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Resistance_Functions.py
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import os
import numpy
import math
from scipy.interpolate import interp1d
from scipy.optimize import fsolve
from ResistanceFunctionsDef import ResistanceSoil,CanopyResistanceAnEvolution,PhotosynthesisBiochemical
from Soil_Functions import Soil_Calculations
import copy
'''
Resistance Functions:
Developed by Mohsen Moradi
Atmospheric Innovations Research (AIR) Laboratory, University of Guelph, Guelph, Canada
Last update: June 2021
'''
class Ressitance_Calculations(object):
def Canopy_Resistance_An_Evolution(self,PAR_sun,PAR_shd,LAI,Kopt,Knit,Fsun,Fshd,Citm1_sun,Citm1_shd,Ca,ra,rb,Ts,Pre,
Ds,Psi_L,Psi_sto_50,Psi_sto_99,CT,Vmax,DS,Ha,FI,Oa,Do,a1,go,e_rel,e_relN,gmes,rjv):
"""
------
INPUT:
PAR_sun: Absorbed direct and diffuse shortwave radiation of the sunlit surface [W m^-2]
PAR_shd: Absorbed direct and diffuse shortwave radiation of the shaded surface [W m^-2]
LAI: Leaf area index [-]
Kopt: optical depth of direct beam perunit plant area [-]
Knit: Canopy nitrogen decay coefficient [-]
Fsun: Partitioning of radiation into sunlit area
Fshd: Partitioning of radiation into shaded area
Citm1_sun: Leaf Interior CO2 mixing ratio [umolCO2 mol^-1]
Citm1_shd: Leaf Interior CO2 mixing ratio [umolCO2 mol^-1]
Ca: Atmospheric CO2 mixing ratio [umolCO2 mol^-1]
ra: Aerodynamic resistance [s m^-1]
rb: Leaf boundary resistance [s m^-1]
Ts: Temperature of the vegetation [C]
Pre: Pressure at the height of trees [mbar]
Ds: Vapor Pressure Deficit [Pa]
Psi_L: Soil water potential for first/second layer of vegetation [MPa]
Psi_sto_50: Water Potential at 50% loss conductivity [MPa]
Psi_sto_99: Water Potential at PLCs loss conductivity [MPa]
CT: Photosyntesis Typology for Plants, Photosynthetic pathway
Vmax: Maximum Rubisco Capacity [umolCO2 s^-1 m^-2]
DS: Plant Dependent, Activation Energy in Photosynthesis for Rubisco Capacity [kJ mol^-1]
Ha: Plant Dependent, Activation energy. [kJ mol^-1 K^-1]
FI: Intrinsic quantum Efficiency [umolCO2 umolPhotons^-1]
Oa: Intercellular Partial Pressure Oxygen [ppm] - [umolO2 mol^-1]
Do: Empirical coefficient for the role of vapor pressure in the biochemical model of photosynthesis [Pa]
a1: Empirical parameter connecting stomatal aperture and net assimilation [-]
go: minimal Stomatal Conductance [mol s^-1 m^-2]
e_rel: Relative Efficiency of the photosynthesis apparatus due to Age/Day-length [-]
e_relN: Relative efficiency of the photosynthesis apparatus due to N limitations [-]
gmes: Mesophyll conductance, not used [mol CO2 s^-1 m^-2]
rjv: Scaling factor between Jmax and Vmax
-------
OUTPUT:
rs_sun: stomatal resistence [s m^-1]
rs_shd: stomatal resistence [s m^-1]
Ci_sun: Leaf Interior CO2 mixing ratio of sunlit [umolCO2 mol^-1]
Ci_shd: Leaf Interior CO2 mixing ratio of shaded [umolCO2 mol^-1]
An: Net Assimiltation Rate [umolCO2 s^-1 m^-2 ]
Rdark: Surface Leaf Concentration [umolCO2 s^-1 m^-2 ]
"""
# Re-define input parameters which are overwritten in this function
PAR_sun_local = copy.copy(PAR_sun)
PAR_shd_local = copy.copy(PAR_shd)
Citm1_sun_local = copy.copy(Citm1_sun)
Citm1_shd_local = copy.copy(Citm1_shd)
Vmax_local = copy.copy(Vmax)
if Citm1_sun_local < 200:
Citm1_sun_local = 200
if Citm1_shd_local < 200:
Citm1_shd_local = 200
# ANSW_SCA is assumed to be one
# Relative efficiency for age
Vmax_local = Vmax_local*e_rel*e_relN
# Scaling from leaf to canopy
# To be recomputed for Vmax only for LAI and with Kopt to avoid issue with SAI LAIdead
FsunV = (1 - numpy.exp(-Kopt * (LAI))) / (Kopt * (LAI))
if FsunV < 0.01:
FsunV = 0
if FsunV > 1:
FsunV = 1
FshdV = 1 - FsunV
# Two big leaves with Kn
Can_sun = (1 - numpy.exp(-(Kopt + Knit) * LAI)) / (Kopt + Knit)
Can_shd = (1 - numpy.exp(-Knit * LAI)) / Knit - (1 - numpy.exp(-(Kopt + Knit) * LAI)) / (Kopt + Knit)
# Two big leaves with Kn
Vmax_sun = Vmax_local * Can_sun / (LAI * FsunV)
Vmax_shd = Vmax_local * Can_shd / (LAI * FshdV)
