**2. Below-ground processes**

Water uptake from the soil by roots is determined by soil water content and other related soil physical properties, the root architecture system and soil and root resistances to water flow as well as demand from the canopy. As more fine-scale detailed data on the physiology of water absorption become available, the integration of this information into a quantitative framework from the root to the soil layer level can be established by coupling of the soil water dynamics and root architecture. Finally, water flow resistance from soil to root as well as for radial and axial resistances in the roots can be calculated for the entire root system.

The amount of water that a tree can remove from the soil depends on the volume of soil exploited by the root system of the tree (Landsberg and McMurtrie, 1984). Water absorption by roots from the soil is determined by three main factors:


Although the water movement in soil is well represented by microscopic or macroscopic approaches, a detailed description of the root system is lacking (Molz, 1981). Water movement through plant roots has been theoretically analyzed by Landsberg and Fowkes (1978) who derived an analytical solution for a single root with a constant hydraulic resistance. Alm *et al*. (1992) extended Landsberg and Fowkes' approach to a case of

a great diversity of models of soil-tree-atmosphere transfers in which the processes are partly or fully integrated. However, these models all share a common characteristic: they infer canopy functions from leaf observations. Based on how the inference is achieved, models of soil-tree-atmosphere transfers can be divided into four groups: the big-leaf models, direct scaling models, multi-layer canopy models and multi-layer soil-canopy models. Multi-layer models can be further divided into two types: incomplete multi-layer models and complete multi-layer models. This chapter describes these methods and

To overcome shortcomings of previous multi-layer models, accurate descriptions of basic biophysical processes are given first priority in the modeling process. Complex models, whose wide-range application is often hampered by the lack of specific data, should have their processes simplified in order to be accommodated into spatial frameworks where appropriate. Biophysical processes within simple models should consider new data sources and understanding to gain more accurate predictions (Ranatunga et al., 2008). Simplicity can be pursued only when the reality of basic biophysical processes is not compromised. The process of water movement through the soil-plant-atmosphere continuum has been widely and sucessfully used (Molz, 1981; Jarvis *et al*., 1981; Boyer, 1985; Eckersten, 1991; Ranatunga and Murty, 1992; Nobel and Alm, 1993; Cienciala *et al*., 1994; Friend, 1995; Williams *et al*., 1996). The integrated modeling process is designed to cover both below- and above-ground processes, which can be combined through a methodology that links the soil-water status with the atmospheric weather conditions through canopy exchangeable water storages and

Water uptake from the soil by roots is determined by soil water content and other related soil physical properties, the root architecture system and soil and root resistances to water flow as well as demand from the canopy. As more fine-scale detailed data on the physiology of water absorption become available, the integration of this information into a quantitative framework from the root to the soil layer level can be established by coupling of the soil water dynamics and root architecture. Finally, water flow resistance from soil to root as well as for radial and axial resistances in the roots can be calculated for the entire root system.

The amount of water that a tree can remove from the soil depends on the volume of soil exploited by the root system of the tree (Landsberg and McMurtrie, 1984). Water absorption

1. soil properties such as soil water content, hydraulic conductivity and water potential,

3. the absorption capacity of roots is dependent on water flow resistances, i.e., the soil and

Although the water movement in soil is well represented by microscopic or macroscopic approaches, a detailed description of the root system is lacking (Molz, 1981). Water movement through plant roots has been theoretically analyzed by Landsberg and Fowkes (1978) who derived an analytical solution for a single root with a constant hydraulic resistance. Alm *et al*. (1992) extended Landsberg and Fowkes' approach to a case of

presents the background of the soil-tree-atmospheric water transfer.

estimated tree water uptake by roots and water loss by transpiration.

by roots from the soil is determined by three main factors:

2. root-system architecture as a network of absorbing organs, and

**2. Below-ground processes** 

root resistances.

variable resistance along the root by separating roots into segments, each with homogeneous resistance. With recent advances, the physiological characteristics of water uptake by roots such as radial and axial resistances can now be examined at the centimeter scale (North and Nobel, 1995). Simulation models for the below-ground tree are needed because of the difficulty in observing and quantifying the architecture of roots, the soil-water dynamics and the soil and root resistances to water flow. By coupling soil water dynamics and root architecture along with soil and root hydraulic resistances, the process of water transport from the soil to the roots and through the roots can be described. The below-ground processes and their interactions in relation to water movement are illustrated in Fig.1.

Fig. 1. A concise structural diagram of the below-ground processes and their interactions

#### **2.1 Modelling soil water dynamics**

Although soil-water fluxes are difficult to measure, the relative capabilities of existing models and the credibility of their results are still an important concern because soil water dynamics is sometimes inadequately represented in models of the soil-plant-atmospherewater interactions and processes (Clemente *et al*., 1994). Analytical solutions to the soilwater flow equation (Richards' equation (Richards, 1931)) are not possible for dynamic field

Soil-Tree-Atmosphere Water Relations 161

*VPD Rnet*

1

where *FE* is the evaporation rate (kg m-2 h-1), *Rnetsoil* is the net radiation at the soil surface (MJ m-2 h-1), *ρ* is the air density (kg m-3), ε is the dimensionless slope of the saturation specific humidity, λ is the latent heat of vaporization (MJ kg-1), *VPD* is the vapor pressure deficit, *Ra soil* is the boundary-layer resistance to the transfer of water vapor and heat (between the soil surface and bottom of the canopy) (s m-1) and *Rss* is soil surface resistance (s m-1). *Rss* that restricts the transfer of water from the soil surface by evaporation can be calculated as

<sup>1</sup> *R R f SWC SS SS*(min)

where *Rss*(*min*) is the minimum value of the soil-surface resistance under optimal conditions

The architecture and space filling properties of the below-surface tree organs are necessary for a mechanistic understanding of water uptake. Pioneering work in numerical simulations of root systems was done by Lungley (1973). Based on theoretical study, Claasen and Barber (1974) assumed uniform root distributions. Rengel (1993) pointed out that the assumption of uniformity of root distribution used in models based on Claasen and Barber (1974) is an oversimplification, since the way roots fill the soil matrix is important for nutrient and water acquisition (Sattelmacher *et al.*, 1990). Fitter *et al*. (1991) also described a simulation model of root growth that simulated the development of root systems varying in several important architectural features including root link length. Root link length refers to a distance

Some attempts have been made to relate root biomass to stem diameter at a standardised height (Santantonio *et al*., 1977; Brown *et al*., 1989), but such relationships probably depend on tree species and site. Soumar *et al*. (1994) found in a study on *Sclerocaryea birrea*, that the root diameter at 1 m from the stem base is an appropriate parameter for predicting horizontal root distribution. However, relating the size of the root system to proximal root diameter may be

Fitter *et al.* (1991) have suggested that it is not possible to derive a simple analytical relationship between root system architecture and water and resource acquisition due to the complexity of the spatial arrangement of the roots within the soil. Fractal geometry is a system of geometry that is more suited to the description of many biological objects than is standard Euclidean geometry (Mandelbrot, 1983). Fractal geometry provides useful perspectives on root branching patterns (Shibusawa, 1994; Van Noordwijk *et al*., 1994) and it is reasonable to accept that fractals may provide quantitative summaries and functional

The branching pattern has been identified as an important characteristic of root systems that strongly influences patterns of foraging within the soil matrix (Fitter, 1987; Hetrick, 1988)

more successful than relating it to stem diameter (Van Noordwijk *et al*., 1994).

 

*<sup>E</sup> ss*

*Ra <sup>F</sup> <sup>R</sup>*

(Walker and Langridge, 1996):

**2.2 Modelling root architecture** 

and *SWC* is the soil water content in the topmost layer.

between two branching points in the root (Bernston, 1994).

insights into root architecture (Bernston *et al*., 1995).

*soil*

1

(3)

(2)

*soil*

*soil*

*Ra*

situations (Zeng and, Decker, 2009) and most efforts have been concentrated on seeking numerical solutions (e.g. Feddes *et al.*, 1978; Broadbridge and White, 1988; Ranatunga and Murty, 1992).

A numerical solution for the unsaturated vertical soil water flow with varying water supply (irrigation and rainfall) is typically used to estimate soil water content in a verticallystructured soil profile (Ranatunga and Murty, 1992). This numerical solution also gives an implicit finite difference solution to the soil water flow equation. The assumption of only one-dimensional vertical flow is quite accurate for agricultural rooting depths (Bresler, 1991). Soil evaporation can be based on a modified version of Penman-Monteith equation (Raupach, 1991) and vapor-flow flux (within the soil profile) is neglected except at the top layer of the soil. To examine the water flow in soils, a volumetric sink term may be added to the classic Richard's flow equation for one-dimensional flow under gravity (Feddes *et al.*, 1978).

At the soil-air interface the soil can lose water to the atmosphere by evaporation of soil water where the potential rate of soil evaporation depends only on atmospheric conditions. During evaporation the requirement is that (Feddes *et al.*, 1978):

$$
\psi(\mathbf{z}, \mathbf{t}) \ge \psi\_{\mathcal{U}} \quad \mathbf{t} \ge \mathbf{0}, \; \mathbf{z} = \mathbf{0}
$$

where ψ is the pressure head and ψl is the minimum pressure head allowed under air-dry conditions. Typically, ψl is estimated from the mean temperature and relative humidity of the surrounding air, and if it is assumed that the pressure at the soil surface is in equilibrium with the atmosphere, then, ψl can be derived from the following relationship (Feddes *et al.,* 1974):

$$\dot{\nu}\_l = \frac{RT}{Mg} \ln(Rh) \tag{1}$$

where *R* is the universal gas constant (8.314 J mol-1K-1 ), *T* is the absolute temperature (K), *g* is the acceleration of gravity (9.81 m s-2), *M* is the molecular weight of water (0.018 kg mol-1) and *Rh* is the relative humidity as a fraction. The simplest option and one which is often used in modeling soil-water flow, is to neglect vapor flow except at the soil surface (Campbell, 1985). Janz and Stonier (1995) incorporated the evaporation rate into the sink term of the top soil layer and were able to make reasonable predictions of soil water dynamics.

The soil water deficit is the amount of water needed to bring the soil moisture content back to field capacity, which is the amount of water the soil can hold against gravity. The soil water deficit is calculated by subtracting the total water content of the soil layer from the water content at the field capacity of the root zone. In standard models, rainfall and the amount of irrigation applied are assumed to be equally distributed among the soil layers, and adjusted by adding to the water content of the soil layer. The numerical procedure gives the soil-water content in vertically structured soil layers.

The Penman-Monteith equation provides good predictions of evaporation of forest surfaces (Saugier, 1996). It uses a modification of the form employed by Raupach (1991) as applied by Walker and Langridge (1996):

situations (Zeng and, Decker, 2009) and most efforts have been concentrated on seeking numerical solutions (e.g. Feddes *et al.*, 1978; Broadbridge and White, 1988; Ranatunga and

A numerical solution for the unsaturated vertical soil water flow with varying water supply (irrigation and rainfall) is typically used to estimate soil water content in a verticallystructured soil profile (Ranatunga and Murty, 1992). This numerical solution also gives an implicit finite difference solution to the soil water flow equation. The assumption of only one-dimensional vertical flow is quite accurate for agricultural rooting depths (Bresler, 1991). Soil evaporation can be based on a modified version of Penman-Monteith equation (Raupach, 1991) and vapor-flow flux (within the soil profile) is neglected except at the top layer of the soil. To examine the water flow in soils, a volumetric sink term may be added to the classic Richard's flow equation for one-dimensional flow under gravity (Feddes *et al.*,

At the soil-air interface the soil can lose water to the atmosphere by evaporation of soil water where the potential rate of soil evaporation depends only on atmospheric conditions.

