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SPARSE CANOPY SYMPOSIUM INTRODUCTION

Modeling Energy Fluxes from Sparse Canopies and Understorys

The Horticulture and Food Research Institute of New Zealand Ltd., P.O. Box 23, Kerikeri 0470, New Zealand

cdaamen{at}skm.com.au

	ABSTRACT

TOP ABSTRACT INTRODUCTION Theory Data from sparsely vegetated... Preliminary comparison of models Model comparison using land... Discussion Conclusion Appendix REFERENCES

 
Land surfaces are an assemblage of component surface types, for instance overstory vegetation species, understory vegetation species, and bare soil. Often two or more surface types absorb a significant fraction of the available energy to the land surface as a whole. In these cases the interaction of fluxes from the component surfaces may be important to the total land surface energy balance. We compare three models of land surface energy balance: a Penman-Monteith model; a model with two component surfaces that don't interact (patch model); and a model with interacting component surfaces (Shuttleworth-Wallace model). Data from six published studies are used to investigate which models best represent a particular land surface taking account of water supply to the component surfaces and overstory canopy architecture. Flux interaction between component surfaces was only found to be important when there was a large difference between the surface resistances (i.e., water availability to the surfaces). Also, all three models were found to estimate the same land surface energy fluxes (to within 50 W m^-2) when both surface resistances were >300 s m^-1. The ratio of (aerodynamic resistance between the canopy air space and the reference height) to (mean component surface boundary layer resistance) was useful for indicating the level of interaction between component surfaces.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Theory Data from sparsely vegetated... Preliminary comparison of models Model comparison using land... Discussion Conclusion Appendix REFERENCES

 
MANY LAND SURFACES have a canopy of vegetation which does not completely cover the understory or the ground surface beneath, that is, a sparse overstory. Such surfaces include cultivated land (e.g., orchards, vineyards, and many row crops) and natural vegetation. Natural vegetation has a sparse overstory in many semiarid environments (e.g., Nichols, 1992; Massman, 1992; Brenner and Incoll, 1997) although these canopies do occur in other environments (e.g., subarctic wetland, Lafleur and Rouse, 1990). Any land surface with a sparse overstory has at least two significant sources of heat and water vapor: the canopy itself and the understory or soil. These two component surfaces are often very different in their control of water and heat fluxes; both need to be accounted for when calculating a flux from the land surface as a whole. Also, interaction of fluxes from the two component surfaces may be important. For example sensible heat rising from a dry soil understory can increase transpiration from the overstory canopy by raising its temperature.

A simple approach to estimating land surface fluxes is provided by the Penman-Monteith equation which was developed to estimate fluxes from a closed canopy of vegetation (Monteith, 1965). It assumes a homogeneous `big leaf' surface with a single value of vapor density deficit at the surface (Jarvis and McNaughton, 1986). The surface resistance can be considered to be the equivalent resistance of all canopy leaves in parallel (Monteith, 1981). The Penman-Monteith equation can be applied in the more general case of an overstory canopy of varying sparseness if the surface resistance of the big leaf is an equivalent resistance of all contributing surface elements from overstory and understory.

A much more detailed description of turbulent transport processes within plant canopies was presented by Raupach (1989) using Lagrangian theory. McNaughton and van den Hurk (1995) applied Raupach's approach in the case of a sparse canopy and showed that a model with the resistance structure given in Fig. 1a is consistent with Raupach's work. This model allows for two surfaces with different properties and different values of vapor density at the surface (i.e., an overstory and an understory). The resistance network in Fig. 1a has been used by Shuttleworth and Wallace (1985), Choudhury and Monteith (1988) and several studies following on from their papers. The network in Fig. 1 a allows interaction of fluxes from the understory and overstory in the canopy air stream and is called the "interactive" model in this study.

View larger version (24K): [in this window] [in a new window]   Fig. 1 Resistance networks for the (a) interactive, (b) patch and (c) Penman-Monteith models. Where it is used in the text the vapor density deficit D is marked against the relevant network node, subscripts a, c, and s denote the deficit at reference height, for the in-canopy air and at the surface respectively. The resistance subscript h denotes an aerodynamic resistance between a component surface and the in-canopy air and subscript c denotes a surface (or stomatal) resistance