if FsunV == 0:
Vmax_sun = 0
# minimum canopy conductance
go_sun = go
# Canopy Boundary layer resistance
rb_sun = rb
# minimum canopy conductance
go_shd = go
# Canopy Boundary layer resistance
rb_shd = rb
gmes_sun = gmes
gmes_shd = gmes
PAR_sun_local = PAR_sun_local / (LAI * Fsun)
PAR_shd_local = PAR_shd_local / (LAI * Fshd)
# sunlit fraction
if Fsun > 0:
Ci_sun = fsolve(self.CO2_Concentration,Citm1_sun_local,args=(PAR_sun_local,Ca,ra,rb_sun,Ts,Pre,Ds,Psi_L,Psi_sto_50,
Psi_sto_99,CT,Vmax_sun,DS,Ha,FI,Oa,Do,a1,go_sun,gmes_sun,rjv),xtol=1)
if Ci_sun.size == 1:
Ci_sun = Ci_sun[0]
else:
print('Size of Ci_shd is greater than 1')
self.Photosynthesis_Biochemical(Ci_sun,PAR_sun_local,Ca,ra,rb_sun,Ts,Pre,Ds,Psi_L,Psi_sto_50,Psi_sto_99,CT,
Vmax_sun,DS,Ha,FI,Oa,Do,a1,go_sun,gmes_sun,rjv)
CiF_sun = self.PhotoBiochem.CcF
An_sun = self.PhotoBiochem.An
rc_sun = self.PhotoBiochem.rs
Rdark_sun = self.PhotoBiochem.Rdark
SIF_sun = self.PhotoBiochem.F755nm
else:
Ci_sun = 0
CiF_sun = 0
An_sun = 0
Rdark_sun = 0
rc_sun = numpy.inf
SIF_sun = 0
# Shadowed fraction
if Fshd > 0:
Ci_shd = fsolve(self.CO2_Concentration,Citm1_shd_local,args=(PAR_shd_local,Ca,ra,rb_shd,Ts,Pre,Ds,Psi_L,Psi_sto_50,
Psi_sto_99,CT,Vmax_shd,DS,Ha,FI,Oa,Do,a1,go_shd,gmes_shd,rjv),xtol=1)
if Ci_shd.size == 1:
Ci_shd = Ci_shd[0]
else:
print('Size of Ci_shd is greater than 1')
self.Photosynthesis_Biochemical(Ci_shd,PAR_shd_local,Ca,ra,rb_shd,Ts,Pre,Ds,Psi_L,Psi_sto_50,Psi_sto_99,CT,
Vmax_shd,DS,Ha,FI,Oa,Do,a1,go_shd,gmes_shd,rjv)
CiF_shd = self.PhotoBiochem.CcF
An_shd = self.PhotoBiochem.An
rc_shd = self.PhotoBiochem.rs
Rdark_shd = self.PhotoBiochem.Rdark
SIF_shd = self.PhotoBiochem.F755nm
else:
Ci_shd = 0
CiF_shd = 0
An_shd = 0
Rdark_shd = 0
rc_shd = numpy.inf
SIF_shd = 0
DCi_sun = Ci_sun - CiF_sun
DCi_shd = Ci_shd - CiF_shd
DCi = (DCi_sun + DCi_shd) / 2
An = An_sun * (LAI * Fsun) + An_shd * (LAI * Fshd)
Rdark = Rdark_sun * (LAI * Fsun) + Rdark_shd * (LAI * Fshd)
SIF = SIF_sun * (LAI * Fsun) + SIF_shd * (LAI * Fshd)
# stomatal resistence [s m^-1]
rs_sun = rc_sun
rs_shd = rc_shd
lanp = 0.469 # [J umol^-1 CO2]
Lpho = (An + Rdark) * lanp # [W m^-2]
self.CanopyResEvl = CanopyResistanceAnEvolution()
self.CanopyResEvl.rs_sun = rs_sun
self.CanopyResEvl.rs_shd = rs_shd
self.CanopyResEvl.Ci_sun = Ci_sun
self.CanopyResEvl.Ci_shd = Ci_shd
self.CanopyResEvl.An = An
self.CanopyResEvl.Rdark = Rdark
self.CanopyResEvl.Lpho = Lpho
self.CanopyResEvl.SIF = SIF
self.CanopyResEvl.DCi = DCi
return rs_sun,rs_shd,Ci_sun,Ci_shd,An,Rdark,Lpho,SIF,DCi
def CO2_Concentration(self,Cc,IPAR,Csl,ra,rb,Ts,Pre,Ds,Psi_L,Psi_sto_50,Psi_sto_99,CT,Vmax,DS,Ha,FI,Oa,Do,a1,go,gmes,rjv):
self.Photosynthesis_Biochemical(Cc,IPAR,Csl,ra,rb,Ts,Pre,Ds,Psi_L,Psi_sto_50,Psi_sto_99,CT,Vmax,DS,Ha,FI,Oa,Do,a1,go,gmes,rjv)
CcF = self.PhotoBiochem.CcF
self.DCi = Cc - CcF
return self.DCi
def Photosynthesis_Biochemical(self,Cc,IPAR,Csl,ra,rb,Ts,Pre,Ds,Psi_L,Psi_sto_50,Psi_sto_00,CT,Vmax,DS,Ha,FI,Oa,Do,
a1,go,gmes,rjv):
# Re-define input parameters which are overwritten in this function
Cc_local = copy.copy(Cc)
IPAR_local = copy.copy(IPAR)
Csl_local = copy.copy(Csl)
ra_local = copy.copy(ra)
rb_local = copy.copy(rb)
Pre_local = copy.copy(Pre)
DS_local = copy.copy(DS)
Ha_local = copy.copy(Ha)
Oa_local = copy.copy(Oa)
go_local = copy.copy(go)
Ta = Ts
Pre0 = 101325 # [Pa]
Tf = 273.15 # [K]
# Conversion factors
Pre_local = Pre_local * 100 # [Pa]
IPAR_local = IPAR_local * 4.57 # [umolPhotons s^-1 m^-2]
ra_local = ra_local * (0.0224 * (Ta + 273.15) * Pre0 / (Tf * Pre_local)) * 10 ** (-6) # [m^2 s umolH2O^-1]
rb_local = rb_local * (0.0224 * (Ta + 273.15) * Pre0 / (Tf * Pre_local)) * 10 ** (-6) # [m^2 s umolH2O^-1]
Cc_local = Cc_local * 10 ** (-6) * Pre_local # Partial Pressure [Pa * molCO2 molAIR^-1]
Oa_local = Oa_local * 10 ** (-6) * Pre_local # [Pa]
Csl_local = Csl_local * 10 ** (-6) * Pre_local # Leaf surface CO2 concentration [Pa]
# Mesophyl Conductance [s m^2 umolCO2^-1]
rmes = 1 / (1e+6 * gmes)
