Optimal LAI

ZhouOptimalLAIModel{FT, OLPT <: OptimalLAIParameters{FT}, GD, RDTH, HTH} <: AbstractBiomassModel{FT}

An implementation of the optimal LAI model from Zhou et al. (2025) as a biomass model.

This model computes LAI dynamically based on optimality principles, balancing energy and water constraints. LAI is prognostic, in Y.canopy.biomass.LAI, and is mirrored into p.canopy.biomass.area_index.leaf for the rest of the canopy, consistent with PrescribedBiomassModel.

Fields

• parameters: Required parameters for the optimal LAI model
• optimal_lai_inputs: NamedTuple with the spatially varying inputs of the LAImax and steady-state-LAI formulas: GSL (growing season length in days), vpdgs (growing-season mean VPD in Pa), and f0 (fraction of precip for transpiration). Typically created using optimal_lai_inputs, which reads them alongside the initial conditions.
• SAI: Prescribed stem area index (m^2 m^-2)
• RAI: Prescribed root area index (m^2 m^-2)
• rooting_depth: Rooting depth parameter (m) - a characteristic depth below which 1/e of the root mass lies
• height: Canopy height (m) - can be scalar (uniform) or spatially-varying Field

References

Zhou et al. (2025) "A General Model for the Seasonal to Decadal Dynamics of Leaf Area" Global Change Biology. https://onlinelibrary.wiley.com/doi/pdf/10.1111/gcb.70125

OptimalLAIParameters{FT<:AbstractFloat}

The required parameters for the optimal LAI model based on Zhou et al. (2025).

Water limitation is handled through the f0*P/A0 term following Zhou et al. (2025) Equation 11, where P is annual precipitation and A0 is annual potential GPP.

References

Zhou et al. (2025) "A General Model for the Seasonal to Decadal Dynamics of Leaf Area" Global Change Biology. https://onlinelibrary.wiley.com/doi/pdf/10.1111/gcb.70125

• k: Light extinction coefficient (dimensionless), typically 0.5

• z: Unit cost of constructing and maintaining leaves (mol m^-2 yr^-1), globally fitted as 12.227 mol m^-2 yr^-1

• sigma: Dimensionless parameter representing departure from square-wave LAI dynamics, globally fitted as 0.771

• alpha: Smoothing factor for exponential moving average (dimensionless, 0-1). Set to 0.067 for ~15 days of memory

• f0: Fraction of annual precipitation available for transpiration (dimensionless, 0-1). Following Zhou et al. (2025), f0 = 0.65 at the energy-water limitation transition. In arid regions, f0 can be lower: f0 = 0.65 * exp(-0.604 * ln^2(AI/1.9)) where AI is aridity index. Default value 0.65 assumes optimal water use efficiency.

• tau_long_term: Long-term memory timescale (s) of the A0 and precipitation running-mean annual totals that set LAI_max and the steady-state LAI. Default 2 years; a longer value filters the seasonal cycle more strongly, avoiding aliasing of the annual cycle.

OptimalLAIParameters{FT}(toml_dict::CP.ParamDict) where {FT}

Creates an OptimalLAIParameters object from a TOML parameter dictionary.

compute_A0_and_χ(
    fractional_c3::FT,
    parameters::PModelParameters{FT},
    constants::PModelConstants{FT},
    T_air::FT,
    P_air::FT,
    VPD::FT,
    ca::FT,
    PPFD::FT,
    βm::FT,
    vpd_gs::FT,
) where {FT}

Compute potential GPP (A0) and ci/ca ratio used in the optimal LAI model (Zhou et al. 2025).

This function computes the potential GPP assuming fAPAR = 1 (full light absorption), which represents the maximum carbon assimilation possible under given environmental conditions.

A0 responds to the instantaneous vapor pressure deficit, while χ is the growing-season value the water-limitation term of LAI_max calls for, and so is evaluated at the growing-season mean VPD vpd_gs. Both share the same optimal ξ, which depends on temperature and pressure only.

Arguments

• fractional_c3::FT: Photosynthesis mechanism (1 for C3, 0 for C4)
• parameters: PModelParameters containing cstar, beta, etc.
• constants: PModelConstants containing physical constants
• earth_param_set: Additional physical constants
• T_air::FT: Air temperature (K)
• P_air::FT: Atmospheric pressure (Pa)
• q_air::FT: Atmospheric specific humidity (kg/kg)
• ca::FT: Ambient CO2 concentration (mol/mol)
• PPFD::FT: Downwelling radiative flux in PAR band at the surface (moles photons/m^2/s)
• βm::FT: Soil moisture stress factor (dimensionless, 0-1)
• vpd_gs::FT: Growing-season mean vapor pressure deficit (Pa)

Returns

• NamedTuple with A0, the potential GPP with fAPAR=1 (mol C m^-2 s^-1), and χ = ci/ca at the growing-season VPD (unitless)

compute_L_steady_target(A0_daily, k, A0_annual, z, GSL, sigma, precip_annual, f0, ca_pa, chi, vpd_gs)

Compute the steady-state LAI target L_steady (Zhou et al. 2025 Eqs. 11-15) from the daily and annual potential GPP and the water-limitation inputs, without the Eq. 16 acclimation lag. The prognostic LAI relaxes toward this target in the biomass tendency, so the acclimation alpha is applied there rather than here.

compute_L_max(Ao_annual, k, z, precip_annual, f0, ca_pa, chi, vpd_gs)

Compute seasonal maximum leaf area index (LAI_max) based on annual potential GPP and water availability, following Zhou et al. (2025) Equation 11.

