Main Features of General Oceanic Circulation and their Physical Exploration 705 



In the two-layered oceanic model there are as usual two equations of motion for 

 each layer, one for the w-component and the other for the f -component of the velocity, 

 and the continuity equation. The equations of motion for the upper layer therefore 

 take into account the wind stress acting on the sea surface. This then gives three pairs 

 (2 times 3) of differential equations. By cross-wise differentiation, and taking into 

 account the variation of the Coriolis parameter with latitude, one obtains for each 

 layer a vorticity equation. As a first assumption the movements are taken as indepen- 

 dent of the >'-direction. As a consequence, the problem is thus one-dimensional and 

 the equations are considerably simplified. This gives two equations which permit 

 a study of the reponse of the physical sea level and the internal boundary surface to 

 the variable shearing stress of the wind. It is interesting from the mathematical point 

 of view that these equations can be combined to give an equation with a single variable 

 without raising the order. It is of the fourth order and has the form 



1 R T' 



A k Rxxxt 72 ^xttt P^xt "T HA k Rxx 7^ ^tt ^= ~7~ • (XXI. 1) 



R has a fixed numerical relationship to the displacement of the sea surface and the 

 internal boundary surface and can have two values, 7?^ and i?2- In the same way k has 

 the numerical values ki and k^ corresponding to the values R^ and Ro. Moreover, 

 ^^ (= gDJf^) where Dg is the equilibrium thickness of the lower layer and A is a 

 quantity termed by Rossby the "deformation radius". The solution of the differential 

 equation (XXII. 1) gives the "normal values of motion" (equation of normal modes) 

 and makes it possible to determine all the desired quantities of the model such as the 

 displacement of the boundary surfaces and the velocity in the different layers. 



This equation can be used to derive the free waves of the system and their depen- 

 dence on the dimensions of the system, when the wind stress is omitted in the equation. 

 A knowledge of the free waves is of considerable value because of its great importance, 

 since in view of resonance phenomena they may have considerable influence on the 

 forced waves which are generated by the action of the wind. Assuming a normal mode 

 of the form 



Ri = Si sin (Ix + ojit ) (/ = 1 , 2) ; (XXI.2) 



that is, in form of a wave progressing in the negative x-direction with a frequency 

 coi, then the equation (XXII. 1) transforms into 



(/)' 



fl\f 



with 



[j-^ +(l+^.)_!+^ = (XXI.3) 



The solution of this algebraic equation (three positive roots) gives the frequency <u,- 

 as a function of the wave-number l-nll or of the wavelength L. The roots are a»,-,i; 

 ^i,i\ ^1,3 and correspondingly there exist in total six possible modes of wave motion. 

 Figure 341 gives frequencies and periods of the waves for D^ = 500 km, D^ = 3500 km, 

 /= 10-* sec-\ /S ^ 2 X 10-"m-i sec-^ and for wavelengths 10 < L < 12,000 km 

 covering the entire region under consideration. 



2Z 



