Error in calculation with soil

[baugetfa]

There is potentially an error in the calculation of psi_in when psi_ext is not uniform so with a soil. for the solver using Ohm's law analogy. Note that it does not affect the publications because they were about hydroponic condition where the potential is uniform.

To calculate psi_in we use the equivalent conductance of the children and brother and the external potentiel which is psi_e[v] the external pressure of the actual vertex. If psi_e is uniform this correct but not if psi_e changes. See for instance fig 2b in Prusinkiewicz et al. 2007.

To illustrate that let's consider the tip with a psi_e=Pe1 and its parent at Pe2, the masse conservation leads to:

 K2 (Pout2 - Pin2) = k2 (Pin2-Pe2) + Keq1 (Pin2 - Pe1)

Actually in hydroroot we have:

K2 (Pout2 - Pin2) = k2 (Pin2-Pe2) + Keq1 (Pin2 - Pe2)

So we should have for a vertex (i) an equivalent psi_e corresponding to the Keq of the children:

Peq(i) = (Peq(i-1) k(i) + Pe(i) keq(i-1)) / (k(i) + keq(i-1))