in search of a gust definition.pdf

IN SEARCH OF A GUST DEFINITION

L. KRISTENSEN, M. CASANOVA,” M. S. COURTNEY and I. TROEN

Rise, National Laboratory. JO00 Roskilde. Denmark

(Received in final form 11 October. 1990)

Abstract. We propose a simple gust definition based on the theory of excursions by Rice (1944 and 1945). We discuss the relation to the distribution of extreme events and demonstrate theoretically and experimentally that the most probable extreme event is very close to being identical to the gust according to our definition. We demonstrate how it is possible to predict the gust on the basis of the measured mean wind and variance rather than rely on actually measured extreme excursions. Our gust definition also allows us to predict the average duration of a gust.

1. Introduction

Even in the simplest possible situation when the turbulence is homogeneous and stationary, there are at least two fundamental problems in the definition of a gust which, in a loose sense, is a large (positive) wind-speed deviation from the mean wind. Firstly, wind speeds are, as any other continuous signal, always observed through a low-pass filter and secondly, this filter is either mainly temporal or mainly spatial. An ideal temporal filter can be characterized by a number of wind- speed-independent time constants, whereas an ideal spatial filter has a number of wind-speed-independent distance constants assigned to each of the three spatial dimensions. To define a gust, it is necessary to specify a filter. Examples of filters are: anemometers including the signal processing; buildings: bridges; landing airplanes; and wind turbines. Beljaars (1987) has discussed the effects of filtering and also addressed the problem of discretely sampled signals. Here we shall not discuss this last complication.

The filtered wind-speed signal can be analyzed for extreme excursions in many ways and this arbitrariness is probably the main reason that a satisfactory interna- tional convention has still to be established. All methods have in common that the signal or time series in question is subdivided into a number of time series of the same duration. These are considered realizations in an ensemble, which is used to derive statistical information about extreme excursions. From this point of view, Davenport (1964), for example, derived the cumulative distribution of the maxima from each realization for a time series having a normal distribution. Panofsky and Dutton (1984), on the other hand, discuss in great detail extreme excursions in terms of the so-called exceedance statistics (Rice 1944, 1945).

In the following, we shall discuss two different approaches to a gust definition and show in what way they can be made equivalent. With this gust definition, it

* Supported by the Catalonian Research Council (CIRIT), Spain

Boundary-Layer Meteorology 55: 91-107, 1991. 0 1991 Kluwer Academic Publishers. Printed in the Netherlands

92 L. KRlSTENStN ET AL.

Fig. 1. Symbolic illustration of the two approaches, “Rice” and “Gumbel”. To the left, M realizations with the maxima Ou, and the level q/ indicated. The upper right frame shows the mean number of excursions as defined by (1) and the lower left frame the probability density p(%) (8) for

(%)/[021] = 1.2.

is possible from general knowledge of the atmospheric surface layer to predict the magnitude of the wind-speed gust as seen through a cup anemometer. Data from the Lammefjord Experiment (LAMEX) (Courtney, 1987; 1988) are used for validation.

2. Two Approaches

As stated above, there are two different approaches to obtain information about extreme events: one discussed by Rice (1944, 1945) and one by Gumbel (1958). The first deals with the frequency of excursions beyond different thresholds and the second with the probability distribution of maxima. We discuss them in that order. In both, we consider an ensemble of M realizations of a stationary time series u(t) with zero ensemble mean. Figure 1 illustrates the two different approaches.

2.1. THE “RICE" APPROACH

Let n = ~(“ll) be the average frequency with which the level %1 is exceeded. Then N(%) = VT is the average number of times the level % is exceeded during the reference period T. If N,(q) is the actual number of excursions beyond g4 in the i’th realization. then

1N SEARCH OF A GUST DEFINITION 93

(1)

If we assume that the individual excursions are statistically independent in each individual realization, then the excursions will be Poisson distributed, i.e., the probability for IZ excursions will be

P&l = e-NW -Iv”(%). I (2)

n.

