def profit_psi(p, photo='Farquhar', res='low', inf_gb=False, deriv=False):
"""
Finds the instateneous profit maximization, following the
optmization criterion for which, at each instant in time, the
stomata regulate canopy gas exchange and pressure to achieve the
maximum profit, which is the maximum difference between the
normalized photosynthetic gain (gain) and the hydraulic cost
function (cost). That is when d(gain)/dP = d(cost)/dP.
Arguments:
----------
p: recarray object or pandas series or class containing the data
time step's met data & params
photo: string
either the Farquhar model for photosynthesis, or the Collatz
model
res: string
either 'low' (default), 'med', or 'high' to run the optimising
solver
onopt: boolean
if True, the optimisation is performed. If Fall, fall back on
previously performed optimisation for the value of the maximum
profit.
inf_gb: bool
if True, gb is prescrived and very large
Returns:
--------
E_can: float
transpiration [mmol m-2 s-1] at maximum profit across leaves
gs_can: float
stomatal conductance [mol m-2 s-1] at maximum profit across
leaves
An_can: float
net photosynthetic assimilation rate [umol m-2 s-1] at maximum
profit across leaves
Ci_can: float
intercellular CO2 concentration [Pa] at maximum profit across
leaves
rublim_can: string
'True' if the C assimilation is rubisco limited, 'False'
otherwise
"""
# hydraulics
P, trans = hydraulics(p, res=res)
cost, __ = hydraulic_cost(p, P)
# look for the most net profit
gain, Ci, mask = photo_gain(p, trans, photo, res, inf_gb=inf_gb)
expr = gain - cost[mask]
if deriv:
expr = np.abs(np.gradient(expr, P[mask]))
# deal with edge cases by rebounding the solution
gc, gs, gb, __ = leaf_energy_balance(p, trans[mask], inf_gb=inf_gb)
try:
if inf_gb: # check on valid range
check = expr[gc > cst.zero]
else: # further constrain the realm of possible gs
check = expr[np.logical_and(gc > cst.zero, gs < 1.5 * gb)]
idx = np.isclose(expr, max(check))
if deriv:
idx = np.isclose(expr, min(check))
idx = [list(idx).index(e) for e in idx if e]
if inf_gb: # check for algo. "overshooting" due to inf. gb
while Ci[idx[0]] < 2. * p.gamstar25:
idx[0] -= 1
if idx[0] < 3:
break
# optimized where Ci for both photo models are close
Ci = Ci[idx[0]]
trans = trans[mask][idx[0]] # mol.m-2.s-1
gs = gs[idx[0]]
Pleaf = P[mask][idx[0]]
# rubisco- or electron transport-limitation?
An, Aj, Ac = calc_photosynthesis(p, trans, Ci, photo, inf_gb=inf_gb)
rublim = rubisco_limit(Aj, Ac)
# leaf temperature?
Tleaf, __ = leaf_temperature(p, trans, inf_gb=inf_gb)
if (np.isclose(trans, cst.zero, rtol=cst.zero, atol=cst.zero) and
(An > 0.)) or (idx[0] == len(P) - 1) or any(
np.isnan([An, Ci, trans, gs, Tleaf, Pleaf])):
An, Ci, trans, gs, gb, Tleaf, Pleaf = (9999., ) * 7
elif not np.isclose(trans, cst.zero, rtol=cst.zero, atol=cst.zero):
trans *= conv.MILI # mmol.m-2.s-1
return An, Ci, rublim, trans, gs, gb, Tleaf, Pleaf
except ValueError: # no opt
return (9999., ) * 8
(abs(Tleaf - new_Tleaf) <= threshold_conv)
and not np.isclose(gs, cst.zero, rtol=cst.zero, atol=cst.zero))):
break
# no convergence, iterate on leaf temperature
Tleaf = new_Tleaf
iter += 1
if case == 1: # infer leaf water potential, MPa
Pleaf = P[bn.nanargmin(np.abs(E - trans))]
else:
Pleaf = P[iopt]
ksc = p.kmaxS2 * cost[iopt]
rublim = rubisco_limit(Aj, Ac) # lim?
if ((np.isclose(trans, cst.zero, rtol=cst.zero, atol=cst.zero) and
(An > 0.)) or np.isclose(Ci, 0., rtol=cst.zero, atol=cst.zero)
or (Ci < 0.) or np.isclose(Ci, p.CO2, rtol=cst.zero, atol=cst.zero)
or (Ci > p.CO2) or (real_zero is None) or (not real_zero)
or any(np.isnan([An, Ci, trans, gs, new_Tleaf, Pleaf]))):
An, Ci, trans, gs, gb, new_Tleaf, Pleaf = (9999., ) * 7
elif not np.isclose(trans, cst.zero, rtol=cst.zero, atol=cst.zero):
trans *= conv.MILI # mmol.m-2.s-1
if case == 1:
return An, Ci, rublim, trans, gs, gb, new_Tleaf, Pleaf
def Tuzet(p, photo='Farquhar', res='low', inf_gb=False):
"""
Checks the energy balance by looking for convergence of the new leaf
temperature with the leaf temperature predicted by the previous
iteration. Then returns the corresponding An, E, Ci, etc.
