How Close is Close Enough When Measuring Scalar Fluxes with

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For example, at a height of z. 10 m with a sensor displacement of D. 0.2 m, less than 1% of the flux is lost, whereas at z. 1 m the same instrument configuration ...
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How Close is Close Enough When Measuring Scalar Fluxes with Displaced Sensors? L. KRISTENSEN

AND

J. MANN

Risø National Laboratory, Roskilde, Denmark

S. P. ONCLEY National Center for Atmospheric Research,* Boulder, Colorado

J. C. WYNGAARD The Pennsylvania State University, University Park, Pennsylvania (Manuscript received 19 September 1996, in final form 4 December 1996) ABSTRACT To improve the quality of scalar-flux measurements, the two-point covariance between the vertical velocity w˜ and a scalar s˜, separated in space both horizontally and vertically, is studied. The measurements of such twopoint covariances between vertical velocity and temperature with horizontal and vertical separations show good agreement with a symmetric turbulence model when the displacement is horizontal. However, a similar model does not work for vertical displacements because up–down asymmetry exists; that is, there is a lack of reflection symmetry of the covariance function. The second-order equation for conservation of two-point covariance of w˜ and s˜ reveals the reason for this up–down asymmetry and determines its character. On the basis of our measurements, the ‘‘loss of flux’’ for a given lateral displacement decreases with increasing height of the sensors. For example, at a height of z 5 10 m with a sensor displacement of D 5 0.2 m, less than 1% of the flux is lost, whereas at z 5 1 m the same instrument configuration gives rise to a loss of 13%. Also, when the displacement is vertical, the ‘‘flux loss’’ decreases with height if the displacement is kept constant, but in this case the asymmetry causes the loss to be much smaller if the scalar sensor is positioned below the anemometer: if the mean height is 1 m and the displacement 0.2 m, the loss is 18% with the scalar sensor over the anemometer and only 2% if the instrument positions are interchanged. The authors conclude that when measuring close to the ground, the separation should be vertical with the scalar sensor below the anemometer. In this way a symmetric (omnidirectional) configuration with a minimum of flux loss is obtained.

1. Introduction When measuring turbulent fluxes of scalar quantities in the atmospheric surface layer, the eddy-correlation method is the most basic, relying on first principles only. Eddy-correlation measurements typically are carried out using a fast sensor for measuring the scalar (temperature, gas, aerosol) and a fast anemometer, for example, a sonic anemometer, for measuring the fluctuating vertical wind component. This method relies on the assumption that the scalar signal and the velocity signal can be considered as pertaining to the same point in space. Often this assumption is violated (the sonic an-

* The National Center for Atmospheric Research is sponsored by the National Science Foundation. Corresponding author address: Dr. Leif Kristensen, Wind Energy and Atmospheric Physics, Risø National Laboratory, Building 125, P.O. Box 49, Frederiksborgvej 399, DK-4000 Roskilde, Denmark. E-mail: [email protected]

q 1997 American Meteorological Society

emometer-thermometer is an exception), so we want to determine the importance of the anemometer and scalar sensor not being collocated. Qualitatively, the systematic error in the measured flux must be an increasing function of the ratio of the sensor displacement and the scale of the turbulence. This implies that the effect of a given displacement is more important close to the ground than at larger heights. Lee and Black (1994) addressed this problem in the case of a horizontal displacement and presented a useful review of the subject. Their own contribution was a convenient equation for the measured covariance of the scalar and the vertical velocity component as a function of the true flux F, the horizontal displacement of magnitude, the measuring height, the Obukhov length L O, and the angle from the mean-wind direction to the displacement vector. Their result was derived on the basis of the similarity cospectrum, obtained by Wyngaard and Cote´ (1972). Lee and Black (1994) showed that their equation was within 6% agreement, with the result from measurements of temperature fluxes carried out under unstable to neutral conditions.

