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 ϵ(k) as
∂tϵ(q)=νϵ(q)+F(q)+T(q)=0.
ν is the viscosity, F(q) is the term coming from the forcing ⟨→f(t,→q)⋅→v(t,−→q)⟩, and T(q) is the energy transfer that arises because of the non-linearity,
T(q)=i2∫→p→p⋅{⟨→v(t,→q−→p)[→v(t,→p)⋅→v(t,−→q)]⟩+⟨→v(t,−→q−→p)[→v(t,→p)⋅→v(t,→q)]⟩}≡∫→pT(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)≠0 only when p≅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+ϵ,p)≅−T(p−ϵ,p).
Can some one give me a reference about this?