Leaf dark respiration (R_d) and effect on solution of ci

[itowers1]

Reporting an issue which was not apparent in the recent manuscript but will occur in stand simulations.

Net assimilation in the leaf submodel is currently implemented as:

$$ A_n = \frac{A_c + A_j - \sqrt{(A_c + A_j)^2 - 4hA_cA_j}}{2h} - R_d,~(1) $$

where A_c is the vcmax limited photosynthetic rate, A_j is the light-limited photosynthetic rate, h is a curvature factor and R_d is the leaf-level respiration. $A_c$ and $A_j$ are functions of internal $c_i$, which is found through a root-solving problem based on the following equation.

$$ A_n(c_i) = g_c(c_a - c_i).~(2) $$

However, finding that root-solving breaks when A_j is extremely low on account of practically zero light availability, because R_d, which is a fixed value, causes A_n to become negative ranging from the compensation point ($c_i$ ~ 4Pa) to $C_a$ (40Pa), meaning that there is no valid solution, because right side of equation 2 = 0 at $c_i$ = 40.

This is shown in the below figure.

Wondering how to solve this problem. Should the ci be found via root-solving, and then R_d is removed? This would cause the solved $c_i$ to be slightly more negative emerging from upward translation of the $A_n$ curve.