Anisotropy of Electric Field Fluctuations Spectrum of Solar Wind Turbulence
MMNRAS , 1–7 (2020) Preprint 22 June 2020 Compiled using MNRAS L A TEX style file v3.0
Anisotropy of Electric Field Fluctuations Spectrum ofSolar Wind Turbulence
Deepali, (cid:63) Supratik Banerjee † Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh, India
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
To investigate the power and spectral index anisotropy in the inertial range of solarwind turbulence, we use 70 intervals of electric field data accumulated by Clusterspacecraft in the free solar wind. We compute the electric field fluctuation powerspectra using wavelet analysis technique and study its spectral index variation with thechange in angle between the heliocentric radial direction and the local mean magneticfield. We find clear power and spectral index anisotropy in the frequency range 0.01Hz ˆa ˘A¸S 0.1 Hz, with more power in parallel fluctuations than perpendicular. We alsoreport our study of anisotropy as a function of solar activity.
Key words: (Sun:) solar wind – (magnetohydrodynamics) MHD – turbulence –methods: data analysis
The solar wind (SW) provides a natural laboratory for thestudy of fully developed plasma turbulence through in-situmeasurements from spacecraft. Turbulence is believed to beone of the central characters responsible for the accelerationand heating of the SW. Despite being a weakly collisionalplasma, solar wind turbulence (SWT) can be studied in theframework of magnetohydrodynamics (MHD) for the lengthscales superior to the ion gyroscale and ion inertial scale( ∼ km). Similar to hydrodynamic turbulence, here toothe energy is injected at some large length scale and finallydecays due to viscosity at molecular diffusion length scale.Between those two scales, energy is neither added nor re-moved from the system but simply gets transferred to ad-jacent smaller length scales through local nonlinear interac-tions in k-space, often called the turbulent energy cascade.This intermediate range of scales, known as inertial zone,is usually characterized by a constant energy flux rate ( ε )from one scale to the subsequent one. In practice, the turbu-lent fluctuations of the plasma fluid and the electromagneticfield are directly obtained in the form of a time series. Sinceboth the spacecraft speed ( ∼ ∼
50 km/s) of the solar wind at 1 AU are much lower thanthe solar wind speed ( v ∼
600 km/s), one can use Taylor’shypothesis (reference of thesis) to map length scales ( (cid:96) ) intocorresponding time scales ( τ ) as (cid:96) = v τ or different frequen-cies ( f ) to equivalent wave numbers ( k ) as ω = v k (Taylor (cid:63) E-mail: [email protected] † E-mail: [email protected] − − − Hz , where both the magneticenergy and density power spectrum are found to scale as f − / (or ∼ k − / ), velocity power spectrum is found to fol-low a shallower f − / spectrum (Podesta et al. 2006). Tur-bulence power spectrum for higher frequencies are also in-vestigated systematically (Alexandrova et al. 2009; Sahraouiet al. 2009). It is only recently that the first measured powerspectrum of electric fluctuations has been reported (Baleet al. 2005). The spectrum is shown to have a spectral expo-nent of − / and follows the magnetic field fluctuation spec-trum until around 0.45 Hz (breakpoint) at the spacecraftmeasured frequencies. Beyond this breakpoint, the magneticspectrum becomes steeper whereas the electric spectra be-comes enhanced.All these power spectra have been obtained under theassumption of isotropy. In MHD, unlike velocity, Galileantransformation cannot eliminate the effect of mean magneticfield ( B ) which leads to unequal deformation of the Alfv´enicwave packets along and perpendicular to B . Due to a non-negligible B , anisotropy in SWT is normally expected inthe turbulence power level as well as in the spectral indices.However, in the studies conducted using a global mean mag-netic field (Tessein et al. 2009; Sari & Valley 1976; Chenet al. 2011), only power anisotropy and no spectral indexanisotropy were reported. This puzzle can be addressed us-ing several numerical studies of MHD turbulence showingthat the deformation of wave packets of a specific lengthscale will be regulated by the mean field of comparable © a r X i v : . [ phy s i c s . s p ace - ph ] J un Deepali, Supratik Banerjee length scale rather than that of the largest scale(Cho & Vish-niac 2000; Milano et al. 2001). For solar wind, it is found thatthe minimum variance direction of the inertial range fluctu-ations closely follows the scale dependent local mean mag-netic field direction (Horbury et al. 1995). Recently, usingwavelet method, prominent anisotropies both in the powerlevels and the spectral indices of magnetic power spectrawere observed for polar and ecliptic solar winds (Horburyet al. 2008; Podesta 2009). The parallel power is found tobe at least 5 times less than the perpendicular power andthe spectral index varies from around -2 to -5/3 as the anglebetween the local mean magnetic field and the flow direc-tion changes from ◦ to ◦ , thereby being consistent withthe critically balanced MHD turbulence theory by Goldre-ich & Sridhar (1995). Clear signatures of anisotropy for (i)magnetic power spectra at MHD and sub-ion scales and (ii)inertial range velocity spectra were also found for near eclip-tic solar wind using