ferent in the two cases and the latter agreed well with the observations The deepening rate accelerated during the initial rise in wind stress but decreased abruptly as 8V was reduced during the second half of the inertial period even though u2 continued to increase They thus found no evidence of deepening driven by wind stress alone on this time scale although turbulence generated near the surface must still have contributed to keeping the surface layer stirred A particularly clear cut series of observations on convective deepening was reported by Farmer 1975 He has also given an excellent account of the related laboratory and atmospheric observations and models in the convective situation The convection in the case considered by Farmer was driven by the density increase produced by surface heating of water which was below the temperature of maximum density in an icecovered lake Thus there were no horizontal motions and no contribution from a wind stress at the surface From successive temperature profiles he deduced the rate of deepening and showed that this was on average 17 greater than that corresponding to nonpenetrative mixing into a linear density gradient Thus a small but not negligible fraction of the convective energy was used for entrainment The numerical values of the energy ratio derived in this and earlier studies will not be discussed here but note that the relevance of the usual definition has been called into question by Manins and Turner 1978 In certain well documented cases models developed from that of Kraus and Turner 1967 using a parameterization in terms of the surface wind stress and the surface buoyancy flux have given a good prediction of the time dependent behavior of deep surface mixed layers Denman and Miyake 1973 for example were able to simulate the behavior of the upper mixed layer at ocean weather station P over a 2 week period They used observed values of the wind speed and radiation and a fixed ratio between the surface energy input and that needed for mixing at the interface On the seasonal time scale Gill and Turner 1976 have systematically compared various models with observations at a North Atlantic weathership They concluded that the Kraus Turner calculation modified to remove or reduce the penetrative convective mixing during the cooling cycle gives the best agreement with the observed surface temperature Ts of all the models so far proposed In particular it correctly reproduces the phase relations between the dates of maximum heating maximum surface temperature and minimum depth and it predicts a realistic hysteresis loop in a plot of Ts versus total heat content H i e it properly incorporates the asymmetry between heating and cooling periods This behavior is illustrated in figure 8 3 The model also overcomes a previous difficulty and allows the potential energy to decrease during the cool J t I T i dz 1400 1200 1000 1 I 180 I I 200 220 I 240 I 260 Figure 8 3 The heat content in the surface layer as a function of surface temperature T at ocean weather station Echo After Gill and Turner 1976 The reference temperature T is the mean of the temperature at 250 m and 275 m depth and the months are marked along the curve ing period instead of increasing continuously as implied by the earlier models The mixed layer depth and the structure of the thermocline are not however well predicted by these models this fact points again to the factors that have been neglected Niiler 1977 has shown that improved agreement is obtained by empirically allowing the energy available for mixing to decrease as the layer depth increases though a similar behavior is implied by the use of 8 23 see Thompson 1976 for a comparison of the two types of model Direct measurements of the decay of turbulent energy with depth in the mixed layer will clearly be important In many parts of the ocean it may also be necessary to consider upwelling Perhaps the most important deficiency is the neglect of any mixing below the surface layer There is now strong evidence that the density interface is never really sharp but has below it a gradient region that is indirectly mixed by the surface stirring At greater depths too the density profile is observed to change more rapidly than can be accounted for by advection so that mixing driven by internal waves alone or in combination with a shear flow must become significant These internal processes are the subject of the following section 8 4 Mixing in the Interior of the Ocean The overall properties of the main thermocline apparently can be described rather well in terms of a balance between upwelling w and turbulent diffusion K in the vertical Munk 1966 for example after reviewing earlier work summarized data from the Pacific that show that the T and S distributions can be fitted by exponentials that are solutions of diffusion equations for example 245 Small Scale Mixing Processes d 2T Kz dT z 0 8 24 with the scaleheight K w 1 km By using distributions of a decaying tracer 4C he also evaluated a scale time K w 2 and the resulting upwelling velocity w 1 2 cm day and eddy diffusivity K 1 3 cm 2 s 1 have been judged reasonable by modelers of the large scale ocean circulation chapter 15 Munk found the upwelling velocity consistent with the quantity of bot tom water produced in the Antarctic but he was not able to deduce K using any well documented physical model The most likely candidate seemed to be the mixing produced by breakdown of internal waves but other possibilities are double diffusive processes and quasi horizontal advection following vertical mixing in limited regions such as near boundaries or across fronts Some progress has been made in each of these areas in the past 10 years and they will be reviewed in turn First however we shall discuss a set of interrelated ideas about the energetics of the process that are vital to the understanding of all types of mixing in a stratified fluid 8 4 1 Mechanical Mixing Processes a EnergyConstraints on Mixing The overall Richardson number Rio defined by equation 8 8 based on the velocity and density differences over the whole depth of the ocean is typically very large implying that the associated flow is dynamically very stable But a second important fact is that the profiles of density and other properties are now known to be very nonuniform with nearly homogeneous layers separated by interfaces where the gradients are much larger Is it possible that a discontinuous structure of this kind
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