Exchange Processes al the Land Surface for a Range of Space and Time Scales (Proceedings of Ihe Yokohama Symposium, July 1993). IAHS Publ. no. 212, 1993. 343

The averaging of surface flux densities in heterogeneous landscapes

M. R. RAUPACH CS1RO Centre for Environmental Mechanics, GPO Box 821, Canberra, ACT 2601, Australia

Abstract Two Convective Boundary Layer (CBL) length scales are defined, to classify heterogeneous "patchwork-quilt" landscapes according to the nature and extent of the advective processes causing modification of the energy balance of any one patch by its surroundings. The two scales are a dynamic length scale Ut* (where U is the mean wind speed in the CBL and t* the convective eddy time scale) characterizing the downwind distance required for the CBL to become fully mixed after a transition, and UT (where Tis the time required for complete CBL equilibration, approximately the time since dawn) which characterizes the size of a patch which is energetically independent of its neighbours. These length scales define three kinds of heterogeneity, according to the typical along-wind patch dimension X: microscale heterogeneity (X < Ut*) for which inter-patch advection occurs in the surface layer and is maximum; mesoscale heterogeneity (Lfr* < X < UT) for which the CBL is well mixed but CBL-scale advection occurs; and macroscale heterogeneity (X > UT) for which advection is negligible and patches are energetically independent. Averaging schemes for surface fluxes are presented for the limiting cases of microscale and macroscale heterogeneity. The effect of changing X on the spatially averaged latent heat flux (over a heterogeneous landscape with specified area fractions of different surface types) is to reduce the flux with increasing X, but by a modest amount of less than 20% in most cases.

INTRODUCTION

The problem of realistically describing land surface exchanges of heat, water vapour and momentum in Global Climate Models (GCMs) has stimulated the development of numerous land surface parameterization schemes over the last decade; see, for example, the reviews by Dickinson et al. (1991), Schmugge & Andre (1991), Dickinson (1992) and Sellers (1992). The earlier (and still the majority) of these schemes assume land surfaces to be homogeneous within GCM grid cells, typically from 100 x 100 to 500 x 500 km2. The inevitable heterogeneity of the real world poses two main problems: (1) the subgrid-scale variability of the input meteorological fields of downward short and long wave radiation, wind speed, temperature, humidity and (̂ especially) precipitation; and (2) the subgrid-scale heterogeneity of land surfaces themselves, on a wide spectrum of length scales. These problems do not yield to the "brute force" approach of increasing the resolution of the GCM, because the spatial

344 M. R. Raupach

variability of both the input meteorological fields (especially precipitation) and the land surface itself is often almost fractal in character: at any nominated grid scale, subgrid-scale variability still exists.

The solution to the first problem (meteorological field variability) is being sought in statistical means of introducing realistic field variations across grid cells (e.g. Entekhabi & Eagleson, 1989; Hutchinson, 1991). This is particularly important for rainfall, but is also significant for other fields (e.g. temperature, wind, radiation). Such methods are not treated here. Instead, the focus is on the second problem, subgrid-scale heterogeneity. This can be tackled by considering a large, heterogeneous area, such as a GCM grid cell, to be made up of a number of different land surface types which together constitute a "quilt" of "patches" (or a "mosaic" of "tiles" - I prefer the former analogy!). Three tasks then exist: (1) to define the surface characteristics of the patches, either deterministically (e.g. from remotely sensed data at high resolution) or statistically (e.g. by assuming spatial spectral distributions of the surface properties, fractal distributions being one possible assumption); (2) to produce a simple aggregation (or "upscaling") scheme that yields the spatially averaged flux densities over the entire quilt or GCM grid cell; and (3) to produce a simple disaggregation (or "downscaling") scheme that determines the land-atmosphere exchanges over each patch individually. Here I avoid the first of these three tasks (although it is far from trivial) by assuming that the surface characteristics are already specified. The focus is on the second and third tasks of aggregation and disaggregation, respectively.

It has been recognized for some time that the outcome of both flux aggregation and disaggregation depends strongly on the length scale of the surface heterogeneities, or "terrain length scale". Two limiting cases have been defined in previous work: "microscale" heterogeneity, for which the terrain length scale is so small that the atmospheric boundary layer (ABL) responds only to the composite surface structure; and "mesoscale" heterogeneity, such that the ABL behaves independently over each patch in the quilt, without regard for the presence of other patches. The nomenclature is that of Raupach (1991), but others use different words for a similar idea: Shuttleworth (1988) spoke of "disordered" and "ordered" heterogeneity, while Koster & Suarez (1992) refer to the resulting averaging schemes as the "mixture" and "mosaic" approaches.