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Recently Tong and Wyngaard (1996) used a modification of the two-dimensional spectral tensor model by Peltier et al. (1996), replacing one velocity component by a scalar, and predicted the entire cospectrum of a scalar and the vertical velocity component with horizontal displacement between the two sensors. They predicted the ratio of cospectra, where the sensors were displaced 0.53 and 0.63 m, to cospectra with zero displacement and found excellent agreement with experiments carried out with two sonic anemometers. In this article, we discuss the covariance loss due to both horizontal and vertical displacements and compare these with experimentally obtained covariances between temperature and vertical velocity. Our purpose is of a practical nature; namely, to answer the question: how close do the sensors need to be if the systematic flux error due to the separation has to be smaller than a specified amount?

Close to the ground, there is no a priori reason to make a similar assumption of symmetry when the displacement is vertical. In this case, we are forced to consider Rws as depending on both heights z and z9, so we use the notation F ↑(z, z9). Obviously, F →(0) 5 F↑(z, z) 5 F in the constant flux layer near the ground. a. Horizontal displacement We now develop an expression for the flux for horizontal displacement. Circular symmetry implies that the quadrature spectrum of w and s is zero and that there is a simple relation between the measured flux F→(D) and the cospectrum cows(k):

E

`

F → (D) 5

co ws (k) cos(kD) dk.

(5)

2`

Equation (5) can be written in the form

2. Basic theoretical considerations

E

`

We start by discussing some characteristics of displaced fluxes and then use some simple models of turbulence to see the effect of horizontal and vertical displacements. We use the convention that ensemble averaging is symbolized by angle brackets surrounding the turbulent quantity h˜ in question and denote the ensemble mean as the corresponding capital letter, that is, ˜ [ H. ^h& (1) We assume stationarity and horizontal homogeneity of all turbulent quantities, which implies that H is a function only of the height z, whereas h˜ must be considered a function of the position vector x and time t. Applying the Reynolds decomposition h˜ 5 H 1 h (2) to all turbulent quantities, we write the turbulent vertical flux of the scalar s˜ F 5 ^w(x, t)s(x, t)&.

(3)

F → (D) 5 F 2

co ws (k)[1 2 cos(kD)] dk.

(6)

2`

In this form, the integral converges if we use the cospectrum in its asymptotic form (Wyngaard and Cote´ 1972) co(k) 5 Bzzkzz27/3 , F

(7)

where B is a stability-dependent parameter. This is relevant when zDz is much smaller than the scale of the turbulence, where the low-wavenumber contribution to the covariance is negligible. Substituting (7) in (6), we get

[

12 1 2

9 2 zDz F → (D) ø F 1 2 G B 4 3 z

4/3

]

.

(8)

Apart from the directional dependence, this result is equivalent to that obtained by Lee and Black (1994) for small displacements.

The covariance of w(x, t) and s(x9, t9) is defined as Rws(x, x9, t, t9) 5 ^w(x, t)s(x9, t9)&,

(4)

and the assumption of horizontal homogeneity implies that Rws is a function only of the horizontal part of the displacement vector x9 2 x, the time lag t 5 t9 2 t, and the two heights z and z9. Since we measure w and s simultaneously, there is no need to be concerned with the temporal structure of the turbulence. Consequently, the temporal argument in the covariance will be omitted in the following. If the displacement is horizontal, then the covariance Rws is assumed to be a function only of the magnitude D 5 zx9 2 xz of the displacement and the height z. In other words, we postulate that Rws possesses circular symmetry. We denote this covariance as F→(D).

b. Vertical displacement We know of no way to obtain an equation similar to (8) for vertical displacements. Therefore, we present two arguments for the qualitative behavior of the displaced flux. 1) SPECTRAL

MODEL

It is easy to show that we should not expect up–down symmetry. Figure 1 is a symbolic, log-linear, area-conserving plot of the autospectra of s˜ and w˜ at two different heights z and z9, with the first smaller than the last. To be consistent with the notation so far, and to emphasize the height, we denote the four spectra Fs(z; k), Fw(z; k),

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FIG. 1. Symbolic plot of the autospectra of s˜ and w˜ at two different heights z and z9. Since the scales of s˜ and w˜, and consequently the wavelengths of the spectral maxima, are approximately proportional to the height, the upper spectra are displaced to the left with respect to the corresponding lower spectra. The two doubly pointed arrows symbolize the products Fw(z; k)Fs(z9; k) and Fw(z9; k)Fs(z; k), respectively. The first product shows a smaller areal overlapping than the last.

