the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# SurEau-Ecos v2.0: a trait-based plant hydraulics model for simulations of plant water status and drought-induced mortality at the ecosystem level

### Julien Ruffault

### François Pimont

### Hervé Cochard

### Jean-Luc Dupuy

### Nicolas Martin-StPaul

A widespread increase in tree mortality has been observed around the globe, and this trend is likely to continue because of ongoing climate-induced increases in drought frequency and intensity. This raises the need to identify regions and ecosystems that are likely to experience the most frequent and significant damage. We present SurEau-Ecos, a trait-based, plant hydraulic model designed to predict tree desiccation and mortality at scales from stand to region. SurEau-Ecos draws on the general principles of the SurEau model but introduces a simplified representation of plant architecture and alternative numerical schemes. Both additions were made to facilitate model parameterization and large-scale applications. In SurEau-Ecos, the water fluxes from the soil to the atmosphere are represented through two plant organs (a leaf and a stem, which includes the volume of the trunk, roots and branches) as the product of an interface conductance and the difference between water potentials. Each organ is described by its symplasmic and apoplasmic compartments. The dynamics of a plant's water status beyond the point of stomatal closure are explicitly represented via residual transpiration flow, plant cavitation and solicitation of plants' water reservoirs. In addition to the “explicit” numerical scheme of SurEau, we implemented a “semi-implicit” and “implicit” scheme. Both schemes led to a substantial gain in computing time compared to the explicit scheme (>10 000 times), and the implicit scheme was the most accurate. We also observed similar plant water dynamics between SurEau-Ecos and SurEau but slight disparities in infra-daily variations of plant water potentials, which we attributed to the differences in the representation of plant architecture between models. A global model's sensitivity analysis revealed that factors controlling plant desiccation rates differ depending on whether leaf water potential is below or above the point of stomatal closure. Total available water for the plant, leaf area index and the leaf water potential at 50 % stomatal closure mostly drove the time needed to reach stomatal closure. Once stomata are closed, resistance to cavitation, residual cuticular transpiration and plant water stocks mostly determined the time to hydraulic failure. Finally, we illustrated the potential of SurEau-Ecos to simulate regional drought-induced mortality over France. SurEau-Ecos is a promising tool to perform regional-scale predictions of drought-induced hydraulic failure, determine the most vulnerable areas and ecosystems to drying conditions, and assess the dynamics of forest flammability.

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Forests across many regions worldwide are experiencing record-breaking droughts followed by widespread increase in climate-driven disturbance events, including tree mortality (Allen et al., 2015; Fettig et al., 2019; Schuldt et al., 2020), wildfires (Ruffault et al., 2020; Abram et al., 2021) and insect outbreaks (Jactel et al., 2012). Droughts are likely to become more frequent and more intense over the next decades because of the global increase in temperatures and heat waves, which are coupled in some regions to changes in the hydrological cycle (Trenberth et al., 2014). Given the importance of forests for biochemical cycles and ecosystem services (Seidl et al., 2014), there is a growing need for the development of models that can simulate the response of forests to extreme drought. Process-based vegetation models can help to address these issues because they represent the mechanisms governing plant physiological responses to drought and account for the interspecific and intraspecific variations of tree traits and their acclimation to a rapidly changing climate.

The science of plant hydraulics seeks to understand the physical and physiological mechanisms driving water transport in plants. This research field has proven to be a relevant theoretical framework to study the effect of global changes on plant and the terrestrial water cycle (Choat et al., 2018; Brodribb et al., 2020). Advances in plant hydraulic modeling have accelerated over the last 2 decades (Mencuccini et al., 2019; Fatichi et al., 2016) and are used as mean to tackle diverse prediction challenges, such as tree mortality (Venturas et al., 2020; De Kauwe et al., 2020), water use efficiency (Domec et al., 2017; De Cáceres et al., 2021) or species distribution (Sterck et al., 2011). Many of these models were also designed (or reformatted) to be integrated into land surface models and improve the representation of the feedbacks between land and climate systems (Xu et al., 2016; Li et al., 2021; Kennedy et al., 2019; Christoffersen et al., 2016). Recently, modeling water transport in plants also proved to be a promising way to assess the seasonal dynamics of live fuel moisture (foliage and twigs water content, dead to live fuel ratio), a key variable for fire behavior that could play a major role in raising forests' flammability under climate warming (Ruffault et al., 2018a; Nolan et al., 2020).

Most plant hydraulic models represent water fluxes in plants through the mathematical approach of the soil–plant–atmosphere (SPA) continuum, wherein diffusion laws control the water flow through the soil, roots and leaves (Mencuccini et al., 2019). Water flow through plants is considered to be analogous to the electrical current through a circuit with a series of resistance and/or capacitance factors (Sperry et al., 1998). SPA models, however, vary widely in their complexity, some of them representing trees as a single resistance (Mackay et al., 2003; Williams et al., 1996), while others include multiple resistances and capacitances (Sperry et al., 1998; Tuzet et al., 2017; Couvreur et al., 2018). How physiological processes regulate plant transpiration also differs between SPA models (Mencuccini et al., 2019). Some models describe stomatal conductance through semi-empirical models (Christoffersen et al., 2016; Williams et al., 1996; Li et al., 2021; Feng et al., 2018), while others are based on optimality approaches (Wang et al., 2020; Sperry et al., 2017).

The SurEau SPA model was developed specifically to simulate plant desiccation under extreme drought and heat waves (Martin-StPaul et al., 2017; Cochard et al., 2021). As in other SPA models, SurEau describes the soil–plant–atmosphere system as a network of resistances and capacitances and computes water exchanges until stomatal closure. Additionally, SurEau simulates plant tissue desiccation beyond the point of stomatal closure by accounting for residual plant transpiration and the discharge of internal plant water stores (Fig. 1a). Unlike most current approaches (Xu et al., 2016; Tuzet et al., 2017), SurEau explicitly accounts for the differences in capacitance of the symplasmic and apoplasmic compartments, which can be calibrated from pressure–volume curves for the symplasm and vulnerability curves for the apoplasm. Symplasmic capacitances mostly buffer water fluxes during well-watered conditions, whereas apoplasm capacitances come into play when cavitation occurs (Fig. 1a). Thus, SurEau accounts for the leading role of cavitation in the dynamics of plant desiccation (Mantova et al., 2021) and the probability of plant mortality (Adams et al., 2017). SurEau has been successfully evaluated against field cavitation observations (Cochard et al., 2021; hereafter CPRM21), has been applied in different contexts (Lemaire et al., 2021; López et al., 2021) and has performed well in predicting plant water fluxes when compared to other plant hydraulic models (McDowell et al., 2022).

As noted in CPRM21, two characteristics of SurEau impede its use for large-scale ecological applications or its integration into terrestrial biosphere models. First, SurEau requires a high number of parameters because of its detailed representation of plant architecture and the mechanisms involved in plant water exchanges. The second limitation of SurEau is its high computation time, which is partly due to the use of a first-order “explicit” numerical scheme to compute water flows. This scheme requires that variations in water quantities be computed at very small time steps to avoid numerical instabilities due to the Courant–Friedrichs–Lewy condition (CFL; Dutykh, 2016). A numerical method has been proposed to overcome these instabilities and increases the time step (Xu et al., 2016; Tuzet et al., 2017), but this is not directly compatible with SurEau's specificities regarding capacitances and cavitation. Moreover, knowledge regarding numerical physics and methods for simulation have seldom been applied to plant hydraulics.

We present SurEau-Ecos, a new SPA model meant to improve the predictions of ecosystems' transpiration, desiccation and drought-induced mortality at scales from stand to region. SurEau-Ecos draws on the physiological and physical framework of SurEau while limiting the number of parameters and reducing computational cost. In the following sections, we first describe the principles, functioning, main equations and numerical schemes of SurEau-Ecos. Second, we compare simulations produced with three numerical schemes (explicit, semi-implicit and implicit) in terms of prediction stability and computing time. Third, we further describe the differences in plant hydraulic architecture between SurEau-Ecos and SurEau (CPRM21) and their impacts on simulation results. Fourth, we perform a global sensitivity analysis of tree desiccation dynamics to the main SurEau-Ecos input, i.e., plant hydraulic traits and stand and soil parameters. Fifth, we illustrate the potentialities which SurEau-Ecos will provide by running prospective simulations of hydraulic failure probability at the regional scale under changing climate.

## 2.1 Model overview

SurEau-Ecos is a plant hydraulic model that simulates water fluxes between the soil, plant and atmosphere for a monospecific layer of vegetation. In SurEau-Ecos the soil–plant system is discretized into three soil layers and two plant compartments: a leaf and a “stem” (Fig. 1c). Each of the two plant organs contains an apoplasm and a symplasm. The stem apoplasm and symplasm include water volumes of all non-leaf compartments, i.e., trunk, root and branches.

Water dynamics of the SPA system (represented by nodes in Fig. 1c) are locally governed by a generic partial differential equation for water mass conservation:

where *q* is the water quantity (kg m^{3}), *k* is the conductivity, *ψ* is the
water potential, *k*∇*ψ* is the water fluxes, and *s* is the local
sink term (i.e., a negative sign for soil evaporation or transpiration) or
source term (i.e., a positive sign for precipitation and water released by
cavitation).

A spatially integrated form of Eq. (1) can be specified for each compartment
of the plant (Fig. 1c) to derive the rate of change of its absolute water
quantity (volumetric integration). For convenience, we use the water
quantity per unit of leaf area *Q* (kg m${}_{\mathrm{leaf}}^{-\mathrm{2}})$ as a state variable. To
account for the water fluxes between compartments and the contribution of
internal water stocks (i.e., capacitances), the computations of water fluxes
between two adjacent compartments (*F*_{i→j}) are simulated according to
Darcy's law as the product of compartment's interface conductance (*K*_{ij})
and the gradient of water potential (*ψ*):

These fluxes are described in Sect. 2.3.