In particular, the probability for no excursions beyond 021 (n = 0) becomes

P,[O] = ehNcqU). (3)

This number is also the fraction of realizations in which the level % is not exceeded. The assumption that the individual excursions have to be statistically indepen-

dent for (2) to be valid seems in many cases to contradict the behavior of real time series. Indeed, large excursions most often seem to come in groups, separated by time intervals small compared to the integral time scale. In the experimental validation however, we only use (3), i.e., we include only those realizations in which the level in question % is not exceeded. To us it seems conceivable that (3) is correct without (2) being generally true when the average number of excursions n is greater than (or equal to) one.

2.2. THE “GUMBEL" APPROACH

Here we only record the maximum value %!i of u(t) in each of the M realizations. We can then determine the probability P(<Q) that the maximum value does not exceed the level Q by introducing the index function

B<(a)= 1, fOr%!iS%

0, for %!i > 021

for each value of % to obtain

P(Q%)= Idi B,(a). (5) I 1

Often the left-hand side of P(<%) is approximated by the first of the so-called three asymptotes (Gumbel, 1958):

P(<%) = exp -exp ( (

-C %-[%I 1) VW-[%I ’

where

C = 0.5772156649. . . (7)

is the Euler constant, (Q) the mean of the maxima and [“II] the position of the maximum of the probability density function corresponding to (6):

P(Q) = c

(021) - [“u] exp ( -c q--w x

(W - [%I >

X exp -exp -C ( (

Q - [%I 1) (W- [%I .

We note that the Gumbel probability density function is always skew; if (“II) = [%I, it degenerates into a Dirac delta function. The lower right frame in Figure 1 shows (8) for (%)/[%I = 1.2.

2.3. THE EQUIVALENCE AND THE GUST

We saw in Subsection 2.1 that the probability that the level % is not exceeded is given by (3). If the level % is not exceeded, the maximum value will not exceed this level either.

In Subsection 2.2, we discussed the probability P(<Q) that the maximum value of u(t) does not exceed Q. If this is the case, the time series cannot exceed “11 in its entire duration.

We thus conclude

P(<%) = Pc7/JO]

or, identifying (3) and the approximation (6),

(9)

We see that the “Rice” and the “Gumbel” approaches are equivalent as also pointed out by Davenport (1964). If we know the average frequency of excursions beyond any level %, we can determine the probability function for the maximum values and, if this function is known, then the average frequency of excursions beyond any level is known.

In these considerations, we have, in order to be specific, chosen the asymptotic extreme-probability function (6). This is by no means necessary to obtain the general conclusion (9). However, (6) is widely used because it is simple and believed to be reasonably accurate for most applications. Further, in these considerations it leads to a natural definition of a gust:

The gust is the wind-speed deviation from the mean which, on average, is exceeded once during the reference period T.

Since (10) gives us the average number of excursions beyond %, we can determine the gust by setting the left-hand side of (10) equal to one. We conclude that the gust in our approximation becomes equal to the most probable value, the mode

IN SEARCH OF A GUST DEFINITION 95

[“u], of the Gumbel probability density function (8). With this definition we can, with the aid of (2), assign probabilities as follows:

No excursion: &j[Ol = e -1 -37% One excursion: P[wJ[~] = e -I -37% More than one excursion: 1 - 2eF’ -26%.

3. Exceedance Statistics

We now give a brief account of what has been termed the Rice-theory (Rice, 1944 and 1945).

Let u(t) be an ergodic time series and ti(t) its time derivative. If P(u, ti) du dci is the joint probability that the signal and its derivative have values in (u, u + du) and (zi, zi + dti), then the probability that u(t) in the (short) time interval At performs one crossing of the level (3 with a time derivative in the interval (zi, zi + dzi) is

?/

dci P(u, zi) du . (11)

Consequently, the probability that u(t) performs one up-crossing (zi > 0) of Q in the time interval At becomes

r ‘u r

i i dti P(u, ti) du ^I At

I tiip(Q, ti) dzi . (12)

0 0*-G At

0

The average rate of excursion over % is thus

(13)

This means that the average number of times the level % is exceeded during the period T is qqlT.