*ψ*(z,t) *ψl*; t > 0, z = 0 where ψ is the pressure head and ψl is the minimum pressure head allowed under air-dry conditions. Typically, ψl is estimated from the mean temperature and relative humidity of the surrounding air, and if it is assumed that the pressure at the soil surface is in equilibrium with the atmosphere, then, ψl can be derived from the following relationship (Feddes *et al.,*

> *<sup>l</sup> RT In Rh Mg*

where *R* is the universal gas constant (8.314 J mol-1K-1 ), *T* is the absolute temperature (K), *g* is the acceleration of gravity (9.81 m s-2), *M* is the molecular weight of water (0.018 kg mol-1) and *Rh* is the relative humidity as a fraction. The simplest option and one which is often used in modeling soil-water flow, is to neglect vapor flow except at the soil surface (Campbell, 1985). Janz and Stonier (1995) incorporated the evaporation rate into the sink term of the top soil layer and were able to make reasonable predictions of soil water

The soil water deficit is the amount of water needed to bring the soil moisture content back to field capacity, which is the amount of water the soil can hold against gravity. The soil water deficit is calculated by subtracting the total water content of the soil layer from the water content at the field capacity of the root zone. In standard models, rainfall and the amount of irrigation applied are assumed to be equally distributed among the soil layers, and adjusted by adding to the water content of the soil layer. The numerical procedure gives

The Penman-Monteith equation provides good predictions of evaporation of forest surfaces (Saugier, 1996). It uses a modification of the form employed by Raupach (1991) as applied

(1)

During evaporation the requirement is that (Feddes *et al.*, 1978):

the soil-water content in vertically structured soil layers.

by Walker and Langridge (1996):

Murty, 1992).

1978).

1974):

dynamics.

$$F\_E = \left(\frac{\varepsilon Rnet\_{soil} + \frac{\rho \lambda VPD}{Ra\_{soil}}}{\varepsilon + 1 + \frac{R\_{ss}}{Ra\_{soil}}}\right) \frac{1}{\lambda} \tag{2}$$

where *FE* is the evaporation rate (kg m-2 h-1), *Rnetsoil* is the net radiation at the soil surface (MJ m-2 h-1), *ρ* is the air density (kg m-3), ε is the dimensionless slope of the saturation specific humidity, λ is the latent heat of vaporization (MJ kg-1), *VPD* is the vapor pressure deficit, *Ra soil* is the boundary-layer resistance to the transfer of water vapor and heat (between the soil surface and bottom of the canopy) (s m-1) and *Rss* is soil surface resistance (s m-1). *Rss* that restricts the transfer of water from the soil surface by evaporation can be calculated as (Walker and Langridge, 1996):

$$R\_{SS} = R\_{SS(\text{min})} f \left( \text{SWC} \right)^{-1} \tag{3}$$

where *Rss*(*min*) is the minimum value of the soil-surface resistance under optimal conditions and *SWC* is the soil water content in the topmost layer.

#### **2.2 Modelling root architecture**

The architecture and space filling properties of the below-surface tree organs are necessary for a mechanistic understanding of water uptake. Pioneering work in numerical simulations of root systems was done by Lungley (1973). Based on theoretical study, Claasen and Barber (1974) assumed uniform root distributions. Rengel (1993) pointed out that the assumption of uniformity of root distribution used in models based on Claasen and Barber (1974) is an oversimplification, since the way roots fill the soil matrix is important for nutrient and water acquisition (Sattelmacher *et al.*, 1990). Fitter *et al*. (1991) also described a simulation model of root growth that simulated the development of root systems varying in several important architectural features including root link length. Root link length refers to a distance between two branching points in the root (Bernston, 1994).

Some attempts have been made to relate root biomass to stem diameter at a standardised height (Santantonio *et al*., 1977; Brown *et al*., 1989), but such relationships probably depend on tree species and site. Soumar *et al*. (1994) found in a study on *Sclerocaryea birrea*, that the root diameter at 1 m from the stem base is an appropriate parameter for predicting horizontal root distribution. However, relating the size of the root system to proximal root diameter may be more successful than relating it to stem diameter (Van Noordwijk *et al*., 1994).

Fitter *et al.* (1991) have suggested that it is not possible to derive a simple analytical relationship between root system architecture and water and resource acquisition due to the complexity of the spatial arrangement of the roots within the soil. Fractal geometry is a system of geometry that is more suited to the description of many biological objects than is standard Euclidean geometry (Mandelbrot, 1983). Fractal geometry provides useful perspectives on root branching patterns (Shibusawa, 1994; Van Noordwijk *et al*., 1994) and it is reasonable to accept that fractals may provide quantitative summaries and functional insights into root architecture (Bernston *et al*., 1995).

The branching pattern has been identified as an important characteristic of root systems that strongly influences patterns of foraging within the soil matrix (Fitter, 1987; Hetrick, 1988)

Soil-Tree-Atmosphere Water Relations 163

where *LRZi* is the radius of the rooting zone in the soil layer *i* and z is the depth of the soil layer. The architecture of a root system can be dissected into a number of measurable variables of which the most important is root density (*Dt*) which is defined as the root

> 0 0 *m l*

*Vt* 

*n n m l n n L L*

where *nm* and *nl* are the number of branching events in the main axis and lateral axes respectively, *Lmn* is the length of the root link in the main axis and *Ll n* is the length of root

In the soil-plant-atmosphere continuum, water flow is driven through a series of hydraulic resistances that can be identified as soil resistance, root radial resistance, and root axial resistance pertaining to the pathways of conduction through xylem vessels, and stomatal

Root resistance is partitioned into radial and longitudinal (axial) components and the relative importance of these has been stressed (Passioura, 1984; St Aubin *et al*., 1986). However, reasonable quantitative data of axial resistance (*Rl*) and radial resistance (*Rr*) is rare, even though these quantities are a prerequisite for the proper modeling of water flows through roots. The procedure for resistance calculations is mechanistically more appropriate than steady-state models because it includes the effect of the soil-water influence on

Assuming that the soil hydraulic conductivity (*Ks*) is constant within the particular soil layer, the soil resistance associated with the root *Rs* (kg-1 m4 s-1) was developed by Moldrup *et al*. (1992). Water uptake by roots in wet soil is generally determined by the root hydraulic resistance, which is composed of the radial resistance (*Rr*) from root surface to xylem, and axial resistance (*Rl*) along the xylem (Landsberg and Fowkes, 1978; Passioura, 1988). The sum of *Rr* and *Rl* are often called the plant resistance that is found to be independent of the water potential gradient (Abdul-Jabbar *et al.*, 1984) and transpiration rate (Neumann *et al.*, 1974). Therefore, regardless of changes in soil and plant water status with time, *Rr* and *Rl* are

Assuming that the membranes of cells to be crossed before water reaches the xylem form concentric cylinders, it can be easily shown that radial resistances would be related to each

> *r cell <sup>c</sup> <sup>k</sup> <sup>k</sup> <sup>r</sup> R R*

the *k* cell layers to be crossed by water, *n*c is the average number of cells to be crossed before water reaches the xylem and *r*r is the radius of the root. Thornley and Johnson (1990)

where *R*r is radial resistance (kg-1 m2 s-1), *R*cell is cell resistance (kg-1 m2 s-1), *r*<sup>c</sup>

1

*r*

*<sup>c</sup> n r*

(8)

k is the radius of

*n n*

(7)

length per unit volume and *Dt* is calculated as (Tardieu, 1988):

**2.3 Water flow resistances in the soil-root system** 

resistance with a boundary layer adjustment to the leaf surface.

normally assumed to be constants (Frensch and Steudle, 1989). .

other by (Steudle and Brinnkman, 1989):

links in lateral axes*.* 

resistance.

*Dt*

and nutrient and water acquisition (Robinson *et al.*, 1991). There have been a number of attempts to produce architectural classifications of root systems (Weaver, 1968; Krasilnikov, 1968) but none have been particularly successful, partly because of the great variability of root systems. However, following Coupland and Johnson (1965), Van Noordwijk *et al*., 1994 identified two major classes of fractal branching patterns, i.e., herringbone and dichosyntomous. Most monocots follow the herringbone root branching pattern comprising of a main root and laterals whereas most dicots have a dicho-syntomous branching pattern that spawns the parent branch into two daughter branches (Zamir, 2001).

The most important and easily-measurable parameter for root modelling is the proximal root diameter. The relationship between proximal diameter and the total root length of all root links obviously depends on the root branching pattern. Leonardo da Vinci (quoted by Mandelbrot, 1983 and the quoted by Van Noordwijk *et al*., 1994) claimed that the cross sectional area of the main stem is equal to the sum of the cross sectional areas of tree branches. A relationship can be sought between the root diameter at the stem base, and functionally important root parameters (Van Noordwijk and Brouwer, 1995) such as the total root length. This approach, known as a pipestem model (Shinozaki *et al*., 1964) states that each unit of foliage requires a unit pipe of wood to connect it to the root system, and has been used by Van Noordwijk *et al*. (1994) to understand the geometry of root branching in plants, giving the relation between proximal diameter of the root base and the diameter of the divided axes:

$$\frac{\pi D o^2}{4} = \frac{\alpha \pi}{4} \sum\_{j=1}^{\mathcal{N}k} D i\_j^2 \tag{4}$$

where *D*<sup>o</sup> is proximal diameter at the root base (m), *Di* is diameter of the divided axes in the link *i* (m), α is proportionality of cross sectional area before a branching event and the sum of the cross sectional areas after branching and *Nk* is the number of branch roots per branching event. In many previous studies (summarized by Van Noordwijk *et al*., 1994) on root architecture, root link length was assumed to be a constant for a whole root system or a part of root system. A recursive programming procedure can be adopted for divisions of root branching. The number of branching steps in each lateral branch can also be in the recursive programme (Van Noordwijk *et al*., 1994).

The length of the root axis (*L*), which is an important parameter for understanding how the root system fills the soil matrix can then be calculated as:

$$L = mL\_m \tag{5}$$

where *Lm* is the mean length of the root link in the main or lateral axis and n is the number of root segments in the root system. With an estimated *Lm* value for each root link, it is possible to calculate the length of the root axis. Values of *L* decrease with increasing diameter and the longest roots occur in the small diameter classes (Kodrik, 1995).

The soil volume explored by roots in each soil layer (*Vti*) can be determined (Pregitxer et al, 1997):

$$\text{Vt} = \pi \text{L}\_{\text{RZ}}^2 \text{z} \tag{6}$$

and nutrient and water acquisition (Robinson *et al.*, 1991). There have been a number of attempts to produce architectural classifications of root systems (Weaver, 1968; Krasilnikov, 1968) but none have been particularly successful, partly because of the great variability of root systems. However, following Coupland and Johnson (1965), Van Noordwijk *et al*., 1994 identified two major classes of fractal branching patterns, i.e., herringbone and dichosyntomous. Most monocots follow the herringbone root branching pattern comprising of a main root and laterals whereas most dicots have a dicho-syntomous branching pattern that

The most important and easily-measurable parameter for root modelling is the proximal root diameter. The relationship between proximal diameter and the total root length of all root links obviously depends on the root branching pattern. Leonardo da Vinci (quoted by Mandelbrot, 1983 and the quoted by Van Noordwijk *et al*., 1994) claimed that the cross sectional area of the main stem is equal to the sum of the cross sectional areas of tree branches. A relationship can be sought between the root diameter at the stem base, and functionally important root parameters (Van Noordwijk and Brouwer, 1995) such as the total root length. This approach, known as a pipestem model (Shinozaki *et al*., 1964) states that each unit of foliage requires a unit pipe of wood to connect it to the root system, and has been used by Van Noordwijk *et al*. (1994) to understand the geometry of root branching in plants, giving the relation between proximal diameter of the root base and the diameter of

2

recursive programme (Van Noordwijk *et al*., 1994).

root system fills the soil matrix can then be calculated as:

<sup>1</sup> 4 4

where *D*<sup>o</sup> is proximal diameter at the root base (m), *Di* is diameter of the divided axes in the link *i* (m), α is proportionality of cross sectional area before a branching event and the sum of the cross sectional areas after branching and *Nk* is the number of branch roots per branching event. In many previous studies (summarized by Van Noordwijk *et al*., 1994) on root architecture, root link length was assumed to be a constant for a whole root system or a part of root system. A recursive programming procedure can be adopted for divisions of root branching. The number of branching steps in each lateral branch can also be in the

The length of the root axis (*L*), which is an important parameter for understanding how the

where *Lm* is the mean length of the root link in the main or lateral axis and n is the number of root segments in the root system. With an estimated *Lm* value for each root link, it is possible to calculate the length of the root axis. Values of *L* decrease with increasing

The soil volume explored by roots in each soil layer (*Vti*) can be determined (Pregitxer et al,

<sup>2</sup> *Vt L z* 

diameter and the longest roots occur in the small diameter classes (Kodrik, 1995).

