350 The Representation of Oceanic Movements and Kinematics 



The magnitude of the current constancy is only affected to any large extent by varia- 

 tions in the direction of the flow, variations in the velocity have little influence. 

 Wagner (1932) has found, for example, that if the velocity was assumed to be the 

 same for the individual values and the directions were scattered within an angle of 

 90°, then the stability was 90-100%, while if the directions were scattered within 180° 

 the current stability was still 60-90% ; individual values with greater velocities could, 

 however, aff'ect these stability values strongly in either direction. 



A more accurate description of the distribution of a larger number of obser\'ations 

 requires the use of statistical theory (Thorade, 1936). If the measured velocities of 

 the current are m\, w^, Wg, . . ., vv„ for ^-observations and a^, og, a^, . . ., a„ are the 

 corresponding directions (taken clockwise from north from 0° to 360°) then the 

 corresponding ^-components will be m^ = m,\ sin a^ and the A^-components will be 

 Vi = Wi cos ttj, where / = 1, 2, . . ., n. The arithmetic mean of the ^-components 

 will be a, and that of the A^-components v ; then the vectorical velocity is 



w^^ = u} + y2 

 and the vectorial mean direction will be 



u 

 tan a„ = -. 



V 



The deviations of the individual values from the vecto^'ial mean are 



^i — Ui — u and t^j- = f, — v. 



Comparison of the frequency distribution with a Gaussian distribution will then allow 

 us to judge whether the deviations are generally random, so that statistical laws are 

 applicable. 



The mean scatter of the Mj- and ^j-values is then given by the mean error (standard 

 deviation) 



w,/ = e* and m^ = if. 



For a case similar to that of Fig. 142 (150 observations over an interval of 10 sec) 

 Thorade found a point distribution given in Fig. 144 for the frequencies of the devia- 

 tions for intervals of 1 cm/sec; the curves show a Gaussian distribution indicating 

 the completely random nature of the deviations, and show that in spite of the small 

 number of observations the deviations approximate very closely to a random distribu- 

 tion. In this way, the direction varies between 270° and 318° and the velocities between 

 7-4 cm/sec and 21 -8 cm/sec. The vectorial mean gave a current N. 66° W., 14-5 cm/sec, 

 the scalar mean was 14-7 cm/sec, and the current constancy (stability) was therefore 

 98-6%; m spite of the rather large variations in direction and speed of the current 

 this is a surprisingly high current stability value. The mean scatter gave a considerably 

 better idea of these variations: m„ = ± 2-68, m„ = ±2-64 cm/sec, which indicates 

 that for a random distribution of the deviations about 68% of all the deviations ej 

 ofthe£'-component lie between +2-68 cm/sec and —2-68 cm/sec; analogous conditions 

 apply for the 77^ for the A^-component. 



According to statistical theory of scattering, the direction and velocity can be charac- 

 terized most accurately by the "mean error ellipse" which must include half of all 

 the individual values. Considerable numerical work is required for calculating this 