Within the framework of the Rice-theory, it is also possible to estimate the average duration of an excursion beyond “21. Since the total time u(t) is greater than 021 during the period T is

P(u, ti) dzi , (14)

the average duration of an excursion beyond “II can be estimated as

96 L. KRISTENSEN ET AL.

Equations (13) and (14) imply that the gust duration is independent of the reference period T.

With our definition of a gust, with ~l~l T = 1, the average gust duration of course is (exactly)

3c cc

6,%, = O,“ul = T I I

du P(u, ti) dti . W

[“ul --r

The joint probability density P(u, ti) is the important ingredient in these considerations and unfortunately, little is known about it. However, it is easily seen that u(t) and C(t) are uncorrelated. Since u(t) is assumed ergodic, it is also stationary. This means that the ensemble wind speed U = (u(t)) is constant in time and that the covariance and consequently the correlation coefficient is zero. The last statement follows from

Id ((u(f) - Lpi(t)) = - - (z?(t)> = 0 .

2 dt

If P(u, ti) were joint Gaussian, (17) would imply that u(t) and ti(t) would also be statistically independent, i.e., that P(u, ti) can be written as

P(u, zi) = PI(U) x Pz(ti) . (18)

We consider (17) circumstantial evidence that it is reasonable to assume that (18) is fulfilled, albeit not that Pi(u) or P2(ti) are necessarily Gaussian.

It is, in particular for very large and infrequent excursions, important to have a realistic model for Pi(u). Such a model, however, is extremely difficult to establish since only if the unfiltered signal is Gaussian is it easy to verify that the probability density is form-invariant to filtering. In this case, Pi(u) is also Gaussian. We shall leave the discussion of this point to future research and illustrate our findings assuming P(u, ti) joint Gaussian, viz.

1 P(u, ti) = ____ exp

( _ (u - w c2 --

2%-u,(Tfi 24 > 2a; ’

where a: and af are the variances of u and ti, respectively. The rate of excursions and the average duration of one excursion then become

1 CT* Vu=rn;exp -

( ((42 - U)’

u 202, )

and

(20)

97

(21)

where x

erfc(x) = - vz;i e-” dt (22)

is the complementary error function. In the limit when the excursion is very large, i.e., when “II - U % u,, we get the simple result

(23)

which has the straightforward interpretation that the average duration of an excursion is inversely proportional to the deviation from the mean. We must emphasize that this simple result is derived under the assumption that P,(u) is Gaussian and may consequently not generally be true.

4. Cup-Anemometer Gust

We want to apply t’he ideas in the preceeding section to an ordinary cup anemometer signal, subjected to a running means filter of duration T,,.

Let C(t) be the unfiltered wind speed and u(t) the cup-anemometer response. The cup-anemometer dynamics can then be described by means of the first-order differential equation

G(t) = G(ii, u) . (24)

If 6 is kept constant equal to 0, the cup-anemometer response will reach a constant value U, which can be found by solving

G(o, U) = 0. (25)

Experience has shown that the steady-state calibration equation of cup anemometers is very close to being linear with a gain which we, without loss of gen- erality, can set equal to one. The last equation therefore implies that

dG dG 1 -=- ao=-au 7’

If the relative turbulence intensity is not too large, the right-hand side of (24) can be reasonably approximated by a first-order expansion in the deviations fi’ and U’ from the means (Li) = (u) = U. The equation of motion (24) reduces to

L. KRISTENSEN ET AI

, -, ~f+!C=U, (27)

7 7

where (26), which defines a time scale 7, has been used. From quite general considerations, it follows that G(ti, u) is a second-order

polynomial in C and u and from (26), we can therefore conclude that T is inversely proportional to U and that 1 = Ur is independent of U. We call 1 the cup-anemometer distance constant. For a good standard instrument, it is typically 2 m and in general much smaller than the length scale of the turbulence.