go_local = go_local * 10 ** 6 # [umolCO2 s^-1 m^-2]
# Temperature dependence
# Maximum Rubisco Capacity Vm
Ts_k = Ts + 273.15 # [K]
# Reference Temperature [K]
Tref = 25 + 273.15
# Gas Constant [kJ K^-1 mol^-1]
R = 0.008314
# ANS_TEMP is assumed t be one
# Deactivation Energy [kJ mol^-1]
Hd = 200
# Mix of Temperature Function and High Temperature Inhibition
kT = numpy.exp(Ha_local * (Ts_k - Tref)/(Tref * R * Ts_k)) * (1+numpy.exp((Tref*DS_local-Hd) / (Tref*R))) / \
(1+numpy.exp((Ts_k * DS_local - Hd)/(Ts_k * R)))
Vm = Vmax * kT # [umolCO2 s^-1 m^-2]
# Maximum Electron Transport Rate Jm
# Deactivation Energy [kJ mol^-1]
Hd = 200
# Activation Energy [kJ mol^-1]
Ha_local = 50
# entropy factor [kJ mol^-1 K^-1]
DS_local = 0.646
kT = numpy.exp(Ha_local * (Ts_k - Tref) / (Tref * R * Ts_k)) * (1 + numpy.exp((Tref * DS_local - Hd) / (Tref * R))) / \
(1 + numpy.exp((Ts_k * DS_local - Hd) / (Ts_k * R)))
# [umol electrons s^-1 m^-2]
Jmax = Vmax * rjv
Jm = Jmax * kT # [umol electrons s^-1 m^-2]
# Triose Phosphate Utilization
# Activation Energy [kJ mol^-1]
Ha_local = 53.1
# entropy factor [kJ mol^-1 K^-1]
DS_local = 0.490
# Deactivation Energy [kJ mol^-1]
Hd = 150.65
TPU25 = 0.1182 * Vmax # [umolCO2 s^-1 m^-2]
kT = numpy.exp(Ha_local * (Ts_k - Tref) / (Tref * R * Ts_k)) * (1 + numpy.exp((Tref * DS_local - Hd) / (Tref * R))) / \
(1 + numpy.exp((Ts_k * DS_local - Hd) / (Ts_k * R)))
TPU = TPU25 * kT # [umolCO2 s^-1 m^-2]
if CT == 4:
s1 = 0.3 # [1 K^-1]
s3 = 0.2 # [1 K^-1] 0.3 (Cox 2001)
Tup = 40 # [C]
Tlow = 15 # [C]
# Temperature Function 1 for Maximum Rubisco Capacity
f1T = 1 / (1 + numpy.exp(s1 * (Ts - Tup)))
# Temperature Function 2 for Maximum Rubisco Capacity
f2T = 1 / (1 + numpy.exp(s3 * (Tlow - Ts)))
fT = 2**(0.1 * (Ts - 25))
Vm = Vmax * fT * f1T * f2T # [umolCO2 s^-1 m^-2]
ke25 = 20000 * Vmax
ke = ke25 * fT
# CO2 concentration point
# ANSG is assumed to be 2
# Activation Energy [kJ mol^-1]
Ha_local = 37.83
kT = numpy.exp(Ha_local * (Ts_k - Tref) / (Tref * R * Ts_k))
GAM25 = 42.75 # [umol mol^-1]
GAM25 = GAM25 * 10 ** (-6) * Pre_local # [Pa]
# Michaelis - Menten Constant for CO2 [Pa]
GAM = GAM25 * kT
if CT == 3:
# Michaelis-Menten Constants for CO2 and O2
# Activation Energy [kJ mol^-1]
Ha_local = 79.43
Kc25 = 404.9 # [umol mol^-1]
Kc25 = Kc25 * 10 ** (-6) * Pre_local # [Pa]
kT = numpy.exp(Ha_local * (Ts_k - Tref) / (Tref * R * Ts_k))
Kc = Kc25 * kT
# Activation Energy [kJ mol^-1]
Ha_local = 36.38
Ko25 = 278.4 # [umol mol^-1]
Ko25 = Ko25 * 10 ** (-3) * Pre_local # [Pa]
kT = numpy.exp(Ha_local * (Ts_k - Tref) / (Tref * R * Ts_k))
# Michaelis-Menten Constant for O2
Ko = Ko25 * kT
# Dark Respiration
if CT == 3:
Ha_local = 46.39
DS_local = 0.490
Hd = 150.65
Rdark25 = 0.015 * Vmax
kT = numpy.exp(Ha_local * (Ts_k - Tref) / (Tref * R * Ts_k)) * (1 + numpy.exp((Tref * DS_local - Hd) / (Tref * R))) / \
(1 + numpy.exp((Ts_k * DS_local - Hd) / (Ts_k * R)))
Rdark = Rdark25 * kT
elif CT == 4:
fT = 2.0 ** (0.1 * (Ts - 25))
# Temperature Function 3 for Respiration
fT3 = 1 / (1 + numpy.exp(1.3 * (Ts - 55)))
Rdark25 = 0.025 * Vmax
# Leaf Maintainance Respiration / Dark Respiration [umolCO2 s^-1 m^-2]
Rdark = Rdark25 * fT * fT3
# Photosynthesis factors
# Light Absorbed by Photosystem II in CO2 units [umolCO2 s^-1 m^-2]
Q = FI * IPAR_local
d1 = 0.7
d2 = -(Q + Jm / 4)
d3 = Q * Jm / 4
# Electron Transport Rate
J = min((-d2 + numpy.sqrt(d2 ** 2 - 4 * d1 * d3)) / (2 * d1), (-d2 - numpy.sqrt(d2 ** 2 - 4 * d1 * d3)) / (2 * d1))
if CT == 3:
# Gross Assimilation Rate Limited by Rubisco [umolCO2 s^-1 m^-2]
JC = Vm * (Cc_local - GAM) / (Cc_local + Kc * (1 + Oa_local / Ko))
# Light Limited
# Gross Assimilation Rate Limited by Light [umolCO2 s^-1 m^-2]
JL = J * (Cc_local - GAM) / (Cc_local + 2 * GAM)
# Capacity of the leaf to export or utilize the products of photosynthesis
# Gross Assimilation Rate Limited by Export [umolCO2 s^-1 m^-2]
JE = 3 * TPU
elif CT == 4:
# Rubisco Limited
# Gross Assimilation Rate Limited by Rubisco [umolCO2 s^-1 m^-2]
JC = Vm
