Radial evolution of the wave-vector anisotropy of solar wind turbulence between 0.3 and 1 AU
Jiansen He, Chuanyi Tu, Eckart Marsch, Sofiane Bourouaine, Zhongtian Pei
aa r X i v : . [ a s t r o - ph . S R ] F e b Radial evolution of the wave-vector anisotropy of solar wind turbulencebetween 0.3 and 1 AU
Jiansen He , , Chuanyi Tu , Eckart Marsch , Sofiane Bourouaine , Zhongtian Pei ABSTRACT
We present observations of the power spectral anisotropy in wave-vector space ofsolar wind turbulence, and study how it evolves in interplanetary space with increas-ing heliocentric distance. For this purpose we use magnetic field measurements madeby the Helios-2 spacecraft at three positions between 0.29 and 0.9 AU. To derive thepower spectral density (PSD) in ( k k , k ⊥ )-space based on single-satellite measurementsis a challenging task not yet accomplished previously. Here we derive the spectrumPSD (k k , k ⊥ ) from the spatial correlation function CF (r k , r ⊥ ) by a transformationaccording to the projection-slice theorem. We find the so constructed PSDs to be dis-tributed in k-space mainly along a ridge that is more inclined toward the k ⊥ than k k axis, a new result which probably indicates preferential cascading of turbulent energyalong the k ⊥ direction. Furthermore, this ridge of the distribution is found to graduallyget closer to the k ⊥ axis, as the outer scale length of the turbulence becomes largerwhile the solar wind flows further away from the Sun. In the vicinity of the k k axis,there appears a minor spectral component that probably corresponds to quasi-parallelAlfv´enic fluctuations. Their relative contribution to the total spectral density tends todecrease with radial distance. These findings suggest that solar wind turbulence un-dergoes an anisotropic cascade transporting most of its magnetic energy towards largerk ⊥ , and that the anisotropy in the inertial range is radially developing further at scalesthat are relatively far from the ever increasing outer scale. Subject headings: solar wind — turbulence — anisotropy
1. Introduction
Solar wind fluctuations are considered as the genuine and prominent example of magnetohy-drodynamic (MHD) turbulence (e.g., Tu & Marsch 1995; Goldstein et al. 1995; Bruno & Carbone Department of Geophysics, Peking University, Beijing, 100871, China; E-mail: [email protected] State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing 100190 Institute for Experimental and Applied Physics, Christian Albrechts University at Kiel, 24118 Kiel, Germany Space Science Center and Department of Physics, University of New Hampshire, Durham, NH 03824, USA > . × − Hz in the spacecraft frame) range decays with radialdistance (like r − . ) faster than that in the lower-frequency (f < . × − ) range (with radialscaling like r − . ).The radial evolution of the lower-frequency magnetic power spectra can be reproduced by theWKB-theory of Alfv´en wave propagation (Whang 1973; Hollweg 1974), which predicts a similar ra-dial evolution. Whereas the higher-frequency magnetic power spectra, which show a steeper profile(Kolmogrov-like) with its spectral break frequency shifting towards lower values during the radialevolution, were successfully reproduced by Tu’s turbulence model (Tu et al. 1984; Tu 1988), whichtook into account (together with the WKB description) the nonlinear interaction between counter-propagating imbalanced Alfv´en waves. Moreover, the normalized cross-helicity (Alfv´enicity) wasshown to decrease with increasing heliocentric distance (Roberts et al. 1987; Marsch & Tu 1990;Grappin et al. 1990), which to explain was beyond the scope of Tu’s model. To self-consistentlydescribe the radial evolution of turbulent energy, cross-helicity, and Alfv´en ratio, substantial theo-retical efforts had to be made, which finally resulted in general transport equations (Marsch & Tu1989; Tu & Marsch 1990; Zhou & Matthaeus 1989) for the related spectra.In numerical simulations of MHD turbulence, the assumed background magnetic field ( B )was found to cause spatial anisotropy of the turbulent fluctuations along and across the mean field,with the parallel scale generally being larger than the perpendicular scale (e.g., Shebalin et al. 1983;Biskamp & M¨uller 