370 FOFONOFF [SKC'T. 3 



knlLUo and kvlHWo where L and H are the horizontal and vertical scales of 

 the motion and Uo and Wo the characteristic horizontal and vertical velocities. 

 Major variations of density in the real ocean are generally confined to the upper 

 one to two kilometres of the ocean and vertical velocities are estimated to be 

 in the range 10~4 to 10~5 cm sec~i. Hence, for the diffusion terms to be im- 

 portant in (159), but not dominant, the ratios knlLUo, kvlHWo must be of 

 unit magnitude. Thus, in order to get non-trivial solutions of the convective 

 equations, we must assume that ky is about 1 to 10 cm^/sec and kn is in the 

 range 10'^ to 10^ cm^/sec^^. For eddy viscosities of the same order of magnitude, 

 internal stresses due to gradients of the mean flow are negligible except possibly 

 in the intensified boundary currents. Therefore, we can use the geostrophic 

 approximation to evaluate the steady convective flow in the interior of the 

 ocean. This approximation has been utilized recently by Robinson and Stommel 

 (1959) and Welander (1959) to set up models of the oceanic thermocline pro- 

 duced by convective circulation. The stresses due to velocity shear can be 

 important only if we assume that the eddy viscosity is very much larger than 

 the diffusivity. Such a model has been examined by Ichiye (1958) for zonally 

 uniform flow. However, unless the ratio Avjky is of the order of 10^ or greater, 

 the model gives unrealistically high zonal velocities. 



The geostrophic model of convective circulation can best be formulated in 

 terms of the potential energy function 



E'j, = r Pdz^ r p {pB-p)gdz'dz + {z-Zs)[PB-hPB9{^-ZB)]- (164) 



For 2 = 77, the function E' ^p becomes identical with the potential energy, E^, of 

 the water column relative to the ocean bottom introduced in (44). If the 

 convective flow is assumed to be entirely baroclinic, both the bottom pressure, 

 'Pq, and the bottom density, p^, are functions of z^ only. Therefore, we can write 

 (164) in the form 



E\ = E'j,o + X> (165) 



where 



x' = {pB-p)gdz' dz (166) 



is an anomaly of potential energy and E'po is a function of 2 only. The potential 

 energy anomaly, x\ is defined relative to level surfaces rather than isobaric 

 surfaces and, although similar, is not equivalent to the potential energy 

 anomaly defined in (46). We shall not carry the primes on the potential energy 

 functions in the remainder of this section but the diff"erence in the definitions 

 of X must be kept in mind. A function essentially identical with (166) was 

 introduced by Welander (1959) in the study of convective flow with no 

 diffusion. 



