# Iteration

## Iterative solution to Ball-Berry stomatal conductance simultaneous equations (Simile V4+)

ModelId:
BallBerry4a
SimileVersion:
5.9

The Ball-Berry stomatal conductance poses a problem for conventional System Dynamics modelling tools because it depends upon the solution of the following pair of simultaneous equations.

Gs = g0 + g1 * A * H / Ca
A = Gs * AQ

Equations:

Equations in BallBerry4aP

Equations in Environment

Variable C_a : Carbon dioxide concentration (umol CO2 (mol air)^-1)
C_a = graph(time())
Comments:
Typical diurnal curve in forest canopy

Variable H : Relative humidity (proportion)
H = graph(time())
Comments:
Typical diurnal graph (24 hour)

Variable Q : Photon flux density (umol m^-2 s^-1)
Q = graph(time())
Comments:
Graph for a sunny day (24 hours)

Equations in Ball-Berry

Variable Gs_start
Gs_start = if time()==0 then g_0 else last(Gs_0)
Where:
Gs_0=Iteration time step/Gs_0

Variable g_0 : Stomatal conductance in the dark (mol m^-2 s^-1)
g_0 = 0.01

Variable g_1 : Ball-Berry stomatal conductance coefficient
g_1 = 23

Equations in Iteration time step

Alarm
Variable Gs
Gs = if loop_count==0 then Gs_start else Gs_0
Where:
Gs_start=../Gs_start

Variable Gs_0 : Stomatal conductance (mol m^-2 s^-1)
Gs_0 = g_0+g_1*A*H/C_a
Where:
A=Assimilation/A
H=../../Environment/H
C_a=../../Environment/C_a
g_0=../g_0
g_1=../g_1
Comments:
Ball-Berry equation

Variable loop_count
loop_count = iterations(al1)

Equations in Assimilation

Variable A : Assimation (umol CO2 m^-2 s^-1)
A = A_Q*Gs
Where:
Gs=../Gs

Variable A_Q : Assimilation light response curve
A_Q = graph(Q)
Where:
Q=../../../Environment/Q
Comments:
Relationship of Assimilation with photon flux density (light) when stomatal conductance (Gs) is maximum

Results:

Plot of stomatal conductance versus time            Plot of assimilation versus time