- Technical report
- Open Access

# Spectral moments for the analysis of frequency shift, broadening, and wavevector anisotropy in a turbulent flow

- Y. Narita
^{1, 2, 3}Email author

**Received:**9 January 2017**Accepted:**16 May 2017**Published:**26 May 2017

## Abstract

## Keywords

- Turbulence energy spectra
- Moment calculation
- Random sweeping model

## Introduction

Turbulent fluctuations appear commonly in various fluid and gaseous media in geophysical and space science applications, e.g., ocean turbulence, atmospheric turbulence, and plasma turbulence in near-Earth space. The fluctuating fields (flow velocity, density or pressure variation, or electromagnetic field) develop both spatially and temporally. One method to study the turbulent fluctuations is to determine the correlation in the space–time domain (He and Zhang 2006; Zhao and He 2009). The other method, complementary to the space–time correlation, is to determine the energy spectrum in the wavevector and frequency domain as the Fourier transform of the space–time correlation (Kraichnan 1964; Wilczek and Narita 2012).

The wavevector–frequency spectra are accessible by numerical simulations or multi-point measurements using a properly placed sensor array. In the lowest-order picture, the frequencies of the turbulent field are subject to the Doppler shift and broadening caused by the mean flow and the large-scale variation of the flow (referred to as the random sweeping field), and the wavevector anisotropy (caused, for example, by a large-scale magnetic field in plasma) can be fitted by a set of elliptic energy contour levels.

Here we propose a spectral moment method to characterize the wavevector–frequency spectra of turbulence using a smallest set of parameters: the shift velocity, the sweeping velocity, and the eigenvalues and eigenvectors of the spectral moment tensor. The shift velocity is associated with the phase speed in the observer frame in the turbulent medium. In the case of supersonic or super-Alfvénic media such as in the solar wind, the shift velocity should sufficiently be close to the mean flow speed (Doppler shift). The sweeping velocity is associated with the frequency broadening of the energy spectrum around the Doppler shift or the frequency shift when the spectrum is sliced over the frequencies at a given wavenumber. In the limit of hydrodynamic treatment, the sweeping velocity can be of the order of root-mean-square of the large-scale flow variations (Doppler broadening), or can be influenced by changes in the spatial structure. The eigenvalues and the eigenvectors are obtained from the second-order spectral moment tensor in the wavevector domain and can be used to determine the direction of maximum and minimum variation directions (or spectral extension directions) and to characterize the elliptic shape of the spectral anisotropy in the wavevector domain. The algorithm for the spectral moments is presented along with examples of the wavevector–frequency spectra using synthetic data and the spacecraft data in the solar wind.

*L*(Forman et al. 2011) needs to be determined from the observation. For the non-elliptic wavevector anisotropy model (Narita 2015), the shape parameter (the ratio of the coefficients on the parallel and perpendicular wavenumbers) needs to be determined from the observation. In the case of the one-dimensional probability distribution

*f*over a domain of \(\chi\) (e.g., the frequencies and the wavenumber in this study; other choices such as particle velocities or fluctuation amplitudes are also possible), the zeroth-order moment is estimated as

Normalization of the *n*th-order moments is possible by dividing by the zeroth-order moment.

The goal of the article is to develop an algorithm to determine the shift velocity (the phase speed of the major fluctuation component), the sweeping velocity, and the sense of wavevector anisotropy using the measurement of the energy spectrum in the wavevector–frequency domain. The essence of the analysis is to extract the information on the principal axis (or the maximum and minimum variance directions) using the spectral moments, and to interpret the spectra in the frame of random sweeping model. Using the assumption of Gaussian statistics (which requires the smallest set of free parameters to characterize the energy spectrum in the wavenumber–frequency domain) and a power-law elliptic shape of the wavevector anisotropy, it is possible to reconstruct the energy spectra in the wavevector and frequency domains.

