Doppler shift and broadening in solar wind turbulence
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2013
Received: 31 October 2012
Accepted: 13 December 2012
Published: 31 January 2013
We propose a method to evaluate the Doppler shift and broadening using magnetic field data in solar wind turbulence. A model of the energy spectrum is constructed in the wavenumber-frequency domain in the direction of the mean flow, and it is compared with that determined directly from four-point magnetic field data of Cluster spacecraft using the method of least square fitting. The model assumes the frequencies and wavelengths are low enough, and long enough, to be in the inertial range of turbulence. The comparison gives the result that the Doppler shift is consistent with that expected from the measurement of ion bulk speed, whereas the Doppler broadening is not small but far larger than the root-mean-square of ion bulk speed fluctuation possibly due to large-scale counter-propagating waves. Measurement of the Doppler shift can be used as an independent method to determine the mean flow speed, and that of Doppler broadening as a method to evaluate Taylor’s frozen-in flow hypothesis. The finite Doppler broadening also implies that Taylor’s hypothesis is invalid even at low frequencies in the solar wind.
Since the 1960s, the solar wind has repeatedly been interpreted to be in a fully-developed turbulent state (Coleman, 1968; Matthaeus and Goldstein, 1982; Marsch and Tu, 1990; Podesta et al., 2007), as the energy spectrum in the frequency domain often exhibits a power-law with an index close to −5/3, the slope of the inertial-range spectrum for fluid turbulence (Kolmogorov, 1941). In addition, the solar wind is a supersonic and super-Alfvénic flow with the respective Mach numbers being typically 8 to 10. For these reasons, the frozen-in flow hypothesis proposed by Taylor (1938), interpreting the measured frequencies as pure Doppler shift due to the mean flow, is thought to be a valid assumption in the study of solar wind turbulence.
Taylor’s hypothesis is essentially a unique mapping from the measured frequencies into the streamwise wavenumbers, but the quality of mapping diminishes if Doppler broadening associated with large-scale random fluctuations of the flow is present in the measurement. Therefore, it is important to study how large the Doppler broadening is in the solar wind with respect to the Doppler shift, and by comparing the two effects, it is possible to validate Taylor’s hypothesis. The impact of Doppler broadening on Taylor’s hypothesis was discovered in studies of fluid turbulence by generalizing Taylor’s hypothesis for the two-time, two-point correlation function (He and Zhang, 2006) and for the energy spectrum in the wavenumber-frequency domain (Wilczek and Narita, 2012).
Here, we propose an analysis method to evaluate the Doppler shift and broadening from magnetic field measurements in the solar wind and to validate the use of Taylor’s hypothesis. Such an analysis requires both the construction of an energy spectrum model and the development of a high-resolution spectral analysis technique for multi-spacecraft data (Narita, 2012), and it has not been possible to perform the analysis before. The model used here assumes that the frequencies and wavelengths are low enough, and long enough, to be in the inertial range of solar wind turbulence. Magnetic field data from four Cluster spacecraft (Escoubet et al., 2001; Balogh et al., 2001) in the solar wind provide an excellent opportunity for this study. The presented method is useful not only in the validation of Taylor’s hypothesis (scientific use) but also in estimating the flow speed in the solar wind only from magnetic field measurements (experimental use).
2. Doppler Effect in the Energy Spectrum
Equation (1) is a model claiming that the inertial-range spectrum is constructed on the assumption of a Gaussian character of the plasma velocity fluctuation coupled to Kolmogorov’s −5/3 law on the wavenumber dependence. The model exhibits several interesting properties. First, it is analytically integrable both over frequencies and wavenumbers. The one-dimensional (1D) spectra are obtained as a power-law with the same spectral index, −5/3, in both the wavenumber and the frequency domain, and, furthermore, this property holds independently from the choice of U and V. Second, Eq. (1) gives a more exact meaning of Taylor’s hypothesis: it is recovered by taking the limit V → 0 in Eq. (1) such that the fluctuation energy is constrained to the Doppler relation as the spectrum is characterized by the Dirac delta function: E(k,ω) ∝ δ(ω −kU). By doing so, a mapping is valid from the frequencies ω into the streamwise wavenumbers by the Doppler shift kU. A finite Doppler broadening makes the mapping inaccurate and Taylor’s hypothesis invalid.
There is no direct observation or confirmation of the Gaussian frequency distribution in space or laboratory plasma turbulence, but indirect evidence is the fact that Eq. (1) predicts the same power-law index for both wavenumber and frequency spectra. The 1D frequency spectrum for solar wind turbulence is known to be well fitted by the power-law with the index −5/3, and this observational fact motivates the use of our spectral model. We assume no time evolution for the small-scale turbulent field itself. Matthaeus et al. (2010) argue, based on Eulerian time correlation, that Taylor’s hypothesis begins to be inaccurate beyond above 3 hours, which gives a constraint in our analysis method concerning the time-length of the measurement.