# Light Limited
# Gross Assimilation Rate Limited by Light [umolCO2 s^-1 m^-2]
JL = Q
# PEP Carboxylase Limited
JE = ke * Cc_local / Pre_local
# First Polynomium
if CT == 3:
b1 = 0.98
b2 = -(JC + JL)
b3 = JC * JL
elif CT == 4:
b1 = 0.80
b2 = -(JC + JL)
b3 = JC * JL
# Smoothed Minimum between JC and JE [umolCO2 s^-1 m^-2]
JP = min((-b2 + numpy.sqrt(b2 ** 2 - 4 * b1 * b3)) / (2 * b1), (-b2 - numpy.sqrt(b2 ** 2 - 4 * b1 * b3)) / (2 * b1))
# Second Polynomium
if CT == 3:
c1 = 0.95
c2 = -(JP + JE)
c3 = JP * JE
elif CT == 4:
c1 = 0.95
c2 = -(JP + JE)
c3 = JP * JE
# Gross Assimilation Rate Potential [umolCO2 s^-1 m^-2]
A = min((-c2 + numpy.sqrt(c2 ** 2 - 4 * c1 * c3)) / (2 * c1), (-c2 - numpy.sqrt(c2 ** 2 - 4 * c1 * c3)) / (2 * c1))
# New Water Stress Function
Rgsws = 0.02
p2 = math.log((1 - Rgsws) / Rgsws) / (Psi_sto_00 - Psi_sto_50) # [MPa^-1]
q2 = -p2 * Psi_sto_50 # [-]
Rgsw = 1 / (1 + numpy.exp(p2 * Psi_L + q2))
fO = (1 - Rgsw)
if fO > 1:
fO = 1
if fO < 0:
fO = 0
# Solar-induced chlorophyll fluorescence (SIF)
# Je is the actual electron transport rate calculated from the CO2 exchange data
if CT == 3:
Jfe = A * (Cc_local + 2 * GAM) / (Cc_local - GAM)
elif CT == 4:
Jfe = A
fiP0= FI*4 # [umol Electrons umolPhotons^-1]
fiP = fiP0 * Jfe / Q # [0.4 max - stress decrease ]
# degree of light saturation
dls = 1 - fiP / fiP0
kf = 0.05
kd = max(0.03 * Ts + 0.0773, 0.087)
kn = (6.2473 * dls - 0.5944) * dls
fiF = kf / (kf + kd + kn) * (1 - fiP) # [umol Electrons umolPhotons^-1]
SIF = IPAR_local * fiF # [umol electrons s^-1 m^-2]
# k theoretically a function of Vmax and Chlorophyll content
k = 0.0375 * Vmax + 8.25 # [umol m^-2 s^-1 / W m^-2 sr^-1 um^-1]
F755nm = SIF / k # [W m^-2 sr^-1 um^-1]
# Gross Assimilation Rate [umolCO2 s^-1 m^-2]
A = A * fO
# Net Assimilation Rate [umolCO2 s^-1 m^-2]
An = A - Rdark
# Stomatal Conductance
gsCO2 = go_local + a1 * An * Pre_local / ((Cc_local - GAM) * (1 + Ds / Do))
if gsCO2 < go_local:
gsCO2 = go_local
# Stomatal resistance or Canopy [s m^2 umolCO2^-1]
rsCO2 = 1 / gsCO2
CcF = Csl_local - An * Pre_local * (rsCO2 + rmes + 1.37 * rb_local + ra_local) # [Pa]
if CcF < 0:
CcF = 0
# Stomatal resistance or canopy [s m^2 molH2O^-1]
rsH20 = (rsCO2 / 1.64) * (10 ** 6)
# Net Assimilation Rate [umolCO2 s^-1 m^-2]
An = (Csl_local - CcF) / (Pre_local * (rsCO2 + rmes + 1.37 * rb_local + ra_local))
CcF = CcF / (Pre_local * 10 ** (-6)) # [umolCO2 molAIR^-1]
rs = rsH20 * (Tf * Pre_local) / (0.0224 * (Ts + 273.15) * Pre0) # Stomatal resistance or Canopy [s m^-1]
self.PhotoBiochem = PhotosynthesisBiochemical()
self.PhotoBiochem.CcF = CcF
self.PhotoBiochem.An = An
self.PhotoBiochem.rs = rs
self.PhotoBiochem.Rdark = Rdark
self.PhotoBiochem.F755nm = F755nm
self.PhotoBiochem.GAM = GAM
self.PhotoBiochem.gsCO2 = gsCO2
def Soil_Resistance(self,T_soil,Pre,Ws,ea,q_runon,O,Ks,Osat,Ohy,L,Pe,O33,alpVG,nVG,SPAR):
"""
------
INPUT:
T_soil: Soil temperature [C]
Pre: Pressure [mbar]
Ws: Wind speed [m s^-1]
ea: Vapor pressure at T_canyon [Pa]
q_runon: Intercepted water on the surface [mm]
O: Water Content []
Ks: Hydraulic conductivity at saturation [mm s^-1]
Osat: Water content at saturation, saturation moisture 0 kPa [-]
Ohy: Hygroscopic Moisture Evaporation cessation []
L: Slope of logarithmic tension-moisture curve [-]
Pe: Tension at air antry (bubbling pressure) [kPa]
O33: Soil water content at -33 [kPa] of water potential
alpVG: Alpha parameter Van-Genuchten soil water retention curve [mm^-1]
nVG: n parameter Van-Genuchten soil water retention curve [mm^-1]
SPAR: Soil parameter type
-------
OUTPUT:
r_soil: Soil resistance [s m^-1]
b_soil: beta factor [0-1]
alp_soil: relative humidity [0-1]
"""
# -------------------------
# Calculate soil resistance
# -------------------------
# Soil Temperature [K]
Ts_k = T_soil + 273.15
# Water density [kg m^-3]
row = 1000
g = 9.81
# water vapor gas constant [J kg^-1 K^-1]
Rd = 461.5
