root mean square error (_RMSE_) because the residuals between simulations and measurements are non-normally distributed ([PERSON], 2014; [PERSON] et al., 2018). _MAE_ is given as:
\[MAE=\frac{1}{T_{cl}}\sum_{t=1}^{T_{cl}}\left|M_{0}(t)-M_{m}(t)\right| \tag{43}\]
### Model Validation
To validate MOIST, we conducted sensitivity analysis on various temporal and spatial discretization, through theoretical, semi-analytical, and long-term field tests. However, we mainly reported the findings of sensitivity under various temporal and spatial discretization, semi-analytical tests under unsaturated and non-isothermal conditions, and a long-term field test. This is because the results from other semi-coupled models are readily available for comparison. Detailed information about the theoretical tests, and the semi-analytical test under saturated isothermal conditions, can be found in Supporting Information S1.
The current test assumes a 1 m soil column containing Yolo light clay. The relationships between soil water content, pressure head, and unsaturated hydraulic conductivity for the Yolo light clay were described by the Brooks-Corey ([PERSON] & Corey, 1964) model:
\[S=\frac{\theta-\theta_{res}}{\theta_{sat}-\theta_{res}}=\left\{\begin{array}{ ll}\left(\frac{h}{h_{c}}\right)^{-\lambda},h\leq he\\ 1\quad h\geq he\end{array}\right. \tag{44}\]
\[S^{\eta}=\left\{\begin{array}{ll}K_{sat},h\leq he\\ 1\quad h\geq he\end{array}\right. \tag{45}\]
where \(S\) is the effective saturation; \(h_{c}\) is the air-entry value (m); and \(\lambda\) and \(\eta\) are the shape coefficients, where \(\eta=2\gamma\)\(\lambda+3\) (Table 1).
The initial conditions assume the soil water content as 70% of its saturated value and the net radiation as a constant value of 200 W m\({}^{-2}\) at the soil surface. The air temperature and relative humidity during the simulation period remained at 30\({}^{\circ}\)C and 0.2, respectively. Water in the soil column can escape only through evaporation from the top of the column. The upper boundary conditions for soil water and isotope transport were calculated by Equations 19, 20 and 25. For the lower boundary, the rate of water supply from the bottom of the profile is equivalent to the evaporation rate at each time step. The simulation length was 700 hr.
Three spatial steps, \(\Delta z=0.01\) m, \(\Delta z=0.02\) m, and \(\Delta z=0.005\) m were used. The fixed initial temporal step was 25 s. By contrast, for the various temporal steps, three initial steps, \(\Delta t=100\) s, \(\Delta t=50\) s, and \(\Delta t=25\) s, were used and the spatial step was fixed at 0.01 m. Note that if a large temporal step is used, the result may be affected by the time-adaption, which could reduce the difference between semi-coupled and fully coupled methods. Thus, we used small temporal steps to approximately transform the time-adaptive solver into a fixed time step solver. This is because the solver can obtain a satisfactory solution within the provided initial time step without the need for time adjustments when boundary conditions are stable.
#### 2.2.2 Semi-Analytical Test Under Unsaturated Non-Isothermal Conditions
[PERSON] and [PERSON] (1984) developed a semi-analytical solution to predict \(\delta^{2}\)H and \(\delta^{18}\)O profiles under unsaturated, non-isothermal conditions as:
\[\frac{d\delta_{i}}{dz}+\frac{\delta_{i}-\delta_{i,\text{sup}}}{z_{i}+R_{i}z_{ e}}=\frac{R_{i}\sigma_{z}(\alpha_{k}-\alpha_{i}^{\ast})}{z_{i}+R_{i}z_{e}}\, \frac{d\left[ln(\rho R_{tt}c_{v,\text{sat}}(\alpha_{k}-\alpha_{i}^{\ast})) \right]}{dz} \tag{46}\]
\[z_{i}=\frac{D_{i,t}^{l}}{q_{\text{crop}}} \tag{47}\]
\[z_{v}=\frac{D_{i,t}^{l}c_{v,\text{sat}}}{q_{\text{crop}}} \tag{48}\]where \(z_{i}\) and \(z_{v}\) are the liquid and vapor characteristic lengths (m), respectively. Note that all modeling conditions were the same as that described in Section 2.2.1, except the 1 m soil column was divided uniformly at a step of 0.01 m and the total simulation length was 250 days to ensure that steady state is achieved.
#### 2.2.3 Long-Term Simulations at HBLFA Raumberg-Gumpenstein, Austria
#### 2.2.3.1 Site Description
Precipitation and seepage water from lysimeters were collected from May 2002 to February 2007 by [PERSON] et al. (2012) at the HBLFA Raumberg-Gumpenstein, Austria. During the experiment, the air temperature exhibited ainuous variation, ranging between \(-15^{\circ}\)C and \(27^{\circ}\)C (Figure 3a), with a mean of 8.2\({}^{\circ}\)C. Similarly, the atmospheric relative humidity showed seasonal fluctuations and varied between 0.30 and 0.99 (Figure 3b), with a mean of 0.89. In addition, most precipitation events occurred during the summer (Figure 3c), with a daily mean rainfall of 2.8 mm day\({}^{-1}\).
Five lysimeters were used to investigate the influence of land cover and fertilization on soil water and solute transport by [PERSON] et al. (2012). For simplicity, the current study used only lysimeter-3 for the comparison of numerical simulations. Lysimeter-3 had a surface area of 1 m\({}^{2}\), depth of 1.5 m, and was filled with three soil horizons (0-0.25 m, 0.25-1.0 m, 1.0-1.5 m) of undisturbed Dystric Cambisol ([PERSON] et al., 2012) (Figure 4).
Each year the lysimeter was planted with winter rye, which had a maximum rooting depth of 1 m. Weekly precipitation and drainage water samples from the bottom of Lysimeter-3 were collected between May 2002 and February 2007. Isotopic compositions were analyzed by using dual-inlet mass spectrometry. Further information about the site and experimental procedures can be obtained from [PERSON] et al. (2012).
#### 2.2.3.2 Model Setup
The MOIST simulations adopted the solute transport and soil hydraulic parameter values provided by [PERSON] et al. (2012). The soil hydraulic properties were described by the [PERSON] (1980) model. Because the measured saturated hydraulic conductivities varied greatly within different soil horizons ([PERSON] et al., 2012), water flux at the interface of different soil layers may vary drastically. Therefore, to minimize the oscillation of numerical solutions, the ode23 tb solver, which is designed for stiff problems, was used in this simulation.
The initial soil water content and \(8^{13}\)O profiles were provided by [PERSON] et al. (2012). The upper boundaries of soil water and heat transport were calculated by Equations 14 and 16, while the upper boundary of isotope transport was calculated by Equation 25. The lower boundary condition for soil water flow was defined as seepage
Figure 3: Air temperature, relative humidity, and precipitation events at HBLFA Raumberg-Gumpenstein in Austria: (a) air temperature (T), (b) air relative humidity (RH), and (c) daily precipitation (P). Data from [PERSON] et al. (2012).
surface. Zero temperature and zero isotopic concentration gradients were set as the lower boundary conditions of heat and isotope transport, respectively.
The root distribution of winter ye at the site varied during the growing season and has been described in HYDRUS-1D ([PERSON] et al., 2012), along with the water stress function described by the [PERSON] model ([PERSON] et al., 1978). Following this approach, the water stress function parameters \(h_{t}\), \(h_{2}\), \(h_{t}\), and \(h_{t}\) were set as 0, 0.01, 5, and 160 m, respectively ([PERSON] et al., 2012).
Environmental parameters such as air temperature, air relative humidity, and precipitation were assumed to be constant within each hour. Regarding the isotopic compositions of atmospheric vapor, \(\delta_{\alpha}\), it is expressed as following equation when rainfall occurs ([PERSON] et al., 2015):
\[\delta_{\alpha}=\frac{\delta_{min}\ -\left(\alpha^{+}+1\right)1000}{\alpha^{+}} \tag{49}\]
where \(\alpha^{+}\) is the equilibrium fractionation factor.
Obviously, Equation 49 cannot be used when there is a rain-free period. However, the field measurements of \(\delta_{\alpha}\) (oxygen-18) in Austria (where the lysimeter study conducted) ranged between \(-27\%\epsilon\) and \(-13\%\epsilon\) annually ([PERSON] et al., 2012), with an average of \(-20\%\epsilon\)([PERSON] et al., 2016). Therefore, we used \(-20\%\epsilon\) as the value of \(\delta_{\alpha}\) for the rain-free period. Values \(-27\%\epsilon\) and \(-13\%\epsilon\) were used to explore the influence of \(\delta_{\alpha}\) values on the simulated outflow isotopic signals.
Soil hydraulic parameters were calibrated by both [PERSON] et al. (2012) and [PERSON] et al. (2022) with this data set (Table 2). We obtained a new set of hydraulic parameters by calibrating MOIST using the same data set. Consequently, simulations were conducted in three scenarios: Scenario 1, calibrated parameters from [PERSON] et al. (2012) were utilized in MOIST; Scenario 2, calibrated parameters from [PERSON] et al. (2022) were employed in MOIST; Scenario 3, parameters calibrated by MOIST were used. In the first two scenarios, we aimed to highlight the advantages of the coupling scheme because the parameters are independent of MOIST, allowing for a comparison that reflects model differences. In Scenario 3, we aimed to demonstrate that model calibration can further enhance the performance of MOIST.
The soil column was divided into 150 layers with a spatial step of 0.01 m in the model simulation. The simulation length was 1,736 days, and the initial temporal step was 86,400 s. The minimum time step was \(1\times 10^{-8}\) s and the maximum time step was 86,400 s. The time step can be automatically adjusted within the range between minimum and maximum. Lastly, the lysimeter simulation (including Sections 2.2.1 and 2.2.2) was completed by MATLAB R2022a on a personal computer with an Intel processor (i7-10700k).
\begin{table}
\begin{tabular}{l c c c c c c} \hline & Layers (m) & \(\alpha\) (m\({}^{-1}\)) & \(n\) & \(\theta_{max}\) (m\({}^{3}\) m\({}^{-3}\)) & \(\theta_{max}\) (m\({}^{3}\) m\({}^{-3}\)) & \(K_{max}\) (m s\({}^{-1}\)) \\ \hline [PERSON] et al. (2012) & 0–0.30 & 2.3 & 1.14 & 0.30 & 0 & \(1.27\times 10^{-5}\) \\ & 0.31–0.90 & 7.6 & 1.07 & 0.32 & 0 & \(6.94\times 10^{-4}\) \\ & 0.91–1.50 & 1.6 & 1.07 & 0.32 & 0 & \(1.27\times 10^{-5}\) \\ [PERSON] et al. (2022) & 0–0.30 & 2.0 & 1.15 & 0.30 & 0 & \(2.55\times 10^{-5}\) \\ & 0.31–0.90 & 30.0 & 1.11 & 0.41 & 0 & \(3.32\times 10^{-5}\) \\ & 0.91–1.50 & 8.2 & 1.10 & 0.30 & 0 & \(2.55\times 10^{-5}\) \\ \hline \end{tabular}
\end{table}
Table 2 Calibrated Soil Hydraulic Parameters by [PERSON] et al. (2012) and [PERSON] et al. (2022)
Figure 4.— Illustration of the Lysimeter-3 setup from the long-term simulations at HBLFA Raumberg-Gumpenstein, Austria.