This paper sets out rational flux aggregation and disaggregation schemes for both microscale and large-scale heterogeneity, following Raupach (1991), and then presents some new results in which the two averaging schemes are compared. To do this, we consider the surface energy partition into sensible and latent heat within a growing daytime Convective Boundary Layer (CBL) over a heterogeneous landscape. The reason for considering a CBL is to provide a simple but fairly realistic means of coupling the ambient atmospheric temperature and humidity to the surface energy balance. This same coupling is incorporated into a GCM by much more complex means, through the prognostic equations for wind, temperature and humidity (though some GCMs now also include explicit CBL descriptions). The plan of the following sections is: (1) to outline the CBL model, initially considering the conventional case of homogeneous terrain; (2) to define the distinction between microscale and two kinds of larger-scale heterogeneity (in contrast with previous work); (3) to describe the flux aggregation and disaggregation schemes for the limiting cases of small-scale and large-scale heterogeneity; and (4) to present some new model results for flux aggregation and disaggregation in both cases.

The averaging of surface flux densities 345

CBL SLAB MODEL FOR HOMOGENEOUS TERRAIN

McNaughton & Spriggs (1986) and McNaughton (1989) argued that a CBL model provides a rational link between the surface energy balance and larger-scale atmospheric phenomena, at least in the common case of a convectively dominated daytime ABL. This is the spirit in which a CBL model is used here. The particular model used - the simplest available - is the "slab" or "well-mixed" CBL model. The form outlined here accounts for the feedback between CBL development and the partitioning of energy into sensible and latent heat at the surface, as first described by McNaughton & Spriggs (1986). The model assumes a growing, horizontally homogeneous CBL, capped by a thin inversion at height h(t) and well mixed by convective turbulence (except possibly within a surface layer, thin compared with h). In the well-mixed bulk of the CBL, the concentrations of conserved scalar entities are essentially constant with height z up to z = h, where concentrations change sharply. The CBL scalar conservation equation, height-integrated from z = 0 to z = h in an air column moving with the mean wind field, is then:

dt h h d/z dt

-W. (1)

where CJf) is the mixed-layer concentration, W the mean vertical velocity, the + subscript denotes conditions in the free atmosphere just above z = h, and

Fc = vvc(O) is the scalar flux density at the surface (w and c being vertical velocity and concentration fluctuations and the overbar an average over several convective eddy time scales). The W+ term in equation (1) accounts for large-scale atmospheric subsidence or uplift; henceforth it is assumed that W+ - 0. For the scalars 9(potential temperature) and Q (specific humidity), equation (1) then becomes:

dt h h dh dt dt h

QM-Qn h

dh{2)

dt

The flux densities FQand F^are essentially the surface sensible and latent heat flux

densities H = pc FQ = pcJwBand E = p\FQ = p\wq, where pis air density, c the

specific heat of air at constant pressure and Xthe latent heat of vaporization of water. These flux densities are determined by two constraints at the surface (assumed to be vegetated). The first is the surface energy balance

R„ G = H (3)

where FA = Rn — G is the available energy flux density, Rn the net irradiance and G the ground heat flux density. The second is a specification of the surface energy partition into sensible and latent heat, conveniently provided by a single-layer model of the vegetated surface, the Penman-Monteith evaporation equation:

E = p\FQ = eFA+p\DJra

e + 1 +rjr„ (4)

346 M. R. Raupach

where e = (k/c )dQsal/dTis the dimensioriless slope of the saturation specific humidity Qsa[(T)&s a function of temperature T, ra the bulk aerodynamic resistance for transfer from the surface to the CBL (through the thin surface layer), rs the bulk surface resistance for water vapour transfer, and Dm the mixed-layer potential saturation deficit, defined as the saturation deficit of an air parcel brought adiabatically to the surface. [An objection to equation (4) is that the available energy flux density FA

depends on surface temperature through the outgoing long-wave radiation component, thus compromising the prime virtue of the Penman-Monteith equation, that the surface energy partition is expressed independently of the surface temperature and humidity. However, if the outgoing long-wave radiation is linearized about the air temperature 0m , the problem is avoided and an equation similar to (4) is recovered, with a slightly different definition of FA (Raupach, 1991).]

