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Effects of friction in two-dimensional turbulence

In order to study the properties of fully developed three-dimensional turbulence, it can be interesting to consider a portion of the fluid far from the boundaries, such that their interaction with the fluid can be neglected. On the contrary, when a thin layer of fluid is considered, it is important to include the effects of the friction between the two surfaces of the layer and the surrounding three-dimensional environment.

In many physical situations, the incompressible flow of a shallow layer of fluid can be described by the two-dimensional Navier-Stokes equations supplemented by a linear damping term which accounts for friction. An important instance, among others, relevant to geophysical applications is the rotating flow subject to Ekman friction [27]. Other well known examples are the Rayleigh friction in stratified fluids, the Hartmann friction in Magneto-Hydro-Dynamics [28] and the friction induced by surrounding air in soap films [29].

Understanding the effects of the friction term is thus a fundamental issue because of their physical relevance. Moreover, in numerical study of forced two-dimensional turbulence, in order to obtain a statistically steady state, it is necessary to introduce a large scale sink of energy to stop the inverse cascade and to prevent the Bose-Einstein condensation phenomenon [20]. The linear friction term is a natural candidate for this purpose, but its presence can produce strong effects not only on the large scales, but also on scales smaller than forcing correlation length, in the inner core of the direct enstrophy cascade.

In this chapter I will study these effects focusing on the influence of friction on the statistical properties of small-scale vorticity fluctuations.

I will first show the analogies and differences between this problem and the transport of a passive scalar with finite lifetime [30], presenting some results recently obtained for the passive scalar statistics.

Using a Lagrangian description of the vorticity cascade I will then obtain the condition for the equivalence between small-scale vorticity and passive scalar statistics, which allows to extend to vorticity the results of the passive case.

Finally I will present the results of direct numerical simulations which confirms the predictions obtained using the Lagrangian approach. In particular I will show that the statistics of small-scale vorticity fluctuations is intermittent and that intermittency arises from the competition between the stretching properties of the flow and the exponential decay induced by friction.



Subsections
next up previous contents
Next: Origin of the friction Up: tesi Previous: Coherent vortices   Contents
Stefano Musacchio 2004-01-09