 

2

(4)

*L nL <sup>m</sup>* (5)

*RZ* (6)

*j*

*Nk*

*j Do Di*

spawns the parent branch into two daughter branches (Zamir, 2001).

the divided axes:

1997):

where *LRZi* is the radius of the rooting zone in the soil layer *i* and z is the depth of the soil layer. The architecture of a root system can be dissected into a number of measurable variables of which the most important is root density (*Dt*) which is defined as the root length per unit volume and *Dt* is calculated as (Tardieu, 1988):

$$Dt = \frac{\left(\sum\_{n=0}^{n\_m} L\_{m\_n} + \sum\_{n=0}^{n\_l} L\_{l\_n}\right)}{Vt} \tag{7}$$

where *nm* and *nl* are the number of branching events in the main axis and lateral axes respectively, *Lmn* is the length of the root link in the main axis and *Ll n* is the length of root links in lateral axes*.* 

#### **2.3 Water flow resistances in the soil-root system**

In the soil-plant-atmosphere continuum, water flow is driven through a series of hydraulic resistances that can be identified as soil resistance, root radial resistance, and root axial resistance pertaining to the pathways of conduction through xylem vessels, and stomatal resistance with a boundary layer adjustment to the leaf surface.

Root resistance is partitioned into radial and longitudinal (axial) components and the relative importance of these has been stressed (Passioura, 1984; St Aubin *et al*., 1986). However, reasonable quantitative data of axial resistance (*Rl*) and radial resistance (*Rr*) is rare, even though these quantities are a prerequisite for the proper modeling of water flows through roots. The procedure for resistance calculations is mechanistically more appropriate than steady-state models because it includes the effect of the soil-water influence on resistance.

Assuming that the soil hydraulic conductivity (*Ks*) is constant within the particular soil layer, the soil resistance associated with the root *Rs* (kg-1 m4 s-1) was developed by Moldrup *et al*. (1992). Water uptake by roots in wet soil is generally determined by the root hydraulic resistance, which is composed of the radial resistance (*Rr*) from root surface to xylem, and axial resistance (*Rl*) along the xylem (Landsberg and Fowkes, 1978; Passioura, 1988). The sum of *Rr* and *Rl* are often called the plant resistance that is found to be independent of the water potential gradient (Abdul-Jabbar *et al.*, 1984) and transpiration rate (Neumann *et al.*, 1974). Therefore, regardless of changes in soil and plant water status with time, *Rr* and *Rl* are normally assumed to be constants (Frensch and Steudle, 1989). .

Assuming that the membranes of cells to be crossed before water reaches the xylem form concentric cylinders, it can be easily shown that radial resistances would be related to each other by (Steudle and Brinnkman, 1989):

$$R\_r = R\_{cell} \sum\_{k=1}^{n^c} \frac{r^r}{r\_k^c} \tag{8}$$

where *R*r is radial resistance (kg-1 m2 s-1), *R*cell is cell resistance (kg-1 m2 s-1), *r*<sup>c</sup> k is the radius of the *k* cell layers to be crossed by water, *n*c is the average number of cells to be crossed before water reaches the xylem and *r*r is the radius of the root. Thornley and Johnson (1990)

Soil-Tree-Atmosphere Water Relations 165

architectural models. A fairly straightforward method can be introduced to formulate the three-dimensional canopy in terms of the amount and spatial distribution of leaf area within the crown. Thus, each vertical canopy layer has geometrically-defined sub-canopies, assuming a vertically heterogeneous canopy consisting of horizontally homogeneous layers. The radiation, stomatal and leaf boundary-layer resistances, micro-environmental variables and finally the energy balance vary within the canopy. Consequently, the vertical and horizontal variations within the model become apparent. Far-field variation of wind speed and air temperature can normally be simulated with depth outside the canopy, but a uniform profile of vapor pressure outside the canopy is normally assumed (these

The prediction of the vertical distributions of micrometeorological variables is regarded as a necessary step in the prediction of sub-canopy level fluxes. The adoption of integrated stomatal resistance has received theoretical (Finnigan and Raupach, 1987; Paw U and Meyers, 1989; Kelliher *et al.*, 1994; Raupach, 1995; Leuning *et al.*, 1995) and experimental (Baldocchi *et al*., 1987) criticism over the years. As the radiation intercepted and scalar concentrations can be calculated in each sub-canopy within the crown, it is possible to estimate the stomatal resistances individually for each sub-canopy and hence, avoid the integration of the stomatal resistance for the entire canopy. Similarly, the boundary layer resistance can be calculated for each sub-canopy and it is not required to integrate over the total leaf area of the canopy.

 the canopy architecture and leaf area - a three-dimensional analysis to estimate the beam path length of each and every volume of the sub-canopies within a paraboloidshaped-canopy, and to estimate the vertical leaf-area distribution using a Weibull

the solar position and day length - a methodology to calculate the zenith and azimuth

the radiative transfer through the canopy - a comprehensive methodology for radiative

the energy-balance process – an application of the energy balance equation to each

the scalar variation - to estimate wind speed, air temperature, relative humidity and

the canopy-atmosphere processes for resistance calculations - to estimate the stomatal

These canopy processes are discussed in detail in the following sections. The above-ground

The geometric form of the crown is one of the main factors that govern the productive potential of vegetation (Jahnke and Lawrence, 1965). The shape of the crown is strongly correlated with the volume and height of the stand. Biging and Dobbertin (1995) found that the geometric space occupied by the crown is highly correlated with growth. This raises the

assumptions are discussed in detail later).

statistical pattern,

solar radiation, and

and leaf boundary-layer resistances.

**3.1 Crown architecture and leaf area estimates** 

The above-ground canopy processes would normally include;

angles of the sun on an hourly basis and hence the day length,

volume of the sub-canopies in order to estimate leaf temperature,

transfer (short-wave direct, long-wave, and diffuse) through the crown,

processes and their interactions in relation to water movement are given in Fig.2.

question, therefore, whether canopy form can be readily quantified.

explained how the resistance from root surface to the plant vary with the number of roots per unit area in a vertically-structured soil profile.

Melchior and Steudle (1993) stated that *R*l can contribute substantially to the overall hydraulic resistance or even can be the limiting component. Passioura (1972), Newman (1976) and Meyer and Ritchie (1980) used a modified Poiseuille-Hagen equation to calculate *Rl* from root xylem dimensions:

$$R\_l = \frac{4\eta L}{\rho\_w^2 \pi \sum\_{i=1}^{n^v} r\_i^4} \tag{9}$$

where *R*l is axial resistance (kg-1 m4 s-1) associated with the root link, *ri* is mean effective diameter of the xylem vessels (m), η is the viscosity of water (0.001002 kg m-1 s-1)1 , *L* is the length of the xylem vessel (m), *n*v is average number of conducting xylem vessels in the root and ρw is the density of water (kg m-3).

#### **3. Above-ground processes**

Canopy processes such as the leaf-energy balance and transpiration are relatively well understood. However, a sound understanding of how these processes integrate spatially and temporally within trees remains elusive. One approach to tackle this problem is the use of models that allow scaling of canopy processes at the leaf level to the whole canopy (Jarvis *et al*., 1985; Running and Coughlan, 1988; McMurtrie, 1993; Jarvis, 1995). Both aggregated (e.g. 'big-leaf') and distributed (e.g. multi-layer or three-dimensional) approaches are commonly applied in modeling canopy processes. There are costs and benefits to both (Raupach and Finnigan, 1988): Simulations with distributed modeling approaches require assumptions about the distribution of key parameters in space (and time), but allow model parameterization using fine-scale data. Aggregation avoids the need for spatial details by building the effects of non-linearities into the model parameters. These parameters must be estimated directly from coarse-scale data (e.g. canopy rather than leaf-level data).

Tree canopy structure is a complex and dynamic outcome of the evolutionary and ecological interactions and feedbacks between vegetation and environment. Since the transpiration from a single tree is a non-linear function of absorbed solar radiation and other related environmental variables (e.g. temperature, humidity, and wind speed), it is necessary to be able to simulate the radiation regime and other related environmental variables within the canopy and before, the transpiration rate can be adequately calculated. Therefore, the aboveground modeling presented in most of the literature (Myneni and Impens, 1985; Wang and Jarvis, 1990; Whitehead *et al.*, 1990; Ryel *et al.*, 1993) is largely based on hypothesized spatial independence structures from which the canopy is made. It, therefore, precludes an aggregated approach.

Three-dimensional models of the radiative transfer and canopy architecture represent a compromise between the simplicity of the uniform canopy models and the complexity of

<sup>1</sup> The temperature dependence of the viscosity of water is not taken into account. The temperature of water is assumed to be 20 0C.

explained how the resistance from root surface to the plant vary with the number of roots

Melchior and Steudle (1993) stated that *R*l can contribute substantially to the overall hydraulic resistance or even can be the limiting component. Passioura (1972), Newman (1976) and Meyer and Ritchie (1980) used a modified Poiseuille-Hagen equation to calculate

> 2 4 1

*r*

(9)

4 *<sup>l</sup> <sup>v</sup> <sup>n</sup> w i i*

*<sup>L</sup> <sup>R</sup>*

 

where *R*l is axial resistance (kg-1 m4 s-1) associated with the root link, *ri* is mean effective diameter of the xylem vessels (m), η is the viscosity of water (0.001002 kg m-1 s-1)1 , *L* is the length of the xylem vessel (m), *n*v is average number of conducting xylem vessels in the root

Canopy processes such as the leaf-energy balance and transpiration are relatively well understood. However, a sound understanding of how these processes integrate spatially and temporally within trees remains elusive. One approach to tackle this problem is the use of models that allow scaling of canopy processes at the leaf level to the whole canopy (Jarvis *et al*., 1985; Running and Coughlan, 1988; McMurtrie, 1993; Jarvis, 1995). Both aggregated (e.g. 'big-leaf') and distributed (e.g. multi-layer or three-dimensional) approaches are commonly applied in modeling canopy processes. There are costs and benefits to both (Raupach and Finnigan, 1988): Simulations with distributed modeling approaches require assumptions about the distribution of key parameters in space (and time), but allow model parameterization using fine-scale data. Aggregation avoids the need for spatial details by building the effects of non-linearities into the model parameters. These parameters must be

estimated directly from coarse-scale data (e.g. canopy rather than leaf-level data).

Tree canopy structure is a complex and dynamic outcome of the evolutionary and ecological interactions and feedbacks between vegetation and environment. Since the transpiration from a single tree is a non-linear function of absorbed solar radiation and other related environmental variables (e.g. temperature, humidity, and wind speed), it is necessary to be able to simulate the radiation regime and other related environmental variables within the canopy and before, the transpiration rate can be adequately calculated. Therefore, the aboveground modeling presented in most of the literature (Myneni and Impens, 1985; Wang and Jarvis, 1990; Whitehead *et al.*, 1990; Ryel *et al.*, 1993) is largely based on hypothesized spatial independence structures from which the canopy is made. It, therefore, precludes an

Three-dimensional models of the radiative transfer and canopy architecture represent a compromise between the simplicity of the uniform canopy models and the complexity of

1 The temperature dependence of the viscosity of water is not taken into account. The temperature of

per unit area in a vertically-structured soil profile.