In addition, solving Eq. (1) needs to describe the link between *Q* and *ψ*.
This is handled using the notion of capacitance for the plant compartments
and water retention curves for the soil compartments. Plant capacitances
(*C*) are defined as follows:

For any plant compartments a generic equation of the water balance can now be written:

According to the type of compartment, *S* includes cuticular or stomatal
transpiration losses or water release from cavitation, which is also
accounted as a source term in the apoplasm
(Cruiziat et al., 2002). Cuticular or
stomatal transpiration fluxes are computed differently for each compartment
(leaf symplasm includes stomatal transpiration, whereas stem symplasm only
include cuticular transpiration). The contribution of capacitance (*C*) to the
plant compartment water balance is related to the saturated (or initial)
water quantity (*Q*^{sat}) in that compartment and takes different
formulation for symplasm and apoplasm. A pressure–volume curve is used for
the symplasmic capacitance (Tyree and Hammel, 1972), whereas a
constant capacitance is used for the apoplasm (Sect. 2.5). To the best of our
knowledge, this is the first formulation of symplasmic *C* and cavitation flux
as Darcy's law (see details in Sect. 2.3.3. and 2.5.1). These generic forms
are needed for the numerical resolution of water balance at each plant node
(described in Sect. 2.2.1).

For soil compartments, the water balance of a soil layer *j* is computed using
a generic equation following Eqs. (1) and (2), such as

where ${K}_{{\mathrm{soil}}_{j}-\mathrm{Sapo}}$ is the conductance from the soil layer *j* to the
stem apoplasm (Sect. 2.3.1). *S* represents a source (when *S*>0) or
sink (when *S*<0) term that can include soil water inputs from soil
infiltration; drainage from other layers; or outputs such as deep drainage,
soil evaporation, or capillarity depending on the soil layer (Sect. 2.2.2). A
water retention curve for the soil (van
Genuchten, 1980) is used to link ${Q}_{{\mathrm{soil}}_{j}}$ and ${\mathit{\psi}}_{{\mathrm{soil}}_{j}}$
and solve Eq. (5) (Sect. 2.5.2).

In addition to the core soil–plant hydraulic processes driving transpiration
and plant water status (*Q* and *ψ*), SurEau-Ecos also includes an empirical module for
leaf phenology that controls leaf area growth and decreases during senescence
(described in Appendix A) and different modules to represent the stand water
balance (interception, water transfers between soil layers and drainage;
described in Ruffault et al. (2013). The list of
input variables and their respective units is given in Table 1.

Temporal resolution varies according to each type of process (Fig. 1b). Phenology and stand water balances are computed at a daily time step. Soil–plant hydraulic processes (i.e., soil water uptake, transpiration and hydraulic redistribution) are computed at the finer time step (from 0.01 to 1800 s depending on the resolution scheme) and driven by hourly interpolated climate, which is derived from daily climate following (De Cáceres et al., 2021) (see Table B1 for the list of daily input weather variables). The three different numerical resolution schemes currently implemented in SurEau-Ecos are described in Sect. 2.6.

All variables and processes related to stand water balance processes
(precipitation, interception, drainage) are expressed per unit of ground
surface area, while plant hydraulic processes are expressed per unit of leaf
surface area, in accordance with usual practices in each research field.
This implies that initial water volumes of the soil and the plant (leaf
and stem) are expressed per unit of soil area. Following this, leaf area index (LAI)
permits the conversion of quantities from a soil area basis to a leaf area basis.
If the parametrization is performed from individual tree dimensions or from
forest inventories and allometries, an additional parameter is needed, the
average plant foot print (aPFP, in m^{2}), in order to scale individual plant
dimensions on leaf or a soil area basis.

SurEau-Ecos was implemented in the *R* programming language (R Core Team, 2020).
The following sections describe the equations and resolution of the model in
more details.

## 2.2 Water balance in each compartment

### 2.2.1 Plant

The water balance of each of the four-plant compartments (leaf and stem symplasm and apoplasm, Fig. 1c) is determined according to the generic Eq. (4) and solved at each time step.

For the leaf apoplasm, the water balance equation is as follows.

The first term represents the change in water quantity related to the leaf
apoplasmic capacitance (*C*_{LApo}, mmol m${}_{\mathrm{leaf}}^{-\mathrm{2}}$ MPa^{−1}), which
releases or absorbs water according to volume changes due to water potential
changes (*ψ*_{LApo}, MPa). Contrary to symplasmic compartments, this term
is very limited in the apoplasm because the xylem wall is inelastic. Note also
that cavitation is not included in this capacitance. The second and third
terms are the water exchanges between the leaf apoplasm and stem apoplasm
and between the leaf apoplasm and leaf symplasm, respectively. *ψ*_{SApo} is the water potential of the stem apoplasm, *ψ*_{LSym} is the
water potential of the leaf symplasm, *K*_{SApo−LApo}
(mmol m${}_{\mathrm{leaf}}^{-\mathrm{2}}$ s^{−1} MPa^{−1}) is the conductance from the stem
apoplasm to leaf apoplasm and *K*_{LSym} is the conductance of the leaf
symplasm. This equation applies to the non-cavitated part of the xylem,
which receives water from the cavitated part. This source is represented by
the fourth term ${F}_{\mathrm{L}}^{\mathrm{cav}}$ (mmol), which corresponds to the water release by
the cavitated vessels towards the non-cavitated leaf apoplasm
(Hölttä et al., 2009). This term is further described in
Sect. 2.3.2, where we explain how it can be expressed as a function of *ψ*_{LApo}.

Water balance for the stem apoplasm is calculated as follows.

The first term represents the water flux related to the stem apoplasmic
capacitance (*C*_{SApo}) and water potential (*ψ*_{SApo}) changes during the time step. As with the leaf
apoplasm, this term is in general very limited for the stem apoplasm. The
second term represents the water exchange between the stem apoplasm and the
three soil layers. For each soil layer *j*, ${K}_{{\mathrm{soil}}_{j}-\mathrm{SApo}}$ is the
conductance from the soil to the stem apoplasm and ${\mathit{\psi}}_{{\mathrm{s}}_{j}}$ the
soil water potential. The third and fourth terms represent flux to the leaf
apoplasm and stem symplasm, respectively. *ψ*_{SSym} is the water
potential of the stem symplasm and *K*_{SSym} is the stem–symplasm
conductance. The fifth term ${F}_{\mathrm{S}}^{\mathrm{cav}}$ corresponds to the water
released from cavitation to the non-cavitated stem apoplasm water reservoir.

Water balance for the leaf symplasm is as follows.

The first term represents the water flux related to *C*_{LSym} and water
potential changes of the leaf symplasm (*ψ*_{LSym}) during the time
step. The second term is the exchange between leaf apoplasm leaf symplasm.
The third and fourth terms represent the losses of water from the plant to
the atmosphere through leaf stomatal transpiration (*E*_{stom}) and
cuticular leaf transpiration (${E}_{{\mathrm{cuti}}_{\mathrm{L}}}$). Note that with this
formulation, leaf water losses from leaf transpiration remains lower bounded
by ${E}_{{\mathrm{cuti}}_{\mathrm{L}}}$ even when stomata are fully closed
(E_{stom}=0).

Water balance for the stem symplasm is as follows.

The first term represents the water flux related to *C*_{SSym} and water
potential changes of the stem symplasm (*ψ*_{SSym}) during the time
step. The second term is the flux to the stem apoplasm. The third term
represents the losses of water from the plant to the atmosphere through
minimum cortical transpiration (${E}_{{\mathrm{cuti}}_{\mathrm{S}}})$.

### 2.2.2 Soil

The water balance of each of the three soil layers (Fig. 1c) is determined according to the generic Eq. (5) and solved at each time step.

For the first soil layer, the following equation is required.

The first term ($\frac{\mathrm{d}{Q}_{{\mathrm{soil}}_{\mathrm{1}}}}{\mathrm{d}t}$, mmol m${}_{\mathrm{soil}}^{-\mathrm{2}}$)
represents the change in soil water quantity between two consecutive time
steps. The second term is the flux to the stem apoplasm. This flux is
multiplied by LAI to convert water quantities from a leaf area basis to a
soil area basis. ppt_{soil} (mmol m${}_{\mathrm{soil}}^{-\mathrm{2}}$) is the precipitation
reaching the soil, *D*_{1→2} is the drainage (mmol m${}_{\mathrm{soil}}^{-\mathrm{2}}$) of the
first to the second layer, and *E*_{soil} (mmol m${}_{\mathrm{soil}}^{-\mathrm{2}}$) is soil
evaporation that occurs only from this layer.

Similarly, for the second layer the following equation is required.

For the third soil layer, the following equation is required.

Dd is the deep drainage (mmol m${}_{\mathrm{soil}}^{-\mathrm{2}}$). For any layer,
drainage occurs when the field capacity of the soil layer (*θ*_{fc})
is overpassed. Lateral water transfer processes and upward capillary
transfers between layers are neglected. At the time step of the hydraulic
model (*δ**t*) the water balance of each soil layer is treated according
to the losses from transpiration and from evaporation (only for the first
layer). Incoming fluxes from precipitation, drainage and transfers between
soil layers are treated at a daily time step (Fig. 1b). Rainfall
interception and drainage are treated as in SIERRA
(Mouillot et al., 2001; Ruffault
et al., 2013) and follow the design principles of several other water
balance models
(Rambal,
1993; De Cáceres et al., 2015; Granier et al., 1999).