2000; Cho et al. 2002). For balanced strong MHD turbulence with vanishingcross-helicity, the anisotropy is predicted to reveal a scaling relation obeying k k ∼ k / ⊥ , which wasderived in a phenomenological theory (Goldreich & Sridhar 1995) based on the conjecture of criticalbalance, i.e. the rough equality between the linear wave-propagation time and nonlinear eddy-interaction time. Numerical simulations further showed that balanced strong turbulence behaves forstrong or weak B differently in its scaling properties across B : Iroshnikov-Kraichnan scaling wasfound for strong B and Goldreich-Shridar scaling for weak B (M¨uller et al. 2003). These differentscalings are argued to be probably attributed to an increase of dynamic alignment as the cascadeproceeds to smaller scale, which may also induce scaling anisotropy in the plane perpendicular to B (Boldyrev 2005).However, in the solar wind and particular in fast streams one usually observes imbalancedturbulence with outgoing waves dominating over incoming waves. This imbalanced turbulenceimplies different nonlinear interaction time scales for the oppositely propagating waves, and ismore complex than the balanced one. Its physical nature remains a controversial issue, althoughseveral theories have been proposed (Lithwick et al. 2007; Beresnyak & Lazarian 2008; Chandran2008; Podesta & Bhattacharjee 2010). 3 –The spatial anisotropy of solar wind turbulence was studied by means of data analysis employ-ing various tools, such as correlation function (Matthaeus et al. 1990; Dasso et al. 2005; Osman & Horbury2007; N´emeth et al. 2010), structure function (Luo & Wu 2010; Chen et al. 2010, 2012), power ofmagnetic components (Bieber et al. 1996), and power law scaling (Horbury et al. 2008; Podesta2009; Wicks et al. 2010, 2011). Scaling anisotropy becomes more clearly visible if one uses a scale-dependent local mean magnetic field ( B , local ) (which was first introduced by Horbury et al. (2008)applying the wavelet technique) instead of a constant global mean magnetic field (Tessein et al.2009). Some efforts have been also made to reconstruct magnetic PSD in multi-dimensional wave-vector space by means of the k-filtering method (Sahraoui et al. 2010; Narita et al. 2010), whichwas developed originally to distinguish a limited number of plane waves from multi-position mea-surements (Pincon & Lefeuvre 1991). The integrated PSD (k ⊥ ) with a spectral index ∼ − . (k ⊥ ) as derived from thedirect wavelet transformation. However, in some cases studied with the k-filtering method, the inte-grated PSD (k k ) shows a spectral index ∼ − . (k k ) ( ∼ k − k ) as obtained by the wavelet method (Horbury et al. 2008). Thereliability of k-filtering method for estimating turbulent power spectra may need further validation,e.g. by applying it to numerically simulated turbulence with known scaling.Previous studies have revealed the evolution of reduced 1D-PSD in the inner heliosphere, andhave presented evidence of wave-vector anisotropy at specific positions [e.g., 1 AU]. However, theturbulence anisotropy pattern at 0.3 AU (the innermost distance reached in-situ so far) and itsevolution trend between 0.3 and 1.0 AU has not yet been investigated. To do this is an importanttask, because it will provide the needed information about the evolution of the energy cascadingroute in k-space, and reveal possible ways of turbulent energy dissipation required for sustainedsolar wind heating. This work is dedicated to a study of MHD turbulence anisotropy and willprovide new knowledge on its spectral characteristics. The data analysis to achieve these goals isbriefly described as follows.Firstly, we estimate the second-order structure function as a function of θ RB (the angle betweenthe radial direction and the local mean magnetic field vector). Accordingly, the angular distributionof the spatial correlation function is obtained, using the relation between structure function andcorrelation function. Secondly, we fit the measured structure function with a compound fit function,resembling a power-law dependence at short scale and giving an exponential trend at large scale.The fitted angular correlation function is subsequently derived. Thirdly, under the assumption ofa statistically time-stationary state, the relative 2D-PSD in (k k , k ⊥ ) space is constructed from thefitted angular distribution of the correlation function, whereby we make use of the projection-slicetheorem which is fundamental for image processing in medical tomography (see Bovik (2000) for adetailed review). 4 –
