2.1.1 Eddy covariance site (EUCFLUX)

The EUCFLUX site is located within a 200 ha plantation located in south-eastern Brazil (São Paulo State; 22^∘58^′04^′′ S and 48^∘43^′40^′′ W, 750 m a.s.l.) and is managed under a cooperative project of the IPEF (Instituto de Pesquisas e Estudos Florestais) (Nouvellon et al., 2019). The precipitation was on average 1536 mm yr⁻¹ (from 2008 to 2017), with a drier season between June and September, and the mean annual temperature was 19.3 ^∘C. Soils are deep Ferralsols (> 15 m). A clonal plantation of a fast-growing Eucalyptus grandis × urophylla hybrid was established in November 2009 and harvested in June 2017. At the centre of the stand, a flux tower continually measured meteorological variables as well as the fluxes of CO₂ and water vapour between the plantation and the atmosphere, with the eddy covariance method (Baldocchi, 2003). The study area was described in detail in Christina et al. (2017), Nouvellon et al. (2010), Nouvellon et al. (2019), and Vezy et al. (2018). The stand was fertilised in November 2019 with 3.0 g m⁻² of K₂O, 3.3 g m⁻² of P₂O₅, 1.8 g m⁻² of N, 400 g m⁻² of dolomitic lime, and trace elements; then at 3 months with 3.6 g m⁻² of K₂O and 3.12 g m⁻² of N; at 10 months with 6.72 g m⁻² of K₂O and 3.08 g m⁻² of N; and at 20 months of age with 15.12 g m⁻² K₂O. This amounted to a total of 23.60 gK m⁻² from fertilisation and resulted in non-limiting nutrient availability for tree growth during a rotation cycle. This value was higher than the typical 12 gK m⁻² added on average in commercial plantations during a rotation cycle (Cornut et al., 2021).

2.1.2 Fertilisation experiments (Itatinga)

A 2 ha split-plot fertilisation trial (three blocks with three fertilisation treatments per block) was installed at the Itatinga experimental station (23^∘02^′49^′′ S and 48^∘38^′17^′′ W, 860 m a.s.l.; University of São Paulo–ESALQ). It is located 12 km next to the EUCFLUX site, under similar climate and soil conditions. A fast-growing Eucalyptus grandis clone was planted in June 2010, and the soil–tree relationships were studied over the entire rotation of 6 years (from planting to harvesting). The experimental design was described in detail in Battie‐Laclau et al. (2014b). Six treatments (three fertilisation regimes and two water supply regimes) were applied in three blocks. In the present study, we focus on the +K and oK treatments with the undisturbed rainfall regime, which consisted of a non-limiting fertilisation +K (17.55 gK m⁻² applied as KCl at planting, with 3.3 gP m⁻² and 200 g m⁻² of dolomitic lime and trace elements as well as 12 gN m⁻² at 3 months of age) and an oK omission treatment where the same fertilisers as in the +K treatment were applied except for the K fertiliser. The area of each individual plot in the experiment was 864 m².

The concentrations of different elements (N, P, K) in the organs (leaves, trunks, branches, and roots) were measured at an annual time step in eight individuals of each fertilisation treatment and upscaled to the whole stand using allometric relationships (not shown). Biomass and nutrient contents were calculated (using upscaling) from inventories, biomass, and nutrient concentration measurements conducted at 1, 2, 3, 4, 5, and 6 years in each fertilisation treatment. Atmospheric deposition (0.55 gK m⁻² yr⁻¹) and canopy leaching fluxes (0.42 gK m⁻² yr⁻¹) were measured in a nearby experiment from Laclau et al. (2010).

The clones that were planted at the Eucflux and Itatinga stands were different. This has an impact on the response of the trees to environmental conditions, canopy functioning (Attia et al., 2019; Le Maire et al., 2019), and more importantly stand GPP (Attia et al., 2019; Epron et al., 2012). Distinct model parameter values were used for the more sensitive parameters in the model, when differences were observed in their measurements.

2.4.1 Overview of the leaf cohort model

Highly productive tropical eucalypt plantations in Brazil grow from seedlings to 25–30 m high trees in the span of 6–7 years. The plantations present a continuous foliar phenology with leaf production and leaf fall throughout the year. This has previously led to the development of a canopy dynamics model (Sainte-Marie et al., 2014). While this model was sufficient to explain leaf production and leaf fall dynamics, we found it necessary to develop a new cohort-based canopy dynamics model (summarised in Fig. 1), as there was a need for the simulation of both K cycling in the canopy and the effects of K on foliar ontogeny (Laclau et al., 2009; Battie-Laclau et al., 2013). A daily time step was necessary for the simulation of expansion and fall of the leaves of each cohort. All leaves within a cohort were considered to have the same physiological characteristics, growth, and lifespan. A cohort was characterised by a number of leaves per square metres of ground, individual leaf area, and mass. Leaf fall occurred when leaves reached a certain K minimum threshold or the end of their lifespan. This new leaf cohort model is described in the next sections, in the case of no limitation by K.

