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  • View in gallery

    Dimensionless shear (ϕm) and lapse rate (ϕh) vs stability (z/Λ) for CASES-99 observations. Error bars indicate one standard error around the mean.

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    Effect of errors in the momentum flux and the heat flux (δH/H = δτ/τ) on the position of a reference point in the ϕ–z/Λ space. Application to the Businger–Dyer relations (β = 5).

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    The impact of errors in the momentum flux and the heat flux on plots of ϕh and ϕm vs z/Λ for different values of the ratio between δH/H and δτ/τ : (a) δτ/τ = δH/H, (b) δτ/τ = 0 and |δH/H| > 0, (c) |δτ/τ| > 0 and δH/H = 0, and (d) δτ/τδH/H. Open triangles (Δ) indicate ϕh, solid circles (•) indicate ϕm. The solid line represents the BH91 function for momentum. The maximum error margin amounts to |20%|.

  • View in gallery

    Observed stability dependence of the ratio α = δH/H/ δτ/τ for CASES-99; (a) 0 < z/Λ < 0.1, (b) 0.1 < z/Λ < 0.5, (c) 0.5 < z/Λ < 1, (d) z/Λ > 1. The solid lines indicate the 1:1 line (α = 1), the dashed lines indicate the least square fit based on perpendicular offsets for each stability regime. As a measure of scatter we use the 75th percentile of the deviations from the fit. The respective values are (a) 0.036, (b) 0.029, (c) 0.060, and (d) 0.109.

  • View in gallery

    Flux–gradient relationships for real and randomized data for (left) ϕm and (right)ϕh vs z/Λ. Black dots represent real data, gray squares randomized data.

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    (a) Standard plot of ϕm vs z/Λ and (b) an alternative plot from which the slope of ϕm vs z/Λ can be estimated independent of self-correlation. The gray line in (b) is a least square fit to the data. The slope of this fit is passed on to (a) to illustrate that this slope gives indeed a reasonable fit for the ϕm(z/Λ) data.

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Exploring Self-Correlation in Flux–Gradient Relationships for Stably Stratified Conditions

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  • 1 Royal Netherlands Meteorological Institute, De Bilt, and Meteorology and Air Quality Group, Wageningen University, Wageningen, Netherlands
  • | 2 Meteorology and Air Quality Group, Wageningen University, Wageningen, Netherlands
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Abstract

In this paper, the degree of scatter in flux–gradient relationships for stably stratified conditions is analyzed. It is generally found that scatter in the dimensionless lapse rate ϕh is larger than in the dimensionless shear ϕm when plotted versus the stability parameter z/Λ (where Λ is the local Obukhov length). Here, this phenomenon is explained to be a result of self-correlation due to the occurrence of the momentum and the heat flux on both axes, measurement uncertainties, and other possibly relevant physical processes left aside. It is shown that the ratio between relative errors in the turbulent fluxes influences the orientation of self-correlation in the flux–gradient relationships. In stable conditions, the scatter in ϕm is largely suppressed by self-correlation while for ϕh this is not the case (vice versa for unstable stratification). An alternative way of plotting is discussed for determining the slope of the linear ϕm function.

Corresponding author address: Dr. P. Baas, Atmospheric Research, Royal Netherlands Meteorological Institute, Wilhelminalaan 10, P.O. Box 201, 3730 AE, De Bilt, Netherlands. Email: peter.baas@knmi.nl

Abstract

In this paper, the degree of scatter in flux–gradient relationships for stably stratified conditions is analyzed. It is generally found that scatter in the dimensionless lapse rate ϕh is larger than in the dimensionless shear ϕm when plotted versus the stability parameter z/Λ (where Λ is the local Obukhov length). Here, this phenomenon is explained to be a result of self-correlation due to the occurrence of the momentum and the heat flux on both axes, measurement uncertainties, and other possibly relevant physical processes left aside. It is shown that the ratio between relative errors in the turbulent fluxes influences the orientation of self-correlation in the flux–gradient relationships. In stable conditions, the scatter in ϕm is largely suppressed by self-correlation while for ϕh this is not the case (vice versa for unstable stratification). An alternative way of plotting is discussed for determining the slope of the linear ϕm function.