*Rl* from root xylem dimensions:

and ρw is the density of water (kg m-3).

**3. Above-ground processes** 

aggregated approach.

water is assumed to be 20 0C.

architectural models. A fairly straightforward method can be introduced to formulate the three-dimensional canopy in terms of the amount and spatial distribution of leaf area within the crown. Thus, each vertical canopy layer has geometrically-defined sub-canopies, assuming a vertically heterogeneous canopy consisting of horizontally homogeneous layers.

The radiation, stomatal and leaf boundary-layer resistances, micro-environmental variables and finally the energy balance vary within the canopy. Consequently, the vertical and horizontal variations within the model become apparent. Far-field variation of wind speed and air temperature can normally be simulated with depth outside the canopy, but a uniform profile of vapor pressure outside the canopy is normally assumed (these assumptions are discussed in detail later).

The prediction of the vertical distributions of micrometeorological variables is regarded as a necessary step in the prediction of sub-canopy level fluxes. The adoption of integrated stomatal resistance has received theoretical (Finnigan and Raupach, 1987; Paw U and Meyers, 1989; Kelliher *et al.*, 1994; Raupach, 1995; Leuning *et al.*, 1995) and experimental (Baldocchi *et al*., 1987) criticism over the years. As the radiation intercepted and scalar concentrations can be calculated in each sub-canopy within the crown, it is possible to estimate the stomatal resistances individually for each sub-canopy and hence, avoid the integration of the stomatal resistance for the entire canopy. Similarly, the boundary layer resistance can be calculated for each sub-canopy and it is not required to integrate over the total leaf area of the canopy.

The above-ground canopy processes would normally include;


These canopy processes are discussed in detail in the following sections. The above-ground processes and their interactions in relation to water movement are given in Fig.2.

#### **3.1 Crown architecture and leaf area estimates**

The geometric form of the crown is one of the main factors that govern the productive potential of vegetation (Jahnke and Lawrence, 1965). The shape of the crown is strongly correlated with the volume and height of the stand. Biging and Dobbertin (1995) found that the geometric space occupied by the crown is highly correlated with growth. This raises the question, therefore, whether canopy form can be readily quantified.

Soil-Tree-Atmosphere Water Relations 167

used to describe the development of form (Halle *et al.*, 1978), but these have had limited ecological applicability in understanding how specific architectures influence the resource

Unfortunately, quantifying crown geometry is difficult because the profiles are usually irregular and simple geometric representations may not be adequate (Biging and Dobbertin, 1995). The occurrence of vertical layering or stratification in a tree canopy has been a matter of considerable attention, not least because of inconsistent definitions and methodology (Bourgeron, 1983). Richards (1983) referred to the stratification of leaf mass and of individual tree heights for which the evidence is weak. Stratification of species refers to the vertical aggregation of average mature heights of species. Extensive quantitative documentation of this phenomenon is also lacking, but there is some support for its

In systems with discontinuous canopies, a more detailed canopy description is necessary, since the diurnal variation of radiation availability and interception are much larger than in closed forests (Palmer, 1977). Array models have been developed that account for horizontal differences (Palmer, 1977), and these canopies were often modeled as a series of onedimensional layers that are horizontally homogeneous. But because individual crowns are not distinguished, their application is limited to canopies which can be defined in simple

Two dimensional models have been used to explore how branch angle and branch length influence leaf overlap and hence the efficiency of irradiance capture (Fisher, 1992). Two dimensional models for parallel rows have been described for rectangular (Goudriaan, 1977; Sinoquet and Bonhomme, 1992), elliptical (Charles-Edwards and Thorpe, 1976) and triangular (Jackson and Palmer, 1979) cross-sections. These models are adequate for infinite hedgerows. They consider the interception of light from only a single direction, whereas, in reality, diffuse radiation comes from many directions and the direction of the solar beam changes during the day (Bristow *et al*., 1985). Modeling an isolated tree presents more difficulty than uniform canopies or row crops where it is possible to reduce the dimension of the problem to a single coordinate. For row crops it is still possible to develop this

Recently three-dimensional models have been used to assess light interception by individual trees. These models have generally been focused on either relatively simple canopy structures or are species-specific, or have considered the crown as a series of layers, or cells with particular foliage characteristics (Myneni and Impens, 1985). Myneni *et al.* (1990) formulated a three-dimensional leaf canopy model using a modified discrete ordinates method. Myneni (1991) used fractal models of trees to simulate leaf area distributions for modeling the radiative transfer and photosynthesis in a forest canopy. However, most of these studies have simply discussed the adaptive significance of species-specific tree crown architectures as simple allometries between crown dimensions (crown depth, width, or area) and individual sizes (mass, stem diameter at breast height), and have limited the investigations on the effects of individual crown architecture as vertical foliage distributions on the interactions between

analysis in a fairly straightforward manner (Gijzen and Goudriaan, 1989).

them (Mohren *et al*., 1984; Biging and Wensel, 1990; Sinoquet and Bonhomme, 1992).

Typically, crown architecture is described in terms of the amount and spatial distribution of leaf area within the crown and defined as the set of features delineating the shape, size and

capture.

existence (Pukkala *et al*., 1991).

geometrical terms.

Fig. 2. A concise structural diagram of the above-ground processes and their interactions

A number of simplifying assumptions were usually made about canopy architecture to model light penetration and to assess canopy leaf area. The most elementary canopy description applied in mechanistic models is the stand-oriented approach; the canopy is assumed horizontally continuous and radiation absorption is calculated per hectare. This approach is sufficient when simulating regularly constructed, mono-species stands with closed canopies (Mohren, 1987; Nikinmaa, 1992; Bossel and Krieger, 1994). However, statistical models describing horizontally homogeneous stands are not applicable in stands where the foliage is grouped into individual tree crowns (Oker-Blom, 1986) and tree crowns display a great variety of aerial structure, which, in response to the environment, show different solutions to the problems of resource capture. As all trees depend on the radiation energy, it is evident that the vegetation structure of trees is regulated by the spatial and temporal variation of irradiance.

Models with one-dimensional canopy layers have been proposed by many authors following Monsi and Saeki (1953). Models of homogeneous continuous canopies of forests or crops have been used to provide insight into the role of specific characteristics such as leaf angle and leaf area index or vertical distribution of leaf characteristics such as leaf mass per unit area or volume (Gutschick and Weigel, 1988). Discontinuous canopies such as those formed by trees need more dimensions to analyze the vertical and horizontal variations of the radiation distribution. For individual trees, qualitative models of branching have been

Fig. 2. A concise structural diagram of the above-ground processes and their interactions

temporal variation of irradiance.

A number of simplifying assumptions were usually made about canopy architecture to model light penetration and to assess canopy leaf area. The most elementary canopy description applied in mechanistic models is the stand-oriented approach; the canopy is assumed horizontally continuous and radiation absorption is calculated per hectare. This approach is sufficient when simulating regularly constructed, mono-species stands with closed canopies (Mohren, 1987; Nikinmaa, 1992; Bossel and Krieger, 1994). However, statistical models describing horizontally homogeneous stands are not applicable in stands where the foliage is grouped into individual tree crowns (Oker-Blom, 1986) and tree crowns display a great variety of aerial structure, which, in response to the environment, show different solutions to the problems of resource capture. As all trees depend on the radiation energy, it is evident that the vegetation structure of trees is regulated by the spatial and

Models with one-dimensional canopy layers have been proposed by many authors following Monsi and Saeki (1953). Models of homogeneous continuous canopies of forests or crops have been used to provide insight into the role of specific characteristics such as leaf angle and leaf area index or vertical distribution of leaf characteristics such as leaf mass per unit area or volume (Gutschick and Weigel, 1988). Discontinuous canopies such as those formed by trees need more dimensions to analyze the vertical and horizontal variations of the radiation distribution. For individual trees, qualitative models of branching have been used to describe the development of form (Halle *et al.*, 1978), but these have had limited ecological applicability in understanding how specific architectures influence the resource capture.

Unfortunately, quantifying crown geometry is difficult because the profiles are usually irregular and simple geometric representations may not be adequate (Biging and Dobbertin, 1995). The occurrence of vertical layering or stratification in a tree canopy has been a matter of considerable attention, not least because of inconsistent definitions and methodology (Bourgeron, 1983). Richards (1983) referred to the stratification of leaf mass and of individual tree heights for which the evidence is weak. Stratification of species refers to the vertical aggregation of average mature heights of species. Extensive quantitative documentation of this phenomenon is also lacking, but there is some support for its existence (Pukkala *et al*., 1991).

In systems with discontinuous canopies, a more detailed canopy description is necessary, since the diurnal variation of radiation availability and interception are much larger than in closed forests (Palmer, 1977). Array models have been developed that account for horizontal differences (Palmer, 1977), and these canopies were often modeled as a series of onedimensional layers that are horizontally homogeneous. But because individual crowns are not distinguished, their application is limited to canopies which can be defined in simple geometrical terms.

Two dimensional models have been used to explore how branch angle and branch length influence leaf overlap and hence the efficiency of irradiance capture (Fisher, 1992). Two dimensional models for parallel rows have been described for rectangular (Goudriaan, 1977; Sinoquet and Bonhomme, 1992), elliptical (Charles-Edwards and Thorpe, 1976) and triangular (Jackson and Palmer, 1979) cross-sections. These models are adequate for infinite hedgerows. They consider the interception of light from only a single direction, whereas, in reality, diffuse radiation comes from many directions and the direction of the solar beam changes during the day (Bristow *et al*., 1985). Modeling an isolated tree presents more difficulty than uniform canopies or row crops where it is possible to reduce the dimension of the problem to a single coordinate. For row crops it is still possible to develop this analysis in a fairly straightforward manner (Gijzen and Goudriaan, 1989).

Recently three-dimensional models have been used to assess light interception by individual trees. These models have generally been focused on either relatively simple canopy structures or are species-specific, or have considered the crown as a series of layers, or cells with particular foliage characteristics (Myneni and Impens, 1985). Myneni *et al.* (1990) formulated a three-dimensional leaf canopy model using a modified discrete ordinates method. Myneni (1991) used fractal models of trees to simulate leaf area distributions for modeling the radiative transfer and photosynthesis in a forest canopy. However, most of these studies have simply discussed the adaptive significance of species-specific tree crown architectures as simple allometries between crown dimensions (crown depth, width, or area) and individual sizes (mass, stem diameter at breast height), and have limited the investigations on the effects of individual crown architecture as vertical foliage distributions on the interactions between them (Mohren *et al*., 1984; Biging and Wensel, 1990; Sinoquet and Bonhomme, 1992).

Typically, crown architecture is described in terms of the amount and spatial distribution of leaf area within the crown and defined as the set of features delineating the shape, size and

Soil-Tree-Atmosphere Water Relations 169

surfaces; savannah, bog, pasture land and coniferous forest. For all these surfaces, the anisotropy (ratio of the highest to the lowest reflectance value) increases with increasing zenith angle of incidence from about a factor of three to about a factor of ten or more due to the shadowing effects produced by the vertical structure of the canopies (Kriebel, 1978).