2.4.2 Leaf cohort production

A new cohort was initialised daily. The number of leaves N produced in the cohort was a function of the height increase of the trees. Indeed, in these fast-growing plantations, most of the new leaves are produced in the top-most part of the crown. The increase in tree height can be computed in the CASTANEA-MAESPA model as the result of increase in trunk biomass and with allometric parameters relating stand biomass and stand height (see the following companion paper: Cornut et al., 2022).

The relationship between daily height increase and leaf production was corrected by a flattening factor. This means that even if the daily height increase was close to zero or even null, leaf production would still happen at a slower but positive rate. The model generated a number of new leaves per m² at a daily time step following this function:

where N was the daily number of leaves produced in number of leaves per metre squared of ground, and ΔH (m) was the increase in tree height. f_p was the flattening factor, meaning that if f_p=0, then leaf production was linearly related to height increase, and as f_p increased, N tended towards a constant function. κ (n_leaves m${}_{\mathrm{ground}}^{-\mathrm{2}}$ m${}_{\mathrm{height}}^{-\mathrm{1}}$) is a conversion factor from height increment to number of new leaves. The parameters used here were fitted using experimental data from the fully fertilised stand. The calibration was a systematic exploration of parameter space using multiple normalised RMSE (addition of the normalised RMSE for multiple variables; see Eq. S1) as a goodness-of-fit indicator. The data used for calibration were destructive leaf biomasses (eight trees and upscaling using a stand inventory at a yearly time step over the rotation), leaf area (same as leaf biomass), and leaf fall (12 L traps at a monthly time step over the rotation) measurements. The calibration was done on cumulative leaf production and leaf fall to maintain consistency in the long-term carbon fluxes rather than focusing on their instantaneous changes.

2.4.3 Leaf cohort lifespan

As long as K was not limiting, the lifespan of a cohort was considered to be constant, since the leaf lifespan deduced from leaf biomass and leaf fall measurements in fully fertilised stand did not show major trends along the rotation, and the amplitude of seasonal changes in lifespan was limited (Fig. S4e). Since no mechanistic explanation was available, we refrained from implementing it in the model. For the sake of simplicity, we did not consider in the present simulations the fall of leaves resulting from extreme events (drought, frost, or heatwave). Indeed, in the studied sites, no large leaf fall due to extreme events was observed. Hence, the leaf lifespan (days), LLS, in non-limiting K conditions was fixed to the average measured value of 480. This average leaf lifespan was estimated as the ratio of the measured annual average leaf biomass (measured annually on eight trees and upscaled using whole-stand inventories) and the annual sum of litterfall (measured monthly).

2.4.4 Leaf expansion in area in the cohort

For a given cohort, individual leaf area (LA) expands from a virtually null area at initialisation of the cohort up to an area of LA_max (mm²). The leaf area dynamics followed a sigmoid function (Fig. 2a; Battie-Laclau et al., 2013). Leaf area was a function of time and not thermal time (as for instance in the original CASTANEA model), since no calibration data were available and it was not deemed necessary for this model. Therefore, the daily leaf area expansion was forced to follow the sigmoid derivative function as

where $\frac{\mathrm{\Delta }\mathrm{LA}}{\mathrm{\Delta }t}$ (mm² d⁻¹) was the daily growth in the area of an individual leaf within a given cohort; t (days) was the number of days since leaf cohort creation; LA_max (mm²) was the (non-limited) maximum leaf area; k_LA (d⁻¹) was a slope parameter; and t50_LA (days) was the inflexion point of the original sigmoid of leaf area increase, meaning that it was the date on which leaves increased the most and on which the leaf area (LA) reached half of the maximum leaf area (LA_max). The parameters LA_max, k_LA, and t50_LA were fitted from 70 measured expanding leaves (Battie-Laclau et al., 2013) in non-limited fertilisation conditions using RMSE as a goodness-of-fit indicator. Parameters k_LA and t50_LA were assumed not to vary along the stand rotation. LA_max was also assumed to be constant, since the leaf scans did not show any explainable trends of mean leaf area during the rotation (Fig. S5).

The total leaf area of a given cohort was given by the product of LA, the area of an individual leaf, and N, the number of leaves in the cohort. The total leaf area of the stand at a given date was calculated by adding up all the cohort areas.

