ERT 355: Biosystems Modelling and Simulation

ERT 355: Biosystems Modelling and Simulation

Part 1: What is mathematical modelling?

Definition of a mathematical model
Definition of a model simplified representation of real systems Types of models pictorial, conceptual/verbal, physical, mathematical Definition of a mathematical model represents a real system in a mathematical form (one or more equations)

Uses of mathematical models
help us to understand, predict and control a system identify areas of deficient knowledge less experimentation by trial-and-error answer various “what if?” scenarios add value to experiments may replace experiments (in rare cases) encourage collaboration among researchers from various disciplines

Characteristics of models
incomplete description of real systems models built from assumptions model simplicity vs. model accuracy no one best model for all circumstances not about computers or ICT

Modelling methodology: an example
How to determine the number of leaves in this tree (or any in trees)? Most accurate method: manually count each and every leaf Problem: tedious and time-consuming Alternative: N = nl x nb where N is no. of leaves; nl is average no. of leaves per branch; and nb is no. of branches nl = 153, nb = 99, so N = 15,147 leaves

Let’s develop our own leaf count model:
Maximum number of small boxes that could fit into the large box is the volume of the large box (Vlarge) divided by the volume of the small box (Vsmall) So max. no. of small boxes that can fit into the large box is:

So applied to canopy: So no. of leaves in canopy (N) is:
Vc = vol. of canopy Vl = average vol. of one leaf The geometrical shape of an ellipsoid is used to represent the canopy volume of a tree and the volume a single leaf

H = 3.5m; W = L = 2.5m; l = 0.14m, so N is 7,185 leaves Why the error?
vs. 15,147 leaves model error of 47% (large!) Why the error? check our assumptions

Revised model: Vol. of canopy occupied by leaves is:
The canopy of a tree is represented by two ellipsoids, where the inner ellipsoid (or shell) is devoid of any leaves, and the outer shell marks the canopy edge. The space between the inner and outer shell is where the leaves are located in the canopy.

So number of leaves (N) is now:
Vo = vol. of canopy occupied by leaves Vl = average vol. of one leaf But, let’s account for leaf density (how closely packed are the leaves): where 1/pl is the fraction of a full ellipsoid volume occupied by a single leaf; the larger the pl , the more closely packed the leaves are together.

Finally, we have: Measuring, we get: fH = fW = fL = 0.5; and pl = 2, so N is 13,951 leaves. Error of only 8% (acceptable).

Our model disadvantages:
Our model advantages: no manual leaf counting faster and less tedious Our model disadvantages: accurate determination of pl difficult assumes uniform distribution of leaves assumes ellipsoidal volumetric canopy and leaves

REVIEW OF MODELLING METHODOLOGY

Types of mathematical models
Mechanistic (process-based) and Empirical Static and Dynamic Discrete and continuous Deterministic and stochastic

Static models Dynamic models Discrete models Continuous models
no time factor Dynamic models has time factor Discrete models time is an integer (1, 2, 3, …) Continuous models time is a real value (1.1, 2.5, 3.0, …)

Deterministic models Stochastic models no element of randomness
has elements of randomness (probabilities)

Mechanistic models Empirical models process-based models
describes and explains the processes more difficult to build Empirical models correlative or statistical models describes but does not explain the processes easier to build

Properties of empirical models
1) easier to build; curve-fitting exercise linear logarithmic sine quadratic exponential

mechanistic models difficult to build because we need to know which and how the factors interact with one another to produce the system process

2) empirical models cannot imply causality (cause-and-effect)
describes how variable are related but does not explain why Examples: - sale of ice cream and reservoir level - electricity bill and weather

3) empirical models are highly environment-specific
can only be used in conditions from which they were derived only its use more limited than mechanistic models (applicable over wider range of environments or conditions) but when used in their environment, empirical models are often very accurate, more so than mechanistic models

Crop production levels
Four levels of production: Level 1: potential growth, limited only by solar radiation Level 2: additionally limited by water Level 3: additionally limited by nitrogen Level 4: additionally limited by other nutrients Helps us to focus on developing our models

Overview of the Gg (Generic crop growth) model

Part 2: Meteorology

Agriculture strongly affected by meteorology (weather) Water (rain) RH
Importance Agriculture strongly affected by meteorology (weather) Water (rain) photosynthesis, respiration, nutrient supply RH 40-80% desirable; too high, high disease and pests low crop yield because high vegetative growth

Solar radiation Air temperature
major energy supplier; photosynthesis, water uptake length of day: flowering soybean (short day); groundnut (long day) Air temperature every plant has an optimum temp. range regulates chemical reactions in photosynthesis, flowering, germination, transpiration, respiration

Solar position Solar position with respect to the observer
 = azimuth (-ve, before solar noon)  = elevation (or solar height)  = site latitude = solar declination  = hour angle

Location of observer on the Earth’s surface
Location of observer on the Earth’s surface. Np and Sp are the North Pole and South Pole, respectively. Solar position with respect to the Earth. Np and Sp are the North Pole and South Pole, respectively.

Solar declination  varies depending on the Earth’s position in orbit around the Sun. Np is the North Pole. Cyclical change in solar declination with the day of year td = day of year (1=Jan 1, …, 365=Dec 31)

Hour angle, : Earth rotates 360 every 24 hours, so every 1 hour, Earth rotates 15 th = local solar time  is –ve before solar noon, +ve after solar noon Local solar time vs. local time local time is determined by - Standard Meridian - political boundaries (West Malaysia & Singapore is actually GMT +7)

Therefore: solar elevation (from horizontal) solar azimuth
(from South) solar inclination (from vertical) solar azimuth (from North in an eastward direction)

Daylength and sunrise and sunset time
Sunrise time Daylength number of hours between sunrise and sunset

Solar radiation Longer the wavelength, lesser the energy:
PAR (photosynthetically active radiation, nm, same as visible light) UV too high energy NIR too low energy

Terminology: radiant flux = amount of energy emitted or received per unit time (W or J s-1) radiant flux density = radiant flux per unit area (W m-2) irradiance = radiant flux density received (incident) radiant emittance = radiant flux density emitted (transmitted)