Two different methods are introduced in the article: (1) computation of the spectral moments in the wavenumber–frequency domain and in the wavevector domain and (2) reconstruction of the spectrum using Gaussian statistics or a power-law distribution. The former is an objective, mathematical exercise and the method itself does not require any specific type of fluctuation data as far as the fluctuations belong the same statistical family (so, discontinuities should be avoided in the data). The latter is a subjective, interpretation work and the results depend on the choice of physics model of interest. We use a Gaussian shape of the spectrum both in the wavenumber–frequency domain, and an elliptic power-law spectrum in the wavevector domain. The corresponding physics model is the random sweeping model in the wavenumber–frequency domain (Kraichnan 1964; Narita 2017; Wilczek and Narita 2012) and the random spatial variation model (Gaussian statistics) with an elliptic sense of wavevector anisotropy (Carbone et al. 1995) for power-law spectrum of elliptic wavevector anisotropy.

It is worth noting that turbulence inherits intermittency (Sorriso-Valvo et al. 1999); otherwise, fully Gaussian fluctuations (no intermittency) imply that there is no wave–wave interaction or fluid nonlinearity (like eddy distortion, vortex line entanglement). In the reconstruction method in the wavenumber–frequency domain assuming Gaussian statistics, the analytic derivation was originally proposed for random sweeping field by Kraichnan (1964), for random sweeping field in a mean flow by Wilczek and Narita (2012), and for random sweeping hydromagnetic field in a mean flow by Narita (2017). Intermittency is renormalized into small-scale fluctuating fields and that fields are regarded as frozen-in into the mean flow and the random sweeping flow in an advected fashion without intrinsic turbulent evolution.

## Frequency shift and broadening

### Random sweeping model

The lowest-order picture of turbulence frequency spectra (in the Eulerian frame) is a combination of frequency shift and broadening of the energy spectra in the streamwise wavenumber domain (primarily in the mean flow direction). The frequency shift is caused by the mean flow (Doppler shift) or by a linear mode wave such as the Alfvén wave. The frequency shift maps uniquely the streamwise wavenumbers onto the frequencies through the relation \(\omega = k_{\mathrm{s}} U_{\mathrm{sft}}\), where \(\omega\), \(k_{\mathrm{s}}\), and \(U_{\mathrm{sft}}\) denote the Eulerian frequencies, the streamwise wavenumbers, and the frequency shift velocity, respectively. The shift velocity essentially represents the Doppler shift, which is the basis of Taylor’s frozen-in flow hypothesis (Taylor 1938). In addition, the frequency broadening occurs whenever (1) the flow has large-scale variations with the zero mean such that the root-mean-square of the frequency deviation (measured from the Doppler-shifted frequency) is linearly proportional to the root-mean-square of the flow velocity variation, so we associate the frequency broadening with the relation \(\sqrt{\langle \delta \omega ^2 \rangle } = k_{\mathrm{s}} U_{\mathrm{swp}}\), by regarding the sweeping velocity \(U_{\mathrm{swp}}\) as the root-mean-square of the flow velocity fluctuation \(U_{\mathrm{rms}}\). Here the angular bracket denotes the operation of statistical averaging, or (2) wave–wave interactions produce waves with deviating frequencies from the Doppler shift such as linear modes, sideband waves, nonlinear modes (Howes and Nielson 2013). In the spectral moment method, we determine the shift velocity and the broadening velocity from the measurements, and compare with the Doppler shift and broadening.

### Spectral moments

### Frequency shift and broadening in the solar wind

The analysis method for the shift velocity \(U_{\mathrm{sft}}\) and the sweeping velocity \(U_{\mathrm{swp}}\) is tested against the four-point magnetic field data in the solar wind obtained by the Cluster spacecraft (Balogh et al. 2001; Escoubet et al. 2001) for a time interval of March 19, 2005, 0300–0600 UT. The four Cluster spacecraft formed a nearly regular tetrahedron with a spacecraft separation of about 1000 km. The spacecraft are near the apogee and are almost standing in the solar wind at a distance of about 20 Earth radii ahead of the Earth. Figure 3 displays the magnetic field magnitude measured by the fluxgate magnetometer (Balogh et al. 2001) and the ion bulk velocity and the ion number density measured by the ion electrostatic analyzer (Rème et al. 2001) as time series plots.