2.4.5 Leaf expansion in mass in the cohort

Individual leaf mass increase within a cohort was similar in shape to the leaf area increase but with a temporal shift, since leaf mass per area continues to increase when the maximum leaf area is attained as follows:

where $\frac{\mathrm{\Delta }\mathrm{BF}}{\mathrm{\Delta }t}$ (g d⁻¹) was the daily growth in mass of an individual leaf in a given cohort, t (days) was the number of days since leaf cohort creation, BF_max (g) was the maximum individual leaf mass, k_BF (d⁻¹) was a slope parameter, and t50_BF (days) was the inflexion point of the original sigmoid of leaf mass increase; therefore it was the date of maximum leaf area increase and also the date when half BF_max was reached. The parameters k_BF and t50_BF were calibrated using individual leaf biomass data and results from Laclau et al. (2009).

Specific leaf area (SLA) of individual leaves showed a decreasing relationship with tree height (Fig. S6a), while LA_max was more constant as described before (Fig. S5). We thus assumed that BF_max increased with tree height:

where BF_max (gC) was the maximal mass of an individual leaf of a cohort at the end of leaf expansion in mass, BF${}_{\max }^{\mathrm{rotation}}$ (gDM) was the maximum mass of an individual leaf throughout the rotation, s_BF and P were the parameters of the power function between leaf mass and tree height H (m), and TC (gC gDM⁻¹) was the leaf carbon content.

2.4.6 Leaf water content

In non-limited nutrient conditions, leaf cell expansion in area was associated with a leaf water inflow in order to maintain an optimum leaf turgor. This water inflow was computed as

where W_xylem→leaf (mL d⁻¹) was the water inflow into the expanding leaf (this was “structural” water associated to the creation of new tissues, not to be confounded with the water used for leaf transpiration), S the leaf area of the cohort (mm²), computed in Eq. (2), and Γ (mL mm⁻²) was the surfacic water content, i.e. the amount of leaf water per leaf area at full turgor. Γ was assumed to be a constant.

Experimental data have shown that at the end of leaf area expansion, when the leaf has reached its maximum area, there was some water outflow, defined hereafter as water expulsion. This is an assumption made from observations of a slight decrease in K leaf content following the end of leaf expansion (Laclau et al., 2009). This leaf water (containing ions) expulsion, probably corresponding to a loss of cell wall extensibility (Pantin et al., 2012) during the maturation of leaf tissue, was limited in quantity and in duration. Hence, the overall leaf water content dynamic starts increasing until a maximum at the end of the leaf area expansion, followed by a small decrease until a constant plateau. This plateau corresponds to the water content necessary to maintain a constant leaf turgor in optimal conditions. The water expulsion flux was computed as

where W_{leaf→phloem} (mL d⁻¹) was the flux of water leaving the leaf at the end of leaf expansion, α (mL d⁻¹) was the constant rate of water expulsion fitted using fine-scaled leaf K concentrations (Laclau et al., 2009), W_leaf (mL) was the amount of water in an individual leaf on the previous day, and $W_{\mathrm{leaf}}^{\mathrm{turgor}}$ (mL) was the amount of water found in the leaf at the final plateau. $W_{\mathrm{leaf}}^{\mathrm{turgor}}$ was computed as Γ×LA_max.

Finally, the variation of leaf water content for an individual leaf in a cohort (W_leaf; mL) was computed by adding the daily net flow $\frac{\mathrm{\Delta }{W}_{\mathrm{leaf}}}{\mathrm{\Delta }t}$ given by

2.5.1 Soil K

Soil K content (K_soil; gK m⁻²) was initialised in the model at the tree-planting date (EUCFLUX: 7 October 2009, Itatinga: 1 June 2010) with a measured value $K_{\mathrm{soil}}^{t_{\mathrm{0}}}$, calculated using K concentration in soil, and soil bulk density at different depths (Maquère, 2008). Then, this value was updated daily with incoming and outgoing fluxes.

The K that is added daily to the K litter pool (K_litter) is the K reaching to the ground through leaf fall (Eq. 27), bark fall (Cornut et al., 2023), branch fall (Cornut et al., 2023), and entering the soil litter pool through fine root turnover (Cornut et al., 2023). Instead of a fixed decomposition rate of K in litter, the model considered K release from litter to be mainly coming from leaching with water, since K is a cation that is not strongly adsorbed on organic surfaces. Litter K release measurements done at the experimental site (Maquère, 2008) showed very similar K release rates for branches, bark, and leaves, further confirming this hypothesis. Moreover, K is released faster than either C, N, or P contained in the litter, suggesting a leaching process independent of litter decomposition. Since we assumed the leaching rate was independent of the litter type, all simulated K litter has been pooled into a unique K litter compartment (K_litter). The following equation was used for K leaching from the litter to the soil:

where K_{litter→soil} (gK m⁻² d⁻¹) was the litter K leaching flux, P_ground (mm d⁻¹) was the daily amount of precipitation reaching the ground, σ (mm⁻¹) was the conversion factor between the K litter leaching rate and throughfall precipitation, and K_litter (gK m⁻²) was the amount of K in the litter. σ was estimated on annual data by dividing the measured K leaching rate (Maquère, 2008) by the annual precipitation that falls on the ground (throughfall).

