SEl^T. 3] nVNAMU'S OF OCEAN CURRENTS 333 



be primarily between pressure gradient and Coriolis forces with the accelera- 

 tions serving as a perturbation of the motion. 



The interaction terms re])resenting convection by either the steady or time- 

 dependent flow become of the same order of magnitude as the pressure gradient 

 and Coriolis forces if the Rossby number of the steady flow approaches unity. 

 AVe have already suggested that the Rossby number is comparable to unity in 

 comparatively swift currents such as the Gulf Stream and Kuroshio. Con- 

 sequently, the interactions between the steady and time-dependent modes 

 cannot be negligible in these concentrated currents. Unfortunately, we do not 

 have a clear understanding of the consequences of the interactions. 



The convection terms become comparable to local accelerations if Rq is of 

 the same order of magnitude as I/'/qTo, i.e. if LJTq^ Uq. If we interpret L as 

 the wave-length and Tq as the period of the time-dependent motion, we can 

 consider LITq to be the velocity (in the sense of phase velocity). Consequently, 

 the non-linear convection terms cannot be neglected in comparison to local 

 accelerations if the steady flow approaches the phase velocity of the varying 

 niotion. 



The divergence of momentum flux, containing the term r'a, will become 

 important if the Rossby number for the time-dependent flow, JRoUojUo, ap- 

 proaches unity. The Reynolds stresses enter into the time-dependent equations 

 with opposite sign to that of the steady-state equations. Consequently, we 

 cannot interpret r"ij as a dissipative term in the equations and cannot simulate 

 its action in terms of eddy viscosity. In fact, the divergence of the Reynolds 

 stresses represents the rate at which momentum is being added to the time- 

 dependent flow from the steady flow. If (22) is averaged to eliminate high- 

 frequency components, we see that the contributions to the average of r"i} 

 come primarily from velocity fluctuations the period of which is greater than 

 the averaging time. Thus, we do not have a term corresponding to "eddy" 

 dissipation in the time-dependent equations. 



As in the steady-state equations, the terms representing molecular friction 

 are negligible except at extremely small scales of motion. In the vertical 

 component of the momentum equations, the coefficient F'r can approach unity 

 for ocean currents. However, the factor HjL is small so that the vertical 

 accelerations can become appreciable only at high frequencies. The vertical 

 component of Coriolis force is small and may be neglected. Thus, the variations 

 of pressure in time-dependent motion result primarily from variations of 

 density and surface slope and may be adequately approximated by a hydrostatic 

 equation except at high frequencies. 



4. Steady-State Circulation 



Ocean currents exhibit fluctuations of a wide range of frequencies and size 

 scales. Surveys of the Gulf Stream, for example, have revealed complex struc- 

 ture both in space and time (Fuglister, 1951 ; Wertheim, 1954). Recent measure- 

 ments of deep currents (Swallow, 1955, 1957 ; Swallow and Hamon, 1960) 