2. Analysis method
In this section, we describe the applied methods and the data analysis, which includes: howto derive angular distributions of the structure function SF( τ, θ VB ) and the correlation functionCF( τ, θ VB ); how to fit SF( τ, θ VB ) and CF( τ, θ VB ) appropriately; and how to obtain PSD (k k , k ⊥ )as transformed from CF( τ, θ VB ), which is in turn obtained from CF(r k , r ⊥ ) by assuming a quasi-steady state with r k ≃ V sw τ cos θ VB and r ⊥ ≃ V sw τ sin θ VB , corresponding to the Taylor assumptionof fluctuations being frozen into the flow, and thus being simply convected by the wind past thespacecraft. Here V sw is the solar wind speed.The second-order magnetic structure function is defined as the ensemble average of the squaredmagnetic field vector difference. It can be written asSF( τ ) = D ( B ( t + τ − B ( t − τ E , (1)where the angular bracket denotes in practice a time average in our subsequent data analysis.This time average permits one to quantify the global scaling of the magnetic fluctuations, withoutdistinguishing a possible scaling-law difference for different angles ( θ VB ) between the samplingdirection and the local mean magnetic field vector ( B , local ). The local mean magnetic field is knownto be changing in time and depend on scale ( B , local ( t, τ )), leading to a scale-dependent variationof the angle θ VB with time. For solar wind with a radial speed much larger than the velocityfluctuation amplitude, the quantity θ VB can be approximated by θ RB (i.e., the angle between theradial direction and the B , local direction), which is used hereafter. To estimate the structurefunction value at a certain time scale τ ′ and for a certain angle θ ′ RB , one needs to pick out thevalues of SF( t, τ ′ ) at those times when θ RB ( t, τ ′ ) = θ ′ RB , and then make an average over all the sopicked samples. Therefore, the corresponding angular distribution of the structure function can beexpressed as SF( τ ′ , θ ′ RB ) = R T ( B ( t + τ ′ ) − B ( t − τ ′ )) dt (cid:12)(cid:12)(cid:12) θ RB ( t,τ ′ )= θ ′ RB R T dt (cid:12)(cid:12)(cid:12) θ RB ( t,τ ′ )= θ ′ RB . (2)Here the time period for the whole chosen data set is indicated as T . It should be much larger thanthe time scale τ ′ , and thus we may formally take the limit T → ∞ .Expressing the ensemble average used in equation (1) explicitly as a time average, the relationbetween the structure function SF( τ ) and the correlation function CF( τ ) can be obtained from thesubsequent calculation:SF( τ ) = 1 T Z T ( B ( t + τ − B ( t − τ dt = 1 T (cid:20)Z T B ( t + τ dt + Z T B ( t − τ dt − Z T B ( t + τ · B ( t − τ dt (cid:21) = 2CF( τ = 0) − τ ) . (3) 5 –Using the above definition (2), the angular distribution of the correlation function can also beapproximated by the angular distribution of the structure function, yielding on the basis of (3) thefollowing relation:SF( τ, θ ′ ) = − D B ( t + τ · B ( t − τ E | θ RB = θ ′ + D B ( t + τ E | θ RB = θ ′ + D B ( t − τ E | θ RB = θ ′ ≃ − τ, θ ′ ) + 2 CF( τ = 0 , θ ′ ) ≃ − τ, θ ′ ) + 2 CF( τ = 0) , (4)where angular isotropy of CF at τ = 0 was assumed in the derivation. Under Taylor’s hypothesisthe solar wind fluctuations can be considered time stationary, as the wave phase speed is small incomparison to the supersonic convection speed, and then CF( τ, θ ′ ) can be rewritten as a spatialcorrelation function in the 2D r-space,CF( τ, θ ′ ) ∼ CF (r k , r ⊥ ) , (5)with r k = V sw τ cos θ ′ and r ⊥ = V sw τ sin θ ′ . This completes the derivation of the two-dimensionalcorrelation function from the structure function. We note that the frozen-in-flow Taylor’s hypothesismay be slightly weakened for smaller heliocentric distance with smaller Alfv´en Mach number, whichdrops from higher than 10 at 1 AU to 3-4 near 0.29 AU. The quantity of main interest is thepower spectral density PSD (k k , k ⊥ ), which in can in principle be obtained