K fertilisation was applied at the beginning of the rotation at several dates, in a solid form (crystals of KCl), and located close to the Eucalyptus plants. The flux of K from this solid fertiliser compartment (K_fertiliser; gK m⁻²) to the soil K compartment was simulated using the following equation:

where K_{fertiliser→soil} (gK m⁻² d⁻¹) was the flux of K from the fertiliser compartment to the accessible soil K pool, and s_fertiliser was the decomposition rate of K fertiliser in d⁻¹. Observations in the fields showed that the KCl fertiliser dissolved quickly at EUCFLUX and Itatinga (less than 2 months).

Atmospheric K deposition is modelled as a constant flux. We used the values measured at Itatinga (Laclau et al., 2010). They amounted to a mean input of K_{atmosphere→soil} of 0.55 gK m⁻² yr⁻¹ distributed uniformly throughout the year. This amount feeds directly into the total K_soil pool.

Deep soil K leaching was included in the model but was parameterised to be a null flux, as was measured in the plantations under study (Maquère, 2008). The K entrance to the soil pool from mineral weathering was simulated as a constant flux. K flux from weathering is directly added to the accessible soil, since this process mainly takes place in the rhizosphere (Pradier et al., 2017). However, as for deep leaching, there is no clear evidence of this flux in the soils under study, where values between 0 and 0.3 gK m⁻² yr⁻¹ are given (Cornut et al., 2021): we therefore also set this flux to zero.

Only a portion of K_soil was accessible to the roots at the beginning of the rotation because of the time spent for root horizontal and vertical expansion. Because K was mainly located in the top soil layers (Maquère, 2008), and because root growth in depth was very fast (Christina et al., 2011), only the horizontal root exploration was considered in the model. An empirical relationship between tree height and area root radius around individual trees was described in Gonçalves (2000) as follows:

where root_radius (m) was the average radius of the horizontal root front around a tree, and H (m) was the tree height (Cornut et al., 2023). Since the planting density was 1666 trees per hectare, a full exploration of the soil was obtained when tree had explored a circle of 6 m² area:

where $K_{\mathrm{soil}}^{\mathrm{accessible}}$ (gK m⁻²) was the soil K accessible for plant uptake, and K_soil (gK m⁻²) was the total bioavailable soil K. The fraction is the ratio of root accessible soil to total soil, bounded between 0 and 1.

Because of the root exploration dynamics, the initial K in the system $K_{\mathrm{soil}}^{t_{\mathrm{0}}}$ was progressively available to roots, at a proportion following the increase in the root explored area. Following the same logic, the amount of K coming from the litter decomposition and atmospheric deposition entered the total soil K pool K_soil, but only a part of this K_soil was available for plant uptake (called $K_{\mathrm{soil}}^{\mathrm{accessible}}$). However, the three other incoming fluxes of K to the soil were considered to be directly accessible for root uptake (i.e. they enter directly in the $K_{\mathrm{soil}}^{\mathrm{accessible}}$ pool) as follows: (1) the fertiliser flux, since fertilisers are applied close to trees at planting or when the root system is exploring the whole volume of the upper soil layers for other fertiliser applications, (2) the K flow coming from soil weathering because most of the weathering takes place in the rhizosphere (Pradier et al., 2017; de Oliveira et al., 2021), and (3) the canopy leaching flux because it enters the soil mostly below the crown foliage. K foliar leaching (K_{leaves→soil}) was computed within the leaf K sub-model, described below (Eq. 28).

2.5.2 Uptake of soil K and cycling in xylem and phloem

To calculate the K uptake by trees in the soil and the fluxes of K in the plant, it was necessary to calculate the optimal quantity of K in the phloem sap. Furthermore, K in phloem sap is essential to a wide range of processes (e.g. loading and unloading of sugars) (Cornut et al., 2023). For these processes, the plant maintains a fairly constant K phloem sap concentration [K]_phloem. To compute this K quantity in the phloem, values of optimal K concentration in the phloem sap ($[K{]}_{\mathrm{phloem}}^{\mathrm{opti}}$), minimum K concentration in the phloem sap ($[K{]}_{\mathrm{phloem}}^{\min }$), and phloem sap volume (V_phloem) per unit surface were needed.

$[K{]}_{\mathrm{phloem}}^{\mathrm{opti}}$ was considered to be the maximum concentration of K in the phloem sap measured in the fully fertilised stand (Battie‐Laclau et al., 2014b). $[K{]}_{\mathrm{phloem}}^{\min }$ was assumed to be the minimum concentration of K in the phloem sap measured in the K omission stand of the same experiment (Battie‐Laclau et al., 2014b).