Daily irradiance s = sunshine hours (no. of hours when solar irradiance >120 W m-2) DL = daylength Iet,d = extra-terrestrial solar irradiance (W m-2) b0 and b1 = empirical coefficients The Angstrom coefficients b0 and b1 used for calculating daily solar radiation for different climate zones (Frere and Popov, 1979) Climate zones b0 b1 Cold or temperate Dry tropical Humid tropical

Hourly ET irradiance: Daily ET irradiance: where a = sinsin b = coscos

Hourly irradiance a = sinsin b = coscos
Typical diurnal trend for solar irradiance

Net radiation net = incoming - outgoing
 = Stefan-Boltzmann constant (5.67 x 10-8 W m-2 K-4) p = surface albedo (typically 0.15) Ta,K4 = air temperature (K) b = 0.2

Direct and diffuse solar radiation
Direct (dr) component from a single direction causes shadows Diffuse (df) component from all directions does not cause shadows Need to distinguish the two interception by plants is different for the two components

Daily direct and diffuse radiation
A set of empirical equations: Idr,d = It,d – Idf,d

Hourly direct and diffuse radiation
A set of empirical equations: Idr = It – Idf

Air vapour pressure RH = relative humidity (%)
Ta = air temperature (C) Relationship between saturated vapor pressure and air temperature

Serdang RH and vapour pressure
Air vapor pressure (ea) and saturated air vapor pressure (es) for Serdang town ( N; E), Malaysia on 31 October 2004 Relative humidity for Serdang on 31 October 2004

Wind speed Idealized daily trend for mean hourly wind speed
actual hourly wind speed can be erratic and difficult to simulate

Air temperature Measured air temperature for Serdang on 31 October 2004

Pattern of change before sunrise, air. temp gradually drops due to heat loss from ground when sun rises, air temp. still drops because heat gain from sun is not enough to overcome heat loss from ground 1-2 hours after sunrise, air temp. begins to rise as heat supplied from sun increases that’s why it is often the coldest just after dawn

For simulation, divide the day into 3 sections
1-2 hours after solar noon (about 14:00 hours), air temp. reaches maximum then starts to drop because ground radiation (heat loss) now exceeds solar radiation For simulation, divide the day into 3 sections Section I – before (sunrise h) Section II – from (sunrise h) to sunset Section III – after sunset

Tmin & Tmax: min. and max. air temperature (C)
Tset : air temperature at sunset (C) – determine from Section II (th = tss) tsr & tss : sunrise and sunset time (hour) th : local solar hour

Serdang air temperature
Mean error 2% Comparison between actual and simulated air temperature

Part 3: Plant-radiation regime

Interception Irradiance above (I0) and below (I) the plant canopies
Intercepted = I0 - I Intercepted radiation potentially available for transpiration and photosynthesis

Hypothetical plant canopy
A randomly placed leaf of area (a) over a ground area (A) probability light intercepted is a/A probability light not intercepted is (1 - a/A) A second randomly placed leaf (same area a) over ground area (A) probability light not intercepted is (1 - a/A)2 So, N randomly placed leaves (all having area a) over ground area (A) probability light not intercepted is (1 - a/A)N

For small leaf area a << A,
(1 - a/A)N  exp(-Na/A) exp(-L) where L is the leaf area index (m2 m-2) or the total leaf area in a unit ground area exp(-L) is known as the penetration function But exp(-L) is for horizontal leaves only leaves are in all angles horizontal leaves, maximum light interception more vertical leaves, light interception decreases

Interception of solar radiation depends on the solar direction and leaf angle. Note: a is the area of the leaf shadow on the ground, and aL the area of the leaf (one side)

so we reduce light penetration by
exp(-kdrL), where kdr is a value between 0 and typically less than 1 kdr is known as the canopy extinction coefficient for direct light kdr = 1 means horizontal leaves, kdr < 1 means leaves are not horizontal the smaller the kdr, the smaller the leaf angle 90 horizontal leaf; 0 erect leaf

where a is shadow area on ground; A
where a is shadow area on ground; A* is exposed canopy area (sunlit leaves) Most canopies have random (spherical) leaf distribution leaves are facing all directions equally (like the surface of a sphere)

Extinction coefficient is calculated as the ratio between the area of canopy shadow on the ground and the exposed surface area of the canopy Thus,

Direct light Direct light below canopies: Direct light intercepted:
Attenuation of irradiance through a canopy according to Beer’s law

Scattering incoming scattering = reflected + transmitted reflected
scattering contributes to radiation regime within canopies transmitted (through the leaf) where  is the scattering coefficient; 0.8 for PAR, 0.2 for NIR (near infrared), and 0.5 for total solar radiation (mean of both PAR and NIR). LEAF

Canopy reflection reflected out of canopy incoming into the canopy
where p is the canopy reflection coefficient, with it being equal to 0.04, 0.25 and 0.11 for PAR, NIR and total solar radiation, respectively CANOPY

Diffuse light Canopy extinction coefficient for diffuse solar radiation kdf at leaf area index (L) from 0.01 to 10 (random leaf distribution only). Constant kdf at a given L.

Discontinuous canopies
Beer’s law assumes closed, homogenous canopies When early growth periods or sparse planting, canopies are opened violates Beer’s law Discontinuous canopies violate one of the assumptions of Beer’s law which require a uniform distribution of canopies

For discontinuous canopies, modify Beer’s law by introducing a clump factor :

PAR absorption In photosynthesis modelling, we are interested in the PAR irradiance incident on leaves, rather than PAR intercepted When direct solar beams enter the canopy, a fraction of it will be intercepted by the leaves and be scattered. Thus, the direct solar component within the canopies is segregated into a component that is scattered and the other that remained direct. The other component is the diffuse component

So, within the canopies, there are 3 components
total direct component of PAR, Qp,dr (unintercepted beam plus scattered beam) direct component of the total direct PAR component, Qp,dr,dr (unintercepted beam without scattering) diffuse PAR component, Qp,df

Hence, the scattered component only is:
Average diffuse component within canopies:

Sunlit leaves absorption:
Shaded leaves absorption: where  is the leaf absorption coefficient (0.8 for PAR)

Sunlit and shaded leaves
Recall: where a is shadow area on ground; A* is exposed canopy area (sunlit leaves), so applied to canopies, since we are taking ground area as 1 m2, a is 1 - exp(-kdrL) => fraction of ground covered by shadows Sunlit LAI: Shaded LAI:

W m-2 convert to mol (photons) m-2 s-1
Conversion W m-2 convert to mol (photons) m-2 s-1 1 W m-2 is 4.55 mol (photons) m-2 s-1

Example Determine the sunlit and shaded leaves PAR absorption for a canopy with a spherical leaf distribution and LAI of 3.0. The incident total solar radiation above the canopies is 800 W m-2, with the diffuse and direct solar components comprising 40% and 60% of the total solar radiation, respectively. Solar inclination is 40º.

