Mathematical description of mini-CO2fix
Differential equations
d(wf) / dt = pn * nf – rgf * wf – nh * wf
d(ws) / dt = pn * ns – rgs * ws – nh * ws
d(wr) / dt = pn * nr – rgr * wr – nh * wr
d(wp) / dt = hu * np – rgp * wp
d(wt) / dt = hu * nt – rgt * wt
d(ww)/dt = ht – hu
Initial values state variables
wf 0.02 kg C m–2 weight foliage
ws 0.06 kg C m–2 weight stems
wr 0.02 kg C m–2 weight roots
wp 0.0 kg C m–2 weight paper
wt 0.0 kg C m–2 weight timber
ww 0.0 kg C m–2 weight waste
Auxiliary equations
pgs = (1/(2cf)) * (lue * ia + pc – ( (lue*ia+pc)2 – 4*cf*lue*ia*pc )0.5)
gross photosynthesis per second [kg CO2 m–2 s–1]
pg = pgs * csph * cdlh * cdgs * mwC/(mwC +2*mwO)
gross photosynthesis per year [kg C m–2 y–1]
pn = pg – (rf * wf + rr * wr +rs *ws) net photosynthesis per year [kg C m–2 y–1]
ia = ie * (1– exp(–k * lai)) ia is absorbed irradiation; ie is external irradiation
lai = wf * sla / fcl leaf area index
ht = (wf + wr + ws) * nh total harvest [kg C m–2 y–1]
hu = ws * nh useful harvest [kg C m–2 y–1]
nr = 1 – nf – ns allocation fraction for roots [-]
nt = 1 – np constant allocation fraction to timber
Wtot = wf + wr + ws total (standing) biomass [kg C m–2]
Parameters
k = 0.7 m2 soil (m2 leaf)–1 extinction coefficient (0.5–0.9)
sla = 15 m2 kg–1 specific leaf area (10-20)
fcl = 0.5 kg C (kg Leaf)–1 fraction carbon in leaf
ie = 200 J PAR m–2 s–1 external irradiation (above canopy)
rf = 0.1 kg C kg–1 C yr–1 mass-based respiration rates foliage
rr = 0.1 kg C kg–1 C yr–1 mass-based respiration rate roots
rs = 0.1 kg C kg–1 C yr–1 mass-based respiration rate stem
nf = 0.2 – constant allocation fraction for foliage
ns = 0.6 – constant allocation fraction for stem
nh = 0.1 yr–1 relative harvesting rate
np = 0.2 – constant allocation fraction to paper
rgf = 1.0 kg C kg–1 C yr–1 turnover rate or relative death rate for foliage
rgr = 0.2 kg C kg–1 C yr–1 turnover rate or relative death rate for roots
rgs = 0.05 kg C kg–1 C yr–1 turnover rate or relative death rate for stem
rgp = 0.5 kg C kg–1 C yr–1 turnover rate for paper
rgt = 0.02 kg C kg–1 C yr–1 turnover rate for timber
cf = 0.9 – dimensionless photosynthesis curvature factor of NRH
lue = 1. 10–8 kg CO2 (J PAR)−1 light use efficiency
pc = 1. 10–6 kg CO2 m−2 s−1 photosynthesis capacity
mwC = 12 g C mol−1 molecular weight Carbon
mwO = 16 g O mol−1 molecular weight Oxygen
csph = 3600 seconds per hour constant
cdlh = 12 daylight hours, hour/day can be varied
cdgs = 200 days per growing season since growth ceases between the growing seasons, we
can also interpret cdgs as days per year.
Forcing functions
At the moment parameters such as for example ie, cdgs and cdlh (external conditions), nf, ns, nr (internal tree physiology) and nh, nt, np (management) are taken as constants. These could be forcing functions. However, this is not needed for the following assignments.
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Mathematical description of mini-CO2fix MFNC2008 PAGE 1