directly from Fouriertransformation of CF (r k , r ⊥ ) as follows:PSD (k k , k ⊥ ) = Z + ∞−∞ Z + ∞−∞ CF (r k , r ⊥ ) exp( − i(k k r k + k ⊥ r ⊥ ))dr k dr ⊥ . (6)However, we take here a new route to estimate PSD (k k , k ⊥ ). It can also be derived from theprojected (integrated) 1D correlation function on the basis of the projection-slice theorem (Bovik2000) with help of the following formula:PSD (k , θ k ) = Z + ∞−∞ Z + ∞−∞ CF (r k , r ⊥ ) exp( − i(k(r k cos θ k + r ⊥ sin θ k )))dr k dr ⊥ = Z + ∞−∞ Z + ∞−∞ CF (r ′ cos θ k − u ′ sin θ k , r ′ sin θ k + u ′ cos θ k ) exp( − i(kr ′ ))dr ′ du ′ = Z + ∞−∞ CF (r ′ ; θ k ) exp( − i(kr ′ ))dr ′ , (7)where θ k is the angle between k and B , local , and CF (r ′ ; θ k ) is the 1D projection (integration) ofCF (r k , r ⊥ ) along the direction normal to k ,CF (r ′ ; θ k ) = Z + ∞−∞ CF (r ′ cos θ k − u ′ sin θ k , r ′ sin θ k − u ′ cos θ k )du ′ . (8)Therefore, there are two approaches to calculate PSD (k k , k ⊥ ), one may adopt either Equa-tion 6 or 7. In practice, the estimation of the 2D correlation function with help of Equation 6 6 –introduces some uncertainty, as the noise involved in the data may destroy the required positiv-ity of the PSD in the entire ( k k , k ⊥ ) space. To guarantee this positivity of PSD everywhere, oneneeds to approximate the CF with some kind of positive-definite fit function before the Fouriertransformation. It is hard to find an adequate function that globally fits the observed CF (r k , r ⊥ )well, whereas it is relatively easy to choose a proper fitting function for the projected CF (r; θ k ).Therefore, in our work, we will use a fitted CF (r; θ k ) to reconstruct reliably the PSD (k k , k ⊥ )according to Equation 7.To provide the reader with an intuitive impression about the relations between CF (r k , r ⊥ ),CF (r , θ k ), and PSD (k k , k ⊥ ), we present the schematic illustration shown in the upper panelof Figure 1, which explains the two roads from CF to PSD (direct 2D Fourier transform andindirect method based on the projection-slice theorem). Similarly, one slice of CF at certainangle θ r is also the 1D inverse Fourier transform of PSD as projected from PSD onto thecorresponding direction k with θ kB = θ r , an approach which is displayed in the lower panel ofFigure 1. The relation between CF and PSD is the basic method for calculating CF , whichwas used in previous studies (Matthaeus et al. 1990; Dasso et al. 2005; Osman & Horbury 2007).In principle, it is also possible to derive PSD from PSD according to the method of inverseRadon transform (filtered back-projection) (private communication with M. Forman). However,this method fails in a typical benchmark test due to extreme large PSD at small | k | , which blursthe entire reconstructed PSD thereby destroying its original pattern.Speaking of the fitting function for the CF, we need to mention also the fitting function for theSF, which is used to reproduce the key features of the SF. For example, people usually adopt anexponential function to fit the profile of the SF at large scale, while they use a power-law functionfor the small-scale trend. However, as far as we know, there exists no attempt to describe boththe small-scale power-law trend and the large-scale exponential trend simultaneously with a singlefitting function. To fulfill this task, we suggest a compound function,SF( τ ) = 2R · [1 − exp( − ( ττ c ) p )] , (9)which interpolates between these limits. There are three parameters to be fitted: R means theauto-covariance at τ = 0, τ c represents the correlation time at large scale, and the index p describesthe power-law scaling at short scale. Generally, for SF( τ, θ RB ) at different θ RB , the parameters R and τ c do not change a lot, while p remains variable. Therefore, in our practice, R and τ c areobtained by fitting the time-averaged SF( τ ), and then p is determined at various θ RB by fittingSF( τ, θ RB ), but only after R and τ c were set. Another practical reason for presetting R and τ c before fitting SF( τ, θ RB ) is that for every θ RB the calculated SF( τ, θ RB ) is usually unable to reachto the outer scale. 7 –
3. Data analysis results