both single and multi spacecraft data(Chen et al. 2010, 2011; Wicks et al. 2011).Unlike velocity and magnetic field, the anisotropy ofelectric field fluctuations are studied occasionally. The first(and probably the only) study of power and spectral in-dex anisotropy in the electric field power spectra was car-ried out by projecting the field along and perpendicular tothe global mean magnetic field (Mozer & Chen 2013). Af-ter following stringent selection criteria, they could find onlythree suitable intervals from the THEMIS and Cluster data.From the analysis, it was found that the parallel power spec-trum magnitudes were comparable to or greater than theperpendicular powers. Both the spectra had similar shapeand inertial range power-law slopes of about -5/3. But theresults obtained could not be validated over more intervalsdue to lack of data. In this paper, following the similar tech-nique of (Horbury et al. 2008), we investigate the powerand spectral index anisotropy of electric field fluctuationsin the ordinary MHD scale. To measure the anisotropy, westudy how the spacecraft-frame spectrum of electric fluctu-ations varies with the angle between average flow directionand local mean magnetic field. Following the wavelet trans-form technique as implemented in Horbury et al. (2008) formagnetic field data. here we can decompose a time series ofelectric field data into wavelet coefficients those are local-ized in both time and frequency (or wavelet scale). In orderto capture the local mean magnetic field, we use Gaussianwindows of different standard deviations thereby giving ameasure of the corresponding length scale. In this paper,We also study the effect of solar activity on the anisotropyof electric field fluctuations.The paper is organized as follows. In § § § § Cluster is the first multi-spacecraft mission (2000, ESA) offour spacecraft flying in tetrahedral configuration. The si-multaneous in-situ measurements of these four spacecraft areused to investigate the small-scale structures and dynamicsin solar wind in three-dimensions. This makes Cluster, an excellent choice to study various physical processes occur-ring in the solar wind (Escoubet et al. 2001). To study theanisotropy in SWT and the effect of sun’s activity on thedistribution of energy in space around earth, we select thedata obtained from Cluster during the solar cycle 23, whichextends from 1996 to 2008, reaching its maximum in Novem-ber 2001. We use the data intervals when the spacecraft is inthe pristine fast solar wind in the ecliptic plane, outside theearth’s magnetic environment at a distance of 1 AU. In thisstudy, we analyze 70 such intervals after searching through5 years of data from January to May each year from 2001-04and 2007. The selected intervals are from parts of the orbitwhen the spacecrafts are at geocentric distances of between R E and R E . The time intervals studied here contain nodata gaps. The time series were visually ensured to be ap-proximately time-stationary and we tried to find the streamsof longest durations possible with sustained high speeds.We choose a few physical quantities for the study ofanisotropy of turbulent solar wind flow. Electric field mea-surements are made by the electric field and wave (EFW) ex-periment (Gustafsson et al. 1997) and magnetic field is mea-sured by fluxgate magnetometer (FGM) instrument (Baloghet al. 1997) on board the cluster spacecraft. We use the mag-netic field data with a resolution of 22 vectors per second andelectric field with a resolution of 25 vectors per second, mea-sured in geocentric solar ecliptic (GSE) coordinate system.Ion moments (velocity, density and temperature) of the solarwind are obtained from cluster ion spectrometry (CIS) ex-periment (Reme et al. 1997). The electric field and magneticfield data are taken from Cluster 4 (C4) and plasma data aretaken from Cluster 1 (C1) spacecraft. As mentioned above,we use 70 intervals for our study whose mean parameters areas follows : (i) Solar wind velocity - (− . , . , . ) kms − (ii) Ion Number Density - . cm − (iii) Ion perpendiculartemperature - . eV . In the table 1, we mention parametersfor two of the longest duration streams that we could find,one each from the period of sun’s maximum and minimumactivity. For the study of anisotropy in the period of so-lar maximum, we have used the data interval from the year2002 when spacecraft C1 and C4 both were at a distanceof . R e from the earth. For the investigation during thesun’s minimum activity, we choose the data interval from theyear 2007 when the spacecrafts were at a distance of R e from the earth, in the free solar wind. Rest all the intervalswere analyzed to test the statistical robustness of the resultsobtained. In the table’s last column, S.I. stands for spectralindex obtained from fast fourier transform (FFT) scheme. As discussed in the introduction, we can use Taylor’s hy-pothesis to study SWT near earth and hence the time se-ries data, measured by the spacecraft, correspond to a sim-ple one-dimensional spatial sample (Taylor 1938). In thisstudy, we measure the reduced spectrum which is defined as(Fredricks & Coroniti 1976): P( f ) = ∫ d k P ( k ) δ ( π f − k . V ) (1)Anisotropy of electric field fluctuation spectrum with respectto the mean magnetic field can be measured by studying how MNRAS , 1–7 (2020) hort title, max. 45 characters Table 1.