Solution PAR is typically 50% of total solar radiation
so PAR flux density is half of the total 800 W m-2; that is, 400 W m-2 or 400 x 4.55 = 1820 mol (photons) m-2 s-1. Of this total PAR, diffuse and direct components are 0.4 x 1820 = 728 and 0.6 x 1820 = 1092 mol (photons) m-2 s-1, respectively. The three flux components within the canopies that must be calculated are: the total direct component Qp,dr; the direct component of the total direct component, Qp,dr,dr and the mean diffuse component

Part 4: Plant water uptake and soil evaporation

Rn = H + ET + G + M (all in W m-2)
Energy balance Rn = H + ET + G + M (all in W m-2) Rn = net radiation main energy supplier H = sensible heat density ET = latent heat flux density G = ground heat flux density energy into or out of soil sub-surface M = miscellany energy density small term (usually less than 5% of Rn) often neglected (M = 0) main energy consumers

Latent heat (ET) all energy supplied is to break bonds and phase change water no temperature change energy to convert 1 kg liquid water to vapour is J latent heat of vapourisation of water () ET (kg water m-2 ground s-1) x  (J kg-1 water) gives ET (J m-2 s-1 or W m-2) ET also known as evapotranspiration

Evapotranspiration water loss from both soil and plant
soil = evaporation plant = transpiration often equals plant water uptake so knowing ET, we know water uptake plants can reduce transpiration conditions of water stress, stomata openings are reduced or closed bad in long term, photosynthesis reduced

Potential vs. actual ET PET AET
maximum ET that can occur under current conditions if there was no water stress AET actual ET occurring under current conditions may equal or be less than PET due to water stress

Sensible heat (H) Heat transfer
energy supplied is to raise temperature thus, can be “sensed”; can be measured by thermometer Heat transfer Rn = radiative ET, H and G = non-radiative by conduction and convention only +ve for energy flow from surface to air -ve for energy flow from air to surface

Energy balance for day and night
Day = ground gains heat Night = ground loses heat +ET = evaporation -ET = condensation

K-theory transport Vertical transport of a generic property 

Latent heat transport equation
is the absolute humidity of air, which is defined as the mass of water vapor contained in a given volume of air (kg m-3) The problem with using absolute humidity is that the volume of air is sensitive to changes in both the air temperature and pressure. Absolute humidity changes when the volume changes, even though the mass of water vapor has not changed. For instance, a 1 m3 of air parcel which contains 2 g of water has an absolute humidity of 2 g m-3. But if that air parcel is expanded to double its volume (2 m3), this means the absolute humidity is now 1 g m‑3 even though the air parcel still contains the same weight of water in it (2 g). Given the way absolute humidity is calculated it appears the amount of water in the air parcel has decreased

so express it using air vapour pressure ea:
where KET is the atmospheric transfer coefficient for sensible heat (m2 s‑1);  is the density of moist air (1.209 kg m-3); cp is the specific heat capacity of moist air which is the amount of heat per unit mass of air required to raise its temperature by one Kelvin (1010 J kg‑1 K‑1); and  is known as the psychometric constant, and it has a value of mbar K-1. cp is known as the volumetric heat capacity: the amount of heat required to raise the temperature of a unit volume of air by one K ( J m-3 K-1)

Sensible heat transport equation
where KH is the atmospheric transfer coefficient for sensible heat (m2 s‑1); and cp gives the volumetric heat capacity of air. Multiplication by cp is so that the above equation gives the amount of heat transferred per unit area ground area per unit time (which is the heat flux density).

Penman-Monteith equation
Uses the electrical network analogy to explain heat transfers Penman-Monteith evapotranspiration (potential) model. Key: ET and H are the latent and sensible heat fluxes, respectively; Tr and To are the temperatures for the reference height and canopy, respectively; er and e0 are the vapor pressure at the reference height and canopy, respectively; ra is aerodynamic resistance; rc is the canopy (or soil surface) resistance.

OR which, incidentally, has the same form as Ohm’s law used to describe electrical current flow: Potential difference (V) = Current (I) x Resistance (R). So analogously, where the current (latent heat flux density) is driven by the potential difference between two points (their vapor pressure difference) but is opposed by resistance (the distance between the two points, and the reciprocal of the atmospheric transfer coefficient KET). Recall that the atmospheric transfer coefficient KET is the ease of atmospheric transfer, or its atmospheric “conductance”. So taking its reciprocal 1/KET denotes the opposite: the atmospheric resistance to transfer.

Evaporation from a saturated environment (within stomata):

But T0 as well as e0 are unknown
But T0 as well as e0 are unknown. To eliminate them from calculations, a vapour pressure deficit D is introduced which is defined as the difference (deficit) between the current amount of moisture in the air and the maximum amount of moisture the air can hold (i.e., saturation), or where es(Tr) is the saturated vapor pressure (mbar) at temperature Tr (C).

 is the slope of the saturated vapor pressure curve (mbar K-1)
for small differences between Tr and T0: Relationship between saturated vapor pressure and air temperature

So introducing D and , and after some algebraic manipulations:

Problem with PM equation
assumes either evaporation or transpiration but not both occurring simultaneously not applicable for open canopies (e.g., early growth periods or sparse planting density)

Shuttleworth-Wallace equation
SW equation: extension of the PM equation ET occurs from both soil and plant simultaneously good for early growth periods or for sparse planting densities

Shuttleworth-Wallace evapotranspiration (potential) model
Shuttleworth-Wallace evapotranspiration (potential) model. raa is the aerodynamic resistance between the mean canopy flow and reference height; rsa is the aerodynamic resistance between the soil and mean canopy flow; rca is the bulk boundary layer resistance; rcs and rss are the canopy and soil surface resistance, respectively.

