756 cox [chap. 22 



at frequency / of y2 over 2/1 and R represents the degree to which a single 

 constituent of y\ remains at a constant phase difference with respect to a 

 similar constituent of 2/2. Perfect coherence is indicated by i?= 1, no coherence 

 by R = 0. The estimated values of R shown in Fig. 23 are based on limited 

 records and, therefore, are subject to statistical errors. Even short sections of 

 completely non-coherent records can give estimates of R greater than zero. 

 The 95% confidence limit for estimates of R (when the true value 's zero) is 

 shown dashed in the figure. The estimates of 6 are obviously meaningless if the 

 true value of R is zero. Therefore estimates of 9 should be discarded unless the 

 estimated value of R is larger than the confidence limit. 



An interesting feature of the recorded temperature oscillations is that there 

 appears to be no statistically significant coherence between the oscillations at 

 50 and 500 m (estimates of phase shift are therefore completely unreliable). As 

 an explanation the author considers the possibility that the oscillations about 

 the seasonal and permanent thermoclines (Fig. 24) are not closely coupled. In 

 this case the observations at 50 and 500 m, which are near the top of the 

 respective thermoclines, would be poorly correlated. But this could only 

 happen if no single mode of internal motion were dominant. For example, if 

 only the first mode were present, observations at all depths (along a vertical 

 line) would be in phase. Similarly, if any other single mode were dominant, the 

 relative phase of the oscillations would be fixed. Little is known about the 

 distribution of internal waves by mode. It would be surprising, however, if any 

 but the first mode were dominant since higher modes involving higher shear 

 would be expected to be more easily damped than the first. Furthermore, the 

 low phase velocities of high modes would be expected to make them easily 

 perturbed and destroyed by irregular propagating conditions in the sea. 



Another cause of reduced coherence between horizontally separated observing 

 points will be treated more thoroughly in the next section. When internal 

 waves of a single mode arrive from a wide variety of directions, the phase 

 relations between observing points become variable and the coherence reduced. 

 Coherence generally decreases with both increases of angular beam, through 

 which the waves come, and separation (in wavelengths) between points of 

 observation. The extreme case occurs for an angular beam 360° broad, that is, 

 isotropic radiation. Then the coherence is unity at zero separation and drops to 

 zero at a separation of 0.38 wavelengths (Fig. 27). 



The horizontal separation between thermometers off Castle Harbor was 

 1.5 km. A rough estimate of the phase velocity of nth mode internal waves is 

 Cn={hln7T){N^ — cu2)'2, where h is the depth of water and N the mean value of 

 Vaisala's frequency. Appropriate values are A = 200 m and A'^^^ 13 radians per 

 hour, leading to a low frequency phase velocity of SAn~'^ km h~i. The co- 

 herence would then be expected to be large for frequencies below 1.5 km/ 

 (0.38 X 8.4^-1 km h.-^) = Aln-^ c/h. The observations show no appreciable 

 coherence even at much lower frequencies. We are forced to conclude either 

 that there are many interfering modes of internal waves or that the temperature 

 fluctuations are not due to free internal waves at all. 



