The Three-dimensional Temperature Distribution and its Variation in Time 1 05 



effective in the horizontal direction caused by currents moving side by side carrying 

 small masses of water at greater or lesser velocity and by eddies of varying size with 

 vertical axes, that is, by the lateral turbulence in the flow. 



In the horizontal direction the disturbances are of greater dimensions than in the 

 vertical direction, particularly those due to atmospheric effects (wind, squalls and 

 rapid changes of pressure), which affect the surface layer of the sea and to some extent 

 the deepei layers also. Disturbances due to coastal and bottom topography are also 

 able to produce turbulence in an horizontal direction with turbulence elements which 

 must obviously develop on a much larger scale than the vertical turbulence. The 

 corresponding exchange coefficient will be much larger than for vertical mixing. In a 

 certain sense there is an analogy with the large-scale lateral turbulence in the atmos- 

 phere which is also quasi-horizontal (isentropic). In this case the coefficient is on the 

 average of the order of 10^ g cm~^ sec~^ as compared with an average value of 50- 

 100 of ordinary vertical turbulence. That lateral large-scale turbulence is also im- 

 portant in oceanic phenomena was first pointed out by Defant (1926), who determined 

 the order of magnitude of this exchange coefficient as about 5 x 10^. Later, Witting 

 (1933) discussed both vertical mixing and lateral mixing, and has attempted the de- 

 termination of the exchange coefficient by large-scale coloration experiments. Rossby 

 and co-workers (1936) have clearly shown that there occurs in the ocean, as in the 

 atmosphere, a lateral mixing of this type along the isotropic surfaces, which is essen- 

 tially in the ocean the same as along the or^-surface. Parr (1938) has shown the large 

 effect of this lateral mixing on the distribution of temperature and salinity in the water 

 masses around Newfoundland; Sverdrup and Fleming (1941) have found the same 

 effect in the coastal water off California and Stommel (1950) has determined the lateral 

 mixing coefficient Ajp in the Gulf Stream to be 2-3 x 10^ cm^/sec. 



For a given horizontal gradient in any of the properties of a mass of water the 

 horizontal convection will play a large part in the long-period equalization of this 

 gradient. This presupposes a transport of the property along the direction of the gra- 

 dient. Furthermore, if a small mass of water has a property s (for instance, temperature) 

 present in amount S (for instance, heat), then the horizontal transport of S across the 

 horizontal turbulent flow in the direction n is as before, Sn = —An(Ssldn). An is now the 

 horizontal exchange coefficient. Its order of magnitude is several times larger than that 

 of the coefficient for vertical mixing A^. Since, in general, the vertical gradient of a 

 water property (such as temperature, sahnity) dsjdz is considerably larger than that in 

 the horizontal direction dsjdn, the horizontal transport Sn may still be of the same order 

 as the vertical transport S^, since in the above equation the product of the two quan- 

 tities is essential. This appears to be the case in reality so that lateral mixing is no less 

 important than the vertical. 



Consider a volume element dx, dy, dz through which there is a turbulent flow with 

 velocity components u, v, w; the exchange coefficients in the three directions A a;, Ay, 

 Ay. Then, for the individual change with time in the property s the following equation 

 will apply 



ds 8s 8s 8s 8s \ f 8 / 8s\ 8 / 8s\ 8 / 8s] 



dI = 8t-^''8x-^'8y+''8-z--p[8x [^^8xj^8y l^^a^j + ^:^ l^^FrJ 



If the .Y-axis is taken as the direction of the turbulent flow (positive in the flow 