Entire energy balance of the system can be described in 8 equations:
or or

After some algebraic manipulations:

Once we know total ET, we determine its components:
OR

Soil heat flux (G): G is 35% of net radiation reaching the ground, and this fraction varies according to the cosine of the solar inclination 

Aerodynamic resistances
Wind speed decreases exponentially with height, and equals zero at a certain height above ground/surface. How high above the ground/surface the wind speed falls to zero is a measure of how rough the surface is: roughness length (z0) Roughness length (z0) and zero-plane displacement (d) In the presence of objects (e.g., canopies) the roughness length is displaced by d (zero-plane displacement)

Roughness lengths z0 for some surface types (Hansen, 1993)
z0 (m) Grass, closely mowed 0.001 Bare soil, tilled Thick grass, 0.5 m tall 0.09 Forest, level topography Coniferous forest 1.10 Alfalfa 0.03 Potato, 0.6 m tall 0.04 Cotton, 1.3 m tall 1.30 Citrus orchard

For crops with crop height h:
Wind speed above canopy: Wind speed below canopy: k is the von Karman constant (0.40) u* is the friction velocity (m s-1): the effectiveness of air turbulence transfer  is the wind speed attenuation coefficient (unitless)

Wind attenuation coefficients α for some vegetation types (Cionco, 1972)
Immature corn 2.8 Sunflower 1.3 Oats Pine trees 2.4 Wheat 2.5 Larch trees 1.0 Corn 2.0 Citrus orchard 0.4

h = crop height (m); zs0 = roughness length of soil surface (m); z0 = roughness length of crop (m); d = zero-plane displacement (m); zr = reference height (m); n = eddy diffusivity coefficient (unitless, n=2 to 3) K(h) = eddy diffusivity transfer coefficient (m2 s-1) = ku*h

Boundary layer resistance
Every surface has a thin boundary layer of still air thicker the layer, the more resistance to transfer of heat or vapour flow can be laminar, turbulent or mixed when turbulence is suppressed, transfer occurs solely due to molecular diffusion (very slow)

where L = leaf area index; u(h) = wind speed at canopy top of height h;
w = mean leaf width (m);  = wind speed attenuation coefficient Equation includes turbulent effects

Stomatal resistance Canopy resistance is:
Stomatal resistance decreases with increasing PAR (photosynthetically active radiation) irradiance, following the relationship by Jarvis (1976) where Lcr is critical LAI, typically taken as maximum LAI (=4)

Soil surface resistance
dry layer thickness, l water water wet layer assume that a soil is always made up of two layers: a thin upper layer that is completely dry, and a thicker, lower layer that is wet. water vapor traverses from the lower wet layer through the upper dry layer to reach the soil-atmosphere boundary. This traversal by vapour through the dry layer is by molecular diffusion. vapor flux in the soil is controlled by four factors: vertical vapor pressure gradient between the dry and wet soil layers molecular diffusion coefficient of vapor in the soil soil porosity (fraction of soil that is made up of pores) soil tortuosity (ratio of the actual path length to the straight path length of flow) (  1)

rss rssdry v / v,sat Dm,v is the molecular diffusion coefficient of vapor in the soil (24.7 x 10‑6 m2 s‑1);  is the tortuosity for soils (2); l is the upper dry layer depth (0.15 m); p is the soil porosity; and  is the pore size distribution index

Pore size distribution index is the slope of the linear line of relative saturation
to soil suction ln (v / v,sat)  ln e higher the suction, drier the soil and smaller the relative saturation  

Conversion W m-2 to mm (water) day-1 (ET / ) x (60 x 60 x 24)
e.g., (120 / ) x = 4.2 mm day-1 1 mm water is equivalent to 1 kg or 1 liter of water in a 1-m2 ground area

Part 5: Water balance

Expressions of soil water content
usually expressed as depth of water (mm) or volumetric water content (m3 m-3) depth of water (mm): 1 mm water depth is equivalent to 1 kg m‑2 ground area or 1 liter m-2 ground area

Volumetric water content is the volume of water per unit volume of soil:

R = rainfall; I = irrigation; CR = capillary rise
Water balance R + I + CR = P + OF + ETa +  (all in mm day-1) WATER INPUT: R = rainfall; I = irrigation; CR = capillary rise WATER OUTPUT: P = percolation; OF = overland flow; ETa = actual evapotranspiration;  = change in soil water content

equation looks deceptively simple, but in practice, the individual components can be difficult to determine/measure use some assumptions: no irrigation supplied, so I = 0 deep water table (> 1 m deep), so CR = 0 flat, levelled land, so OF = 0 therefore water balance equation becomes: R = P + ETa +  or  = R - P - ETa

Two-layered soil Downward flow of water out of the top layer (i=1) is denoted by P1,t, and this component subsequently becomes the percolation of water into the root zone below (i=2). In other words, the infiltration of water into the second layer is P1,t which is the percolation of water from the top layer. Within the root zone, water will still flow downward, and any water leaving this zone is denoted by the component P2,t which is regarded as deep percolation or drainage

Percolation drainage (loss) of water from a soil layer/zone
consists of two components: percolation due to excess water pe percolation due to redistribution pd

Excess water percolates below if the amount of water in soil and amount of water (due to rainfall R) received exceed the soil saturation level: pour 150 ml pe = – 100 = 90 ml amount overflow, pe? initial amount = 40 ml max can hold = 100 ml

Redistribution occurs due to gravity and matric potentials, as defined by Darcy’s Law
gravity potential / energy flow due to gravity (downward) matric potential / energy flow due to differences in water content (wet to dry) Darcy’s Law flow is proportional to differences in potential (or head) and inversely proportional to distance

Flow, q  H/L or q = K H/L where L is distance (m); H is potential difference (m); and K is hydraulic conductivity (m day-1) H is total head which is the sum of matric and gravity heads flow is faster if the difference in potentials is larger, or the distance to flow is smaller

If the depth difference between two soil layers is z, then Hg = z, and
Assuming uniformly wetted soil means no differences in matric potential any where in that soil layer, so which gives: Thus, flow is controlled only by the soil’s K

K depends on soil texture, soil structure and soil water content
K increases with increasing water content until maximum at soil saturation where  is for most soils

Soil texture Ksat (m day-1) v,sat (m3 m-3)
Sand Loamy sand Sandy loam Silty loam Loam Sandy clay loam Clay loam Sandy clay Silty clay loam Clay Silty clay Silt

Law of mass conservation
q/z is the change of water flux density q over the vertical distance z. If q/z increases then this is the same as saying that qin < qout, and that water storage in the volume element must decrease because more water is lost from outflow qout than that gained by inflow qin. Stated more specifically: the rate of increase of q with z must equal the rate of decrease of volumetric water content v with time t.