Corresponding author address: Dr. P. Baas, Atmospheric Research, Royal Netherlands Meteorological Institute, Wilhelminalaan 10, P.O. Box 201, 3730 AE, De Bilt, Netherlands. Email: peter.baas@knmi.nl

1. Introduction

Flux–gradient relationships are used to relate gradients of mean atmospheric profiles to turbulent fluxes. The concept of flux–gradient relationships has proven to be very useful in estimating surface fluxes both in atmospheric models and from observed profiles. The relevant quantities to relate fluxes and gradients are obtained from dimensional analysis. Consequently, the functional form of the flux–gradient relationships must be found by experiment. Some of the current problems in atmospheric boundary layer modeling (e.g., Holtslag 2006), may be associated with uncertainties in the form of the flux–gradient relationships. In the near neutral regime, different studies on the determination of the flux–gradient relationships show similar results with little scatter. In contrast, the proposed functional forms diverge considerably for stronger stability (Högström 1988, 1996; Andreas 2002).

For the stable boundary layer (SBL) a systematic difference in scatter between the dimensionless shear ϕm and the dimensionless lapse rate ϕh is found when plotted versus the stability parameter z/Λ (where Λ is the local Obukhov length). Typically, scatter in ϕh is larger than in ϕm for a large majority of field experiments (see e.g., Oncley et al. 1996; Forrer and Rotach 1997; Duynkerke 1999; Howell and Sun 1999; Yagüe et al. 2001; Cheng and Brutsaert 2005; Steeneveld et al. 2006). Figure 1 illustrates this point for observations from the Cooperative Atmosphere–Surface Exchange Study 1999 (CASES-99).

In this paper we explain the difference in scatter between ϕm and ϕh to be a result of self-correlation. Self-correlation is also referred to in literature as spurious correlation or the shared variable problem and arises when one (dimensionless) group of variables is plotted against another, and the two groups under consideration have one or more common variables (e.g., Hicks 1978; Kenney 1982; Aldrich 1995). In that case, correlation is partly caused by artificial, mathematical reasons, apart from physical and observational aspects. For plots suffering from self-correlation, the amount of scatter is not directly related to the quality of the data or the validity of the physical relationship under consideration.

In general, the influence of self-correlation depends on two factors. First, on the relative variation of the fundamental variables, Vxi = σxi/. Here σxi denotes the standard error of a certain variable xi. When Vxi of the common variable is large compared to Vxi of the other variables, the degree of self-correlation is considerable (Kim 1999; Klipp and Mahrt 2004). Second, the relationship between the orientation of the self-correlation on the one hand and of the true physical correlation on the other hand is important. Depending on this relationship, the influence of self-correlation can be either negligible or very large.

Dimensional analysis is a widely applied tool in boundary layer meteorology (e.g., Stull 1988). However, often not sufficient independent scaling variables are present to construct completely independent dimensionless groups (Andreas and Hicks 2002). As a consequence, some amount of self-correlation is often inevitable.

The aim of this paper is to explore the influence of self-correlation in ϕm and ϕh for stable conditions. Furthermore we discuss the influence of self-correlation as a function of atmospheric stability and we discuss an alternative way of plotting for determining the slope of the linear ϕm function independent from self-correlation. While earlier studies mainly examined the effect of the shared momentum flux, here we investigate the combined effect of the momentum flux and the heat flux on the influence of self-correlation.

In section 2 the background of Monin–Obukhov similarity theory (MOS) is briefly summarized. Section 3 gives a short description of the observational data and section 4 presents a sensitivity analysis on the influence of self-correlation on the flux–gradient relationships. In section 5 the stability dependence of self-correlation will be investigated and in section 6 we will apply an alternative technique to illustrate the influence of self-correlation. In section 7 we suggest some characteristics we think a new scaling approach should meet. Finally in section 8 we draw conclusions.