Arguments:
----------
p: recarray object or pandas series or class containing the data
time step's met data & params
photo: string
either the Farquhar model for photosynthesis, or the Collatz
model
threshold_conv: float
convergence threshold for the new leaf temperature to be in
energy balance
iter_max: int
maximum number of iterations allowed on the leaf temperature
before reaching the conclusion that the system is not energy
balanced
inf_gb: bool
if True, gb is prescrived and very large
Returns:
--------
trans_can: float
transpiration rate of canopy [mmol m-2 s-1] across leaves
gs_can: float
stomatal conductance of canopy [mol m-2 s-1] across leaves
An_can: float
C assimilation rate of canopy [umol m-2 s-1] across leaves
Ci_can: float
average intercellular CO2 concentration of canopy [Pa] across
leaves
rublim_can: string
'True' if the C assimilation is rubisco limited, 'False'
otherwise.
"""
# stability pre-requisites
fw = fLWP_stable(p, photo=photo, res=res, inf_gb=inf_gb)
# run the energy balance sub-routine
An, Aj, Ac, Ci, trans, gs, gb, Tleaf, Pleaf = gas_exchange(p, fw,
photo=photo,
res=res,
inf_gb=inf_gb)
rublim = rubisco_limit(Aj, Ac) # lim?
if not np.isclose(trans, cst.zero, rtol=cst.zero, atol=cst.zero):
trans *= conv.MILI # mmol.m-2.s-1
return An, Ci, rublim, trans, gs, gb, Tleaf, Pleaf
def Cmax_gs(p, photo='Farquhar', res='low', inf_gb=False):
"""
Finds the instantaneous optimal C gain for a given C cost.
First, the C gain equation is derived for gs, beta, Ci unknown.
Then, the derived form of the equation is solved for Ci over a range of
possible betas, gs, all of which are directly or indirectly leaf
water potential P dependent.
A check (check_solve) is performed to verify that the optimization satisfies
the zero equality criteria and, finally, results are bound via a range of
physically possible Ci values.
N.B.: there can be several possible optimizations
Arguments:
----------
p: recarray object or pandas series or class containing the data
time step's met data & params
photo: string
either the Farquhar model for photosynthesis, or the Collatz model
inf_gb: bool
if True, gb is prescrived and very large
Returns:
--------
gsOPT: float
stomatal conductance [mol.m-2.s-1] for which the A(gs) is maximized
AnOPT: float
maximum C assimilation rate [μmol.m-2.s-1] given by the diffusive supply
of CO2
transOPT: float
transpiration rate [mmol.m-2.s-1] for which the A(gs) is maximized
CiOPT: float
intercellular CO2 concentration [Pa] for which the A(gs) is maximized
"""
# energy balance
P, trans = hydraulics(p, res=res, kmax=p.kmaxCM)
# expression of optimization
Ci, mask = Ci_sup_dem(p, trans, photo=photo, res=res, inf_gb=inf_gb)
gc, gs, gb, __ = leaf_energy_balance(p, trans[mask], inf_gb=inf_gb)
expr = np.abs(
np.gradient(A_trans(p, trans[mask], Ci, inf_gb=inf_gb), P[mask]) -
dcost_dpsi(p, P[mask], gs))
try:
if inf_gb: # check on valid range
check = expr[gc > cst.zero]
else: # further constrain the realm of possible gs
check = expr[np.logical_and(gc > cst.zero, gs < 1.5 * gb)]
idx = np.isclose(expr, min(check))
idx = [list(idx).index(e) for e in idx if e]
if inf_gb: # check for algo. "overshooting" due to inf. gb
while Ci[idx[0]] < 2. * p.gamstar25:
idx[0] -= 1
if idx[0] < 3:
break
# optimized where Ci for both photo models are close
Ci = Ci[idx[0]]
trans = trans[mask][idx[0]] # mol.m-2.s-1
gs = gs[idx[0]]
Pleaf = P[mask][idx[0]]
# rubisco limitation or electron transport-limitation?
An, Aj, Ac = calc_photosynthesis(p,
trans,
Ci,
photo=photo,
inf_gb=inf_gb)
rublim = rubisco_limit(Aj, Ac)
# leaf temperature?
Tleaf, __ = leaf_temperature(p, trans, inf_gb=inf_gb)
if (np.isclose(trans, cst.zero, rtol=cst.zero, atol=cst.zero) and
(An > 0.)) or (idx[0] == len(P) - 1) or any(
np.isnan([An, Ci, trans, gs, Tleaf, Pleaf])):
An, Ci, trans, gs, gb, Tleaf, Pleaf = (9999., ) * 7
elif not np.isclose(trans, cst.zero, rtol=cst.zero, atol=cst.zero):
trans *= conv.MILI # mmol.m-2.s-1
return An, Ci, rublim, trans, gs, gb, Tleaf, Pleaf
except ValueError: # no opt
return (9999., ) * 8
def solve_std(p, sw, photo='Farquhar', res='low', iter_max=40,
threshold_conv=0.1, inf_gb=False):
"""
Checks the energy balance by looking for convergence of the new leaf
temperature with the leaf temperature predicted by the previous
iteration. Then returns the corresponding An, E, Ci, etc.