If we take the soil layer thickness as L, then
Earlier, we established q = K, so re-arranging:

So at time t2, the volumetric water content is:
Therefore, percolation due to redistribution is t2 - t1 = R – (pe + pd) pd = t2 - t1 - R - pe t2 is now available for evapotranspiration ETa

Actual ET When water is limiting, evapotranspiration is not at maximum but is reduced to a rate known as actual ET PET is scaled down to AET by a reduction factor dependent on the amount of water in the soil

Actual soil evaporation
Potential soil evaporation is reduced to actual evaporation by a reduction factor that is dependent on the relative water content

Actual transpiration all transpiration is from water in the
second soil layer only Potential plant transpiration is reduced to actual transpiration by a reduction factor that is dependent on the relative water content

Plant cannot use the water below the soil wilting point level
Most agricultural crops are C3 plants; only three are C4: sugar cane, maize and sorghum C3 plants photosynthesize to produce a 3-C compound (3-phosphoglyceric acid) and C4 a 4-carbon compound (oxaloacetic acid). C4 are more efficient in using water and solar radiation to convert into biomass. Critical water point for C3 and C4 plants are 50% and 30% of relative water content, respectively. C4 more efficient in using water.

Photosynthesis (C3)

Empirical approaches de Wit (1965): Goudriaan and van Laar (1978):
Gross photosynthesis as a function of absorbed PAR, determined using the equations by de Wit (1965), and Goudriaan and van Laar (1978)

n,canopy is the net canopy assimilation rate of CO2; e is the RUE; It,d is the total solar radiation incident on the canopy; and f is the fraction of It,d intercepted by the canopy. RUE for C3 and C4 plants are typically measured at about 1.9 and 2.5 g MJ-1 of intercepted PAR Total dry matter as a function of cumulative total solar radiation intercepted (Monteith, 1977)

Mechanistic approaches
Today, increasingly more mechanistic photosynthesis models are being used. Over the years numerous mathematical models have been developed that take into account the underlying mechanism of photosynthesis such as the diffusion of CO2 into the chloroplast, enzyme kinetics, and the biochemical reactions in the carbon-reduction cycle. Two most significant work to build upon this progress are by Farquhar et al. (1980) and Collatz et al. (1991)

Light and dark reactions
Photosynthesis equation: Photosynthesis involves two chemical reaction steps: light and dark reactions. The light reactions involve the absorption of sunlight by chlorophyll and carotenoids to produce the energy carriers, NADPH and ATP, via a process called photophosphorylation occur in the thylakoid membranes inside the chloroplast, and NADPH and ATP are subsequently transferred from there to the stroma for use in the dark reactions

Term “dark reactions” is misleading because these reactions do not occur in the dark
so-called only because they do not require light for their reactions. Dark reactions are the biochemical reduction of CO2 to carbohydrates using the high energy carriers NADPH and ATP. occur in the stroma inside the chloroplast. Depending on the plant type, there are three pathways of dark reactions: C3, C4 and CAM (Crassulacean acid metabolism)

Photosynthesis occurs in the chloroplast, an organelle in the mesophyll cells

The C3 pathway converts atmospheric carbon into a chemical compound with three carbon atoms (3-phosphoglyceric acid or PGA), The C4 pathway produces a compound with four carbon atoms (oxaloacetic acid) The CAM pathway so-called after the plant family in which it was first found (Crassulaceae) CO2 is stored in the form of an acid (malic acid) before use in photosynthesis. Unlike C3 and C4 plants, CAM plants open their stomata at night and close them during the day. Most agricultural crops are C3 types, whereas sugar cane, maize and sorghum are C4 types. CAM plants include pineapple and dragon fruit (pitaya), and many succulents such as cacti.

Calvin cycle. Shaded area denotes the carboxylation cycle.

The major processes and their equations in the Calvin cycle
a) Carboxylation of RuBP: RuBP + CO2 + H2O  2PGA b) Regeneration of RuBP: c) Oxygenation of RuBP: RuBP + O2  PGA + PGIA d) Regeneration of PGA: PGIA O2  0.5PGA + 0.5CO2

Cycle begins by the enzyme Rubisco (ribulose biphosphate-carboxylase/oxygenase) fixing one mol of CO2 to one mol of RuBP (ribulose 1,5-biphosphate) to produce two mol of PGA. This step is the start of the carboxylation cycle. Regeneration of RuBP requires two mol of PGA and energy from NADPH and ATP to yield one mol of RuBP and one mol each of a carbohydrate compound and oxygen.

Rubisco is an enzyme that catalyzes both the fixation of CO2 and O2 with RuBP.
CO2 competes with O2 for RuBP. The fixation of O2 to RuBP is known as photorespiration (or oxygenation), and it is regarded as a wasteful process because about 10% of assimilated carbon is lost. In photorespiration, Rubisco fixes one mol of oxygen with one mol of RuBP to give one mol each of PGA and PGIA (2-phosphoglycolate). PGA is regenerated when one mol of PGIA reacts with a quarter mol of oxygen to yield half mol each of PGA and CO2. So the final result of oxygenation is the release of CO2 which is a loss of assimilated carbon in making carbohydrates.