Finally, equations (2) to (4) are closed with an "entrainment hypothesis" to specify the growth of h(t). The entrainment is related directly to the buoyancy flux density, or flux density of buoyant potential energy from the surface into the CBL: this is gFQvl T0, where g is gravitational acceleration, T0 a reference absolute temperature and FGvthe flux density of virtual potential temperature 6V = 0(1 + 0 .610 . The buoyancy flux density defines a velocity scale w, = (hgF$v/TQ)m for the convective turbulence in the CBL. The entrainment hypothesis used here is based on the turbulent energy budget in the CBL (Rayner & Watson 1991):

d/? = CKWI ( 5 )

d ' CTwl+gh(e+-Qm)/T0

where CK = 0.18 and CT = 0.8 are empirical constants. Equations (2) to (5) form a closed set in the five unknowns 0m(O, Qm(t), H{f), E{t)

and h(t). They can be solved readily by standard numerical methods as an initial value problem, given external prescriptions of FA(t), rjt), rs(t), 0+(z)and Q+(z). A useful way of viewing the entire model is in a semi-Lagrangian framework: the model describes changes in the properties of a column of air, filling the depth of the CBL, which moves across the land surface in response to the mean wind field. In an inhomogeneous landscape, FA(t), ra(t) and rs(t) change as the air column traverses different surfaces.

It is straightforward to include C02 in this model by writing equation (1) for C02

and including a surface boundary condition to specify the C02 flux. A suitable form, consistent with the single-layer equations (3) and (4) for the energy fluxes, is (e.g. Raupach et ah, 1992):

Fc - (Mc/V)S0!L (6) l.6rs + ra

where Fc is the C02 mass flux (positive upward), Mc is the molecular weight of C02, V the molar volume, C, the intercellular C02 concentration in stomatal cavities (typically 250 ppm for C3 plants) and the factor 1.6 accounts for the differing molecular diffusivities of C02 and H20 in air.

The averaging of surface flux densities 347

KINDS OF HETEROGENEITY

Kinds of heterogeneity can be defined from simple scaling arguments which identify two critical CBL length scales. The first is a dynamic scale specifying how long the CBL take to become fully mixed, defined thus: the turbulence in the CBL has a length scale h (around 102 m early, growing to around 103 m later in the day) and a velocity scale w* (typically 1 to 2 m s"1). The turbulence time scale is therefore t* = h/w* (typically 103 s), so the CBL is well mixed over time scales > t*, and poorly mixed on time scales of order or less than t*. Consider a CBL moving at mean wind speed U over a "patchwork-quilt" landscape in which the patches have along-wind length scale X: downwind of any one patch boundary, the distance required for dynamic equilibration of the turbulence to the new conditions, and for the CBL to approach a well mixed state, is Ut% (typically 1 to 5 km). This is the first critical CBL length scale. Microscale heterogeneity occurs when the patch length scale X < Ut*. In this case the turbulence and mixed-layer properties in the bulk of the CBL (especially the potential saturation deficit Dm) can never adjust to alterations in surface conditions from patch to patch. The response of the air flow to surface conditions is confined to the thin surface layer, above which all CBL properties take uniform, average values. The surface energy exchange processes over individual patches are determined by this average CBL state, and therefore interact through microscale (leading-edge, local, surface-layer) advection between patches. Microscale heterogeneity is the surface geometry in which advective interactions are maximum.

Conversely, if X > Ut*, then the CBL turbulence adjusts fully and Dm becomes fully mixed from one patch to the next. This requirement defines mesoscale heterogeneity. At this scale, individual patches can be treated with a one-dimensional CBL slab model of the kind outlined above. However, the patches are not necessarily energetically independent, because Dm (although well-mixed vertically through the CBL) continues to adjust to the new surface state for distances much larger than Ut*\ this is essentially advection at CBL scale (rather than at the surface-layer scale which dominates when X « Ut*). In the limit of large X, the patches become fully independent energetically, with negligible CBL-scale advection between patches; this occurs only after the CBL "forgets" its upwind Dm and h, and acquires Dm and h values identical with those for a CBL which had grown ab initio over the new patch. This condition can be labelled macroscale heterogeneity. Work in progress (to be described separately) shows that the time scale needed for energetic independence can be estimated fairly well as the time of CBL growth or time since dawn, T. The requirement of energetic independence defines the second critical CBL length scale, UT; this is typically 102 km and much larger than Ut*.