Estimating V_phloem was done through relationships between phloem sap volume and xylem sap volume, since no direct measurements or estimates were available. Xylem sap volume was considered to be a function of basal area, sapwood area at DBH (Guillemot et al., 2021), height of the tree, and branch and root biomasses. The trunk cross section was divided into sapwood area and heartwood area. The trunk (respectively heartwood) volume was modelled as a cone with a base disc of area equal to the basal area (respectively equal to the heartwood area). Trunk sapwood volume was estimated as the difference between trunk volume and heartwood volume. Branch and root sapwood volume were deduced from their biomass, considering that branches and root biomass are entirely composed of sapwood. Their volume were computed using the density of eucalypt sapwood. The lumen volume of the xylem (i.e. the xylem sap volume) was considered to be 13.6 % of total xylem volume as reported in general for angiosperms (Zanne et al., 2010), since no eucalypt-specific data were available. Following Hölttä et al. (2013), and considering the relatively similar lumen proportion between both xylem and phloem (Nobel, 2005), phloem sap volume was considered to be 2 % of the total xylem sap volume.

Uptake of K from the soil by the trees was a function of demand by growing organs, remobilisation of K from senescent organs, and soil supply. The amount of K available for uptake was computed in Eq. (11). K demand by the trees needs to be calculated. To that end, the following was calculated.

First, the target amount of K in the phloem was calculated as

where $K_{\mathrm{phloem}}^{\mathrm{tar}}$ was the target amount of K in the phloem sap (gK m⁻²), $[K{]}_{\mathrm{phloem}}^{\mathrm{opti}}$ was the optimal K concentration in the phloem sap (gK L⁻¹), V_phloem was the volume of phloem sap (dm ³ m⁻²), K_NPP was the optimal quantity of K needed for organ growth, and $K_{\mathrm{leaf}}^{\mathrm{demand}}$ was the optimal quantity of K needed for leaf development.

Finally the demand for K uptake from the soil is the following:

where $K_{\mathrm{soil}\to \mathrm{xylem}}^{\mathrm{demand}}$ (gK m⁻² d⁻¹) was the quantity of K uptake necessary for optimal tree functioning, $K_{\mathrm{phloem}}^{\mathrm{tar}}$ was from Eq. 12, $K_{\mathrm{xylem}}^{\mathrm{tar}}$ (gK m⁻²) was the target amount of K in the xylem sap, K_phloem (gK m⁻²) was the amount of K in the phloem sap, K_remob (gK m⁻²) was the amount of K remobilised from the woody organs (Cornut et al., 2023), and K_xylem (gK m⁻²) was the amount of K in the xylem.

Uptake of K from the soil to the xylem sap is the minimum between the soil “offer”, i.e. what can be taken up from the soil knowing the soil K content and the soil-to-root K resistance, and the xylem K “demand”:

where K_soil→xylem (gK m⁻² d⁻¹) was the uptake flux, K_soil (gK m⁻²) was the amount of K in the accessible soil, R_soil→xylem (days) was the resistance to absorption by plant roots, and $K_{\mathrm{soil}\to \mathrm{xylem}}^{\mathrm{demand}}$ (gK m⁻² d⁻¹) was the uptake demand from Eq. (13).

In the model, internal K cycling (Marschner et al., 1996) was a necessary process that provides feedback for the uptake of K from the soil, maintaining K homeostasis in the phloem sap, and linking organ remobilisation and allocation of K for growth. In the K circulation model (Fig. 1), two K fluxes are represented, one from the phloem sap to the xylem sap (representing a flux mainly happening in roots in plants) and one from xylem sap to phloem sap (mainly happening in the shoots). These representations allowed the phloem sap to maintain a K content of phloem close to optimal values (Eq. 12).

Firstly, the flux of K from the xylem sap to the phloem sap was calculated. It was a function of phloem “demand” and xylem sap K of the previous time step. We assumed that all the K available in the xylem sap could potentially be transferred to the phloem sap the next day as follows:

where K_{xylem→phloem} (gK m⁻² d⁻¹) was the flux of K from the xylem to the phloem, K_xylem (gK m⁻²) was the amount of K in the xylem sap, K_xylem (gK m⁻²) was the amount of K in the phloem sap, and $K_{\mathrm{phloem}}^{\mathrm{tar}}$ was from Eq. (12).

The transport of K from the phloem to the xylem took place if K concentration in the phloem sap was higher than its optimal value (e.g. following leaf resorption) as follows:

where K_{phloem→xylem} (gK m⁻² d⁻¹) was the flux of K from the phloem to the xylem, and $K_{\mathrm{phloem}}^{\mathrm{tar}}$ was from Eq. (12).

