Canopy Cover and Height

$fr_{LAImx}=\frac{fr_{PHU}}{fr_{PHU}+exp(\Box_1 - \Box_2 * fr_{PHU})}$ 5:2.1.10

where $fr_{LAImx}$ is the fraction of the plant’s maximum leaf area index corresponding to a given fraction of potential heat units for the plant, $fr_{PHU}$ is the fraction of potential heat units accumulated for the plant on a given day in the growing season, and $\Box_1$and $\Box_2$ are shape coefficients. The fraction of potential heat units accumulated by a given date is calculated:

$fr_{PHU}=\frac{\sum_{i=1}^d HU_i}{PHU}$ 5:2.1.11

where $fr_{PHU}$ is the fraction of potential heat units accumulated for the plant on day d in the growing season, $HU$ is the heat units accumulated on day $i$ (heat units), and $PHU$ is the total potential heat units for the plant (heat units).

The shape coefficients are calculated by solving equation 5:2.1.10 using two known points ($fr_{LAI,1}$,$fr_{PHU,1}$) and ($fr_{LAI,2}$,$fr_{PHU,2}$):

$\Box_1=1n[\frac{fr_{PHU,1}}{fr_{LAI,1}}-fr_{PHU,1}]+\Box_2*fr_{PHU,1}$ 5:2.1.12

$\Box_2=\frac{(1n[\frac{fr_{PHU,1}}{fr_{LAI,1}}-fr_{PHU,1}]-1n[\frac{fr_{PHU,2}}{fr_{LAI,2}}-fr_{PHU,2}])}{fr_{PHU,2}-fr_{PHU,1}}$ 5:2.1.13

where $\Box_1$ is the first shape coefficient, $\Box_2$ is the second shape coefficient, $fr_{PHU,1}$ is the fraction of the growing season (i.e. fraction of total potential heat units) corresponding to the 1st point on the optimal leaf area development curve, $fr_{LAI,1}$ is the fraction of the maximum plant leaf area index (i.e. fraction of $LAI_{mx}$) corresponding to the 1st point on the optimal leaf area development curve, $fr_{PHU,2}$ is the fraction of the growing season corresponding to the 2nd point on the optimal leaf area development curve, and $fr_{LAI,2}$ is the fraction of the maximum plant leaf area index corresponding to the 2nd point on the optimal leaf area development curve.

$h_c=h_{c,mx}*\sqrt{fr_{LAImx}}$ 5:2.1.14

where $h_c$ is the canopy height for a given day (m), $h_{c,mx}$ is the plant’s maximum canopy height (m), and $fr_{LAI,mx}$ is the fraction of the plant’s maximum leaf area index corresponding to a given fraction of potential heat units for the plant. As can be seen from Figure 5:2-1, once the maximum canopy height is reached, $h_c$ will remain constant until the plant is killed.

$h_c=h_{c,mx}*(\frac{yr_{cur}}{yr_{fulldev}})$ 5:2.1.15

where $h_c$ is the canopy height for a given day (m), $h_{c,mx}$ is the plant’s maximum canopy height (m), $yr_{cur}$ is the age of the tree (years), and $yr_{fulldev}$ is the number of years for the tree species to reach full development (years).

The amount of canopy cover is expressed as the leaf area index. For annuals and perennials, the leaf area added on day $i$ is calculated:

$\Delta LAI_i=(fr_{LAImx,i}-fr_{LAImx,i-1})*LAI_{mx}*(1-exp(5*(LAI_{i-1}-LAI_{mx})))$

while for trees, the leaf area added on day $i$ is calculated:

$\Delta LAI_i=(fr_{LAImx,i}-fr_{LAImx,i-1})*(\frac{yr_{cur}}{yr_{fulldev}})*LAI_{mx}*(1-exp(5*(LAI_{i-1}-(\frac{yr_{cur}}{yr_{fulldev}})*LAI_{mx})))$

$LAI_i=LAI_{i-1}+\Delta LAI_{i}$ 5:2.1.18

where $\Delta LAI_i$ is the leaf area added on day $i$, $LAI_i$ and $LAI_{i-1}$ are the leaf area indices for day $i$ and $i-1$ respectively, $fr_{LAImx,i}$ and $fr_{LAImx,i-1}$ are the fraction of the plant’s maximum leaf area index calculated with equation 5:2.1.10 for day $i$ and $i-1$,$LAI_{mx}$ is the maximum leaf area index for the plant, $yr_{cur}$ is the age of the tree (years), and $yr_{fulldev}$ is the number of years for the tree species to reach full development (years).

Leaf area index is defined as the area of green leaf per unit area of land (Watson, 1947). As shown in Figure 5:2-2, once the maximum leaf area index is reached, $LAI$ will remain constant until leaf senescence begins to exceed leaf growth. Once leaf senescence becomes the dominant growth process, the leaf area index for annuals and perrenials is calculated:

$LAI=LAI_{mx}*\frac{(1-fr_{PHU})}{(1-fr_{PHU,sen})}$ $fr_{PHU}>fr_{PHU,sen}$ 5:2.1.19

$LAI=(\frac{yr_{cur}}{yr_{fulldev}})*LAI_{mx}*\frac{(1-fr_{PHU})}{(1-fr_{PHU,sen})}$ $fr_{PHU}>fr_{PHU,sen}$ 5:2.1.20

where $LAI$ is the leaf area index for a given day, $LAI_{mx}$ is the maximum leaf area index, $fr_{PHU}$ is the fraction of potential heat units accumulated for the plant on a given day in the growing season, $fr_{PHU,sen}$ is the fraction of growing season ($PHU$) at which senescence becomes the dominant growth process, $yr_{cur}$ is the number of years of development the tree has accrued (years), and $yr_{fulldev}$ is the number of years for the tree species to reach full development (years).