In order to estimate the light extinction coefficient, the leaf inclination function and the solar radiation, knowledge of solar position and day length at the time of interest is required. Running and Coughlan (1988) used the latitude of the modeled location and the day of the year as input to the day length calculations of transpiration and energy balance. Nikolov and Zeller (1992) employed the work done by Running and Coughlan (1988) to model solar radiation. Wang and Jarvis (1990) also incorporated the solar position into their radiative transfer model: MAESTRO, using the method developed by Barkstrom (1981). West and Wells (1992) also used the same method (Barkstrom, 1981) to estimate the solar position in a

Radiation availability is one of the main driving forces behind water uptake by trees. The geometrical architecture of a tree, one of the main factors determining the radiative transfer within the canopy, defines the size, shape and geometry of the tree. The position and the size of trees and their component parts, and the orientation of leaves all play significant roles in the interaction of the tree with the incident radiation. Leaf area and the spatial arrangement of the foliage, branches, and stems determine the transmission of radiation

Radiative transfer in vegetation canopies has been studied for decades ( Ross, 1981; Myneni and Ross, 1990; Law et al., 2000). A variety of models have been developed, ranging from models using the simple Beer's law to complex computer simulation models (Wang and Baldocchi, 1989; Myneni *et al.*, 1990; Andrieu *et al.*, 1995). Theoretical basics and reviews of different models can be found in Ross (1981), Goel (1988) and Myneni and Ross (1990).

The leaf-angle distribution which is a fundamental property of plant canopy structure is needed for computing distributions of leaf irradiance (Campbell, 1990) along with radiation transfer through canopies and extinction coefficients. The leaf inclination angle (θL) is defined as the angle between the vertical and a normal to the leaf surface, and the leaf azimuthal angle (θK) is the angle about the points of the compass between a normal to the leaf surface and the solar beam. Idealized leaf inclination density functions have been widely used to approximate actual leaf angle distributions (Ross, 1981). Several formulae have been given for uniform leaf inclination angles, but randomly distributed azimuthal angles (Lang, 1986; Goudriaan, 1988; Campbell, 1990). The leaf inclination described by one mean leaf angle over-evaluates the effect of the leaf orientation on radiative exchanges (Lemeur, 1973). A two-dimensional probability function *g(*θLθK*)d*θL*, d*θK can be used to describe the fraction of the leaf area

The mathematical description of the leaf orientation distribution is a troublesome part in modeling radiative transfer through a vegetation canopy. Yet it is important because it is the only representation of the canopy geometrical structure, and directly related to the *G*function (Ross, 1981) and the canopy scattering phase function - the two most important functions in the radiative transfer theory. Early practices were either to classify the leaf

oriented with the inclination, θL and the azimuth θK (Lemeur, 1973).

light interception model.

through a forest canopy.

**3.3 Radiative transfer process within the canopy** 

geometry of the tree, and described in terms of the amount and spatial distribution of leaf area within the crown. The crown is assumed to be symmetrical around the tree stem. The radial distribution of leaf area can be approximated by a Gaussian-like pattern (Morales *et al.,* 1996); i.e. the greatest leaf area occurred half the distance between the stem axis and the edge of the crown. However, the direct measurement of leaf area in a three-dimensional canopy is almost impossible (Castro and Fetcher, 1998). Therefore, in order to simplify the modeling task, leaf area in vertically-structured canopy layers can be assumed to be homogeneous, ignoring radial distribution of leaf area around the stem (Kinerson et al., 1974). Furthermore, the vertical distribution of leaf area can be considered to fellow a Weibull statistical pattern (Yong et al., 1993).

Not only are leaves at the top of canopies (or on different sides of isolated trees) normally subject to different energy loads, but the path-lengths along which water must move to different parts of the tree are different (Landsberg, 1986). Procedures to predict the vertical profile of leaf area in the crown were developed (Kinerson and Fritchen, 1971; Schreuder and Swank, 1974; Gray, 1978; Beadle et al., 1982; Massman, 1981; Hagihara and Hozumi, 1986). The beam-path length for each sub-canopy within the canopy can be estimated for varying zenith angles of the sun. A rigorous description of the problem which involves setting up a system of mathematical equations to describe the radiation field at all points in the space within the crown should be considered in the calculation of the radiative transfer process.

It is well established that crown structure has important implications for light interception (Horn, 1971; Givnish, 1984; Kohyama, 1991). Isolated trees, in contrast to those growing in rows, often tend to form in the following shapes - cone, intermediate, cylinder or parabola. These shapes result from the typical growth habits of trees (Sinoquet and Bonhomme, 1992).

The canopy architecture is a complex and dynamic outcome of the evolutionary and ecological interactions and feedbacks between vegetation and environment. As such, canopy structure is a key feature of forest ecosystem processes (Campbell and Norman, 1989; Norman and Campbell, 1989). Thus, understanding and quantifying canopy characteristics is critical in modeling processes in the canopy and in predicting ecosystem responses (Meyers and Paw U, 1987).

#### **3.2 Solar position and daylength estimates**

Transpiration and water-use models require knowledge of the solar position (defined in terms of zenith and azimuth angles) and day length. The zenith angle of the sun has been shown to influence canopy and soil albedo (Stewart, 1971; Pinker *et al*., 1980) and absorption (Otterman *et al*., 1993). The absorption of solar irradiance by trees and soil under trees is significantly larger than the absorption of the bare soil surface, especially at large zenith angles. The absorption of solar irradiance by trees is larger than that by soil under trees at high zenith angles (Otterman *et al*., 1993) so the efficiency of radiation transfer within canopies increases with increasing zenith angle. Therefore, the zenith angle dependence enters the formulation of radiation transfer to express the partitioning of absorbed solar energy between the tree and the soil components of the surface.

Kock *et al*. (1990) found that the sun zenith angle has a decisive influence on the spectral reflectance measured above forest trees, even for solar angle variations of less than 10%. Kriebel (1978) has reported the reflectance values at 0.52μm wavelength of four natural

geometry of the tree, and described in terms of the amount and spatial distribution of leaf area within the crown. The crown is assumed to be symmetrical around the tree stem. The radial distribution of leaf area can be approximated by a Gaussian-like pattern (Morales *et al.,* 1996); i.e. the greatest leaf area occurred half the distance between the stem axis and the edge of the crown. However, the direct measurement of leaf area in a three-dimensional canopy is almost impossible (Castro and Fetcher, 1998). Therefore, in order to simplify the modeling task, leaf area in vertically-structured canopy layers can be assumed to be homogeneous, ignoring radial distribution of leaf area around the stem (Kinerson et al., 1974). Furthermore, the vertical distribution of leaf area can be considered to fellow a

Not only are leaves at the top of canopies (or on different sides of isolated trees) normally subject to different energy loads, but the path-lengths along which water must move to different parts of the tree are different (Landsberg, 1986). Procedures to predict the vertical profile of leaf area in the crown were developed (Kinerson and Fritchen, 1971; Schreuder and Swank, 1974; Gray, 1978; Beadle et al., 1982; Massman, 1981; Hagihara and Hozumi, 1986). The beam-path length for each sub-canopy within the canopy can be estimated for varying zenith angles of the sun. A rigorous description of the problem which involves setting up a system of mathematical equations to describe the radiation field at all points in the space within the

It is well established that crown structure has important implications for light interception (Horn, 1971; Givnish, 1984; Kohyama, 1991). Isolated trees, in contrast to those growing in rows, often tend to form in the following shapes - cone, intermediate, cylinder or parabola. These shapes result from the typical growth habits of trees (Sinoquet and Bonhomme, 1992). The canopy architecture is a complex and dynamic outcome of the evolutionary and ecological interactions and feedbacks between vegetation and environment. As such, canopy structure is a key feature of forest ecosystem processes (Campbell and Norman, 1989; Norman and Campbell, 1989). Thus, understanding and quantifying canopy characteristics is critical in modeling processes in the canopy and in predicting ecosystem responses

Transpiration and water-use models require knowledge of the solar position (defined in terms of zenith and azimuth angles) and day length. The zenith angle of the sun has been shown to influence canopy and soil albedo (Stewart, 1971; Pinker *et al*., 1980) and absorption (Otterman *et al*., 1993). The absorption of solar irradiance by trees and soil under trees is significantly larger than the absorption of the bare soil surface, especially at large zenith angles. The absorption of solar irradiance by trees is larger than that by soil under trees at high zenith angles (Otterman *et al*., 1993) so the efficiency of radiation transfer within canopies increases with increasing zenith angle. Therefore, the zenith angle dependence enters the formulation of radiation transfer to express the partitioning of absorbed solar

Kock *et al*. (1990) found that the sun zenith angle has a decisive influence on the spectral reflectance measured above forest trees, even for solar angle variations of less than 10%. Kriebel (1978) has reported the reflectance values at 0.52μm wavelength of four natural

crown should be considered in the calculation of the radiative transfer process.

Weibull statistical pattern (Yong et al., 1993).

(Meyers and Paw U, 1987).

**3.2 Solar position and daylength estimates** 

energy between the tree and the soil components of the surface.

surfaces; savannah, bog, pasture land and coniferous forest. For all these surfaces, the anisotropy (ratio of the highest to the lowest reflectance value) increases with increasing zenith angle of incidence from about a factor of three to about a factor of ten or more due to the shadowing effects produced by the vertical structure of the canopies (Kriebel, 1978).

In order to estimate the light extinction coefficient, the leaf inclination function and the solar radiation, knowledge of solar position and day length at the time of interest is required. Running and Coughlan (1988) used the latitude of the modeled location and the day of the year as input to the day length calculations of transpiration and energy balance. Nikolov and Zeller (1992) employed the work done by Running and Coughlan (1988) to model solar radiation. Wang and Jarvis (1990) also incorporated the solar position into their radiative transfer model: MAESTRO, using the method developed by Barkstrom (1981). West and Wells (1992) also used the same method (Barkstrom, 1981) to estimate the solar position in a light interception model.

#### **3.3 Radiative transfer process within the canopy**

Radiation availability is one of the main driving forces behind water uptake by trees. The geometrical architecture of a tree, one of the main factors determining the radiative transfer within the canopy, defines the size, shape and geometry of the tree. The position and the size of trees and their component parts, and the orientation of leaves all play significant roles in the interaction of the tree with the incident radiation. Leaf area and the spatial arrangement of the foliage, branches, and stems determine the transmission of radiation through a forest canopy.

Radiative transfer in vegetation canopies has been studied for decades ( Ross, 1981; Myneni and Ross, 1990; Law et al., 2000). A variety of models have been developed, ranging from models using the simple Beer's law to complex computer simulation models (Wang and Baldocchi, 1989; Myneni *et al.*, 1990; Andrieu *et al.*, 1995). Theoretical basics and reviews of different models can be found in Ross (1981), Goel (1988) and Myneni and Ross (1990).

The leaf-angle distribution which is a fundamental property of plant canopy structure is needed for computing distributions of leaf irradiance (Campbell, 1990) along with radiation transfer through canopies and extinction coefficients. The leaf inclination angle (θL) is defined as the angle between the vertical and a normal to the leaf surface, and the leaf azimuthal angle (θK) is the angle about the points of the compass between a normal to the leaf surface and the solar beam. Idealized leaf inclination density functions have been widely used to approximate actual leaf angle distributions (Ross, 1981). Several formulae have been given for uniform leaf inclination angles, but randomly distributed azimuthal angles (Lang, 1986; Goudriaan, 1988; Campbell, 1990). The leaf inclination described by one mean leaf angle over-evaluates the effect of the leaf orientation on radiative exchanges (Lemeur, 1973). A two-dimensional probability function *g(*θLθK*)d*θL*, d*θK can be used to describe the fraction of the leaf area oriented with the inclination, θL and the azimuth θK (Lemeur, 1973).