Parameter values for the streams analyzed near solar maximum and minimumYear From To Avg. solar wind velocity Ion number density Ion perpendicular Temperature S.I. (FFT) kms − ( n i ) cm − ( T i ⊥ ) eV f (Hz) E ( m V / m ) f Figure 1.
Power spectral density of electric fluctuations δ E y asa function of frequency computed from FFT algorithm. its spacecraft-frame power spectrum P( f ) varies with anglebetween the average flow direction and the local mean mag-netic field. In practice, the electric and magnetic fields aremeasured at different resolutions by the spacecraft. In orderto compute the power spectra, electric field data is subsam-pled onto the time tags of the magnetic field data by linearinterpolation. This allows both the field measurements tohave a cadence of 22 Hz. The power spectral density(PSD)plot of E y obtained using FFT algorithm for one of the in-tervals from 02 Feb, 2002 1400 hrs to 03 Feb, 2002 0006 hrsis shown in Fig. 1. The regular peaks in the high frequencyregion of the plot correspond to the spin tone data and itsharmonics (Bale et al. 2005). We find the spectral index forFig. 1 to be f − . when measured in the inertial range 0.01Hz to 0.1 Hz by performing linear least-squares fitting, con-sistent with the Kolmogorov value of − / . Similarly, thePSD plots were computed for all the intervals analyzed forthis study and only those with well-defined power law spec-tra were retained. Wavelet analysis is a tool for studying localized variationsof power within a time series. Decomposition of time seriesinto time-frequency space allows us to study both scales ofvariability and how those scales vary in time (Torrence &Compo 1998). We have time series for each component i ofelectric field, E i ( t k ) , where t k = t + k δ t , with equal time spac-ing δ t = / s and k = , , ... N − . The wavelet function, ψ ( η ) , used here is the morlet wavelet. It consists of a plane wave modulated by a Gaussian and is given as: ψ ( η ) = π − / e i ω η e − η / (2)where η is a nondimensional time parameter and ω is thenondimensional frequency taken as 6 for it to construct anearly orthonormal set of wavelets (Farge 1992). The con-tinuous wavelet transform of a discrete sequence E i ( t k ) isdefined as the convolution of E i ( t k ) with a scaled and trans-lated version of ψ ( η ) : w i ( t j , f l ) = N − (cid:213) k = E i ( t k ) ψ (cid:18) t k − t j s l (cid:19) (3)where t j is time and f l is frequency. To obtain the set offrequencies, one needs to choose a set of wavelet scales s de-pending on the wavelet function chosen. For Morlet waveletit is convenient to have: s m = s m δ m , m = , , ..., M (4)where smallest resolvable scale, s = δ t /( . ) . For ade-quate sampling of scales, we choose δ m as 0.5 and conductthe analysis for a total of 25 scales. For ω = , the waveletscale s l is related to the frequency f l as f l = . / s l fromwhich we get a set of 25 frequencies. Instead of computingthe wavelet coefficients from equation (3), we calculate themin fourier space using fast fourier transform (FFT) and con-volution theorem, which is considered to be faster. We cal-culate the wavelet coefficients at 25 frequencies ranging from2.7 mHz to 11 Hz (nyquist frequency for the data used).Wavelet coefficients w i are now used to compute powermagnitudes to form the wavelet power spectrum. Power incomponent i at time t j and f l is defined as: P ii ( t j , f l ) ∝ | w i ( t j , f l ) | . (5)For the vector field E (t), the trace power is the sum of thepower of the three orthogonal components, such that:Trace power, P = P xx + P yy + P zz . In this study, variation of power-law index of electric fluctu-ations spectrum (obtained using wavelet analysis) is studiedwith the angle between flow direction and local mean mag-netic field. We now calculate the scale dependent magneticfield at time t j and wavelet scale s l . To measure this, eachcomponent i of magnetic field time series is weighted with aGaussian curve centered at time t j and scaled with s l , givenas: ¯ b i ( t j , s l ) = N − (cid:213) k = B i ( t k ) exp (cid:34) − ( t k − t j ) s l (cid:35) . (6)This results in a time series of vectors ¯ b ( t j , s l ) for each scale s l pointing in the direction of local mean magnetic field. MNRAS000