#### 2.2.3.3 Model Calibration
Three soil horizons were defined in this lysimeter study ([PERSON] et al., 2012). Limited by the computation time of MOIST, we did not calibrate all 13 parameters (4 parameters of van Genuchten model of each horizon plus the longitude dispersivity). Instead, we calibrated the saturated hydraulic conductivity of each horizon because \(K_{sat}\) are generally prone to large uncertainties ([PERSON] and [PERSON], 2019) and highly sensitive to the choice of models (Table 2). Moreover, the longitude dispersivity was also calibrated because it directly affects the isotope transport ([PERSON] et al., 2022). One dispersivity for three horizons can be estimated because only the isotopic composition of the drainage water was measured in the [PERSON] et al. (2012) data set. The remaining soil hydraulic parameters, such as \(\alpha\), \(n\), \(\theta_{sat}\), and \(\theta_{res}\) were identical to that from [PERSON] et al. (2012).
The calibrated saturated hydraulic conductivities from [PERSON] et al. (2012) served as base values, and each was assigned a coefficient ranging between 0.1 and 10 for scaling purposes. These coefficients were uniformly sampled through a Monte Carlo simulation. Then, the Nash-Sutcliffe Efficiency (_NSE_) of simulated outflow isotope signals, along with the disparity between modeled and measured total outflow amount, was calculated. In cases where MOIST failed to produce results under specific coefficients, a negative _NSE_ was assigned. The set of coefficients yielding the highest _NSE_ and the smallest difference between modeled and measured total outflow was selected. The calibration was conducted on Cedar, a heterogeneous high-performance cluster (HPC) within the Digital Research Alliance of Canada. Note that the validation is simplified, but it provided valuable insights into the potential impact of calibration on the performance of MOIST when applied to real-world situations.
## 3 Results
MOIST was validated against semi-coupled solutions under various temporal and spatial discretization, theoretical solutions, semi-analytical solutions under saturated isothermal conditions, semi-analytical solutions under unsaturated non-isothermal conditions, and long-term field measurements. Here we focused primarily on the validation of MOIST against semi-coupled solutions under various temporal and spatial discretization, semi-analytical tests under unsaturated non-isothermal conditions and the long-term field measurements. Validations against theoretical tests and semi-analytical tests under saturated isothermal conditions can be found in the Supporting Information S1.
Comparison Between Fully Coupled and Semi-Coupled Methods Under Various Spatial and Temporal Discretization
The feasibility and stability of fully coupled and semi-coupled methods were assessed. The initial \(\delta^{2}\)H profile was uniformly distributed with a value of 0%. Given that water (\(\delta^{2}\)H = 0%) was continuously supplied from the bottom of the profile, and water escaped the column through evaporation only at the soil surface, \(\delta^{2}\)H of soil water in the column at any point in space and time is unlikely to be \(\gtrsim\)0%, especially at its lower boundary. However, considering numerical errors, a negative value of \(-\)0.3% was set as the threshold for the largest acceptable deviation from 0% (\(\delta^{2}\)H of the bottom-supplied water) because the largest error of simulated \(\delta^{2}\)H from MOIST in the semi-analytical test is 0.31% (Table 4). Additionally, a 0.1% threshold was used as the maximum acceptable absolute deviation between the fully coupled and semi-coupled methods, recognizing that the two methods cannot produce identical numerical solutions.
\begin{table}
\begin{tabular}{l c c} \hline & MOIST & Semi-analytical solution \\ \hline Maximum \(\delta^{2}\)H (\%) & 40.34 & 40.65 \\ Maximum \(\delta^{10}\)O (\%) & 21.15 & 22.13 \\ Slope of HDO/H\({}_{1}\)SO in vapor dominated region & 3.55 & 3.31 \\ Slope of HDO/H\({}_{2}\)SO in liquid dominated region & 1.91 & 1.90 \\ \hline \end{tabular}
\end{table}
Table 3: Comparison of Maximum \(\delta^{2}\)H, \(\delta^{10}\)O, and the Slope of \({}^{2}\)H/\({}^{10}\)O in Liquid Dominated Regions Estimated From MOIST and the Analytical Solution Under Unsaturated and Non-isothermal Conditions
\begin{table}
\begin{tabular}{l c c} \hline & MOIST & Semi-analytical solution \\ \hline Maximum \(\delta^{3}\)H (\%) & 40.34 & 40.65 \\ Maximum \(\delta^{10}\)O (\%) & 21.15 & 22.13 \\ Slope of HDO/H\({}_{1}\)SO in vapor dominated region & 3.55 & 3.31 \\ Slope of HDO/H\({}_{2}\)SO in liquid dominated region & 1.91 & 1.90 \\ \hline \end{tabular}
\end{table}
Table 4: Difference of Maximum Isotopic Composition Between Numerical and Analytical Solutions With a Layer Thickness of 0.01 m
#### 3.1.1 Various Temporal Discretization
Under different temporal discretization, the fully coupled method consistently displayed \(\delta^{2}\)H values greater than \(-0.3\%\)e (see Figures 5a-5c). By contrast, the semi-coupled method exhibited unreasonable negative \(\delta^{2}\)H values (depicted by dark blue areas in Figures 5d-5f) across all temporal step lengths. This discrepancy can be attributed to errors originating from soil water and heat equations being incorporated into the isotope transport equation. Consequently, noticeable disparities in the estimated transient \(\delta^{2}\)H between the two methods emerged (highlighted by pink areas in Figures 5g-5i), and these differences became more pronounced with increasing temporal steps (Figures 5g-5i). This highlights the robustness of the fully coupled method, which is less sensitive to temporal discretization and capable of accommodating larger time steps compared to the semi-coupled method.
Solver CPU times of both fully coupled and semi-coupled methods were presented in Figure 6. For the fully coupled method, solver CPU times were 14.4 s, 7.2 s, and 3.5 s under temporal discretization of 100 s, 50 s, and 25 s, respectively (Figure 6). By contrast, the corresponding times for the semi-coupled method were 29.2 s, 15.2 s, and 7.7 s, respectively (Figure 6). Notably, the fully coupled method reduced solver CPU time by approximately 50% compared to the semi-coupled method. This gained efficiency can be attributed to the fact that all equations were solved simultaneously in the fully coupled method, requiring the solver to be called only once within a single iteration. On the other hand, the semi-coupled method involved solving water and heat transport equations before addressing the isotope equation,
Figure 5: Comparison of simulated transient hydrogen isotope ratio (\(\delta^{2}\)H) profiles using fully coupled and semi-coupled numerical methods under unsaturated non-isothermal conditions at three spatial discretization, including 100 s (left), 50 s (center), and 25 s (right). Panels (a) to (c) show the \(\delta^{2}\)H profiles from the fully coupled method. Panels (d) to (f) show the \(\delta^{2}\)H profiles from the semi-coupled method. Panels (g) to (i) show the absolute differences in the \(\delta^{2}\)H profiles between the fully coupled and semi-coupled method. FC, SC, and D represent fully coupled solutions, semi-coupled solutions, and the differences between two solutions, respectively. A threshold of \(-0.3\%\)e was set as the largest acceptable deviation from 0 to 0.1% as the maximum deviation between the fully coupled and semi-coupled methods because they cannot produce identical numerical solutions.
Figure 6: The solver CPU time of fully coupled and semi-coupled version of MOIST under a spatial discretization of 0.01 m and temporal discretization of 25 s, 50 s, and 100 s, respectively.
leading to the solver being called twice within one iteration (Figure 6). Additionally, the solver CPU time approximately decreased linearly with the initial step increased (Figure 6), which supports that employing a small initial time step can effectively alleviate the need for a time-adaptive solver.
#### 3.1.2 Various Spatial Discretization
The comparison between fully coupled and semi-coupled method was also conducted under various spatial discretization (Figures 7a-7f). The semi-coupled method exhibited similar \(\delta^{\mathrm{5}}\)H transit profiles as the fully coupled method. The difference between these two methods was diminished as spatial step decreased (Figures 7g-7i). However, the unreasonable negative \(\delta^{\mathrm{2}}\)H values can be observed again at the bottom from semi-coupled method (Figures 7d-7f), as appeared under various temporal discretization (Figure 5). This illustrated that the semi-coupled method is more sensitive to spatial discretization than the fully coupled method.
Again, the solver CPU times for both fully coupled and semi-coupled methods under various spatial discretization were presented in Figure 8. Specifically, with spatial discretization of \(\Delta z\) = 0.005 m, 0.01 m, and 0.05 m, the solver CPU times for the fully coupled method were 21.0 s, 14.4 s, and 14.5 s, respectively (Figure 8). By contrast, the corresponding times for the semi-coupled method were 29.9 s, 29.2 s, and 28.9 s under the same conditions. The consistently lower CPU times from the fully coupled method than the semi-coupled method confirms the enhanced efficiency of the fully coupled approach.
Figure 8: The solver CPU time of fully coupled and semi-coupled version of MOIST under a temporal discretization of 25 s and spatial discretization of 0.005 m, 0.01 m, and 0.02 m, respectively.
Figure 7: Comparison of simulated transient hydrogen isotope (\(\delta^{\mathrm{5}}\)H) profiles using fully coupled and semi-coupled solute transport methods under unsaturated, non-isothermal conditions at three spatial discretization, including 0.02 m (left), 0.01 m (center), and 0.005 m (right). Panels (a) to (c) show the \(\delta^{\mathrm{5}}\)H profiles from the fully coupled method. Panels (d) to (f) show the \(\delta^{\mathrm{5}}\)H profiles from the semi-coupled method. Panels (g) to (i) show the absolute differences in the \(\delta^{\mathrm{5}}\)H profiles between the fully coupled and semi-coupled methods. FC, SC, and D represent fully coupled solutions, semi-coupled solutions, and the differences between two solutions, respectively. A threshold of \(-0.3\%\) was set as the largest acceptable deviation from 0 to 0.1\(\%\) as the maximum deviation between the fully coupled and semi-coupled methods because they cannot produce identical numerical solutions.
