Wave-number–frequency spectrum for turbulence from a random sweeping hypothesis with mean flow

Abstract

We derive the energy spectrum in wave-number-frequency space for turbulent flows based on Kraichnan's idealized random sweeping hypothesis with additional mean flow, which yields the instantaneous energy spectrum multiplied by a Gaussian frequency distribution. The model spectrum has two adjustable parameters, the mean flow velocity and the sweeping velocity, and has the property that the power-law index of the wave-number spectrum translates to the frequency spectrum, invariant for arbitrary choices of the mean velocity and sweeping velocity. The model spectrum incorporates both Taylor's frozen-in flow approximation and the random sweeping approximation in a natural way and can be used to distinguish between these two effects when applied to real time-resolved multipoint turbulence data. Evaluated in real space, its properties with respect to space-time velocity correlations are discussed, and a comparison to the recently introduced elliptic model is drawn.

Error estimate of Taylor's frozen-in flow hypothesis in the spectral domain

“…Furthermore, the spectral index is the same both in the frequency domain and in the wavenumber domain (Wilczek and Narita, 2012). The effect of the sweeping velocity appears in the coefficient of the frequency spectrum.…”

“…The existence of Doppler broadening is confirmed both theoretically for a random sweeping model (Wilczek and Narita, 2012) and observationally in interplanetary space (Narita, 2014). Starting with the hydrodynamic treatment for Taylor's hypothesis, we develop a magnetohydrodynamic approach to Taylor's hypothesis with an application to the solar wind measurements of Doppler shift and broadening.…”

“…According to the random sweeping model, the power-law index of the energy spectrum is the same between the frequency domain and the wavenumber domain even though there is significant frequency broadening by the sweeping velocity and Taylor's hypothesis is violated (Wilczek and Narita, 2012). The difference between a valid case and an invalid case of Taylor's hypothesis appears as a coefficient in the frequency spectrum such that the whole spectral curve is lifted, depending on the ratio of the sweeping velocity to the mean flow velocity (Wilczek et al, 2014).…”

“…Here we develop a method to estimate the error of Taylor's hypothesis in the spectral domain using the random sweeping model (Kraichnan, 1964;Wilczek and Narita, 2012) assuming a Gaussian statistics of the large-scale flow variation. The existence of Doppler broadening is confirmed both theoretically for a random sweeping model (Wilczek and Narita, 2012) and observationally in interplanetary space (Narita, 2014).…”

Non-elliptic wavevector anisotropy for magnetohydrodynamic turbulence

“…Various models have been proposed to explain or reproduce the anisotropic energy spectra in the wavevector domain: the density fluctuation model (Higdon, 1984), the critical balance model by regulating the Alfvén wave interaction time with the eddy turnover time Sridhar, 1995, 1997;Forman et al, 2011), generalization of critical balance by adding damping (von Papen and Saur, 2015); two-component model consisting of slab geometry and quasi-two-dimensional turbulence (Matthaeus et al, 1990;Bieber et al, 1996), the three-component model as an extension of the two-component model by adding the compressible fluctuation component (Matthaeus and Ghosh, 1999), the cross-helicity model showing asymmetry in the energy spectrum between the parallel and the anti-parallel directions to the mean magnetic field (Chandran, 2008), and the elliptic anisotropy model parameterized by the shape coefficients (Carbone et al, 1995).…”