600 



FIGURE 4. Shadowgraph of a thickened "finger" inter- 

 face, formed between a layer of sugar solution on top 

 of salt solution. (Scale: the tank is 150mm. wide.) 



shows a shadowgraph picture of such an interface 

 between a layer of sugar solution (S) above heavier 

 salt solution (T) . This is bounded by sharp edges, 

 where the fingers break down and feed an unstable 

 buoyancy flux into the convecting layers on either 

 side. 



Finger interfaces between two such layers have 

 been shown to thicken linearly in time [Stern and 

 Turner (1969), Linden (1973)]. They have also been 

 observed in plan by Shirtcliffe and Turner (1970) 

 who showed that the convection cells have a square 

 cross section, with upward and downward ifiotions 

 alternating in a close-packed array. The initial 

 stability of an interface has been examined quanti- 

 tatively by Huppert and Manins (1973) . When a 

 layer of S is placed on a layer of T, the sharp 

 boundary thickens by diffusion; the condition for 

 formation of fingers depends on the magnitude of 

 the gradients and the ratio of dif fusivities, x, 

 and is related to the overall differences by 



BAS/aAT > t/t 



(5) 



These results can be extended to three components, 

 as can the earlier linear stability theories 

 [Griffiths (1978)]. For heat and salt, (5) shows 

 that fingers should form with very small destabiliz- 

 ing salinity differences, and suggests that they 

 will be ubiquitous phenomena in the ocean. 



Our confidence in applying these results on a 

 geophysical scale has recently been increased greatly 

 by the direct observations of fingers (using an 

 optical method) by Williams (1974, 1975) under 

 conditions close to those predicted by Linden (1973) 

 on the basis of laboratory results. Magnell (1976) 

 has also measured horizontal conductivity variations 

 with the right scale (=:2cm.) to support this inter- 

 pretation. 



As mentioned above, there is not as big a differ- 

 ence between the "diffusive" and "finger" cases as 

 there appears to be when we simply compare the 

 interfaces illustrated in Figures 2 and 4. Layers 

 can be produced from a smooth gradient in the latter 

 case too, by supplying a flux of S at the edge of 

 a gradient of T; this was first demonstrated, using 

 a sugar flux above a salt gradient, by Stern and 



Turner (1969). When viewed on the scale of the 

 convecting layers, there is in fact a close corres- 

 pondence between the two systems. The inequality 

 of diffusivities results in an unstable buoyancy 

 flux across statically stable interfaces in both 

 cases, and this maintains convection above and below. 

 Only the mechanism of interfacial transport differs, 

 and it is here that the detailed structure of the 

 interface enters. Across a finger interface the 

 buoyancy flux is dominated by the destabilizing 

 component, S, and salt is transported faster than 

 heat, whereas the opposite holds a diffusive inter- 

 face. 



Corresponding laboratory measurements of the 

 two coupled fluxes have been made for finger inter- 

 faces. Again there is a strong dependence on the 

 density ratio across the interface, and the ratio 

 of heat to salt fluxes is constant over a consider- 

 able range. Turner (1967) has shown in the heat 

 salt case that the salt flux is about 50 times as 

 large at Rp* = aAT/6AS ->- 1 as it would be if the 

 same salinity difference were maintained at solid 

 boundaries, and falls slowly as R_* increases. 

 He also obtained a value for the flux ratio aFfj./SFg 

 =0.56 over the range 2 < Rp* < 10. Linden (1973) 

 has made direct observations of the structure of 

 salt fingers and the velocity in them that support 

 these estimates of the salt flux. His estimate of 

 the flux ratio was much lower, but recent experiments 

 in our laboratory have supported the earlier value. 

 These new experiments have concentrated on achieving 

 as small a value of Rp* as possible, but measurements 

 in the "variable" range of flux ratio are still 

 elusive. This range could, however, be of great 

 importance in the ocean, where Rp* is often close 

 to unity. 



It is also of interest to mention the experiments 

 of Linden (1974b) who applied a shear across a 

 salt-finger interface. He showed that a steady 

 shear has little effect on the fluxes , though it 

 changes the fingers into two-dimensional sheets 

 aligned down shear. Unsteady shears (i.e., stirring 

 on both sides of the interface) can, on the other 

 hand, rapidly disrupt the interface, and actually 

 decrease the salt flux. 



There are now many examples of layering in the 

 ocean which are consistent with the "fingering" 

 process. These are observed in situations where 

 both the mean salinity and the temperature decrease 

 with increasing depth, and often occur under warm 

 salt intrusions of one water mass into another. 

 The first observations were made by Tait and Howe 

 (1958, 1971) under the Mediterranean outflow, and 

 a good summary of other measurements is to be found 

 in Fedorov (1976) . For reasons which will be dis- 

 cussed more fully in later sections, it is difficult 

 to find cases where one can be sure that the forma- 

 tion of layers bounded by finger interfaces has 

 been the result of one-dimensional processes, 

 strictly analogous to those studied in the labora- 

 tory. Once layers have formed, however, the effects 

 'of the fluxes through the finger interfaces between 

 them can properly be discussed in these terms , and 

 two practical examples will be given. 



The first arises in the context of sewage disposal 

 in the sea. Fischer (1971) has discussed the case 

 where effluent, which can be regarded for this 

 purpose as nearly fresh (though polluted) water, is 

 ejected from a pipe laid along the bottom, and 

 rises as a line plume into sea water which is strat- 

 ified in temperature. Careful design of the outfall 