## 2.3 Conductances and fluxes

### 2.3.1 Plant and soil conductances

The model includes four apoplasmic conductances (three root-to-stem and one
stem-to-leaf conductance), two symplasmic conductances (one for the stem and one for the
leaves) and three soil-to-root conductances (${K}_{{\mathrm{soil}}_{j}-Rj}$, one per
soil layer *j*) (Fig. 1c). Symplasmic conductances of the leaves (*K*_{LSym})
and stem (*K*_{SSym}) drive the fluxes between the symplasmic and apoplasmic
compartments. These conductances are set to a constant value throughout the
simulation. Xylem (i.e., apoplasmic) conductances are composed of three
root-to-stem conductances in parallel (${K}_{{\mathrm{R}}_{j}-\mathrm{SApo}}$, one per soil layer
*j*) and one stem-to-leaf conductance (*K*_{SApo−LApo}). These conductances can
vary throughout the simulation from their initial value down to 0 according
to the level of cavitation (expressed by the percent loss in conductance).

In practice, it is also useful to define the total plant conductance
*K*_{Plant} as follows:

The stem-to-leaf apoplasmic conductance (*K*_{SApo−LApo}) is expressed as
a function of the percent loss of conductance due to xylem embolism in the
leaf:

where ${k}_{\mathrm{SApo}-\mathrm{LApo},\mathrm{max}}$ is the initial (maximum) root-to-leaf conductance
and PLC_{L} (%) is the percent loss of conductance. PLC_{L} is
proportional to the level of xylem embolism. It occurs when the water
potential drops below the capacity of the leaf xylem to support negative
water potential and is computed by using the sigmoidal function
(Pammenter and Vander Willigen, 1998):

where *P*_{50,L} (MPa) is the water potential causing 50 % loss of
plant hydraulic conductance and slope_{L} (% MPa^{−1}) is the slope of
linear rate of embolism spread per unit of water potential drop at the
inflection point *P*_{50,L}.

The apoplasmic conductance from each root *j* to the stem apoplasm
(${K}_{{\mathrm{R}}_{j}-\mathrm{SApo}}$) is expressed as a function of the level of embolism
computed at the node of the stem apoplasm:

where PLC_{S} is computed as PLC_{L} with the stem apoplasmic
potential (*ψ*_{SApo}) and vulnerability curves parameters specific to
the stem (slope_{S} and *P*_{50,S}). ${K}_{{\mathrm{R}}_{j}-\mathrm{SApo},\mathrm{max}}$ is the maximal
root-to-stem apoplasmic conductance of layer *j*. It is derived from fine-root
area of the layer *j* such as

where *K*_{R−SApo} is the total conductance of the root system. RAI_{j}
is the fine-root area of the layer *j*:

where RAI is the total fine-root area that is computed from the stand
leaf area index and the root-to-leaf area ratio (RaLa) and *r*_{i} the
root fraction in each soil layer, which is determined according to the
equation from Jackson et al. (1996):

where *z*_{h,j} is the depth (m) from the soil surface to the interface
between layers *j* and *j*+1, the factor of 100 converts from meters to centimeters and
*β* is a species-dependent root distribution parameter (Jackson et al.,
1996). Following this, the conductance between each soil layer *j* and the stem
apoplasm (${K}_{{\mathrm{soil}}_{j}-\mathrm{Sapo}}$) is determined as the result of two
conductances in series, ${K}_{{\mathrm{R}}_{j}-\mathrm{SApo}}$ and the conductance from soil to
root (${K}_{{\mathrm{soil}}_{j}-{R}_{j}}$):

The conductance of the soil to fine roots ${K}_{{\mathrm{soil}}_{j}-{R}_{j}}$ for each
soil layer *j* is computed as follows:

with *L*_{a} and *L*_{v} the root length per soil area and soil
volume for each soil layer, respectively, with both computed from soil depth
and RAI_{j}, whereas *r* is the radius of fine absorbing roots. *k*_{sat}
is the soil hydraulic conductivity at saturation, *m* is a parameter of
shape from the van Genuchten equation and REW is the relative extractable
water content computed as follows:

where *θ* is the relative water content (soil water content per unit of soil
volume) changing dynamically with changes in absolute soil water reserve in
the rooting zone, *θ*_{s} is the relative soil water content at
saturation and *θ*_{r} is the relative soil water content at wilting
point. *θ*_{s} and *θ*_{r} are parameters measured in the
laboratory or derived from soil surveys with pedotransfer functions.

The total available water (TAW) for the plant can also be computed as the
difference between the water quantity at field capacity (*θ*_{fc})
and the water quantity at *θ*_{r} summed over the three soil layers
as follows:

where rfc_{j} and th_{j} are the rock fragment content (%) and
thickness (m) of the soil layer *j*, respectively. TAW is not a parameter in
SurEau-Ecos but is an integrative value resulting from the interaction between soil
characteristics and rooting depth.

### 2.3.2 Cavitation

SurEau-Ecos also considers the capacitive effect of cavitation
(Hölttä et al., 2009), i.e., the water released to the
streamflow when cavitation occurs. The non-cavitated part of the xylem
receives a water flux from the cavitated part, corresponding to
${F}_{\mathrm{L}}^{\mathrm{cav}}$ in Eq. (6) (${F}_{\mathrm{L}}^{\mathrm{cav}}>\mathrm{0}$), and is then transferred to adjacent
compartments. The amount of water corresponding to a new cavitation event is
derived from the quantity of water in the apoplasm at saturation
(${Q}_{\mathrm{LApo}}^{\mathrm{Sat}}$) and the temporal variations in PLC_{L} as follows:

This flux is linearized in temporal variations in *ψ*_{LApo} in order to
express this flux in the form of a Darcy's law to match the generic form of
Eq. (2). For that purpose, we introduce an equivalent conductance
(${K}_{\mathrm{L}}^{\mathrm{cav}}$) as follows:

where ${K}_{\mathrm{L}}^{\mathrm{cav}}=-\frac{{Q}_{\mathrm{LApo}}^{\mathrm{Sat}}{\text{PLC}}^{\prime}\left({\mathit{\psi}}_{\mathrm{LApo}}\right)}{\mathrm{d}t}{\text{PLC}}^{\prime}$ is the derivative of the PLC with respect to *ψ*,
which is computed from the cavitation curve, and ${\mathit{\psi}}_{\mathrm{LApo}}^{\mathrm{cav}}$ is the minimal
value of potential ever reached over time, which controls the current
cavitation level (${\text{PLC}}_{\mathrm{L}}={\text{PLC}}_{\mathrm{L}}\left({\mathit{\psi}}_{\mathrm{LApo}}^{\mathrm{cav}}\right)$).
PLC^{′} is computed as follows:

Following the same approach, the flux derived from the stem when cavitation occurs is defined as follows:

## 2.4 Sources and sinks

### 2.4.1 Stomatal and cuticular plant transpiration

Plants lose water through stomatal transpiration (*E*_{stom}), cuticular
transpiration of the leaf (${E}_{{\mathrm{cuti}}_{\mathrm{S}}}$) and cuticular transpiration of
the stem (${E}_{{\mathrm{cuti}}_{\mathrm{S}}}$). Cuticular transpiration of the roots is
considered to be negligible and is not taken into account. The total plant
transpiration *E*_{Plant} is decomposed as the sum of the leaf
(E_{L}) and wood transpiration (${E}_{{\mathrm{cuti}}_{\mathrm{S}}}$):

where *E*_{leaf} is computed as follows:

and ${E}_{{\mathrm{cuti}}_{\mathrm{S}}}$ is computed as follows:

where VPD_{L} (MPa) is the vapor pressure deficit of the leaf, *P*_{atm} is the
atmospheric pressure (MPa), *g*_{stom} is the stomatal conductance,
${g}_{{\mathrm{cuti}}_{\mathrm{L}}}$ is the cuticular conductance of the leaf, *g*_{bound} is the
conductance of the leaf boundary layer and *g*_{crown} is the conductance of
the tree crown.

VPD_{L} is a function of leaf temperature (*T*_{L}). *T*_{L} is computed
at the leaf surface by solving the energy budget as in CPRM21. *g*_{bound}
and *g*_{crown} are computed following Jones (2013). *g*_{bound}
varies with leaf shape, size (*d*_{leaf}) and wind speed; *g*_{crown} is a
function of wind speed.

${g}_{{\mathrm{cuti}}_{\mathrm{L}}}$ is a function of *T*_{L}, which is based on a single or double
*Q*_{10} equation depending on whether leaf temperature (*T*_{L}) is above
or below the transition phase temperature (*T*_{Phase}) (Cochard, 2021):

if

if

where *g*_{stom} is the stomatal conductance taking into account the
dependence of *g*_{stom} on light, temperature and CO_{2} concentration, as well as water status:

where *g*_{stom,max} is the stomatal conductance without water stress and is determined as a
function of light, temperature and CO_{2} concentration following
Jarvis (1976). *γ* is a regulation factor that
varies between 0 and 1 to represent stomatal closure according to *ψ*_{LSym} and an empirical sigmoid function depending on the potential at 50 % of stomatal closure (*ψ*_{gs,50}) and a shape parameter
(slope_{gs}) describing the rate of decrease in stomatal conductance per
unit of water potential drop.

### 2.4.2 Soil evaporation

*E*_{soil} depends on the maximum soil conductance (*g*_{soil0}) and the REW
of the first soil layer as follows:

## 2.5 Capacitances

As described in Sect. 2.1, the link between *Q* and *ψ* are not
represented in the same way for the soil and plant compartments. The notion
of capacitance is used for the plants, while water retention curves are used
for the soil.

### 2.5.1 Plant compartments

The contribution of capacitance (*C*) to the plant compartment water balance is
related to the saturated (or initial) water quantity (*Q*) in that
compartment. Symplasmic and apoplasmic capacitances are not modeled in the
same way, but both require the water volume at saturation (*Q*^{sat}) of the
considered reservoir. For the leaves, the volume of symplasmic and
apoplasmic reservoirs at saturation (${Q}_{\mathrm{LSym}}^{\mathrm{Sat}}$ and ${Q}_{\mathrm{LApo}}^{\mathrm{Sat}}$,
respectively) are defined as follows:

with

where DM is the dry matter per unit of leaf area. The leaf dry matter
content (LDMC), fraction of apoplasmic tissue in the leaves (*α*_{LApo}) and leaf mass per area (LMA) are all input parameters.