2. Monin–Obukhov similarity theory

Turbulent motions in the atmospheric surface layer can be well described in terms of MOS (e.g., Stull 1988). This theory states that under homogeneous and stationary conditions every dimensionless group is a universal function of z/L, where z is the height above the surface and
i1520-0469-63-11-3045-e1
is the Obukhov length. Here k is the Von Kármán constant (taken as 0.4; Högström 1996), g is the acceleration due to gravity, and θ is the mean potential temperature;
i1520-0469-63-11-3045-e2
is the friction velocity and is the surface kinematic turbulent heat flux. The momentum flux is denoted by τ and ρ (1.2 kg m−3) is the air density. Overbars indicate mean values; primes indicate deviations from the mean.
The dimensionless shear ϕm and the dimensionless lapse rate ϕh are defined as
i1520-0469-63-11-3045-e3
i1520-0469-63-11-3045-e4
Here U is the mean wind speed and θ* = −/u* is a turbulent temperature scale.
In principle MOS is only valid in the surface layer. Above this layer the magnitude of the fluxes generally decreases with height in the SBL (e.g., Stull 1988) and Monin–Obukhov scaling is no longer appropriate. Alternatively, Nieuwstadt (1984) used local fluxes instead of surface fluxes (local scaling). All dimensionless groups now depend on z/Λ, where
i1520-0469-63-11-3045-e5
is the local Obukhov length at height z (Nieuwstadt 1984; Holtslag and Nieuwstadt 1986). MOS can be considered as a special case of local scaling theory.

3. Observational data

To illustrate our results, we use observations from the CASES-99 measurement campaign (Poulos et al. 2002), which was organized in October 1999, 50 km east of Wichita, Kansas (37.65°N, 96.74°W; ∼440 m asl), over gently rolling terrain with slopes from 0.1 to 0.8 degrees. The experimental area was covered with prairie grass with a roughness length of 0.03 m.

A 60-m tower was equipped with a dense vertical array of either 20-Hz Campbell Scientific (CSAT3) or Applied Technologies K-style sonic anemometers at 10, 20, 30, 40, 50, and 55 m. Temperature profiles are obtained at six levels from slow response aspirated temperature/humidity sensors at 5, 15, 25, 35, 45, and 55 m. (All the data were obtained from http://www.atd.ucar.edu/rtf/projects/cases99/asciiDownload.jsp.)

Around the main mast a network of flux stations was set up. Hartogensis and De Bruin (2005) operated a 10 m mast with a CSAT3 sonic anemometer and a KH20 Krypton hygrometer at 2.6 and 10.2 m. To investigate the impact of self-correlation as a function of atmospheric stability, we use eddy-correlation data from the 10.2-m level. To calculate turbulent fluxes, the raw data are processed by a software package (Van Dijk and Moene 2004; see also http://www.met.wau.nl/projects/jep/report/ecromp.pdf). The package provides a statistical error for each flux variable. This error is mainly based on the measurement accuracy and the number of independent samples from which the averaged flux value is composed. For normally distributed samples the statistical error equals 2 times the standard deviation.

4. Influence of self-correlation on flux–gradient relationships

The impact of self-correlation can be investigated by imposing errors on the common variables. When the resulting contaminated points move roughly along the reference curve, self-correlation has significant influence. By reference curve we mean a first-order guess of the relationship, based on earlier studies. To be more specific, for the flux–gradient relationships in the SBL this is a log-linear relationship in the near-neutral regime. With increasing stability, the functions gradually deviate from log linear. Of course, in principle, even this first-order guess in itself suffers from self-correlation, but this second-order effect is discarded.

By imposing a 10% error on u*, Andreas and Hicks (2002) show that self-correlation indeed causes a difference in scatter between ϕm and ϕh for unstable conditions. For stable conditions, Klipp and Mahrt (2004) found that self-correlation explains 65% of the variance between ϕm and z/Λ, owing to the occurrence of u* in both quantities.

