~ Pergamon Renewabh, Energy, Vol. 4. No. 2, pp. 261 263. 1994

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TECHNICAL NOTE

Predictability of wind speed patterns

ERNEST C. NJAU

Physics Department, University of Dar es Salaam, P.O. Box 35063, Dares Salaam, Tanzania

(Received 12 December 1992: accepted 8 July 1993)

Abstract--Theoretical work done earlier by the author shows that variations in all the basic meteorological parameters are generally represented by expressions having only a particular mathematical formulation. We report in this paper that this theoretical conclusion has been found to apply quite well to observed wind speed variations. Consequently, it is apparently possible to use the mathematical formulation just mentioned in predicting wind speed patterns at any given location for projected applications in wind- energy systems for long-term strategies for protecting crops and/or buildings from damaging winds.

I. INTRODUCTION

The output of windmills and other wind-driven systems essentially depends in part on the prevailing wind speed patterns. On this basis, estimates of future outputs of wind- driven systems can hardly be worked out without sufficient knowledge of the corresponding wind speed patterns which have themselves to be predicted. Attempts to predict wind speed patterns should take into account the existing physical relationships and hence correlations between wind speed as a meteorological parameter on one hand and other meteoro- logical parameters on the other hand. In 1987, Mosetti [1] concluded that variations in different meteorological par- ameters have mostly similar periodicities but differ only in amplitudes and phases. Subsequent theoretical work by Njau [2, 3, 4] has shown that any meteorological parameter (rep- resented by P(t), where t denotes time) at a given location may he expressed in the time interval from t = 0 to t T by the equation :

P(t) =C~+,~a,,{[l+~,bkf~(t)]Dr(t)}" (1)

where Co = a constant, a° = a coefficient for a fixed value of n, bk = a coefficient for a fixed value of k, N = a positive integer, fk(t) = a variation caused by the k ~h variable com- ponent of solar energy incident onto the lower atmosphere, Dr(t) - a function dependent upon both T and t and whose details are given in Njau [2, 3], and T/> 2 days.

The periodicities in f~(t) for k = 1 N are similar to the periodicities in the solar energy incident onto the tropos- phere. The latter periodicities include: 7.0 days, 23.4 days, about 27 days, 51.4 days, I year, 13 months, about 11 years, 35 40 years, 8(~90 years, 180 years, about 250 years, 19,000 years, 23,000 years, 41,000 years and 100,000 years as well as harmonics of these periodicities.

It has already been shown that eq. (1) is obeyed quite well by observed air and surface temperature [5, 6], as well as observed monthly, seasonal and annual rainfall [6, 7]. In this paper we show that observed wind speed data also obey eq. (1) quite well. Additionally we briefly indicate practical applications of the realisation that wind speed patterns may be computed on the basis ofeq . (1).

2. ANALYSIS

Several wind speed records have been analysed by us in order to find out whether or not they obey eq. (1). The results for this exercise showed clearly that each record involved did obey eq. (l) fairly well in such a way that all the summation terms [on the extreme right-hand side of eq. (1)] for which n > 3 or sometimes even n >/2 had negligible contributions and hence were ignored. Typical representative examples of the results are illustrated by Figs 1~, over different fime- lengths. Variations in observed hourly average wind speed at Dar es Salaam (6~5YS, 39°12%) during part of July 1983 are plotted in Fig. 1 using solid lines. The procedure described in Njau [3] has been used to develop the following expression for f(t) which represents these variations :

f(t) = [1 + A c o s (cod+rt)](2.40+ 1.58cosoJ,t) (2)

where A-0 .246+0 .036 t , ~ - 0 . 5 2 radians per day,

ol l1 •

~ 2

o 4 10 16 22 28

Day of month

Fig. 1. A plot of observed hourly average wind speed at Dar es Salaam during part of July 1983 (solid lines). Hourly average wind speed calculated from eq. (2) has also been

plotted in the figure using dotted lines.

261

262 Technical Note

4I ,.~ 2 -

._.5

0 J I

0 4 8 12

Month of year

Fig. 2. A plot of observed daily average wind speed at Dar es Salaam during 1979 (solid lines). Daily average wind speed calculated from eq. (3) has also been plotted in the figure

using dotted lines.

c~J2 = 6.28 radians per day, and t = 0 at midday of 15 July 1983.

It is clear that eq. (2) is a first-order version ofeq. (l). Function f(t) has been plotted in Fig. 1 using dotted lines

for comparison with the observed variations. As may be noted even visually, there is a reasonable agreement between the solid line and the dotted line plots in Fig. 1. We have used solid lines to plot in Fig. 2 variations of observed daily average wind speed at D a r e s Salaam during 1979. On the basis of reference [3], these variations are represented by V(t) such that :

V(t) = 1.75 + [1 + 0.53 sin ~o 3 t](l +0.43 sin o~ t)O.44s(t),

before the end of April 1979;

2.22+ [1 + 0.64cos (o5t](0.42 + 0.47 sin ((o3t +2.41)

+0.72s(t)), (3)

after end of April 1979 ;

where ~03 - 2.09 radians per month, ~o4 = 4.18 radians per month, uJ5 = 1.05 radians per month, and s(t) = a com- bination of unity-amplitude sinusoids at periods 7 days, 14 days, l0 days and 20 days (see references [4, 6]).

t t 19180 1970 1975

Fig. 3. A plot of observed monthly mean wind speed at Moshi from 1969 up to 1980 (solid lines). The envelope formed by the solid lines has been sketched using dis-

continuous lines.