The magnetic data (with a time resolution of about 0.25 s) used here is from measurements byHelios-2 spacecraft at three radial positions (0.29, 0.65, and 0.87 AU) during three time intervals(day of year: 106-109, 76-78, and 49-51 in 1976). The solar wind streams explored during thesetime intervals are known to be recurrent streams emanating from a common source region on theSun (Bavassano et al. 1982). The corresponding radial evolution of 1D reduced magnetic PSD waspresented in that paper, which observationally promoted the development of the WKB-like solarwind turbulence model (Tu et al. 1984). Three decades later, we analyse the same data set again,but for the purpose of revealing the evolution of solar wind turbulence in terms of its wave-vectoranisotropy.We use Equation 2 to estimate the second-order structure function SF( τ ). It is defined asthe magnetic vector difference squared, which is averaged respectively over the three time intervalsof our data set. During the estimation, the data gaps are excluded without making any type ofinterpolation. The top three panels of Figure 2 illustrate the estimation results as red curves. Theblue lines are fitting results based on Equation 9, which basically match the estimates at bothsmall and large scales. The fitting parameters ( R [nT ], t c [s], p ) at three radial positions are foundto be: (827, 116, 0.61), (53, 465, 0.61), and (25, 857, 0.67), respectively. The fitting parameter t c (corresponding to the correlation time) increases with heliographic distance. The values of theexponent p relate to the power-law index ( ∼ − ( p + 1)) of the corresponding PSDs, which is foundto be around − .
6, i.e. near the Kolmogorov value of − /
3. The bottom three panels of Figure 2show the corresponding correlation function CF( τ ) as derived from Equation 3.We calculate the structure functions in the angular dimension as a function of θ RB accordingto Equation 2, and display them in the first row of Figure 3. Apparently, the distribution of SF isnot uniform in the angle range between 0 ◦ and 90 ◦ , with a lower level near 0 ◦ . The non-uniformangular distribution is more significant at short scales [e.g., <
100 s]. For SF( τ, θ RB ) at larger scales( τ >
100 s), it gradually changes from uniformity at 0.29 AU to non-uniformity at 0.87 AU. Thisangular non-uniformity is a feature hinting at anisotropy of the power spectrum in the wave-vectorspace. Likewise, the extension of the angular non-uniformity towards larger scales indicates thatthe wave-vector anisotropy of larger-scale fluctuations evolves as heliocentric distance increases.We also fit the estimated structure function by the function SF( τ, θ RB ) of Equation 9. To makesure the fitting process converges for every angle, we restrict the number of fitting parameters to p , while we fix the other two parameters ( R and t c ), both of which may be regarded as constantwithout angular dependence. The fitted angular distributions are illustrated in the second row ofFigure 3, which look similar to the observations. The angular dependence of the fit parameter p is plotted in the third row, showing that the angular variation of SF( τ, θ RB ) is non-uniform notonly in magnitude (first row in Figure 3) but also in the scaling index (third row). We note thatSF( τ, θ RB ) as shown in Figure 3 relates to the squared module of the magnetic-vector difference( δB x + δB y + δB z ). The structure function SF( τ, θ RB ) for the component δB k (parallel to B , local ) 8 –shows a similar non-uniform angular dependence. However, the calculated SF for δB k has a plainsegment starting at small τ , and cannot be fitted well by the function of Equation 9.The angular distribution of the correlation function CF( τ, θ RB ) is derived from the fit functionSF( τ, θ RB ) according to Equation 4. In the light of the projection-slice theorem as applied tothe relationship between the 2D functions CF and PSD (lower panel in Figure 1), the quantityCF( τ, θ RB ) is essentially a 2D correlation function CF(r k , r ⊥ ), which is in principle an inverse Fouriertransform of the 2D PSD(k k , k ⊥ ) yet not known. In Figure 4, we plot the resulting CF(r k , r ⊥ ).The coordinates of the abscissa ( r k ) and ordinate ( r ⊥ ) are estimated by r k = V sw τ cos θ RB and r ⊥ = V sw τ sin θ RB , respectively. The