2.5.3 K cycling in the leaves

The leaf K balance equation of the individual leaf of each leaf cohort was given by the following sum of fluxes:

where $\frac{\mathrm{\Delta }{K}_{\mathrm{leaf}}}{\mathrm{\Delta }t}$ (gK d⁻¹) was the daily variation of the quantity of K in an individual leaf of a given cohort; K_{phloem→leaf} was the amount of K entering the leaf during leaf expansion (see Eq. 22); K_leaf→soil was the canopy leaching flux (see Eq. 28); K_{leaf→phloem} was the sum of K following water expulsion at the end of leaf expansion (Eq. 24), the maximum between K resorption driven by the phloem demand (Eq. 25), and the K resorption at leaf senescence (Eq. 26); and K_{leaf→litter} was the K flux occurring the last day of the cohort, when the leaf was simulated to fall. Leaf K inflow (K_{phloem→leaf}) was computed as a function of the K offer by the phloem and K demand for leaf growth at the canopy scale and organ growth at the tree scale.

The calculation of the water inflow in the leaf during leaf expansion was calculated first in the case of no K limitation (W_xylem→leaf in Eq. 5). This allowed the calculation of a theoretical optimal K flux entering the expanding leaf, $K_{\mathrm{phloem}\to \mathrm{leaf}}^{\mathrm{nonlimited}}$, computed considering an optimal concentration of K in the water entering the leaf, $[K{]}_{\mathrm{leaf}}^{\max }$ (gK mL⁻¹). This value was approximated as the maximum concentration found in the leaf water on different measurement campaigns (Battie-Laclau et al., 2013; Laclau et al., 2009). The resulting K flux was the non-limited rate of K entrance in the expanding leaf:

where $K_{\mathrm{phloem}\to \mathrm{leaf}}^{\mathrm{nonlimited}}$ (gK d⁻¹) was the maximum entrance of K⁺ ions in the expanding leaf.

However, restriction of this flux occurs due to the phloem limitation of K supply at canopy scale that may not attain the K demand for optimal growth. A reduction of the K inflow in the leaf was therefore applied if the leaf demand at canopy scale $K_{\mathrm{leaf}}^{\mathrm{demand}}$ was higher than the available K_phloem (the “offer”).

$K_{\mathrm{leaf}}^{\mathrm{demand}}$ (gK m⁻²) was the K demand of all expanding leaves of the stand and was computed as the sum of $K_{\mathrm{phloem}\to \mathrm{leaf}}^{\mathrm{nonlimited}}\times N$ for all leaf cohorts (with N the number of leaves of each cohort; see Eq. 1) as follows:

To calculate the phloem sap “offer”, the following relationship was used:

where K_{phloem→organs} (gK m⁻²) was the amount of K available for leaf expansion and organ growth in the phloem sap, K_phloem (gK m⁻²) was the total amount of K in the phloem sap, $[K{]}_{\mathrm{phloem}}^{\min }$ (gK L⁻¹) was the minimal concentration of K in the phloem sap, V_phloem (L) was the phloem sap volume, K_NPP (gK m⁻²) was the optimal amount of K for organ growth (Cornut et al., 2023), and $K_{\mathrm{leaf}}^{\mathrm{demand}}$ (gK m⁻²) was the demand for optimal leaf expansion.

Then the limitation of K for leaf expansion was calculated as a ratio of available (“offer”) K-to-K demand:

where L_K was the ratio of available K in the phloem sap to demand of K from organ growth and leaf expansion, K_{phloem→organs} (gK m⁻²) was available phloem K (Eq. 20), and K_NPP and $K_{\mathrm{leaf}}^{\mathrm{demand}}$ were organ growth and leaf expansion demands respectively (both gK m⁻²; see above).

The quantity of K entering the expanding leaf was thus defined as the following:

where K_{phloem→leaf} (gK d⁻¹) was the amount of K⁺ ions that enter the expanding leaf in limited K conditions, $K_{\mathrm{phloem}\to \mathrm{leaf}}^{\mathrm{nonlimited}}$ was computed in Eq. (18), and L_K was computed in Eq. (21).

The K outgoing flux from leaf to phloem (Fig. 2b) can be decomposed into

where K_expulsion (gK m⁻² d⁻¹) was the K flux leaving the leaf during leaf maturation (Eq. 24), $K_{\mathrm{resorption}}^{\mathrm{phloem}}$ (gK m⁻² d⁻¹) was the resorption flux driven by phloem sap demand (Eq. 25), and $K_{\mathrm{resorption}}^{\mathrm{senescence}}$ (gK m⁻² d⁻¹) was the resorption flux driven by leaf senescence (Eq. 26).

Here, W_{leaf→phloem} was calculated in Eq. (6); W_leaf was the previous day's leaf water content, calculated in Eq. (7); and K_leaf was the previous day's leaf K content of the cohort.