In this section we will impose errors on both the momentum flux τ and the sensible heat flux H. The erroneous fluxes can be expressed as
i1520-0469-63-11-3045-e6
where, for example, Δτ and ΔH can be considered as random or systematic measurement errors. In this study we ignore uncertainties in the gradients, although these can be considerable (Akima 1970; Oncley et al. 1996; Frentzen and Vogel 2001). When we reduce the errors Δτ and ΔH to infinitesimal values, we obtain from Eq. (6)
i1520-0469-63-11-3045-e7
At first we assume the relative errors in τ and the H to be equal, so δτ/τ = δH/H. In this case, the direction in which a reference point shifts as a result of the imposed errors reads
i1520-0469-63-11-3045-e8
This result is found by differentiating ϕm, ϕh, and z/Λ with respect to both τ and H and using the assumption δτ/τ = δH/H (see the appendix). Equation (8) states that the effect of an imposed error δτ/τ = δH/H on a reference point in the ϕ, z/Λ space causes a deviation from the reference point along a line through the reference point and the origin. Equation (8) indicates that the response of ϕm and ϕh on imposed errors is exactly equal. Consequently, when δτ/τ = δH/H, any difference in observed scatter between ϕm and ϕh cannot be explained as an effect of self-correlation. For the Businger–Dyer relationships (ϕm = ϕh = 1 + 5z/Λ; Dyer 1974), this result implies that for increasing stability, the deviation becomes more aligned with the slope of the ϕ function, β (Fig. 2).

In reality the assumption δτ/τ = δH/H is not generally valid (see section 5). Because often the exact correlation between δτ/τ and δH/H is unknown, we examine four limit situations of the error ratio parameter α = (δH/H)/(δτ/τ):

  • (a) δτ/τ = δH/H (α = 1; as above)
  • (b) δτ/τ = 0 and |δH/H| > 0 (α → |∞|)
  • (c) |δτ/τ| > 0 and δH/H = 0 (α = 0)
  • (d) δτ/τδH/H (−∞ < α < ∞)

The magnitude of the imposed relative errors is randomly taken from a uniform distribution and is between −20% and +20%. Starting from a reference point, for each of the four limit situations 100 artificial data points are generated. Figure 3 shows the results of this numerical simulation. The solid lines correspond to the reference curve, in this case the Beljaars–Holtslag (1991, hereafter BH91) formulation for momentum. The results of the current study do not depend on the choice of this specific function. Other realistic functions lead to similar conclusions.

Figure 3a confirms our analytical approach of Eq. (8): ϕm and ϕh behave similarly and all points are on a straight line through the reference point and the origin. When the relative errors in τ and the H are not equal as in the cases (b), (c), and (d), the flux–gradient functions indeed scatter differently. In these cases the random ϕm data scatters approximately along the reference curve, while for ϕh the data scatters roughly perpendicular to this curve. For case (c) this leads to the interesting paradoxical result that when τ contains an error and H is perfect, ϕm(z) shows little scatter while ϕh(z) shows large scatter. Except for Fig. 3d all points are on straight or slightly bent lines. Figure 3d shows much more scatter, because contrary to the cases (a), (b), and (c), there is no fixed correlation between δτ/τ and δH/H in this case (−∞ < α < ∞). In fact, in case (d) the three other cases are enclosed. Figure 3 also infers that the magnitude of the deviations depends on α. For example, the maximum deviation in case (a) is much smaller than in case (d) for both ϕm and ϕh.

In general, when δτ/τδH/H, ϕh will always have more scatter around the reference curve than ϕm for a given dataset in stable conditions. This is a nonphysical effect that rises from the mathematical expressions for ϕm, ϕh and z/Λ. For ϕm the shift along the reference curve demonstrates that self-correlation has substantial influence, while for ϕh self-correlation only is important when δτ/τ = δH/H. The different behavior of ϕm and ϕh illustrates the fact that just common variables on both axes does not automatically imply that self-correlation is important. Only when the mathematical relationship of the common variables is roughly in line with the physical relationship (like in ϕm), scatter remains hidden and self-correlation will have significant influence.

For a given α the direction in which a certain reference point will shift due to errors in the fluxes reads for ϕm(z)
i1520-0469-63-11-3045-e9
and for ϕh(z)
i1520-0469-63-11-3045-e10

These expressions are derived in the appendix and agree with the limit situations of Fig. 3. For example, when α = 1 as in Fig. 3a, Eqs. (9) and (10) both reduce to Eq. (8). In fact, these equations predict the orientation of self-correlation for a given value of α in a certain point (z/Λ, ϕm,h).

The above analysis can also be applied to the unstable regime. As such, it can be shown that the effect on scatter in ϕm and ϕh is then mostly opposite: in the unstable regime ϕm exhibits more scatter than ϕh (see also Andreas and Hicks 2002; Johansson et al. 2001).