The mean magnetic field is nearly dawnward, \((1.0, -6.4, 1.4\;{\mathrm{nT}})\) in the GSE (geocentric, solar, ecliptic) coordinate system, with a mean magnitude of about 6.7 nT. The mean ion bulk velocity is nearly anti-sunward \((-438, 34, 51\;{\mathrm{km/s}})\) in GSE with a mean magnitude of about \(442.2\;{\mathrm{km/s}}\), and \((433, 0, -90\;{\mathrm{km/s}})\) in the mean field-aligned (MFA) coordinate system with the *z* axis pointing in the direction of the mean magnetic field and the *x*–*z* plane spanning the mean magnetic field and the mean ion bulk velocity. The mean ion number density is \(7.4\,\hbox {cm}^{-3}\). The standard deviation of the magnetic field vector fluctuations is \((2.2, 2.2, 3.3\;{\mathrm{nT}})\) and that of the ion bulk velocity fluctuations is \((15, 17, 16\;{\mathrm{km/s}})\) in the GSE coordinate system. The observation is made in a slow solar wind.

The shift velocity and the sweeping velocity are determined as a function of the wavenumbers, and are shown as histograms in Fig. 6. The distributions maximize at a shift velocity of about 400 km/s and sweeping velocities between 80 and 100 km/s. The mean ion bulk speed is 443 km/s, so the shift velocity obtained from the multi-point magnetometer data is smaller than the flow speed by about 10%. The sweeping velocity is larger than the root-mean-square of the ion bulk speed fluctuation 17 km/s and is closer to the Alfvén speed 54 km/s estimated from the mean magnetic field and the mean ion density. The sound speed \(c_\mathrm{s} = \sqrt{\gamma p_{\mathrm{th}}/\rho } = \sqrt{\gamma k_{\mathrm{B}} T /m}\) is estimated as 56 km/s, where \(\gamma =5/3\) is the polytropic index, \(p_{\mathrm{th}} = n k_{\mathrm{B}} T\) the thermal pressure, \(\rho = m n\) the ion mass density, *m* the proton mass, *n* the ion number density, \(k_{\mathrm{B}}\) the Boltzmann constant, and \(T = 0.23\) MK the ion temperature. Magnetosonic speed is \(\sqrt{V_{\mathrm{A}}^2 + c_{\mathrm{s}}^2} = 78\) km/s.

- 1.The uncertainty in the shift velocity estimate is taken from the standard error by using the central limit theorem. That is, the true shift velocity \(U_{\mathrm{sft(0)}}\) is within a interval ofat a probability of 95%. Here$$\begin{aligned} U_{\mathrm{sft}} - \frac{2 U_{\mathrm{swp}}}{\sqrt{n}}< U_{\mathrm{sft(0)}} < U_{\mathrm{sft}} + \frac{2 U_{\mathrm{swp}}}{\sqrt{n}} \end{aligned}$$(7)
*n*stands for the degree of freedom. We use the number of time subintervals in the analysis (\(n=24\)). Note that the sweeping velocity is proportional to the standard deviation of the frequency-sliced spectrum with the difference in the factor of the streamwise number \(k_{\mathrm{s}}\). The standard error of the shift velocity isand it is estimated as \(\Delta U_{\mathrm{sft}} = 17 \, {\mathrm{km/s}}\) at \(k_{\mathrm{s}} = 1.0 \times 10^{-3} \, {\mathrm{rad/km}}\), \(\Delta U_{\mathrm{sft}} = 17 \, {\mathrm{km/s}}\) at \(k_{\mathrm{s}} = 1.5 \times 10^{-3}\, {\mathrm{rad/km}}\), and \(\Delta U_{\mathrm{sft}} = 15 \, {\mathrm{km/s}}\) at \(k_{\mathrm{s}} = 2.0 \times 10^{-3}\, {\mathrm{rad/km}}\), respectively.$$\begin{aligned} \Delta U_{\mathrm{sft}} = \frac{U_{\mathrm{swp}}}{\sqrt{n}} , \end{aligned}$$(8) - 2.The uncertainty in the sweeping velocity is estimated using the chi-squared distribution with \(n-1\) degrees of freedom, \(\chi ^2_{n-1}\). The chi-squared distribution has a variance of \(2(n-1)\) (Lehmann and Casella 1998). The standard error of the squared sweeping velocity isfor a sufficiently large value of$$\begin{aligned} \Delta U_{\mathrm{swp}} = U_{\mathrm{swp}} \sqrt{\frac{1}{2(n-1)}} \end{aligned}$$(9)
*n*(typically \(n > 10\)). See, for example, Eq. 19 in Ref. Harding et al. (2014). From Eq. (9), we obtain the standard error of the sweeping velocity estimate as \(\Delta U_{\mathrm{swp}} = 12\,{\mathrm{km/s}}\) at \(k_{\mathrm{s}} = 1.0 \times 10^{-3} \, {\mathrm{rad/km}}\), \(\Delta U_{\mathrm{swp}} = 13\,{\mathrm{km/s}}\) at \(k_{\mathrm{s}} = 1.5 \times 10^{-3} \, {\mathrm{rad/km}}\), and \(\Delta U_{\mathrm{swp}} = 11\,{\mathrm{km/s}}\) at \(k_{\mathrm{s}} = 2.0 \times 10^{-3} \, {\mathrm{rad/km}}\) using \(n=24\).