The apoplasmic and symplasmic water quantities of the stem at saturation (${Q}_{\mathrm{SSym}}^{\mathrm{Sat}}$ and ${Q}_{\mathrm{SApo}}^{\mathrm{Sat}}$, respectively) includes the volume of the roots, trunk and branches. They are computed based on the volume of the woody compartment and the water fraction of this volume as follows:

where *V*_{S} is the volume of tissue of the stem compartment (including the
root, trunk and branches), ${M}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}$ is the water molar mass,
*α*_{Water} is the proportion of water in this volume, and *α*_{SApo} and *α*_{SSym} are the apoplasmic and symplasmic fraction
of this water volume, respectively.

Symplasmic reservoirs behave as variable plant capacitances related to the pressure volume curve, which corresponds to the water quantity changes in symplasmic cells ($\frac{\mathrm{d}Q}{\mathrm{d}t}$). Symplasmic conductances are functions of the ${Q}_{\mathrm{LSym}}^{\mathrm{Sat}}$ and the temporal change in the symplasmic relative water content (RWC) (illustrated here for the leaf, but similar equations apply for the trunk):

with this formulation the capacitance of the leaf symplasm (*C*_{LSym}) can
be written as follows:

where RWC^{′} is the derivative of the RWC with respect to *ψ*_{LSym},
derived from pressure–volume curves
(Tyree and
Hammel, 1972; Bartlett et al., 2012).

We used the following formulation for RWC^{′} (see the justification below for
the expression above *ψ*_{tlp}):

with

when ${\mathit{\psi}}_{\mathrm{LSym}}\ge {\mathit{\psi}}_{\mathrm{tlp}}=\frac{\mathrm{1}}{\frac{\mathrm{1}}{\mathit{\u03f5}}+\frac{\mathrm{1}}{{\mathit{\pi}}_{\mathrm{0}}}}$.

In the above equation the formulation for *ψ*_{LSym}<*ψ*_{tlp} simply
results from the fact that $\text{RWC}=\frac{{\mathit{\pi}}_{\mathrm{0}}}{{\mathit{\psi}}_{\mathrm{LSym}}}$.

The case *ψ*_{LSym}≥*ψ*_{tlp} was obtained from basic manipulations
of the derivation of the following form of the pressure–volume curve:

which with a derivative with respect to *ψ*_{LSym} becomes

and thus ${\text{RWC}}^{\prime}=\frac{\text{RWC}}{-{\mathit{\pi}}_{\mathrm{0}}-{\mathit{\psi}}_{\mathrm{LSym}}-\mathit{\u03f5}+\mathrm{2}\mathit{\u03f5}\text{RWC}}$.

Apoplasmic capacitance is constant and is computed as the product between
${Q}_{\mathrm{Apo}}^{\mathrm{Sat}}$ and the specific apoplasmic capacitance (*C*_{Apo}). Note
that, given the very low elasticity of the xylem, this contribution is very
weak.

### 2.5.2 Soil compartments

Capacitances for soil are not explicitly computed in SurEau-Ecos. Rather, soil water
potentials for the different soil layers (*ψ*_{soil}, MPa) are directly
computed according to the van Genuchten parametric formulation
(van Genuchten, 1980):

where *m*, *n* and *α* are empirical parameters describing the typical
sigmoidal shape of the function and REW is the relative extractable water
(see Eq. 21).

## 2.6 Numerical resolution

### 2.6.1 Plant compartments

The resolution of the plant hydraulic part of SurEau-Ecos is to solve the water balance
for the four hydraulic compartments (i.e., nodes in Fig. 1c), whose equation
are presented in Sect. 2.2.1. Three different numerical resolution schemes
were implemented to solve water balances of plant compartments. For these
three schemes, water potentials were discretized between two consecutive
time steps, *ψ*^{n} and *ψ*^{n+1}, separated by *δ**t*. Thanks to
cautious hypotheses, these equations were linearized at the first order in
*ψ*, to lead to a four-equation linear system. Specifically, we
neglected all variations of capacitances and conductances during a given
time step (*C*≈*C*^{n} and *K*≈*K*^{n}), as these variations are
expected to be marginal with respect to weather changes, stomatal
regulation or water release by cavitation.

The simpler explicit scheme, also implemented in SurEau, assumes that water fluxes can be expressed
from the current time step *n* (see Appendix B1 for details). From the generic
water balance Eq. (4), it leads to

Rearranging this equation, the potential at the next time step *ψ*^{n+1} can be simply computed as follows:

While the implementation of the explicit time integration scheme is
undoubtedly the most straightforward numerical solution, it suffers from a well-known numerical constraint referred to as the CFL,
which imposes very small time steps (*δ**t*) to avoid numerical
instabilities:

This constraint implies that the smaller the *C*, the smaller the *δ**t*.
An intuitive interpretation of this limitation is that the time step needs
to be small enough to avoid water movements between non-adjacent cells. This
constraint is particularly strong in plant xylem that is inelastic (i.e., *C*
is very small) such that apoplasmic compartments cannot absorb water fluxes
from their adjacent compartments when the time step is too large. This
typically imposes *δ**t* to be smaller than 10 ms (CPRM21).

A common option to avoid these numerical instabilities is to use an
implicit scheme, where fluxes are estimated from the values of *ψ* at time *n*+1 (*ψ*^{n+1}) as follows:

This numerical scheme is unconditionally stable, meaning that an increase in
*δ**t* will not induce numerical instabilities but might induce
a loss of numerical accuracy. One very important limitation of this scheme
is that the equations of the different compartments now correspond to a
system of four equations that are coupled. Such a system can be linearized
(by pieces to account for thresholds such as cavitation) and solved. In
general, it implies the inversion of the matrix of the linear system, but
the resolution can also be done analytically when the equations are not too
many, as it is the case with SurEau-Ecos (see details in Appendix B2).

An alternative scheme, based on a semi-implicit approach, has also been recently proposed to solve water balances in plant hydraulic models while overcoming the numerical instabilities associated with an explicit formulation (Xu et al., 2016; Tuzet et al., 2017; Li et al., 2021; De Kauwe et al., 2020). Although not usual in numerical resolution approaches, this scheme has been shown to have great performance and has led to convergence in simulations with time steps on the order of 10 min (Xu et al., 2016).

This approach consists of solving the differential equation of each
compartment assuming that *ψ*_{j} and *S* remain constant (respectively
equals to ${\mathit{\psi}}_{\mathrm{j}}^{n}$ and *S*^{n}) as follows:

After linearization of the coefficient, this ordinary differential equation has the following solution:

Therefore, *ψ*^{n+1} can be estimated by its value at *u*=*δ**t*:

which implies that

with

and

One can notice here that $\stackrel{\mathrm{\u0303}}{\mathit{\psi}}$ is the steady-state solution of the
equation, typically valid when *C*=0 (fully elastic media).

In practice, this formulation is equivalent to the corresponding numerical
scheme (provided that *δ**t* is very small):

This formulation allows for comparing this scheme to the explicit and implicit
schemes proposed above. This scheme uses *ψ*^{n+1} as a value
for *ψ* (so that it remains stable) and ${\mathit{\psi}}_{j}^{n}$ as a
value of *ψ*_{j} (so that the equations of the four compartments are
decoupled) and can be seen as an intermediate between the explicit and the
implicit scheme. For that reason, it will be referred to as
semi-implicit (Appendix B3). In theory, the water fluxes computed from values of water
potentials evaluated at different time steps should be less accurate than
the implicit scheme, especially when water potential changes are fast. It is
thus expected that simulations require a larger time step to converge than
the implicit scheme.

For the three different numerical schemes, we assume that soil potentials
were estimated at the current time step *n* (i.e., ${\mathit{\psi}}_{{\mathrm{S}}_{j}}\approx {\mathit{\psi}}_{{\mathrm{S}}_{j}}^{n}$) as in the explicit formulation (instead of *n*+1, as normally
expected in an implicit scheme). This assumption is supported by the very
small variations in soil potentials occurring during a single time and
avoids the linearization of soil potential equations, which would have
required unnecessary complex developments.

Source and sink fluxes *S*^{n+½} are computed for the climate at
the middle of the time step (mid-climate between the current and next time step
at $n+\frac{\mathrm{1}}{\mathrm{2}})$. For the implicit scheme, to account for the quick
adjustment of stomatal regulation to climate variations, *E*^{n+½}
accounts for linear variations in water potential *ψ*_{LSym} over the time
step, thanks to the derivative of transpiration function
${{S}^{\prime}}^{n+\frac{\mathrm{1}}{\mathrm{2}}}\left({\mathit{\psi}}_{\mathrm{LSym}}^{n}\right)$, also estimated for
the mid-climate, but the current regulation of ${\mathit{\psi}}_{\mathrm{LSym}}^{n}$ is as follows:

### 2.6.2 Soil compartments

Soil water balance in SurEau-Ecos is solved for each soil layer (Sect. 2.2.2) following
a simple explicit scheme assuming that water fluxes can be expressed from
the current time step *n*. From the generic soil water balance Eq. (5), it leads
to

In this section, we explore the benefits and limitations of the three
numerical schemes implemented in SurEau-Ecos to solve water fluxes, namely an
explicit, semi-implicit and implicit scheme. As mentioned above,
the minimal time step required for accurate simulations is determined by
computational limitations that depend on the chosen scheme. First, unlike
the implicit and semi-implicit scheme, the explicit scheme is limited by
the CFL, which causes numerical instabilities. We explored how much
computation time can be gained by using implicit or semi-implicit schemes
compared to the explicit scheme. In addition, in the case of the implicit and
semi-implicit scheme, reducing the temporal resolution (i.e., increasing the
time step) can also limit the accuracy of the simulation. The magnitude of
corresponding errors then depends on the physiological processes at play in
the plant and on the precision of the numerical scheme. We also assessed the
sensitivity of model outputs to the temporal resolution (time step *δ**t*) for the implicit and semi-implicit schemes.