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FIG. 2. Parameter A defined by Eq. (9) as a function of kz.

Since the covariance F↑ (z, z9) is equal to the integral of the cross-spectra xws(z, z9; k), that is,

E

`

F ↑ (z, z9) 5 Fs(z9; k), and Fw(z9; k). It is essential for the argument that 1) neutral surface scaling applies so that all scales (wavelengths of the spectral peaks) are proportional to height and all second-order statistics, including spectral shapes, are invariant with height, and 2) at a particular height, the scalar has a larger scale than the vertical velocity component. (This is usually the case because the magnitudes of the streamwise fluctuations of w˜, in contrast to those of s˜, are limited by the presence of the ground.) We have indicated these scale relations in Fig. 1. We write the cross-spectrum xws(z, z; k) between s˜ and w˜ at the height z as zxws(z, z; k)z 5 A[Fw(z; k)Fs(z; k)]1/2.

(9)

In the ‘‘energy-containing’’ wavenumber range, A is a weak function of kz. An indication of this functional dependence can be obtained from the surface-layer spectra of w and the potential temperature u by Kaimal et al. (1972). Figure 2 shows A as a function of kz.1 If z/z9 is not too far from one, we may assume zxws(z, z9; k)z ø A[Fw(z; k)Fs(z9; k)] . 1/2

(10)

Figure 1 indicates that the product Fw(z9; k)Fs(z; k) is larger than Fw(z; k)Fs(z9; k) and, consequently, that zxws(z, z9; k)z , zxws(z9, z; k)z when z , z9.

1 Of course, the parameter A cannot be larger than unity since the curve seems to show at very low values of kz. We may consider this inconsistency a result of the experimental uncertainty in the spectral expressions by Kaimal et al. (1972).

x ws (z, z9; k) dk [F 5 F ↑ (z, z)],

2`

(11) we expect that F↑(z, z9)/F , F↑(z9, z)/F when z , z9. 2) SECOND-ORDER

BUDGET METHOD

We can investigate the origin of this asymmetry by deriving and examining the conservation equation for ^w(x)s(x9)&, which is applicable for any displacement in three dimensions. The equations for the fluctuating parts of w˜ and s˜ are ]w(x) ]w(x) d 1 U(z) 1 = · [u(x)w(x)] 2 ^w 2 (x)& ]t ]x dz 52

1 ]p(x) g 1 u (x) 1 n¹ 2 w(x) r0 ]z T0

(12)

and ]s(x) ]s(x) 1 U(z) 1 w(x)S9(z) 1 = · [u(x)s(x)] ]t ]x 2

d ^w(x)s(x)& 5 g s¹ 2 s(x), dz

(13)

where u(x) is the fluctuating velocity vector, S9(z) 5 dS/dz, r0 is the air density, p is the fluctuating part of the pressure, g is the acceleration of gravity, T0 is the mean temperature, u is the fluctuating potential temperature, n is the kinematic viscosity, and gs is the diffusivity of s˜. We derive the conservation equation for the displaced flux ^w(x)s(x9)& by multiplying (12), evaluated at x, by s(x9) and multiplying (13), evaluated at x9, by w(x). Adding and averaging these two equations yields

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] ] ] ^w(x)s(x9)& 5 2 [U(z)^w(x)s(x9)&] 2 [U(z9)^w(x)s(x9)&] 2 = · [^u(x)w(x)s(x9)&] 2 =9 · [^u(x9)w(x)s(x9)&] ]t ]x ]x9 2 ^w(x)w(x9)&S9(z9) 2

