In turbulence kinetic energy is transferred from scale to scale through the turbulent cascade. There is a lot of phenomenological description of this process such as (Please complain if you do not agree with this list.)

- The energy is transfer is local in Fourier space.

- The energy transfer is directed (from large to small scales for a direct cascade).

- The same amount of energy is coming from large scales as is going to the small ones.

My question is the following: Is there a formal definition of an energy cascade? If 'yes', can you please give me some references? I expect that such a definition will amount to a list of requirements for the kinetic energy transfer kernel. If I write the time derivative of the kinetic energy spectrum $\epsilon(k)$ as

$$ \partial_t \epsilon(q) = \nu \epsilon(q) + F(q)+T(q) = 0\, .$$

$\nu$ is the viscosity, $F(q)$ is the term coming from the forcing $\langle \vec{f}(t,\vec{q}) \cdot \vec{v}(t,-\vec{q})\rangle$, and $T(q)$ is the energy transfer that arises because of the non-linearity,

$$T(q) = \frac{i}{2} \int_{\vec{p}} \vec{p} \cdot \left\{ \langle \vec{v}(t,\vec{q}-\vec{p}) \left[\vec{v}(t,\vec{p}) \cdot \vec{v}(t,-\vec{q}) \right]\rangle + \langle \vec{v}(t,-\vec{q}-\vec{p}) \left[\vec{v}(t,\vec{p}) \cdot \vec{v}(t,\vec{q})\right]\rangle\right\} \\ \equiv \int_{\vec{p}} T(q,p) \, .$$

Is there a definition of an energy cascade in terms of a list of properties that $T(q,p)$ must satisfy? Thinking about it, it is easy to guess something like,

- $T(q,p) \neq 0$ only when $p\cong q$.

- $T(q,p)$ is positive for $p<q$ and negative for $p>q$.

- $T(q,p)$ is anti-symmetric around the point $p=1$, $T(p+\epsilon, p) \cong − T(p-\epsilon, p)$.

Can some one give me a reference about this?