A New Procedure Based on Surface Renewal Analysis to Estimate Sensible Heat Flux:
A Case Study over Grapevines
F. CASTELLVI´
Departament de Medi Ambient i Ciencie`s del So` l, Universitat de Lleida, Lleida, Spain R. L. SNYDER
Department of Land, Air and Water Resources, University of California, Davis, Davis, California (Manuscript received 26 January 2009, in final form 30 September 2009)
ABSTRACT
Grapevines are grown on a range of soils and in different climates. Depending on the desired final product and method of harvesting, the trellis system, and hence vineyard architecture, varies dramatically. Conse-quently, the crop coefficients used to estimate vineyard evapotranspiration from reference evapotranspiration are less universal than for other crops. Evapotranspiration measurements are often limited because of a lack of fetch and unavailability of vine parameters required as input. In this paper, a new procedure based on surface renewal (SR) analysis was derived to estimate sensible heat flux over the grapevines. A two-dimensional sonic anemometer operating close to the canopy can provide the inputs required to estimate the sensible heat flux and the work of expansion of the air parcels under constant pressure. Regardless of the stability conditions, the SR estimates were comparable to those determined using the eddy covariance (EC) method providing a root-mean-square error of 14 W m22for all data. Thus, the procedure is low cost and makes SR a simpler method than EC to estimate latent heat flux as the residual of the energy balance equation in moderately tall and heterogeneous crops. The experiment was carried out at a site influenced by regional advection of sensible heat flux.
1. Introduction
The ability to estimate grapevine water use is impor-tant for worldwide production (Mullins et al. 1992) in climates where water resources are limited. Vineyard architecture is variable because the trellis systems uti-lized to produce the grapes depend both on the final product desired (wine, raisins, or table grapes) and the method of harvest (by hand or machine). The spacing between rows and vines is adjustable, with canopy cover as low as 10% (Jacobs et al. 1996) or as high as 100%, depending on the trellising. Consequently, using crop coefficients and reference evapotranspiration to esti-mate crop evapotranspiration is not as straightforward as for other crops (Williams and Ayars 2005). Large lysimeters, which are capable of accurately measuring evapotranspiration on an hourly basis, are extremely
expensive (Scott et al. 2005), so it is hard to justify their use by farmers.
A large variation in seasonal water use of mature grapevines has been reported, and it is unknown how much of the variability results from production practices or the measurement method (Williams et al. 2003a). Paraphrasing Williams et al. (2003b), the main limitations to determining evapotranspiration from vineyards are that ‘‘(1) soil and water balance techniques require dif-ficult assessments of various soil and/or water parameters, (2) reliability of sap flow sensors is questionable; espe-cially on large vines, and (3) micrometeorological tech-niques require expensive instrumentation and large areas of uniform fetch, which are uncommon in many grape production areas.’’ Because the surface renewal (SR) method may operate close to the canopy (Paw U et al. 1995, 2005), fetch requirements may be minimized, which makes it a useful micrometeorological method for small grape vineyards where fetch requirements limit the ap-plication of other methods. Spano et al. (2000), Castellvı´ et al. (2002), and Castellvı´ (2004) have used the SR method to estimate sensible heat flux H over grape vineyards. Corresponding author address: F. Castellvı´, E.T.S.E.A. C. Rovira
Roure, 191, Departament de Medi Ambient i Ciencie`s del So`l, University of Lleida, Lleida 25198, Spain.
The methodology used in Spano et al. (2000) included high-frequency temperature measurements at several heights within the canopy. In their method, thermocouple damage was a possible limitation. The Castellvı´ et al. (2002) and Castellvı´ (2004) methods require temperature measurements only at a single level. The main difference between methods is that Castellvı´ (2004) is a half-order closure model (i.e., scalar flux depends on the square root of the eddy diffusivity), whereas Castellvı´ et al. (2002) is a semiempiric first-order closure model. The Castellvı´ (2004) procedure gave a better match to sonic anemo-meter H data than Castellvı´ et al. (2002). When measure-ments are taken within the roughness sublayer, however, the Castellvı´ (2004) method requires (z* 2 d) as an input, where z* is the roughness sublayer depth and d is the zero-plane displacement.
This paper reports on a new SR procedure based on Castellvı´ (2004) that is used to estimate sensible heat flux over a mature vineyard. An experiment was con-ducted during the summer of 2007 at the University of California in the Kearney Agricultural Research and Extension Center (KAC) near Parlier, California. The local climate is characterized by light winds and regional advection of sensible heat flux. The new procedure over-comes limitations derived from 1) fetch requirement restrictions, because it operates close to the canopy, 2) potential thermocouple damage and data acquisition system limitations, because required measurements are taken at a single level, and 3) the need to measure the zero-plane displacement, the roughness length for mo-mentum, the plant area index (PAI), etc. It is indirectly shown that it is difficult to a priori guess realistic z* and d values based on observed weather conditions.