Carboxylation and oxygenation are highly temperature dependent.
Moreover, the ratio between carboxylation and oxygenation depends primarily on concentrations of CO2 and O2 at the carboxylation site and temperature. With increasing temperature, for example, the solubility of CO2 relative to O2 decreases; thus, this favors oxygenation and results in a greater loss of assimilated CO2. Oxygenation is also favored over carboxylation in high solar irradiance.

Carbon losses, however, are not solely due to photorespiration, but also due to dark respiration.
occurs all the time regardless of the presence or absence of light. This is in contrast to photorespiration which occurs only in light. occurs in the cell organelle mitochondria it is a process whereby energy is released from the oxidation of organic compounds, and this energy is used by the plants for their living maintenance (upkeep of cell activities), and growth (synthesis of structural compounds). Both photorespiration and dark respiration are components of plant respiration, and both result in losses of assimilated carbon

Enzyme kinetics: Michaelis-Menten equation
Kc is defined as the substrate concentration at half the maximum velocity

Competitive inhibition
and In competition with O, velocity of C reaction is

Rubisco-limited In plants, there is a minimum amount of internal CO2 below which there is no CO2 assimilation. This critical or minimum amount of CO2 is known as the CO2 compensation point (*). applied to photosynthesis: vc is the carboxylation velocity (per unit leaf area) (mol m-2 s-1); Vc,max is the maximum carboxylation velocity (per unit leaf area) (mol m-2 s‑1); Ci is the concentration of internal (intercellular) CO2 in air (mol mol‑1); Oa is the concentration of ambient O2 in air (mol mol‑1); and Kc and Ko are the Michaelis-Menten constants for CO2 and O2, respectively (mol mol‑1) * is the CO2 compensation point (mol mol‑1) so that when Ci = *, vc = 0

Kc(25) Michaelis-Menten constant for CO2 (300 mol mol-1)
Ko(25) Michaelis-Menten constant for O2 ( mol mol-1) (25) CO2 / O2 specificity factor (2600 mol mol-1) Vc,max(25) Rubisco capacity rate (per unit leaf area) (200 mol m-2 s-1) em Quantum efficiency (0.06 mol mol-1) α Leaf absorption for PAR (0.8) Oa Ambient concentration of O2 in air ( mol mol-1) Ci Intercellular CO2 concentration in air (C3 plant) (245 mol mol-1)  reflects the ability of Rubisco to discriminate between CO2 and O2. The higher the value, the better the discrimination and the higher productivity for CO2.

Sensitivity to temperature
Kc, Ko, , and Vc,max are highly sensitive to temperature. It is assumed that these parameters depend on temperature in the same way as the activity of enzyme depends on temperature. The reaction rates of enzyme typically increase by a factor of 2 for every 10 C increase in temperature, or where  and (25) are the enzyme reaction rates at temperatures T and 25 C, respectively. To generalize where (25) is the model parameter value at 25 ºC (Kc(25), Ko(25), (25) or Vc,max(25)); Tf is the actual leaf temperature (ºC); and Q10 is the relative change in the parameter  for every 10 ºC change.

Q10 values Kc(25) Michaelis-Menten constant for CO2 (2.1) Ko(25) Michaelis-Menten constant for O2 (1.2) (25) CO2 / O2 specificity factor (0.57) Vc,max(25) Rubisco capacity rate (per unit leaf area) (2.4)

Additionally, Vc,max is sensitive to inhibition at temperatures exceeding 35 ºC. Vc,max needs a high temperature cutoff, after which it decreases rapidly. Studies have shown that the critical temperature value for Vc,max is actually at about 40 ºC. The high temperature responses of Vc,max is determined empirically by

Photosynthesis can be limited by amount of light and amount of sink (e
Photosynthesis can be limited by amount of light and amount of sink (e.g., sucrose) already produced light can be limited, so photosynthesis can be limited by light too much sink may have already been produced, so photosynthesis becomes limited by the sink

Light-limited Qt is the PAR flux density based on per unit leaf area, and not per unit ground area. Additionally, Qt is expressed in the unit mol (photons) m-2 s-1, rather than the usual unit W m-2 for solar radiant energy em is the intrinsic quantum efficiency (also known as quantum yield) for CO2 uptake (mol mol-1), which is the number of moles of CO2 fixed per unit mole of absorbed PAR C4 plants = mol mol-1 C3 plants:

Sink limited vs is the sink-limited rate of CO2 assimilation (per unit leaf area) (mol m-2 s-1)

Gross assimilation rate
What we have now is three equations to calculate the gross CO2 assimilation rate. Which equation to select depends on which factor (Rubisco, light or sink) that is most limiting to gross assimilation: where MIN() denotes the minimum of the enclosed values.

Photosynthesis response to temperature at three levels of PAR flux densities (Qp, mol m-2 s-1)

Photosynthesis response to PAR at three levels of leaf temperature

Photosynthesis response to CO2 concentration at three levels of PAR flux densities (Qp, mol m-2 s-1). Leaf temperature is fixed at 30 ºC.

Leaf temperature Tf Solve energy balance first to obtain total sensible heat (H): Determine the sensible heat for crop (Hc), then calculate Tf:

Scaling up to canopy photosynthesis
Divide the canopy into sunlit and shaded leaf classes and calculate the assimilation rate for each leaf class. The gross assimilation rate for each leaf class is then summed based on the fraction of leaf area for each class:

Example Determine the gross canopy photosynthetic rate at 20 ºC (leaf temperature) for a canopy with a spherical leaf distribution and LAI of 3.0. The incident total solar radiation above the canopies is 800 W m-2, with the diffuse and direct solar components comprising 40% and 60% of the total solar radiation, respectively. Solar inclination is 40º.

Solution PAR is typically 50% of total solar radiation
so PAR flux density is half of the total 800 W m-2; that is, 400 W m-2 or 400 x 4.55 = 1820 mol (photons) m-2 s-1. Of this total PAR, diffuse and direct components are 0.4 x 1820 = 728 and 0.6 x 1820 = 1092 mol (photons) m-2 s-1, respectively. The three flux components within the canopies that must be calculated are: the total direct component Qp,dr; the direct component of the total direct component, Qp,dr,dr and the mean diffuse component

Determine the gross leaf assimilation rate for the sunlit and shaded leaves that have absorbed Qsl = and Qsh = mol m‑2 s-1, respectively. Referring to the chart, sl = 27.8 mol m‑2 s-1 and sh = 8.7 mol m‑2 s-1. So, gross canopy assimilation rate of CO2 (per unit ground area) is: Homework: Use the equations as shown earlier to determine sl and sh.