5. Stability dependence of self-correlation

In section 4 we found that α influences the orientation of scatter in flux–gradient plots. In practice however, it is hard to estimate the actual value of α. In this section we show how α may be influenced by atmospheric stability.

For our analysis we use eddy-correlation data from CASES-99, but similar results were obtained with routine flux observations from the Wageningen University weather field (Jacobs et al. 2003). With the statistical error described in section 3, relative errors of τ and H can be obtained according to
i1520-0469-63-11-3045-e11
where x is τ or H and se(x) is the corresponding statistical error. Note that the statistical error is not equal to the actual measurement error and does also not provide information about the sign of it. Therefore, it is impossible to determine unambiguously the effect on the orientation of scatter in the flux–gradient plots. However, it is still possible to indicate that a stability dependence is present. For example, when se(τ) ≪ se(H), it is reasonable to assume that most of the true errors in τ will also be much smaller than those in H.

We analyze α as a function of stability and anticipate which of the four cases from Fig. 3 is more likely to occur. Figure 4a shows that in the near neutral situation, the value of δH/H is likely to be larger than the value of δτ/τ. This is caused by the fact that in these conditions τ has much larger values than H. In general, smaller values of a certain variable are accompanied by larger relative errors. In the weakly stable regime (Figs. 4b,c), the relative errors are slightly related, especially for small error values. To quantify the spread, we made a linear fit to the data (using perpendicular offsets) and calculated for each data point the absolute deviation from the fit. The 75th percentile of these deviations is used as an indication for the spread. In the very stable regime (Fig. 4d), this measure is increased compared to the weakly stable regime demonstrating that the relationship between the relative errors in the fluxes disappears. The difference in behavior between the stability regimes demonstrates that α depends on stability. With respect to our analysis on self-correlation this implies that, in a certain flux–gradient plot, the orientation of scatter due to errors in the turbulent fluxes also depends on z/Λ.

To summarize, the near neutral situation of Fig. 4a compares to Fig. 3b; the weakly stable situation of Figs. 4b,c roughly compares to Fig. 3a; and the very stable situation of Fig. 4d compares to Fig. 3d.

6. Randomizing real observations

In this section we illustrate our earlier findings by applying an alternative method to examine self-correlation. We randomize the original dataset, by mixing all available values for each of the different variables (τ, H, dU/dz, etc.) at random. In this way the characteristic statistical distributions of the variables are conserved, which is important because only then a good comparison with the real data is possible. From the randomized data we recompute z/Λ, ϕm and ϕh. It is clear that the new data points do not have any physical meaning at all (Hicks 1981; Andreas and Hicks 2002; Klipp and Mahrt 2004; Mahrt and Vickers 2003).

Figure 5 shows the result of our analysis. For ϕm, the unphysical randomized points are close to the original data. This occurs because the common variable u* appears in the denominator in both ϕm and z/Λ, combined with the fact that the physical relationship between ϕm and z/Λ has a positive slope from itself. This result corresponds to the parallel shift we found in section 4 and demonstrates again that ϕm(z) is heavily influenced by self-correlation.

Contrary to ϕm, plots of ϕh versus z/Λ are not affected by self-correlation in the sense that erroneous or even randomized data can still give misleadingly good results. However, the randomized data do not look completely chaotic for ϕh. Just as in the case of ϕm, the statistics of the common variables are reflected. The difference is that in ϕh u* appears in the nominator, which causes the randomized ϕh data to show some hyperbolic relationship. This behavior corresponds to the normal shift of a reference point relative to the reference curve that we found in section 4.

The result of Fig. 5a raises the question if there is any information in a plot of ϕm versus z/Λ. The small differences between the randomized data and the real data cannot result from limited data ranges: Table 1 shows that for each variable the range amounts at least a factor of 10, which is considerable. Fortunately, as is also known from practice, ϕm indeed contains useful information. We can determine the slope of ϕm versus zindependent from self-correlation by anticipating a linear relationship of the type ϕm = 1 + β z/Λ. Noting the definitions of ϕm and z/Λ and multiplying both sides with u3*/(kz), we can plot u2*dU/dzu3*/(kz) versus –g/θ (Fig. 6b). The slope of this plot provides an independent estimate for the slope β of the standard ϕm plot as given in Fig. 6a. We limit our analysis to data for which z/Λ < 1.5, since only then a linear relationship can be assumed (e.g., Holtslag and De Bruin 1988; BH91). Comparing Figs. 6a and 6b shows that the slopes of both figures agree well. This indicates that, contrary to the scatter, the shape of ϕm is not very sensitive to self-correlation. Note that scatter in Fig. 6b is small, despite the fact that we plot a difference term on the y axis.