Arguments:
----------
p: recarray object or pandas series or class containing the data
time step's met data & params
sw: float
mean volumetric soil moisture content [m3 m-3]
photo: string
either the Farquhar model for photosynthesis, or the Collatz
model
threshold_conv: float
convergence threshold for the new leaf temperature to be in
energy balance
iter_max: int
maximum number of iterations allowed on the leaf temperature
before reaching the conclusion that the system is not energy
balanced
inf_gb: bool
if True, gb is prescrived and very large
Returns:
--------
trans_can: float
transpiration rate of canopy [mmol m-2 s-1] across leaves
gs_can: float
stomatal conductance of canopy [mol m-2 s-1] across leaves
An_can: float
C assimilation rate of canopy [umol m-2 s-1] across leaves
Ci_can: float
average intercellular CO2 concentration of canopy [Pa] across
leaves
rublim_can: string
'True' if the C assimilation is rubisco limited, 'False'
otherwise.
"""
# initial state
Cs = p.CO2 # Pa
Tleaf = p.Tair # deg C
Dleaf = np.maximum(0.05, p.VPD) # gs model not valid < 0.05
# hydraulics
P, E = hydraulics(p, res=res)
if sw >= p.fc:
g1 = p.g1
else:
g1 = p.g1 * fwWP(p, p.Ps)
# initialise gs over A
g0 = 1.e-9 # g0 ~ 0, removing it entirely introduces errors
Cs_umol_mol = Cs * conv.MILI / p.Patm # umol mol-1
gsoA = g0 + (1. + g1 / (Dleaf ** 0.5)) / Cs_umol_mol
# iter on the solution until it is stable enough
iter = 0
while True:
An, Aj, Ac, Ci = calc_photosynthesis(p, 0., Cs, photo, Tleaf=Tleaf,
gs_over_A=gsoA)
# stomatal conductance, with moisture stress effect
gs = np.maximum(cst.zero, conv.GwvGc * gsoA * An)
# calculate new trans, gw, gb, mol.m-2.s-1
trans, real_zero, gw, gb, Dleaf = calc_trans(p, Tleaf, gs,
inf_gb=inf_gb)
new_Tleaf, __ = leaf_temperature(p, trans, Tleaf=Tleaf, inf_gb=inf_gb)
# update Cs (Pa)
boundary_CO2 = p.Patm * conv.FROM_MILI * An / (gb * conv.GbcvGb)
Cs = np.maximum(cst.zero, np.minimum(p.CO2, p.CO2 - boundary_CO2))
Cs_umol_mol = Cs * conv.MILI / p.Patm
# new leaf-air vpd, kPa
if (np.isclose(trans, cst.zero, rtol=cst.zero, atol=cst.zero) or
np.isclose(gw, cst.zero, rtol=cst.zero, atol=cst.zero) or
np.isclose(gs, cst.zero, rtol=cst.zero, atol=cst.zero)):
Dleaf = np.maximum(0.05, p.VPD) # kPa
# update gs over A
gsoA = g0 + (1. + g1 / (Dleaf ** 0.5)) / Cs_umol_mol
# force stop when atm. conditions yield E < 0. (non-physical)
if (iter < 1) and (not real_zero):
real_zero = None
# check for convergence
if ((real_zero is None) or (iter >= iter_max) or ((iter >= 1) and
real_zero and (abs(Tleaf - new_Tleaf) <= threshold_conv) and not
np.isclose(gs, cst.zero, rtol=cst.zero, atol=cst.zero))):
break
# no convergence, iterate on leaf temperature
Tleaf = new_Tleaf
iter += 1
Pleaf = P[bn.nanargmin(np.abs(trans - E))]
rublim = rubisco_limit(Aj, Ac) # lim?
if ((np.isclose(trans, cst.zero, rtol=cst.zero, atol=cst.zero) and
(An > 0.)) or np.isclose(Ci, 0., rtol=cst.zero, atol=cst.zero) or
(Ci < 0.) or np.isclose(Ci, p.CO2, rtol=cst.zero, atol=cst.zero) or
(Ci > p.CO2) or (real_zero is None) or (not real_zero) or
any(np.isnan([An, Ci, trans, gs, new_Tleaf, Pleaf]))):
An, Ci, trans, gs, gb, new_Tleaf, Pleaf = (9999.,) * 7
elif not np.isclose(trans, cst.zero, rtol=cst.zero, atol=cst.zero):
trans *= conv.MILI # mmol.m-2.s-1
return An, Ci, rublim, trans, gs, gb, new_Tleaf, Pleaf