CO2 to carbohydrates Every gram of CO2 is equal to 30/44 g of CH2O. This is because one mol of CH2O and CO2 are equivalent to their molecular weights of 30 and 44 g of CH2O and CO2, respectively. Likewise, one mol of CO2 is 44  10-6 g of CO2 So this translates to every one mol CO2 m-2 day-1 being equivalent to 30  10‑6 g CH2O m-2 day-1.

Part 7: Respiration

Plant respiration is separated into two components: photorespiration and dark respiration
Photorespiration is the oxygenase reaction between O2 and Rubisco, resulting in the loss of assimilated carbon Dark respiration is the oxidation of carbohydrates (plant food) to release energy, which is used for living maintenance and growth

Conceptual model of plant respiration

A maintenance tax (RM) is subtracted from the total substrate supply (i.e., gross photosynthesis, canopy) leftover (canopy - RM) is used for tissue production with a growth efficiency Yg Yg is the plant’s ability to convert substrate into new plant structures. the plant’s ability to convert substrate into new plant structures. What is actually used for the production of new plant materials is Yg(canopy - RM), and the leftover RG = (1-Yg)(canopy - RM) is lost via growth respiration

Maintenance respiration
Maintenance respiration is required to sustain living tissues. Even the maintenance of electrical potentials across the cell membranes requires energy. Maintenance respiration varies widely, depending on the plant species, and even differs among different parts of the same plant

R’M is the maintenance respiration rate (g CH2O m-2 ground area day-1); kM is the maintenance respiration coefficient (g CH2O g-1 dry matter day-1); and W is the plant weight (g dry matter m-2 ground area). Maintenance respiration is taken to be proportional to the plant weight W Maintenance respiration coefficient (at 25 C air temperature) for various plant parts Plant part g CH2O g-1 dry matter day-1 green leaves, kM,greenleaves 0.030 stem, kM,stem 0.015 roots, kM,roots storage organs, kM,storage 0.010

Maintenance respiration rates, however, are highly dependent on temperature. It is assumed that the maintenance respiration rates depend on temperature in the same way as the enzyme activities depend on temperature, where the Q10 value for maintenance respiration is taken as 2. Maintenance rates additionally need to be corrected for plant age. As the plant ages, its protein content decreases but the amount of stable components such as support tissues and reserve compounds increases. This, in turn, decreases the maintenance respiration requirement. Wleaves is the weight of both green and dead leaves (g dry matter m-2 ground area). As the plant ages, the proportion of dead leaves to green leaves increases (due to leaf death). This lowers the fraction of green leaves to total leaves (Wgreenleaves/Wleaves) and decreases the maintenance respiration rate accordingly.

Growth respiration The amount of glucose required to synthesize a new material depends on the chemical make up and its amount in the material. It has been found that this production process, unlike plant maintenance, is independent of environmental conditions, and dependent only on the nature of the plant component formed.

Carbon content (fraction)
Glucose requirement G and the carbon content for major biochemical groups in plant tissues Biochemical group G (g CH2O g-1 dry matter) Carbon content (fraction) Carbohydrates 1.242 0.450 Proteins 2.700 0.532 Lipids 3.106 0.773 Lignin 2.174 0.690 Organic acids 0.929 0.375 Minerals 0.050 0.000

Fractions of major biochemical groups in several plant parts, and the calculated glucose requirement for the production of the plant parts Group Young leaf Wheat seed Broad bean Oil-rich seed Wood stem Sugar beet roots Carbohydrates 0.53 0.79 0.54 0.15 0.49 0.78 Proteins 0.25 0.12 0.33 0.30 0.02 0.05 Lipids 0.01 0.48 0.00 Lignin 0.03 0.04 0.38 Organic acids 0.06 Minerals G 1.656 1.452 1.719 2.572 1.569 1.271

Total glucose requirement for growth:
G is the total glucose requirement (g CH2O g-1 dry matter); Gi is the glucose requirement for each plant part i (i.e., green leaves, stem, roots and storage organs) (g CH2O g-1 dry matter); and Fi is the fraction of dry matter of the individual plant parts Shortcut: nitrogen content is equal to the protein content multiplied by 0.16 (the average nitrogen N content of proteins is about 16%): only need to analyse total C and N content – faster, cheaper and simpler

Since (canopy – RM) is the amount of assimilates potentially available for growth (expressed in g CH2O m-2 day-1), (canopy - RM) / G is the total weight of dry matter actually produced in a unit ground area per day (expressed as g dry matter m-2 day-1). Thus, the weight of a plant part is incremented by where Wi,t and Wi,t+1 are the weights (g m-2 ground area) of a given plant part i (e.g., green leaves, stem, roots or storage organs) at the current time step t and next time step t+1, respectively; t is the interval for each time step (days); and Fi is the fraction of dry matter of plant part i

Typical glucose requirement for the synthesis of various plant parts
g CH2O g-1 dry matter green leaves, Ggreenleaves 1.463 stem, Gstem 1.513 roots, Groots 1.444 storage organs, Gstorage 1.415* * highly variable

Growth development stage
Phenology is the study of the timing of life cycle events, and how they respond to their environment, in particular to weather The current phase in the growth of a plant is known as the growth development stage (s), and it is defined by its physiological age and the formation of its various organs and their appearance

The growth development of a plant is punctuated by several milestones or points of significance such as the point of seed emergence, flowering, tuber initiation, bulking and maturity, ripening, plant maturity, and senescence Very often, the most important point in the growth development stage is the switch from vegetative to reproductive stage

Adopt a one-dimensional and irreversible scale to denote the growth development stage
no “standard” to follow – up to us to decide seed emergence (s = 0), flowering (s = 1), and plant maturity (s = 2)

