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hydroshoot/doc/ecophysio_solver.rst

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+======================
+Processes interactions
+======================
+
+Interactions among the physiological processes occur.
+
+.. figure:: figs/ecophysio_solver_1.svg
+    :align: center
+
+    Solving the interactions between leaf's water potential, transpiration rate and temperature !
+
+From the one hand. Leaf water potential (:math:`\Psi`) exerts a control on stomatal conductance to water vapor
+(:math:`g_{s, \ H_2O}`) and, consequently, on transpiration flux (:math:`E`). However, the latter also determines how much
+water will flow through the hydraulic segments to be withdrawn from the soil, which affects the distribution of
+water potential across those segments (Ohm's law). Hence, we have a reciprocity between the distribution of water
+potential across the shoot (hydraulic structure) and transpiration fluxes from individual leaves.
+
+From the other hand, leaf temperature (:math:`T`) determines the rate of leaf photosynthesis (Arrhenius functions)
+and stomatal conductance to both :math:`CO_2` (:math:`g_{s, \ CO_2}`) and water vapor (:math:`g_{s, \ H_2O}`).
+This means that :math:`T` affects also the transpiration flux :math:`E` which also means that :math:`T` exerts a further
+control on the hydraulic structure.
+
+Putting it all together:
+
+For a given initial temperature (:math:`T_0`) and soil water potential (:math:`\Psi_0`), a leaf can fix :math:`CO_2`
+with a rate :math:`A_{n, \ 0}` while having a stomatal conductance rate of :math:`g_{s, \ H_2O, \ 0}`.
+The corresponding water flux transpired by this leaf will be :math:`E_0`. Summed up across all leaves, plant
+transpiration will withdraw water from the soil, reducing thus soil water potential and resulting in a new :math:`\Psi`
+value (:math:`\Psi_1`).
+:math:`\Psi_1`, will hence impose a new stomatal conductance :math:`g_{s, \ H_2O, \ 1}` and consequently a new
+transpiration flux :math:`E_1`. Yet, this new :math:`E_1` means that leaf temperature is :math:`T_1`... **REWIND**!
+
+HydroShoot resolve this interactions iteratively until reaching steady rate values of :math:`A_n,` and :math:`E` (
+implying steady state distribution of water potential cross the shoot). :numref:`fig_2` shows how interactions between
+the hydraulic, energy and gas-exchange processes is handled in HydroShoot.
+
+.. _fig_2:
+
+.. figure:: figs/ecophysio_solver_2.png
+
+    Schematic representation of the numerical resolution of HydroShoot. Meteorological inputs that are common
+    to all leaves are air temperature (:math:`T_{air}, \ K`), air relative humidity (:math:`RH, \ -`), air
+    :math:`CO_2` concentration :math:`[\mu mol\ {mol}^{-1}]`, wind speed (:math:`u, \ m \ s^{-1}`), and atmospheric
+    pressure (:math:`P_a, \ kPa`). Inputs per individual leaves are the absorbed photosynthetic photon flux density
+    (:math:`PPFD, \ \mu mol \ m^{-2} \ s^{-1}`) and :math:`{PPFD}_{10}` the absorbed :math:`PPFD` during the last
+    10 days. :math:`\Psi_u \ [MPa]` is xylem water potential at the nodes between each pair of hydraulic segments.
+    :math:`\Psi_{u, \ init} \ [MPa]` is initial :math:`\Psi_u`. :math:`\Psi_{soil} \ [MPa]` is soil water potential.
+    :math:`T_i \ [K]` is leaf temperature. :math:`T_{i, \ init} \ [K]` is initial :math:`T_i`.
+    :math:`K_{init} \ [kg \ s^{-1} \ m \ {MPa}^{-1}]` is initial hydraulic conductivity of each segment.
+    :math:`\epsilon_x \ [MPa]` is the maximum allowable error of the estimation of xylem water potential. Finally,
+    :math:`\epsilon_T \ [K]` is the maximum allowable error of the estimation of leaf temperature.
+    Circles inside module boxes indicate internal iteration loops. Symbols between curly brackets represent spatially
+    structured variables.
+
+