The mathematical description of the leaf orientation distribution is a troublesome part in modeling radiative transfer through a vegetation canopy. Yet it is important because it is the only representation of the canopy geometrical structure, and directly related to the *G*function (Ross, 1981) and the canopy scattering phase function - the two most important functions in the radiative transfer theory. Early practices were either to classify the leaf

Soil-Tree-Atmosphere Water Relations 171

angle distribution consists of two angles i.e., the leaf inclination angle (θL) and the leaf azimuthal angle (θK). The θL is defined as the angle between the vertical and a normal to the leaf surface, and the θK is the angle about the points of the compass between a normal to the leaf surface and the solar beam (Campbell, 1990) (Fig. 3). Idealized leaf inclination density functions have been widely used to approximate actual leaf angle distributions. If all leaves are inclined at a constant angle (mean leaf angle)*,* θ0, then, the inclination angle distribution,

<sup>0</sup> ( ) ( )sin *LL L g*

where δ(θL - θ0) is the Dirac delta function. A horizontal distribution results when θ<sup>0</sup> *=* 0, a vertical or cylindrical distribution when θ<sup>0</sup> *=* π*/2*, and a conical distribution when θ0 is between these values. In most plants the upper (adaxial) surfaces of the leaves face the upper hemisphere so that θL varies between 0 and 90 whereas θK varies between 0 and 360. The leaf inclination described by one mean leaf angle over-evaluates the effect of the leaf orientation on radiative exchanges (Lemeur, 1973). A two-dimensional probability function *g(*θLθK*)d*θL*, d*θK can be used to describe the fraction of the leaf area oriented with the inclination, θL and the azimuth θK (Lemeur, 1973). This function expresses the probability that a leaf has an inclination within θL and θL*+d*θL and an azimuth within θK and θK*+d*θK.

 

(,) 1

   

 (12)

(11)

 

*KL KL g dd*

It is possible to estimate the value of light extinction coefficient for different sets of orientation of the leaves and the direction of incoming radiation by considering the projected area of solids having the angle distributions for the given distribution function (Monteith, 1975). The value of the light extinction coefficient for direct beam radiation is related to its mean cosine of incidence on the leaf surfaces. For a single flat leaf, the cosine of the angle of irradiance is described by θL, θK, and φL, and the cosine *t*(φL,θL*,*θK) is determined

' ' ( ) cos cos sin sin cos *LLK L L L L K t*

 

The distribution of *t* is uniform with θK if no preferred azimuthal angle is assumed. The cumulative distribution function of *t* can be found by increasing θK from *-*π to 0. With symmetrical geometry , the other half of the azimuthal circle (from 0 to π) can be omitted. θK, at the range from *-*π to 0, is equivalent to the range of 0 - 1 for the cumulative distribution probability *S*. The average value of *t* can be found as the integral of *t* with respect to *S*. This is the same quantity as the average projection of the leaves into the direction of the solar beam called the *O* function by De Wit (1965) and the *G* function by Ross (1975) and

As pointed out by Myneni *et al*. (1988), the canopy scattering phase function determines how realistically the physics of the radiation transport process in vegetation is represented (Ross,

  (10)

 

2 /2

*K L*

*K L*

0 0

 

Therefore, integration over θL and θK leads to (Ross, 1975):

 

 

*g*(θL) is given by (Ross, 1981):

as (Ross, 1975):

Goudriaan (1989).

**3.3.2 Radiation scattering** 

orientation distributions as erectophile, planophile, spherical, extremophile, or plagiophile, and give a mathematical description for each of these leaf orientation distributions (De Wit, 1965), or to represent a leaf orientation distribution by frequencies in three or nine leaf angle classes (Goudriaan, 1977, 1988).

## **3.3.1 Radiation transfer through the canopy**

Several studies have been done to examine the influence of canopy architecture on penetration of radiation in theoretical models (Kira *et al*., 1969; Oker-Blom and Kellomaki, 1983; Campbell and Norman, 1989). Many of these studies used the Beer-Lambert model, which defines canopy light penetration as proportional to the cumulative leaf area index (e.g. Monsi and Saeki, 1953). Small within-species variations in light extinction have often been attributed to changes in solar altitude during sampling of under-canopy light flux density (Campbell and Norman, 1989) rather than to intra-specific variability. Recent results, however, suggested substantial variation in light extinction for several tree species (Gholz *et al*., 1991). Models have often been used to estimate light penetration in forest stands of different *LAI* (Dalla-Tea and Jokela, 1991; Gholz *et al*., 1991).

Fig. 3. Diagrammatic representation of the orientation of the normal to a leaf surface

The leaf angle distribution, which is a mathematical description of the angular orientation of the leaves in the canopy, is needed for computing distributions of leaf irradiance along with radiation transfer through canopies and extinction coefficients (Campbell, 1990). The leaf

orientation distributions as erectophile, planophile, spherical, extremophile, or plagiophile, and give a mathematical description for each of these leaf orientation distributions (De Wit, 1965), or to represent a leaf orientation distribution by frequencies in three or nine leaf angle

Several studies have been done to examine the influence of canopy architecture on penetration of radiation in theoretical models (Kira *et al*., 1969; Oker-Blom and Kellomaki, 1983; Campbell and Norman, 1989). Many of these studies used the Beer-Lambert model, which defines canopy light penetration as proportional to the cumulative leaf area index (e.g. Monsi and Saeki, 1953). Small within-species variations in light extinction have often been attributed to changes in solar altitude during sampling of under-canopy light flux density (Campbell and Norman, 1989) rather than to intra-specific variability. Recent results, however, suggested substantial variation in light extinction for several tree species (Gholz *et al*., 1991). Models have often been used to estimate light penetration in forest stands of

Fig. 3. Diagrammatic representation of the orientation of the normal to a leaf surface

The leaf angle distribution, which is a mathematical description of the angular orientation of the leaves in the canopy, is needed for computing distributions of leaf irradiance along with radiation transfer through canopies and extinction coefficients (Campbell, 1990). The leaf

classes (Goudriaan, 1977, 1988).

**3.3.1 Radiation transfer through the canopy** 

different *LAI* (Dalla-Tea and Jokela, 1991; Gholz *et al*., 1991).

angle distribution consists of two angles i.e., the leaf inclination angle (θL) and the leaf azimuthal angle (θK). The θL is defined as the angle between the vertical and a normal to the leaf surface, and the θK is the angle about the points of the compass between a normal to the leaf surface and the solar beam (Campbell, 1990) (Fig. 3). Idealized leaf inclination density functions have been widely used to approximate actual leaf angle distributions. If all leaves are inclined at a constant angle (mean leaf angle)*,* θ0, then, the inclination angle distribution, *g*(θL) is given by (Ross, 1981):

$$\log(\theta\_L) = \delta(\theta\_L - \theta\_0)\sin\theta\_L \tag{10}$$

where δ(θL - θ0) is the Dirac delta function. A horizontal distribution results when θ<sup>0</sup> *=* 0, a vertical or cylindrical distribution when θ<sup>0</sup> *=* π*/2*, and a conical distribution when θ0 is between these values. In most plants the upper (adaxial) surfaces of the leaves face the upper hemisphere so that θL varies between 0 and 90 whereas θK varies between 0 and 360. The leaf inclination described by one mean leaf angle over-evaluates the effect of the leaf orientation on radiative exchanges (Lemeur, 1973). A two-dimensional probability function *g(*θLθK*)d*θL*, d*θK can be used to describe the fraction of the leaf area oriented with the inclination, θL and the azimuth θK (Lemeur, 1973). This function expresses the probability that a leaf has an inclination within θL and θL*+d*θL and an azimuth within θK and θK*+d*θK. Therefore, integration over θL and θK leads to (Ross, 1975):

$$\int\_{\theta\_{\mathbb{K}}=0}^{\theta\_{\mathbb{K}}=2\pi} \int\_{\theta\_{\mathbb{L}}=0}^{\theta\_{\mathbb{L}}=\pi/2} g(\theta\_{\mathbb{K}'}, \theta\_{\mathbb{L}}) d\theta\_{\mathbb{K}} d\theta\_{\mathbb{L}} = 1 \tag{11}$$

It is possible to estimate the value of light extinction coefficient for different sets of orientation of the leaves and the direction of incoming radiation by considering the projected area of solids having the angle distributions for the given distribution function (Monteith, 1975). The value of the light extinction coefficient for direct beam radiation is related to its mean cosine of incidence on the leaf surfaces. For a single flat leaf, the cosine of the angle of irradiance is described by θL, θK, and φL, and the cosine *t*(φL,θL*,*θK) is determined as (Ross, 1975):

$$\text{At}(\varphi\_{\text{L}}\,\theta\_{\text{L}}\,\theta\_{\text{K}}) = \cos\varphi\_{\text{L}}\cos\theta\_{\text{L}} + \sin\varphi\_{\text{L}}\sin\theta\_{\text{L}}\cos\theta\_{\text{K}} \tag{12}$$

The distribution of *t* is uniform with θK if no preferred azimuthal angle is assumed. The cumulative distribution function of *t* can be found by increasing θK from *-*π to 0. With symmetrical geometry , the other half of the azimuthal circle (from 0 to π) can be omitted. θK, at the range from *-*π to 0, is equivalent to the range of 0 - 1 for the cumulative distribution probability *S*. The average value of *t* can be found as the integral of *t* with respect to *S*. This is the same quantity as the average projection of the leaves into the direction of the solar beam called the *O* function by De Wit (1965) and the *G* function by Ross (1975) and Goudriaan (1989).

#### **3.3.2 Radiation scattering**

As pointed out by Myneni *et al*. (1988), the canopy scattering phase function determines how realistically the physics of the radiation transport process in vegetation is represented (Ross,

Soil-Tree-Atmosphere Water Relations 173

radiative transfer and photosynthesis in a forest canopy. Recently, three-dimensional models such as Myneni *et al*.'s (1990) have been used to assess light interception by individual trees. These models have generally focused on either relatively simple canopy structures or are species-specific, or have considered the crown as a series of layers, or cells with particular foliage characteristics (Myneni and Impens, 1985; Ryel *et al.*, 1993). The foliage intercepts most of the radiation captured by the canopy, and an increased foliage area decreases the penetration of radiation. It is normally assumed that only the crown intercepts radiation; stems, branches and fruits are 'invisible' for the light beams (Jarvis and

Short-wave direct radiation (*Rbs*) and diffuse radiation (*Rds*) are the main spectral characteristic for many radiative transfer models Myneni (1991). However, Kjelgren and Montague (1998) stated that trees over asphalt had consistently higher leaf temperatures than those over turf due to the interception of the greater upwards long-wave radiation fluxes due to the higher surface temperatures of the asphalt over turf. When bare soil between trees exits, upwards long-wave radiation may contribute substantially to the energy balance in the canopy. Therefore, the radiation above, within and below the canopy can be separated in terms of short-wave radiation (*Rb*) and long-wave radiation (*Rlw*), and

The characterization of the magnitude and direction of the diffuse and direct components of incident radiation is required as the light distribution within the canopy depends on the spectral optical properties of the leaves and the soil. Therefore, the optical properties of the vegetation elements and the soil are important factors to determine radiative transfer within

Radiant energy received from the sun and the atmosphere is exchanged for latent and sensible heat by plants and the soil surface. Sensible heat is transferred because of the temperature difference between trees and surrounding air, and it moves by convection, advection and diffusion. The relative amounts of latent and sensible heat exchanged by plants and soil surfaces are understood to be mediated by resistance to water movement in

Gates (1962) pioneered leaf energy budget investigations in plant ecology. Although the interaction between a canopy and its radiant energy environment can be extremely complex, a simplified approach that serves to introduce basic concepts is now well-established. According to energy conservation, the heat gain by radiation should be equal to the heat

The leaf temperature of trees is higher at the upper part than at the lower in daytime (He *et al*., 1996; Zermeno and Hipps, 1997) and the difference between leaf temperatures at a depth of 20 m and 11 m is about 12 0C around noon (Miyashita and Maitani, 1998). Therefore, it is clear that energy balance cannot be applied uniformly throughout the canopy. Describing the long-wave radiative transfer is further complicated in plant canopies because leaf elements are also long-wave radiation sources (Rotenberg et al., 1999). This fact complicates the description of long-wave radiative transfer in plant canopies. To make the situation even worse, in order to solve the leaf temperature from

aqueous and vapour phases between the plant or the soil and the atmosphere.

losses in the canopy when leaf temperature is constant.

Leverenz, 1983; Cannel *et al*., 1987).

*Rb* is separated into *Rbs* and *Rds* radiation.

the canopy.