In grape vineyards, negligible heat storage may be assumed in the canopy. Based on Wilson et al. (2002), storage was considered insignificant for woody canopies with height less than 8 m. In Meyers and Hollinger (2004) storage in maize and soybean was less than 5% of the net radiation when the canopy was fully developed. Eddy covariance (EC) measurements of sensible and latent heat fluxes over pear and mango orchards with PAIs that are considerable greater than those of grapevines showed a lack of energy balance closure of 10%–12% (Conceicxa˜o et al. 2008; Teixeira et al. 2008). Therefore, the lack of closure observed was similar to that of other canopies with negligible storage, such as grasses (Twine et al. 2000; Wilson et al. 2002; Castellvı´ et al. 2008). For irrigation assessment and routinely hydrological appli-cations, it is generally assumed that the simplified surface energy balance equation applies and closure is forced (Brutsaert 1988). At sites influenced by regional advec-tion of sensible heat flux, however, closure forced via the Bowen ratio using similarity theory should be discarded
(Motha et al. 1979; Todd et al. 2000; Lee et al. 2004). To use robust, easy to maintain, and affordable instrumen-tation in regional advection conditions, forcing closure to estimate latent heat flux as the residual of the sim-plified surface energy balance equation is an alternative if reasonable sensible heat flux estimates can be de-termined. In this paper, it is shown that the inputs re-quired for estimating the sensible heat flux and the work of the expansion of air parcel under constant pressure can be provided by a two-dimensional sonic anemometer.
2. Theory
a. Surface renewal analysis for estimating sensible heat flux densities
scalar flux conservation equation for a steady and planar homogeneous turbulent flow, and assuming that 1) an air parcel comes from aloft with a volume that covers all the sources, V 5 zS (where z $ h, with h the canopy top and S the unit ground area), and remains in contact with the sources (Fig. 1b); 2) the flux at z 5 0 is negligible, 3) during the contact the scalar transfers from the sources to the parcel changing its concentration with no losses from the parcel top; and 4) turbulent diffusion dominates the mixing within the parcel and, as a result, the parcel is uniformly mixed with no molecular diffusion. There-fore, one arrives to the following expression to estimate the sensible heat flux density at height z (Paw U et al. 1995):
H 5 rCp(az)A
t (1)
where A and t are the temperature ramp amplitude and period determined at z, respectively, and a is a param-eter that corrects for all four assumptions made, though it mainly corrects for unequal mixing because a large volume of the air parcel is considered when measure-ments are taken at one height. Note that traces mea-sured above the canopy are representative of the upper part of the canopy, whereas the sources at the ground were presumed negligible. Therefore, a must correct for the gradient from the bottom to the top of the parcel. Currently, there is ample evidence that half-hourly a values depends on the measurement level, stability conditions, and canopy architecture (Snyder et al. 1996; Katul et al. 1996; Castellvı´ 2004; Castellvı´ and Snyder 2009). In the appendix it is shown that when measure-ments are taken within the roughness sublayer, half-hourly a estimates can be obtained by the following expression: a 5 k p (z* d) z2 tu*f 1 h (z) 1/2 h # z # z*, (2)
where u* is the friction velocity, k ’ 0.4 is the von Ka´rma´n constant, fh(z) is the stability function for heat
transfer, and z [5(z 2 d)/L] is an stability parameter, with L the Obukhov length
L 5 u
3
*Ty
kgw9Ty9, (3)
where g is the acceleration resulting from gravity and Ty
is the virtual temperature. To be consistent with deri-vations the near-neutral conditions made in the appen-dix, the following formulation for fh(z) is considered
(Ho¨gstro¨m 1988):
fh(z) 5 1 1 7.8z 0 # z (1 12z)1/2 z ,0
. (4)
b. Determination of the roughness sublayer depth The roughness sublayer depth extends from the ground to the bottom of the inertial sublayer. Because turbu-lence is characterized by the presence of distinct co-herent structures mainly generated by shear near the canopy top and thermal plumes for convective condi-tions, it is well known that similarity does not apply in the roughness sublayer and that z* varies with rough-ness elements, stability, and turbulence intensity (Thom 1971; Oliver 1971; Antonia et al. 1979; Garrat 1980; Shaw and Pereira 1982; Dolman 1986; Meyers and Paw U 1986; Raupach et al. 1991; Gao et al. 1989; Paw U et al. 1992; Cellier and Brunet 1992; Physick and Garratt 1995; Raupach et al. 1996; Mo¨lder et al. 1999; Pingtong and Takahashi 2000; Finnigan and Shaw 2000; Graefe 2004; Harman and Finnigan 2007). Determination of z* re-quires knowledge of detailed profiles of wind speeds and temperatures to discriminate the height were similarity starts to hold. According to analysis shown in the ap-pendix, it may be estimated as
(z*N d) 51 na 2 N(z) p kz 2
å
n j51(tu*) 1 j , (5)where aN(z)is the mean a value for near-neutral cases and n denotes the number of samples taken around neutral conditions.
3. Materials and methods a. The field experiment
The three wind speed components and the sonic tem-perature were recorded at 10 Hz at a height of 2.8 m above the ground using a 3D sonic [31000RE, R. M. Young, EXW (EX Works) Traverse City, Michigan]. The raw data were stored in binary format using a CR1000 logger (Campbell Scientific, Inc., Logan, Utah). Postprocessing consisted of conversion of the half-hourly data files to ASCII and processing the data using the protocol from Mauder et al. (2007) to determine means, variances, and covariances. The 3D classic coordinate rotation method was used instead of the planar rotation method.