1 ] g ^p(x)s(x9)& 1 ^u(x)s(x9)& 1 molecular terms. r0 ]z T0

Since we are only seeking a qualitative insight, we assume that we can simplify (14) by dropping terms that cannot cause asymmetry under all stratifications. The first four terms on the right, being divergences, do not globally create or destroy displaced flux; they only move it around in space. The first pair, which represents mean advection, vanishes in free convection. However, since up–down asymmetry exists in free convection, these terms cannot be dominant. The second pair represents turbulent transport, which Wyngaard (1982) showed is generally small in single-point Reynolds flux equations within the surface layer. The buoyancy term is negligible for conditions sufficiently close to neutral, where the up–down asymmetry also exists; therefore, it is not a dominant term. Finally, the molecular terms are negligible, as pointed out by Wyngaard (1982). Thus, we drop all but the fifth and the sixth terms on the right side and write the lowest-order budget as ] 1 ] ^w(x)s(x9)& . 2^w(x)w(x9)&S9(z9) 2 ^p(x)s(x9)&. ]t r0 ]z (15) We can get physical insight into (15) by considering its zero-displacement form:

7 8

] dS 1 ]p ^ws& . 2^w 2 & 2 s . ]t dz r0 ]z

(16)

This is the observed lowest-order form of the temperature flux budget (Wyngaard et al. 1971). The terms on the right side of (16) represent mean-gradient production and pressure destruction, respectively. Under quasi-steady conditions in the surface layer, the pressure term behaves as (Wyngaard 1982) 2

7 8

1 ]p ^ws& s .2 , r0 ]z t

(17)

where t is a large-eddy timescale. With (17) and the quasi-steady assumption, the lowest-order flux budget (16) is 0 . 2^w 2 &S9(z) 2

^ws& , t

(18)

which is equivalent to the observed flux-gradient relation (Businger et al. 1971)

^ws& 5 2^w 2 &t

(14)

dS , dz

(19)

where ^w2&t is the eddy diffusivity. Subtracting the conservation equation for ^w(x9)s(x)& [(15) with x and x9 interchanged] from (15) yields a conservation equation for scalar flux asymmetry: ] [^w(x)s(x9)& 2 ^w(x9)s(x)&] ]t 5 2^w(x)w(x9)&[S9(z9) 2 S9(z)] 2

[

]

1 ] ] ^p(x)s(x9)& 2 ^p(x9)s(x)& . r0 ]z ]z9

(20)

The experience with the zero-displacement flux budget (16) suggests that the terms on the right side of (20) represent the principal production and destruction mechanisms, respectively. If so, the flux asymmetry stems from the production term and therefore from a difference in the mean scalar gradient at the two measuring points. To determine the sign of the flux asymmetry, we qualitatively represent the destruction term in (20) with the counterpart of its observed form (17). In quasi-steady conditions, (20) then yields a flux-gradient relation for the flux asymmetry: ^w(x)s(x9)& 2 ^w(x9)s(x)& . 2^w(x)w(x9)&t(x, x9)[S9(z9) 2 S9(z)].

(21)

Here t 5 t (x, x9) is again a large-eddy timescale. Equation (21) indicates that to the lowest order, horizontal displacement does not cause flux asymmetry. For vertical displacement it indicates that F ↑ (z, z9) 2 F ↑ (z9, z) 5 ^w(z)s(z9)& 2 ^w(z9)s(z)& . 2^w(z)w(z9)&t(z, z9)[S9(z9) 2 S9(z)].

(22)

This equation predicts the rather surprising result that the absolute value of the measured flux is larger if the scalar sensor is placed under the anemometer. To see this, first let the flux F 5 F↑(z, z) be positive. Then S9(z) must be negative and increasing with height. This means that the right-hand side of (22) has the same sign as z 2 z9 and, consequently, that F↑ (z, z9) . F↑(z9, z) if z . z9. If F is negative, the sign of right-hand side must be opposite to that of z 2 z9 so that F↑(z, z9) ,

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FIG. 3. (a) Displaced flux versus flux from the experiment at Risø with z 5 1 m and z9 5 2.5 m. The slope of the fitted line is 0.45. (b) Same as (a) but from NCAR and with z 5 4 m and z9 5 3 m and a slope of 1.01.

