N uptake
2022-11-16
Full version
Theory
This is based on McMurtrie & Näsholm, 2017 New Phytologist (doi: 10.1111/nph.14927).
The root length density per unit soil volume is a function of root biomass \(C_r\) and morphological parameters (density: \(\rho_r\), rooting depth \(z_r\), root radius \(r_0\)). \[ l_r = \frac{C_r}{2 \pi r_0 \rho_r \ z_r} \] The half-distance between root hairs follows from \(l_r\): \[ r_x = \sqrt{\frac{1}{\pi l_r} + r_0^2} \] And the total root surface area is \[ A_r = 2 \pi r_0 l_r \]
The uptake flux through a unit root surface area can be modelled for a steady-state condition where a demand and supply are set equal. The “demand” flux can be modelled based on Eq. 2 in McMurtrie & Näsholm as \[ j_r = \frac{j_\text{rmax}}{1 + \frac{j_\text{rmax}}{k c_{s0}}} \] The supply flux is the sum of a diffusive and an advective component. The diffusive component is taken here as being driven by the difference between inorganic N concentrations at the root surface (\(c_{s0}\)) and in the bulk soil \(c_x\), divided by the “travel distance” of N in the soil, i.e., the half-distance between roots \(r_x\). The advective component is determined by the water uptake velocity \(v_0\) and the N concentration in the bulk soil. \[ j_s = \frac{D(c_x - c_{s0})}{r_x} + v_0 c_x \] Setting \(j_r = j_s\) yields a solution fo \(c_{s0}\) from which the root area-specific N uptake flux can be calculated either from the supply or the demand function.
The N concentration in the bulk soil is taken to be determined by the amount of inorganic N per unit soil surface area (the units corresponding to the CN-model formulation) divided by the root surface area and soil depth: \[ c_x = \frac{N_\text{inorg}}{z_r A_r} \] (The last point is confusing to me).
This is implemented below.
Implementation
par_fnup <- list(
D = 3, #D = 0.0005, # m d-1, rhizosphere diffusivity
j = 2000, #j = 0.0056, # Jrmax = maximum root-N influx, g N m2 d-1
k = 1, #k = 0.025, # initial slope of MM for N uptake flux = root absorbing power, m d-1, McMurtrie & Näsholm, 2017
f = 0 # transpiration velocity
)
calc_conc <- function(N, root_distance, par){
(sqrt((par$D*par$j - par$D*par$k*N - par$f*par$k*N*root_distance + par$j*par$k*root_distance)^2
- 4*par$D*par$k*(-par$D*par$j*N - par$f*par$j*N*root_distance)) - par$D*par$j + par$D*par$k*N
+ par$f*par$k*N*root_distance - par$j*par$k*root_distance)/(2*par$D*par$k)
}
calc_fnup_demand <- function(conc, par){
par$k * par$j * conc / (par$k * conc + par$j)
}
calc_fnup_supply <- function(conc, N, root_distance, par){
par$D * (N - conc) / root_distance + par$f * N
}
calc_fnup <- function(ninorg, root_distance, par){
conc <- calc_conc(ninorg, root_distance, par)
nup_demand <- calc_fnup_demand(conc, par)
nup_supply <- calc_fnup_supply(conc, ninorg, root_distance, par)
# if (abs(nup_demand - nup_supply) > 0.001){
# warning("N uptake based on supply and demand equations not identical.")
# out <- NA
# } else {
# out <- nup_demand
# }
out <- nup_demand
return(out)
}Express it all as a function of root biomass.
# rough values based on Weemstra et al., 2020 https://doi.org/10.1111/1365-2435.13520
par_rootmorph <- list(
zroot = 1,
rho = 0.2 * 100^3, # g m-3
r0 = 0.3 / (2 * 1000) # m, radius
)
calc_root_length_density <- function(croot, par){
croot / (par$rho * par$zroot * 2 * pi * par$r0)
}
calc_a_root <- function(croot, par){
2 * pi * par$r0 * calc_root_length_density(croot, par)
}
calc_root_distance <- function(root_length_density, par){
sqrt(1/(pi * root_length_density) + par$r0^2)
}
calc_nup <- function(croot, ninorg, par_fnup, par_rootmorph){
a_root <- calc_a_root(croot, par_rootmorph)
n_conc <- ninorg /( par_rootmorph$zroot * a_root)
root_length_density <- calc_root_length_density(croot, par_rootmorph)
root_distance <- calc_root_distance(root_length_density, par_rootmorph)
fnup <- calc_fnup(n_conc, root_distance, par_fnup)
a_root * fnup
}Visualisations
ggplot() +
geom_function(fun =
function(x)
calc_fnup(x,
root_distance = 1,
par = list(
D = 0.0005,
j = 0.0056,
k = 0.025,
f = 0
)
)) +
geom_function(fun =
function(x)
calc_fnup(x,
root_distance = 1,
par = list(
D = 0.0005,
j = 0.056,
k = 0.025,
f = 0
)
), color = "blue") +
labs(title = "N uptake as a function of Ninorg") +
xlim(0, 100)
Root distance as a function of root mass.
