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## Numerical Implementation

The article discusses the numerical implementation of the CO2 partial pressure and vapor pressure calculations at the leaf surface, as well as the internal leaf CO2 partial pressure, which are required for the stomatal resistance and photosynthesis models.

Key points:

1. Equations are derived to calculate the CO2 partial pressure at the leaf surface (c_s) and the vapor pressure at the leaf surface (e_s), assuming negligible capacity to store CO2 and water vapor at the leaf surface.

2. The transpiration fluxes are related through equations linking the differences in vapor pressures and the leaf boundary layer and stomatal resistances.

3. An expression for the stomatal resistance (r_s) is derived as a function of photosynthesis (A_n) using a quadratic equation.

4. The internal leaf CO2 partial pressure (c_i) is calculated iteratively until convergence, using a hybrid algorithm that combines the secant method and Brent's method.

5. The equations are solved separately for sunlit and shaded leaves, resulting in different values for A_n and r_s.

The article provides the detailed mathematical formulations and equations used in the numerical implementation of these key leaf-level processes within the land surface model.