a theoretical and experimental study of the self-similarity concept

A theoretical and experimental study of the self-similarity concept 11Using (15), (21) and (22) we get the following equation for the coefficient of vertical turbulent diffusionν:1 − ∂∂υ=− −ς= − −ς2 1 2 H h h( )∂01 h. *( H h) (∂1 )a tt2If the diurnal change in the upper mixed layer thickness is 1 m, and the thermocline thickness is 30 m,then:ν=0.3 cm 2 /sat the top of the thermocline.For practical use, equations (23) and (9) are the most important ones.The non-dimensional vertical fluxes of temperature as a function of the non-dimensional vertical coordinateς are presented in Figure 4. The theoretical curve corresponding to (21) is marked Q 2in caseA (Fig. 4a). The experimental non-dimensional vertical flux is marked as Q 1. The corresponding approximatecurves for detrainment (case B) are shown in Figure 4b. A comparison of the experimentalresults with theory has the same main features as in the case of the traditional self-similarity concept. Incase A, the observations fit well with theory.3. DISCUSSION AND CONCLUSIONSThe traditional self-similarity concept as well as the extended theory of self-similarity of vertical fluxeswere compared with calculations based on an analysis of observations. In general, the observationssupported the theory. However, the four-day expedition was far too short to gather an adequate data setof CTD profiles in different stability conditions of the air-sea interface with respect to the evolution of themixed layer thickness.The self-similar profiles for temperature based on the observations showed that different kinds of profilesexist depending on the evolution of the mixed layer, with different profiles being found for cases ofentrainment (case A) and detrainment (case B). The observational proof for the self-similar structure oftemperature was better in case A, while the profiles representing case B gave less satisfactory evidence.This is partly because of the small number of case-B profiles measured. On the other hand, most of thetime case A conditions dominated, and profiles representing case B were not always pure; i.e. someprofiles represented the transition between detrainment and entrainment. However, self-similarity oftemperature becomes visible if a time-integration over an inertial period (about 14 h ) is carried out (seeMälkki & Tamsalu 1985).The profiles of the vertical temperature fluxes showed a clear difference between cases A and B. Thenon-dimensional fluxes of temperature also showed differences in the profiles based on the evolution ofthe mixed layer depth. In case A, the fit of theory with observations was better than in case B, thereasoning for this being the same as in the case of the self-similarity profile for temperature.In the case of an entrainment type of profile, the coefficient of vertical turbulent diffusion can be solvedin the stratified layer using the flux-self-similarity concept. Thus, the self-similarity concept can be usedas a tool to parameterize the vertical turbulence in numerical modelling.2(23)