### Semi-Analytical Tests Under Unsaturated Non-Isothermal Conditions
Under unsaturated conditions, a drying layer appeared at the soil surface (Figure 9a) and water flow in this region was dominated by upward vapor diffusion (Figure 9b). The total water flux within the column was uniform with depth (Figure 9b) because a steady state flow was achieved. \(\delta^{2}\)H and \(\delta^{18}\)O increased sharply with depth until reaching their peak values at approximately 0.02 m below the soil surface, which is the location of the bottom of the drying layer (Figures 9c and 9d). This differs from the saturated conditions (Supporting Information S1), where the maximum \(\delta^{2}\)H and \(\delta^{18}\)O appeared at the soil surface. This is because in the unsaturated system the soil water content at the air-soil interface was close to residual soil water content: when a drying layer forms, the air invades into the drying layer, resulting in a downward shift of the isotope peak values as compared to the saturated system.
Visually, the analytical solution and the estimates from MOIST agreed very well across the entire soil profile (Figures 9c and 9d). If numeric models are biased, they tend to be biased at locations where sharp changes of isotope occur. Therefore, we compared the simulated peak concentrations obtained from the MOIST and the analytical solutions. Quantitatively, the approaches agreed well at the maximum \(\delta^{2}\)H and \(\delta^{18}\)O (Table 3). They also agreed in terms of the slopes of HDO/H\({}_{2}\)\({}^{18}\)O in the liquid and vapor dominated regions (Table 3). This demonstrates that MOIST is robust with respect to the accuracy of calculating isotope transport within soil.
[PERSON] et al. (2021) presented the simulated maximum \(\delta^{3}\)H and \(\delta^{18}\)O from a revised HYDRUS-1D model (rHZ for short) under unsaturated non-isothermal conditions with fine (\(\Delta z=1\times 10^{-6}\) m), medium (\(\Delta z=0.005\) m), and coarse (\(\Delta z=0.01\) m) spatial discretization. However, their spatial intervals were not uniformly distributed, except in the coarse spatial discretization scenario. Because MOIST currently does not support adaptive spatial steps (which will be addressed in the future), its results were compared with those of rHZ under the coarse scenario.
Table 4 showed that the largest differences in \(\delta^{2}\)H and \(\delta^{18}\)O between MOIST and the analytical solution were 0.31 and 0.98\(\%\)e, respectively. By contrast, these values were 34.68 and 24.88\(\%\)e from rHZ. Compared to rHZ, MOIST significantly reduced the numerical error of the maximum isotopic compositions between numerical and analytical solutions (Table 4), illustrating that MOIST offered better accuracy under the same spatial
Figure 9: Comparison of the semi-analytical test results from MOIST after steady state evaporation was reached under unsaturated and non-isothermal conditions. (a) Volumetric soil water content; (b) liquid and vapor flux; (c) simulated \(\delta^{2}\)H as a function of depth from numerical and semi-analytical solutions; and (d) simulated \(\delta^{18}\)O as a function of depth by numerical and semi-analytical solutions.
discretization scheme as compared to rHZ. Many factors could lead to the superior performance of MOIST. The most striking difference is that MOIST is based on the fully coupled method while rHZ is based on the semi-coupled method. In the semi-coupled method, mass balance errors from soil water and heat transport can be inflated by coarse spatial resolutions and transferred into isotope transport at each time step. This is reflected in the fact that fine spatial discretization is needed for rHZ to pass this analytical test. By contrast, there was little error transfer from the soil water and heat equations to the isotope equation in MOIST, because all the equations were integrated by the fully coupled method. These findings illustrated that the fully coupled method has the potential to improve numerical accuracy and stability under large spatial steps.
### Validation by the Long-Term Experiment at HBLFA Raumber-Gumpenstein
In the long-term validation, MOIST modeled the isotopic signals of drainage water under three scenarios: Soil hydraulic parameters calibrated by [PERSON] et al. (2012) (Scenario 1, Figure 10a), [PERSON] et al. (2022) (Scenario 2, Figure 10b), and MOIST (Scenario 3, Figure 10c).
In scenario 1, the MOIST model, as well as the HYDRUS-1D revised by [PERSON] et al. (2012) (rHS for short), reproduced temporal variations of measured \(8^{18}\)O in drainage water (Figure 10). However, the _NSE_, _KGE_, and _MAE_ of rHS were 0.35, 0.60, and 1.00%, respectively. By contrast, these indices from MOIST were 0.47, 0.58, and 0.92%, respectively, suggesting that MOIST slightly outperformed rHS when using parameters calibrated by
Figure 10: \(8^{\mathrm{rd}}\)O from seepage water over the course of the experiment at HBLFA Raumberg-Gumpenstein in Austria. Included are the measurement values, (a) simulations from MOIST (with and without fractionation) based on the parameters calibrated by [PERSON] et al. (2012) and simulations from [PERSON] et al. (2012), (b) simulations from MOIST based on parameters calibrated [PERSON] et al. (2022) under various isotopic compositions of atmospheric water vapor and results from [PERSON] et al. (2022); (c) simulations from MOIST after calibration. _NSE_, _KGE_, and _MAE_ are Nash-Sutcliffe Efficiency, Kling-Guppa Efficiency, and Mean Absolute Error, respectively.
rHS. Interestingly, although fractionation would typically result in more enriched isotopic compositions, this was not observed between the curves generated by rHS (without fractionation, represented by the blue solid line in Figure 10) and MOIST (with fractionation, represented by the red solid line in Figure 10). This discrepancy may be attributed to the possibility that the calibrated parameters do not precisely match MOIST. However, when the fractionation process was removed from MOIST, as expected, the temporal isotopic compositions became more depleted (red dash line in Figure 10). This underscores the significance of incorporating fractionation when modeling isotope transport within soil. In scenario 2, parameters calibrated by another version of revised HYDRUS-1D ([PERSON] et al., 2022, rHZ for short) were employed in MOIST for comparison. It can be observed that rHZ and MOIST generated comparable temporal patterns of drainage water isotopic compositions, especially after 1,500 days (Figure 10b). Quantitively, _NSE_, _KGE_, and _MAE_ of rHZ were 0.19, 0.58, and 1.15%, respectively. By contrast, these indices from MOIST were 0.33, 0.52, and 1.04%, respectively. The smaller errors (_MAE_) from MOIST supported the fully coupled method outperformed the semi-coupled method. Moreover, simulated isotopic signals of outflow from MOIST under various \(\delta_{u}\) values illustrated a marginal influence of \(\delta_{u}\) on the time series of outflow isotopic signals (Figure 10b).
Results from scenario 3 demonstrated that accuracy of predictions from MOIST benefited significantly from calibration (Figure 10c). The temporal trend of 8\({}^{18}\)O of drainage water fitted well to the measurements, especially from the first 600 days. Moreover, the highest _NSE_ (0.48), _KGE_ (0.76), and the smallest _MAE_ (0.90) among the three scenarios not only confirmed the outperformance of MOIST, but also underscored the importance of calibration when applying a hydrological model into real conditions.
### Calibrated Soil Hydraulic Conductivities and the Longitude Dispersivity
Only saturated hydraulic conductivity (\(K_{sat}\)) and longitudinal dispersivity were calibrated in MOIST (Table 5) and the remaining parameters were maintained as calibrated by [PERSON] et al. (2012). This is because MOIST exhibited better accuracy with [PERSON] et al. (2012) calibrated parameters compared to those calibrated by [PERSON] et al. (2022) (Figure 10).
MOIST yielded \(6.08\times 10^{-2}\) m for longitudinal dispersivity, which is close to initial values (\(4.70\times 10^{-2}\) m, Table 5). For saturated hydraulic conductivities (\(K_{sat}\)), initial values (from rHS) were \(1.27\times 10^{-5}\), \(6.94\times 10^{-4}\), and \(1.27\times 10^{-5}\) m s\({}^{-1}\) for three horizons, with corresponding _MAE_ values of estimated \(K\) were \(1.83\times 10^{-6}\), \(4.7\times 10^{-5}\), and \(3.47\times 10^{-6}\) m s\({}^{-1}\), and the overall _MAE_ is \(1.8\times 10^{-5}\) m s\({}^{-1}\). By contrast, MOIST calibrated \(K_{sat}\) values for three horizons to \(4.20\times 10^{-6}\), \(5.80\times 10^{-4}\), and \(5.72\times 10^{-6}\) m s\({}^{-1}\), with respective _MAE_ values of estimated \(K\) were \(5.37\times 10^{-7}\), \(3.92\times 10^{-5}\), and \(3.89\times 10^{-7}\) m s\({}^{-1}\), and the overall _MAE_ is \(1.48\times 10^{-5}\) m s\({}^{-1}\). In comparison to initial values from rHS, MOIST exhibited smaller errors of \(K\) that calculated from calibrated \(K_{sat}\). This suggested that the results from MOIST were acceptable, although the calibration of MOIST was simplified (fewer objective functions than rHS).
\begin{table}
\begin{tabular}{c c c c c c c} \hline & Layers (m) & Longitude dispersivity (m) & \(K_{sat}\) (m s\({}^{-1}\)) & _MAE_ of \(K\) (m s\({}^{-1}\)) & Overall _MAE_ of \(K\) (m s\({}^{-1}\)) & Objective functions \\ \hline [PERSON] et al. (2012) & 0–0.30 & \(4.70\times 10^{-2}\) & \(1.27\times 10^{-5}\) & \(1.83\times 10^{-6}\) & \(1.80\times 10^{-5}\) & _SW_; \(\delta_{Bt}\); \(K\); \(BC\); \(BF\)-\(t\) \\ & 0.31–0.90 & & \(6.94\times 10^{-4}\) & \(4.70\times 10^{-5}\) & & \\ & 0.91–1.50 & & \(1.27\times 10^{-5}\) & \(3.47\times 10^{-6}\) & & \\ MOIST & 0–0.30 & \(6.08\times 10^{-2}\) & \(4.20\times 10^{-6}\) & \(5.37\times 10^{-7}\) & \(1.48\times 10^{-5}\) & \(\delta_{Bt}\); \(BF\)-\(T\) \\ & 0.31–0.90 & & \(5.80\times 10^{-4}\) & \(3.92\times 10^{-5}\) & & \\ & 0.91–1.50 & & \(5.72\times 10^{-6}\) & \(3.89\times 10^{-7}\) & & \\ \hline \end{tabular} _Note._ The calibration of longitude dispersivity is not evaluated because the measurements are not available. _SW_, \(\delta_{Bt}\), \(K\), _RC_, \(BF\)-\(t\), and _BF-\(T\)_ are soil water content measurements, isotopic compositions of bottom flow, unsaturated hydraulic conductivities, soil water retention curve measurements, time series bottom flow, and total bottom flow, respectively. _MAE_ of \(K\) is calculated by averaging errors between modeled \(\mathcal{K}\) (estimated by the measured \(K(h)\) and calibrated \(K_{sat}\)) and measured \(\mathcal{K}\).
