[lucamirtanini]

Attachment 68071Hi all,
I cannot understand how the author (figure attached) passes from the correlation tensor Rii to the three-dimensional spectrum E(k).

In particular, I cannot understand how it can arrive to the definition of E(k) as integral on the shell surface.

Can you help me to understand the passages?

PS
Sorry for all my question, but it is difficult to me to find someone that can help me in the real life

[LuckyTran]

Because the trace of phi is the kinetic energy at a given wavelength


You define the correlation tensor. Then you take its Fourier transform. And then you recognize that the trace of this thingy is the kinetic energy at a given wavelength (because you are a spectral space). That is, you recognize that since 8.1.3 gives the mean kinetic energy when you integrate phi over the entire wavenumber space. That means, integrating over a spherical shell gives you the kinetic energy at a specific wavenumber.

[lucamirtanini]

Originally Posted by LuckyTran

That means, integrating over a spherical shell gives you the kinetic energy at a specific wavenumber.

Why integrating over a spherical shell gives me the kinetic energy at a specific wavenumber?

[LuckyTran]

Originally Posted by lucamirtanini

Why integrating over a spherical shell gives me the kinetic energy at a specific wavenumber?

Well obviously integrating over a single shell means you are integrating at a specific wavenumber. What's not obvious is that this gives the kinetic energy.

Integrating over the entire sphere (or integrating over all wavenumbers) gives you the mean kinetic energy (via 8.1.3). Well not exactly, because it's off by a factor (the 3 instead of 1/2). The shell is one slice of this sphere (i.e. at one wavenumber). Hence it follows that the shell contains the contribution to the kinetic energy due to one wavenumber.

Going from 8.1.3 to 8.1.4 is natural. Maybe you need some calculus but understanding of high-school level geometry is all that is needed. The trick is recognizing the truth of 8.1.3. Getting to 8.1.3 is hard. Going from 8.1.3 to 8.1.4 should be easy.

[lucamirtanini]

I have understood. Thank you for your kind help