F↑ (z9, z) if z . z9. In general, (22) shows that F↑(z, z9)/ F . F↑(z9, z)/F when z . z9.

of these sensors and the data acquisition system are part of the NCAR ASTER (Atmosphere–Surface Turbulent Exchange Research) facility (Businger et al. 1990).

3. Experimental investigations We carried out experiments to compare the covariance between temperature and vertical velocity from vertically separated sensors to the actual vertical temperature flux. These experiments were at two different locations: the Marshall field site near Boulder, Colorado, and the RIMI site (the Risø Integrated Milieu project) (Hummelshøj 1994) near Roskilde, Denmark. At the Marshall field site, covariances with lateral separations were also measured. a. Marshall measurements The Marshall field site has been used primarily for development of radar systems. However, it has a fetch of approximately 500 m over wild grassland from a few wind directions. Since the wind does not frequently come from these directions, only 31 hours of data were collected over a period of 36 days from the end of August of 1993. The instrumentation consisted of one three-dimensional and one one-dimensional sonic anemometer (Applied Technologies, Inc., models 3K and 1K) and five fast-response platinum-wire thermometers (Atmospheric Instrumentation Research, Inc., model FT). The anemometers were mounted on a mast and were vertically separated with the three-dimensional sonic at a height of 4 m and the one-dimensional sonic 1 m below. A thermometer was located with each of the anemometers, slightly behind the vertical, acoustical paths, and one thermometer was placed initially at 3.5 m between the anemometers and later moved vertically to 3.33 m. Two thermometers were placed on either side of the sonic at 3 m, one at a horizontal distance of 0.25 m the other at 0.50 m. The latter was later moved to 0.75 m. Most

b. RIMI measurements The RIMI site, approximately 1 km from Risø National Laboratory, was designed to study exchange of anthropogenic nitrogen compounds in rural Denmark. It is situated on a grass field in gently rolling terrain with height variations of less than 5 m in a 1-km radius from the site. A permanent 10-m mast with vanes, cups, and sonics at several heights gives information on stability, wind speed, and direction. For this experiment, an additional mast was erected 100 m from the permanent mast. For directions between 1908 and 3588, the fetch is uninterrupted grass for 300– 500 m. The mast was instrumented with two one-dimensional sonics (METEK Meteorologische Messtechnik GmbH, Hamburg, Germany) capable of measuring vertical velocity and temperature. The sonics were mounted on a vertical sliding mechanism allowing the measurement heights to vary between 1 and 2.5 m. To avoid ‘‘cross-talk,’’ it was necessary to put a small sound-absorbing plate between the sonics. The experiment ran for 34 days from February to April 1995, and since the above mentioned wind directions are quite common, 643 hours of data were useful. 4. Analysis To get a relation between the actual and the displaced flux, we compute a least squares fit of the former to the latter for each vertical or horizontal displacement as the examples in Fig. 3a for the Risø data and Fig. 3b for the NCAR data show. Each slope is then plotted versus D/z or z/z9 in Figs. 5 and 6. Because each week of data from Risø is fitted separately, several estimates of the slope appear at each z/z9 in Fig. 6.

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KRISTENSEN ET AL. TABLE 1. Parameters of cospectral fit to Eq. (23).