\end{table}
Table 5: _Initial Longitude Dispersivity and Saturated Hydraulic Conductivities From [PERSON] et al., (2012) and Calibrated by MOIST_
## 4 Discussion
### Difference Between the Fully Coupled Method and the Semi-Coupled Method
The semi-coupled variation of MOIST exhibited unfeasible \(\delta^{2}\)H values across various spatial and temporal discretization, a problem not encountered by the fully coupled version of MOIST (Figures 5 and 7). This discrepancy can be attributed to the fact that mass balance errors arising from solving soil water and heat transport equations are introduced into the isotope transport equation within the semi-coupled method. By contrast, the fully coupled method avoids this issue by analytically integrating the soil water and heat transport equations into the isotope transport equation (Equations 36 and 37). This aligns with the findings of [PERSON] and [PERSON] (2019), who also observed that the fully coupled method excels in capturing coupled physical processes, mitigating errors, and delivering more precise outcomes when compared to the semi-coupled approach.
Moreover, it is worth noting that the disparities between the fully coupled and semi-coupled methods diminished as finer temporal and spatial steps were employed (Figure 5). This phenomenon can be attributed to the fact that the finer spatial and temporal discretization leads to reduced mass balance errors in solving the soil water and heat transport equations ([PERSON] et al., 2005; [PERSON] et al., 2018; [PERSON] et al., 2021). However, a surprising trend can be observed that as the layer thickness decreased in the semi-coupled simulations, a longer dark blue band appeared at the bottom of the column (Figures 6(d)-6(f)). This phenomenon can be attributed to numerical instability in the differential equations. To ensure numerical stability, adherence to CFL (Courant-Friedrichs-Lewy) condition is essential ([PERSON] et al., 1967):
\[\left|\frac{\zeta\Delta t}{\Delta z}\right|<\ \varepsilon_{max} \tag{50}\]
where \(\varepsilon_{max}\) is typically set to 1. This equation stipulates that the spatial step (\(\Delta z\)) should not be excessively small compared to the temporal step (\(\Delta t\)), as excessively small \(\Delta z\) values can lead to numerical oscillations. In the case of the semi-coupled method, where \(\Delta t\) was 25 s, decreasing \(\Delta z\) resulted in a larger \(\left|\frac{\zeta\Delta t}{\Delta z}\right|\) value, increasing the likelihood of instabilities. However, in the fully coupled method, as shown in Figures 5 and 7, the relationship between instability and spatial step size was less evident. This could be due to the scaling down of \(\varsigma\) by coefficients from the soil water and temperature equations (Equations 26 and 27), which may reduce the value of \(\left|\frac{\zeta\Delta t}{\Delta z}\right|\). Consequently, compared to the semi-coupled method, the fully coupled method demonstrated greater stability across a range of spatial step sizes, highlighting its superior reliability in predicting temporal variations in the isotopic compositions of soil water.
While our tests were conducted under ideal boundary conditions, the results aligned with the findings presented in [PERSON] and [PERSON] (2004). They further underscore the fact that the fully coupled method effectively minimizes errors and displays reduced sensitivity to variations in spatial and temporal step sizes. This attribute enhances the accuracy of one-dimensional isotope transport simulations. However, the semi-coupled method has advantages in steady-state systems. By assuming linear changes in the water and heat solution within the time step, the isotope equation in the semi-coupled method can use a smaller time step, improving the stability of the isotope solution, although at the cost of increased computation time. In our simulations (Sections 2.2.1), to ensure consistency, the time step for the isotope equation in the semi-coupled method was kept the same as that for the water and heat equations, but the time consumption of semi-coupled method was doubled compared to the fully coupled method, as shown in Figure 6. This contrasts with the theoretical expectation that the semi-coupled method should generally be faster due to its lower memory and computational requirements per iteration ([PERSON] and [PERSON], 2021; [PERSON] and [PERSON], 2019). Several factors can contribute to this discrepancy.
First, the choice between using the semi-coupled method and the fully coupled method depends on the degree of coupling exhibited by the variables involved ([PERSON] and [PERSON], 2019). In our study, all variables are tightly coupled due to intricate interactions between water and heat transport, which concurrently influence the movement of isotopes. Solving a coupled system in a segregated manner may lead to increased computational time, as it may require more iterations for convergence to occur ([PERSON] and [PERSON], 2004). Notably, the semi-coupled method consumed more time because it necessitates two solver calls within a single time step, whereas thefully coupled method requires only one call. This observation is supported by the solver CPU time (Figures 6 and 8).
Second, it is essential to consider the scale of the problem. In our tests, we examined a relatively small-scale problem, with the minimum spatial discretization (\(\Delta z\)) is 0.005 m. This means that the semi-coupled method processed two matrices with sizes of \(400\times 400\) and \(200\times 200\), respectively (Figure 10(a)). By contrast, the fully coupled method handled a single matrix with a size of \(600\times 600\) (Figure 10(a)). Both matrix size and quantity are critical factors influencing solver CPU times. With a smaller matrix size (resulted from a larger \(\Delta z\)), the fully coupled method can be faster than the semi-coupled method because both methods require similar CPU time per solver call, but the semi-coupled method calls the solver twice. However, for a larger matrix size (resulted from a smaller \(\Delta z\)), the efficiency of the fully coupled method may decrease, as processing a single large matrix can significantly increase RAM usage and processing time compared to handling two sub-matrices in the semi-coupled method (Figure 10(a)). For example, the solver CPU times of the semi-coupled method were 2 times larger than that of the fully coupled method when \(\Delta z\) = 0.02 and 0.01 m, but 1.4 times larger when \(\Delta z\) = 0.005 m (Figure 8). This suggested that the efficiency of fully coupled method depends on the scale of the problem (e.g., the number of soil layers) ([PERSON] & [PERSON], 2019).
To better illustrate the influence of problem scale (\(\Delta z\)) on the solver CPU time, we attempted finer simulations, such as \(\Delta z\) = 0.001 m, but non-convergence of the upper boundary solutions occurred in MOIST. To resolve this, \(\Delta t\) must be reduced but reducing the initial \(\Delta t\) alters the simulation conditions, making the solver CPU time non-comparable. Although the current version of MOIST cannot explicitly explore larger simulation problems without
Figure 11: Illustration of matrices sizes from fully coupled method and semi-coupled method (Panel a); Panel b is the solver CPU time comparison between fully coupled method and semi-coupled method across various spatial discretization.
modifying initial \(\Delta t\), we developed a simplified example (codes are available from the Supporting Information S1) with six different \(\Delta z\) levels (0.0005 m, 0.001 m, 0.005 m, 0.01 m, 0.05 m, and 0.1 m) to compare time consumption of the fully coupled method and semi-coupled method across different problem scales (\(\Delta z\)). It can be found that the solver CPU times for the fully coupled and semi-coupled methods followed the same trend as seen in Figure 8 when \(\Delta z>0.005\) m (Figure 11b). For smaller \(\Delta z\) values, the solver CPU time for both methods increased sharply due to larger matrix sizes. When \(\Delta z\) was 0.0005 m, the time consumption of both methods was similar (Figure 11b). Consequently, it is expected that for even finer discretization, such as \(\Delta z=0.0005\) m, the fully coupled method may spend more time than the semi-coupled method ([PERSON] & [PERSON], 2021; [PERSON] & [PERSON], 2019). However, such a fine spatial discretization is uncommon in practice, and the resulting matrix size is extremely large (60,000 \(\times\) 60,000 for the fully coupled method, 40,000 \(\times\) 40,000 and 20,000 \(\times\) 20,000 for the semi-coupled method), potentially causing an \"out of memory\" error. Presently, research on isotope transport in soil and plant root water uptake typically focuses on depths around 2 m ([PERSON] et al., 2020; [PERSON] et al., 2021; [PERSON] et al., 2004; [PERSON], 2017) with a spatial interval of \(\Delta z=0.01\) m (e.g., the lysimeter simulation in this study). Under these conditions, the matrix to be solved typically has a size of 600 \(\times\) 600. Therefore, the fully coupled method offers advantages in terms of computation time when compared to the semi-coupled method.
### Outperformance of MOIST Under the Semi-Analytical Test and the Long-Term Lysimeter Simulation
The fully coupled approach empowers MOIST to exhibit superior accuracy in theoretical tests when compared to the revised HYDRUS-1D (rHZ, [PERSON] et al., 2021). There are several distinctions between MOIST and rHZ, including model structure and algorithms. However, theoretical testing scenarios with fixed boundary conditions and parameters, as well as known true values, provide a controlled environment that isolates model differences and underscores the impact of the fully coupled and semi-coupled methodologies on the results.
It is worth noting that numerical errors originating from the soil water flow equation can be propagated to the isotope transport equations, potentially leading to the observed numerical oscillations (Figures 5 and 7). This issue has prompted previous models like SiSPAT-Isotope ([PERSON] et al., 2005) and the revised HYDRUS-1D ([PERSON] et al., 2021) to necessitate an extremely small thickness for the first soil layer (e.g., 1 \(\times\) 10\({}^{-6}\) m) to pass theoretical tests (Supporting Information S1).
However, in semi-analytical tests where spatial discretization is 0.01 m, error control of rHZ may not be as effective as it is under a finer spatial discretization of 1 \(\times\) 10\({}^{-6}\) m. This discrepancy resulted in a significant difference between rHZ and the semi-analytical solution. Therefore, it is crucial for a semi-coupled method to rigorously manage mass balance errors ([PERSON] et al., 2019). By contrast, when equations are solved using the fully coupled method, fewer mass balance errors are transferred between equations ([PERSON] & [PERSON], 2004; [PERSON], 2004; [PERSON] et al., 1996), a fact that has been verified through temporal and spatial discretization tests (Figures 5 and 7). Consequently, MOIST outperformed rHZ under a coarse spatial discretization in the semi-analytical test (Table 3).
Under the long-term lysimeter simulation, the fully coupled method outperforms the semi-coupled method in simulating isotope transport. Both MOIST and rHZ consider fractionation. Notably, MOIST utilizes parameters that have been calibrated by rHZ, and it exhibits a slight but discernible advantage in performance over rHZ (Figure 10). Moreover, the calibrated parameters from [PERSON] et al. (2012) were applied to rHZ, the resulting Nash-Sutcliffe Efficiency (NSE) was 0.19 ([PERSON] et al., 2021), while MOIST achieved a notably higher NSE of 0.47. This discrepancy, with rHZ yielding a smaller NSE compared to MOIST, supports the idea that the semi-coupled method may introduce more mass balance errors into the process of isotope transport when compared with the fully coupled method. Additionally, as demonstrated by the theoretical test, under the spatial step of 0.01 m (used in the lysimeter simulation), the maximum errors between the semi-analytical solution and the semi-coupled method from rHZ exceeded those from MOIST (Table 4). This evidence, along with the above-mentioned discrepancy underscores the robustness and effectiveness of the fully coupled approach as adopted in MOIST.