The K resorption flux K_resorption from the leaf to the phloem could be activated by low phloem K content. This was a mechanism to maintain homeostasis in the phloem, since K was essential for many phloem functions (Cornut et al., 2021). Evidence was also provided by leaves losing K during their lifespan, especially in K-deficient trees (Battie-Laclau et al., 2013). Another piece of evidence was the high concentrations of K in the petiole compared to other leaf parts (Fig. S9d). This was not the case for N (Fig. S9c) and suggests an intense circulation of K to and from the leaf. The resorption of the leaf towards the phloem was

where $K_{\mathrm{resorption}}^{\mathrm{phloem}}$ (gK d⁻¹) was the cohort phloem-driven resorption, K_leaf (gK) was the K content of leaves in the cohort, R_{leaf→phloem} (days) was the resistance to resorption, and L_K was the K limitation computed in Eq. (21).

The leaf K resorption flux during leaf senescence $K_{\mathrm{resorption}}^{\mathrm{senescence}}$ followed a sigmoid function:

where $K_{\mathrm{resorption}}^{\mathrm{senescence}}$ (gK d⁻¹) was the resorption flux occurring at leaf senescence, just before leaf fall. LLS (days) was the leaf lifespan, which was also the inflexion point of the sigmoid, and k_r was the parameter corresponding to the speed of the resorption flux at the inflexion point. We approximated the time it took for active K resorption to be 1 w, as K⁺ ions are highly mobile, and evidence from chlorophyll degradation at senescence suggests extremely fast dynamics (Mattila et al., 2018).

The K flux from leaves to litter was the sum of each falling cohort multiplied by the K content of the respective leaf cohort at leaf fall:

where K_{leaves→litter} (gK m⁻² d−1) was the K flux from leaves to litter, n was the total number of falling leaf cohorts, i was the number of each individual falling leaf cohort, and $K_{\mathrm{leaf}}^{\mathrm{fall}}$ (gK m⁻² d−1) was the amount of K from each cohort that fell and reached the K litter pool.

We assumed that the daily canopy leaching flux strength was proportional to the throughfall that occurs during precipitation as observed previously in a eucalypt forest (Crockford et al., 1996):

where λ (mm${}_{\mathrm{throughfall}}^{-\mathrm{1}}$) was the fraction of leaf K that was leached per millimetre of daily throughfall, W_tip (mm) was the throughfall, and K_leaf (gK) was the amount of K in the leaf. λ was calibrated considering the leaf area index and leaf K content of a well fertilised canopy as well as canopy K leaching measurements (Laclau et al., 2010).

Finally, the K flux accompanying the leaf fall, K_{leaf→litter}, happened following one of the two conditions: when leaf cohort lifespan LLS was reached, or when the K concentration in leaf water ($\frac{K_{\mathrm{leaf}}}{W_{\mathrm{leaf}}}$) was below a threshold value [K]_min of 9.25e⁻⁵ gK mL⁻¹. At one of these dates, the leaf cohort was shed and $K_{\mathrm{leaf}\to \mathrm{litter}}=K_{\mathrm{leaf}}$. This [K]_min threshold value was either reached after resorption during senescence or through other processes (phloem demand, Eq. 25; leaching, Eq. 28) thus diminishing the leaf lifespan in K-deficient trees. Indeed, leaf fall was related to strong K deficiency in several studies (Laclau et al., 2009; Battie-Laclau et al., 2013).

2.6.1 Number of leaves produced at cohort initialisation

Since leaf production was a function of tree height which itself is a function of tree trunk biomass, K availability could have an indirect impact on leaf production through its impact on tree trunk production. Briefly, tree trunk production could be affected by a reduced allocation of C due to either a decrease in GPP or an increase in the share of C partitioned to other organs (for more details on trunk production, see Cornut et al., 2023). No specific impact of K deficiency on the number of new leaves generated was included in the model, since experiments have shown that leaf generation speed at the branch level is not impacted by K availability, and leaf biomass production is not substantially different between oK and fully K fertilised stands (Cornut et al., 2021).

2.6.2 Impact of K limitation on individual leaf area

When there was no K limitation, in optimal conditions, leaf expansion in area was computed as in Eq. (2), and the water inflow was simply simulated to follow this leaf expansion as in Eq. (5). However, under K limitation, individual leaf area was strongly affected by K availability (Battie-Laclau et al., 2013). Mechanistically, the increase of leaf area was driven by a water flux entering the leaf, because the turgor pressure participates to the cell expansion, following the logic of the Lockhart model (Lockhart, 1965). The Lockhart model was simplified in the present study due to the important number of parameters of the original model that had not been measured in our context and the difficulty regarding their calibration. This model allowed to relate the K availability in the phloem sap and the expansion of leaves at the individual leaf level on a daily time step. Using the dynamic water content of leaves during expansion, K demand for the cohort at each time step was calculated. The availability of K in the phloem sap then determined a K-limited water flux and thus the leaf expansion rate (Fig. 2d).

First, K availability controls the water entrance flux (W_xylem→leaf; Eq. 5) in the leaf during leaf expansion, since there was a lower limit of osmotic potential required for the entrance of water in the leaf cells.