**3.4 Canopy-energy balance** 

1981). Therefore, the scattering phase function should be one of the most important factors determining the level of accuracy of the radiative transfer process.

With the assumption of isotropic scattering for leaf elements, Sellers (1985) found a simple expression for the scattering phase function, and solved the equations analytically. However, the isotropic scattering assumption is a rather crude approximation for real leaf elements. As a consequence of this assumption, scattering does not change with the relative magnitudes of the reflection and transmission coefficients for a given leaf scattering coefficient. The unrealistic implications of this shortcoming can be seen easily in the expression for the single scattering albedo for a semi-infinite horizontal leaf canopy. It is a quarter of the leaf scattering coefficient (the leaf scattering coefficient equals the sum of the leaf reflection and transmission coefficients). Since only single scattering is considered, radiation which is transmitted through a leaf is never able to escape from the canopy and cannot contribute to the canopy reflection. Thus, the transmission coefficient is unrelated to the canopy single scattering albedo, and the only relevant leaf optical parameter is the leaf reflection coefficient. For most situations, canopy hemispheric reflectance as given by the isotropic two-stream model, proposed by Dickinson (1983) and further developed by Sellers (1985), hardly changes with the relative magnitudes of the reflection and transmission coefficients. Although the isotropic two-stream approximation model has widely been applied (Sellers, 1985), comprehensive testing of the model has been limited.

Although Hassika *et al.* (1997) stated that the radiation rescattered by the crowns can be neglected, estimates using the Kubelka-Munk equations that indicate this term is small, up to secondary scattering may be good enough for the canopy radiative process.

#### **3.3.3 Radiative transfer process with the canopy**

Radiative transfer models are based on more or less abstract representations of reality, usually in relation to the aim and scale of the simulation; models based on a uniform-canopy hypothesis and on the architectural description of tree crowns represent the opposite extremes of this range. Radiative models of the first type can be used to simulate light interception by agricultural crops and uniform forests (Wang and Baldocchi, 1989) and to provide insight into the role of specific characteristics such as leaf angle and leaf area index or vertical distribution of leaf characteristics such as leaf mass per unit area or volume (Gutschick and Weigel, 1988). Models in the second group work at the same scale as leaves or leaf groups and are used to investigate the relationship between canopy architecture and radiative regime in complex heterogenous canopies (Myneni *et al.*, 1990). For individual trees, qualitative models of branching have been used to describe the development of form (Halle *et al.*, 1978), but these model applications have had limited ecological applicability in understanding how specific architectures influence light capture. Two dimensional models have been used to explore how branch angle and length influence leaf-overlap and, hence, the efficiency of light capture (Fisher *et al*., 1981). These approaches were limited in that they considered the interception of light from only a single direction, whereas, in reality, diffuse radiation comes from many directions and the direction of the solar beam changes during the day. Neither of these model types can address some of the research tasks typical for forest ecology (Pukkala *et al*., 1993).

Myneni *et al.* (1990) formulated a modified discrete ordinates method for a threedimensional leaf canopy. Myneni (1991) used fractal models of trees to simulate the

1981). Therefore, the scattering phase function should be one of the most important factors

With the assumption of isotropic scattering for leaf elements, Sellers (1985) found a simple expression for the scattering phase function, and solved the equations analytically. However, the isotropic scattering assumption is a rather crude approximation for real leaf elements. As a consequence of this assumption, scattering does not change with the relative magnitudes of the reflection and transmission coefficients for a given leaf scattering coefficient. The unrealistic implications of this shortcoming can be seen easily in the expression for the single scattering albedo for a semi-infinite horizontal leaf canopy. It is a quarter of the leaf scattering coefficient (the leaf scattering coefficient equals the sum of the leaf reflection and transmission coefficients). Since only single scattering is considered, radiation which is transmitted through a leaf is never able to escape from the canopy and cannot contribute to the canopy reflection. Thus, the transmission coefficient is unrelated to the canopy single scattering albedo, and the only relevant leaf optical parameter is the leaf reflection coefficient. For most situations, canopy hemispheric reflectance as given by the isotropic two-stream model, proposed by Dickinson (1983) and further developed by Sellers (1985), hardly changes with the relative magnitudes of the reflection and transmission coefficients. Although the isotropic two-stream approximation model has widely been

determining the level of accuracy of the radiative transfer process.

applied (Sellers, 1985), comprehensive testing of the model has been limited.

to secondary scattering may be good enough for the canopy radiative process.

**3.3.3 Radiative transfer process with the canopy** 

Although Hassika *et al.* (1997) stated that the radiation rescattered by the crowns can be neglected, estimates using the Kubelka-Munk equations that indicate this term is small, up

Radiative transfer models are based on more or less abstract representations of reality, usually in relation to the aim and scale of the simulation; models based on a uniform-canopy hypothesis and on the architectural description of tree crowns represent the opposite extremes of this range. Radiative models of the first type can be used to simulate light interception by agricultural crops and uniform forests (Wang and Baldocchi, 1989) and to provide insight into the role of specific characteristics such as leaf angle and leaf area index or vertical distribution of leaf characteristics such as leaf mass per unit area or volume (Gutschick and Weigel, 1988). Models in the second group work at the same scale as leaves or leaf groups and are used to investigate the relationship between canopy architecture and radiative regime in complex heterogenous canopies (Myneni *et al.*, 1990). For individual trees, qualitative models of branching have been used to describe the development of form (Halle *et al.*, 1978), but these model applications have had limited ecological applicability in understanding how specific architectures influence light capture. Two dimensional models have been used to explore how branch angle and length influence leaf-overlap and, hence, the efficiency of light capture (Fisher *et al*., 1981). These approaches were limited in that they considered the interception of light from only a single direction, whereas, in reality, diffuse radiation comes from many directions and the direction of the solar beam changes during the day. Neither of these model types can address some of the research tasks typical for forest ecology (Pukkala *et al*., 1993).

Myneni *et al.* (1990) formulated a modified discrete ordinates method for a threedimensional leaf canopy. Myneni (1991) used fractal models of trees to simulate the radiative transfer and photosynthesis in a forest canopy. Recently, three-dimensional models such as Myneni *et al*.'s (1990) have been used to assess light interception by individual trees. These models have generally focused on either relatively simple canopy structures or are species-specific, or have considered the crown as a series of layers, or cells with particular foliage characteristics (Myneni and Impens, 1985; Ryel *et al.*, 1993). The foliage intercepts most of the radiation captured by the canopy, and an increased foliage area decreases the penetration of radiation. It is normally assumed that only the crown intercepts radiation; stems, branches and fruits are 'invisible' for the light beams (Jarvis and Leverenz, 1983; Cannel *et al*., 1987).

Short-wave direct radiation (*Rbs*) and diffuse radiation (*Rds*) are the main spectral characteristic for many radiative transfer models Myneni (1991). However, Kjelgren and Montague (1998) stated that trees over asphalt had consistently higher leaf temperatures than those over turf due to the interception of the greater upwards long-wave radiation fluxes due to the higher surface temperatures of the asphalt over turf. When bare soil between trees exits, upwards long-wave radiation may contribute substantially to the energy balance in the canopy. Therefore, the radiation above, within and below the canopy can be separated in terms of short-wave radiation (*Rb*) and long-wave radiation (*Rlw*), and *Rb* is separated into *Rbs* and *Rds* radiation.

The characterization of the magnitude and direction of the diffuse and direct components of incident radiation is required as the light distribution within the canopy depends on the spectral optical properties of the leaves and the soil. Therefore, the optical properties of the vegetation elements and the soil are important factors to determine radiative transfer within the canopy.

#### **3.4 Canopy-energy balance**

Radiant energy received from the sun and the atmosphere is exchanged for latent and sensible heat by plants and the soil surface. Sensible heat is transferred because of the temperature difference between trees and surrounding air, and it moves by convection, advection and diffusion. The relative amounts of latent and sensible heat exchanged by plants and soil surfaces are understood to be mediated by resistance to water movement in aqueous and vapour phases between the plant or the soil and the atmosphere.

Gates (1962) pioneered leaf energy budget investigations in plant ecology. Although the interaction between a canopy and its radiant energy environment can be extremely complex, a simplified approach that serves to introduce basic concepts is now well-established. According to energy conservation, the heat gain by radiation should be equal to the heat losses in the canopy when leaf temperature is constant.

The leaf temperature of trees is higher at the upper part than at the lower in daytime (He *et al*., 1996; Zermeno and Hipps, 1997) and the difference between leaf temperatures at a depth of 20 m and 11 m is about 12 0C around noon (Miyashita and Maitani, 1998). Therefore, it is clear that energy balance cannot be applied uniformly throughout the canopy. Describing the long-wave radiative transfer is further complicated in plant canopies because leaf elements are also long-wave radiation sources (Rotenberg et al., 1999). This fact complicates the description of long-wave radiative transfer in plant canopies. To make the situation even worse, in order to solve the leaf temperature from

Soil-Tree-Atmosphere Water Relations 175

Sensible and latent heat transfer processes consider the environmental gradients that occur between some reference level outside the canopy and sub-canopies inside the canopy in a three-dimensional modeling framework. Gradients of these quantities exist between the bulk air and sub-canopies within the canopy and drive the fluxes from (or to) the canopy. To evaluate these gradients, for example in a tree canopy, linkages between the strengths of the respective sources and sinks and scalar concentration in the sub-canopy must be considered. These linkages arise because the rate at which material is released (or taken up) affects the local scalar concentration in sub-canopies within the canopy, and the rate of leaf emission, (or uptake), depends on the local scalar concentration. Implicitly, it can be assumed that the atmosphere within the sub-canopy is well mixed and no inter- or intra-layer mixing within the canopy. It can be assumed further that water vapor outside any sub-canopy (but inside the canopy) is removed without affecting the micro-environment along its pathway to the bulk air outside the canopy. However, it is recognized that these assumptions are not accurate, but a reasonable answer to a difficult, if not unquantifiable, situation. Despite these assumptions, the humidity deficit between the various sub-canopies and the bulk air is not

1. surface temperature with respect to the sub-canopy location within the canopy, and

The diurnal variations of wind speed and air temperature and humidity of air can be calculated using empirical equations given in the literature (Landsberg and James, 1971; Dogniaux 1977; Wu, 1990; Eckersten, 1991; Paw U *et al.*, 1995). The shape of the diurnal scalar curves is modeled with a variety of methods with varying degrees of complexity. These methods include linear models and curve-fitting models based typically on sine or

The vertical wind profile affects the boundary layer resistance for heat and water vapor transfer (Landsberg, 1986). Although tree edges can modify the wind speed, and wind direction both inside and outside of the canopy, it is normally assumed that wind speed

Air temperature affects latent and sensible heat flux. A simple method is applied to determine the air temperature inside the canopy from the air temperature outside the canopy and sensible heat flux. Diurnal variation of air temperature can also be estimated using a sinusoidal progression. There are several studies that have been carried out to determine the vertical profile of the air temperature (*Ta*) above the canopy (Wu, 1990), but such knowledge does not exist in the same simple manner for profiles within the canopy. Paw U *et al*. (1995) used a form of surface renewal analysis by assuming that under unstable conditions (canopy warmer than the air), any rise in the temperature profile represents air being heated by the canopy, and under stable conditions (canopy cooler than air) any temperature drop represents air being cooled by the canopy. The air temperature normally changes in a sinusoidal progression during the day and a decreasing exponential curve at

To estimate relative humidity (*Rh*) rather than interpolating it over time using synoptic values, it is more appropriate to use linear interpolation of the absolute air humidity (*Ha*)

2. vertical profiles of air temperature and wind speed outside the canopy.

Fourier analysis (De Wit *et al.*, 1978; Worner, 1988; Fernandez, 1992).

equal as long as the modelling process simulates;

does not change while passing through the canopy.

night (van Engelen and Guerts, 1983).