No rainfall, calm winds, and high-amplitude daily tem-peratures were observed during the experiment. Table 1 shows the maximum, minimum, mean, and standard de-viations observed for the wind speed, air temperature, friction velocity, sensible heat flux for unstable and stable atmospheric surface layer conditions, and the number of samples collected under stable and unstable conditions. Regional advection was common, and the atmospheric surface boundary layer typically became neutral around 1430 PST. Regional advection results from air movement from large nonirrigated areas, which surround the San Joaquin Valley, over the irrigated cropped areas on the valley floor.
b. Sensible heat flux determination
Ramp dimensions, friction velocity, roughness sub-layer depth, the zero-plane displacement, and the sta-bility parameter are required as input to estimate the sensible heat flux iterating Eqs. (1)–(5).
1) RAMP DIMENSIONS
Determination of coherent structures assuming a se-quence of ideal ramps combined with the use of struc-ture functions of different order (Van Atta 1977; Chen et al. 1997a) is an objective technique to estimate ramp dimensions (Paw U et al. 2005) which is desirable to provide an independent method to estimate scalar sur-face fluxes. Scheme 1 (Fig. 1b) was used. It requires a lower frequency than scheme 2 (the microfront time is on the order of few seconds). Once the ramp ampli-tude was determined, several time lags r were used to linearize the relationship, which holds for r Lr,
A35 2f[S3(r)]/rgt, where S3(r) is the third-order
struc-ture function for solving the ramp period. According to Chen et al. (1997a), the shortest time lag to be used for linearization (r1G) is that which produces the first global
maximum of S3(r)/r. To estimate the maximum time lag (rend) to be used for linearization so that r Lr, the
second global maximum of S3(r)/r was determined. Based on a ramp model shown in scheme 2, the second global maximum for scheme 1 occurs at a time lag r2G,
giving r2G’3/4t. According to Qiu et al. (1995), Lq’ 0.25t and, therefore, r2G ’Lr. The last time lag used
for linearization was determined as 1% of r2Gor rend’
0.01Lrto ensure that rend Lr. Therefore, the structure
functions were evaluated within a range of time lags that provides a close ramp amplitude (Chen et al. 1997a) and total period (Chen et al. 1997b) to ramp model in scheme 2.
2) FRICTION VELOCITY
For h , z , z*, the following expression holds (Kaimal and Finnigan 1994):
u* 5 0.4 su, (6)
where su is the horizontal wind speed standard
deviation.
3) ROUGHNESS SUBLAYER DEPTH,ZERO-PLANE DISPLACEMENT,AND SENSIBLE HEAT FLUX ESTIMATION
If one selects samples at near-neutral conditions, (zN* 2 d) can be determined by rearranging terms in Eq. (5) by setting the appropriate aN(z) value recom-mended in the appendix. If available, net radiation minus soil heat flux may help to identify near-neutral cases. If unavailable, the transition from stable to unstable cases is detected when the ramp amplitude changes its sign from negative to positive (Fig. 1b). For simplicity, it was assumed that (zN* 2 d) remains fairly constant, giving (zN* 2 d) 5 (z* 2 d) regardless of the stability conditions. Some scientists found that d depends on the wind speed and stability conditions (Rosenberg 1974; Brutsaert 1988; Raupach 1994; Verhoef et al. 1997; Gao et al. 2003; TABLE1. Maximum, minimum, mean, and standard deviation, observed for the wind speed, air temperature, friction velocity, and buoyant
sensible heat flux for unstable and stable cases (N; number of half-hourly samples).
Unstable (N 5 1159) Stable (N 5 1295)
Max Min Mean Std dev Max Min Mean Std dev
u (m s21) 2.60 0.02 0.84 0.52 3.09 0.02 0.72 0.53
T (8C) 41.0 4.6 21.9 6.72 38.8 2.4 18.2 7.08
u* (m s21) 0.55 0.02 0.21 0.09 0.61 0.01 0.13 0.10
Harman and Finnigan 2007), but others have not found this dependency, which suggests that d likely depends on the canopy architecture (Munro and Oke 1973; Adrie and Van Boxel 1988). Expressions for estimating d that depends exclusively on h, such as d ’ ch, where c is a constant in the 0.5–0.95 range, were reported for uni-form canopies and plants in rows when the wind di-rection was not aligned with the rows (Stanhill 1969; Kondo 1971; Brutsaert 1988; Wieringa 1993; Verhoef et al. 1997; Oue 2001; Takagi et al. 2003). We note that most studies to determine d are based on fitting the wind log law by measuring the wind speed at different heights using cup anemometers. To ensure high-quality mea-surements, samples were typically not used when u , 2 m s21. Consequently, under light wind conditions over a heterogeneous surface, it is difficult to assign a priori a value for coefficient c for the scale d ’ ch.
A procedure based on two steps is proposed to solve H. The procedure avoids d determination or subjective estimation and assumes that it remains constant. The appropriate expression for fh(z) [Eq. (4)] is known after
determining the ramp dimensions for temperature. The two steps are described below.