Scale for the growth development stage for barley and wheat (as defined by the Agrometeorological Centre of Excellence, Canada) Scale Description Planting 1 Emergence (more than half the plants are visible) 2 Jointing (earliest date for 1st internode elongation, appearance of first leaf) 3 Heading (base of head reaches the same height as the base of the short blade) 4 Soft dough (kernel deforms easily but no “milk” exudes) 5 Ripe (kernels can no longer be deformed)

Growth development rate
The progress rate of plant growth is known as the growth development rate (r; day-1) The rate at which the growth development stage advances The higher the rate, the earlier the next milestone in the development stage is reached Temperature is often the most important factor that determines the growth development rate Other factors: vernalization and day length, depending on the plant species

Growth development rate is not possible to measure directly
Determined indirectly by measuring the duration (in days) between two growth development stages setup an experiment in a controlled environment where the air temperature is set constant at a certain value, and the time it takes for a plant to reach a certain growth development stage is recorded then repeat experiment a few more times but using a different constant air temperature for each experiment

Dependence of growth duration on air temperature
Dependence of growth development rate on air temperature Tb = base temperature, below which there is no growth (note: no –ve growth)

Base temperature for some crops
Base temperature (ºC) Field pea, lentils, linseed, oats, spinach 1-2 Barley, rape, wheat 3 Lettuce 4 Asparagus, peas 3-6 Canola and forages 5 General plant growth Potatoes 6-7 Safflower, sunflower 7-8 Beans, cucumbers, maize, soybean 10 Millets 8-14 Cowpea, sorghum 11 Pumpkins, tomatoes 10-13 Castor, peanut, pigeon pea 13 Guar 15 Sesame 16 Melons 15-18

For a given air temperature T, the growth development rate r (day-1) can be determined by
where s,i and s,i+1 are the growth development at stage i and i+1, respectively; and tT is the time period (days) between s,i and s,i+1 at constant growing air temperature T ºC. But if we keep every successive milestone in the growth development stage one unit apart, then

where the subscripts t and t+1 represent the time step at t and t+1, respectively; t represents the time interval between two time steps (days); and r t gives the advance in growth stage that had occurred from the time t to t+1.

Leaf area index growth The increase in leaf area index is determined from the current weight of green leaves and specific leaf area (SLA) SLA is the ratio of leaf area to leaf weight, and it is an important indicator of how much biomass is allocated to the expansion of leaf area. The leaf area index for the next time step t+1is determined by SLA is typically m2 g-1

Leaf death There are two possible reasons for leaf death: 1) leaf age, and 2) self-shading of leaves. Leaf death due to age typically occurs after flowering (post-anthesis): where Dage is the leaf death rate due to leaf age (day-1). Dage is proportional to the growth development rate as well as to leaf age. With increasing leaf age, s approaches 2 (reaching maturity) so that 2- s becomes increasingly small, so that leaf death rate becomes increasingly large. A minimum of 0.1, however, is set as the difference between 2 and s to avoid any excessive leaf death rates at the later growing stages.

Leaf death is also due to self-shading where the shading from the upper parts of the canopy diminishes the solar irradiance within the lower plant canopy; thus, causing leaf deaths in the lower canopy parts. A critical LAI (leaf area index) of 4.0 is typically chosen as the point where self-shading becomes a significant effect. Leaf death rate due to self-shading is determined by where Dshade is the leaf death rate due to leaf self-shading (day-1); L and Lcr are the LAI and critical LAI (m2 m-2), respectively; and MIN[] is the minimum of the enclosed values. Here, it is assumed that leaf death rate due to self-shading begins after LAI exceeds a critical value (typically Lcr = 4.0), after which death rate increases linearly with increasing LAI until a maximum value of 0.03 day-1. This maximum value is set to avoid any excessive leaf death rates at the later growing stages

The actual death rate of leaves Dleaves (day-1) is then the larger of the two rates Dage and Dshade:
where MAX() the maximum of the enclosed values. The weight of dead leaves can then be calculated as where Wdeadleaves,t+1 and Wdeadleaves,t are the weights of dead leaves for the next time step t+1 and current time step t; and (Wgreenleaves Dleaves t) gives the weight of green leaves that have died within the time period between t and t+1.

The weight of green leaves is calculated differently from that for stem, roots and storage organs because the death rate of green leaves must be subtracted from the growth rate of green leaves.

Plant height growth where Tts is the accumulated temperature sum (ºC day); ht is the current height (m) at time t; hm is the maximum possible height of the plant (m); and b0 and b1 are the intercept (unitless) and slope (ºC-1 day-1) coefficients, respectively Typical plant height growth following the logistic function

Temperature sum A plant must accumulate a certain number of temperature sum (or heat units) to advance to the next milestone in the development stage Unit function: Increment temperature sum only when mean temperature is greater than base temperature

Example An example showing the daily temperature sum Tts and accumulated temperature sum Tts (base temperature Tb is 5 ºC) Day 1 2 3 4 5 6 7 8 9 Tavg 10 12 Tts Tts 15 20 23 26 Tavg = average of Tmax and Tmin

Root elongation Roots grow to a certain maximum depth, provided that they are not limited by soil conditions such as a compacted layer The maximum depth depends very much on the plant species and ranges from 0.5 to 1.5 m or more Root growth generally stops at about the flowering stage, and root elongation rate is surprisingly quite independent of root weight

where dr,t and dr,t+1 are the rooting depth (m) at time step t and t+1, respectively; dm is the maximum rooting depth (m); dg is a constant root elongation rate, denoting rooting depth increase per day (m day-1); t is the time step interval (days); s is the growth development stage; and v,t is the volumetric soil water content at time t (m3 m-3); and v,wp is volumetric soil water content at permenant wilting point (m3 m-3). Assume that root elongation growth only occurs before flowering (pre-anthesis or s < 1) and when the soil water content exceeds the permenant wilting point (i.e., v,t > v,wp). No growth occurs if these two conditions are violated (i.e., dr,t+1 = dr,t). Moreover, root elongation cannot exceed the maximum possible root depth (dm).

Water stress effects When water supply is insufficient, plant growth is additionally limited to water. First: we need to determine the level of water stress being experienced. This is equivalent to the ratio between actual and potential plant transpiration: Then reduce plant height growth rate, root elongation rate and gross assimilates produced:

ERT 355: Biosystems Modelling and Simulation

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