Regarding the randomness and random phase approximation, I thought it was rather clear what I’ve meant.

In weak turbulence, the only degrees of freedom that you excite are sound waves. In developed turbulence, there is also vorticity. If you consider a single vortex and decompose its velocity field into Fourier modes, you will clearly see that phases of different modes are correlated with each other. This is pretty much all – you have vortices interacting with each other and sound waves interacting with each other as well as with vortices. Random phase approximation for separate Fourier modes is lost but stochasticity – is not, because both velocity and vorticity fields are random.

Regarding he Brownian motion – thermal effects are clearly subdominant (suppose I take nearly zero temperature and study turbulence in liquid Helium: I’ll still have all these patterns with Kolmogorov spectra etc.) Regarding shear stress and bubbles – also, I think, not that relevant for the problem – these effects are included into the definition of viscosity which is macroscopic. Also, consider a turbulence in convection systems for example, where energy of the flow is low (for turbulence, what is important is not only the energy of the flow per unit volume but also the dissipation of energy).

Cheers,
Dmitry.

Also, what makes it sure that a fluid can be considered a fluid? You have brownian motion and, perhaps, the formation of bubles due the energy released by shear-stress along the boundary of layers of different reynold numbers. It seems they should be taken seriously when going to chaos.

In the first paragraph, you write “Turbulence is a complex behaviour of a dissipative medium or a field, disordered, stochastic in time and space.”. Now, when you define strong turbulance you write “strong or developed turbulence random phase approximation ceases to be valid – different modes strongly interact with each other and there exists large coherent contribution.”. So, there is no random here, but instead, it indicates something completely, different, that is, something of almost of infinite complexity.

Other thing that makes me sad because stochastic inherent to a continuous system, but I keep thinking of the brownian motion and vapor formation inside the liquid. For example, what makes it sure that the transition of chaos is not merely that the flow is cloged, or disturbed, or cause by an amplification of brownian movement. Another thing it is if the irregular distribution of energy in the system could lead to state transtions in some parts, like, formation of microbubbles, and if they wouldn’t contribute significantly to the motion of the fluid.

What do you mean by turbulent behavior?

Kindly see the first paragraph after the subtititle “1. What is turbulence?” Or does this definition make you unhappy anyhow?

Cheers,
Dmitry.