In contradiction to the above result of Blyth and Harding (1995), the patch model has been successfully used at the scale of the sparsely-vegetated land surface. For these land surfaces the component surfaces (overstory and understory) are treated like patch types even though there are no distinct patches of overstory and understory. Norman et al. (1995) showed that differences between the interactive and patch models made little difference to total flux from the land surface at the MONSOON and FIFE sites studied. Norman et al. (1995) preferred the simplicity of the patch model. The ENWATBAL model of crop and soil water balance (Lascano, 2000, this issue; Evett and Lascano, 1993) effectively uses the patch model by partitioning available energy between overstory and understory and not allowing interaction of fluxes from these sources. ENWATBAL has been successfully applied to several sparsely cropped land surfaces (Evett and Lascano, 1993). Huntingford et al. (1995) concluded that energy fluxes over Sahelian savannah (bush overstory, herbaceous understory) are described equally well by Penman-Monteith, patch, or interactive models. Oliver and Sene (1992) concluded that fluxes above a developing vine crop would be well estimated using two non-interacting component surfaces. Brenner and Incoll (1997) indicated that transpiration from sparse shrubland was described equally well using models similar to the interactive and patch models. Brenner and Incoll (1997) used a three component surface: bare exposed soil; soil as shrub understory; and the shrub canopy.

All the above studies have considered a specific land surface or a restricted group of land surfaces and have not made generalized conclusions regarding the conditions under which use of the patch or interactive models is appropriate. In this study, the three evaporation models are compared for six different sparsely-vegetated land surfaces and for a wide range of conditions of water supply. The interactive model is assumed to be the most realistic because it is a simplification of a more detailed Lagrangian-based model. This assumption is supported by the results of this study. We compare the performance of the patch and Penman-Monteith models to the interactive model. The comparison reveals the dominant behavior of a given land surface and how the behavior reflects its physical characteristics and conditions of water supply.

In the following section the three models are fully described. Then data from six sparsely vegetated land surfaces are presented and the adjustment of leaf area index is outlined. Outputs from the models are presented for hypothetical land surfaces and for the six land surfaces taken from recent literature. We discuss the implications for the estimation of energy fluxes from sparsely vegetated land surfaces in the Discussion section.

	Theory

TOP ABSTRACT INTRODUCTION Theory Data from sparsely vegetated... Preliminary comparison of models Model comparison using land... Discussion Conclusion Appendix REFERENCES

 
Evaporation from a vegetated land surface can be calculated by summing the contributions from all homogeneous surface areas or subareas of leaf and soil within a representative land area. There are several possible approaches to this summation that vary in complexity and accuracy. Below we describe three such approaches: the "big leaf", interactive, and "patch" models. Later in the paper we apply these models to sparsely vegetated land surfaces for which we only allow two component surfaces or subareas: the understory and the overstory. We begin with a more general introduction to the summation approaches used in the models.

The Big Leaf or Penman-Monteith Model
The big leaf or Penman-Monteith model is the simplest approach. It was developed to describe the energy fluxes from a fully vegetated land surface. This model ignores any variations of microclimate within the vegetation and assumes that each leaf surface is at the same temperature and sees the same saturation deficit at its surface. Thus all leaves could be considered to be part of a large leaf covering the land surface, hence the name big leaf. Equation [1] gives the evaporation rate from each surface subarea i (E_i).

(1)

where r_s,i is the surface (or stomatal) resistance of subarea i per unit area of i ; a_i is the area of i; for a hypostomatous leaf (or soil) and for amphistomatous symmetrical leaves; and D_s is the saturation vapor density deficit at the surface of all subareas. The land surface total evaporation is the summation over all subareas above a representative unit area of ground _{, i}, (a variable enclosed thus represents an aggregated or "total land surface" value). The summation is equivalent to the parallel arrangement of all the surface resistances in the canopy (Eq. [2]).

(2)

where _c is called the canopy resistance and is a resistance per unit land area. If we define , then _c is the equivalent resistance of all the resistances r_c,i in parallel.

Now D_s is unknown but we do know D_a, the vapor density deficit at a reference point above the land surface. The Penman-Monteith equation is derived by substituting for D_s in the following way. Firstly we assume that the total flux from all the component surfaces i (i.e., from the big leaf) meets aerodynamic resistance r_a,PM (s m^-1) between the big leaf and the reference point, as shown in Fig. 1c. Equations [3] and [4] follow. Conservation of energy is stated in Eq. [5].

(3)

(4)

(5)

where is the psychrometer constant = c_p/; = latent heat of vaporization (J g^-1); c_p is the volumetric specific heat of air (J m^-3 K^-1); H is the sensible heat flux (W m^-2); T_a is the reference temperature (K); T_s is the temperature of the big leaf (K); e_a is the vapor density at the reference point (g m^-3); e_s is vapor density at the surface of the big leaf (K); A is the available energy at the land surface (W m^-3); and the subscript `PM' indicates that the flux or resistance is specific to the Penman-Monteith model. The slope of the saturation vapor density curve with temperature can be considered to be:

(6)

where a superscript * indicates a vapor density at saturation (g m^-3). (In this study the value given to is calculated as a function of T_a.) Now, D_s can be written as:

(7)

and substitution of Eq. [3], [4], [5], and [6] into Eq. [7] allows Eq. [8] to be derived.