Power spectral density of electric fluctuations δ E y asa function of frequency computed from FFT algorithm. its spacecraft-frame power spectrum P( f ) varies with anglebetween the average flow direction and the local mean mag-netic field. In practice, the electric and magnetic fields aremeasured at different resolutions by the spacecraft. In orderto compute the power spectra, electric field data is subsam-pled onto the time tags of the magnetic field data by linearinterpolation. This allows both the field measurements tohave a cadence of 22 Hz. The power spectral density(PSD)plot of E y obtained using FFT algorithm for one of the in-tervals from 02 Feb, 2002 1400 hrs to 03 Feb, 2002 0006 hrsis shown in Fig. 1. The regular peaks in the high frequencyregion of the plot correspond to the spin tone data and itsharmonics (Bale et al. 2005). We find the spectral index forFig. 1 to be f − . when measured in the inertial range 0.01Hz to 0.1 Hz by performing linear least-squares fitting, con-sistent with the Kolmogorov value of − / . Similarly, thePSD plots were computed for all the intervals analyzed forthis study and only those with well-defined power law spec-tra were retained. Wavelet analysis is a tool for studying localized variationsof power within a time series. Decomposition of time seriesinto time-frequency space allows us to study both scales ofvariability and how those scales vary in time (Torrence &Compo 1998). We have time series for each component i ofelectric field, E i ( t k ) , where t k = t + k δ t , with equal time spac-ing δ t = / s and k = , , ... N − . The wavelet function, ψ ( η ) , used here is the morlet wavelet. It consists of a plane wave modulated by a Gaussian and is given as: ψ ( η ) = π − / e i ω η e − η / (2)where η is a nondimensional time parameter and ω is thenondimensional frequency taken as 6 for it to construct anearly orthonormal set of wavelets (Farge 1992). The con-tinuous wavelet transform of a discrete sequence E i ( t k ) isdefined as the convolution of E i ( t k ) with a scaled and trans-lated version of ψ ( η ) : w i ( t j , f l ) = N − (cid:213) k = E i ( t k ) ψ (cid:18) t k − t j s l (cid:19) (3)where t j is time and f l is frequency. To obtain the set offrequencies, one needs to choose a set of wavelet scales s de-pending on the wavelet function chosen. For Morlet waveletit is convenient to have: s m = s m δ m , m = , , ..., M (4)where smallest resolvable scale, s = δ t /( . ) . For ade-quate sampling of scales, we choose δ m as 0.5 and conductthe analysis for a total of 25 scales. For ω = , the waveletscale s l is related to the frequency f l as f l = . / s l fromwhich we get a set of 25 frequencies. Instead of computingthe wavelet coefficients from equation (3), we calculate themin fourier space using fast fourier transform (FFT) and con-volution theorem, which is considered to be faster. We cal-culate the wavelet coefficients at 25 frequencies ranging from2.7 mHz to 11 Hz (nyquist frequency for the data used).Wavelet coefficients w i are now used to compute powermagnitudes to form the wavelet power spectrum. Power incomponent i at time t j and f l is defined as: P ii ( t j , f l ) ∝ | w i ( t j , f l ) | . (5)For the vector field E (t), the trace power is the sum of thepower of the three orthogonal components, such that:Trace power, P = P xx + P yy + P zz . In this study, variation of power-law index of electric fluctu-ations spectrum (obtained using wavelet analysis) is studiedwith the angle between flow direction and local mean mag-netic field. We now calculate the scale dependent magneticfield at time t j and wavelet scale s l . To measure this, eachcomponent i of magnetic field time series is weighted with aGaussian curve centered at time t j and scaled with s l , givenas: ¯ b i ( t j , s l ) = N − (cid:213) k = B i ( t k ) exp (cid:34) − ( t k − t j ) s l (cid:35) . (6)This results in a time series of vectors ¯ b ( t j , s l ) for each scale s l pointing in the direction of local mean magnetic field. MNRAS000 , 1–7 (2020)
Deepali, Supratik Banerjee
