# Is there a formal definition of an energy cascade in terms of the energy transfer kernel?

+ 1 like - 0 dislike
41 views

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 $p=1$, $T(p+\epsilon, p) \cong − T(p-\epsilon, p)$.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverf$\varnothing$owThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.