(8)

where = /. Substitution of Eq. [8] into Eq. [2] gives the Penman-Monteith equation (Eq. [9]).

(9)

In this study we want to extend the range of application of the Penman-Monteith equation from fully vegetated land surfaces to include partially vegetated land surfaces. This can be achieved by restricting evaporation from soil with a soil resistance which is equivalent to the stomatal resistance of a leaf. All surface resistance values r_s,i for both leaf and soil subareas will contribute to calculation of _c. We acknowledge that a soil resistance model has little in common with a model of stomatal resistance. These differences do not hinder the calculation of _c but do demand a cautious approach to its interpretation. More details of the application of this model to partially vegetated land surfaces are given in the section entitled Model Consistency.

The Interactive Model
Clearly the Penman-Monteith equation is based on a rather simplified view of canopy microclimate. In reality, all component surfaces don't see the same surface saturation deficit, D_s, but a whole range of saturation deficits depending on their position and disposition within the canopy. There is a systematic variation of D with height within the canopy and a spread of D_s at each height according to the radiation interception, stomatal resistance, and wind speed past each leaf.

Rather than assuming that vapor density deficit has a single value at all surfaces i (D_s) we assume a single value within the canopy air space (i.e., D_c). Then evaporation from each subarea i can be calculated using Eq. [10] and the summation for the land surface using Eq. [11].

(10)

(11)

where f_i is the fraction of total available energy for the land surface (A) incident on surface i, ; r_h,i is the total resistance to the flow of sensible heat from surface i to the in-canopy air of deficit D_c, the boundary layer resistance (s m^-1); r_v,i is the total resistance to water vapor flow from surface i to the in-canopy air, the sum of boundary layer and surface resistances . Here r_v,i and r_h,i are resistance values of surface i per unit land surface area (s m^-1) and the subscript `N' indicates that the flux is calculated with the interactive model.

Another equation for land surface evaporation can be derived using Eq. [10] and [11] in place of Eq. [2] and then following the approach of Eq. [3] to [8] to substitute for D_c. Shuttleworth and Wallace (1985) gave a solution for two component subareas i: i = 1 was the incomplete or sparse overstory surface (all overstory leaves) and i = 2 the bare soil understory.

Here we use the approach presented by McNaughton (1994) which replaces the summation given in Eq. [10] and [11] by Eq. [12].

(12)

where r_h and r_v are effective resistance values for the whole land surface and are calculated as follows:

(13)

(14)

where . This approach is unrestricted in the number of subareas i input to the summation. Using a continuum of leaves each with its own resistances, Green and McNaughton (1997) calculated a very accurate estimate of evaporation from an apple tree. This result indicates that the effects of variations in D through the overstory can be well accounted for using this model with one value of D_c within the canopy air space.

Using Eq. [11] and the approach of Eq. [3] to [8] to substitute for D_c, leads immediately to Eq. [15]:

(15)

where r_a is the aerodynamic resistance between the in-canopy air space and the reference height as shown in Fig. 1a. Equation [15] has the form of the Penman-Monteith equation. Successive application of the effective resistance approach could be used to cast more complex canopy resistance networks into the same form. Even multi-layer models of canopy evaporation can be cast into this form if care is taken with the algebra (e.g., Shuttleworth, 1979).

As stated earlier we limit our study to a summation of fluxes from the two very different component surfaces, the overstory canopy and the understory (vegetation or bare soil). This allows us to draw on examples from the literature that have followed the work of Shuttleworth and Wallace (1985). McNaughton (1994) did not include the soil surface in his summation. However, a soil surface resistance in series with an aerodynamic resistance between the surface and the in-canopy air is algebraically equivalent to a leaf stomatal resistance in series with a boundary layer resistance.

The interactive model (i.e., Eq. [15]) effectively uses the big-leaf model once to describe the flux between the overstory surface and the in-canopy air, and a second time for the flux from the understory surface. Differences in D at the component surfaces are maintained by boundary layer resistances r_h,i as shown in Fig. 1a. Fluxes from one component surface interact with fluxes from the other component surface by influencing the in-canopy environment (D_c, T_c, and e_c).

A criticism of use of the interactive model is that turbulent diffusion theory (K theory) is assumed to be applicable in the canopy air space where this theory has been shown to be inadequate. However, McNaughton and van den Hurk (1995) showed that the structure of the resistance network in Fig. 1a was consistent with a detailed Lagrangian model after introducing a near-field resistance in series with the boundary layer resistance of the overstory. This resistance was found to be small in comparison to the boundary layer resistance and it can be ignored for many land surfaces with a distinct overstory and understory.