For these simulations, all inputs were set identical to those used in the
section dedicated to the evaluation of SurEau-Ecos (see Sect. 4). Daily weather was
kept constant, without precipitation, and simulations were run until total
hydraulic failure of the plant. To compare the explicit scheme with the two
other schemes, we made two slight simplifications to the model. First, we
neglected the cavitation term in Eqs. (6) and (7). Indeed, the explicit
numerical scheme of SurEau-Ecos cannot account for the flux term associated with water
released by cavitation. This is due to the direct dependence of
${K}_{\mathrm{L}}^{\mathrm{cav}}$ and ${K}_{\mathrm{S}}^{\mathrm{cav}}$ on *δ**t* (Sect. 2.3.2) that prevents
the CLF from being satisfied at any time step. Second, the values for stem and leaf
of apoplasmic capacitances (*C*_{SApo} and *C*_{LApo}) were increased (from
about $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{3}}$ to 10 mmol m^{2} MPa) to decrease computational costs and ease the
comparison between the numerical schemes. The CFL constraint imposed very
small time steps (on the order of $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{5}}$ s) with the original values of plant
apoplasmic capacitance, which caused unaffordable computation times under
most CPUs. Preliminary analyses showed that the impact of *C*_{SApo} and
*C*_{LApo} were negligible on simulation results for values up to 50–100 mmol m^{2} MPa.

When using the implicit or semi-implicit schemes with a relatively
small time steps (*δ**t*=10 s) our results show that these schemes
yielded identical plant dynamics to those obtained with the explicit mode
(Fig. 2). However, the gains in computation time were considerable.
Computation time was divided by about 10 for the implicit and semi-implicit
scheme compared to the explicit scheme. This is because *δ**t* had to be
set to 1 s for the explicit theme because of the CFL. Any attempt to set a
*δ**t* above this 1 s threshold caused (as expected by the CFL) critical
numerical instabilities (Fig. B1). Since some modifications to the model had
to be performed for this comparison, the differences in computation times
were solely indicative and were reported to illustrate the benefits of the
semi-implicit and implicit schemes compared to the explicit scheme. Our
results showed that the semi-implicit scheme was less accurate than the
implicit scheme. Smaller time steps were required for the convergence of the
model. Numerical explorations show that the semi-implicit scheme requires
time steps on the order of 1 min (which is slightly slower than described in
Xu et al. , 2016, which stated that 10 min was enough), whereas the time step
can be generally larger than 30 min with the implicit scheme (Figs. B2
and B3).

* This computational time is for indicative purposes only as several changes had to be made to the model to run it with the explicit scheme (see details in main text).

For the implicit and semi-implicit schemes, two adaptive time steps were further implemented to reduce computation times. This improvement was based upon the assumption that smaller time steps were only required when changes in two critical processes, stomatal regulation and cavitation, were the highest. In a “normal” mode, the base time step is at 10 min but is automatically and gradually refined up to 1 min in periods of intense regulation changes, based on a criterion aimed at preventing variation in stomatal regulation and cavitation of more than 1 % between two consecutive time steps. In a “fast” mode, the base time step is at 1 h and refined up to 10 min. The implementation of adaptive time steps allowed for further increasing this gain in computing time (Table 2) without affecting plant dynamics.

Due to the reduction of plant compartments, SurEau-Ecos requires fewer parameters than SurEau. However, the parametrization of plant hydraulics models can be problematic, especially for large-scale applications (i.e., for many species and stands). In order to facilitate the parametrization of SurEau-Ecos, we provided a table where we listed the most important parameters and where to find relevant datasets (Table 3). We also proposed some procedures to estimate the value of the parameters not directly available in current databases. We distinguished four different types of parameters: (i) the species-specific parameters, (ii) the plant (or stand) morphological parameters, (iii) the soil parameters and (iv) the parameters linked to hydraulic conductance.

Species-specific parameters (leaf, stomatal and hydraulic traits) can be
derived from direct ecophysiological measurements or traits databases. This
includes the parameters related to stomatal conductance, now available in
several databases
(Klein, 2014;
Lin et al., 2015), and parameters of the *p*−*v* curves and vulnerability curves to
cavitation both for the leaves and stems. The *p*−*v* curves are generally available for
leaves (Bartlett et al., 2016), but very few data are
available for the stem (but see Tyree and
Yang, 1990; Meinzer et al., 2008). Until the release of additional datasets
for these traits, we recommend to use the same value for the leaf and the
stem symplasm. Vulnerability curves to cavitation are increasingly available
at the branch and leaf level. In cases where it would be difficult to find
the data for either the stem or the leaves, some hypotheses regarding the
level of segmentation can be made. However, for vulnerability curves to
cavitation, we recommend paying attention to the method that has been used
to build the curves, as many artifacts are known to influence these values
depending on the tree species (Sergent et al., 2020). Cuticular conductance
at a reference temperature (*g*_{cuti20}) and its dependence on temperature
(*Q*_{10a}, *Q*_{10b}, and *T*_{Phase}) are increasingly recognized as a key
trait for survival time during drought (Duursma
et al., 2019) and heat waves (Cochard, 2021). *g*_{cuti20} is
increasingly available in species trait databases, but the parameters driving
*g*_{cuti} dependence to temperature are far less measured
(Riederer and Schreiber, 2001). Recent methodological
innovations should allow a greater acquisition of this trait
(see Billon et al., 2020).

The plant (or stand) morphological parameters that determine the overall leaf area index (LAI) and the plant internal water stores can be derived from forest inventory and species-specific allometries. LAI can also be derived from vegetation remote sensing data.

The soil parameters determine the total soil available water for plant
(TAW, see equation 23), which depends on the volume of soil explored by
roots on the one hand (i.e., a function rock fragment content and rooting
depth) and the water retention curve on the other hand (i.e., the relationship
between water potential with soil water content). Such parameters can
primarily be derived from soil databases. Note, however, that such databases
generally provide only pedo-physical information (textures, organic matter
content, rock fragment, depth) so that it will be needed to apply
pedotransfer function to compute the parameters (Tóth et al., 2017).
Pedotransfer function can also be used to compute the soil hydraulic
conductance (*K*_{Sat}), although *K*_{Sat} global databases are also
available
(Gupta
et al., 2021).

Finally, the hydraulic conductance of the soil-to-leaf pathway and its repartition within the plant is rarely available (Mencuccini et al., 2019). The easiest way to obtain some values for these parameters is to compute the total maximal plant hydraulic conductance by using flux data (derived from sap flow, leaf gas exchange or remote sensing) and in situ water potential data (Mencuccini et al., 2019). The distribution between compartment can then be done by using average hydraulic architecture maps (Tyree and Ewers, 1991; Cruiziat et al., 2002). Alternatively, this can be computed from the elementary conductivity of plant organs taken from databases and plants sizes derived from inventory (De Cáceres et al., 2021).

SurEau-Ecos relies on the same biological and physical principles of SurEau (CPRM21). The soil–plant–atmosphere system is segmented and described as compartments linked together and exchanging water fluxes according to the gradients of water potential and hydraulic conductances. However, significant disparities between the implementation, parametrization and resolution of water fluxes between the two models led to some major differences in plant architecture and representation of water fluxes. It was therefore essential to confirm that both models provide comparable dynamics of the main state variables under similar conditions. This comparison of model outputs also consists at as an indirect evaluation effort of SurEau-Ecos since SurEau has been evaluated against field data (see details in CPRM21).

We identified three major differences in plant architecture and
representation of hydraulic processes within the models. First, plant
architecture is simpler in SurEau-Ecos than in SurEau. SurEau-Ecos represents the plant as two leaf
cells (leaf apoplasm and leaf symplasm) and two stem compartments that
include the woody volume of branches, trunk and roots. In contrast, SurEau offers a
detailed plant organ discretization (including roots, trunk, branches,
leaves and buds). Second, while both models represent the belowground stems
by three roots in parallel, the resistance to water flow linked to the root
endoderm (a symplasmic root resistance) is not explicitly included in
SurEau-Ecos contrary to SurEau. Instead, only one resistance per root, from the root entry to
the stem, is accounted to mimic all possible resistance (root symplasm and
apoplasm). Finally, in *SurEau-Ecos,* all leaf level fluxes to the atmosphere – i.e., the
stomatal and the cuticular fluxes – pass through the symplasm, whereas in
SurEau stomatal fluxes pass through the apoplasm and cuticular fluxes.

To compare model outputs, we performed an equivalent parameterization of the
two models (see details in Fig. B4) and ran simulations until total
hydraulic failure of the plant. We started the comparison with a typical plant
fully described in CPRM21 whose parameters are given for each organ in Table B2. We then aggregated the values of SurEau parameters to match the following
input parameters of SurEau-Ecos: water quantities of the leaf and stem compartments
(${Q}_{\mathrm{LApo}}^{\mathrm{sat}}$, ${Q}_{\mathrm{LSym}}^{\mathrm{sat}}$, ${Q}_{\mathrm{SApo}}^{\mathrm{sat}}$ and ${Q}_{\mathrm{SSym}}^{\mathrm{sat}}$),
the symplasmic conductance of the stem (*K*_{SSym}), the apoplasmic root-to-stem conductance (*K*_{R−SApo}) and the apoplasmic stem-to-leaf conductance
(*K*_{SApo−LApo}). We also set the cuticular conductance of non-leaf organs
to 0 in both models. All other submodels, parameters and environmental
forcing (weather and soil) were also set equal, including stomatal, boundary
layer and crown conductance, linear approximation for the leaf energy
balance, soil parameters, and hourly climatic inputs. This ensured that any
divergence between models could only come from either the numerical scheme
or plant hydraulic architecture.