It provides the time- and frequency- localized mean fielddirection. We now find the inclination ( θ vb ) of this mean fieldwith respect to the average flow direction V sw (samplingdirection) of solar wind (sw) using: cos ( θ vb ) = V sw · ¯ b | V sw | | ¯ b | . (7)For the inertial range turbulence that is axisymmetric aboutthe magnetic field direction, the power spectra are indepen-dent of the azimuthal angle φ and hence we consider thepower values averaged over all φ (Horbury et al. 2008). Af-ter obtaining the directions of the local mean magnetic field,we now bin the square amplitude of the wavelet coefficientsat a given scale according to the scale-dependent field to findthe power distribution in different directions.In this study, we obtain different angle sets for differ-ent intervals, such as ◦ to ◦ , for the distribution of bincounts shown in Fig. 2(a) and ◦ to ◦ in Fig. 2(b). We con-struct bins of equal spacing of ◦ , say, for the 2002 intervalwe get 16 bins. The distributions of bin counts was qual-itatively and quantitatively similar at all frequencies from2.7 mHz to 11 Hz. For each bin and for a given frequency f l , we average the trace power values corresponding to allthe angles that lie in the bin. By doing this, we obtain theaverage trace power value P( f l , θ vb ) corresponding to thefrequency f l for that particular angle bin. As a result, weobtain the electric power spectrum when field is pointingin that particular direction. The number of times the fieldpoints in a particular direction gives us the measurements ofthe bin. Many bins have thousands of such measurements.A reasonable statistical sample is the one where we havesufficient number of measurements, so we reject those anglebins which have less than 200 contributing power levels. Here we present the results of our study on anisotropy ofelectric fluctuations spectrum with respect to the local meanmagnetic field direction. We analyzed 30 data intervals from2001, 2002 and 2004 to achieve statistical robustness of ourresults. Fig. 3 is obtained by conducting analysis on thedata from one of the 30 intervals, that is, from 1400 hrsof February 2, 2002 to 0006 hrs of February 3, 2002. Wechoose to present this interval as it is found to be the onlyone among the 30 which has the angle spread from ◦ to ◦ covering both parallel and perpendicular directions tothe local mean magnetic field. For most of the selected in-tervals the spread was found to be between ◦ and ◦ .The figure shows the trace power spectral density P( f l , θ vb ) plotted with frequency f l for two angle bins, ◦ to ◦ (blue) and ◦ to ◦ (orange). The power values havebeen computed using Morlet Wavelet scheme for a total of25 different frequencies ranging from 2.7 mHz to 11 Hz.The power law exponents found are -1.49 and -1.64 forperpendicular and parallel direction respectively, calculatedby performing linear least-squares fitting in the same fre-quency range as used for the calculation from fourier spec-trum of the electric fluctuations for the same interval inFig. 1, that is, 0.01 Hz to 0.1 Hz. The wavelet electric powerspectrum in the direction perpendicular ( ◦ - ◦ ) to the local mean magnetic field is observed to have less powerthan that in the parallel direction ( ◦ − ◦ ) in the iner-tial range but similar values otherwise. The outliers in thespectra are probably because of the inadequate count in theangle bin at those scales.In Fig. 4, we present the spectral indices plotted withrespect to θ vb fitted over the inertial range 0.01 Hz to 0.1Hz. The power law exponents have been averaged over the30 intervals and plotted for outward and inward sectors sep-arately in Fig. 4(a) and (b), respectively. It illustrates thedistribution of spectral exponents with the increase in an-gle between the flow direction and the local mean magneticfield. We observe that the power law indices are confined inthe range − . ± . for both the sectors and found to beapproximately the mirror image of each other. In this section, we try to find the dependence of anisotropyon solar activity, if any. We aim to observe if there is anyvariation in anisotropy during solar maximum or minimum.Solar cycle 23 lasted from 1996 to 2008 allowing the clus-ter spacecraft to make measurements during both the solarmaximum and solar minimum period. Sun’s activity peakedin late 2001. After searching through 4 years of Cluster datafrom the years 2001-03 (near