The Patch Model
The patch model uses two non-interacting big-leaf models side by side. Therefore it acknowledges that the taller vegetation of the overstory may see a different saturation deficit D at its surface than does the shorter vegetation or bare soil of the understory. The canopy microclimate develops separately over the two types of vegetation when they occupy separate areas as complete ground covers. The patch model is appropriate for this situation, effectively using the Penman-Monteith model separately for each component, then averaging the results for the landscape by weighing them according to the fractional areas covered by each. A model with the same structure (Fig. 1b) has also been used for sparse canopies (e.g., Norman et al., 1995), though it is not strictly appropriate for this purpose. The motivation in these studies is that the patch model is claimed to be algebraically more convenient for some applications, and the results better than the Penman-Monteith model, though perhaps not as good as the interactive model. It is not clear why this should be so.

The patch model has been developed for simplicity. Because the direct linkage to more accurate models has been abandoned it is not clear how resistance values should be assigned. In its correct application, when the patches are so extensive that the microclimate of one does not affect the microclimate of the other, the different patches would be characterized by quite different roughness lengths, and so quite different r_a,Pi values. The rules for constructing the aerodynamic resistance of the understory are not clear when it is partially covered by an overstory. For instance Norman et al. (1995) equated the aerodynamic resistance between the reference height and the overstory (i.e., the sum r_a,P1 + r_h,1 in Fig. 1b) with the resistance usually used in the Penman-Monteith equation (r_a,PM). For the understory our sum ( r_a,P2 + r_h,2) was equated with something similar to r_a,PM + r_h,2. Below we select values for resistances r_a,P1 and r_a,P2 by requiring that the models predict the same flux for a chosen reference case.

Model Consistency
Later in this study we compare six land surfaces which have been studied previously using the interactive model. With this in mind the resistances of the interactive model (Fig. 1a) are used to assign values to the resistances r_a,P1, r_a,P2, and r_a,PM of the patch (Fig. 1b) and Penman-Monteith (Fig. 1c) models. This departs from the usual application of the Penman-Monteith model in which r_a,PM is given by Eq. [16] under conditions of neutral stability (Monteith, 1965).

(16)

where z is the reference height; d is the displacement height of the canopy; z₀ is the roughness length; k is von Karman's constant; and u is the windspeed at the reference height. Equation [16] implicitly includes boundary layer resistances by integrating the resistance from the big leaf surface at (z₀ + d) to the reference height. We do not use Eq. [16] here because we want as much consistency as possible between the Penman-Monteith model and the interactive model. In the case of the patch model, the equations for resistances differ between authors (e.g., Norman et al., 1995; Blyth and Harding, 1995) and we extend this diversity by giving our own definitions below.

We require that all three models estimate the same land surface energy flux in the following reference case. The reference land surface has two component surfaces that are completely symmetric (i.e., they have the same surface resistance , the same boundary layer resistance , and the same available energy . Under these conditions the fluxes from the component surfaces will be equal. The patch model will estimate the same land surface flux as the interactive model when the resistances r_a,Pi are given by Eq. [17].

(17)

where r_a with no extra subscript is the resistance from the interactive model shown in Fig. 1a. With the definition in Eq. [17], a short circuit across the cut wire in Fig. 1b would make Fig. 1a and 1b equal. To equate the Penman-Monteith model with the interactive model in the reference case, it is easiest to use the effective resistances . Here _c is equal to the equivalent resistance of parallel resistances r_c,1 and r_c,2 thus in the reference case . The resistance r_a,PM is given in Eq. [18].

(18)

where _h is equal to the equivalent resistance of parallel resistances r_h,1 and r_h,2 and in the reference case .

This choice of a reference case seems logical for agreement between the Penman-Monteith model and the interactive model because two identical homogeneous component surfaces can equally well be described as a single surface. The choice is not so clear for the patch model. The interactive model has two component surfaces that always influence each other so there is no condition under which total agreement with the patch model would be expected. However, in the reference case flux interaction is at a minimum because both component surface temperatures are equal and there can be no energy flux from one surface to the other. So the reference case is also appropriate for agreement between the patch and interactive models and furthermore using one case for all three models makes the model comparison consistent.