(i) Step 1
Equation (1), with a constant a parameter, is used to estimate H for near-neutral cases. The aN(z) value to solve for (zN* 2 d) in Eq. (5) applies for the actual a at near-neutral conditions.
(ii) Step 2
Assuming d ’ ch and coefficient c is known regard-less of the wind direction (i.e., flow direction variability did not sense drastic roughness changes), Eqs. (1)–(4) can be iterated to solve for H. Starting with z 5 0 and (z* 2 d), the first approximations for a, H, and L are found for the actual atmospheric surface layer. Using L, the first approximation for z is determined. Then, the process is iterated until convergence is achieved for z. Coefficient c is solved by trial and error to match H from steps 1 and 2; H estimates from step 1 are taken as a reference. Thus, coefficient c is estimated for near-neutral conditions by forcing agreement between Eqs. (2) and (5) through Eq. (4) via z. A method based on sim-ulating annealing (SA) was used to solve for c. SA is a global optimization method that distinguishes between different local optima. By giving the different bound-aries for the unknown coefficients involved in the func-tion to be optimized, starting from an initial point, the algorithm takes a random step and the function is eval-uated. After minimizing (or maximizing) the function, any downhill step is accepted; however, some uphill steps may also be accepted. Acceptability implies that the
initial process starts. The uphill decision is made by the Metropolis criteria (Metropolis et al. 1953). As the opti-mization process proceeds, the length of the steps decline until it closes in a global optimum. SA is recommended as a local optimizer for difficult functions because it has proven superior to multiple restarts and conventional optimization routines for difficult problems. SA is even capable of providing solutions out of initial boundaries for very complicated functions, but it requires a pow-erful computer. Because boundaries for coefficient c can clearly be restricted between 0 and 1, a standard com-puter can accomplish the SA calculations (Goffe et al. 1994). The root-mean-square error (rmse) was the target function to be minimized.
c. Performance evaluation
Linear regression analysis (slope and intercept), co-efficient of determination R2, and the rmse were used to compare the SR sensible heat flux HSRagainst the EC
method HEC. The coefficient D 5
å
y/å
x with x thees-timate and y the reference value was also computed as an evaluation parameter (Marth 1988). The coefficient D is used to determine the percentage ( p) being over- or underestimated as p 5 100(1 2 D), which provides an integrated evaluation of the bias on daily, weekly, and monthly time scales by averaging out random errors in the half-hourly estimates. The bias is (D 2 1) times the mean value determined from the observations.
4. Results
a. Estimating (zN* 2 d)
According to vineyard architecture, the degree of het-erogeneity was considered moderate, and thus according to the appendix aN(z) 50.55 was set in Eq. (5). Post noon, half-hourly samples of near-neutral stability cases were not included because convection may be important during that period and Eq. (4) may differ from unity. Selection was constrained to two half-hourly samples before sunrise and two after sunrise when the estimated friction velocity was u* $ 0.2 m s21. It was assumed that the u* threshold resulted in good ventilation of the canopy and rows (i.e., the transfer of momentum was capable of renewing the entire air layer from the ground to above the measurement height providing a near-zero temperature vertical gradient). The latter is required to ensure that Eq. (4) is close to unity and that the ramp frequency remains fairly constant with height (Paw U et al. 2005). They are both leading hypotheses in eval-uating the recommended mean aN(h,z,1.2h)given in the
(zN* 2 d) of 1.0, 5.7, 3.0, and 1.1 m, respectively. If u* $ 0.3 m s21 was chosen as the threshold (i.e., more con-servative), then the respective (zN* 2 d) statistics were 1.2, 4.6, 3.0, and 0.7 m for the 40 samples observed. Thus, the same mean was obtained for (zN* 2 d).
b. Estimating the zero-plane displacement and sensible heat flux
The optimum SA downhill gave the scale d 5 0.62h. The H estimates comparing step 2 HSR2and step 1 HSR1
are shown in Fig. 2. The linear fit had the following: slope 5 1.00, intercept 5 1.0 W m22, and R2 5 0.98.
Figure 3 shows HSR2versus HECfor all of the data. The
performance was generally excellent, with some spuri-ous samples observed for a few stable cases when the sign of the ramp did not match the sign of HEC, or when
HSR2was close to zero. The worst estimates corresponded
to samples gathered around 1500 (early September) to 1700 (October) PST during the transition from un-stable to un-stable atmospheric conditions and during the night under calm conditions. However, the total number of spurious data was small, regardless of the stability case. Table 2 shows slope, intercept R2, rmse, and D obtained for HSR2versus HECfor the unstable and stable cases
and for all data. No bias was observed, and the rmse was small.
