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Turbulence generation of waves:

The scale function has a fourier-space counterpart: it therefore can define a wave dispersion relation. This makes it possible to make multifractal models with a physical basis similar to that used in Lumley-Shur type models of atmospheric turbulence in which tubulence generates gravity waves. Here, we model the conservative multifractal using the standard method, however the cloud density is then obtained by a wavelike fractional integration which is equivalent to a fractional integral of attentuating waves. In other words, the group wave velocity is well defined and the overall model respects all the classical turbulence symmetries (Kolmogorov law, Corrsin-Obhukov law), except that the vertical structure follows Bolgiano-Obukhov statistics (in order to take into account the differential stratification induced by the buoyancy forces).

Below we show a series of six simulations with increasing wavelike character, the latter controlled by the paramter Hwav. When Hwav=0, the strucutres are localized, as Hwav increases, they become increasingly unlocalized, wave-like. All the simulations follow anisotropic, mulitfractal extensions of classical Corrsin-Obukov statistics for passive scalars. The false colour renditions are single scatter radiative transfer.

This figure shows the effect of increasing Hwav with Hwav+Htur=H=0.33, Ht=0.66, C1=0.1, a=1.8. There is a small amount of differential anisotropy (d=1, c=0.05, e=0.02, f=0). The random seed is the same in all cases so that one can see how structures become progressively more and more wave-like while retaining the same scaling symmetries, close to observations.

The pictures below show vertical cross-sections with sphero-scale increasing from 1/8 to 128 by factors of 2 (left to right, top to bottom).

Below the false colour renditions are single scatter radiative transfer.

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