A comparison of models should reflect the requirements of a particular study and/or the accuracy of the measurements used to verify model results. Here we take the standpoint of studies of watershed hydrology or larger scale climate and focus on the models' estimates of latent and sensible heat flux from the land surface as a whole. Where necessary we also compare the models in terms of their allocation of fluxes to overstory and understory. The accuracy of flux measurement methods and the resultant errors in land surface energy balance are usually of the order of 50 W m^-2 for any given half hour period in the authors' experience. Thus differences between models of <50 W m^-2 are not considered to be significant. Other possible standpoints are not considered in this paper. These standpoints include focusing on overstory canopy water use, or component surface temperatures, and may require a different emphasis and different criteria for model comparison.

The interactive model is taken to be the standard against which the patch and Penman-Monteith models are compared because it is consistent with the application of Lagrangian theory to turbulent transport in sparse canopies (McNaughton and van den Hurk, 1995). In particular we use the differences E_P - E_N and E_PM - E_N . For an input available energy A these differences also indicate the difference in land surface sensible heat flux between models because all three models conserve energy.

	Data from sparsely vegetated land surfaces

TOP ABSTRACT INTRODUCTION Theory Data from sparsely vegetated... Preliminary comparison of models Model comparison using land... Discussion Conclusion Appendix REFERENCES

 
Published Studies
Many published studies have used the interactive model of Shuttleworth and Wallace (1985) or its subsequent developments. In Table 1 we draw on some of this work to provide an indication of typical values of resistances and state variables found for several sparsely vegetated land surfaces. These studies use two component surfaces: i = 1 is the overstory canopy surface; and i = 2 is the underlying bare soil surface. Boundary layer resistances (r_b,1 and r_b,2) and surface resistances (r_s,1 and r_s,2) are given as resistance per unit surface area for one side of the surface. Using this convention the stomatal resistance, r_s,1, is the value that would be measured at a leaf surface using a porometer. Parameter n indicates that the overstory vegetation is amphistomatous or hypostomatous . Thus for a canopy with leaf area index = a₁, the resistance to sensible heat flow from the canopy surface to canopy air per unit land area, whereas the associated resistance to water vapor flow and is dependent on n. The surface area of the understory a₂ is set to a constant value of 1.0 and the surface is taken to be one-sided, thus and . The extinction coefficient for net radiation, , is taken from the source paper or calculated from . Values of are often assumed to be constant for a given species and reflect the leaf angle distribution.

Adjustment of Parameters n and a₁
The very different land surfaces listed in Table 1 span a range of values for resistances. The resistances and r_h,2 are implicitly dependent on land surface characteristics including canopy height, over-story canopy architecture, and leaf width, and thus describe the nature of the land surface. In contrast r_s,1 and r_s,2 describe water availability at each component surface. Because water supply is variable, r_s,1 and r_s,2 are given a wide range of values in the following analysis. This range of values used for r_s1 and r_s2 makes differences in n less relevant and it is assumed here that .

The fraction of available energy at the land surface absorbed by the overstory, f₁, is often expressed as a function of leaf area index a₁:

(19)

This equation is unlikely to be accurate when a canopy is clumped, for instance where single bushes or rows of trees are separated by bare soil. Despite this, many studies applying the Shuttleworth-Wallace approach resort to use of Eq. [19]. In this study Eq. [19] is used to aid comparison of the land surfaces listed in Table 1.

The land surfaces in Table 1 have different fractions of total available energy absorbed at the overstory and understory surfaces (i.e., different f₁ and f₂ values). In many cases evaporation from a land surface will be strongly influenced by the f₁ and f₂ values. To avoid confounding our comparison of land surfaces with comparisons of energy disposition, are set to the same value of 0.5 for all surfaces in Table 1. The leaf area index, a₁, was then adjusted using Eq. [19]. This allows a comparison of canopy architecture with the same distribution of energy between overstory and understory. In this process it is assumed that the resistances r_bi and r_a are unchanged which is not unreasonable if changes in a₁ are not large and an open canopy is maintained (for the land surfaces in Table 1 the final value of a₁ ranges between 1 and 2). (N.B. The r_b,1 are the boundary layer resistances of the component surfaces per unit area of component surface. The r_s,i values are used as variables in the analysis below.)

Leaf area index is a canopy characteristic which reflects the nature of the land surface at a given site, however we think it is not unreasonable to adjust the measured value for the following reasons. Firstly leaf area index is seldom constant for a given land surface and often increases during the growth of a crop or as a rainy season progresses. Furthermore, for a given species a₁ can change markedly between sites with similar weather conditions (e.g., as a result of different soil conditions or rainfall distribution). Thus the values of a₁ in Table 1 are particular to the time and location of measurement. The resistances r_a, r_b,1, and r_b,2 are more general in their description of the nature of the land surface and the species present.