In summary, three kinds of heterogeneity, characterized by different ranges of X, have been defined: (1) microscale heterogeneity (X < Ut*), for which inter-patch advection is very strong and occurs at surface-layer scales, with Dm uniform in the overlying CBL; (2) mesoscale heterogeneity (Ut* < X < UT), for which Dm is fully mixed so that a one-dimensional (CBL slab) description applies, but where Dm

continues to adjust downwind of the surface transition, so that CBL-scale advection is occurring; (3) macroscale heterogeneity (X > UT), for which CBL-scale advection is negligible so that the CBL evolution and energy balance of individual patches can be calculated with complete disregard for the presence of other patches.

348 M. R. Raupach

These simple considerations need two qualifications. First, it is assumed that the ABL remains convective (that is, a CBL); thus, large patches of stable flow are not considered. Second, no account is taken of dynamical processes such as the formation of locked-in convective circulations, with possible associated "venting" of the CBL, at patch boundaries. The question of whether these circulations have a significant influence on regional energy balances remains open. However, even if such circulations are present, the scale arguments above remain applicable.

In the following section, the effect of scale heterogeneity X on the spatially averaged CBL energy balance will be investigated by considering the opposite limits of microscale and macroscale heterogeneity. These represent the limits of maximum and negligible inter-patch advective interaction, and therefore place bounds on the extent to which X can influence the spatially averaged energy balance over the heterogeneous surface.

FLUX AGGREGATION AND DISAGGREGATION

Microscale heterogeneity (X < Ut„) Aggregation and disaggregation in the microscale case was treated by Raupach (1991) along the following lines. The analysis begins by recasting the Penman-Monteith equation, (4), in the "equilibrium" form

« ~ (e + l)rD

Here E = eFA/(e + 1 ) is the thermodynamic equilibrium evaporation rate, D = Ersthe corresponding thermodynamic equilibrium saturation deficit (such that E = E whenever D = D ) and rD = ra + rs/(e + 1). This resistance, a weighted sum of ra and rs, can be interpreted as the total resistance to the flux density FD of saturation deficit D = Qsal{T) - ^between the surface and the atmosphere. With linearization of Qsal{T), FD is given by (Raupach, 1991, equations (lib) and (14)):

F - "Jill ~ D">~D<« (8) ° P^

Equations (7) and (8) show that the latent heat flux density E departs from the equilibrium value E and the deficit flux density FD departs in the opposite sense from zero, as Dm departs from Deq.

Suppose that the microscale-heterogeneous landscape is made up of a large number (indexed as i) of patches with specified area fractions^ (summing to 1) and available energy flux densities FAj. It is also assumed that each patch is characterized by unique values of the bulk aerodynamic and surface resistances, rai and rsi. Here, rai is the resistance to scalar transfer from the surface to the well-mixed layer (through the entire surface layer), integrated over the whole patch. Then, for each patch, the quantities Eeqi = eFAAe + !) . Deqi = Eeqi''r»and rDi = rai + r

sA£ + ^ ^ specified from the

given surface properties. In turn, these quantities specify the latent heat flux density Et and the deficit flux density FDl for each patch from equations (7) and (8), respectively.

The averaging of surface flux densities 349

Spatially averaged values of all these quantities are readily found. The averaged flux densities are simple area-weighted means of the flux density contributions from each patch, from the definition of a flux as mass or energy flow through a surface per unit time, and a flux density as flux per unit surface area. Thus, denoting spatial

averages by angle brackets, (FAy = J^ j^^-and \E/ = E ft£;(summing over i). Likewise, the spatially averaged deficit flux density is

rDi rDi

(noting that Dm is well mixed across patches for microscale heterogeneity). This can be equated with (FD\ = (D ) - Dm//rD), the spatially averaged version of equation (8), to define the spatially averaged quantities /£> \and (rD^. Matching the two expressions for (FD term by term, it follows that

TT-^T- ' <P«< - toE^T* (10)

\'D; 'Di 'DI

The spatially averaged latent heat flux density (for microscale heterogeneity) can now be written in terms of spatially averaged (as opposed to individual-patch) surface properties:

^s^.^T»--^ <„> The flux aggregation problem for microscale heterogeneity is now solved in the

sense that spatially averaged flux densities are now determined by suitably spatially averaged surface properties. These spatially averaged surface properties, given by equation (10), enable the diurnal course of the spatially averaged energy balance (H(t) and E(t)) and other CBL properties (especially Dm{t) and h(t)) to be determined using the CBL slab model outlined in the previous section. It is reemphasized that Dm and h are the same for all patches, because of the small scale of the heterogeneities. The disaggregation problem is also solved, in that the CBL deficit Dm and the surface properties for each patch determine Et for that patch, from equation (7).