7. Discussion and future research

Scatter can be misleadingly small in plots were self-correlation is important. Consequently, there is need for an alternative way to relate fluxes and profiles. One way to deal with this problem is to search for alternative scaling approaches that are less vulnerable to self-correlation (e.g., Klipp and Mahrt 2004).

The Buckingham–Pi theorem does not prescribe how to compose the different dimensionless groups. When information on relative errors of the different variables (u*, H, ∂U/∂z, ∂θ/∂z, etc.) is available, it would be possible to construct the dimensionless groups in the most advantageous way. In practice this means that the most uncertain variables (here the turbulent fluxes) are raised to the lowest power in order to minimize the influence of their high relative error. Because in stable conditions fluxes are much smaller than gradients, fluxes should be raised to the lowest possible powers. In the unstable regime the situation is opposite: fluxes are large and gradients are small. Consequently, higher powers for the fluxes (e.g., u3*) combined with lower powers for the gradients are preferable. Based on this arguments, it can be stated that MOS is much more suitable for unstable than for stable conditions. MOS can be regarded as a flux-based scaling approach, while for the SBL a gradient-based scaling might be more suitable.

For the SBL, in many cases the above reasoning leads to a scaling based on the gradient Richardson number, which is given by
i1520-0469-63-11-3045-e12
By using a Ri-based scaling for the stable regime, Klipp and Mahrt (2004) also largely circumvent the problem of self-correlation (see also Sorbjan 2006).

8. Conclusions

Flux–gradient relationships are traditionally used to relate turbulent fluxes and atmospheric mean profiles in the atmospheric boundary layer. However, self-correlation significantly influences the relationships between the dimensionless shear ϕm and the dimensionless lapse rate ϕh versus the stability parameter z/Λ. In general, self-correlation arises when two parameters share some common variable. For ϕh(z) and ϕm(z/Λ), the common variables are the momentum flux τ and the heat flux H. The former appears in ϕm, ϕh, and z/Λ, the latter in ϕh and z/Λ. Here we consider stable stratification only.

The impact of self-correlation on the flux–gradient relationships depends highly on the ratio of relative errors in the momentum flux and the heat flux. This ratio governs both the orientation and the magnitude of the scatter. In general, scatter in ϕm remains unrealistically small, because the data are scattered roughly along the physical curve. Contrary, for ϕh scatter is much larger, because the data are scattered roughly perpendicular to the physical curve; ϕh will therefore always show more scatter than ϕm. This holds only for the stable regime. In the unstable regime the effect of self-correlation is mostly reversed. In this analysis, errors associated with possible violations of Monin–Obukhov similarity theory and errors in ∂θ/∂z and ∂U/∂z are not taken into account, although in reality these are also important.

When imposed relative errors on the fluxes are equal, the difference in behavior between ϕh and ϕm disappears: in both cases a reference point shifts along a straight line through this reference point and the origin. Therefore no difference in scatter between ϕm and ϕh arises in this special case. However, it is not realistic to assume the ratio between the relative errors of the fluxes to be a constant within a single dataset, because this ratio depends strongly on stability. In practice this means that in a certain flux–gradient plot, the influence of self-correlation on the orientation of the scatter also varies with stability. While self-correlation occurs in both ϕm and ϕh, only in case of ϕm the influence is significant. Self-correlation only becomes important when the mathematical relationship of the common variables is roughly in line with the physical relationship of the parameters that include them.

Overall, current practice of evaluating ϕm and ϕh functions from field data is troublesome. The large influence of self-correlation may lead to false confidence in the found relationship, in particular for ϕm. By plotting quantities in an alternative way, we are able to determine the slope of ϕm independent from self-correlation. It seems that, contrary to the scatter, the slope of ϕm is not very sensitive to self-correlation. The evaluation of ϕh does not suffer from artificially enhanced correlation. However, due to the factor u3*, small deviations in u* may lead to large scatter, which can hamper an accurate estimation of ϕh.