	Preliminary comparison of models

TOP ABSTRACT INTRODUCTION Theory Data from sparsely vegetated... Preliminary comparison of models Model comparison using land... Discussion Conclusion Appendix REFERENCES

 
All models were set to estimate equal fluxes in the symmetric reference case (in section Model Consistency). In the section entitled Hypothetical Simulation of a `Symmetric' land Surface we consider the effects of deviating from the reference conditions only by the surface resistance values. We then discuss the value selected for the upper resistor in the patch model.

Hypothetical Simulation of a Symmetric Land Surface
As a first step in comparing the three models described in the Theory section the following standardized inputs were used: . In this first hypothetical run, symmetry of the two component surfaces was assumed by setting . The maximum and minimum values of r_a from Table 1 (70 s m^-1 and 7 s m^-1) were used in turn. The surface resistances r_c,1 and r_c,2 were allowed to range independently between 0 and 10000 s m^-1.

In the section Model Consistency it was a requirement that all three models predicted identical estimates of land surface evaporation (E) when r_c,1 and r_c,2 were equal. However in these symmetric examples, when r_c,1 r_c,2 the patch model always predicted less evaporation than the interactive model and the Penman-Monteith model always predicted more evaporation than the interactive model. The largest differences in lE are shown in Table 2 and occurred when the difference between r_c,1 and r_c,2 was greatest. When , the patch model and interactive models give similar estimates of lE for all values of r_c,1 and r_c,2 and the PM model is much less accurate. When , the difference in evaporation between the PM and interactive models is less than the difference between patch and interactive model estimates |E_PM - _EN| < |_EP - _EN|. We will return to this point later in the paper.

View this table: [in this window] [in a new window]   Table 2 Land surface evaporation (W m-2)

View larger version (21K): [in this window] [in a new window]   Fig. 2 Land surface evaporation when both component surfaces have equal surface resistance. Data shown are from the interactive model and from an altered patch model with upper resistors equal to ra. Data for rc,1 = rc,2 = 0 s m-1 are plotted on the left of the figure

	Model comparison using land surfaces in table 1

TOP ABSTRACT INTRODUCTION Theory Data from sparsely vegetated... Preliminary comparison of models Model comparison using land... Discussion Conclusion Appendix REFERENCES

 
For the land surfaces in Table 1 the symmetry between component surfaces assumed above in the section Preliminary Comparison of Models does not occur. Below, the models are compared firstly over a range of surface resistances for two contrasting land surfaces (in sections Sparse Millet and Shortgrass Steppe) and secondly for all the surfaces listed in Table 1 with given specific surface resistances (section All Land Surfaces).

Sparse Millet
The three models were compared for a land surface with the characteristics of sparse millet (Table 1). The inputs used were: . Figure 3a shows the evaporation from a millet land surface estimated with the interactive model (E_N). Figures 3b and 3c show the differences in evaporation between the patch and interactive models (E_P - E_N) and the Penman-Monteith and interactive models (E_PM - E_N), respectively. For millet the patch model closely resembles the interactive model for most surface resistances except when (wet overstory canopy). The Penman-Monteith model differs greatly from the interactive model when r_c,1 r_c,2.

View larger version (64K): [in this window] [in a new window]   Fig. 3 Evaporation from a millet land surface for a range of component surface resistances. Data shown are evaporation estimated with the interactive model EN (a), the difference between patch and interactive models EP - EN (b), and the difference between Penman-Monteith and interactive models EPM - EN (c). The figures are plotted on a log-log scale except for the leftmost column and the lowest row for which the surface resistance = 0 s m-1 is not consistent with the log scale

View larger version (60K): [in this window] [in a new window]   Fig. 4 Evaporation from a steppe land surface for a range of component surface resistances. See Fig. 3 for a description of data shown

View larger version (47K): [in this window] [in a new window]   Fig. 5 The difference in evaporation between the Penman-Monteith and interactive models for the grass overstory of a steppe land surface EPM,1 - EN,1. EPM,1 was calculated using Eq. [1] and [2]. The figures are plotted on a log-log scale except for the leftmost column and the lowest row for which the surface resistance = 0 s m-1 is not consistent with the log scale

View larger version (19K): [in this window] [in a new window]   Fig. 6 Comparison of the differences in land surface evaporation between the patch and interactive models (EP - EN) and the Penman-Monteith and interactive models (EPM - EN). In this figure rc,1 = 100 s m-1 and rc,2 = 10000 s m-1 for all land surfaces listed in Table 1. The differences are plotted against the resistance ratio where h is the effective boundary layer resistance of both component surfaces in parallel

	Discussion

TOP ABSTRACT INTRODUCTION Theory Data from sparsely vegetated... Preliminary comparison of models Model comparison using land... Discussion Conclusion Appendix REFERENCES

 
In summary, the Penman-Monteith model assumes a land surface can be described by a single big leaf surface with averaged properties and a single value of vapor density deficit at the surface (D_s). The patch model allows for two different component surfaces with different values of D at each surface but does not allow the fluxes from either surface to interact with the other. The interactive model allows fluxes from each component surface to change the in-canopy environment (T_c, D_c, E_c) and in this way influence fluxes from the other component surface. Thus, in terms of complexity, the patch model comes between the Penman-Monteith and interactive models.