Figure 3 shows the dynamics of water potentials, leaf transpiration and
percent loss of conductance obtained when simulations were run from a wet
soil profile until hydraulic failure is reached. Note that for this
comparison the output of the trunk in SurEau was compared to the stem in
SurEau-Ecos. For both models, at the beginning of the simulations when the soil was
wet, leaf and stem water potentials followed the hourly variations in
meteorological conditions, thereby reflecting the response of stomata to
light and response of plant transpiration to *g*_{stom} and VPD. As the soil
reservoir emptied, stomata progressively closed according to the intensity
of foliar water potential. After about 65 d for both models, the stomata
permanently closed and transpiration was limited to cuticular losses that
gradually accentuated the drought stress of the plant (decreased plant water
potentials). Simultaneously, cavitation increased in the different organs,
inducing water release from the apoplasm which partly dampened the decrease
in plant water potentials. These results show that SurEau-Ecos and SurEau yielded very
similar results when parameterized in such a way that plant organs had
similar conductances and water reservoirs.

Despite similar dynamics, we also identified some differences in infra-daily water potentials between the two models. As a result, the time to leaf hydraulic failure was underestimated by 3 d (out of 90 d) in SurEau-Ecos compared to SurEau. These slight differences can be linked to the presence of the higher number of compartments in SurEau that increase the seasonal dampening effect of water potential compared to SurEau-Ecos where a lower number of compartments are represented. Notably, we observed some differences between the short-term (infra-daily) variations in the water potential dynamics of the trunk symplasmic compartment of SurEau and the stem compartment of SurEau-Ecos (including the volume of roots, trunk and branches; see Table B2). The daily magnitude of the fluctuation in SurEau-Ecos appeared more dampened (Figs. 3, B5 and B6). The most plausible explanation for this difference is that the volume of the stem compartment in SurEau-Ecos is greater than the volume of the trunk compartment in SurEau. This is likely to lead to greater water discharge and lower water potential fluctuations in SurEau-Ecos (Fig. B6). Ongoing developments of a modular version of SurEau within the Capsis modeling platform (Dufour-Kowalski et al., 2012) will allow us to more deeply evaluate the effects of plant hydraulic architecture on the dynamics of plant desiccation.

## 6.1 Model sensitivity to input parameters

We carried out a variance-based sensitivity analysis to gain insights into the species traits that influence plant water dynamics in SurEau-Ecos and explore the main drivers of tree response to extreme drought. Variance-based approaches can measure sensitivity across the whole input space (i.e., it is a global method) and quantify the effect of interactions that can be unnoticed on a local sensitivity analysis approach (i.e., when moving one parameter at a time). Here, we used the Sobol's sensitivity analysis method (Sobol, 2001) and reported “total order indices” that quantify the contribution of each parameter to the variance of the model output.

Two different physiological phases control the dynamics of plant desiccation
under extreme drought, according to whether *ψ*_{LSym} is above or below
the point of stomatal closure (Fig. 1a). Three time-based metrics were
therefore considered to explore the sensitivity of plant desiccation to
input parameters: (i) the time to hydraulic failure, (ii) the time to
stomatal closure, and (iii) the survival time, defined as the time
difference between hydraulic failure and stomatal closure (see an
illustration in Fig. 4). We performed a sensitivity analysis for three
different tree species with contrasting ecology and which exhibited various
combinations of input parameters (Table 4). For each parameter, we randomly
sampled a value within a range of ±20 % of the observed value.
Starting from a wet soil, and without further precipitation, we ran
simulations until hydraulic failure of the plant, defined as the moment when
leaves reach 99 % loss of hydraulic conductivity (${\text{PLC}}_{\mathrm{L}}>=\mathrm{99}$ %). This threshold guarantees that plant water pools were almost
empty and that no other water reservoirs are available for the plant. The
water content of plant tissues is probably a better indicator of plant
mortality than the percent loss of conductivity (Martinez-Vilalta et al.,
2019; Mantova et al., 2021). However, an accurate prediction of moisture
content would require the integration of carbon metabolism (Martinez-Vilalta
et al., 2019) that is currently not implemented in SurEau-Ecos. Daily climate inputs
were set constant according to the simulations shown in Sect. 4. In total,
we ran 700 000 simulations in the sensitivity experiment.

We based our selection of parameters used in the sensitivity analysis on the
results from preliminary analyses and from the findings by CPRM21. To ease
the interpretation of the results, we grouped the parameters according to
several families, representing different processes: “water use”
(LAI_{max}, *K*_{Plant}, TAW and *g*_{stom,max}), “regulation” (*ψ*_{gs,50}), “water leaks” (*g*_{cuti20}, *Q*_{10a}), “safety” (*P*_{50}) and
“plant internal stores” (*V*_{S}) (see definition in Table 1). The total
available water (TAW) for the plant is not an input parameter in
SurEau-Ecos, but it is an integrative index resulting from the interaction between soil
characteristics and rooting depth. TAW is determined as the difference between
the water quantity at field capacity and the water quantify at residual
water content cumulated over the three soil layers. To make TAW vary in
simulations without affecting soil physical properties, we adjusted rooting
depth to match the targeted TAW.

Our results showed that a few parameters explained most of the variability
in the response of trees to extreme drought (Fig. 4), although their importance
largely depended on the physiological phase under study. The parameters
related to “water use” (LAI_{max}, TAW and *K*_{Plant}) and “regulation” (*ψ*_{gs50}) mainly explained the variance in time to stomatal closure, i.e.,
the first physiological phase. It suggests that, in this phase, interactions
between how much water is available in the soil (TAW) and how fast plant
transpiration will empty that reservoir (LAI_{max}, *ψ*_{gs,50} and
*K*_{Plant}) determine the time to stomatal closure. The surprisingly
relative low influence of *g*_{stom,max} on the time to stomatal closure
could be explained by the fact that, with that set of parameters and
environmental conditions, *K*_{Plant} has a more limiting impact on plant
transpiration than *g*_{stom,max}. In the second phase (after stomatal
closure), survival time was mostly driven by parameters related “water use”
(LAI, *P*_{50}), “water leaks” (*g*_{cuti20}), “safety” (*P*_{50}) and
“plant internal stores” (*V*_{S}). In that phase, the importance of TAW
and *ψ*_{gs,50} decreased to the benefit of traits related to the rate
of water losses through cuticular transpiration (*g*_{cuti20} and
*Q*_{10a}); the volume of water reservoirs in the root, trunk, and branches
(*V*_{S}); and plant resistance to cavitation (*P*_{50}). When both phases
were considered jointly, we observed that the variability in the time to
hydraulic failure was mainly associated with stand parameters (LAI and
TAW) and to a lesser extent with *ψ*_{gs,50} and *g*_{cuti20}.

We also observed that the patterns described here above were almost
identical regardless of the vegetation type under study. In particular, the
parameters controlling “time to hydraulic failure” and “survival time” were
similar among the three studied vegetation types, suggesting a similarity of
plant adaptation strategies to avoid hydraulic failure in a changing
climate. The one exception to this pattern is the importance of varying
plant resistance to cavitation (*P*_{50}) in survival time. The influence of
*P*_{50} ranged from low for *Quercus ilex* (about 0.05) to very important for *Quercus petraea* (about
0.37). This observation suggests that less drought-resistant species (with
higher *P*_{50}) receive a more direct benefit when lowering their *P*_{50} to
increase their survival time than drought-resistant species (with lower
*P*_{50}). This might be due to the nonlinear response of water potential
to soil and plant water content, which implies that the rate of change of
plant water potential increases as soil and plant water content decreases.

Our results shed some light on our understanding of plant functioning under
extreme drought. We highlighted the prominent role of stand traits, namely
LAI_{max}, TAW and *ψ*_{gs,50}, in determining the time
needed to reach stomatal closure. In contrast, physiological variables,
namely *g*_{cuti20}, *Q*_{10a}, *V*_{S} and *ψ*_{50,L}, played a more
important role in determining “survival time”. Two improvements to the
present analyses may strengthen these findings. First, numerous correlations
exist between those traits, reflecting trade-offs and plant functioning
strategies
(Christoffersen et
al., 2016; Martin-StPaul et al., 2017) that we did not take into account.
Similarly, it has been shown that LAI_{max} and TAW covary because
trees with a higher amount of available water tend to develop a higher leaf
surface value (Hoff and Rambal, 2003). Second, the relative importance of
input parameters is likely to be influenced by climate. For instance, we
would expect the influence of *g*_{min20}, *Q*_{10a} and *Q*_{10b} on
survival time to increase when temperature increases, following previous
results showing the vulnerability of trees during heat waves
(Cochard, 2021). Integrating these potential improvements in
future simulations may further help to elucidate the specific spatial and
temporal patterns of drought-induced mortality (Meir et al.,
2015).

## 6.2 Model sensitivity to the inclusion of symplasmic and apoplasmic capacitances

Whether or not plant hydraulic capacitances are explicitly taken into account is one of the key distinctions between current large-scale plant hydraulic models. Some models represent trees as single- or multiple-resistance organisms (e.g., Kennedy et al., 2019), while others like SurEau and SurEau-Ecos also include one or several hydraulic capacitances. SurEau and SurEau-Ecos describe the soil–plant–atmosphere system as a network of resistances and capacitances while introducing a novel distinction between the symplasmic and apoplasmic capacitances. This approach is beneficial for model parametrization and to derive values such as water content, as has already been discussed in Sect. 2. However, the role and importance of both the symplasmic and apoplasmic capacitances for plant survival and water dynamics have not yet been studied.

To further understand the role hydraulic capacitances on plant water dynamics, we conducted sensitivity experiments were capacitances were successively set to zero: first apoplasmic capacitance (leaf and stem), then symplasmic capacitance (leaf and stem), and finally both apoplasmic and symplasmic capacitances. These simulations applied the same experiment settings as the model comparison experiment (see Sect. 4), i.e., similar plant parameters, soil and climate conditions.