### Applications for MOIST
As a one-dimensional isotope transport simulator, MOIST contributes to enhancing the precision of calibrated model parameters, such as saturated hydraulic conductivities and isotopic longitudinal dispersivity, as demonstrated in this study (Table 5). This improvement is attributed to the fact that the fully coupled method employed in MOIST reduces errors of simulated isotopic signals. Previous studies confirmed that incorporating isotopic and soil water information can enhance the accuracy of model calibration ([PERSON] and [PERSON], 2020; [PERSON] et al., 2022). However, model accuracy is a fundamental requirement. When the model itself generates numerical errors, these errors are compensated by calibrated parameters, potentially leading to biased parameter values. For instance, [PERSON] et al. (2022) obtained the calibrated soil saturated hydraulic conductivity was two orders of magnitude smaller than that reported by [PERSON] et al. (2012). The greater \(K_{\text{sat}}\) from [PERSON] et al. (2012) is a result of ignoring fractionation. As is known, fractionation reduces the evaporative isotopic flux. To achieve a satisfactory fit between observations and predictions of outflow isotopic signals, neglecting fractionation forces longitudinal dispersivity to increased and downward convection to decreased ([PERSON] et al., 2022). However, it is worth noting that the spatial step used in [PERSON] et al. (2022) was 0.01 m, which could produce significant numerical errors (Table 3). Such errors can lead to overly enriched isotopic signals, thereby amplifying the influence of fractionation ([PERSON] et al., 2016). Consequently, it remains unclear whether the impact of fractionation on model calibration results from fractionation itself or from compensation for numerical errors. By contrast, the fully coupling scheme of MOIST leads to more accurate isotope simulations and consequently has a smaller impact on parameter calibration compared to the semi-coupled method (Table 5). This not only enables MOIST to derive more accurate model parameters but also potentially offers a better understanding of the impact of evaporative fractionation on model calibration.
Moreover, MOIST can be used to test assumptions and validate methods in water transit time estimations ([PERSON] et al., 2022) and evaporation estimation studies ([PERSON] et al., 2017; [PERSON] et al., 2021; [PERSON] et al., 2021). This is because MOIST can construct high-precision and high-resolution spatial-temporal distributions of isotopic species, which is crucial for studying water transit time ([PERSON] et al., 2022). Understanding water transit time, also known as water age distribution, is essential for comprehending the connectivity between sources of transpiration, streamflow, groundwater recharge ([PERSON] et al., 2022), and the water cycles in the critical zone ([PERSON] et al., 2022). However, challenges persist in obtaining high-precision soil and plant water isotope data ([PERSON] et al., 2022), despite related research efforts ([PERSON] et al., 2016; [PERSON] et al., 2017). Additionally, various methods for water age estimation based on discrete isotopic measurements often exhibit discrepancies and large uncertainties ([PERSON] et al., 2022; [PERSON] et al., 2023). For instance, in the Bruntland Burn catchment, [PERSON] et al. (2018a) simulated a watershed water age of approximately 2 years (using the instantons age mixing method), contrasting with 1.2 years (using flux tracing) reported by [PERSON] et al. (2015) and approximately 1 year (using StorAge Selection) reported by [PERSON] et al. (2017). Due to the absence of true values, the accuracy and effectiveness of different methods remain unknown. Addressing this issue, MOIST can generate high-resolution spatial-temporal isotopic profiles, enabling the derivation of transit time from these profiles to serve as ground truth, especially incorporating with recent developed continuous plant water sourcing models, such as CrisPy ([PERSON] et al., 2024) and PRIME ([PERSON] et al., 2024). While other semi-coupled method-based numerical models can perform similar tasks, the accuracy of isotopic profiles may be affected by numerical oscillations (Figures 5 and 7), leading to increased uncertainties. Although MOIST is a one-dimensional model and water transit time studies are primarily conducted at the watershed scale ([PERSON] et al., 2022), smaller-scale experiments coupled with accurate modeling serve as the foundation for upscaling ([PERSON] et al., 2021; [PERSON] et al., 2019). Therefore, MOIST holds promise for transit time studies and cross-validating various water transit time estimation methods.
Regarding the evaporation estimation, MOIST can validate methods for calculating short-term and long-term evaporation based on measured isotopic compositions of soil water. For example, [PERSON] et al. (2017) determined evaporation rates from measured soil water isotopic compositions, while [PERSON] et al. (2021) computed long-term evaporation rates from deep soil isotopic compositions. Both approaches rely on the assumption that sampled soil isotopic compositions at specific depths (shallow or deep soil) retain the evaporation signal. Understanding how much evaporation signal is preserved by the isotopic composition at a given depth is crucial for assessing the accuracy of calculated evaporation rates from methods such as those of [PERSON] et al. and [PERSON] et al. This assessment necessitates high-resolution (both temporal and spatial) data on soil evaporation rates and soil water isotopic compositions, which can be challenging to obtain. MOIST can bridge this gap through simulations.
### Limitations and Future Work
First, in the long-term lysimeter study simulation (Figure 10), a notable observation emerged regarding the simulated \(\delta^{18}\)O values. Regardless of whether the model is calibrated or not, or whether it considers fractionation, there was a consistent underestimation of \(\delta^{18}\)O between the 1,200 th and 1,500 th days. This discrepancy can likely be attributed to the phenomenon of preferential flow. During this period, there was a significant influx of precipitation that was relatively enriched in \(\delta^{18}\)O, occurring between the 1,050 th and 1,300 th days. This enriched rainfall contributed to the \(\delta^{18}\)O enrichment of water reaching the bottom of the lysimeter. Consequently, a portion of this enriched water was transported to the bottom of the soil column via preferential flow pathways, resulting in an overall increase in \(\delta^{18}\)O of drainage water. It is worth noting that in the MOIST, rHZ, and rHS models, preferential flow is not considered for isotope transportation. This omission leads to the systematic underestimation of \(\delta^{18}\)O.
Previous research has demonstrated that different modes of soil water mobility, such as preferential and piston flow, as well as mobile and immobile soil water, can significantly influence the isotopic composition of bulk soil water ([PERSON] et al., 2022; [PERSON] et al., 2018). This effect becomes particularly pronounced when substantial rainfall infiltration occurs, potentially giving rise to significant preferential flow events that closely mirror the isotopic signals of the input water. By contrast, the current version of MOIST does not account for preferential flow, resulting in incomplete simulations. To rectify this limitation and better understand hydraulic processes, it is imperative to incorporate preferential flow and its impact on isotopic variations in bulk soil water. Neglecting these factors in isotope simulations can lead to biased interpretations of runoff generation and evaporation ([PERSON] et al., 2015). Therefore, future iterations of MOIST should consider implementing a two-pore domain model to provide a more comprehensive simulation of isotope transport processes within the soil. Such enhancements are likely to result in improved simulation accuracy ([PERSON] et al., 2018).
Second, the current version of MOIST relies on field measurements from [PERSON] et al. (2012) to represent \(\delta_{a}\) during rain-free period when simulating the long term lysimeter study. By contrast, the approach presented by [PERSON] et al. (2021) for estimating \(\delta_{a}\) is locally suitable. However, this local method was unable to be applied in MOIST because in [PERSON] et al. (2021), the isotope value of topmost soil layer can be computed from the solution of each time step and used in this local method directly. By contrast, MOIST operates on a cell-centered scheme, resulting in the surface soil water isotopic compositions being unknown at the commencement of each time step. Deriving the isotope concentrations of surface soil water in MOIST necessitates computation based on isotope mass balance between the top-half layer and the isotope value in the atmosphere (refer to Equation 25). Consequently, the formula proposed by [PERSON] et al. (2021) is inapplicable in MOIST as one equation cannot resolve two unknowns simultaneously. In the future iterations of MOIST, we aim to incorporate the equations for local \(\delta_{a}\) to better describe isotope transport processes.
Lastly, the primary objective of this study is to assess the performance of MOIST under various conditions and demonstrate the performance of the fully coupled method. Consequently, during the model comparison, we refrained from conducting an exhaustive calibration for MOIST. Notably, with the inclusion of calibrated saturated hydraulic conductivities, our model exhibited smaller errors compared to other models. Furthermore, the calibrated ranges of saturated hydraulic conductivities were smaller than the initial values (as shown in Table 5), aligning with the findings of [PERSON] et al. (2022) under the same data set. This alignment underscores the acceptability of the calibrated saturated hydraulic conductivities and longitudinal dispersivity. It instills confidence that a more comprehensive calibration process could further enhance the capacity of MOIST for accurate simulations.
We concur with the notion that for practical model applications, comprehensive calibration using independent data is indispensable ([PERSON] et al., 2018; [PERSON] et al., 2021). This necessity arises due to the inherent uncertainties associated with many parameters. Calibration serves the purpose of refining the model by imposing constraints, resulting in a more accurate representation of potential hydrological processes. Therefore, in forthcoming iterations, we intend to implement comprehensive calibration as an integral step toward practical applications.
## 5 Conclusions
We developed MOIST, a novel soil water and isotope transport model using MATLAB programming language. MOIST is unique in that it solves water, vapor, heat, and isotope transport simultaneously with a fully coupledmethod and with a cell-centered numerical approximation to the derivatives. MOIST successfully passed well-known theoretical tests, and semi-analytical solutions. Due to its adoption of fully coupled solution method, MOIST can predict isotope profiles under various temporal and spatial discretization with greater accuracy and stability than the existing semi-coupled models. We also tested MOIST against well-controlled long-term lysimeter studies at HBLFA Raumberg-Gumpenstein in Austria. Results showed good agreement between measured and predicted values and outperformed HYDRUS-1D. Given the evidence accrued in this study, it can be concluded that MOIST can be a robust tool for simulating one-dimensional isotope transport within soil. Moreover, the model is open-source and thus can be customized according to different requirements. As such, the adopted equations and parameters can be easily updated as our knowledge about isotope fractionation and transport continues to expand, making MOIST a suitable tool for current and future exploration of isotope transport in the soil-vegetation-atmosphere continuum.
