What is the correct direction of turbulence energy cascade? I have learned from a fluid mechanics textbook [1]  that the turbulence energy is cascaded from the largest eddy to the smallest eddy and is then dissipated by the molecular viscous effect. But recently I was reading Chapter 3 of a thermodynamics textbook [2], Prof. A Bejan claimed that such a classic Richardson picture is wrong and the turbulence is ALWAYS cascaded from the smaller scale to the larger scale, which is supported by Prof. C H Gibson [3]. While I found the terminology "inverse cascade" from this thread and it seems that there is already a lot of research on this topic, e.g. [4]. Hence, I got confused if Prof. A Bejan and Prof. C H Gibson are talking about this "inverse cascade" phenomenon and what is the correct direction of turbulence energy cascade?
[1] Pope, Stephen B., Turbulent flows. Cambridge university press, 2000.
[2] Bejan, Adrian. Entropy generation minimization: the method of thermodynamic optimization of finite-size systems and finite-time processes. CRC press, 2013.
[3] https://thejournalofcosmology.com/APSPittsGibson.pdf
[4] Chen, Shiyi, Robert E. Ecke, Gregory L. Eyink, Michael Rivera, Minping Wan, and Zuoli Xiao. "Physical mechanism of the two-dimensional inverse energy cascade." Physical review letters 96, no. 8 (2006): 084502.
 A: The answer depends on what kind of fluid theory you are considering.

*

*In the 3D viscous incompressible flow the kinetic energy is transferred from large scale eddies to small scale eddies, in particular if you inject energy at a wavenumber $k_F$ kinetic energy will be transferred to the wavenumbers k s.t. $k>k_F$. This is well know fact both numerically and theoretically, see the Kolmogorov41 model (which doesn't rely on NS) theory which prescribes an energy spectrum: $E(k)= C \epsilon^{2/3}k^{-5/3}$, where $\eta$ is the dissipation rate.


*In 2D visc. incompr. flow the kinetic energy is transferred
from small to large eddies, in particular if you inject energy at $k_F$ your energy will be transferred both to the larger and smaller wavenumbers, this fact can be flagged as an inverse canscade. See in this respect Kraichnan-Leith-Batchelor phenomenology according which $E(k) \approx \epsilon^{2/3} k^{-5/3}\theta(k<k_f) + \eta^{2/3}k^{-3} \theta(k>k_F)$ where $\eta$ is the Kolmogorov scale. In other words, you have two different power law behaviors. This double cascade is due to the presence of two inviscid quadratic invariants: energy and enstrophy.
Let's go back to Gibson's point. In an older paper he states:

Everything that wiggles is not turbulence

From this point we should be aware that his defintion of turbulence is different from the standard one. Moreover he states that:

Eddies form at the Kolmogorov scale (Fig. 1), pair with neighboring
eddies, and these pairs pair with neighboring pairs, etc. ...

I believe he is referring to the large structures formation rather than the energy carried from them. The point of the energy cascade is to define how energy is transferred across the scales not how energy transfer determines the vortex merging or structure formation. In addition the presence of backscatter can be emergent in other more complicated theories, in fact Gibson presented an exotic topic like dark matter planets.
A: Depends on the flow! In homogeneous isotropic turbulence, energy may go in both directions, though there is a clear preference for it going to the smallest eddies (called forward cascade), in line with the theories of Kolmogorov and many others. When energy goes the other way, it is called backscatter.
In this paper (arXiv link), they found that forward cascade happens twice as often as backscatter.
Read more here.