Measured shift velocity and sweeping velocity in comparison to the theoretical estimate for the ion bulk speed, Alfvén speed, and sound speed using the mean values of plasma and magnetic field data

Streamwise wavenumber \(k_{\mathrm{s}}\) (rad/km) | Shift velocity measurement \(U_{\mathrm{sft}} \pm \Delta U_{\mathrm{sft}}\) (km/s) | Ion bulk speed with Alfvén speed \(U_{\mathrm{flow}} \pm V_{\mathrm{A}}\) ( km/s) | |
---|---|---|---|

\(1.0\times 10^{-3}\) | \(396\pm 17\) | ||

\(1.5\times 10^{-3}\) | \(423\pm 17\) | \(443\pm 54\) | |

\(2.0\times 10^{-3}\) | \(401\pm 15\) |

Streamwise wavenumber \(k_{\mathrm{s}}\)(rad/km) | Sweeping velocity measurement \(U_{\mathrm{swp}} \pm \Delta U_{\mathrm{swp}}\) (km/s) | Alfvén speed \(V_{\mathrm{A}}\) (km/s ) | Sound speed \(c_{\mathrm{s}}\) (km/s) |
---|---|---|---|

\(1.0\times 10^{-3}\) | \(82\pm 12\) | ||

\(1.5\times 10^{-3}\) | \(85\pm 13\) | 54 | 56 |

\(2.0\times 10^{-3}\) | \(76\pm 11\) |

## Wavevector anisotropy

### Spectral moment matrix

Turbulent fluctuations can be anisotropic in the wavevector domain imposed by a boundary effect or a large-scale magnetic field. If the Eulerian frequencies are limited to the positive domain and the turbulent fluctuations are measured in a flow, the spectral extension reasonably reflects the direction of the mean flow (hereafter the maximum extension direction).

### Anisotropy analysis

*x*axis by 30\(^{\circ }\).

The eigenvalues and the eigenvectors are determined for the spectrum model. We obtain eigenvalues of \(\lambda _1 \simeq 0.084\) and \(\lambda _2 \simeq 0.040\) and a tilt angle of about 33\(^{\circ }\) between the maximum extension direction (the eigenvector for the largest eigenvalue) and the *x* axis (reference axis). The wavevector anisotropy for the example case is thus characterized by an elliptic ratio of \(\epsilon = \frac{\lambda _2}{\lambda _1} \simeq 2.1\) and a tilt angle of \(\theta \simeq 33^\circ\). Both the eigenvalue ratio and the tilt angle reasonably reproduce the true anisotropy properties (\(c_{{y}}/c_{{x}} = 1/2\) and \(\theta = 30^\circ\)) though the results are not exactly the same as the true values. The systematic error is estimated in more detail below.