FIG. 4. Example of fit of cowT(k) to Eq. (23) with z 5 2 m, L 5 2.9 m, and m 5 0.24.

a. Horizontal displacement In order to predict the relative attenuation of fluxes due to the lateral displacement in general, it is necessary to get an accurate description of the cospectrum of w and temperature. We do that by fitting cowT(k) } 1/[1 1 (kL) ]

2m 7/6m

(23)

to spectra from the Risø experiment at z 5 1, 1.5, 2, and 2.5 m. Here 0 , m # 1 is a dimensionless constant and L is a length scale. In this way we obtain in the limit Lk → ` an expression equivalent to (7) and are at the same time able to account for the cospectral behavior at lower wave numbers. The experimental spectra are simple average wavenumber spectra of all runs at the given height. The spectra with negative fluxes are multiplied by 21 in the averaging. This procedure will give most weight to spectra with large (positive or negative) fluxes. Figure 4 shows the fit at z 5 2 m and Table 1 gives the parameters at the other heights. For future use, we take L 5 1.2z and m 5 0.23. Compared to the cospectra of Kaimal et al. (1972), our spectral fit has a broader peak and the corresponding wavenumber is lower by a factor of 2. Figure 5 shows the measured loss of flux compared to the loss predicted according to (5) when using the spectral shape (23). The integration in (5) has been carried out numerically. As Fig. 5 shows, the measured F→(D)/F is systematically larger than our prediction based on (5). The reason is that the unavoidable sonic anemometer path filtering reduces F more than F→ by high-wavenumber attenuation; at wavenumbers larger than the reciprocal sonic pathlength, the one-point cospectrum is reduced more than the two-point cospectrum, which is already reduced due to the displacement. As mentioned in the introduction, Lee and Black (1994) suggested and justified experimentally an equation for F→(D):

[

1 21 2

F → (D) 5 F exp 2b d,

z LO

D z

4/3

]

,

(24)

z (m)

L (m)

m

1.0 1.5 2.0 2.5

1.0 1.5 2.9 3.1

0.23 0.23 0.24 0.22

where the decay factor b depends on the angle d between the wind and displacement vector and the stability z/ LO. In our NCAR measurements, 90% of the runs have stabilities between z/LO 5 0 and 22 and z908 2 dz , 458. Assuming on average d 5 608, Lee and Black would get b 5 1.9 for neutral stability and 0.48 for z/LO 5 22. The corresponding curves are also shown in Fig. 5. As seen from this figure, our measurement and theory are not in direct contradiction with Lee and Black. However, we do not observe any systematic variation of F→(D)/F with d and no large variation with stability. b. Vertical displacement Here F↑(z, z9) is a function of w˜ at z and s˜ at z9. Alternatively, we may, assuming neutral surface-layer scaling, consider F↑ a function of the ratio z/z9, that is, F ↑ (z, z9) [ f

1z 1 z92 [ g 1z92 . z9 2 z

z

(25)

Up–down symmetry, that is, F↑(z9, z) 5 F↑(z, z9), would imply that the function g has the property g

1h2 5 g(h). 1

(26)

Figure 6 clearly shows that the measurements do not support such a symmetry. Fitting the measured ratios with straight lines gives

FIG. 5. Vertical flux of temperature measured with three different values of the lateral displacement between the thermometer and the anemometer. The solid line shows the prediction Eq. (5) with cowT(k) given by Eq. (23). The other lines are from Lee and Black (1994). The dashed has z/LO 5 0, while the dotted has z/LO 5 22.

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FIG. 6. Displaced flux as a function of z/z9. Dots are from the NCAR measurements, circles from Risø. The lines are linear fits to g(z/z9)/F [see Eq. (27)].

12

z g 5 z9

5

[ [

] ]

1 z92 , z F 1 2 0.1 1 2 12 , z9 F 1 2 1.0 1 2

z

z , z9 (27) z . z9.

This interpolation curve is shown in Fig. 6. 5. Conclusions Our purpose has been to discuss and to quantify the ‘‘loss’’ of vertical scalar flux due to an arbitrary separation between the anemometer and the scalar sensor in order to determine how close the two sensors must be to reduce systematic errors in the measured flux below a specified level. Our model predicts that this error is small if the separation is small compared to the scale of the turbulence. This conclusion is supported by our measurements at the NCAR site and at the RIMI site in Denmark shown in Fig. 5 (lateral displacement) and Fig. 6 (vertical displacement). For example, if the ratio of the displacement to the height is 0.1, more than 90% of the flux is recovered. When measuring vertical scalar fluxes close to the ground, the measurements show that it might be important to consider the effect of the sensor configuration. Using the general expression (23) for the cospectrum cowT(k), we are able to determine F→(D) on basis of the parameters L and m in the measured cospectrum cowT. Our equation for F→(D) (5) has no azimuthal dependence since it is derived for circular symmetry. We have verified this symmetry assumption experimentally only for z908 2 dz , 458. Moore (1986) assumes different equations for lateral and longitudinal displacements. Lee and Black (1994) found in their measurements that F→ depends on both the direction of D and the stability parameter z/LO. Our measurements do not allow us to carry out a similar detailed analysis, but we have in-

