Commit last-minute

hydroshoot/doc/hydraulic.rst

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+===================
+Hydraulic structure
+===================
+
+The *hydraulic* module computes the distribution of xylem water potential across plant segments by analogy to Ohm’s law
+(:numref:`fig_1_hydraulic`)
+
+.. _fig_1_hydraulic:
+
+.. figure:: figs/hydraulic.png
+
+    Illustration of the variables required to calculate the hydraulic structure: water flux (:math:`F`),
+    hydraulic conductivity (:math:`K`), water pressure at upper (downstream) and lower (upstream) extremities
+    of the conducting element (respectively :math:`H_u` and :math:`H_l`), and the length of the segment (:math:`L`).
+
+Water flux (:math:`F, \ kg \ s^{-1}`) across the hydraulic segment of length (:math:`L, \ m`)
+is driven by the difference of xylem pressures across this segment (:math:`H_u - H_l, \ MPa`)
+and regulated by segment's hydraulic conductivity (:math:`K, \ kgs^{-1} \ m \ MPa^{-1}`):
+
+.. math::
+    F = - K \cdot \frac{H_u - H_l}{L}
+
+The hydraulic conductivity varies with water potential as a result of xylem cavitation under water deficit
+**(Tyree and Sperry, 1989)**. This relationship is described in HydroShoot as:
+
+.. math::
+    K = K_{max} \cdot \frac{1} {\left( 1 + \left( \frac{\Psi}{\Psi_{crit, \ stem}} \right) ^{c_{x1}} \right)}
+
+where
+:math:`K_{max} \ [kg \ s^{-1} \ m \ MPa^{-1}]` is the maximum conductivity of the segment,
+:math:`\Psi \ [MPa]` is the arithmetic mean of xylem potential of the segment,
+:math:`\Psi_{crit, \ stem} \ [MPa]` and
+:math:`c_{x1} \ [-]` are shape parameters.
+
+:math:`K_{max}` is estimated empirically as proposed by **Tyree and Zimmermann (2002)** as:
+
+.. math::
+    K_{max} = c_{x2} \cdot D^{c_{x3}}
+
+where
+:math:`D \ [m]` is the average diameter of the segment, and
+:math:`c_{x2}` and
+:math:`c_{x3}` are shape parameters, mostly given within the ranges of [2.5, 2.8] and [2.0, 5.0], respectively.
+
+The last two equations apply to all conducting segments (not leaves blades). Water potential of the upper extremity
+of the petiole is assumed equal to that of the lumped leaf water potential :math:`\Psi_{leaf} \ [MPa]`.