Using Taylor’s hypothesis about “frozen turbulence”, the dynamic equation (27) can be reformulated in the spatial domain as

where x is the spatial coordinate in the direction of the mean wind. In order to evaluate the variance of u and ti, as seen through a running-mean

filter of duration r,.,, we need the power spectrum W(w) of C(t), subjected to the instrument filtering expressed in equation (28) as well as the filtering of the running means. We know the streamwise wave-number spectrum F,(k) of Cz quite well and, applying Taylor’s hypothesis and (28), we get’

2L FdwIU) u1+ (zwlu)*’

where

sine(x) = sin X

(29)

(30)

appears squared with the argument (or,.,)/2 as a consequence of the running- mean filtering.

Since the cup-anemometer distance constant 1 is usually much smaller than the scale of the turbulence and since r,, is usually chosen much smaller than this scale divided by the mean-wind speed U, the variance of U is not affected by the combined filtering accounted for in (29).

Since we are mostly interested in strong winds, we shall here limit ourselves to the case of neutral stratification. However, it is possible to generalize the following to include diabatic effects.

Kaimal et al. (1972) give an explicit expression for F,(k) allowing us to relate (T: to the so-called friction velocity u*. We get

* We here use the convention that frequency spectra are “one-sided” so that 0 < o < x whereas wave- number spectra are “two-sided” with --m < k < 2.

r x

2 UC< = I

T(w) dw = i

F,(k) dk = 4.77~;. (31) 0 -zc

The variance of the derivative ti is very sensitive to the filtering. It is easily shown (e.g., see Panofsky and Dutton, 1984) that CT: is given by

* x

2 uir = I

w2’P(o) dw = 1 w1sinc2(yj X

0 0

2 FdwIu) dw

x u 1 + (fwlLq2 ’ (32)

which shows that ai is essentially independent of the detailed behaviour of F,(k) at numerically small values of k. Therefore F,(k) may be approximated by the expression pertaining to locally isotropic turbulence:

F,(k) = $q~2’3~k~-“‘3 , (33)

where E is the rate of dissipation of specific turbulent kinetic energy and LY~ the Kolmogorov constant for the longitudinal velocity spectrum.

For neutral stratification we have

4 E=-. KZ

Substituting (33) and (34) in (32) we obtain

(34)

(35)

where zc

Z(x) = i

(1 - cos(xs)) 5 ds . (36) 0

According to Frenzen and Hart (1983) and a discussion by Kristensen et al. (1989),

QIK 4’3 = 0.165 . (37)

Zhang et al. (1988) found

K = 0.40. (38)

For the evaluation of (20) and (23), we now have (31) and


We get

77% = 0.1047 x L l/3

0 x r x p2 u7,,

2 rrm ( > 1

and the estimated gust duration

-l/3

Finally, the gust as defined in Subsection 2.3 becomes

The integral (36) can be evaluated analytically. The result is

Z(x) = J-1 f V%(cosh(x) - 1) - 27~

lOI(1/3)I(5/6)

(39)

(40)

(41)

(42)

(43)

where iF2(a; b, c; z) is one of the generalized hypergeometric functions (Grad- shteyn and Ryzhik, 1965, p. 1045).

When x S 1, we can use the asymptotic expression

I(x) (44)

In this limit, the dependence of the distance constant 1, i.e., the effect of the anemometer filtering, in (41) and (42) vanishes so that

[%I = u + (T, {21n(0.14g3(?)i13$)r2

and

l/3

(45)

(46)

When x + 1.

IN SEARCH OF A GUST DEFlNlTlON 101

0 2 4 6 8 10 12 14 16 18 20

X

Fig. 2. The square roots of the function I(X) given by (43) (solid line) and the approximations (44) and (47) (dotted lines).

