L. Richardson, D. Proctor, R. C. Smith

Sep 10, 2007

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Journal

Quarterly Journal of the Royal Meteorological Society

Abstract

The variance—that is to say “the mean, for many observations, of the square of the deviation” of the vertical velocity from zero—is about 1/4 metre2 second−2. The difference between a real horizontal wind and the value obtained for it by observing a balloon through a single theodolite has a variance from its mean of about 1 or 2 metres2 second−2 This variance increases with the speed of the mean-wind. The difference between the W-E wind component at two times at the same place had a variance from its mean which changed with the time-interval in the remarkable manner shown in Fig. 3. The asymptotic value for indefinitely long time-intervals is 88 metres2 second−2. The curve starting at no variance for zero time-interval rises irregularly with a peak at 6 hours, it cuts the asymptote at about 1 or two days, attains a maximum at 11 days, and cuts the asymptote again at 19 days. This relates to the summer of 1918, June 6 to September 28; and the features of the curve might well be different for a different season. The difference between the winds at the same time and height at two places on the map has a variance from its mean which has been deduced from Durward's statistics, and also independently for some short distances. The results are shown in Fig. 4. Between 10 and 100 kilometres Durward's results show that (variance of difference of component velocity) = k × (distance of stations apart), where k = 5·4×10−3 cm. sec.−2 for the component parallel to the resultant at one member of the pair, and k = 3·6×10−3 cm. sec.−2 for the component at right angles. Improving the simultaneity of the observations at paired places, by reducing the time-error from 30 minutes to 5 minutes, quite often strangely increases the variance of the difference of wind-components. If there be a square of side l on the map such that 10 km. < l < 100 km., and the wind is accurately known at all points of three sides but only at the centre of the remaining side, then we are uncertain of the fractional rate of change of area of the air which fills the square at one initial instant, the uncertainty being expressed by a “standard deviation” of per second, where k is the constant mentioned above. Thus, in the stated range, large squares give better results than small ones. This behaviour of the wind is contrasted with the behaviour of most of the functions commonly discussed in books on finite differences. When the method of finite differences is to be applied to the dynamics of wind, a necessary preliminary will be to smooth the observational data1.

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