For all data, Eq. (6) provided reliable friction velocity estimates with linear fitting u* 5 0.38suand R250.91. In
Eq. (5), the mean value found for (zN* 2 d) seemed small in comparison with other studies. Cellier and Brunet (1992) reported that the scale (zN* 2 d)/d, where d is the mean clear distance between rows aligned across the flow, fell between 3.0 and 4.0 over a 2.35-m-tall maize canopy, with d 5 0.8 m. For trees (savannah), Garrat (1980) found (zN* 2 d) ’ 3d, where d is the mean horizontal
spacing between trees. Fazu and Schwerdtfeger (1989) reported a factor of 4.6 and Wenzel et al. (1997) found a factor of 8.0 for a coniferous forest. Figure 4 shows the actual a [i.e., by rearranging terms in Eq. (1) and using HECas the actual H] versus L for all of the data. When
turbulence was mainly mechanically driven, the actual a values tended to fall within the interval (0.4, 0.6), so an aN(z)50.5 for z/h 5 1.22 was realistic. The relationship d 5 0.62h was slightly smaller than the typical d 5 (2/3)h value recommended over crops (Brutsaert 1988), and one might expect a higher value for d because of the calm winds at a site (Rosenberg 1974; Gao et al. 2003). It is difficult to quantify the regional advection influence on the H estimates because the uncertainty with Eq. (4) only can be assessed by a parallel experiment as described in Ho¨gstro¨m (1988), where measurements over a nearby homogeneous surface provide fh(z) for the area and d is
determined in the vineyard. ANALYSIS OF SENSITIVITY
If trial and error was directly applied to adjust (zN* 2 d) and d by minimizing the rmse between HSR2and HECfor
all the data, the results (Table 2) would be obtained for FIG. 2. HSR2vs HSR1. FIG. 3. HSR2corresponding to pair (3.0, 0.62) vs HEC.
TABLE2. HSR2corresponding to three distinct [(zN* 2 d), d/h] pairs vs HECfor unstable (Unst) and stable (Stab) cases, and for all data (All). Linear regression analysis: slope a, and intercept b (W m22); R2, Rmse (W m22), and D are defined in the text.
(3.00, 0.62) (2.58, 0.40) (3.26, 0.85) Pair Unst Stab All Unst Stab All Unst Stab All a 1.03 0.99 1.02 1.02 0.87 0.96 1.02 1.07 1.03
b 24 23 22 24 22 24 23 24 24
R2 0.94 0.82 0.97 0.95 0.81 0.96 0.94 0.83 0.96
Rmse 18 8 14 7 7 10 19 10 15
different pairs of (zN* 2 d) and d/h. For example, the local downhill providing the pair (3.5, 0.95) gave the following: slope 5 1.0, intercept 5 25.0 W m22, R25 0.96, rmse 5 15.0 W m22, and D 5 1.01. If LRdenotes
the rmse corresponding to a local downhill and GR
corresponds to the global downhill (i.e., the optimum) rmse, then the total of 46 local downhills gave LR 2
GR#5 W m22. This implies that the two-step process is
useful for reliable H estimates, but the best estimates for (zN* 2 d) and d will remain unknown.
Table 2 shows the HSR2performance for the other two
pairs. The pair (2.58, 0.4) corresponding to aN(z)50.51, which was recommended in the appendix for regular canopies, and the pair (3.26, 0.85), which corresponds to (zN* 2 d) 5 1.42h (the expected scale found in the ap-pendix for moderate heterogeneous canopies). Table 2 shows that all three pairs gave HSR2values close to HEC.
The pair provided using Eq. (5) performed closer to HEC.
However, all pairs provided good estimates. Therefore, the typically reported scale of 3–4 for (zN* 2 d)/d in field crops is unrealistic for the grape vineyard. As a rule of thumb, if one takes the intermediate scale (zN* 2 d)/d ’ 3.5, for our case of study results (zN* 2 d) ’ 8.5 m. Even assuming d 5 h, HSR2overestimated HECby 62% though
the intercept was negligible and R2 5 0.96. Because multiple [(zN* 2 d), d/h] pairs obtained by the two-step process lead to HSR2close to HEC, Eq. (1) implies that
distinct pairs must provide similar half-hourly a values after convergence is achieved. Figure 5a shows the iter-ated half-hourly a values obtained from the pair (2.58, 0.40) versus the pair (3.00, 0.62). The iterated a values nearly matched and the linear regression statistics were the following: slope 5 1.03, intercept 5 0.02, and R25 0.99. Thus, (zN* 2 d) and Eq. (4) through z adjusted Eq. (2) to converge to the same value. Figure 5b shows the iterated a values corresponding to the pair (3.00,
0.62) versus the actual a value. The agreement was good.
Based on Physick and Garratt (1995), it is question-able to consider (zN* 2 d) a constant. The authors sug-gested that (z* 2 d) linearly decays to about 37% of (zN* 2 d) when (zN* 2 d)/L $ 0.2, whereas (z* 2 d) re-mains fairly close to (zN* 2 d) for unstable cases. The same stability correction was implemented in the itera-tion procedure. For stable cases, HSR2underestimated
HECby about 6% (D 5 0.94). The intercept and rmse
were the same as that shown in Table 2, but the re-gression statistics were slightly worse, with slope 5 0.78 and R250.77. In essence, it was found that the H esti-mates captured a higher portion of the HECvariability
when (zN* 2 d) was assumed constant. A variable (z* 2 d) stability correction could provide better estimates in other experiments. The actual z* and d was unavailable. FIG. 4. Actual a vs L.
