Non Axi-symmetric Anisotropy of Solar Wind Turbulence
A.J. Turner, G. Gogoberidze, S.C. Chapman, B. Hnat, W.-C. Mueller
aa r X i v : . [ a s t r o - ph . S R ] A ug Non-axisymmetric Anisotropy of Solar Wind Turbulence
A.J. Turner, ∗ G. Gogoberidze,
1, 2
S.C. Chapman, B. Hnat, and W.-C. M¨uller Centre for Fusion, Space and Astrophysics; University of Warwick, Coventry, CV4 7AL, United Kingdom Institute of Theoretical Physics, Ilia State University, 3/5 Cholokashvili ave., 0162 Tbilisi, Georgia Max-Planck-Institut f¨ur Plasmaphysik, Boltzmannstr. 2, 85748 Garching bei M¨unchen, Germany
A key prediction of turbulence theories is frame-invariance, and in magnetohydrodynamic (MHD)turbulence, axisymmetry of fluctuations with respect to the background magnetic field. Paradoxi-cally the power in fluctuations in the turbulent solar wind are observed to be ordered with respectto the bulk macroscopic flow as well as the background magnetic field. Here, non-axisymmetryacross the inertial and dissipation ranges is quantified using in-situ observations from Cluster. Theobserved inertial range non-axisymmetry is reproduced by a ’fly through’ sampling of a DirectNumerical Simulation of MHD turbulence. Furthermore, ’fly through’ sampling of a linear superpo-sition of transverse waves with axisymmetric fluctuations generates the trend in non-axisymmetrywith power spectral exponent. The observed non-axisymmetric anisotropy may thus simply arise asa sampling effect related to Taylor’s hypothesis and is not related to the plasma dynamics itself.
PACS numbers: 94.05.Lk, 52.35.Ra, 95.30.Qd, 96.60.Vg
Solar wind fluctuations observed by satellites in-situexhibit power law scaling regions identified with an iner-tial range of magnetohydrodynamic (MHD) turbulence,and with a ’dissipation’ range below ion kinetic scales,providing a natural laboratory for plasma turbulence (fora recent review see, e.g., Ref. [1]). In hydrodynamic tur-bulence, any anisotropy in fluctuations at large scaleswill tend to isotropize as the cascade proceeds to smallerscales [2]. The situation is different in plasma turbulencewhere the existence of a mean magnetic field sets a nat-ural preferential direction for anisotropy. Anisotropy isthus a key topic in theoretical [3–5], numerical [6, 7], andobservational studies of plasma turbulence in the solarwind [8–13].The seminal study of Belcher and Davis [8] usedMariner 5 observations to investigate anisotropy of thesolar wind magnetic fluctuations in the low frequency(energy containing) and inertial intervals. They foundthat the fluctuations on average have 5 : 4 : 1 poweranisotropy in an orthogonal coordinate system whose axisare [ e B × e R , e B × ( e B × e R ) , e B ], where e B is a unit vec-tor in the average magnetic field direction and e R is aunit vector radially away from the sun. This conclusion,that solar wind fluctuations are non-axisymmetricallyanisotropic with respect to the magnetic field direction inthe low frequency and inertial intervals was confirmed bydifferent authors [14]. Recent results using k -filtering [10]were consistent with the results of [8] in that the mainpower was found in the plane perpendicular to the localmagnetic field distributed preferentially in the directionperpendicular to both the magnetic field and the solarwind velocity. Dissipation range magnetic fluctuationshave also been found to be non-axisymmetric [15] usingminimum variance analysis [16].The anisotropic expansion of the solar wind can intro-duce a preferred direction as captured by models [17, 18]and as observed on longer timescales (5-12 hrs, see [19]). However, from the perspective of turbulence, orderingof the observed non-axisymmetric power anisotropy withthe direction of the solar wind bulk flow velocity atthe inertial and dissipation scales is rather unexpected.If the macroscopic bulk flow speed is sufficiently largecompared to that of the fluctuations and that of thecharacteristic wave speeds of the plasma, then on thetimescales over which we observe turbulence in-situ thisbulk flow simply acts to advect the fluctuating plasma.This is Taylor’s hypothesis [20], and if it holds, thensince the observed properties of the evolving turbulenceare frame independent they should not correlate withthe macroscopic flow direction. The observation of non-axisymmetry [8, 10] with respect to the macroscopic flowdirection in the high speed solar wind flow is thus para-doxical. In plasma turbulence, one would anticipate ax-isymmetric anisotropy ordered with respect to the localmagnetic field. Indeed, theories of MHD and kineticrange turbulence assume axisymmetry of statistical char-acteristics [3–5, 21, 22] (note that although the modeldeveloped in Ref. [4] implies local non-axisymmetry ofturbulent eddies, it still assumes an axisymmetric en-ergy spectrum). As a consequence, studies of anisotropyof solar wind turbulence using single spacecraft obser-vations often assume axisymmetry (see, e.g., [9, 12, 23]and references therein). Understanding the origin of thisnon-axisymmetry is the subject of this