# vapor molecular diffusivity [m^2 s^-1]
Da = (2.11 * 1e-5) * (((Ts_k) / 273.15)**1.94) * (Pre * 100 / 101325)
esat = 611 * numpy.exp(17.27 * T_soil / (237.3 + T_soil)) # [Pa]
SoilCal = Soil_Calculations()
SoilCal.Conductivity_Suction(SPAR,Ks,Osat,Ohy,L,Pe,O33,alpVG,nVG,O)
Ko = SoilCal.CondSuc.Ko # [mm s^-1]
Po = SoilCal.CondSuc.Po # [mm]
if Po < 0:
Po = 0
alp_soil = numpy.exp(-Po * g / (1000 * Rd * Ts_k))
# 40-200 um Size of the pores -- Particle Size/3 [m]
Psz = (11.12 * nVG**3.286) * 1e-6
# Boundary Layer Thickness
dm = 2.26 * 1e-3 / numpy.sqrt(Ws)
### it is only for ANSW = 4
gammap = (alp_soil * esat - ea) / (row * Rd * Ts_k) # [-]
if gammap < 0:
r_soil = 0
else:
# Internal soil viscous resistance [s m^-1]
rsv = gammap / (4 * Ko / (1000 * 3600))
f_O = (2 / numpy.pi) * (numpy.sqrt(numpy.pi / (4 * O)) - 1) / numpy.sqrt(4 * O) # [-]
# viscous boundary layer resistance [s m^-1]
rvbl = (dm + Psz * f_O) / Da
r_soil = rvbl + rsv
if O <= Ohy:
r_soil = numpy.inf
b_soil = 1
if q_runon > 0:
r_soil = 0
alp_soil = 1
b_soil = 1
self.ResSoil = ResistanceSoil()
self.ResSoil.r_soil = r_soil
self.ResSoil.alp_soil = alp_soil
self.ResSoil.b_soil = b_soil
return r_soil,b_soil,alp_soil
def Leaf_Boundary_Resistance(self,Ws,Ts,Ta,hc,d_leaf,LAI,zatm,disp_h,zom):
# Re-define input parameters which are overwritten in this function
d_leaf_local = copy.copy(d_leaf)
# Wind speed [m s^-1]
u = Ws
d_leaf_local = d_leaf_local / 100 # [m]
# von Karman constant
k = 0.4
# Empirical coefficient [m s^-0.5]
a = 0.01
# Zero plane displacement [m]
d = disp_h
# Domain height [m]
z = zatm
# Hypothesis Logarithmic distribution of wind speed
# Friction Velocity [m s^-1]
us = k * u / math.log((z - d) / zom)
# Wind Speed top Canopy [m s^-1]
u_hc = (us / k) * math.log((hc - d) / zom)
# Attenuation Coefficient
alpha_den = (z / hc - 1)
alpha = math.log(u / u_hc) / alpha_den
alpha = 0.5 * alpha * LAI / 2
# Expression of Leaf Boundary Layer Resistance
gb = (2 * a / alpha) * ((u_hc / d_leaf_local) ** 0.5) * (1 - numpy.exp(-alpha / 2)) \
if d_leaf_local != 0 else (2 * a / alpha) * (numpy.inf ** 0.5) * (1 - numpy.exp(-alpha / 2))
# Expression for free convection
# Molecular diffusivity of heat [m^2 s^-1]
Dh = 1.9e-5
# Grashof number [-]
if Ts > Ta:
Gr = 1.6e8 * (Ts - Ta) * d_leaf_local ** 3
else:
Gr = 0
# The leaf boundary conductance at free convection [m s^-1]
gb_free = 0.5 * Dh * (Gr ** 0.25) / d_leaf_local if d_leaf_local != 0 else numpy.nan
gb = gb + gb_free
# Leaf Boundary Layer Resistance [s m^-1] one-sided for unit leaf
rb = 1 / gb
return rb
def Leaf_BR(self,u_hc,Ts,Ta,d_leaf,alpha):
"""
------
INPUT:
u_hc: wind speed at the height of trees [m s^-1]
Ts: Trees temperature [C]
Ta: Air temperature at the height of trees [C]
d_leaf: Leaf dimension of trees[cm]
alpha: Attenuation Coefficient [-]
-------
OUTPUT:
rb: Leaf boundary layer resistance [s m^-1]
"""
# Re-define input parameters which are overwritten in this function
d_leaf_local = copy.copy(d_leaf)
d_leaf_local = d_leaf_local/100 # [m]
a = 0.01 # [m s^-0.5] (Chodhury and Monteith 1988)
# Expression for Leaf Boundary Layer Resistance [m s^-1]
gb = (2 * a / alpha) * ((u_hc / d_leaf_local)**0.5) * (1 - numpy.exp(-alpha / 2))
# Expression for free convection (Leuning 1995, Monteith 1973)
Dh = 1.9 * 1e-5 # [m^2 s^-1]
Gr = 1.6 * 1e+8 * (Ts - Ta) * (d_leaf_local ** 3)* (Ts > Ta) # [-]
gb_free = 0.5 * Dh * Gr ** (0.25) / d_leaf_local # [m s^-1]
gb = gb + gb_free
# Leaf Boundary Layer Resistance [s m^-1] one-sided for unit leaf
rb = 1 / gb
return rb
def Urban_roughness(self,hc_H,hc_L,Csoil,Croad,Croof):
"""
------
INPUT:
hc_H: Height of high vegetation [m]
hc_L: Height of low vegetation [m]
Csoil: boolean operator for presence (1) and absence (0) of soil
Croad: boolean operator for presence (1) and absence (0) of road
Croof: boolean operator for presence (1) and absence (0) of roof
-------
OUTPUT:
zom: roughness eddy diffusivities for momentum [m]
zoh: roughness eddy diffusivities for heat [m]
disp_h: maximum displacement height [m]