solar maximum) and 2007 (nearsolar minimum), we found 60 intervals of pristine solar winddata, with 30 intervals from each maximum and minimumperiod.For each interval, we computed the electic field fluc-tuations spectrum using wavelet analysis for all the pos-sible angle ranges for the particular interval and measuredtheir spectral indices between the spacecraft frame frequencyranging from 0.01 Hz to 0.1 Hz. We then plotted the his-tograms to observe the distribution of power-law exponentscalculated for parallel ( ◦ - ◦ ) and perpendicular ( ◦ to ◦ ) directions with respect to the local mean magnetic fieldduring the solar maximum and minimum period. As statedabove, we could find very few intervals with angle rangespreading from ◦ to ◦ upon searching through 5 years ofCluster data. So, for parallel and perpendicular directions,few angle bins were clubbed so as to obtain sufficient numberof bin counts from 30 intervals each during sun’s maximumand minimum activity. From the histograms (see fig: 5), itis clearly found that the spectral indices for near perpendic-ular electric power stays approximately − . irrespective ofthe solar activity whereas for small angle ( ◦ − ◦ ) powerspectra, the index is having a mode near − . during solaractivity and − . during solar calm. In this paper, we investigate the spectral index anisotropyof solar wind turbulence in the inertial range with respectto the local mean magnetic field. We present observationalresults from wavelet analysis of Cluster electric field fluctu-ations. Wavelet analysis enabled us to calculate the scale-dependent field which in turn allowed us to study scale-dependent anisotropy. The frequency power spectrums ofelectric field fluctuations were calculated as a function of
MNRAS , 1–7 (2020) hort title, max. 45 characters Figure 2.
Histogram of number of bin counts for each angle bin computed at the frequency 0.04 Hz for ( a ) the data interval fromFebruary 2, 2002 during sun’s maximum activity (in blue) and ( b ) the data interval from January 17, 2007 during solar minimum (inorange). Figure 3.
Electric power spectra at two different angle rangesof the local mean magnetic field to the flow direction: ◦ - ◦ (blue) and ◦ - ◦ (orange) for the data interval 02 Feb, 200214:00 to 03 Feb, 2002 00:06. angles between the heliocentric radial direction and the lo-cal mean magnetic field. Our study clearly shows that elec-tric power undergoes prominent power and spectral indexanisotropies within the frequency range − − − Hz. Incontrast with the magnetic power spectra, parallel electricpower slightly overcomes the perpendicular power. Duringthe maximum solar activity period, the perpendicular powerspectrum is less steeper (-1.5) than the parallel one (-1.7)which is opposite to the case during minimum solar activitywhere the parallel spectra is slightly less steeper (-1.4) thanthe perpendicular power spectra (-1.5).
ACKNOWLEDGEMENTS
The data used for this study are available at the Cluster Sci-ence Archive ( https://csa.esac.esa.int/csa/aio/html/home_main.shtml ). The authors acknowledge the Clusterteams for producing the data. This research has been sup-ported by the grant DST/PHY/2017514.
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Deepali, Supratik Banerjee
Figure 4.
Power-law exponents of the angle-dependent trace power spectrums vs θ vb fitted over 0.01 Hz to 0.1 Hz for (a) outward and(b) inward sectors Figure 5.
Distribution of spectral indices in parallel ( ◦ - ◦ ) and perpendicular directions ( ◦ - ◦ ) during both minimum and maximumactivity of sun.Reme H., et al., 1997, Space Sci. Rev., 79, 303Sahraoui F., Goldstein M. L., Robert P., Khotyaintsev Y. V.,2009, Phys. Rev. Lett., 102, 231102Sari J. W., Valley G. C., 1976, J. Geophys. Res., 81, 5489Taylor G. I., 1938, Proceedings of the Royal Society of LondonSeries A, 164, 476Tessein J. A., Smith C. W., MacBride B. T., Matthaeus W. H., Forman M. A., Borovsky J. E., 2009, ApJ, 692, 684Torrence C., Compo G. P., 1998, Bulletin of the American Mete-orological Society, 79, 61Wicks R. T., Horbury T. S., Chen C. H. K., Schekochihin A. A.,2011, Phys. Rev. Lett., 106, 045001MNRAS , 1–7 (2020) hort title, max. 45 characters This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS000