Spectra Properties:

Variable	Value
Prolongation method	Spectral
Shape of spectra region	Rectangle
Window function	Cosine

Everything else is default, see: ​page

Spectra Region

Following the wiki page on spectra objects, to calculate the spectra region in the simulation domain, you should compute the region in coarse cells first.

I took this to be a constant 40 pc (the diameter of the cylinder) in y & z, and adjusted the length in x to accommodate the shear angle + 10 pc on either side at its widest point:

Computing the number of coarse zones in this rectangular region gives,

Theta	Coarse Zones (Nx,Ny,Nz)
0	(13, 26, 26)
15	(20, 26, 26)
30	(28, 26, 26)
60	(58, 26, 26)

Power Spectrum

To get a power spectrum, one takes a fourier transform (FT) of the data, and then takes the complex conjugate of the resultant fourier transform. This gives the power in each of the component frequencies of the signal.

Kinetic Energy

There seems then to be 2 different ways of computing the power spectrum of the kinetic energy. First, we can take the fourier transform of \sqrt{\rho (v_x + v_y + v_z)^2} and then square this to get the power spectrum. Or, we can take the FT of \sqrt{\rho} ~v_x and then sum up the squares of the various FTs for each dimension.

That is, since the LHS does not strictly equal the RHS,

(FT(\sqrt{\rho(v_x^2 + v_y^2 + v_z^2)}))^2 .DNE. (FT(\sqrt{\rho} v_x))^2 + ((FT(\sqrt{\rho}v_y))^2 + ((FT(\sqrt{\rho}v_z))^2

(but their integrals do - by Parsevel's theorem, stating the total power is equal), yet each give power spectra with the correct units for KE, which do you choose??

Intuitively it seems you want to save the squaring for after taking the FT, since if you do it before, 'you lose information about direction'. For instance, consider the 1D case of the initial condition of colliding flows. If you square the initial condition, you just get a constant function, peaking about k=0.

Further, since,

FT(\sqrt{\rho}v_x ~) ~.DNE.~ FT(\sqrt{\rho})\times FT(v_x)

yet, again, each will have the correct units for KE once squared, which is proper? For the case of the initial condition — what does the LHS and RHS look like? How about the power spectra for the LHS and RHS? What about at some late times when there is lots of structure in the density?