zom_H: high vegetation roughness momentum [m]
zom_L: low vegetation roughness momentum [m]
zoh_H: high vegetation roughness heat [m]
zoh_L: low vegetation roughness heat [m]
d_H: displacement height of high vegetation [m]
d_L: displacement height of low vegetation [m]
zom_other: roughness momentum for the other urban surfaces [m]
"""
# bare soil roughness momentum [m]
if Csoil == 1:
zom_soil = 0.003
else:
zom_soil = 0.0
# road roughness momentum
if Croad == 1:
zom_road = 0.003
else:
zom_road = 0.0
# roof roughness momentum Wang et al. (2013) [m]
if Croof == 1:
zom_roof = 0.01
else:
zom_roof = 0.0
# vegetation roughness momentum [m] Brutsaert (1975) high vegetation
zom_H = 0.123 * hc_H
# vegetation roughness momentum [m] Brutsaert (1975) low vegetation
zom_L = 0.123 * hc_L
zom_other = [zom_soil, zom_road, zom_roof]
# roughness eddy diffusivities for momentum [m]
zom_other = max(zom_other)
# Heat Roughness [m]
zoh_L = zom_L * 0.1
zoh_H = zom_H * 0.1
# roughness eddy diffusivities for heat [m] [Brutsaert (1975)]
zoh_other = 0.1 * zom_other
# PATCH SCALE ROUGHNESS
zom = max(max(zom_H, zom_L), zom_other)
zoh = max(max(zoh_H, zoh_L), zoh_other)
zom_ground = max(zom_L, zom_other)
zoh_ground = max(zoh_L, zoh_other)
# Displacement height
d_L = 2 / 3 * hc_L
d_H = 2 / 3 * hc_H
disp_h = max(d_H, d_L)
return zom,zoh,zom_ground,zoh_ground,disp_h,zom_H,zom_L,zoh_H,zoh_L,d_H,d_L,zom_other
def WindProfile_Canyon(self,Hcan,Htree,R_tree,Wcan,Wroof,Kopt,LAI_t,Zatm,WindSpeed_top,Zp,trees,Zref_und,zom_und):
"""
------
INPUT:
Hcan: canyon height [m]
Htree: Tree height [m]
R_tree: Tree radius [m]
Wcan: Canyon width [m]
Wroof: Roof width [m]
Kopt: Optical transmission factor [-]
LAI_t: Leaf area index of tree [-]
Zatm: Height of the domain [m]
WindSpeed_top: wind speed at the top of the domain [m s^-1]
Zp: Height of interest within the canyon [m]
trees: Presence of trees [0: No, 1: Yes]
Zref_und: Refrence height [m]
zom_und: Aerodynamic roughness length [m]
-------
OUTPUT:
dcan: Urban displacement height including trees [m]
zomcan: Urban momentum roughness height including trees [m]
u_Hcan: Wind speed at canyon height [m s^-1]
u_Zpcan: Wind speed within canyon at height Zpcan [m s^-1]
w_Zpcan: Vertical wind speed within canyon [m s^-1]
"""
# Re-define input parameters which are overwritten in this function
Htree_local = copy.copy(Htree)
R_tree_local = copy.copy(R_tree)
LAI_t_local = copy.copy(LAI_t)
if trees == 0:
Htree_local = 0
R_tree_local = 0
LAI_t_local = 0
# Best fit for staggered arrays , or a = 3.59 for square arrays
a = 4.43
# Von Karman constant
k = 0.4
# b=1, no incorporation for drag correction factors. Good fit for staggered arrays
b = 1.0
# nominal drag for cubical obstacles
CDb = 1.2
# Plan area fraction of buildings and vegetation
Ap_build = Wroof
Ap_tree = 4 * R_tree_local
Ap_urb = Wcan + Wroof
# Frontal area fraction of vegetation and buildings: assumption infinite urban canyon perpendicular to the
# wind direction (Length of building and plot equals infinity)
Af_build_s = Hcan
Af_veg_s = 2 * R_tree_local
# Tree canopy transmittance (optical = P2D)
P2D = numpy.exp(-Kopt * LAI_t_local)
# Guan et al. 2003
P3D = P2D ** 0.40
# Guan et al. 2000
Pv = (-1.251 * P3D ** 2 + + 0.489 * P3D + 0.803) / CDb
# Plan area fraction of buildings and Calculation of structural parameters and wind profile in the city
Lp_tot = (Ap_build + (1 - P3D) * Ap_tree) / Ap_urb
H_tot = (Hcan * Ap_build + (Htree_local + R_tree_local) * (1 - P3D) * Ap_tree) / (Ap_build + (1 - P3D) * Ap_tree)
# Urban displacement height and roughness length with incorporation of trees (Kent 2017), (MacDonald 1998)
# displacement height of canyon [m], eq. 23
dcan = H_tot * (1 + a ** (-Lp_tot) * (Lp_tot - 1))
Af_build = H_tot / (H_tot - dcan) * Af_build_s
Af_veg = H_tot / (H_tot - dcan) * Af_veg_s
zomcan = H_tot * (1 - dcan / H_tot) * numpy.exp( -(1 / k ** 2 * 0.5 * b * CDb * (1 - dcan / H_tot) *
(Af_build + Pv * Af_veg) / Ap_urb) ** (-0.5))
