SEt^T. 3J DYNAMICS OF 0(!EAN CURRENTS 369 



The equation for density in terms of T* is 



p = poll-^'(^*-To*)l (161) 



and (159) is 



Q/Ti* a/77* P|27^* 



_ + „,__ = i„V^r. + i,_. (162) 



The boundary condition at the ocean surface for the apparent temperature is 



h~ = g* = -^--So{E-P), (163) 



cxs pCp a 



where q is the flux of heat across the surface, E — P the net rate of evaporation 

 and *S^o the salinity at the surface. At a solid boundary, the flux may be taken 

 to be zero. In the steady state, the integral of the apparent surface source Q* 

 over the entire ocean surface must vanish. Hence, if Q* is different from zero, 

 both positive and negative sources must be present. However, (162) yields un- 

 stable distributions of density if Q* is negative. Thus, the diffusion equation can 

 only be applied in regions where heating is dominant, or where net precipitation 

 is sufficient to maintain stability in the presence of cooling. As mentioned 

 earlier, Stommel (1957) has suggested that regions of instability, in which 

 downward flow is present, may be confined to relatively small areas at high 

 latitudes in the real oceans. 



Lineikin (1955) carried out an analysis of (159) together with the momentum 

 and continuity equations, (25) to (28). He linearized the problem by con- 

 sidering perturbation velocities about a state of rest in which the density 

 increased linearly with depth. Expressing the surface- wind stress in terms of a 

 Fourier series, he examined the rate of decay of the perturbations of velocity 

 and density with depth for the general term of the series. Lineikin assumed the 

 Coriolis parameter to be constant and the eddy viscosity coefficients to be equal 

 to the diffusivity coefficients. Under these assumptions, he was able to show 

 that the characteristic depth of penetration of the perturbations was given by 

 fH\/E, where L is the horizontal scale of the wind stress component and E is 

 the gravitational stability assumed in the unperturbed state. The depth of 

 penetration is of the order of 1000 metres for horizontal scales of 100 km. For 

 oceanic scales of the order of several thousand kilometres, the indicated depth 

 of penetration is unreasonably large. Stommel and Veronis (1957) suggested 

 that a more realistic depth could be obtained by taking into account the varia- 

 tion of the Coriolis parameter. They examined simple models in which rotation 

 was neglected altogether, taken as constant, and allowed to vary with latitude. 

 The results obtained indicated that the depth was limited much more sharply 

 if /S is taken to be different from zero. It is interesting to note that the depth 

 was least in the case of no rotation. 



It can be seen by converting (159) to non-dimensional form that the magni- 

 tude of the diffusion terms in comparison with the convective terms is given by 



13— s. I 



