Extracting the Spectrum by Spatial Filtering

The spectrum of a flow quantifies the energy content of spatial scales and can yield valuable information about the cascade ranges, dissipation, and turbulence intensity. It is often calculated using Fourier series, which are formally justified only in periodic domains, and are inherently global in space, incapable of characterizing the flow locally. We will show that the spectrum of a flow field can be extracted within any local region by straightforward filtering in physical space. This can be as simple as unweighted averaging of adjacent grid-cells. We derive the conditions under which such a "filtering spectrum" is meaningful, and discuss its close connections and important differences with the wavelet spectrum. We demonstrate the approach using synthetic fields, 2D turbulence from a Direct Numerical Simulation, and 3D turbulence from the JHU Database.