Letter, and is es-sential if solar wind observations are to be employed inthe study of turbulence, in particular in the context ofdirect comparisons between theoretically predicted andobserved statistical properties and scaling exponents.Here, we show that the observed non-axisymmetricanisotropy can arise as a data sampling effect ratherthan as a physical property of the turbulence. Wefirst sample the output of a direct numerical simulation(DNS) of MHD turbulence with a ’fly through’ emulat-ing single spacecraft in-situ observations using Taylor’shypothesis. We will see that this is sufficient to re-produce the observed non-axisymmetry in the inertialrange of the solar wind. To understand how this non-axisymmetry can arise, we consider the simplest scenario-a ’fly through’ sampling of a fluctuating field composedof linearly superposed transverse waves with axisymmet-ric power anisotropy. The only free parameter in thismodel is the power spectral exponent of perpendicularfluctuations. This model reproduces the observed trend- that the non-axisymmetry increases with the perpen-dicular power spectral exponent as we move from theinertial to the dissipation range of scales. The observednon-axisymmetric anisotropy may thus simply arise as asampling effect related to Taylor’s hypothesis.We present the analysis of a sample interval [January20, 2007, 1200-1315 UT] of fast quiet solar wind ob-served by Cluster spacecraft 4 whilst the magnetic fieldinstruments FGM and STAFF-SC were in burst mode,providing a simultaneous observation across the inertialand dissipation ranges. FGM (sampled at 67 Hz) andSTAFF-SC data (sampled at 450 Hz) are combined bythe same procedure as in [23, 24], where a discrete wavelettransform is applied to both instrument data sets. Thismerging procedure generates one time series containingfrequencies ranging from the highest frequency of theSTAFF-SC data and the lowest frequency of the FGMdata. This interval is of fast solar wind with a flow speedof ∼
590 km / s with plasma parameters: average mag-netic field B ≃ β ≃ .
5, protondensity ρ p ≃ − , proton temperature T p ≃
29 eVand Alfv´en speed V A ≃
60 km / s.We use the continuous wavelet transform (CWT), asoutlined in [25, 26], to select fluctuations on a specificscale, τ . The fluctuations are resolved at each τ by aCWT performed on each component of the magnetic fielddata, B ( t j ), using the Morlet wavelet, to give a fluctu-ation vector, δ B ( t j , τ ). The vector fluctuation are thenprojected onto the local field. At each scale τ the localmagnetic field is defined for every time, t j , by the con-volution of a Gaussian window of width 2 τ centred on t j with the data, such that B ( t j , τ ) = [ B ( t j ) ∗ g ( t j , τ )]where g is the Gaussian window. The scale τ is relatedto frequency f , (in Hz) of the central frequency of theMorlet wavelet. This allows the local magnetic field andthe fluctuations to be rewritten as a functions of timeand frequency B ( t j , f ) and δ B ( t j , f ), respectively.We define the local system of unit vectors following [8].The unit vector in the direction of the local magneticfield is e z ( t j , f ) = B ( t j , f ) / (cid:12)(cid:12) B ( t j , f ) (cid:12)(cid:12) . The other twoperpendicular unit vectors are ordered with respect tothe macroscopic flow velocity direction, such that e x ( t j , f ) = e z × ˆ V (cid:12)(cid:12)(cid:12) e z × ˆ V (cid:12)(cid:12)(cid:12) , e y ( t j , f ) = e z × e x , (1)where ˆ V is the unit vector of the bulk flow velocity direc- tion time-averaged over the interval. During this intervalthe spacecraft is in a fast and steady stream of the solarwind, thus the local velocity is close to the time averagedmacroscopic velocity in (1).It can be shown that the results obtained are not sen-sitive to the small level of observed variations in the solarwind direction. The magnetic field fluctuations are thenprojected to the new basis (1) and the Power SpectralDensity (PSD) of the corresponding components are de-fined by: P SD z,x,y ( f ) = 2∆ N N X j =1 δB z,x,y ( t j , f ) (2)where N is the sample size at each frequency and ∆ issampling interval of the data.We plot the PSD of these components for this Clus-ter interval in Fig. 1. In the Figure, P SD z , P SD y and P SD x are (from bottom to top) represented by black,red and blue lines, respectively. The frequency rangecaptured by this interval of data covers both the dissi-pation range and the high frequency part of the inertialrange. The average ratio of the total perpendicular tothe parallel PSD in the inertial range does not vary sig-nificantly with frequency and is ∼
17 : 1. This is con-sistent with prior observations [14] that the power in theinertial range is predominately perpendicular to the lo-cal magnetic field, implying Alfv´enic fluctuations. In thedissipation range the power isotropises [27, 28]. The ra-tio of the PSD of the perpendicular components is shownin the inset of the Figure. The mean value of the ratio ofthe PSD of the perpendicular components in the inertialrange is
P SD y /P SD x ≃ .