(Eckersten, 1991).

the energy balance equation, the absorbed long-wave radiation, which in turn depends on the temperature of the leaf, must be known in advance. Solving the long-wave radiative transfer problem relies on iterative search methods which can be computationally intensive for a canopy process model. Complicating this, the energy balance equation has (potentially up to) four roots because long wave radiation is proportional to the fourth power of temperature. At least one of the roots should be bio-physically sensible. Unfortunately there is no guarantee that iterative search methods will converge on the correct root and sometimes yield bifuricated or chaotic solutions (Baldocchi, 1994). To overcome this difficulty, a polynomial expression can be introduced in order to solve the energy balance equation analytically for leaf temperature. Doing so avoids the involvement of a complex algorithm with numerous partial derivatives and iterations in energy balance calculations such as that used by Cienciala *et al*. (1994). Applying the energy balance calculation in a horizontally- or/and vertically-structured canopy allows us to investigate spatial variations of leaf temperature.

The sensible heat flux can be considered to be proportional to the difference between the radiometric temperature which is identical to canopy surface temperature and the air temperature at a reference height, and inversely proportional to aerodynamic resistance.

The large amount of standing biomass in a forest causes a substantial amount of net radiation to be transformed into canopy heat storage, further affecting evapotranspiration (Jarvis, 1976). The heat storage, which is mostly in biochemical reactions represents a fraction of radiation over a very short period (Montheith, 1975) is about 8% of the *Rnet* (Jones, 1992).

#### **3.5 Scalar variations within the canopy**

Many canopy models have been developed to describe the exchange of sensible and latent heat between plant canopies and the atmosphere (Baldocchi, 1993). An important function of these models is to predict the mean profile of humidity and temperature of the air in the canopy, because transpiration at each canopy level depends on the air temperature and humidity at that level.

To calculate these profiles, some assumptions are made about the turbulent transport processes within the canopy. The most common assumption has been that turbulent convection conveys scalars (heat and vapour pressure), down local concentration gradients by a turbulent diffusion process. As such, these models have been based on *K*-theory (Waggoner and Reifsnyder, 1968). The *K*-theory has been challenged by observations of fluxes of scalars moving in directions opposed to their local concentration gradients within plant canopies (Denmead and Bradley, 1985). New theories have been developed to explain counter-gradient transport, but are yet to produce an accurate method to demonstrate scalar transport processes within the canopy. Raupach (1989), for instance, stated that because of the strong influence of source distribution and the relatively weaker influence of the fine details of the inhomogeneous wind field, a rather simple model for the wind field in the canopy is probably all that is necessary to calculate scalar concentrations in the vertical dimension quite accurately. Simple, half-order closure modelling which assumes a uniform scalar profile does not yield large errors in the computation of flux densities because the source-sink formulation of fluxes is relatively insensitive to changes in scalar concentrations in the profile and the scalar gradients are small (Baldocchi, 1992).

the energy balance equation, the absorbed long-wave radiation, which in turn depends on the temperature of the leaf, must be known in advance. Solving the long-wave radiative transfer problem relies on iterative search methods which can be computationally intensive for a canopy process model. Complicating this, the energy balance equation has (potentially up to) four roots because long wave radiation is proportional to the fourth power of temperature. At least one of the roots should be bio-physically sensible. Unfortunately there is no guarantee that iterative search methods will converge on the correct root and sometimes yield bifuricated or chaotic solutions (Baldocchi, 1994). To overcome this difficulty, a polynomial expression can be introduced in order to solve the energy balance equation analytically for leaf temperature. Doing so avoids the involvement of a complex algorithm with numerous partial derivatives and iterations in energy balance calculations such as that used by Cienciala *et al*. (1994). Applying the energy balance calculation in a horizontally- or/and vertically-structured canopy allows

The sensible heat flux can be considered to be proportional to the difference between the radiometric temperature which is identical to canopy surface temperature and the air temperature at a reference height, and inversely proportional to aerodynamic resistance.

The large amount of standing biomass in a forest causes a substantial amount of net radiation to be transformed into canopy heat storage, further affecting evapotranspiration (Jarvis, 1976). The heat storage, which is mostly in biochemical reactions represents a fraction of radiation

Many canopy models have been developed to describe the exchange of sensible and latent heat between plant canopies and the atmosphere (Baldocchi, 1993). An important function of these models is to predict the mean profile of humidity and temperature of the air in the canopy, because transpiration at each canopy level depends on the air temperature and

To calculate these profiles, some assumptions are made about the turbulent transport processes within the canopy. The most common assumption has been that turbulent convection conveys scalars (heat and vapour pressure), down local concentration gradients by a turbulent diffusion process. As such, these models have been based on *K*-theory (Waggoner and Reifsnyder, 1968). The *K*-theory has been challenged by observations of fluxes of scalars moving in directions opposed to their local concentration gradients within plant canopies (Denmead and Bradley, 1985). New theories have been developed to explain counter-gradient transport, but are yet to produce an accurate method to demonstrate scalar transport processes within the canopy. Raupach (1989), for instance, stated that because of the strong influence of source distribution and the relatively weaker influence of the fine details of the inhomogeneous wind field, a rather simple model for the wind field in the canopy is probably all that is necessary to calculate scalar concentrations in the vertical dimension quite accurately. Simple, half-order closure modelling which assumes a uniform scalar profile does not yield large errors in the computation of flux densities because the source-sink formulation of fluxes is relatively insensitive to changes in scalar concentrations

over a very short period (Montheith, 1975) is about 8% of the *Rnet* (Jones, 1992).

in the profile and the scalar gradients are small (Baldocchi, 1992).

us to investigate spatial variations of leaf temperature.

**3.5 Scalar variations within the canopy** 

humidity at that level.

Sensible and latent heat transfer processes consider the environmental gradients that occur between some reference level outside the canopy and sub-canopies inside the canopy in a three-dimensional modeling framework. Gradients of these quantities exist between the bulk air and sub-canopies within the canopy and drive the fluxes from (or to) the canopy. To evaluate these gradients, for example in a tree canopy, linkages between the strengths of the respective sources and sinks and scalar concentration in the sub-canopy must be considered. These linkages arise because the rate at which material is released (or taken up) affects the local scalar concentration in sub-canopies within the canopy, and the rate of leaf emission, (or uptake), depends on the local scalar concentration. Implicitly, it can be assumed that the atmosphere within the sub-canopy is well mixed and no inter- or intra-layer mixing within the canopy. It can be assumed further that water vapor outside any sub-canopy (but inside the canopy) is removed without affecting the micro-environment along its pathway to the bulk air outside the canopy. However, it is recognized that these assumptions are not accurate, but a reasonable answer to a difficult, if not unquantifiable, situation. Despite these assumptions, the humidity deficit between the various sub-canopies and the bulk air is not equal as long as the modelling process simulates;


The diurnal variations of wind speed and air temperature and humidity of air can be calculated using empirical equations given in the literature (Landsberg and James, 1971; Dogniaux 1977; Wu, 1990; Eckersten, 1991; Paw U *et al.*, 1995). The shape of the diurnal scalar curves is modeled with a variety of methods with varying degrees of complexity. These methods include linear models and curve-fitting models based typically on sine or Fourier analysis (De Wit *et al.*, 1978; Worner, 1988; Fernandez, 1992).

The vertical wind profile affects the boundary layer resistance for heat and water vapor transfer (Landsberg, 1986). Although tree edges can modify the wind speed, and wind direction both inside and outside of the canopy, it is normally assumed that wind speed does not change while passing through the canopy.

Air temperature affects latent and sensible heat flux. A simple method is applied to determine the air temperature inside the canopy from the air temperature outside the canopy and sensible heat flux. Diurnal variation of air temperature can also be estimated using a sinusoidal progression. There are several studies that have been carried out to determine the vertical profile of the air temperature (*Ta*) above the canopy (Wu, 1990), but such knowledge does not exist in the same simple manner for profiles within the canopy. Paw U *et al*. (1995) used a form of surface renewal analysis by assuming that under unstable conditions (canopy warmer than the air), any rise in the temperature profile represents air being heated by the canopy, and under stable conditions (canopy cooler than air) any temperature drop represents air being cooled by the canopy. The air temperature normally changes in a sinusoidal progression during the day and a decreasing exponential curve at night (van Engelen and Guerts, 1983).

To estimate relative humidity (*Rh*) rather than interpolating it over time using synoptic values, it is more appropriate to use linear interpolation of the absolute air humidity (*Ha*) (Eckersten, 1991).

Soil-Tree-Atmosphere Water Relations 177

The turbulent transport between canopies and the bulk of the atmosphere depends on the turbulent nature of the planetary boundary layer (Meyers and Paw U, 1986 and 1987). In the transfer of water vapor to and from trees, some exchanges occur by molecular diffusion such as the passage of water through stomata. The flux of diffusing gas (kg m-2 s-1) can then be equated to the concentration difference (kg m-3) over a diffusion resistance (s m-1) as given by Fick's law. In the process of the diffusion of water vapor away from the leaves, the stomatal resistance (*R*t) accounts for diffusion from the evaporation sites within the leaf to the leaf surface, while the leaf boundary layer resistance (*R*a) accounts for diffusion from the surface to the well-mixed surrounding air. Both the stomatal resistance and the leaf boundary layer resistance are highly dependent on the size, shape and surface properties of

Empirically established stomatal resistance, which is the most important factor determining transpiration from high vegetation especially forests, has marked variation within the canopy as well as over the day and the season. Stomatal resistance (*R*t) depends both on soil and atmospheric factors. These factors are short-term changes in leaf water potential, vapour pressure deficit, solar flux, leaf temperature, ambient carbon dioxide concentration and significant drying of the soil (very negative soil water potential). Therefore, there is clearly a need to describe the response of *R*t to atmospheric factors as well as soil water

a. serially integrating the stomatal resistance of individual layers, weighted by leaf area,

b. using measured values of latent heat flux and other relevant variables in a stand-level

Baldocchi *et al*. (1991) presented an excellent overview of the strengths and weaknesses of different approaches for estimating canopy stomatal resistance. As discussed in their paper, the above mentioned approaches may not yield the same results. The former is primarily a physiological parameter whereas the latter involves additional eco-physiological factors within the canopy. The latter also includes the contribution of soil evaporation. Baldocchi *et al.* (1991) developed a multi-layer canopy stomatal conductance model in which the spatial variation of canopy structure and the radiative transfer within the canopy were taken into

Jarvis (1976) has modelled the stomatal resistance as a function of solar radiation, temperature, specific humidity deficit, leaf water potential, and ambient CO2, using a nonlinear least squares technique. Based on the Jarvis work, Stewart (1988) developed a model in which the stomatal resistance was related to solar radiation, temperature, specific humidity deficit, soil moisture deficit and leaf area index. A semi-empirical stomatal resistance model was proposed by Ball *et al.* (1987). After analyzing Ball's model, Aphalo and Jarvis (1993) have proposed a new model, which views the model as a description of the relationship between CO2 flux rate and stomatal conductance, rather than as a model of stomatal conductance alone. Ball's empirical model was later modified

status. Estimation of stomatal resistance has generally involved two approaches;

**3.6 Water flow resistance process in the canopy-atmosphere process** 

the leaves, and wind velocity.

**3.6.1 Stomatal resistance** 

or

account.

equation.

by Leuning (1995).

A knowledge of solar radiation is of interest in studies relating to crop evapotranspiration, forest transpiration and for solar energy applications. The incident global radiation (350- 3000 nm) on the earth's surface (*Rnet*) is a product of incident radiation outside the atmosphere (*Ro*) and the atmospheric transmissivity, which is dependent upon the degree of cloudiness.

If uniform cloudiness is assumed over the day, the mean irradiance (*Rn* M) is computed from the ratio of the daily sum of actual global radiation to the daily sum of global radiation from a clear sky. The effects of cloudiness on radiation within the day can be approximated by introducing a factor that varies according to cloudiness.
