SECT. 5] INTERNAL WAVES 753 



A. Limiting Frequencies 



Internal waves have maximum amplitude below the surface ; therefore, at 

 this depth, d^ip/dz^ must be of opposite sign to ip. Neglecting a term p-'^{dpjdz) 

 {difjjdz), which can be shown to be negligible for ocean water (Groen, 1948), one 

 finds from equation (1) that the ratio of i/j" to ip is given by —k^{N^ — cxj^) 

 {co" — Q")~^. Hence, internal waves can exist only if Q<\a>\<Nm or 

 Njn <\co\<Q, where Nm is the maximum value of N{z) in the water column. 



Observed values of Nm are of order IO-2 sec~i while Q<1.5x 10~'* sec~i, 

 consequently the limitation Q< |a»| < Nm is appropriate to the sea. 



The limits Q and Nm are therefore the lowest and highest frequencies per- 

 mitted to running internal waves of the form of (1). 



B. Modes 



At any frequency within these limits, solutions to (2-5) can be found for an 

 infinity of discrete values of k. With each value kn there is a corresponding 

 amplitude function ifjn (Fjeldstad, 1933). These form the normal modes of the 

 w^ater. The lowest value of k corresponds to a surface wave ; all higher values 

 to internal waves. The second smallest value of k corresponds to an internal 

 mode which has a single maximum of amplitude at some depth in the water 

 column. The next higher value of k represents a second-mode internal wave. 

 This has two maxima of amplitude with a node between, and so on. 



5. Spectrum 



For a complete statistical description of the state of internal waves in the 

 sea a knowledge of the directional spectrum (compare Chap. 15, page 571) 

 for each mode of internal waves is needed. One requires a time series of internal 

 motions at many depths and at many horizontal positions in the sea — more 

 information than is at present available. Except for some near-shore experi- 

 ments (reported below) recent measurements of sufficient duration to provide 

 estimates of the spectrum of internal fluctuations have been made without the 

 possibility of finding the distribution of internal waves in modes or direction. 



A. Temperature Fluctuations off Castle Harbor, Bermuda 

 Haurwitz, Stommel and Munk (1959) report temperature observations from 

 depths of 50 m and 500 m off-shore from Castle Harbor, Bermuda Islands. This 

 remarkable set of observations, extending from December, 1954, to October, 

 1955, is not only the longest, nearly continuous time series of internal tempera- 

 ture data, but also is from a locality which is well situated to represent thermal 

 conditions far from effects of continental borders. 



The spectra of thermal oscillations (Fig. 23) extend from below the inertial 

 frequency Q (one cycle per 22.4 h at the latitude of Bermuda) to 2.5 cycles per 

 hour (c/h). At low frequencies, the spectra possibly show small peaks near the 

 semi-diurnal tidal frequency and (at 500 m) near the inertial or diurnal tidal 