Acknowledgments

We thank our colleagues Oscar Hartogensis and Henk de Bruin for providing their eddy-correlation data, as well as all other researchers and technicians who contributed to CASES-99. We also thank two anonymous reviewers for their constructive comments and suggestions. The first author has been supported by the Netherlands Organization for Scientific Research (NWO), in particular through the project “land surface climate and the role of the stable boundary layer.”

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APPENDIX

Sensitivity of Flux–Gradient Relationships to Changes in the Momentum and the Heat Flux

The dimensionless wind shear, ϕm, and the dimensionless lapse rate, ϕh, are given by
i1520-0469-63-11-3045-ea1
According to Monin–Obukhov similarity theory these dimensionless groups are universal function of
i1520-0469-63-11-3045-ea2
We are interested in ∂ϕm,h/∂ζ as a result of imposed errors on the common variables, the momentum flux τ and the sensible heat flux H.
We assume the relative errors in the fluxes to be related:
i1520-0469-63-11-3045-ea3
First we calculate δϕm, δϕh, and δζ separately, after which we perform the division. Using formal differentiation rules and Eq. (A3) we obtain
i1520-0469-63-11-3045-ea4
i1520-0469-63-11-3045-ea5
i1520-0469-63-11-3045-ea6
Using Eqs. (A4), (A6), and (A3) we can now calculate ∂ϕm/∂ζ:
i1520-0469-63-11-3045-ea7
The same exercise can be done for ∂ϕh/∂ζ. Using Eq. (A5), (A6), and (A3) we obtain
i1520-0469-63-11-3045-ea10

Fig. 1.
Fig. 1.

Dimensionless shear (ϕm) and lapse rate (ϕh) vs stability (z/Λ) for CASES-99 observations. Error bars indicate one standard error around the mean.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3778.1

Fig. 2.
Fig. 2.

Effect of errors in the momentum flux and the heat flux (δH/H = δτ/τ) on the position of a reference point in the ϕ–z/Λ space. Application to the Businger–Dyer relations (β = 5).

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3778.1

Fig. 3.
Fig. 3.

The impact of errors in the momentum flux and the heat flux on plots of ϕh and ϕm vs z/Λ for different values of the ratio between δH/H and δτ/τ : (a) δτ/τ = δH/H, (b) δτ/τ = 0 and |δH/H| > 0, (c) |δτ/τ| > 0 and δH/H = 0, and (d) δτ/τδH/H. Open triangles (Δ) indicate ϕh, solid circles (•) indicate ϕm. The solid line represents the BH91 function for momentum. The maximum error margin amounts to |20%|.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3778.1

Fig. 4.
Fig. 4.

Observed stability dependence of the ratio α = δH/H/ δτ/τ for CASES-99; (a) 0 < z/Λ < 0.1, (b) 0.1 < z/Λ < 0.5, (c) 0.5 < z/Λ < 1, (d) z/Λ > 1. The solid lines indicate the 1:1 line (α = 1), the dashed lines indicate the least square fit based on perpendicular offsets for each stability regime. As a measure of scatter we use the 75th percentile of the deviations from the fit. The respective values are (a) 0.036, (b) 0.029, (c) 0.060, and (d) 0.109.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3778.1

Fig. 5.
Fig. 5.

Flux–gradient relationships for real and randomized data for (left) ϕm and (right)ϕh vs z/Λ. Black dots represent real data, gray squares randomized data.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3778.1

Fig. 6.
Fig. 6.

(a) Standard plot of ϕm vs z/Λ and (b) an alternative plot from which the slope of ϕm vs z/Λ can be estimated independent of self-correlation. The gray line in (b) is a least square fit to the data. The slope of this fit is passed on to (a) to illustrate that this slope gives indeed a reasonable fit for the ϕm(z/Λ) data.

Citation: Journal of the Atmospheric Sciences 63, 11; 10.1175/JAS3778.1

Table 1.

Data ranges for the relevant variables in Fig. 5.

Table 1.
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