## Data Availability Statement
Source codes of MOIST ([PERSON] & Si, 2023) used for semi-analytical tests, theoretical tests, calibration, and the long term lysimeter study tests were developed by MATLAB R2022a and available via Creative Commons Attribution 4.0 International license through [[https://zenodo.org/records/8397416](https://zenodo.org/records/8397416)]([https://zenodo.org/records/8397416](https://zenodo.org/records/8397416)). The long term lysimeter measurements are available in [PERSON] et al. (2012).
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wiley/ffea1893_168d_4360_9077_d9071300eca6.md | ([PERSON] & [PERSON], 1981; [PERSON] & [PERSON], 2018) is inconsistent with the elastic properties reported so far for solid iron phases ([PERSON] et al., 2007; [PERSON] et al., 2022; [PERSON], 2007; [PERSON] & [PERSON], 2022; [PERSON] et al., 2001), an observation that requires the introduction of other mechanisms such as interstitial impurities ([PERSON] et al., 2022), melt ([PERSON], 2007) and/or polycrystallinity ([PERSON] et al., 2007; [PERSON] & [PERSON], 2022) in quantitative microscopic models of the inner core.
On the theory side, atomistic simulations have attempted to shed light on the phase diagram of solid iron at core conditions, but simulations are challenged by the large sizes and long times required to explore the complex phase space of possible configurations and by the necessity to retain _ab-initio_ energetic accuracy in order to distinguish between the different structures. The \(T_{m}\) determined by atomistic methods at ICB pressures ranges from 5,400 ([PERSON] et al., 2000) to 6,900 K ([PERSON] & [PERSON], 2009), depending on the simulation size, method, and interatomic potential employed. The dependence of the thermodynamical properties on the size of the simulation is especially problematic in the case of bcc iron, whose thermodynamical and mechanical stability (including elastic stability, i.e. stability against strain distortions) are still debated. Although bcc iron is mechanically unstable at low temperature ([PERSON] et al., 2003), _ab-initio_ lattice dynamics ([PERSON] et al., 2010) and molecular dynamics (MD) ([PERSON] et al., 2003; [PERSON] et al., 2003) studies with small simulation cells found that vibrational anharmonicities make bcc iron mechanically stable at high temperatures, and contribute to lower its free energy to values that are close to that of hcp iron at core conditions ([PERSON] & [PERSON], 2023; [PERSON] et al., 2023; [PERSON] et al., 2003). A recent _ab-initio_ study with 2,000 atoms has confirmed that bcc iron is mechanically stable at 360 GPa and 6,600 K ([PERSON] et al., 2022). However, earlier simulations conducted with up to 180 atoms ([PERSON] et al., 2015) found that bcc iron violates the Born stability criteria, and thus is elastically unstable at Earth's core conditions. Therefore, it appears that the mechanical stability of bcc iron is remarkably sensitive to the simulation size.
More importantly, anomalously fast self-diffusion has been observed in large-scale MD simulations for bcc iron ([PERSON] et al., 2017), while no self-diffusion has been reported in simulations involving small cells ([PERSON] & [PERSON], 2023; [PERSON] et al., 2023; [PERSON] et al., 2003). Large self-diffusion, if confirmed, invalidates the use of most theoretical methods used so far to determine the free energies of solid iron at core conditions. For example, phonon-based methods are based on the principle that the direct use of the ideal crystal acts as a reference state, but the principle is clearly violated in the presence of substantial atomic self-diffusion. In addition, it casts doubt on the accuracy of simulations carried out with cell sizes and time-scales that may not be sufficiently large to show self-diffusion ([PERSON] & [PERSON], 2023; [PERSON] et al., 2023; [PERSON] et al., 2003), as already pointed out in [PERSON] et al. (2017).
In this Letter, we determine the properties and relative stability of different structures of iron at Earth's core conditions based on an approach that retains the accuracy of _ab-initio_ methods but where the statistical sampling is accelerated with the help of a deep-learning model. The approach allows us to include atomic self-diffusion in the crystalline phases and more generally to overcome the limits on the system size and temporal scales faced by standard _ab-initio_ methods. Our results indicate that Gibbs free-energy differences between competing structures for pure Fe are small enough to open the possibility that the stability ordering of phases could be overturned by the presence of impurities, whose role is therefore crucial not only to account for the density deficit and for geochemical observations, but also to stabilize the correct crystal structure of Fe in the Earth's inner core. Different competing phases however differ substantially in terms of elastic properties and seismic velocities, with bcc Fe coming much closer to seismic data for the shear wave velocity of the inner core than all other considered structures.
## 2 Methodology
### The Deep-Learning Interatomic Potential
Although the accuracy of _ab-initio_ methods based on density-functional theory (DFT) ([PERSON], 1964; [PERSON], 1965; [PERSON], 1965) in describing the potential energy surface of a large system of atoms is unsurpassed, _ab-initio_ simulations are challenged by the large simulation sizes and times that are required to obtain free energies and elastic constants that are precise enough to solve the most relevant geophysical questions such as the crystal structure of the inner core and its shear wave velocities. Deep-learning interatomic potentials (DLP), based on deep neural networks, have emerged as an efficient method to accurately describe the DFT potential energy surface in a computationally cost-effective manner. In our recent work ([PERSON] & [PERSON], 2024),we constructed and validated a set of potentials for iron that cover hcp, fcc, and bcc structures as well as the liquid phase. These potentials exhibit accuracy comparable to DFT within the temperature range of 4,000-7,600 K and in the pressure range of 75-650 GPa. Due to the significant role played by temperature-dependent thermal electronic excitation, five DLP models have been established at temperatures of 4,000, 5,000, 6,000, 7,000, and 7,600 K. We exclusively utilize DLP-6000, trained at 6,000 K, for simulations conducted at the same temperature. Rigorous validation tests demonstrate their accuracy in large-scale simulations and in the presence of extended defects. Additionally, we benchmarked their Gibbs free energy difference compared to the DFT potential, which is found to be less than 6 meV/atom or 1% of the thermal energy. Remarkably, this difference is shown to be independent of pressure, temperature, and the number of atoms, further confirming the applicability of these models to large-scale simulations. In addition, we obtain elastic constants and self-diffusion coefficients that are essentially indistinguishable from the _ab-initio_ results, in those situations where a comparison is possible. More details on the construction of the deep-learning models and their validation can be found in [PERSON]. [PERSON] and [PERSON] (2024) and in the Supporting Information (Texts S1-S8 in Supporting Information S1).
### Molecular Dynamics Simulations
Molecular dynamics (MD) simulations with the DLP models were performed using LAMMPS([PERSON], 1995). Except for the thermodynamic integration calculations to determine the free energy (see below), the Nose-Hoover thermostat ([PERSON], 1985) and Martyna-Tobias-Klein barostat ([PERSON] et al., 1994) were used to control temperature and pressure, respectively. The timestep was set to 1 fs below 7,000 K and 0.5 fs at 7,600 K. Simulations lasted for more than 100 ps to reduce the statistical uncertainty. We used the Langevin thermostat in these thermodynamic integration calculations since the Nose-Hoover thermostat has difficulties in achieving ergodicity when the system is close to be harmonic, as discussed below.
### Free Energy Calculations
To determine the phase stability and melting line, we calculated free energies for hcp, fcc, bcc and liquid iron using the thermodynamic integration method (TDI) method, which involves computing the free energy difference between the system of interest and a reference system with known free energy (also see Texts S3-5 in SI). We used the Einstein crystal as the reference for solids and the ideal gas for the liquid phase. The free energy calculations followed a three-step sequence. First, we applied TDI to compute the free energy difference between the reference system and a Lennard-Jones (LJ) model developed using the force-matching algorithm ([PERSON] et al., 2004). Next, we determined the free energy difference between the LJ and DLP models. Finally, we applied a correction from DLP based on free energy perturbation theory to achieve DFT accuracy, with the correction typically as low as 6 meV/atom ([PERSON] et al., 2000; [PERSON] & [PERSON], 2024). Error bars on melting lines include all statistical uncertainties arising from the calculation of free energies.
Special consideration is required when determining the free energy for the bcc phase. Since direct TDI simulations from a diffusive system to the Einstein crystal are not physically feasible, we tuned the parameters of the LJ potential to ensure that it has no self-diffusion and then determined its free energy. Subsequently, we transitioned from LJ to the DLP model for the bcc phase. We have thoroughly checked the continuity of the transformation and found the absence of first-order transitions by monitoring self-diffusivity as the coupling constant varied from the LJ to DLP sides. It should be noted that a similar method has been employed to determine the free energy of a superionic phase ([PERSON] & [PERSON], 2018). We have performed extra tests based on the coexistence method ([PERSON] et al., 1994) to obtain the free energy for bcc iron, and the resulting free energy shows good agreement with the TDI method (see Text S4 in SI). Additionally, we have verified the DFT correction based on perturbation theory is converged with system size by using a simulation cell up to 1,024 atoms, where the correction is typically less than 3 meV/atom ([PERSON] & [PERSON], 2024). The effects of self-diffusion have been properly taken into account by the configurations extracted from MD simulations using DLP models that exhibit self-diffusion.
### Mechanical Stability and Elastic Constants
The mechanical stability and the elastic constants of bcc and fcc iron were calculated using the stress-strain method as in our previous studies ([PERSON] & [PERSON], 2022). In this method, the deviatoric stress on a distorted simulation cell after applying a small strain was estimated by performing MD simulations in the canonical diffusion in the bcc structure, independent of the simulation size (Figure 1). However, diffusion events are more rare in the 128-atom system where more than 100 ps are required to observe a few diffusion events. The calculated self-diffusion coefficient for bcc iron \(\left(2\times 10^{-10}\mathrm{m^{2}/s}\right)\) agrees with previous _ab-initio_ results \(\left(1.8\times 10^{-10}\mathrm{m^{2}/s}\right)\)([PERSON] et al., 2019), which further confirms the accuracy of the DLP employed in our study, and is only an order of magnitude smaller than in the liquid ([PERSON] et al., 2000).
We found that diffusion takes place primarily by cooperative atomic hopping to the nearest lattice sites along the \(\left(111\right)\) crystallographic direction. In rare cases, iron atoms also move along the \(\left(100\right)\) or \(\left(110\right)\) directions (see Figures S2-2 in Supporting Information S1). It should be noted that the observed diffusion direction differs from that reported in ([PERSON] et al., 2017), where diffusion was observed only along the \(\left(110\right)\) direction.
Large and cooperative self-diffusion at high temperatures has been observed in other bcc metals (Kadkhodaei and Davariashtiyani, 2020). We find that atomic diffusion is the result of rapid and temporally well-defined concerted events during which all atoms exchange their positions cooperatively in a loop. In some cases the diffusion event starts with the creation of a vacancy-interstitial pair and terminates with the annihilation of the defect (see Figures S2-2 in Supporting Information S1) We stress that, from a practical point of view, the presence of self-diffusion invalidates the use of approaches for the calculation of free energies directly based on the ideal crystal as a reference, such as anharmonic phonon approximations ([PERSON] et al., 2017) and direct thermodynamic integration from the [PERSON] crystal ([PERSON] et al., 2003).