8, in close agreement with [8].This ratio increases significantly in the dissipation rangeto
P SD y /P SD x ≃ .
45, coincident with a steepening ofthe PSD power spectral exponents. We will explore theorigin of these average values and trends. Ratios of thePSD can emphasize fluctuations in the PSD that are lo-calized in frequency, these can be observational artefactssuch as spacecraft spin tones, or can be of physical ori-gin such as power enhancements at the cross-over fromthe inertial to dissipation range. The PSD ratio can beseen to oscillate at the transition between the inertial anddissipation range due to these effects. We performed theabove PSD analysis described on a second Cluster inter-val; a fast stream (January 30, 2007, 0000-0100 UT) andobtained similar results.We now see how this non-axisymmetry can arise. Spa-tial snapshots of the three-dimensional velocity and mag-netic fields of developed incompressible MHD turbulenceprovide a model for the inertial range fluctuations, thesewere obtained from the DNS described in [6]. We willconsider two simulations: Case I is a globally isotropicfreely decaying turbulence with resolution 512 and CaseII corresponds to 1024 ×
256 forced turbulence simula-tion with strong background magnetic field B , such that b rms /B ∼ . V inthe components defined in (1). From each ’fly through’we can then calculate the time averaged ratio of thepower in the two perpendicular components in the samemanner as above for the solar wind data. These simu-lations offer two contrasting, and informative, numericalexperiments. In Case I the background field direction isfree to vary and a wide range of angles between the localbackground field and the pseudo-macroscopic flow canbe realized. For Case II the ’fly through’ is restricted tothe plane perpendicular to the applied background mag-netic field, to ensure that the local background magneticfield is nearly perpendicular to the pseudo-macroscopicflow direction- this will provide us with a simple test casein what follows. The results are presented in Figure 2.The blue dotted line corresponds to the solar wind datashown in the inset of Figure 1. The crosshatched andhatched areas indicate the range of values obtained froma number of simulation ’fly throughs’ for Case I and CaseII, respectively. Looking at the inertial range, we thensee a remarkable agreement between the average of thepower ratio in the inertial range in the solar wind, andCase I. Case II also shows non-axisymmetry but tends tooveremphasize the value as compared to that observedby Cluster in the solar wind.A full understanding of the origin of power non-axisymmetry in the solar wind should capture both thatit is seen in the inertial and dissipation range of scalesand that it is strongest in the dissipation range. We nowshow that these features can be qualitatively capturedby quite simple considerations. Consider a linear super-position of waves transverse to a constant backgroundmagnetic field B . For simplicity we assume V sw ⊥ B .To fix coordinates, the solar wind velocity V sw and B are directed along y and z , respectively so that the mag-netic fluctuations δ B associated with the waves are inthe x, y plane. If the Fourier amplitudes of the fluc-tuations are δ B ( k ) then under Taylor’s hypothesis theenergy densities E y ( ω ) and E x ( ω ) observed for the com-ponents parallel and perpendicular to V sw are given by(see also [29]) E y,x ( ω ) = 18 π Z d k | δ B y,x ( k ) | δ ( ω − k · V sw ) . (3)If we now assume that there exists some scaling relationbetween the parallel and perpendicular wave numbers(for example, critical balance [3]) and given δ B ( k ) ⊥ k ,after integration over k k we obtain E y ( ω ) = Z d k ⊥ E D ( k ⊥ ) δ ( ω − k y V sw ) sin α, (4) E x ( ω ) = Z d k ⊥ E D ( k ⊥ ) δ ( ω − k y V sw ) cos α, (5)where E D ( k ⊥ ) is two-dimensional spectrum of fluctua-tions [3] and α is the angle between k ⊥ and V sw . Ex-pressions (4) and (5) generally integrate to give E y ( ω ) = E x ( ω ). In particular, for axi-symmetric fluctuations E D ( k ⊥ ) = E D ( k ⊥ ) = Ck − γ − ⊥ , where γ correspondsto a one-dimensional spectrum, Eqs. (4-5) yield E y ( ω ) = C ω V sw Z (cid:18) ω V sw + k x (cid:19) − γ − d k x , (6) E x ( ω ) = C Z k x (cid:18) ω V sw + k x (cid:19) − γ − d k x . (7)This can be integrated in terms of Beta functions andafter straightforward manipulation (for γ >
0) we obtain E y ( ω ) /E x ( ω ) = 1 /γ. (8)Thus the ratio E y ( ω ) /E x ( ω ) is a decreasing functionof the spectral index in qualitative agreement withthe results presented above. The values of the ra-tio E y ( ω ) /E x ( ω ) for different values of γ are indicatedon Figure 2. Specifically, for γ = 1 . ± . E y ( ω ) /E x ( ω ) =0 . ± .