Figure 5 shows the results of the simulations of the sensitivity experiment. Overall, hydraulic capacitances induced significant differences in both the dynamics of plant water potentials and the time to hydraulic failure. More specifically, we observed that symplasmic capacitance can buffer short-term variations in plant water potentials and therefore induced fewer negative values at midday. Apoplasmic capacitances played a major role in both delaying the time to hydraulic failure and buffering daily variations in plant water potentials by providing water when cavitation occurs. The importance of this effect increases with decreasing water potentials (increasing drought). Our results therefore suggest that the representation of plant water storage greatly affects the simulations of plant water dynamics. Further studies aimed at measuring plant water content will help to validate and affine the role of plant water storage for tree response to extreme drought.

In this section, we aimed to illustrate the potentialities offered by SurEau-Ecos for improving our understanding of forest response to drought. We explored whether the probability of plant hydraulic failure simulated by SurEau-Ecos was related to the distribution of two tree species at their southern distribution margin, and we then used this information to identify future areas at risk of drought-induced tree mortality. Specifically, we hypothesized that hydraulic failure was a significant constraint to tree distribution at the regional level.

We quantified the probability of hydraulic failure over France (544 000 km^{2}) for two different species chosen for their contrasted functioning
strategies: an evergreen Mediterranean oak (*Quercus ilex*) and a temperate deciduous
European beech (*Fagus sylvatica*) (see main parameters in Table 3). *Quercus ilex* is a drought-resistant
species with low LAI, *P*_{50}, and deep root systems to extract water from
cracks in the bedrock during drought. In contrast, *Fagus sylvatica* is characterized by a
higher vulnerability to drought (higher *P*_{50}) and higher LAI values. As
in Sect. 5, we defined hydraulic failure as the point when leaves reach 99 % loss of hydraulic conductivity (PLC_{L}≥99 %).
For each period investigated, we reported the probability of hydraulic
failure as the frequency of years during which PLC_{L}≥99 %.

We ran simulations for present (1991–2020) and future (2071–2100) periods at
an 8 km^{2} resolution over France for both species. Climate data for the
present period (1970–2020) were extracted from the SAFRAN climate reanalysis
database (Vidal et al., 2010), which covers France at an 8 km^{2} resolution. Projections of climate variables for the future climate
period (2071–2100) were obtained from a climate simulation program involved
in the fifth phase of the Coupled Model Intercomparison Project (CMIP5)
and produced as part of the EURO-CORDEX initiative (Kotlarski
et al., 2014). One single global circulation model–regional climate model (GCM–RCM) couple was extracted for these analyses
(i.e., MPI-ESM-REMO2009), which was chosen because of its averaged climate
trajectory over France when compared to an ensemble of GCM–RCM couples
(Fargeon et al.,
2020; Ruffault et al., 2020). Data were extracted at a 0.44^{∘}
spatial resolution for the historical (1990–2005) and future (2006–2099)
periods. Model outputs were bias-corrected and downscaled at the 8 km^{2}
resolution using a quantile–quantile correction approach
(Ruffault et al., 2014).

To apply the model at the landscape scale, we made several simplifying
assumptions. First, we assumed that each 8 km^{2} grid cell was covered by
trees of the same species and that LAI was set to a constant value representative
of observed values for the considered species (Table 3). Second, soil
characteristics were also set constant over the territory. Both assumptions
are unrealistic because stand characteristics vary at the local scale and
have a primordial role in the probability of hydraulic failure (Sect. 5).
However, as we aimed to assess the regional (rather than the local)
vulnerability of tree species to changes in climate, we did not expect this
to be a main limitation, provided that the results of these simulations be
interpreted accordingly to these assumptions. To assess whether the
probability of hydraulic failure was a good proxy of the current southern
range of tree species distribution, we compared the results of our
simulations with presence and absence data for each species. Tree species data
were extracted from the national forest inventory database (available
at http://www.ifn.fr, last access: 12 May 2022) and aggregated to obtain presence–absence on the 8 km studied
grid following Cheaib et al. (2012).

Maps of probability of hydraulic failure (probability of reaching PLC_{L}≥99 %) are shown in Fig. 6. We observed
contrasting regional patterns according to the species under study. We
observed a higher probability of mortality in southeastern France for both
species, but the probability of hydraulic failure was higher for the European
beech than for the holm oak. In the rest of the country, the probability of
hydraulic failure was almost 0 for the holm oak. In contrast, we observed
probabilities up to 50 % for the European beech in the western part and
middle of the country, where the climate is temperate. When comparing these
results with the maps of current species distribution, we observed a
reasonable degree of spatial agreement between our simulations and
presence/absence data. European beech was predominantly present in areas
where our simulations indicated a probability of drought-induced mortality
equal to 0 %. However, we could not interpret the results for the holm
oak in the same way since the current distribution of this species indicates
that the southern climate margin is not reached in the present climate. In
the parts of the country where summer drought is less intense, several other
factors might explain why *Quercus ilex* is currently not observed, including
competition from more productive species, cold resistance or even forest
management policies.

Our projections for the end of the century showed a future increase in the
areas characterized by a high risk of hydraulic failure over France. For
*Fagus sylvatica*, the areas characterized by a high risk of hydraulic failure will extend
towards the northeast and west of the country (i.e., over the major part of the
territory). For *Quercus ilex*, our simulations indicated that the probability of hydraulic
failure should significantly increase in southeastern France, where this
species is currently widespread.

Altogether, these results indicate that future climate conditions might overcome the capacity of the two studied tree species to face drought over France, which might increase the likelihood of tree mortality and wildfires in the future. Adding information about the LAI and soil physical properties might further refine our simulation results. LAI can be estimated from remote sensing indices (see for instance (De Kauwe et al., 2020). However, TAW estimations are more problematic because information about root depth is rarely available (Ruffault et al., 2013; Venturas et al., 2020).

SurEau-Ecos can already be applied as a standalone model to understand plant water dynamics and can be used in a wide of research applications, from stand-scale estimations of water fluxes to regional predictions of drought-induced mortality (see Sect. 7). In addition, the specific distinction between the symplasmic and apoplasmic compartments implemented in SurEau-Ecos provides a solid foundation for predicting and monitoring water storage in the plant, a key factor in ecosystem disturbances such as mortality (Martinez-Vilalta et al., 2019) and wildfires (Ruffault et al., 2018b; Pimont et al., 2019).

The development of several supplementary key processes also warrants future consideration to extend the range of research questions and applications that SurEau-Ecos would be able to address. First, SurEau-Ecos currently simulates plant water dynamics for a single tree species for a homogeneous forest stand, and it therefore neglects the effects of species interactions on tree response to drought. This would, however, require us to affine the current representation of water competition between trees and microclimatic effects. Such developments would not only provide a mechanistic basis for multi-species modeling but could also help us to better understand the processes driving heterogenous mortality in the canopy and integrate the effects of forest management on stand structure microclimatic conditions. Another important limitation of SurEau-Ecos is that it does not simulate the processes related to photosynthesis, respiration, growth and carbon allocation. Future developments will aim at integrating SurEau-Ecos with other forest models that are designed to represent the carbon cycle and vegetation dynamics, including the forest growth models CASTANEA (Dufrêne et al., 2005) and GO+ (Moreaux et al., 2020), as well as the gap model ForCEEPS (Morin et al., 2021) under the Capsis platform (Dufour-Kowalski et al., 2012). These future research projects and developments will also be an opportunity to further evaluate the feedbacks between carbon balance, growth metabolism and hydraulic properties, including the impacts of post-drought growth on the recovery of hydraulic properties and therefore on tree vulnerability to water stress in the long run (Arend et al., 2022).

Drought is arguably one of the most important natural disturbances threatening forest ecosystems in a number of regions worldwide (Allen et al., 2015). The challenges facing our understanding of the role of plant hydraulics in vegetation dynamics are numerous (McDowell et al., 2019), with one being the ability of current vegetation models, including those based on plant hydraulics, to predict plant desiccation dynamics at regional scales (Venturas et al., 2020; De Kauwe et al., 2020; Rowland et al., 2021; Trugman et al., 2021). Here, we presented SurEau-Ecos, a new plant hydraulic SPA model aimed at predicting plant water status and drought-induced mortality at scales from stand to region. SurEau-Ecos was designed to simulate the plant water status of the different plant's compartments, while at the same time balancing for the needs of input parameters and computational requirements. SurEau-Ecos simulates key mechanisms associated with plant desiccation during drought and heat waves, including the dynamics of plant's water status beyond the point of stomatal closure via residual transpiration flow, plant cavitation and the solicitation of plants' water reservoirs. We showed that SurEau-Ecos was able to provide accurate estimations of plant water status dynamics compared to the SurEau model, despite the latter representing plant hydraulics mechanisms in more detail. This confirms that, for large-scale applications, the changes we implemented in SurEau-Ecos largely outweigh a potential loss of accuracy associated with the simplification of plant architecture and hydraulic processes. SurEau-Ecos provides the capability for us to better understand the role of plant hydraulics in vegetation dynamics under climate-change conditions characterized by increased drought frequency.

Leaf area index (LAI) of the stand is updated daily. Species can have
either evergreen or winter deciduous phenology. Evergreen species are
assumed to maintain a constant LAI throughout the year. LAI values of
deciduous plants are adjusted as a function of leaf phenology (∅)
and the maximum of the stand (LAI_{max}) as follows:

∅ is set to 0 until budburst occurs. Budburst is assumed to be
driven by the cumulative effect of forcing temperatures (*R*_{f}) on bud
development (Chuine and Cour, 1999) as follows:

where *t*_{0} is a parameter defining the initial date of the forcing
period, *t*_{f} the budburst date and *F*^{∗} is a parameter defining the amount
of forcing temperature to reach budburst. Once budburst is reached, ∅
increases from 0 to 1 at a rate specified by a parameter describing the LAI
growth rate per day (*R*_{LAI}). In autumn, leaf fall occurs
(∅ starts to decline) when the average daily temperature falls
below 5 ^{∘}C (Sitch et
al., 2003; De Cáceres et al., 2015) and then ∅ declines at a
similar rate to LAI growth in spring.