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cluded, together with our result in Fig. 5, the result by Lee and Black (1994) for lateral displacement for two values of z/LO, namely 0 and 22. We see that their equation is not inconsistent with our simple model and measurements. It should be pointed out that the spectral parameters L and m are obtained at the RIMI site in Denmark, whereas F→(D) is measured at the Marshall site. When the two sensors are displaced vertically, we find both theoretically and experimentally that there is no up–down symmetry in the sense that we get a different value of F↑ just by switching the positions of the sensors. We argued qualitatively, on basis of the length scales of the vertical velocity and the scalar, that we must expect F↑ to be larger when the scalar sensor is below the anemometer than when their positions are interchanged. More detailed second-order budget considerations show that this behavior is due to the dependence of the scalar gradient on height. The measurements of temperature flux, shown in Fig. 6, confirm this result. Neither of the theoretical considerations allows us actually to predict anything more than the nature of the asymmetry, and the solid curve in Fig. 6 is simply the interpolation (27) of the experimental results. As an example of the use of our findings, let us assume that we want to measure a scalar flux at the height z 5 1 m and that we for practical reasons are forced to place the sensors D 5 0.2 m apart. Then from Fig. 5 we see that the ‘‘flux loss’’ is about 13% if the displacement is horizontal. If the displacement is vertical, (27) and Fig. 6 show that the corresponding loss is 18% if the scalar sensor is positioned over the anemometer but just about 2% if the positions are interchanged. On the basis of our theoretical and experimental results, we recommend that the separation between the anemometer to measure the vertical velocity component and the scalar sensor be vertical. Then the sensor configuration has a large azimuthal acceptance angle, and the need to lag one signal with respect to the other if the displacement vector has a streamwise component has been eliminated. We have shown that placing the scalar sensor under the anemometer minimizes the ‘‘loss’’ of covariance due to sensor separation. On the other hand, Wyngaard (1988) showed that an eddy-correlation package with reflection symmetry about a horizontal plane through the measuring point eliminates the cross-talk error in flux caused by flow distortion. Thus, in designing an instrument package, one should balance the inherent flux error due to vertical asymmetry of the array against the minimization of the separation-induced flux loss by placing the scalar sensor under the anemometer. Acknowledgments. We wish to thank Don Lenschow and Tom Horst of NCAR for their interest and support of this project. Also, we very much value Tom Horst’s comments to the manuscript. We appreciate the comments from Chenning Tong of the Pennsylvania State

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University. Jørgen Højstrup of Risø National Laboratory read the manuscript and offered many useful comments and suggestions for improvements. Norbert Szczepczynski of the National Oceanic and Atmospheric Administration’s Environmental Technology Laboratory provided several temperature sensors used in the Marshall experiment. REFERENCES Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux-profile relationship in the atmospheric surface layer. J. Atmos. Sci., 28, 181–189. , W. F. Dabberdt, A. C. Delany, T. W. Horst, C. L. Martin, S. P. Oncley, and S. R. Semmer, 1990: The NCAR Atmosphere–Surface Turbulent Exchange Research (ASTER) facility. Bull. Amer. Meteor. Soc., 71, 1006–1011. Hummelshøj, P., 1994: Dry deposition of particles and gases, Rep. Risø-R-658(EN), 110 pp. [Available from Risø National Laboratory, P.O. Box 49, DK-4000 Roskilde, Denmark.]

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