Z(x) = 2.L x2 2x6

and the dependence on r,, disappears, i.e.,

and

(47)

(48)

(49)

In this case, the running-mean filtering has a time constant so short that the filtering is completely dominated by that of the cup anemometer. The square roots of the function Z(x) in (43) and the two approximations (44) and (47) are shown in Figure 2. If r, = ls, 1= 1.7m and U= lOms-‘, we get x= iJr,/1=6. In this case, Figure 2 shows that we can use (44) instead of the more complicated (43).

5. Experimental Validation

Data from the Lammefjord Experiment (LAMEX) (Courtney, 1987; 1988) offer an excellent opportunity to compare the prediction (42) with real data. Further,

102 L. KRLSTENSEN ET AL.

they make it possible to test the postulated equivalence (9) between the two definitions of a gust.

5.1. SHORT DESCRIPTION OF LAMEX

An array of three masts instrumented with cup anemometers, wind vanes and one three-component sonic anemometer, was erected in Lammefjord, an area of land reclaimed from a shallow, flat-bottomed fjord on the island of Zealand in Denmark. One purpose was to study the lateral spectral coherence between the streamwise, turbulent velocity components at points displaced over a plane (= 30 x 30 m’) normal to the prevailing wind, corresponding to the rotor plane of a wind turbine. The upwind fetch was about 3000 m with a level variation of no more than 1 m. The land is used for agriculture, predominantly growing of root crops such as carrots and potatoes.

The measurements started about the beginning of June 1987 and terminated at the end of June 1988. Data were recorded continuously with a sampling rate of 8 s-i and were stored permanently on 1 Gbyte removable optical disks with a storage capacity corresponding to 34 days of data. The system worked essentially uninterrupted with a data recovery of about 80%.

An independent climatological sensor and data-logging system recorded 10 min averages of mean-wind speed and directions, temperatures and temperature differ- ences between various heights, incoming short-wave radiation, humidity and pres- sure.

In this investigation, we use the signals from cup anemometers at 10, 20 and 30 m heights. These instruments are of the Rise Model 70 type with the distance constant I = 1.7 m (Busch et al., 1980).

5.2. DATA

The criteria for choosing data was focused on the gradient Richardson number Ri because the prediction (42) assumes neutral stratification. In order to obtain the best possible statistical confidence, we searched for a long time series (many hours), which looked stationary and with Ri close to zero.

We found several 20 h periods, all with wind speeds in the neighbourhood of lOms-i. Since the distance constant is 1 = 1.7 m, the temporal resolution before averaging is about 0.2 s.

The time series we use here were from a 20 h uninterrupted period in April 1988 of signals from the three heights 10, 20 and 30m. The mean-wind speeds were 11.4, 12.5 and 12.9 m s-l, respectively. Other periods show almost the same result. Figure 3 displays the time series from 10 m.

Each time series is subdivided into realizations of duration T. We have chosen T = 2, 5, 10 and 30 min. As we see from Figure 3, there are signal variations with periods longer than even 30 min and therefore we prepared a set of new realizations of (U - U)/(+,, where U and cU are the time mean and the time root-mean- square of the signal in each of the realizations of duration T. The old realizations

360

E 270 .- 2 180

.; 90

';; 15 \ E 10

3 5

0

IN SEARCH OF A GUST DEFINITION

Time (hours)

5 10 15 20

103

0 5 10 15 20 25 30

Time (minutes)

Fig. 3. Time series of wind direction and speed u at the height 10 m. Top and middle frame: 20 h of 10 min average of direction and speed, respectively. Bottom frame: 30 min blowup of 2 s averages of

wind speed.

have thus been transformed into a set of realizations of a “stationary” signal of zero mean and unit variance.