zohcan = zomcan / 10
# Calculation of wind profile above and in the canyon with a logarithmic and exponential wind profile.
# Friction Velocity Atmosphere [m s^-1]
us_atm = k * WindSpeed_top / math.log((Zatm - dcan) / zomcan)
self.Ustar_Atm = us_atm
# Wind Speed at canyon top [m s^-1]
u_Hcan = (us_atm / k) * math.log((Hcan - dcan) / zomcan)
# Attenuation Coefficient Canyon not corrected for presence of trees.
alpha = math.log(WindSpeed_top / u_Hcan) / (Zatm / Hcan - 1)
if Zp >= Hcan:
u_Zp = (us_atm / k) * math.log((Zp - dcan) / zomcan)
w_Zp = 0
elif Zp <= Hcan and Zp >= Zref_und:
u_Zp = u_Hcan * numpy.exp(-alpha * (1 - Zp / Hcan))
w_Zp = 0
elif Zp <= Zref_und and Zp >= zom_und:
uref_und = u_Hcan * numpy.exp(-alpha * (1 - Zref_und / Hcan))
usref_und = k * uref_und / math.log(Zref_und / zom_und)
u_Zp = (usref_und / k) * math.log(Zp / zom_und)
w_Zp = 0
else:
u_Zp = 0
w_Zp = 0
print('wind speed calculation height higher than reference height or lower than roughness length')
return dcan,zomcan,u_Hcan,u_Zp,w_Zp,alpha
def WindProfile_Roof(self,Hcan,hveg,VerticalProfUrban,Geometry_m):
# Wind profile from 1-D model
vx = copy.copy(VerticalProfUrban.vx)
vy = copy.copy(VerticalProfUrban.vy)
z_urban = copy.copy(Geometry_m.z[:-1])
vx_intp = interp1d(z_urban, vx)
vy_intp = interp1d(z_urban, vy)
u_Zp = numpy.sqrt(vx_intp(Geometry_m.dz/2+Hcan)**2+vy_intp(Geometry_m.dz/2+Hcan)**2)
u_Hveg = numpy.sqrt(vx_intp(hveg+Hcan)**2+vy_intp(hveg+Hcan)**2)
return u_Zp,u_Hveg
def Wall_Aerodynamic_Resistance(self,VerticalProfUrban,Geometry_m,windMin,Cp,iz_wall,ParCalculation):
"""
------
INPUT:
VerticalProfUrban: Vertical profile of variables obtained from 1-D model
Geometry_m: Geometric parameters
windMin: Minimum wind speed in the urban area [m s^-1]
Cp: Air specific heat [J kg^-1 K^-1]
iz_wall: z index in the urban canyon
ParCalculation: General calculation parameters
-------
OUTPUT:
RES: Aerodynamic resistance [s m^-1]
"""
# Air density [kg m^-3]
rho = copy.copy(VerticalProfUrban.rho[iz_wall])
vett = numpy.sqrt(VerticalProfUrban.vx[iz_wall] ** 2 + VerticalProfUrban.vy[iz_wall] ** 2)
vett = max(vett, windMin)
# Convective heat transfer coefficient [W K^-1 m^-2]
hc = 5.678 * (1.09 + 0.23 * (vett / 0.3048))
# Using energy balance for a control volume inside the urban unit, the convective heat transfer coefficient should be limited
# hc must be less than (rho * cp / dt) * [(1-lambdap) * Hmean / (4 * lambdaf * dz)]
if hc > ((rho*Cp/ParCalculation.dts) * ((1-Geometry_m.lambdap)*Geometry_m.Height_canyon) / (4*Geometry_m.lambdaf*Geometry_m.dz)):
hc = (rho*Cp/ParCalculation.dts) * ((1-Geometry_m.lambdap)*Geometry_m.Height_canyon) / (4*Geometry_m.lambdaf*Geometry_m.dz)
# Term in energy equation [s m^-1]
RES = ((rho * Cp)/hc)
return RES
def Ground_Aerodynamic_Resistance_1D(self,WindSpeed_top,Zatm,VerticalProfUrban,Gemeotry_m,Ta,Ts,hcan,dcan,zomcan,zom_und,Ztree,Rtree,ColParam):
"""
------
INPUT:
WindSpeed_top: Wind speed at the top of the domain [m s^-1]
Zatm: Domain height [m]
VerticalProfUrban: Vertical profile of variables obtained from 1-D model
Gemeotry_m: Geometric parameters
Ta: Air temperature near the ground [K]
Ts: Total ground temperature [K]
hcan: Canyon height [m]
dcan: Displacement height of the canyon [m]
zomcan: Aerodynamic roughness length of the canyon [m]
zom_und: Aerodynamic roughness length of the ground [m]
Ztree: Trees height [m]
Rtree: Trees radius [m]
ColParam: 1-D model parameters
-------
OUTPUT:
rap_can: Aerodynamic resistance near the ground [s m^-1]
rap_Ztree_In: Aerodynamic resistance between trees and canyon air [s m^-1]
