# Strong and weak turbulence

As far as I understand, in general, turbulence describes the process of transfer of energy from a large scale, at which a system is perturbed, to a small scale, where dissipation happens. At intermediate scales, universal and scale invariant behaviour is observed. This is called turbulence.

In the context of turbulence, one often comes across the term "weak" and "strong" turbulence. Both lead to energy cascades, but the difference between the two seems to involve the time scales on which turbulence happens. While weak turbulence is an asymptotical phenomenon, strong turbulence occurs within a finite time span.

Can somebody explain the difference in more detail and correct me if my statements above are wrong? Do the energy cascades, that are found in both cases, agree? Which models (purely mathematical or experimental) show the two types of behaviour?

• Do you have references that refer to strong or weak turbulence? I've never heard of those classifications in any of my work so maybe it's a term that isn't as common. – tpg2114 Feb 7 '15 at 19:13
• For example here, here and here. I came across the term "weak turbulence" here, which made me wonder what the difference is. – physicus Feb 7 '15 at 20:04
• To put links in a comment, the format is [text describing the link](http url to the link) – tpg2114 Feb 7 '15 at 20:05

As far as I understood this distinction in plasma physics, you refer to weak/strong in terms of your Rayleigh decomposition of fields:
Any field $X =$, $\vec v$,$\rho$,$T$,$\vec B$ can be decomposed into a background part $\bar X$ and a fluctuating part $X'$, while constructing $X'$ thus that the average $\bar{X'}=0$.
The usual scheme for wave-analysis is now to construct evolution equations for all $X'$s while neglecting nonlinear terms, then fourier-transforming and thus obtaining wave-solutions. Here terms of order $X'/\bar X$ or higher are neglected, as one says the fluctuations to be weak or $X'/\bar X \lt 1$.

Turbulence now comes into play when not neglecting the nonlinear terms. This introduces wave-coupling, but the wavespectrum remains somehow tractable while the weak approximation holds. However for the strong case now fluctuating and mean-components interact introducing additional complications and changes to the physics, thats the basics of the whole distinction.

That was now just written from memory, but tell me if there's some more clarification needed.

• Thanks! So basically for weak turbulence the backreaction of the fluctuations on the background is negligible, while this approximation breaks down in the strong case. That does make sense in the context in which I came across weak turbulence. – physicus Feb 8 '15 at 11:16
• I hope so, else my understanding was flawed ;) – AtmosphericPrisonEscape Feb 8 '15 at 21:43

A simple way to understand this is as follows.

Every system can be linearized and decomposed into linear eigenmodes. If the system was truly linear, the amplitude and phase of each eigenmode would remain constant, and each eigenmode would evolve dynamically by its corresponding eigenfrequency.

In the weak turbulence limit, we allow non-linearities, but assume they are small. The small parameter of expansion is the ratio of energy in the oscillations to the total energy of the plasma. In this limit, the eigenmodes are weakly coupled and the expansion is mathematically valid.

In the strong turbulence limit, the linear eigenmodes are now strongly coupled. What this means is that the amplitude and phase of a given eigenmode can now change on time scales faster than the frequency of the eigenmode in question. This doesn't allow for the expansion in the weak turbulent limit.

For a discussion of this, see Galeev and Sagdeev, Nonlinear Plasma Theory, Reviews of Plasma Physics, Vol. 7, pages 1-55.

I also do not fully understand the notions of weak and strong turbulence.

What is clear to me is that under idealized conditions like homogeneity, isotropy and high Reynolds numbers we may differentiate three regions in the fluctuation wavenumber spectra: the lowest part up to the energy-containing wavenumber, $0 < k < k_L$; then the Kolmogorov region scaling with $k^{-5/3}$ in the interval $k_L < k < k_K$, and the viscous interval, $k_K < k < \infty$. Here $k_K$ denotes the Kolmogorov wavenumber. It is $\infty$ for vanishing viscosity or for $Re = \infty$.

The lowest part has another (reduced) mixing behaviour because the motion is not really turbulent but only to be termed with some good will as "chaotic" and unpredictable - in contrast to the mean flow in a RANS sense. The highest part also has a reduced mixing because there mixing is driven by molecular forces only. But the center piece of the spectrum mixes quite effectively. Maybe this helps a bit forward?

It is important that the range between the energy-containing and the Kolmogorov scale is large enough. Important for many industry issues ...

• Hi Helmut Z. Baumert and welcome to Physics.SE! Please see this help post to learn how to write your equations in a way nicer way i.e. in $\LaTeX$, in order to improve legibility. Thanks! – Gonenc Jul 30 '15 at 9:52