Example: Turbulent Power Spectrum

One of the ways this is used frequently in astrophysics is to compute the power spectrum of a velocity field to look at the turbulence properties.

For a simulation with velocity components \(u\), \(v\), and \(w\), we compute the power spectrum as:

\[E(k) = \int_{k=|k|} dk [ \hat{u}(k)^2 + \hat{v}(k)^2 + \hat{w}(k)^2 ]\]

where \(k\) is the radial wavenumber, \(k = \sqrt{k_x^2 + k_y^2 + k_z^2}\). This gives us the power at a scake \(k\).

Kolmogorov turbulence theory says that homogeneous, isotropic, incompressible turblence should scale like:

\[E(k) dk \sim k^{-5/3}\]

We can see this behavior, for example here: https://ui.adsabs.harvard.edu/abs/2005ApJ…632.1021Z/abstract, which looks at Rayleigh-Taylor unstable flames. Here’s a snapshot of the flame at two points in time along with the power spectrum: