# How to tell if a system has a direct or reverse energy cascade?

We know, in 3D turbulence one observes a direct energy cascade, where the energy flows from the large scales to small scales (see wiki 1,1), usually attributed to vortex stretching. We also know that in 2D turbulence the energy cascade is reverse, and energy flows from small scales to large scales by vortex merging (see here, or a simulation video here).

Is there a way to tell what cascade one is going to observe without resorting to a direct simulation? For instance, if I would like to understand what kind of cascade one expects in the standard non-linear $$\phi^4$$ scalar field, how would I go about it?

Any references would be of help,.

## 1 Answer

Is there a way to tell what cascade one is going to observe without resorting to a direct simulation?

Yes, one usually considers features such as the dimensionality and (ideal/inviscid) invariants of the flow — and the scale invariance in the intermediate scales in inverse cascades can even be linked to universality classes in statistical physics (see the Scholarpedia article).

It has been shown, however, using a description based on the system's Fourier modes and their interactions, that those features alone might not be enough to determine the direction of the cascade:

Inviscid invariants of flow equations are crucial in determining the direction of the turbulent energy cascade. In this work we investigate a variant of the three dimensional Navier-Stokes equations that shares exactly the same ideal invariants (energy and helicity) and the same symmetries (under rotations, reflexions and scale transforms) as the original equations. It is demonstrated that the examined system displays a change in the direction of the energy cascade when varying the value of a free parameter which controls the relative weights of the triadic interactions between different helical Fourier modes.

Which might not be too surprising, if those modes represent "local, non-local, and distant interactions between scales" (from this answer), when we remember that also universality classes are determined by the system dimensionality, symmetries and range of forces.

Real physical systems are bound to be more complicated, with phenomena such as split energy cascades in thin fluids or inverse energy cascades in 3D active fluids popping up. The 2018 review, Cascades and transitions in turbulent flows (arxiv), is probably of interest and this presentation might also be. More recent mathematical results also exist.