Here, $W_{\mathrm{xylem}\to \mathrm{leaf}}^{\mathrm{Klimited}}$ (mL d⁻¹) was the flux of water entering the leaf during leaf expansion reduced with K limitation, K_{phloem→leaf} was from Eq. (22), $K_{\mathrm{phloem}\to \mathrm{leaf}}^{\mathrm{nonlimited}}$ was from Eq. (18), and r was a parameter $r\in [\mathrm{0},\mathrm{1}]$ of the same order of magnitude as the ratio of K-limited individual leaf area compared to non-limited leaf area.

Secondly, leaf water content W_leaf was recalculated using an updated value for the expansion ($W_{\mathrm{xylem}\to \mathrm{leaf}}^{\mathrm{Klimited}}$ instead of W_xylem→leaf).

Finally, the non-limited leaf area expansion increment $\frac{\mathrm{\Delta }S}{\mathrm{\Delta }t}$ computed in Eq. (2) was updated with a new K-limited leaf area expansion increment$\frac{\mathrm{\Delta }{S}_{\mathrm{Klimited}}}{\mathrm{\Delta }t}$, considered to be directly proportional to the water flux entering the leaf:

where $\frac{\mathrm{\Delta }{S}_{\mathrm{Klimited}}}{\mathrm{\Delta }t}$ (mm² d⁻¹) was the area increase of the expanding leaf computed after accounting for K limitation, $W_{\mathrm{xylem}\to \mathrm{leaf}}^{\mathrm{Klimited}}$ was obtained from Eq. (29), and Γ (mL mm⁻²) was the leaf surfacic water content.

Figure 2Outputs of the leaf cohort expansion model over the course of the lifespan of a single leaf from a cohort (produced on day 504 of the rotation at the Itatinga site) in two contrasted K availabilities, +K (a, b) and oK (c, d); leaf water content, leaf K, and leaf surface at the individual leaf scale (a, c); total flux of K at the leaf scale showing the transition from K sink (positive flux to the leaf) to K source (negative flux from the leaf) (b, d). Small negative K fluxes (corresponding to a loss of K from the leaf) during the leaf's existence are K foliar leaching fluxes during precipitation events. The grey line represents the fall of the leaf cohort. The quantity of K that is in the leaf at the moment of leaf fall is added to the K litter pool.

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2.7.1 Leaf K-deficiency symptoms

When leaves experience strong K deficit, they display anthocyanic symptoms (i.e. they turn purple from the leaf margins; Gonçalves, 2000). This has a strong impact on the photosynthetic capacity of affected areas (Battie‐Laclau et al., 2014a). We assumed that leaf K-deficiency symptom area results from the history of K deficiency the leaf has experienced since the beginning of its growth. This was modelled as function of the accumulation of K deficit in the leaves over time, called “deficit days” (DDs). The daily increase in DD was computed as

where $\frac{\mathrm{\Delta }\mathrm{DD}}{\mathrm{\Delta }t}$ (g) was the daily increase of the deficit days, [K_leaf]_max (gK mL⁻¹) was the optimal (maximal) foliar concentration of K, W_leaf (mL) was the amount of water in the individual leaf (after K limitation; Eq. 7), and K_leaf (gK) was the leaf K content.

The proportion of symptoms in a leaf (Fig. 2b, d) was then computed as

where SP was the leaf surfacic symptom proportion, DD was the accumulated deficit days computed in Eq. (31), Θ was a conversion factor from deficit days to symptom proportion, and SP_max was the maximum proportion of symptom area on a single leaf, with $\mathrm{0}<\mathrm{SP}<{\mathrm{SP}}_{\max }$.

2.7.2 Impact of symptoms on leaf photosynthesis

Leaves, even with symptoms, continue to intercept radiations. In the model, it means that the light interception sub-model was not affected by symptom area; i.e. the total leaf area of each cohort was not affected by leaf K symptoms. Note that the total leaf area under K deficiency was reduced through various processes such as lower number of produced leaves because of lower growth in height (Eq. 1), reduction of individual leaf sizes (Eq. 30), and through the shorter lifespan of leaves because of K-deficiency-associated leaf fall (Sect. 2.5.3).

However, leaf symptoms have a strong effect on leaf-scale photosynthesis. Indeed, experimental results (Battie‐Laclau et al., 2014a) have demonstrated that the leaf-scale photosynthesis was strongly reduced when there was K-deficiency symptoms. This decrease was almost linear, suggesting that we could model leaf photosynthesis as fully active in the non-symptomatic areas of the leaves and null in the symptomatic area; i.e. the photosynthesis was reduced by the proportion of symptoms in the leaf.

For the sake of simplicity, this was implemented in the model by reducing the two leaf-scale photosynthetic parameters Vc_max and J_max according to the leaf area proportion affected by symptoms as follows:

where Vc_max and J_max were respectively the maximum carboxylation rate and the maximum rate of electron transport, and SP was described in Eq. (32).