598 



.' I -'.- ..• 



n 



^JLliiAi" iii.i»>iM>Bif iiiiiii li^nttm 



(both expressed in density units) should be a 

 function of Rp alone for given diffusing substances: 



FIGURE 2. Shadowgraph picture of a sharp "diffusive" 

 interface, formed between a layer of salt solution above 

 a denser sugar solution. Note the convective plumes each 

 side of the interface, evidence of strong interfacial 

 transports. (Scale: the tank is 150mm. wide.) 



and sugar the analogue of salt, since in each case 

 the convection is maintained, and the interface 

 kept sharp, by the mote rapid vertical transfer of 

 the faster diffusing component. Such interfaces 

 have been called "diffusive interfaces", for reasons 

 which will become clearer in the following section. 



Fluxes through Diffusive Interfaces 



Quantitative laboratory measurements have been made 

 of the S and T fluxes across the interface between 

 a hot salty layer below a cold fresh layer, and 

 they have been interpreted in terms of an extension 

 of well-known results for simple thermal convection 

 at high Rayleigh number. Explicitly, Turner (1955) , 

 Crapper (1975) , and Marmorino and Caldwell (1976) 

 have shown that the heat flux aFrp (in density units) 

 is described by 



aF^ = A^ (aAT) (2) 



where A]_ has the dimensions of velocity. For a 

 specified pair of diffusing substances, A^ is a 

 function of the density ratio Rp=6AS/aAT, where 8 

 is the corresponding "coefficient of expansion" 

 relating salinity to density differences. The 

 deviation of A, from the constant A obtained using 

 solid boundaries, with a heat flux but no salt flux, 

 is a measure of the effect of AS on F„. When Rp 

 is less than about 2, A^ > A due to the increased 

 mobility of the interface, and when Rp > 2, Ai 

 falls progressively below A as R. increases and 

 more energy is used to transport salt across the 

 interface . The empirical form 



A]^/A =3.8 (SAS/aAT)" 



(3) 



[Huppert (1971)] provides a good fit to the obser- 

 vations over the whole of the measured range 1.3< 

 Rp<7. 



The salt flux also depends systematically on 

 Rp , and has the same dependence on AT as does the 

 heat flux. Thus the ratio of salt to heat fluxes 



BFg/aF^ = f. (gAS/aAT) 



(4) 



The results reproduced in Figure 3 support this 

 relation, and they also reveal the striking feature 

 that the flux ratio is substantially constant (=0.15) 

 for 2<Rp<7. [The more recent experiments of 

 Marmorino and Caldwell (1976) suggest that the flux 

 ratio can be as high as 0.4 with much smaller heat 

 fluxes, but the reason for this discrepancy is not 

 yet resolved] . Experiments by Shirtcliffe (1973) , 

 using a layer of salt solution above sugar solution, 

 have shown a much stronger dependence of Frp on Rp 

 than (3) , but again a constant flux ratio, the 

 measured value (for NaCl and sucrose) being 

 6Fs/aF,p ; 0.60. Note that the flux ratio must 

 always be <1, for energetic reasons: the increase 

 in potential energy of the driven component must 

 always be less than that released by the component 

 providing the energy. This implies that the density 

 difference between two layers must always increase 

 as a result of a double-diffusive transport between 

 them. 



Direct measurements through the interface in 

 Shirtcliffe ' s experiment suggest that this has a 

 diffusive core, in which the transport is entirely 

 molecular, and which is bounded above and below by 

 unstable boundary layers. The "thermal burst" 

 model of Howard (1964) has recently been extended 

 to this two-component case by Linden and Shirtcliffe 

 (1978) , to predict both the fluxes and flux ratios. 

 The constant range of flux ratio can be explained 

 in the following way. Boundary layers of both T 

 and S grow by diffusion to thicknesses proportional 

 to K^h and K^h , and then both break away intermit- 

 tently. If only the statically unstable part at 

 the edge of the double boundary layer is removed 

 (such that aAT=BAS), then the fluxes will be in 

 the ratio T-5, in reasonable agreement with the 

 laboratory results for the two values of t used. 

 Linden (1974a) has given a mechanistic argument to 

 explain the increase of flux ratio at lower values 

 of Rp , which he attributes to the direct entrainment 

 of both properties across the interface. 



It is worth noting in passing that Huppert (1971) 



FIGURE 3. The ratio of the fluxes of salt and heat (in 

 density units) across an interface between a layer of 

 hot, salty water below colder, fresh water, plotted as 

 a function of the density ratio R . [From Turner 

 (1965).] P 