04 (gray area in the low frequency range), for γ = 5 / E y ( ω ) /E x ( ω ) = 0 . γ = 3 / E y ( ω ) /E x ( ω ) = 0 .
67 (red dotted line) and for γ = 2 . ± .
05 (dissipation range spectral index observedhere) E y ( ω ) /E x ( ω ) = 0 . ± .
02 (gray area in the highfrequency range).This result has a simple physical explanation. Whenthe satellite samples three-dimensional fluctuations asa one-dimensional series all the fluctuations with fixed k y contribute to a given frequency ω = k y V sw . Thismixture of different modes does not produce any non-axisymmetric anisotropy if energy density is equally dis-tributed among different scales ( γ = 1), but if this isnot the case, the resultant frequency spectra for theenergy density of different components will be non-axisymmetric. Indeed, consider the case when energydensity is decreasing with scale ( γ > f = k y V sw , since the waves are trans-verse, modes with k x < k y contribute more to E x ( ω )compared to E y ( ω ). For the modes with k x > k y thesituation is reversed, but because the modes in the latterrange have less power, one finally has E x ( ω ) > E y ( ω ).Results from the simple model can be seen from Fig-ure 2 to be consistent with the results from Case II ofthe DNS which share the same restricted geometry, thatis, the background field is perpendicular to the macro-scopic flow velocity. The model also gives the qualitativetrend in the anisotropy with the spectral index of the tur-bulence. On the other hand, the simple model predictsstronger power anisotropy than seen in the solar wind aswell as in the DNS without strong background magneticfield (Case I). This suggests that the overestimation innon-axisymmetry is indeed due to this restricted geom-etry of the simple model. To practically test this ideawould require knowledge of the distribution of the anglebetween V sw and B which, within any given interval ofsolar wind data is non-uniform, and which can vary fromone interval to another.The sampling effects described above have implicationsfor the results obtained by methods used for the studyof turbulence anisotropy that rely upon Taylor’s hypoth-esis. In addition, multispacecraft methods that do notuse Taylor’s hypothesis explicitly, such as k -filtering (see,e.g., [10] and references therein) still require filtering ofthe data in the spacecraft frame - selecting perturbationswithin some frequency interval [ − ω max , ω max ]. As the so-lar wind speed is much larger than the Alfv´en speed, it isclear that this procedure is essentially a filtering with re-spect to the wave vector component parallel to the solarwind velocity. As demonstrated here, this leads to pref-erential filtering of the modes with wave vectors mainlyparallel to the solar wind, and as a result could generatenon-axisymmetric anisotropy.Finally, we have for the restricted case where the back-ground field is perpendicular to the flow, obtained anexplicit relationship between the spectral exponent andthe power non-axisymmetry. For this special case, thenon-axisymmetry in power is quite sensitive to the valueof the exponent. It may be possible to develop this for-malism, at least numerically, to use the observed non-axisymmetry in power to determine the scaling exponentmore accurately. This is central to testing predictions ofturbulence theories.The authors acknowledge the Cluster instrumentteams for providing FGM and STAFF-SC data. Thiswork was supported by the UK STFC. ∗ Electronic address: [email protected][1] R. Bruno and V. Carbone, Living Rev. Sol. Phys. , 4(2005).[2] A.S. Monin and A.M. 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Horbury, Astrophys.J. , 76 (2011). −2 −1 −6 −2 −4 −4 Frequency (Hz) PS D ( n T H z − ) −2 −1 PS D Y / PS D X Figure 1:
P SD s of magnetic field components (from bottomto top) -
P SD z (black), P SD y (red) and P SD x (blue), where e z = B / (cid:12)(cid:12) B (cid:12)(cid:12) , e x = e z × ˆ V / (cid:12)(cid:12) e z × ˆ V (cid:12)(cid:12) and e y = e z × e x . Errorbars are always smaller than 4% and are usually smaller thanthe line width. In the insert - The ratio of perpendicular PSDs P SD y /P SD x . −2 −1 P o w e r R a t i o ( E Y / E X ) Figure 2: The ratio E y /E x , where e x and e y refer to a fieldaligned coordinate system as in Figure 1, for the solar winddata (blue dotted line), the range of values obtained from anumber of simulation ’fly throughs’ for Case I (crosshatchedarea) and Case II (hatched area), as well as predictions ofEq. (8) for the spectral indices γ = 5 / γ = 1 . γ = 1 . ± . γ = 2 . ± ..