As a necessary condition for thermodynamic stability, the local mechanical stability of bcc iron at Earth's core conditions was investigated by performing molecular dynamics simulations with up to 128,000 atoms at different pressure and temperature conditions. In addition to monitoring the presence of bcc long-range order in the simulation cell, the stability of bcc iron against elastic instabilities was checked by determining the elastic constants with the stress-strain method. We found that bcc iron becomes mechanically stable in a narrow range of temperatures close to the melting point (Figure 1). Below the Born stability line, bcc iron transforms, in our simulations, into an fcc structure (see Figures S6-4 and S6-5 in Supporting Information S1), instead of the hcp iron at 360 GPa. In a previous study ([PERSON] et al., 2015), the transformed structure from bcc iron was classified as hcp. However, their analysis was based on a comparison of pair-distribution functions and on the use of a small 216-atom simulation cell. The fcc and hcp structures exhibit nearly identical pair-distribution functions below 5 A ([PERSON] and [PERSON], 2024), and a correct assignment becomes challenging with small cells.
It has been argued, based on self-consistent anharmonic lattice dynamics calculations, that the temperature-induced mechanical stability of bcc iron is due to phonon anharmonicity ([PERSON] et al., 2010). We calculated the
Figure 1: Calculated melting lines for the solid phases considered in this work. Rec iron is mechanically unstable below the instability line (light blue). The inset shows the atomic mean-squared displacements for bcc iron from simulations at 360 GPa and 7,000 K, with 128 and 1,024 atoms, averaged over initial times. The slope of the mean square displacement with 1,024 atoms corresponds to a diffusion coefficient of \(2\times 10^{-10}\mathrm{m^{2}/s}\).
second-order derivative of the free energy of bcc iron with respect to atomic displacements from the ideal crystal positions and found the presence of unstable modes with negative curvature at all temperatures from 4,000 to 7,000 K (see Figures S6-6 in Supporting Information S1). Contrary to previous claims ([PERSON] et al., 2010), and in agreement with more recent MD simulations ([PERSON] et al., 2017), we conclude that anharmonic contributions to the lattice free energy are not sufficient to stabilize bcc iron mechanically, and that therefore self-diffusion is crucial to explain the mechanical stability of bcc iron at 6,000 K.
### Relative Phase Stability at Earth's Core Conditions
Having demonstrated the mechanical stability of bcc iron at core conditions, we calculated the Gibbs free energies for bcc, hcp, fcc and liquid iron, and determined the phase diagram of iron from 100 GPa to 4,000 K up to 550 GPa and 7,600 K. Gibbs free energies were determined by the thermodynamic integration method ([PERSON] et al., 1999). Although the maximal discrepancy between the DLP model and the DFT energies is only 6 meV/atom ([PERSON] & [PERSON], 2024), Gibbs free energies determined with the DLP model were nonetheless corrected to achieve _ab-initio_ accuracy through free energy perturbation theory ([PERSON] et al., 2002; [PERSON] & [PERSON], 2024).
The results, shown in Figure 2, confirm that hcp iron is the only thermodynamically stable solid phase of iron at Earth's core conditions. Consistently, hcp iron melts at higher temperatures than all other solid structures, and the hypothetical melting lines of fcc and bcc iron follow their relative stability order (bcc \(<\) fcc \(<\) hcp). Our calculated melting curve for hcp iron compares well with recent initio studies ([PERSON], 2009; [PERSON] et al., 2002) with a maximal difference of 80 K, which is less than the uncertainty of 100 K in these studies. The melting temperature of \(6450\pm 25\) K at the Earth's inner core boundary (330 GPa), agrees well with previous experimental studies with shock waves (\(6230\pm 540\) K) ([PERSON] et al., 2022) and fast X-ray diffraction (\(6230\pm 500\) K) ([PERSON] et al., 2013) but differs from the resistance-heated diamond anvil cell studies (\(5120\pm 390\) K) ([PERSON] et al., 2019). A large discrepancy is present compared to the previous classical-potential-based MD simulations ([PERSON] et al., 2000; [PERSON] et al., 2000), which we attribute to the inaccuracy of these classical potentials ([PERSON] et al., 2002). There is a good agreement between our calculated hcp melting points at 360 GPa (6,715 \(\pm\) 20 K) and those reported in more recent work by ([PERSON] et al., 2023) (6,692 \(\pm\) 45 K), suggesting that different DFT implementations yield a similar accuracy of the reported thermophysical properties. However, the reported bcc melting point (6,519 \(\pm\) 80 K) in their study is much higher than in our study (6,164 \(\pm\) 30 K). Once again, we believe that the difference might be caused by the large simulation sizes required to converge the free energy of the bcc structure.
The Gibbs free energies at Earth's inner core conditions of the three structures considered here, namely hcp, fcc, and bcc, are all within a narrow range of about 30 meV/atom, more than an order of magnitude smaller than at
Figure 2: (a) Calculated melting line of hcp iron, compared to previous theoretical ([PERSON], 2009; [PERSON] et al., 2002; [PERSON] et al., 2000; [PERSON] et al., 2000; [PERSON] & [PERSON], 2009) and experimental results ([PERSON] et al., 2013; [PERSON] et al., 2022; [PERSON] et al., 2020; [PERSON] et al., 2019). (b) Gibbs free energy difference between different structures of Fe at 400 GPa, with respect to hcp.
ambient temperature (700 meV/atom). We also find that temperature-induced electronic excitations provide a significant contribution to this reduction ([PERSON] & [PERSON], 2024).
There is a strong debate in the literature about the thermodynamic and mechanical stability of bcc iron due to the lack of an accurate and consistent approach to reach DFT accuracy with the large cell sizes required to capture the effects of self-diffusion. In this study, we employ a single, consistent, and extensively validated deep-learning-based approach, as thoroughly described in ([PERSON] & [PERSON], 2024), to clarify the role played by self-diffusion. Thanks to this new approach, we are in a position to rationalize and place on a much stronger and coherent basis a collection of mutually inconsistent earlier results. We found that self-diffusion is not sufficient to stabilize bcc iron thermodynamically over other competing phases. The finding disagrees with the only other study where self-diffusion was reported ([PERSON] et al., 2017). However the entropic contribution to the free energy in [PERSON] et al. (2017) was determined on a qualitative basis only. In addition, [PERSON] et al. (2017) propose that self-diffusion deactivates a soft-mode driven transition from bcc to hcp along the \(\left\langle 110\right\rangle\) direction. In our study, we found that the self-diffusion mechanism is more complicated and occurs not only along the \(\left\langle 110\right\rangle\) direction. Additionally, in our phonon simulations we find a low-temperature soft mode that drives the bcc phase into the fcc structure instead of hcp.
We also found that the melting line of bcc iron rises above its elastic instability limit at pressures above 260 GPa, indicating that bcc iron is locally mechanically stable at core conditions, although in a narrow range of temperatures. In a previous theoretical study by ([PERSON] et al., 2015), bcc iron was found to be mechanically unstable even at 7000 K along the Bain deformation path, suggesting that bcc iron lies at a saddle point rather than at a local minimum in the potential energy surface. With our new data, we can reinterpret their findings. At lower temperatures, the phase transition is driven by mechanical instability, while at 7000 K, it is driven by free energy.
There is a strong debate in the literature about the thermodynamic and mechanical stability of bcc iron due to the lack of an accurate and consistent approach to reach DFT accuracy with the large cell sizes required to capture the effects of self-diffusion. In this study, we employ a single, consistent, and extensively validated deep-learning-based approach, as thoroughly described in [PERSON] and [PERSON] (2024), to clarify the role played by self-diffusion. Thanks to this new approach, we are in a position to rationalize and place on a much stronger and coherent basis a collection of mutually inconsistent earlier results. We found that self-diffusion is not sufficient to stabilize bcc iron thermodynamically over other competing phases. The finding disagrees with the only other study where self-diffusion was reported ([PERSON] et al., 2017). However the entropic contribution to the free energy in ([PERSON] et al. (2017)) was determined on a qualitative basis only. In addition, [PERSON] et al. (2017) propose that self-diffusion deactivates a soft-mode driven transition from bcc to hcp along the \(\left\langle 110\right\rangle\) direction. In our study, we found that the self-diffusion mechanism is more complicated and occurs not only along the \(\left\langle 110\right\rangle\) direction. Additionally, in our phonon simulations we find a low-temperature soft mode that drives the bcc phase into the fcc structure instead of hcp.
## 4 The Origin of the Low Shear Velocity in the Earth's Inner Core
Among the observed physical properties on the Early inner core, the low shear velocity remains difficult to reconcile with the intrinsic elastic properties of hcp iron. It has been proposed that grain boundaries might play an important role, and previous simulations have demonstrated a very low shear velocity due to viscous grain boundaries ([PERSON] et al., 2007; [PERSON]. [PERSON] & [PERSON], 2022) or the premelting effects ([PERSON] et al., 2013). However, it remains unclear whether this mechanism will work for geophysically relevant grain sizes. The role of light impurities in interstitial sites might also be significant ([PERSON] et al., 2022), but future studies are needed to understand whether they affect the stability of the solid phases.
Except for the above possibilities, we have found that a striking feature of bcc iron is its low shear velocity compared to other phases, as shown in Figure 3. The anomaly is caused by soft transverse acoustic phonons that lead to a small value for \(c^{\prime}=c_{11}-c_{12}\) and to the result that the shear velocity is essentially controlled by \(c_{44}\). The calculated shear velocity of bcc iron at its melting point and at 360 GPa is 3.60 km/s, in remarkable agreement with seismic observations (3.68 km/s ([PERSON] & [PERSON], 1981) or 3.58 km/s ([PERSON] & [PERSON], 2018)), which makes of bcc iron a strong candidate for the structure of iron in the inner core. For comparison, the calculated shear velocities of hcp and fcc iron at their melting points and 360 GPa are 4.4 and 4.5 km/s, respectively.
atomic self-diffusion, and it is the only structure whose shear sound velocity matches seismic data. Future work is needed to clarify the role played by light elements in the thermodynamic stability of the different iron phases at core conditions.
## Data Availability Statement
The first-principles calculations were performed using the Quantum ESPRESSO package ([PERSON] et al., 2009). The DeePMD package was used to train the deep-learning interatomic potential ([PERSON] et al., 2018). The raw data supporting the findings of this study are available on Zenodo ([PERSON], 2024b). The DFT data set and interatomic potentials can be accessed on Zenodo ([PERSON], 2024a).