5.3. ANALYSIS

According to (lo), the Gumbel distribution (6) of maxima and the assumption (3) imply that N (“II) is an exponential function of %. Figure 4 shows ln(N (%)) from 10 m in one case. The reference period was T = 10 min and the running-means filter had 7, = 2 s. The curve is not a straight line, and consequently we must admit that one or both of assumptions (6) and (3) are not warranted. However, from the discussions in subsections 2.1 and 2.2, we know that (9) is true in general and it is conceivable that an asymptotic distribution other than (6) might fit the data better. At this point, we assume that the curve in Figure 4 is locally sufficiently straight for our purpose.

We see from Figure 4 that the dimensionless excursion which on average is exceeded once is about 2.8. In a similar way we determined the gust for T = 2, 5, 10 and 30 min and r,., = 1, 2 and 3 s at the height 10 m. We also found the


Fig. 4. In(N(%)) as a function of the dimensionless excursion level (“u - (%))/a,,.

dimensionless gusts for T = 2, 5, 10 and 30 min and T, = 2 s at the heights 10, 20 and 30 m. The normalized gusts

determined by counting excursions (“Rice” approach) are shown together with the prediction (42) and the estimated most probable values of the extreme excursions (“Gumbel” approach) in Figure 5.

This figure shows that the two ways of determining the gust as defined in subsection 2.3 and the prediction (42) are in mutual agreement within 8%. The agreement seems to improve with increasing values of the reference period T so that it is typically about 1% for T = 30 min.

The data we use here are typical, but we have found cases in which the relative deviations are larger. For example in August 1987, we had a case with T, = 2 s, where the prediction (42) compared to a “Gumbel” gust was about 8% too large for T = 2 min and about 12% too large for T = 30 min. In this case, the deviation between the “Rice” gust and the “Gumbel” gust was less than 4%.

Inspection of (45) and (48) reveals that no matter what kind of filtering the wind speed is subjected to, running-means or first-order filtering, the expression for Al. is of the form

(51)

where r. is the time constant pertaining to the filtering and /3 is a dimensionless constant with a value of about 0.1, depending on the actual type of filtering. Since

IN SEARCH OF A GUST DEFINITION 105

35 I

30

rt

25

20

3.0

Y

25

z=20m. 7,,“=2*

z=tOm. r-=15 z=1om. r,,“=zs *= I om. T-=33

0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 5 10 15 20 25 30

T (min) T (min) T (r-r-in)

Fig. 5. The normalized gust p as function of the reference time T. Open squares: “Rice” approach. Open circles: Prediction (42). Filled squares: “Gumbel” approach.

the value of /3 is determined empirically from data reported in the literature, and since such data are being revised from time to time, it is important to know how sensitive (51) is to changes in /?.

We find

The sensitivity to variation in p appears to be quite high if

In the cases that we have studied here, the left-hand side of (53) is greater than one. The sensitivity to p is thus a decreasing function of the reference time T and

106 L. KRlSTENSEN ET AL.

an increasing function of the filter time-constant TV. The relative deviations in Figure 5 seem to support this conclusion.

6. Conclusions

We have proposed a simple gust definition based on the theory of rate of excursions, first discussed by Rice (1944 and 1945). It is repeated here:

The gust is the wind-speed deviation from the mean which, on average, is exceeded once during the reference period T.

We have showed theoretically that the gust defined in this way under certain assumptions is identical to the most probable extreme excursion and predicted its value for a cup anemometer signal from a neutrally stratified atmosphere on the basis of the mean and the variance. The LAMEX data showed that the three ways of determining the gust defined this way are in agreement within about 10%.

Our definition has a number of advantages:

l The gust is a wind speed, measured in m s-i. This is precisely the kind of information, together with the mean-wind speed and direction, an aircraft pilot wants and usually gets from the control tower, when he plans a landing.

l The gust can be determined with the aid of the mean-wind speed and the variance by means of a simple equation. Since the variance is based on information from the whole reference period T, this gust prediction has a higher statistical significance than a prediction based on the largest excursion from the preceeding reference period.

l It is possible to state a probability that the wind speed will exceed the gust during the reference period: viz., 63% that there will be at least one excursion.

l It is possible to estimate the average gust duration.