Macroscale heterogeneity (X > UT) In this opposite extreme, the CBL develops independently over each patch, so each patch has not only its own energy balance FAi

= H: + Eh but also its own mixed-layer saturation deficit Dmi and CBL depth h-v The diurnal evolution of these quantities can must therefore be found by considering each patch independently, say with a CBL slab model. It does not seem possible to provide an averaged treatment of the composite surface along the lines of equations (10) and (ID-

RESULTS FOR HETEROGENEOUS LANDSCAPES

The aim of this section is to compare the averaged flux densities (especially (E))

350 M. R. Raupach

Table 1 Parameters specifying surface types.

surface z0 (m) g0 (m s"1) g, (m s"1)

water 0.0001 1 1

crop 0.1 0.001 0.05

forest 1.0 0.001 0.005

desert 0.01 0.0001 0.0001

Table 2 Profiles above the CBL and initial conditions.

C+ 0 C(0)

0 0.005 K m"1 15 K 10 K

Q 0 kg kg"1 m"1 0 kg kg"1 saturated

C 0 (ppm) m"1 350 ppm 500 ppm

produced by microscale and macroscale averaging as defined above. To do this, we consider a heterogeneous surface consisting of just two kinds of surface, such as crop and desert, overlain by a CBL. This is described by the CBL slab model outlined in equations (2) to (5), with C02 included through equation (6).

In the following examples, the pairs of surface types are chosen from four possibilities, specified by suitable choices for ra and rs: water, well-watered crop, forest and desert. For each surface type, the aerodynamic resistance ra from the surface to the well-mixed layer is estimated (neglecting diabatic influence) by

ra = [ln(zm/Zo)]2/ (k2Um), with k (= 0.4) the von Karman constant, ZQ the surface roughness length (taken to be the same for momentum, heat and water vapour), Um the wind speed in the mixed layer and zm the height at which the logarithmic wind profile yields wind speed Um; this is taken as 100 m, but the choice is not critical as the results are quite insensitive to zm. The bulk surface resistance rs is assumed to depend on FA (Denmead, 1976) such that the bulk conductance \lrs increases linearly with FA, taking values g0 when FA = 0 and gl when FA = 500 W m"2. The parameters specifying ra and rs are given for each of the four assumed surface types in Table 1.

The model is integrated from initial (dawn) conditions G(0), Q(0) and C(0) given in Table 2. Over the entire heterogeneous area, the external weather forcing is assumed spatially uniform and is specified by: (1) concentration profiles 6+(z), £>+(z)and C+(z) above the CBL, determined by the linear form C+(z) = C+0 + Yczwith time-independent coefficients given in Table 2; (2) the available energy flux density FA(t), taken as the same for all patches and given for daylight hours by F , w s i n ( 2 ^ / r < % ) ( w i t h FAmax = 5 0 0 w m"2. 'the t i me from dawn and Tday = 86400 s); and (3) the wind speed U,n in the CBL, assumed steady and equal to 3 m s"1.

The averaging of surface flux densities 351

6 B 10 -12 14 16 18

time ChO

• 0 .0 + 0 .2 O D.5 A 0 .8 X 1.D

Fig. 1 Average latent heat flux E over microscale-heterogeneous desert-crop surfaces with five choices of crop fraction.

6 8 10 -12 14 16 1B

time Crr) • 0 .0 + 0 .2 •> 0 .5 A O.B X 1.0

Fig. 2 CBL saturation deficit Dm over microscale-heterogeneous desert-crop surfaces with five choices of crop fraction.

To show the typical diurnal evolutions predicted by the CBL slab model, Figs 1 to 4 present E, Dm, the C02 flux density Fc and concentration C, for five microscale-heterogeneous surfaces consisting of desert and crop (as defined in Table 1) with crop

352 M. R. Raupach

E Si

Fig. 3 Average C02 flux <FC> over a microscale-heterogeneous desert-crop surface with five choices of crop fraction.