Although the patch model is intermediate in complexity, the land surface fluxes predicted by the patch model are not intermediate to those predicted by the Penman-Monteith and interactive models. In general the patch model predicts a lower land surface evaporation rate and the Penman-Monteith a higher evaporation rate than the interactive model. Furthermore, it can not be assumed that the magnitude of the difference between the patch and interactive models will be smaller than that between the Penman-Monteith and interactive models.

For a given land surface, the extra evaporation predicted by the interactive model above that predicted by the patch model (E_N - E_P) results from the interaction of fluxes between the component surfaces. Using this extra evaporation as a measure of flux interaction, the data indicate that flux interaction was largest when: (i) r_c,1 and r_c,2 were most different; (ii) r_a was larger than the boundary layer resistance r_h,1; and (iii) when the overstory and understory absorb similar fractions of the available energy (i.e., f₁ f₂ 0.5). (For condition (iii) above, a continuum of f₁ and f₂ values was modelled, results are not presented here.) When these conditions occur the patch model is not a good choice for estimating land surface fluxes. The conditions are discussed in greater detail in the Appendix. In Fig. 6 it can be seen that when this measure of flux interaction was largest, the difference between the Penman-Monteith model and the interactive model was smallest. This example suggests that the land surface as a whole is well approximated by a single homogeneous surface with appropriately averaged properties when there is a large interaction between the component surfaces.

Whether the Penman-Monteith or patch model is the better approximation of the interactive model depends on the nature of the land surface being considered. Our results show that the ratio is a good indicator of the best choice in the case of a dry understory and an overstory canopy well supplied with water. As r_a becomes small the difference between the in-canopy vapor density deficit D_c of the interactive model and the deficit at reference height D_a also becomes small, and the patch and interactive models approach each other. When r_a is large in comparison to _h, D_c is closer to the deficit at the component surfaces (D_s,1 and D_s,2) and the Penman-Monteith equation is a good approximation as discussed above.

Finally, under dry conditions the differences between models become less significant. When both r_c,1 and r_c,2 > 300 s m^-1, the patch and Penman-Monteith models estimated energy fluxes to within 50 W m^-2 of the interactive model for the steppe and millet land surfaces (Fig. 4 and 5). Dry conditions like this were dominant at the steppe site described by Massman (1992). At these higher values of r_c,1 and r_c,2 evaporation from both component surfaces is tightly controlled by these resistances and interaction of the fluxes becomes less important.

	Conclusion

TOP ABSTRACT INTRODUCTION Theory Data from sparsely vegetated... Preliminary comparison of models Model comparison using land... Discussion Conclusion Appendix REFERENCES

 
Evaporation and sensible heat flux from a land surface with an incomplete overstory canopy can be modelled using all of the three models in Fig. 1. The three models estimate the same total land surface fluxes to within 50 W m^-2 when r_c,1 r_c,2 or when r_c,1 and r_c,2 > 300 s m^-1 (r_c,2 and r_c,2 indicate the conditions of water supply at the component surfaces). When r_c,j << r_c,k, the differences between the models can be related to the ratio (subscript j = 1 or 2 and subscript k = 3 - j). When this ratio is large the land surface is decoupled from the overhead air stream and component surface j is relatively well coupled to the in-canopy environment. This is likely to occur for low overstory canopies (e.g., steppe or pasture), sheltered canopies (e.g., orchards) or canopies with large leaf area index and narrow leaves. In contrast, when the ratio is small, the component surfaces do not interact much and both surfaces `see' the overhead air stream largely unaffected by the other (e.g., taller canopies with larger leaves).

We have argued that the interactive model is the most widely-applicable because its resistance network (Fig. 1a) is a simplification of more complex and realistic Lagrangian models (McNaughton and van den Hurk, 1995). K theory is not strictly applicable, but resistance networks still provide a good description of land surface fluxes. Sparsely vegetated land surfaces range from those with non-interacting component surfaces to those with highly interacting component surfaces. At the two opposing ends of this scale either the patch or the Penman-Monteith model gives accurate land surface flux estimates. The interactive model is in agreement with either of the two other models at both ends of the scale and is able to simulate the continuum of land surface types between.