Because Eq. (4) is based on similarity theory, the pro-cedure is only recommended for flat terrain. Castellvı´ et al. (2008), however, showed that SR analysis, operating in the inertial sublayer, performed well for sensible heat, latent heat, and carbon dioxide fluxes over moderately sloping grassland located in the foothills of the Sierra Nevada Mountains near Ione, California, where similar-ity did not hold.
In the SR analysis, fetch requirements require further research (i.e., according to the authors’ knowledge; there is no publication on this topic). Therefore, it is difficult to address how the different footprints sensed by SR analysis and the EC method might alter the re-sults shown in Table 2.
It is of interest to remark that the site showed little change in surface roughness with wind direction. Thus, the observed performance shown in Table 2 corresponds only to flows across rows. For wind direction that is parallel to the rows, a new pair [(zN* 2 d), d] might be needed.
5. Summary and concluding remarks
A new procedure to iterate H in SR analysis was presented. The results were close to H from the EC method. Only a few spurious H estimates were obtained, and they mainly corresponded to periods where ramps were not well formed. Because Eq. (6) is a key part of the estimation procedure, reliable measures of su
are crucial. Therefore, use of an affordable, two-dimensional sonic anemometer is recommended to en-sure accurate inputs at low wind speeds of the wind direction and standard deviation. The work of expan-sion of air parcels under constant pressure needs to be included for closure of the surface energy balance. Using the sonic ‘‘near virtual’’ temperature addresses this problem (Paw U et al. 2000).
To conclude, a two-dimensional sonic anemometer operating close to the canopy can provide all of the in-puts required to estimate half-hourly H and the work of expansion of air parcels under constant pressure in the new SR method. The combination of the SR procedure and the simplified surface energy balance equation ap-pears to be an affordable alternative to be considered for estimating water use at sites influenced by regional ad-vection of sensible heat flux.
Acknowledgments. The authors acknowledge the re-view task made by the three rere-viewers. Neil Rambo and Steve Ewert (Dept. of Water Resources, San Joaquin District, Fresno, California), Alfonso Russo (University of Catania, Catania, Italy), Tom Shapland and Frank Anderson (University of California, Davis, Davis,
California), Larry E. Williams (University of California, Kearney Agricultural Center, Parlier, California), James Ayars (USDA ARS, Fresno, California) for helping with data collection and providing the research site, and Asun, Carla, and Ta`nia for their help in using various facilities at the University of Lleida. This work was sup-ported by a grant (4600004549) from the California De-partment of Water Resources (Sacramento, California), the TRANSCLA project (CGL 2006-12474-C01), and the Ministerio de Ciencia y Innovacio´n (Spain).
APPENDIX
The Parameter a: A Qualitative Analysis for Neutral Cases
In the following, though valid for any scalar, deriva-tions are made for temperature. If one may consider that at a point in flow above the canopy the heat transfer is predominantly vertical, it can be described by the one-dimensional diffusion equation
›T ›t 5 › ›z Kh ›T ›z , (A1)
where t denotes time and T is the mean air temperature over a period t 2 t0. To evaluate how T and its vertical
gradient at a measurement height z change with time, a neutral initial condition is assumed. The Laplace trans-form of temperature with respect to time is denoted as Tl(s). Assuming that the turbulent eddy diffusivity for
heat Khis constant over a period (t 2 t0), the Laplace
transform with respect to time in Eq. (A1) leads to
sTl(s)5Kh›
2Tl (s)
›2z . (A2)
The general solution for (A2) is
Tl(s)5A(s)ez ffiffiffiffiffiffiffiffiffis/Kh
p
1B(s)ez ffiffiffiffiffiffiffiffiffis/Kh
p
. (A3)
A realistic physical solution for (A3) requires B(s) 5 0 (a finite temperature value is needed when z is large). The derivative of (A3) versus z is
›Tl(s) ›z 5 ffiffiffiffiffiffiffi 1 Kh s sTl(z,s)pffiffiffiffiffiffiffis1. (A4)
›T(z,t) ›z 5 1 ffiffiffiffiffiffiffiffiffiffi pKh p ðt t0 ›T(z,s) ›s ›s ffiffiffiffiffiffiffiffiffiffiffiffiffi (t s) p . (A5)
For convenience, however, expression (A5) is rewrit-ten as ›T(z,t) ›z 5 2 ffiffiffiffiffiffiffiffiffiffi pKh p ðt t0 ›T(z,s) ›s d ffiffiffiffiffiffiffiffiffiffiffiffiffi (t s) p h i . (A6)
According to Fig. 1b, ramps in temperature traces repre-sent injections of heat into the bulk flow that fits the local temperature gradient. Based on the ramp model in scheme 2, which is more realistic than scheme 1, for un-stable cases dT/ds equals A/Lrwhen the air parcel is in
contact with the surface and 2A/Lfduring the renewal
phase. Thus, when the integration is over a ramp event, the solution for the integral in Eq. (A6) can be expressed as
ðt 0 ›T(z,s) ›s d ffiffiffiffiffiffiffiffiffiffiffiffiffiffi (t s) p h i 5A Lr ffiffiffiffi L p 1 A Lf ffiffiffi t p qffiffiffiffiffiffiLr . (A7)
For a stable case, (A7) has the opposite sign. According to (A2), the integration must begin at an instant t0when
the temperature gradient is negligible. Canopy-scale coherent structures are of interest in SR analysis; thus, the rapid incursion of a ‘‘fresh’’ air parcel located at level ze(position 1 in Fig. 1b) with temperature T(ze)allows for the assumption that T(ze)is close to that observed at the baseline of the ramp. If the fraction of the ramp duration corresponding to the heating period (or cooling period under stable conditions) is denoted as p, then Lr5pt
and Lf5(1 2 p)t. Consequently, from (A6) and (A7)
the temperature gradient can be expressed as ›T(z,t) ›z 5 2A ffiffiffiffiffiffiffiffiffiffiffiffi tpKh p p 1ffiffiffip (1 ffiffiffi p p ) (1 p) . (A8)
The fraction p, expressed as a percentage, ranges be-tween 94% and 99.9% (Chen et al. 1997a). For conve-nience, Eq. (A8) can be simplified to
›T(z,t) ›z 5 A ffiffiffiffiffiffiffiffiffiffiffiffi tpKh p . (A9)
We note that (A9) links the vertical gradient with the ramp amplitude and total ramp period observed in the time series. Thus, because the microfront time period is not required, reasonable frequencies can be used. Above the canopy the flux gradient relationships hold, so sensi-ble heat flux can be expressed as H 5 rCpKh(›T/›z).