We should keep in mind that the gust definition we propose here is meaningful only if we can assume that the signal is stationary. There are, however, many turbulent wind phenomena, which are of interest to aircraft pilots, sailors and others, who must operate in windy environments, and which cannot be described as gusts in the sense discussed here. For example, in connection with storms, it is observed that the “wind level” changes from a low wind speed of a few meters per second to perhaps 12m s-i and stays there for time long compared to the reference period. The problem is mostly how to define such phenomena in a way that can be described and predicted.

Acknowledgements

The Lammefjord Experiment was carried out under contract EN-3W-002-DK (B) with the Commission of the European Directorate-General for Science, Research and Development. We are indebted to Peter Kirkegaard for deriving (43) and the

IN StARCH OF A GUST I>EFINITlON 107

approximation (44). The referee, unknown to us, and our colleagues D. H. Len- schow of the National Center for Atmospheric Research and C. W. Fairall of NOAA/ERL/Wave Propagation Laboratory, both in Boulder, CO, have offered very useful comments. The careful preparation of the figures was to a large extent done by our colleagues Stiren W. Lund and Morten Frederiksen.

References

Busch, N. E., Christensen, 0.. Kristensen, L., Lading, L., and Larsen, S. E.: 1980, ‘Cups, Vanes, Propellers and Laser Anemometers’, in F. Dobson, L. Hasse and R. Davis (eds.), Air-Sea Inrer- action, Instruments and Methods, Plenum Press, New York. pp. 11-46.

Beljaars. A. C. M.: 1987, ‘The Influence of Sampling and Filtering on Measured Wind Gusts’. J. Atmos. Ocean. Technol. 4, 613-626.

Courtney. M. S.: 1987, ‘The Lammefjord Experiment’, Proceedings of the American Wind Energ) Association Annual Conference, October, San Francisco. CA, pp. 124-129.

Courtney, M. S.: 1988, ‘An Atmospheric Turbulence Data Set for Wind Turbine Reseearch’. Proceed- ings of the 1988 Brifish Wind Energy Association Conference, London, England, pp. 89-94.

Davenport. A. G.: 1964, ‘Note on the Distribution of the Largest Value of a Random Function with Application to Gust Loading’, Proc. Insf. Cit.. Eng. (London) 28, 187-196.

Frenzen, P. and Hart. L. H.: 1983. ‘A Further Note on the Kolmogorov-van KBrm;in Product and the Values of the Constants’, Proc. Sixth Symp. OH Tzwbtrfence and DiJfrfsion. March 22-25. Boston. MA. pp. 24-27.

Gradshteyn, I. S. and Ryzhik. I. M.: 1965, Table of Inregruls, Series. and Products. Fourth Edition. Academic Press, New York.

Gumbel, E. J.: 1958, St&tics of Extremes, Columbia University Press, New York. Kaimal, J. C., Wyngaard, J. C., Izumi, Y. and Cote, 0. R.: 1972, ‘Spectral Characteristics of Surface

Layer Turbulence’, Quart. J. Roy. Meieorol. Sot. 98, 563-589. Kristensen, L., Lenschow, D. H., Kirkegaard, P. and Courtney. M.: 1989. ‘The Spectral Velocity

Tensor for Homogeneous Boundary-Layer Turbulence’. Boundary-Layr Meteorol. 41. 149-193. Panofsky, H. A. and Dutton. J. A.: 1984, Atmospheric Turbulence, John Wiley & Sons, New York. Rice. S. 0.: 1944, 1945. ‘Mathematical Analysis of Random Noise’, Bell Syst. Tech. J. 23. 282-332,

24. 46-156. Zhang, S. F., Oncley. P. 0. and Businger, J. A.: 1988. ‘A Critical Evaluation of the von Kdrman

Constant from a New Atmospheric Surface Layer Experiment’. Proc. Eighth Symp. on Turbulence and Diffirsion, April 25-29, San Diego. CA 148-150.