## C1 Explicit scheme

Let

and let

Applying the explicit scheme (Eq. 48 in main text) to the four water balance equations (Eqs. 6 to 9 in main text) gives the following equations.

Equation (6) can be rearranged to determine ${\mathit{\psi}}_{\mathrm{LApo}}^{n+\mathrm{1}}$:

Similarly, Eq. (7) gives ${\mathit{\psi}}_{\mathrm{SApo}}^{n+\mathrm{1}}$:

Equations. (8) and (9) give ${\mathit{\psi}}_{\mathrm{LSym}}^{n+\mathrm{1}}$ and ${\mathit{\psi}}_{\mathrm{TSym}}^{n+\mathrm{1}}$:

## C2 Implicit scheme

By combining Eqs. (6), (7), (8) and (9) with the implicit discretization (Eq. 50),
it is possible to analytically compute the unknown water potentials of each
compartment at time *n*+1.

First, we eliminate ${\mathit{\psi}}_{\mathrm{LSym}}^{n+\mathrm{1}}$ in Eqs. (6) and (8) by summing $\left(\mathrm{6}\right)+\left(\mathrm{8}\right)\times \frac{{K}_{\mathrm{LSym}}}{{K}_{\mathrm{LSym}}+\frac{{C}_{\mathrm{LSym}}}{\mathit{\delta}t}+\frac{{E}^{\prime n+\frac{\mathrm{1}}{\mathrm{2}}}}{\mathrm{2}}}$ and re-organizing the result as follows:

with ${\mathit{\delta}}_{\mathrm{L}}^{\mathrm{cav}}=\left\{\begin{array}{ll}\mathrm{0},& \text{if}{\mathit{\psi}}_{\mathrm{LApo}}^{n+\mathrm{1}}\ge {\mathit{\psi}}_{\mathrm{LApo}}^{\mathrm{mem}}\text{(no new cavitation event)}\\ \mathrm{1},& \text{if}{\mathit{\psi}}_{\mathrm{LApo}}^{n+\mathrm{1}}{\mathit{\psi}}_{\mathrm{LApo}}^{\mathrm{mem}}\text{(new cavitation event)}\end{array}.\right)$

Next, let us define intermediate variables to ease the resolution with the following equations:

Now, Eq. (C5) can be rewritten as follows:

Similarly, eliminating ${\mathit{\psi}}_{\mathrm{SSym}}^{n+\mathrm{1}}$ in Eqs. (7) and (9) by summing $\left(\mathrm{7}\right)+\left(\mathrm{9}\right)\frac{\mathrm{1}}{\mathrm{1}+\frac{{C}_{\mathrm{SSym}}}{{k}_{\mathrm{SSym}}\mathit{\delta}t}}$ and re-organizing the equation leads to the following result:

with ${\mathit{\delta}}_{\mathrm{S}}^{\mathrm{cav}}=\left(\right)open="\{">\begin{array}{ll}\mathrm{0},& \text{if}{\mathit{\psi}}_{\mathrm{SApo}}^{n+\mathrm{1}}\ge {\mathit{\psi}}_{\mathrm{SApo}}^{\mathrm{cav}}\text{(no new cavitation event)}\\ \mathrm{1},& \text{if}{\mathit{\psi}}_{\mathrm{SApo}}^{n+\mathrm{1}}{\mathit{\psi}}_{\mathrm{SApo}}^{\mathrm{cav}}\text{(new cavitation event)}\end{array}.$

Similarly, by defining the following equations:

and

equation (C10) can be rewritten as follows:

Now, we eliminate ${\mathit{\psi}}_{\mathrm{SApo}}^{n+\mathrm{1}}$ from the simplified Eqs. (C9) and (C13) by summing $\left(\mathrm{C}\mathrm{5}\right)+\left(\mathrm{C}\mathrm{9}\right)\times \frac{{K}_{\mathrm{SLApo}}}{{K}_{\mathrm{SLApo}}+\stackrel{\mathrm{\u0303}}{K}}$ and re-organizing it as follows:

Let

These equations can be combined to determine ${\mathit{\psi}}_{\mathrm{LApo}}^{n+\mathrm{1}}$, ${\mathit{\psi}}_{\mathrm{SApo}}^{n+\mathrm{1}},\phantom{\rule{0.125em}{0ex}}{\mathit{\psi}}_{\mathrm{LSym}}^{n+\mathrm{1}}$ and ${\mathit{\psi}}_{\mathrm{SSym}}^{n+\mathrm{1}}$

We can now rearrange this to determine ${\mathit{\psi}}_{\mathrm{LApo}}^{n+\mathrm{1}}$:

Knowing ${\mathit{\psi}}_{\mathrm{LApo}}^{n+\mathrm{1}}$, we can determine ${\mathit{\psi}}_{\mathrm{SApo}}^{n+\mathrm{1}}$ from Eq. (C9):

In practice, because we do not know whether new cavitation events will occur during the time step, Eqs. (C6) and (C7) and (C11) and (C12) are first computed assuming that ${\mathit{\delta}}_{\mathrm{L}}^{\mathrm{cav}}$ and ${\mathit{\delta}}_{\mathrm{S}}^{\mathrm{cav}}$ did not change since the last time step. This will be correct for most time steps, except those when cavitation either starts or ends. At this stage, we should hence check whether solutions ${\mathit{\psi}}_{\mathrm{LApo}}^{n+\mathrm{1}}$ and ${\mathit{\psi}}_{\mathrm{SApo}}^{n+\mathrm{1}}$ are below or above ${\mathit{\psi}}_{\mathrm{LApo}}^{\mathrm{cav}}$ and ${\mathit{\psi}}_{\mathrm{SApo}}^{\mathrm{cav}}$ in order to eventually update ${\mathit{\delta}}_{\mathrm{L}}^{\mathrm{cav}}$ or ${\mathit{\delta}}_{\mathrm{S}}^{\mathrm{cav}}$ if needed. In cases where there is change (for time steps exactly corresponding to begin or end of cavitation events), the computation should be done again with actualized values of ${\mathit{\delta}}_{\mathrm{L}}^{\mathrm{cav}}$ and ${\mathit{\delta}}_{\mathrm{S}}^{\mathrm{cav}}$.

Finally, knowing ${\mathit{\psi}}_{\mathrm{LApo}}^{n+\mathrm{1}}$, we can solve ${\mathit{\psi}}_{\mathrm{LSym}}^{n+\mathrm{1}}$ from Eq. (8):

Knowing ${\mathit{\psi}}_{\mathrm{SApo}}^{n+\mathrm{1}}$, we can solve ${\mathit{\psi}}_{\mathrm{LSym}}^{n+\mathrm{1}}$ from Eq. (9):

## C3 Semi-implicit scheme

Let

and with

By combining Eqs. (6)–(9) with the semi-implicit Eq. (57), it leads to the following equations.

For ${\mathit{\psi}}_{\mathrm{LApo}}^{n+\mathrm{1}}$,

For ${\mathit{\psi}}_{\mathrm{SApo}}^{n+\mathrm{1}}$,

For ${\mathit{\psi}}_{\mathrm{LSym}}^{n+\mathrm{1}}$,

For ${\mathit{\psi}}_{\mathrm{SSym}}^{n+\mathrm{1}}$,

The model code and instructions on how to run the model version presented in this paper are available from https://doi.org/10.5281/zenodo.5878978 (Ruffault et al., 2022).

Weather simulation data of Global-Regional simulation model used in this study is available from the EURO-CORDEX initiative at https://www.euro-cordex.net/index.php.en (last access: 13 March 2021) for noncommercial research and educational purposes.

JR led the writing of the manuscript with input from all authors. JR and NMS coordinated the project. HC and JLD supervised the project. JR, NMS and FP developed the code and conducted the experiments. NMS developed a preliminary version of the code. FP designed the numerical resolutions of the model with inputs from NMS. All authors read and approved the manuscript.

The contact author has declared that none of the authors has any competing interests.

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

Julien Ruffault received funding from ECODIV department of INRAE. We acknowledge the INRAE ACCAF Metaprogram for its financial support of the project Drought&Fire. We thank Miquel De Cáceres and Xiangtao Xu for their careful reading of our original manuscript and their many insightful comments and suggestions.

This research has been supported by the Agence Nationale de la Recherche (grant no. ANR-18-CE20-0005). This study was completed with support from the Environmental Research and Development Program (SERDP) project through Forest Service Agreement 20-IJ-11221637-178.

This paper was edited by Hans Verbeeck and reviewed by Xiangtao Xu and one anonymous referee.

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- Abstract
- Introduction
- Description of SurEau-Ecos
- Impacts of numerical schemes on simulations and computation times
- Model parametrization
- Comparison between SurEau-Ecos and SurEau
- Sensitivity experiments
- Regional prediction of climate-change impacts on tree mortality
- Limitations and future developments
- Conclusion
- Appendix A: Leaf phenology module in SurEau-Ecos
- Appendix B: Additional tables and figures
- Appendix C: Numerical schemes
- Code availability
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References

- Abstract
- Introduction
- Description of SurEau-Ecos
- Impacts of numerical schemes on simulations and computation times
- Model parametrization
- Comparison between SurEau-Ecos and SurEau
- Sensitivity experiments
- Regional prediction of climate-change impacts on tree mortality
- Limitations and future developments
- Conclusion
- Appendix A: Leaf phenology module in SurEau-Ecos
- Appendix B: Additional tables and figures
- Appendix C: Numerical schemes
- Code availability
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References