	ACKNOWLEDGMENTS

 
CCD would like to thank the New Zealand Foundation for Research Science and Technology for funding this research (FRST contract CO6646) and the American Society of Agronomy for assistance with travel costs to the Annual Meetings 1998 where this paper was first presented.

Received for publication September 15, 1999.

	Appendix

TOP ABSTRACT INTRODUCTION Theory Data from sparsely vegetated... Preliminary comparison of models Model comparison using land... Discussion Conclusion Appendix REFERENCES

 
Here we enlarge on the discussion of the conditions that promote the interaction of fluxes from component surfaces. The difference in land surface evaporation estimated by the interactive and patch models is used as a measure of the importance of flux interaction. In this discussion we assume that the sparsely-vegetated land surface has only two component surfaces (overstory and understory) identified by subscripts j and k. Component surface j could be either the understory or the overstory because the structure of all three models is symmetric (j k).

Surface Resistances r_c,j and r_c,k
When the component surface resistances r_c,j and r_c,k were most different (i.e., r_c,j 0 s m^-1 and r_c,k s m^-1) the interaction of fluxes between the surfaces was largest (Table 2, Fig. 4b). This occurs because large differences in r_c,j and r_c,k will be associated with large differences in temperatures and vapor densities at the component surfaces. The resulting gradients between the surfaces may be enough to drive significant fluxes. A common example occurs when a sparse overstory canopy well-supplied with water has an underlying dry soil surface. In this case a net sensible heat transfer from the understory (high surface temperature) to the overstory (lower surface temperature) is likely. In contrast, when both the overstory and understory are water stressed (not well-supplied with water) there is likely to be a positive sensible heat flux away from both surfaces, and the interaction of fluxes is less important (sections Sparse Millet and Shortgrass Steppe).

Aerodynamic Resistances r_a and _h
A land surface with r_c,j << r_c,k will have the largest interaction of fluxes between component surfaces when r_a > (Fig. 6). The relative magnitudes of the two resistances is important to the size of the interaction. When r_a is small (<10 s m^-1) the in-canopy air is well-coupled with the overhead airstream, and the effect of fluxes from component surfaces on the in-canopy environment is small [i.e., (T_c, D_c, E_c) (T_a, D_a, E_a)]. In this case similar fluxes are predicted by the interactive and patch models. In contrast, when _h is small any alteration of the in-canopy environment will have a large effect on fluxes from component surface j.

In Fig. 6 the largest interaction of fluxes from overstory and understory is shown by the shortgrass steppe, kiwifruit orchard, and rangeland surfaces (Table 1). These land surfaces all have values of r_a 30 s m^-1 and are consistent with a land surface largely decoupled from the overhead air stream. The steppe surface had a canopy close to the ground (canopy height not given, but site roughness length = 0.012 m, Massman, 1992). The kiwifruit orchard had a vine canopy height of 2.5 m however it was surrounded by 8 m high shelter trees which increased resistance to energy transport between the vine canopy and the atmosphere. At the rangeland site the greasewood bush height was 0.75 m and the bushes covered 25% of the land surface. A dense and low bush cover may also offer a large resistance to energy fluxes although uncertainty in the accuracy of the r_a estimate was indicated in this study. An expression for the degree of coupling between a land surface and the overhead airstream was introduced by McNaughton and Jarvis (1983). Below we consider briefly whether this formulation is useful for estimating flux interaction at a sparsely vegetated land surface.

Application of the coupling concept to a sparsely vegetated land surface has not been discussed in the literature. Two possible forms for the decoupling coefficient are given below, _ls for the land surface as a whole (Eq. [A1]) and _v for the canopy `patch' (Eq. [A2]).

(A1)

(A2)

Figure 7 uses _ls as the x-coordinate in place of the ratio in Fig. 6. _v was not found to be a good indicator of the importance of flux interaction between component surfaces (data not shown).

View larger version (20K): [in this window] [in a new window]   Fig. 7 As for Fig. 6, with decoupling coefficient ls as the ordinate axis

Absorbed Fractions of Available Energy f_j and f_k
The effect of overstory canopy sparseness on the interaction of fluxes from component surfaces is more implicit. Flux interaction is largest when the overstory and understory absorb similar fractions of the available energy (f₁ f₂ 0.5). In the extreme cases (overstory very sparse or very dense) one component surface will dominate the energy balance and interaction is less important. Even when the overstory canopy is well supplied with water, a low f₁ is usually associated with a low leaf area index a₁ and therefore a higher r_h,1 which will result in decreased flux interaction. A high f₁ is associated with a low f₂ and therefore smaller sensible heat flux from the understory. In the case of a wet understory surface with a dry overstory the result may be different because there is a much weaker link between r_h,2 and f₂.

	REFERENCES

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