*x*–

*y*reference system as:

- 1.
Maximum extension directions are determined for different values of spectral slopes (\(\alpha = \lbrace 1, 2, 3\rbrace\)) at a fixed value of the elliptic shape \(\epsilon _{\mathrm{true}} = c_{{y}}/c_{{x}} = 1/2\). Figure 8 top panel displays the observed (or measured) tilt angle \(\theta _{\mathrm{obs}}\) against the true tilt angle \(\theta _{\mathrm{true}}\). The tilt angle is within an error of about 6\(^{\circ }\) (Fig. 9).

- 2.
Maximum extension directions are determined for different values of elliptic shape (\(\epsilon _{\mathrm{true}} = \lbrace 0.05, 0.1, 0.5\rbrace\)) at a fixed spectral slope \(\alpha = 2\). Again, the observed tilt angle is almost linearly proportional to the true tile angle within an error of about 3\(^{\circ }\) (Figs. 8 middle panel, 9).

- 3.
Elliptic shapes (the ratio of eigenvalues) are evaluated for different values of spectral slopes (\(\alpha = \lbrace 1, 2, 3\rbrace\)) at a fixed tilt angle (\(\theta = 0^\circ\)). The observed eigenvalue ratios \(\epsilon _{\mathrm{true}}\) monotonously increase with the true ratios \(\epsilon _{\mathrm{true}}\). The increase rate is larger at smaller values of the elliptic ratio (Fig. 8 bottom panel).

### Wavevector anisotropy in the solar wind

The spectral moment method is applied to the magnetic energy spectrum in the wavevector domain using the same solar wind data as that shown in Fig. 3. Figure 10 left panel displays the energy spectrum (integrated over the frequencies) in the parallel and perpendicular wavevector components to the mean magnetic field.

We construct the spectral moment matrix \(\kappa _{ij}\) and determine the eigenvalues \(\lambda _1\) (maximum variance) and \(\lambda _2\) (minimum variance) and the associated eigenvectors \(\vec {e}_1\) and \(\vec {e}_2\). The eigenvalue ratio \(\lambda _2/\lambda _2\) is about 0.32. The maximum and minimum variance directions in the wavevector spectrum show an offset angle from the mean magnetic field by 76.9\(^{\circ }\).

## Conclusion

The spectral moment method is robust in characterizing the energy spectra in the wavevector–frequency domain in two aspects. First, the lowest-order picture of the turbulence energy spectra is obtained in the wavevector–frequency domain by estimating the frequency shift velocity, the sweeping velocity, and the eigenvalues and eigenvectors of the second-order spectral moment tensor for the elliptic wavevector anisotropy. Second, the spectral moment method does not require the knowledge on the power-law index when estimating the sense of elliptical shape in the spectrum. The spectral index needs to be specified when one reproduces the wavevector spectrum using a model. The spectral moment method is advantageous in that the shift and sweeping velocities can be determined in a statistical way by repeating the analysis procedure at various wavenumbers. In contrast, the previously known method performs a fitting between the measurement and the spectrum model under a given set of free parameters (the shift velocity and the sweeping velocity) (Narita et al. 2013). In the spectral moment method, one can average the estimated shift and sweeping velocities over the wavenumbers.

The spectral moment method can be applied to various multi-point measurements in geophysical and space plasma measurements. The spectral moment method can also be used in the study of dispersion relation from the spectral measurements. The frequency estimate from the first-order moments can be compared with the theoretical branches of dispersion relation such a sound waves or Alfvén waves. The frequency estimate from the second-order moments can be used as an error or an uncertainty measure of the dispersion relation study. The dispersion analysis in a mean flow, however, requires the knowledge of the flow velocity in order to correct for the Doppler shift.

## Declarations

### Acknowledgements

This work is financially supported by the Austrian Space Applications Programme at the Austrian Research Promotion Agency, FFG ASAP-12, under contract 853994.

### Competing interests

The author declares that he has no competing interests.

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## Authors’ Affiliations

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