u_Hcan: Wind speed at the canyon height [m s^-1]
alpha: Attenuation Coefficient [-]
Ri: Bulk Richardson number
Utot:
"""
# Interpolate wind speed
vx = copy.copy(VerticalProfUrban.vx)
vy = copy.copy(VerticalProfUrban.vy)
z_urban = copy.copy(Gemeotry_m.z[:-1])
vx_intp = interp1d(z_urban, vx)
vy_intp = interp1d(z_urban, vy)
# Wind speed at the canyon height [m s^-1]
u_Hcan = numpy.sqrt(vx_intp(hcan)**2+vy_intp(hcan)**2)
# Make sure that the wind speed at the top of the domain is different from the wind speed at the canyon height
# (if they are equal, then Attenuation Coefficient will be zero)
wind_top = WindSpeed_top if WindSpeed_top != u_Hcan else u_Hcan+0.1
# Attenuation Coefficient Canyon not corrected for presence of trees
alpha = math.log(wind_top / u_Hcan) / (Zatm / hcan - 1)
Ck = 0.4
#---------------------------------------
# Aerodynamic resistance near the ground
#---------------------------------------
zz = Gemeotry_m.dz/2
Utot = numpy.sqrt(vx[0]**2+vy[0]**2)
Utot = max(Utot,ColParam.WindMin_Urban)
# Compute bulk Richardson number
# Ta and Ts should be in [K]
Ri = 2 * 9.81 * zz * (Ta - Ts) / ((Ta + Ts) * (Utot ** 2))
if Ri > 0.16:
Ri = 0.16
# Calculation from Louis, 1979 (eq. 11 and 12)
b = 9.4
cm = 7.4
ch = 5.3
R = 0.74
a = Ck / math.log(zz / zom_und)
if Ri > 0:
fm = 1 / ((1 + 0.5 * b * Ri) ** 2)
fh = fm
else:
c = b * cm * a * a * (zz / zom_und) ** 0.5
fm = 1 - b * Ri / (1 + c * (-Ri) ** 0.5)
c = c * ch / cm
fh = 1 - b * Ri / (1 + c * (-Ri) ** 0.5)
rap_can = R / ((a ** 2) * Utot * fh)
# --------------------------------------
# Aerodynamic resistance above the trees
# --------------------------------------
zz = dcan + zomcan
Utot = numpy.sqrt(vx_intp(zz) ** 2 + vy_intp(zz) ** 2)
# Calculation from Louis, 1979 (eq. 11 and 12)
b = 9.4
cm = 7.4
ch = 5.3
R = 0.74
a = Ck / math.log(zz / zom_und)
if Ri > 0:
fm = 1 / ((1 + 0.5 * b * Ri) ** 2)
fh = fm
else:
c = b * cm * a * a * (zz / zom_und) ** 0.5
fm = 1 - b * Ri / (1 + c * (-Ri) ** 0.5)
c = c * ch / cm
fh = 1 - b * Ri / (1 + c * (-Ri) ** 0.5)
rap_can_AboveTree = R / ((a ** 2) * Utot * fh)
# ------------------------------------------------
# Aerodynamic resistance just underneath the trees
# ------------------------------------------------
zz = Ztree-Rtree
Utot = numpy.sqrt(vx_intp(zz) ** 2 + vy_intp(zz) ** 2)
# Calculation from Louis, 1979 (eq. 11 and 12)
b = 9.4
cm = 7.4
ch = 5.3
R = 0.74
a = Ck / math.log(zz / zom_und)
if Ri > 0:
fm = 1 / ((1 + 0.5 * b * Ri) ** 2)
fh = fm
else:
c = b * cm * a * a * (zz / zom_und) ** 0.5
fm = 1 - b * Ri / (1 + c * (-Ri) ** 0.5)
c = c * ch / cm
fh = 1 - b * Ri / (1 + c * (-Ri) ** 0.5)
rap_Ztree = R / ((a ** 2) * Utot * fh)
# Aerodynamic resistance between trees and canyon air
rap_Ztree_In = max(rap_can_AboveTree - rap_Ztree, 0)
return rap_can,rap_Ztree_In,u_Hcan,alpha,Ri
def Roof_Aerodynamic_Resistance_1D(self,VerticalProfUrban,Geometry_m,z0,Ts):
vx = copy.copy(VerticalProfUrban.vx)
vy = copy.copy(VerticalProfUrban.vy)
th = copy.copy(VerticalProfUrban.th)
z_urban = copy.copy(Geometry_m.z[:-1])
vx_intp = interp1d(z_urban, vx)
vy_intp = interp1d(z_urban, vy)
th_intp = interp1d(z_urban, th)
vx_air = vx_intp(Geometry_m.dz/2 + Geometry_m.Height_canyon)
vy_air = vy_intp(Geometry_m.dz/2 + Geometry_m.Height_canyon)
th_air = th_intp(Geometry_m.dz/2 + Geometry_m.Height_canyon)
Ck = 0.4
zz = Geometry_m.dz/2
Utot = numpy.sqrt(vx_air**2 + vy_air**2)
# Compute bulk Richardson number
# th_air and Ts should be in [K]
Ri = 2 * 9.81 * zz * (th_air - Ts) / ((th_air + Ts) * (Utot ** 2))
# Calculation from Louis, 1979 (eq. 11 and 12)
b = 9.4
cm = 7.4
ch = 5.3
R = 0.74