51D

5 DO

490

4BQ

470

4 BO

450

440

430

420

410

400

390

3B0

370

360

350

340

330

320

310

300

ttme CrrJ

0.2 O 0.5

Fig. 4 C02 concentration C over a microscale-heterogeneous desert-crop surface with five choices of crop fraction.

fractions 0, 0.2, 0.5, 0.8 and l. The properties of the composite surfaces are given by equations (10) and (ll). As the wet, photosynthetically active crop fraction increases relative to the dry desert, E and Fc increase, D decreases and the afternoon C02

The averaging of surface flux densities 353

drawdown C+-C increases. Note the high C02 fluxes in the early morning associated with the large early C02 concentrations imposed by the initial condition C(0) = 500 ppm, chosen to simulate the commonly observed buildup of respired C02 in a shallow, stable, nocturnal boundary layer. The general forms for the diurnal cycles in Fig. 1 are not surprising and are in good qualitative agreement with observations and other, similar CBL slab models, including McNaughton and Spriggs (1986) and Raupach et al. (1992); the latter paper also explores the use of the C02 drawdown C+-C to infer regionally averaged estimates of the flux Fc.

We now examine the effect of the length scale X on spatially averaged fluxes, by comparing spatially averaged fluxes over microscale and (energetically independent) macroscale patches with the same area fractions of their constituent surface types. Figure 5 shows the daytime average of the spatially averaged latent heat flux, for three pairs of surface types (desert-crop, forest-crop and water-crop) under both microscale and macroscale heterogeneity. The abscissa is the area fraction of the crop. Since the surface types are energetically independent in the macroscale case, {E)is obtained simply by averaging the large-area E values for each surface type (calculated with the CBL model, using the external forcing specified in Table 2), with weighting by area fraction. These averages are given by straight lines joining the appropriate end points in Fig. 5. In the microscale case, on the other hand, the spatial averages must be calculated using the response of the CBL to the composite surface defined by equation (10), rather than individual surface types. The resulting averages are (convex) curved lines, because of the nonlinearities in both the averaging rule (equation (10)) and also the CBL model. The extent of the curvature in these lines is a measure of the difference between an average which assumes complete energetic independence between

tuu

350

3DD

250

2 00

150

10D

50

n

-

- '

-

- j

-

• i l l l l l l l l l

0 .4 D.B

a Desert-Crop'micro

A Desert-Crop : macro

crop surface Tract I on

+ Forest-Crop•micro

x Forest-Crop:macro

0 . 0

Water-Crop•mIcro

Water-Crop : macro

Fig. S Daytime average <E> showing the difference between microscale averages (curved lines) and macroscale averages (straight lines).

354 M. R. Raupach

crop surface Tract ion

D E:mlcro-av + E:desert o Ex rop A H:micrc-av x Hidesert V h:crop

Fig. 6 Individual-patch fluxes E and H for a desert-crop microscale-heterogeneous surface, with microscale-average < £ > and <H>.

surface types (macroscale) and maximum energetic interaction via advection (microscale). The microscale-average E exceeds the macroscale average, but the difference is not great: the largest difference, about 20%, occurs for the greatest surface contrast (crop-desert) at median area fractions. The physical reason for the difference is that, in the microscale case, the presence of dry patches among wet ones results in a deeper CBL than would otherwise develop, leading to increased entrainment of dry air from above the CBL and a consequent higher average evaporation rate. Similar modest differences between microscale and macroscale averages occur for the other fluxes, H and Fc (not shown here).

Figure 6 shows, for the desert-crop surface pair, the way that the individual-patch latent heat fluxes over desert and crop depend on area fraction, in the microscale case. (In the macroscale case, the individual-patch fluxes are independent of area fraction). As the crop fraction decreases (or desert fraction increases), the evaporation from the crop surface increases to about 1.6 times its value when the entire surface is covered with crop. This is the enhancement by local advection of the evaporation from small patches of crop in a mainly desert region. The enhancement ratio depends on ra for the crop surface, and would be higher for a surface with lower ra. The individual-patch sensible heat fluxes behave in a complementary way.

Acknowledgements I have gained great benefit in formulating the ideas in this paper from numerous, enjoyable discussions with Keith McNaughton and John Finnigan.

The averaging of surface flux densities 355

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