Mixing Efficiencies in Patchy Turbulence

The efficiency of mixing in stably stratified systems where the turbulent mixing is confined to intermittent patches is investigated theoretically. It is possible to define two different flux Richardson numbers for mixing in such a system. One, the small-scale flux Richardson number, R ft , is based on the initial potential energy increase caused by small-scale turbulent mixing within the patches. This is the parameter that is obtained from laboratory and numerical experiments intended to determine turbulent mixing efficiencies. The other, the large-scale flux Richardson number, R f , is based on the final potential energy increase, obtained after the mixed fluid has spread out laterally in the system. This is the relevant parameter for determining large-scale, irreversible, changes in the stratification caused by mixing. It is shown that the large-scale flux Richardson number is always smaller than the small-scale flux Richardson number, and that the difference can be almost a factor of 2. The commonly used mixing efficiencies, 0.17-0.2, obtained from laboratory and numerical experiments of small-scale homogeneous turbulence, are a measure for the small-scale flux Richardson number R ft rather than the large-scale flux Richardson number R f . If the maximum small-scale flux Richardson number R ft = 0.2 is relevant for mixing in oceanic patches, one should use R f = 0.11 for the large-scale flux Richardson number. The latter value is supported by results from recent microstructure experiments in the ocean.