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Figure 3: Shear velocities of bcc and fcc iron at Earth’s inner core conditions (360 GPa), compared to previous theoretical studies ([PERSON] et al., 2022; [PERSON]. [PERSON] & [PERSON], 2022; [PERSON] et al., 2013, 2015, [PERSON] & [PERSON], 2009). The velocity of shear seismic waves in the Earth’s inner core as reported in the preliminary reference Earth model (PREM) is also shown ([PERSON] & [PERSON], 1981).
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* [PERSON] et al. (2017) [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], et al. (2017). Advanced capabilities for materials modelling with Quantum ESPRESSO. _Journal of Physics: Condensed Matter_, 29(46), 465901. [[https://doi.org/10.1088/1361-648%aaf79](https://doi.org/10.1088/1361-648%aaf79)]([https://doi.org/10.1088/1361-648%aaf79](https://doi.org/10.1088/1361-648%aaf79))
* [PERSON] et al. (2011) [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], et al. (2011). Scikit-learn: Machine learning in Python. _Journal of Machine Learning Research_, 12(85), 2825-2830.
* [PERSON] et al. (1996) [PERSON], [PERSON], & [PERSON] [PERSON] (1996). Generalized gradient approximation made simple. _Physical Review Letters_, 77(18), 3865-3868. [[https://doi.org/10.1103/physrevlett.77.3865](https://doi.org/10.1103/physrevlett.77.3865)]([https://doi.org/10.1103/physrevlett.77.3865](https://doi.org/10.1103/physrevlett.77.3865))
* [PERSON] et al. (2000) [PERSON], [PERSON], [PERSON], [PERSON], & [PERSON] (2000). Finite-size corrections to the free energies of crystalline solids. _The Journal of Chemical Physics_, 12(112), 5339-5342. [[https://doi.org/10.1063/1.48102](https://doi.org/10.1063/1.48102)]([https://doi.org/10.1063/1.48102](https://doi.org/10.1063/1.48102))
* [PERSON] et al. (2013) [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], & [PERSON] (2013). Electronic properties and magnetism of iron at the Earth's inner core conditions. _Physical Review B_, 87(11), 115130. [[https://doi.org/10.1103/physrevb.87.115130](https://doi.org/10.1103/physrevb.87.115130)]([https://doi.org/10.1103/physrevb.87.115130](https://doi.org/10.1103/physrevb.87.115130))
* [PERSON] et al. (2013) [PERSON], [PERSON], [PERSON], & [PERSON] [PERSON] (2013). Impact of magnetism on Fe under Earth's core conditions. _Physical Review B_, 87(11), 014048. [[https://doi.org/10.1103/physrevb.87.014045](https://doi.org/10.1103/physrevb.87.014045)]([https://doi.org/10.1103/physrevb.87.014045](https://doi.org/10.1103/physrevb.87.014045))
* [PERSON] et al. (2021) [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON] [PERSON] (2021). Phase diagram of a deep potential water model. _Physical Review Letters_, 126(23), 236001. [[https://doi.org/10.1103/physrevlett.126.236001](https://doi.org/10.1103/physrevlett.126.236001)]([https://doi.org/10.1103/physrevlett.126.236001](https://doi.org/10.1103/physrevlett.126.236001)) | wiley | Competing Phases of Iron at Earth's Core Conditions From Deep‐Learning‐Aided <i>ab‐initio</i> Simulations | Zhi Li, Sandro Scandolo | https://doi.org/10.1029/2024gl110357 | 2,024 | CC-BY |
wiley/ffe1c238_a022_40fc_9746_b2de1a036adc.md | "In this study, we focus on an end-member type of detachment faulting observed along almost magmatic(...TRUNCATED) | wiley | "How Hydrothermal Cooling and Magmatic Sill Intrusions Control Flip‐Flop Faulting at Ultraslow‐S(...TRUNCATED) | Arne Glink, Jörg Hasenclever | https://doi.org/10.1029/2023gc011331 | 2,024 | CC-BY |
wiley/ffdcd525_85e5_4165_86dc_a6d1ff23b7c0.md | "# Geophysical Research Letters+\nFootnote †: This is an open access article under the terms of th(...TRUNCATED) | wiley | Global Prediction of Flash Drought Using Machine Learning | Lei Xu, Xihao Zhang, Tingtao Wu, Hongchu Yu, Wenying Du, Chong Zhang, Nengcheng Chen | https://doi.org/10.1029/2024gl111134 | 2,024 | CC-BY |
wiley/ffd4c007_fe7f_4e1f_90e1_af8da5bcac6e.md | "# Geophysical Research Letters+\nFootnote †: This is an open access article under the terms of th(...TRUNCATED) | wiley | "Considerable Uncertainty of Simulated Arctic Temperature Change in the Mid‐Holocene Due To Initia(...TRUNCATED) | Jian Shi, Gerrit Lohmann | https://doi.org/10.1029/2023gl106337 | 2,024 | CC-BY |
wiley/ffd08a09_4d50_4741_ac35_08ed72f89c0c.md | "# Earth and Space Science\n\nResearch Article\n\n10.1029/2021 EA001645\n\nThemal and Physical Prope(...TRUNCATED) | wiley | "Thermal and Physical Properties of Deccan Basalt and Neoarchean Basement Cores From a Deep Scientif(...TRUNCATED) | Labani Ray, Ravindra Kumar Gupta, Nishu Chopra, Dhulipudi Gopinadh, Sujeet Kumar Dwivedi | https://doi.org/10.1029/2021ea001645 | 2,021 | CC-BY |
wiley/ffc23f97_b278_4dd3_8af6_c75ce9e3a5db.md | "Illuminating the Arctic: Unveiling seabird responses to artificial light during polar darkness thro(...TRUNCATED) | wiley | "Illuminating the Arctic: Unveiling seabird responses to artificial light during polar darkness thro(...TRUNCATED) | Kaja Balazy, Dariusz Jakubas, Andrzej Kotarba, Katarzyna Wojczulanis‐Jakubas | https://doi.org/10.1002/rse2.425 | 2,024 | CC-BY |
wiley/ffb975e5_3075_47f7_8fad_c899fab33107.md | "# Geophysical Research Letters+\nFootnote †: This is an open access article under the terms of th(...TRUNCATED) | wiley | "Constraints on the Projected Tropical Pacific Sea Surface Temperature Warming Pattern by the Tropic(...TRUNCATED) | Jun Ying, Matthew Collins, Robin Chadwick, Jian Ma, Tao Lian | https://doi.org/10.1029/2024gl111233 | 2,024 | CC-BY |
wiley/ff9c0e3f_cc49_454c_b255_53e4c69833ec.md | "# Earth and Space Science\n\nResearch Article\n\n10.1029/2023 EA002909\n\nReonstructing and Nowcast(...TRUNCATED) | wiley | Reconstructing and Nowcasting the Rainfall Field by a CML Network | Peng Zhang, Xichuan Liu, Mingzhong Zou | https://doi.org/10.1029/2023ea002909 | 2,023 | CC-BY |
wiley/ffb6a6af_0845_4ca2_936e_60eaaf2aa7fd.md | "## 2 Methods\n\n### Emulators and Simulators of Air Quality\n\nWe trained emulators to predict air (...TRUNCATED) | wiley | Emission Sector Impacts on Air Quality and Public Health in China From 2010 to 2020 | "Luke Conibear, Carly L. Reddington, Ben J. Silver, Ying Chen, Stephen R. Arnold, Dominick V. Sprack(...TRUNCATED) | https://doi.org/10.1029/2021gh000567 | 2,022 | CC-BY |
End of preview. Expand in Data Studio
EVE-Corpus
EVE-Corpus is a large-scale, cleaned, and anonymized text corpus of Earth Observation (EO) documents formatted in Markdown.
It is designed to support research in EO and domain-specific LLM training.
The corpus contains 186k Markdown files, sourced from peer-reviewed journals, EO websites and scientific repositories.
Dataset Features
- file_path (
string): the file path of the source document within the s3 bucket. - text (
string): full document content in Markdown format. - publisher (
string): Publisher or source of the document. - title (
string): Title of the document. - authors (
string): Author(s) of the document. - url (
string): Source URL. - year (
int64): Year of publication. - license (
string): Document license information.
Dataset Summary
EVE-Corpus aggregates high-quality EO-related text covering topics such as:
- Satellite missions
- Remote sensing techniques
- Climate and atmospheric science
- Land, ocean, and cryosphere monitoring
- Environmental modelling and geospatial analytics
All documents have been deduplicated, cleaned, and anonymized to remove personal information.
Data Distribution
The dataset includes 2.8B tokens from 22 EO-related sources.
| publisher | tokens | percentage |
|---|---|---|
| MDPI | 1.3B | 46.55 |
| Copernicus | 723M | 25.89 |
| NCBI | 485M | 17.37 |
| ISPRS | 74.1M | 2.65 |
| Wiley | 71.6M | 2.56 |
| Elsevier | 43.7M | 1.56 |
| Cambridge Press | 40.3M | 1.44 |
| Springer | 25.4M | 0.91 |
| Taylor and Francis | 11.8M | 0.42 |
| AMS | 5.7M | 0.2 |
| SAGE | 3.8M | 0.14 |
| NASA | 2.5M | 0.09 |
| Arxiv | 2.1M | 0.08 |
| IEEE | 992K | 0.04 |
| EGUP | 671K | 0.02 |
| Oxford Academic | 576K | 0.02 |
| Iopscience | 507K | 0.02 |
| Frontiers | 307K | 0.01 |
| EOGE | 245K | 0.01 |
| EOS | 96K | 0.003 |
| MIT | 83K | 0.003 |
| Uk Met Office | 7K | 0.0002 |
| Total | 2.8B | 100 |
Source Collection
The dataset was constructed using this scraping pipeline.
Preprocessing Pipeline
The dataset was processed using the following pipeline:
Extraction
- Supports PDF, HTML, XML, Markdown, and nested folder structures.
- Automatically detects file formats unless explicitly specified.
Deduplication
- Performs exact matching using SHA-256 checksums.
- Supports LSH-based near-duplicate detection (configurable: shingle size, permutations, similarity threshold).
Cleaning
- Removes irregularities and noise artifacts.
- Corrects LaTeX equations and tables using LLM assistance.
PII Removal
- Automatically masks names and emails using the Presidio framework.
Metadata Extraction
- Extracts Title, Authors, DOI, URL, Year, Journal, and Citation Count from scientific papers.
Export
- Saves processed content in multiple formats (default: Markdown).
Attribution info
Attributions to the original authors can be found in the attribution.txt. Each dataset entry has the file_path column that maps the document to the corresponding attribution entry
Citation
If you use this project in academic or research settings, please cite:
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