SKdT. 3] DYNAMICS OF OCEAN CURRENTS 331 



By evaluating the coefficients Bq, Re, Fj and s for a given flow, we obtain an 

 appreciation of the magnitudes of the forces involved and can decide which 

 force needs to be taken into account for a satisfactory interpretation of the 

 flow in terms of the equations. Conversely, we can examine the coefficients to 

 determine the horizontal and vertical scales for which any given term of the 

 equations becomes comparable to unity. For example, the ratio of the non- 

 linear accelerations to the Coriolis forces is given by the Rossby number, Rq. If 

 we use the observation that steady velocities in the ocean do not exceed 2 or 

 3 m/sec, we can make the Rossby number comparable to unity only by as- 

 suming the current to be sharply confined horizontally {L small), or by assuming 

 a current near the equator (/o small). At mid-latitudes, the Rossby number 

 approaches unity for horizontal scales of 20 to 30 km for the maximum velocities 

 given above. Such conditions are approached only in concentrated current 

 systems such as the Gulf Stream and Kuroshio. Another way to interpret the 

 Rossby number is to note that the ratio UolL is the characteristic magnitude 

 of the vertical component of relative vorticity, dUzjdxx — dUijdxz. Thus, the 

 Rossby number can be considered as the ratio of the relative vorticity to the 

 Coriolis parameter. Hence, if the relative vorticity approaches the Coriolis 

 parameter, the Rossby number will be near unity and non-linear accelerations 

 will be of the same magnitude as the Coriolis forces. 



The Reynolds stress terms become important if the magnitude of the velocity 

 fluctuations exceed that of the steady flow {uqJUq>\), or if the Reynolds 

 stresses change rapidly over distances that are short compared with the 

 characteristic length of the steady flow {dr'ijjdxj > 1). Thus, if we retain Reynolds 

 stresses and neglect steady-state accelerations in setting up a model of ocean 

 circulation, we are implicitly assuming that the characteristic length over 

 which the stresses act is larger than the length for which the Rossby number is 

 unity. Consequently, the steady flow must be an average of velocity com- 

 ponents whose fluctuations greatly exceed their mean. 



The other coefficients, the Reynolds number, Re, and the Froude number, Fr, 

 approach unity only at extremely small horizontal and vertical scales of 

 motion. Consequently, molecular friction and vertical accelerations may 

 always be neglected in the steady state. The vertical components of the Coriolis 

 forces, characterized by s in equation (18), do not exceed 0.1% of the vertical 

 pressure gradient even at the extreme velocities encountered in the ocean. Thus, 

 the vertical balance of forces is represented to a high degree of accuracy by the 

 hydrostatic equation 



«'|^+1 = 0. (20) 



dx 3 



If eddy viscosity coefficients are used in the momentum equations instead of 

 Reynolds stresses, their magnitudes must be related to the Reynolds stresses 

 according to the non-dimensional equation 



uo\" dr'ij 1 



UoJ dx'j R'e 



^ >..L-' s-'u: 



IxhH^ dx's^ 



(2J) 



