330 



FOFONOFF 



[sect. 3 



produced primarily by slopes of the sea surface and variations of density 

 within the interior of the ocean. Furthermore, the forces due to horizontal 

 gradients of pressure are balanced primarily by the Coriolis forces. The remain- 

 ing forces due to accelerations and friction are generally smaller than the 

 pressure gradient and Coriolis forces, but can become important in some 

 regions of the ocean. The magnitudes of the forces can be compared by con- 

 verting the momentum and continuity equations to non-dimensional form 

 such that the larger forces are of unit magnitude. The magnitude of the remain- 

 ing forces will then be indicated relative to unity by the non-dimensional 

 coefficients of the force terms formed by the conversion. 



To convert the steady-state equations (11) and (12) to non-dimensional 

 form, we introduce a characteristic length, L, and depth, H, to describe the 

 horizontal and vertical scales of the motion. We characterize the velocities by 

 Uq for the mean, uq for the fluctuating horizontal components ; Wq for the mean, 

 M'o for the fluctuating vertical component. From the continuity equations (12) 

 and (14), we define Wq= UqHJL and wq = uqHIL. If we denote a typical slope 

 of the ocean surface by s, we obtain pogs for the characteristic pressure gradient. 

 We can express the balance between pressure gradient and Coriolis forces by 

 choosing Uq so that /o?7o = grs, where /o is the characteristic magnitude of the 

 Coriolis parameter 'IQ-^. 



Each variable divided by its characteristic magnitude becomes a non- 

 dimensional variable of unit magnitude, i.e. its magnitude will range between 

 zero and unity. i If we denote the non-dimensional variable by primes, we 

 obtain the non-dimensional steady-state equations in the form : 



Ro 



U'i 



-4 



dU'i 

 dx'i 



U', 



uqY dr'ij 

 Uol dx'j 



+ ^Hk^'i U/c 



dp' 



dx'i 



Rq 



Re 



V2C7',.+ 



2 d^U'i 



{i= 1,2) (17) 



dx'i 



Uol dx'j 



+ Sesjk^'jU'k 



,dp' 



dx'z 



dp'U'j 



dx'i 



1 + 



Eli 



Re\ 



VWs+ - 



2 dWs 



dx': 



= 0, 



(18) 

 (19) 



where 1/p' = a' = po/p ~ 1 

 r'ij = Rijlpouo^ 

 Ro = UolfoL, Rossby number 

 Re = poUoLjp,, Reynolds number 

 Fr = Uo^/gH, Froude number. 



1 Strictly speaking, this will not be true unless we use maximum magnitudes as for the 

 characteristic values. As these are not generally known beforehand, we must interpret 

 the non-dimensional magnitudes as being near unity, that is, within an order of magnitude 

 of unity. 