3. Application to Solar Wind Turbulence
The wavenumber-frequency spectrum, is determined from the measurement of four-point magnetic field at Cluster, and the spectrum and is used to estimate the parameters U and V by comparison with the spectrum model. In this way, we construct the best-fitted spectrum model for the analyzed time interval. Transformation of the time series data into the frequency domain is performed by a Fast Fourier Transform, and that from the spatial coordinate into wavevectors is performed by the Multi-point Signal Resonator (MSR) technique developed particularly for the analysis of four-point magnetic field data from Cluster (Narita et al., 2011). MSR enables direct determination of the energy spectrum as done by Narita et al. (2010) and Sahraoui et al. (2010). In contrast to their earlier methods, MSR is a very-high-resolution adaptive filter technique, and is constructed as a fusion of the minimum variance projector (the wave telescope technique, see Motschmann et al., 1996; Glassmeier et al., 2001) with the MUSIC algorithm (Multiple Signal Classification) developed by Schmidt (1986) based on the analysis of eigenvalues and eigenvectors of the measured covariance matrix. We determine the four-dimensional (4D) energy spectrum first, and then obtain the two-dimensional (2D) spectrum Eobs(k, ω) in the flow direction by averaging the 4D spectrum over wavenumbers perpendicular to the flow. The ion data are used for the orientation of wavenumber.
3.2 Least square fitting
4. Summary and Discussion
Multi-point measurements of the magnetic field are a useful tool to investigate plasma turbulence properties in the interplanetary space, as they have the potential to determine the mean flow velocity as well as the random sweeping velocity by comparing the measured wavenumber-frequency spectrum with the theoretical model in Eq. (1). We obtain from the fitting procedure that the mean flow velocity (characterized by the parameter U) is in a good agreement with the directly-measured ion bulk velocity 〈u〉. The capability of evaluation of the mean flow velocity using multipoint magnetic field data should be emphasized here. This method is valid as long as the observed plasma is in a fully-developed turbulent state such as in the solar wind, and provides an independent method of flow velocity measurement other than direct ones. For future multi-spacecraft missions, the presented method can be used as an additional means of calibration for plasma experiments on board. We also find that the Doppler broadening is not small but far larger than expected from the ion measurement. The random sweeping velocity is of the same order as the mean flow velocity. In the analyzed time interval the ratio V/U is about 0.44. Taylor’s frozen-in flow hypothesis is therefore not a very valid assumption, at least in the analyzed time interval of solar wind turbulence. It should be interpreted in such a way that the frequency spectrum is Kolmogorov’s power-law, not because Taylor’s hypothesis is valid but because it is one of the properties of the inertial-range spectrum as modeled in Eq. (1).
The spectrum model used here is based on the hydrodynamic treatment of the inertial-range spectrum in the sense that it neglects the magnetohydrodynamic effects of such as large-scale Alfvén waves. In order to correctly obtain the Doppler shifted spectrum from that in the plasma frame, it is necessary to take into account the spectral information of both the wavenumber space and the frequency space. Namely, the Lorentz transformation should be performed for both the frequency and the wavenumber spaces. In other words, in the limit of U = 0, the model equation (1) should be written as a function of the wavenumber and the frequency that represents the dispersion relation for magnetohydrodynamics. If large-scale Alfvén waves were present with the propagation speed Vph = VA cos θ (where VA is the Alfvén velocity and θ is the propagation angle from the mean magnetic field), the spectral peak should split from a single Doppler relation line ω = kU into two branches: ω = k(U + Vph) and ω = k(U − Vph) due to the large-scale waves propagating forward and backward with respect to the flow. Therefore, we interpret our result such that a splitting might exist in the measured spectrum; splitting is so weak that it is recognized as a large frequency broadening. In this scenario, the sweeping velocity V is expected to be about the doubled Alfvén speed, and, in fact, the ratio V/VA is about 2.3 in the analyzed interval (the sweeping velocity at about 195 km/s from the fitting; Alfvén velocity at about 85 km/s, estimated from the magnetic field strength and the ion number density).
The spatial scale of the large-scale fluctuation, either eddies or Alfvén waves, cannot be estimated in the present analysis alone. Different approaches are, however, possible to estimate the spatial scale associated with the sweeping velocity: deriving information on large-scale structures from the frequency spectrum; extending the model spectrum to include the energy-containing range in the wavenumber domain as in von Kármán’s model; Cluster data analysis for large-scale tetrahedron phase; mapping coronal structure scale onto the Cluster orbit (nearly at 1 AU). The combination of the two different studies, one for analysis of random sweeping velocity and the other for analysis of energy-containing eddy scales, would give more complete information on the driving mechanism of turbulence in the interplanetary space.
Ion data from the Cluster CIS HIA instrument team and Cluster Active Archive are acknowledged. This work is financially supported by the Bundesministerium für Wirtschaft und Technologie and the Deutsches Zentrum für Luft-und Raumfahrt, Germany, under contract 50 OC 0901, and Collaborative Research Center 963, “Astrophysical Flow, Instabilities, and Turbulence” of the German Science Foundation.
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