ggplot() +
geom_function(fun =
function(x)
calc_root_distance(
calc_root_length_density(x, par_rootmorph),
par_rootmorph
)) +
xlim(0, 10000)
The total soil column N uptake as a function of total root biomass.
ggplot() +
geom_function(fun =
function(x)
calc_nup(x,
ninorg = 1,
par_fnup,
par_rootmorph
)) +
geom_function(fun =
function(x)
calc_nup(x,
ninorg = 1,
par_fnup,
list(
zroot = 1,
rho = 0.2 * 100^3, # g m-3
r0 = 0.03 / (2 * 1000) # m, radius
)
), col = "royalblue") +
labs(title = "Total N uptake as a function of Croot",
x = "Croot") +
xlim(0, 1000)## Warning: Removed 1 row(s) containing missing values (geom_path).
## Removed 1 row(s) containing missing values (geom_path).
The N uptake fraction as a function of the root volume density.
df <- tibble(croot = seq(1000)) |>
mutate(volume_density = croot / (par_rootmorph$rho * par_rootmorph$zroot),
nup = calc_nup(croot,
ninorg = 0.2,
list(
D = 3,
j = 2000,
k = 1,
f = 0
),
par_rootmorph
),
nup2 = calc_nup(croot,
ninorg = 0.2,
list(
D = 3,
j = 500,
k = 1,
f = 0
),
par_rootmorph
)
) |>
mutate(nup_fraction = nup/0.2,
nup_fraction2 = nup2/0.2)
df |>
ggplot() +
geom_line(aes(croot, nup_fraction)) +
geom_line(aes(croot, nup_fraction2), col = "royalblue")
df |>
ggplot() +
geom_line(aes(volume_density, nup_fraction)) +
geom_line(aes(volume_density, nup_fraction2), col = "royalblue")
The N uptake fraction as a function of (bulk soil) inorganic N.
df <- tibble(ninorg = seq(0.1, 50, by = 0.01)) |>
mutate(nup = calc_nup(croot = 500,
ninorg,
par_fnup,
par_rootmorph
)) |>
mutate(nup_fraction = nup/ninorg)
df |>
ggplot(aes(ninorg, nup_fraction)) +
geom_line()
df |>
ggplot(aes(ninorg, nup)) +
geom_line()
df <- expand.grid(croot = seq(from = 1, to = 1000, by = 10),
ninorg = seq(from = 1, to = 50, by = 0.5)) |>
as_tibble() |>
mutate(fnup = purrr::map2_dbl(croot,
ninorg,
~calc_nup(.x, .y, par_fnup, par_rootmorph)))
gg_full <- ggplot(df, aes(croot, ninorg)) +
geom_tile(aes(fill = fnup)) +
geom_contour(aes(z = fnup), color = "grey50") +
scale_fill_scico(palette = 'roma') +
theme_classic() +
labs(title = "N uptake", x = "Croot", y = "Ninorg")
gg_full
Simple version
Theory
The simplest representation of the fact that N uptake is saturating both with respect to the root biomass and with respect to soil inorganic N (the latter is often ignored in models) is the following: \[ N_\text{up} = V_\text{C}(N_\text{inorg}) \frac{C_r}{k_C + C_r} \] where \(k_C(N_\text{inorg})\) is also saturating: \[ V_C(N_\text{inorg}) = V_\text{max}\frac{N_\text{inorg}}{k_V + N_\text{inorg}} \]
Implementation
calc_vmax <- function(ninorg, par){
par$vmax * ninorg / (par$kv + ninorg)
}
calc_nup_simpl <- function(croot, ninorg, par){
vmax = calc_vmax(ninorg, par)
vmax * croot / (par$kc + croot)
}par <- list(
kc = 800,
kv = 10,
vmax = 20
)
ggplot() +
geom_function(fun = function(x) calc_nup_simpl(croot = 15, ninorg = x, par)) +
xlim(0, 200) +
labs(x = "Ninorg", y = "Nup", title = "Croot = 15")
ggplot() +
geom_function(fun = function(x) calc_nup_simpl(x, ninorg = 10, par)) +
xlim(0, 100) +
labs(x = "Croot", y = "Nup", title = "Ninorg = 10")
par <- list(
kc = 800,
kv = 10,
vmax = 20
)
df <- expand.grid(croot = seq(from = 1, to = 1000, by = 10),
ninorg = seq(from = 1, to = 50, by = 0.5)) |>
as_tibble() |>
mutate(fnup = purrr::map2_dbl(croot,
ninorg,
~calc_nup_simpl(.x, .y, par)))
gg_simpl <- ggplot(df, aes(croot, ninorg)) +
geom_tile(aes(fill = fnup)) +
geom_contour(aes(z = fnup), color = "grey50") +
scale_fill_scico(palette = 'roma') +
theme_classic() +
labs(title = "N uptake", x = "Croot", y = "Ninorg")
gg_simpl
Comparison
library(patchwork)
gg_full + gg_simpl