Intermittency and the multi-mode probability vortex.

In the previous section, we established that one-mode PDF's can deviate from the Rayleigh distributions if the flux of probability in the amplitude space is not equal to zero. However, in the full $N$-mode amplitude space, the flux lines cannot originate or terminate, i.e. there the probability ``sources'' and ``sinks'' are impossible, see (69). Even adding forcing or dissipation into the dynamical equations does not change this fact because this can only modify the expression for the flux (see the Appendix) but it cannot change the PDF continuity equation (65). Thus, presence of the finite flux for the one-point PDF's corresponds to deviation of the flux lines from the straight lines in the $N$-mode amplitude space. The global structure of such a solution in the $N$-mode space corresponds to a $N$ dimensional probability vortex. This probability vortex is illustrated in figure ? which sketches its projection onto a a 2D plane corresponding to one low-wavenumber and one high-wavenumber amplitudes. Taking 1D sections of this vortex one observes a positive one-mode flux at high $k$ and a negative one-mode flux at low $k$, in accordance with the numerical observations of figure 2.

Figure: Projection of the probability flux vortex on a $(s_1,s_2)$ plane where $s_1$ and $s_2$ are the amplitudes at wavenumbers $k_1$ and $k_2$ such that $k_1>k_2$.

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We should say, however, that existence of the probability vortex solutions, although consistent with numerics, remains hypothetical and further work needs to be done to find solutions of (69) with non-zero curl.