When combined with Eqs. (6) and (A9), the a parameter
can be expressed as a 5 [(Kh/p)(t/z2)]1/2. If z . z*,
Monin–Obukhov similarity theory holds, and Kh 5 k(z d)u*f1h (z). When h # z # z*, gradients above the canopy are weaker than in the inertial sublayer, and thus Khis enhanced. Observations support an enhancement
factor of (z* 2 d) over (z 2 d) (Cellier and Brunet 1992; Graefe 2004), so Kh 5k(z* d)u*f1h (z) is valid for h # z # z*. This leads to the following expression to estimate aat height z in the roughness sublayer:
a(z)5 k p (z* d) z2 u*tf 1 h (z) 1/2 . (A10)
The ramp period remains fairly constant with height (Gao et al. 1989; Paw U et al. 2005), and because u* is z-less dependent above the canopy, (A10) gives the re-lationship zaN(z)5haN(z5h)for z ’ 0, where N denotes
near-neutral cases. Therefore, relationships valid at the canopy–atmosphere interface are applicable to relating (zN* 2 d) and aN(z). If one sets in (A10) the relationship
resulting from the mixing layer analogy (1/t) 5 b(uh/h)
[where uh is the wind speed at the canopy top and
b a constant (Raupach et al. 1996)], and the scale u* 5 cuh[where c is a constant (Kaimal and Finnigan 1994)],
the relationship (zN* d) 5 aha2
N(z5h)is obtained, where a 5 (p/K)(b/c). The relationship (zN* d) 5 4.7ha2 N(z5h)54.71za 2 N(z) h # z # zN* (A11)
was found by setting fh(z 5 0) 5 1, which is the most
realistic value from a physical point of view (Ho¨gstro¨m 1988); letting b 5 0.2, which is an intermediate value from the range b 5 0.1 to b 5 0.3 reported by Raupach et al. (1996), Paw U et al. (1992), and Shaw et al. (1995); and letting c ’1/3(Raupach et al. 1996; Graefe 2004).
The actual half-hourly a values are determined after rearranging terms in Eq. (1). Assuming that H is avail-able, one may evaluate aN(z)as a mean value for samples
taken from near-neutral cases; however, it is not straight-forward to check (A11). Determination of (zN* 2 d) requires detailed profile measurements of temperature and wind speed. However, a qualitative analysis is pos-sible from observations as listed in Table A1. One may classify the surfaces in Table A1 from 1 to 4 as regular surfaces. For this set, surfaces 1 and 2 are real and the artificial surfaces 3 and 4 concur within scales observed in crops such as d/h about 0.7, z0/h about 0.13, and z*/h
increasing D/h as a measure of heterogeneity in vege-tation planted in rows. For surface 2 and all WT sur-faces, the flow was across aligned roughness elements. Savannah was included in Table A1 as an example of a natural extreme heterogeneous landscape where (A10) cannot be applied. Instruments are not neces-sarily deployed at the canopy top, so Table A1 shows aN(z)for z/h ratios of 1.0, 1.1, and 1.2, as examples. The
averaged aN(z) values for all regular surfaces were
aN(z5h)50.54, aN(z51.1h)50.51, and aN(z51.2h)50.49,
so the mean values were robust. Regardless of the z/h ratio, the standard deviation was small, as std dev 5 0.03. It is of interest to mention that the mean aNvalue for all
surfaces and heights was aN50.51 with std dev 5 0.02.