LAI_max is determined by the minimum of energy-limited and water-limited fAPAR:

• Energy-limited: fAPAR_energy = 1 - z/(k*A0)
• Water-limited: fAPAR_water = f0P/A0 * (ca(1-chi))/(1.6D)

Arguments

• Ao_annual::FT: Annual total potential GPP (mol CO2 m^-2 yr^-1).
• k::FT: Light extinction coefficient (dimensionless), typically 0.5
• z::FT: Unit cost of constructing and maintaining leaves (mol m^-2 yr^-1), 12.227
• precip_annual::FT: Mean annual precipitation (mol H2O m^-2 yr^-1)
• f0::FT: Fraction of precipitation available for transpiration (dimensionless), 0.65
• ca_pa::FT: Ambient CO2 partial pressure (Pa), typically ~40 Pa at 400 ppm
• chi::FT: Optimal ratio of intercellular to ambient CO2 (dimensionless), typically 0.7-0.8
• vpd_gs::FT: Mean vapor pressure deficit during growing season (Pa)

Returns

• LAI_max::FT: Seasonal maximum leaf area index (m^2 m^-2)

Notes

Following Zhou et al. (2025) Equation 11:

fAPAR_max = min{1 - z/(k*A0), f0*P/A0 * (ca(1-chi))/(1.6*D)}

The first term is energy-limited (carbon gain vs leaf cost trade-off). The second term is water-limited (precipitation constrains transpiration, scaled by intrinsic water use efficiency iWUE = ca(1-chi)/(1.6*D)).

The iWUE factor converts water flux to carbon flux:

• ca(1-chi): CO2 drawdown from ambient to intercellular (Pa)
• 1.6*D: VPD adjusted for CO2/H2O diffusivity ratio (Pa)

References

Zhou et al. (2025) Global Change Biology, Equation 11

compute_m(GSL, LAI_max, Ao_annual, sigma, k)

Compute the parameter m, which represents the ratio of steady-state LAI to steady-state GPP.

This implements Equation 20 from Zhou et al. (2025). The parameter m quantifies the relationship between LAI and GPP dynamics, representing the extent to which seasonal LAI dynamics depart from a "square wave" (where maximum LAI would be maintained throughout the growing season).

Arguments

• GSL::FT: Growing season length (days). Defined as the length of continuous period above 0C longer than 5 days.
• LAI_max::FT: Seasonal maximum leaf area index (m^2 m^-2, dimensionless)
• Ao_annual::FT: Annual total potential GPP (mol m^-2 yr^-1). This is the integral of daily A0 over the year.
• sigma::FT: Dimensionless parameter representing departure from square-wave LAI dynamics. Globally fitted as sigma = 0.771
• k::FT: Light extinction coefficient (dimensionless)

Returns

• m::FT: Parameter relating steady-state LAI to steady-state GPP (dimensionless, units work out as: days * m^2 m^-2 / (mol m^-2 yr^-1 * dimensionless) with implicit conversion)

References

Zhou et al. (2025) Global Change Biology, Equation 20

compute_steady_state_LAI(Ao_daily, m, k, LAI_max)

Compute steady-state LAI from daily potential GPP using the Lambert W function solution.

This implements Equations 13-15 from Zhou et al. (2025). The steady-state LAI (L_s) is the LAI that would be in equilibrium with GPP if weather conditions were held constant. Given daily meteorological conditions, this is computed on a daily basis.

Arguments

• Ao_daily::FT: Daily potential GPP (mol m^-2 day^-1). This is the GPP that would be achieved if fAPAR = 1, calculated from LUE * PPFD.
• m::FT: Parameter relating steady-state LAI to steady-state GPP (dimensionless), from compute_m()
• k::FT: Light extinction coefficient (dimensionless), typically 0.5
• LAI_max::FT: Seasonal maximum LAI constraint (m^2 m^-2, dimensionless)

Returns

• L_steady::FT: Steady-state leaf area index (m^2 m^-2, dimensionless). Always >= 0.

Notes

The solution uses the Lambert W0 function: Ls = min{mu + (1/k)W0[-kmuexp(-k*mu)], LAImax} where mu = m * A0. The result is constrained to be non-negative and below LAI_max.

References

Zhou et al. (2025) Global Change Biology, Equations 13-15

lambertw0(x::T; maxiter::Int = 8) where {T<:AbstractFloat}

Compute the principal branch (W0) of the Lambert W function for x in [-1/e, Inf).

This is a GPU-device-friendly implementation using a fixed number of Halley iterations. The Lambert W function satisfies W(x)*exp(W(x)) = x.

Arguments

• x::T: Input value, must be >= -1/e ~ -0.36788
• maxiter::Int: Maximum number of Halley iterations (default: 8; Halley's method has cubic convergence, so 8 is generous)

Returns

• W::T: Lambert W0(x), the principal branch value, or NaN for invalid inputs

Algorithm

Uses Halley's method with a fixed number of iterations for GPU compatibility:

• No dynamic memory allocation
• No conditional breaks (runs all iterations)
• Broadcastable for use with CuArrays: lambertw0.(cuarray)

Device Compatibility

This implementation is designed to work on both CPU and GPU:

• All operations are scalar and supported on CUDA.jl
• No array allocations or dynamic loops
• Type-generic over AbstractFloat (Float32, Float64)

References

Corless et al. (1996) "On the Lambert W function"