/ • , ' I f "/ ~ \ \ I Il l

4

.E 2

0 i II 19J70 19t75 1980

Fig. 4. A plot of observed monthly mean wind speed at Mombo from 1969 to 1980 (solid lines). The envelope formed by the solid lines has been sketched using discontinuous lines.

The apparent or rapid change that took place near the end of April 1979 (see Fig. 2) was caused by physical processes already given by Njau [4, 6]. Theoretically such a transition normally gives rise to changes in the mathematical expressions representing the meteorological parameter involved as reflected in eq. (3) with regard to daily average wind speed at Dares Salaam. The general form of eq. (3) is clearly identical to a first order version of eq. (1). We have plotted V(t) as given by eq. (3) in Fig. 2 using dotted lines. A look at Fig. 2 clearly shows that both the solid line plot and the dotted line plot generally agree with each other fairly well.

Figures 3 and 4, respectively, display plots of observed monthly mean wind speed at Moshi (3'21'S, 3T20'E) and Mombo (4~'55"S, 38'14'E) from 1969 up to 1980 (solid lines). In each of the two figures, discontinuous lines have been used to sketch the ~envelope' of the wind speed variations. The shape of the 'envelope' in Fig. 3 clearly shows that the wind speed variations in this figure can be represented by a func- tion G(t) that incorporates a slightly distorted amplitude modulation process having sufficiently large modulation index. Thus G(t) may be expressed generally as follows :

G(t) = { 1 + [1 + a0t]fm(t)}[l + b0t]f~(t) +constant (4)

where fo,(t) is a modulating function, f~(t) is a carrier function, both a0 and b0 are constants, and the frequencies in fc(t) are greater than those in fro(t).

An analysis of Fig. 3 shows that the period of the modu- lating signal fro(t) is equal to that of the 11 year solar cycle, and that fro(t) has minima approximately coinciding with sunspot maxima. Of course the apparent distortion shown in Fig. 3 and coinciding with the year 1978 has been caused by overmodulation as explained earlier by Njau [2].

Finally the shape of the variation envelope shown in Fig. 4 using discontinuous lines clearly implies that the wind speed variations shown in this figure may be represented by function R(t) such that:

R(t) = {1 + [ l +kot]y,,(t)}y~(t) +constant (5)

where y,,(t) is a modulating function, y¢(t) is a carrier function, ko is a constant, and the frequencies in yc(t) are much smaller than those in ym(t).

The function ym(t) apparently has a period approximately equal to that of the 11 year solar cycle. Besides, this function has minima approximately coinciding with sunspot maxima. Each of eqs (4) and (5) is clearly a first-order version of eq. (1).

Technical Note 263

3. CONCLUSION

We have established that observed wind speed actually varies in accordance with eq. (1) as already suggested by theory. This finding implies that it is possible for predictions of future wind speed patterns at different locations to be made using eq. (1) as elaborated in Njau [3]. Such predictions would and should help experts in wind technology to predict performance characteristics of wind-driven systems well in advance.

process takes place quasi-continuously in time. Secondly the non-stationary nature of Dr leads to fold-over distortions in P with a result that considerably amplified waves are formed at a range of frequencies including those equal to half of those in f~. I f P represents heat energy then the largest variations in D~ would have amplitudes of about 20% of the constant component of incoming solar energy.

Note added m proo! After completing this technical note, we found out (in a

forthcoming paper by the author) that a generalised form of eq. (1) easily leads to the long-awaited physical mechanism linking the sun to climate/weather variations. Generally both terms P and Dr in eq. (1) are 3-dimensional functions of time, latitude and longitude while each fk is a l-dimensional function of time. If we include the generalised forms of P and D~ into eq. (1), the resultant equation has the following physical interpretation. Firstly the small l-dimensional (i.e. t-dependent) changes in incoming solar energy represented by f~ for k = 1 to N amplitude-modulate the, relatively, much larger 3-dimensional changes represented by D~. This modulation generates a series of waves both in the zonal as well as latitudinal directions. But since the non-stationary term Dr continuously generates variation frequencies and then systematically evolves them towards smaller values, these waves attain considerable amplifications whenever any of the frequencies in Dr equals a frequency in f~. Indeed this

REFERENCES

l. F. Mosetti, On the essence of some climatic cycles. Boll. ocean Teor. Appl. V, 145 148 (1987).

2. E. C. Njau, A generalised theory of sun-climate/weather link and climatic change. Nuovo Cinwnto 12C, 597 611 (1989).

3. Prediction of meteorological parameters: 1. analytical method. Nuovo Cimento 14C, 473 488 (1991).

4. E. C. Njau, Theory of quasi-biennial and some other oscillations in meteorological parameters. Nuot:o Cimenlo 15C, 37 44 (1992).

5. E. C. Njau, Sun-controlled spatial and time-dependent cycles in the climatic/weather system. Nuoeo Cimento 15C, 17 23 (1992).

6. E. C. Njau, Global and regional temperature various predicted by an analytical method. Proc. Ind. nam. Sci. Acad. (in press),

7. E. C. Njau, States and inter-state switching in meteoro- logical parameters. Nuovo Cimento 15C. 25 35 (1992).