For different surfaces, Paw U et al. (1995) consistently found a(z5h) close to 0.5 in Eq. (1) after calibration
against the EC method using half-hourly samples that are collected for all stability cases. Calibration using samples around the neutral case is realistic for (A11), because (A10) predicts higher a(z) than aN(z) for
un-stable cases, and vice versa, as observed in Paw U et al. (1995) and Castellvı´ (2004) for a variety of canopies. The five-WT cylinder surface has canopy characteristics such as ‘‘plant area index’’ (‘‘PAI’’ ’ 2.3), d/h (50.76), and z0/h (about 0.1), and roughness lengths that are similar
to the reference grass (Brutsaert 1988); however, z* seems too short. For homogeneous surfaces, the scales (z* 2 d)/z0and z*/h were reported to range between 20–
30 and 1.5–3.0, respectively (Brutsaert 1988; Verhoef et al. 1997; Graefe 2004). However, the five–surface WT cylinder provided (z* 2 d)/z054.7 and z*/h 5 1.2. For
this surface, aNvalues were much smaller than for the
regular surfaces with an average of 0.29. For the rest of
the WT cylinder surfaces, aN values tend to increase
according to the degree of heterogeneity. For surfaces 6 and 7, D/h 5 2.3 and 3.3, respectively, which are rather large ratios that are rarely found for crops and orchards planted in rows. The mean aNvalues for all heights was
0.55, with std dev 5 0.03. For surfaces 8–10, the degree of heterogeneity was large (D/h ranging from 4.6 to 9.5), and the mean aN value (all surfaces and heights) was
0.65 with std dev 5 0.03. For savannah, the aN(z)values
were shifted about 0.35 from regular surfaces. Strictly, according to assumptions made to arrive to (A11), the aN(z)only applies for homogeneous surfaces, and under
weather conditions and canopy characteristics that are suitable for intermediate scales. SR analysis, however, has given results close to the EC method over homo-geneous and heterohomo-geneous canopies (crops and tree orchards) when realistic z* and d values were chosen (Castellvı´ 2004; Castellvı´ et al. 2006; Castellvı´ and Snyder 2009). Thus, based on aN(z5h)for homogeneous
and moderately heterogeneous surfaces, an initial esti-mate is (zN* 2 d) 5 1.37h, while for moderately hetero-geneous canopies (zN* 2 d) 5 1.42h [i.e., in (A11) aN(z5h)
was set to 0.54 and 0.55, respectively]. The latter (zN* 2 d) scales concur with the Graefe (2004) recommendation of zN* 5 2.4h as an upper limit expected for moderately tall canopies, including homogeneous and heterogeneous vegetation not exceeding 1.7 m. The vegetation height was, however, recognized as being rather uncertain. This issue indicates that the a values obtained from Table A1 are realistic. However, a more objective (zN* 2 d) ex-pression than (A11) that is not relying on observed in-termediate scales is obtained by rearranging terms in (A10) as
TABLEA1. Determined aN(z5h), from (A11), corresponding to different surfaces reported in the literature (Graefe 2004; Table 1). The following notation is used: D, mean distance between roughness elements; l, frontal area index; PAI, plant area index; z0, roughness length for momentum; WT, wind tunnel experiment; and N/a, not available. Remaining terms are defined in the text.
ID, surface h (m) D (m) l PAI (m) d (mm) z0(m) z* aN(z5h) aN(z51.1h) aN(z51.2h)
1, Pine/spruce 24.5 0 2.5 5 21.1 N/a 57 0.56 0.53 0.51 2, Corn* 2.35 0.8 1.7 3.4 1.76 N/a 5.16 0.55 0.53 0.51 3, WT wheat 0.047 0 0.47 0.738 0.033 5.6 0.087 0.49 0.47 0.45 4, WT strips 0.060 0.044 0.23 0.230 0.043 8.7 0.128 0.55 0.52 0.50 5, WT cylinder 0.143 0.010 1.42 2.2292 0.109 13.5 0.173 0.31 0.29 0.28 6, WT cylinder 0.006 0.014 0.179 0.2810 0.005 0.396 0.015 0.59 0.57 0.54 7, WT cylinder 0.006 0.020 0.091 0.1428 0.004 0.346 0.013 0.56 0.54 0.52 8, WT cylinder 0.006 0.028 0.045 0.0706 0.003 0.214 0.014 0.62 0.59 0.57 9, WT cylinder 0.006 0.040 0.023 0.0361 0.002 0.144 0.016 0.70 0.67 0.64 10, WT cylinder 0.006 0.057 0.011 0.0173 0.001 0.054 0.015 0.70 0.67 0.64 11, Savannah 9.5 10 0.2 0.4 7.1 900 37.1 0.82 0.78 0.75 12, Savannah 2.3 5 0.32 0.64 1.8 435 10 0.87 0.83 0.79 13, Savannah 8 20 0.03 0.06 4.8 400 44.8 1.03 0.98 0.94
(z*N d) 51 na 2 N(z) p kz 2
å
n j51(t u*) 1 j , (A12)where index j indicates measurement at a height z of n samples gathered around neutral conditions and aN(z)is an averaged value that suits (A12). Because aN(z) is unknown, (A12) is not straightforward to apply. How-ever, according to Table A1, when instrumentation is deployed within h , z , 1.2h, the averaged aN for
regular canopies (D/h , 0.75) is aN(z)50.51. For
veg-etation planted in rows with a moderate degree of het-erogeneity (0.75 # D/h , 3.5), aN5 0.55. As rule of
thumb, one may use these intermediate aNvalues for
aN(z).
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