main part of CF(r k , r ⊥ ) is elongated along r k , which is similarto the “2D” population of the so-called Maltese cross (Matthaeus et al. 1990). However, the “slab”population, which was reported in previous statistical studies of CF with r k parallel to the directionof interval-averaged (non-local) magnetic field (Matthaeus et al. 1990; Dasso et al. 2005), is not soprominent in our cases.Ideally, the corresponding PSD(k k , k ⊥ ) can be gained directly from 2D Fourier transform ofCF (r k , r ⊥ ). However, in practice, the transformed value might be negative or not certainly posi-tive, thereby restraining the application of the direct 2D Fourier transform. To obtain PSD (k k , k ⊥ ),we then turn to Equation 7 for a step-by-step derivation. Firstly, by integrating CF (r k , r ⊥ ) overthe path normal to the direction with certain angle θ ′ with respect to r k , the reduced 1D CF (r)corresponding to the angle θ ′ is calculated. Secondly, the corresponding PSD as a Fourier trans-form of CF is calculated. To guarantee the positivity of the estimated PSD, CF is fitted beforetransformation with a function related to that for SF as previously described. According to theprojection-slice theorem, the estimated PSD profile is essentially a slice of PSD (k k , k ⊥ ) along k with θ ′ with respect to k k . Thirdly, PSD (k k , k ⊥ ) is formed by assembling various PSD profiles,with different angles ranging from 0 ◦ to 90 ◦ with respect to k k . We note that, in calculation, CF and CF one cannot let r go to infinity. As a result, the transformed PSD may slightly departfrom the real one. Therefore, in Figure 5, we just present the normalized PSD , n rather than theabsolute PSD . The uncertainty (confidence interval) for the estimated PSD is not providedhere, since due to the complexity of the estimation method that was not yet possible.Obviously, the normalized PSD , n shown in Figure 5 is not uniformly distributed at all angles,indicating an anisotropic wave-vector distribution. This anisotropy is mainly characterized by aridge distribution which has a bias towards k ⊥ as compared to k k . Moreover, as the heliographicdistance increases, the ridge distribution becomes more inclined toward k ⊥ at the same | k | , inassociation with lower PSD (darker blue in the figure) around the k k region and higher PSD(brighter blue in the figure) around the k ⊥ region. The discovery of this bent ridge and its radialevolution imply that solar wind turbulent energy cascades preferentially along the k ⊥ as comparedto the k k axis, and the turbulence cascade radially develops with more energy cascading to the k ⊥ region, as the scale (1 / | k | ) is shifting away from the radially-growing outer scale (1 / | k | ). In additionto the major ridge distribution, a minor population seems to exist close to k k (see Figure 5a), andappears to become weaker at farther distances (see Figures 5b,c). The observational fact that PSD 9 –is composed of two populations, with the major one bending more perpendicularly and the minorone becoming weaker, seems compatible with the previously suggested two-component turbulencemodel, which invokes non-damping convective structures (spatially varying across B ) that aresuperposed on damping Alfv´en waves (spatially varying along B ) (Tu & Marsch 1993).To emphasize the trend of the ridge distribution and its radial evolution, we estimate theridge position of every scale by averaging the angles with local lg(PSD) as the weights (i.e., first-order moment centroid method). The estimated ridge positions are shown as black dashed linesin Figure 5 and appear straight. Whether or not the straightness is realistic is yet unknown.Furthermore, we fit the estimated ridge position with following simple formula, k k = α · k / · k / ⊥ , (10)where k (= 2 π/ ( V sw · τ c )) is related to the outer-scale wave-number. α is the coefficient to be fitted,which is ∼ k k − k ⊥ -relation profileaccording to Equation 10 is basically coincident with the observed ridge distribution. However, somedepartures, e.g., the estimated black dashed line looks more straight than the fitted red line, stillremain. Nevertheless, the relation ( k k ∼ k / k / ⊥ ), as predicted by the critical-balance hypothesis(Goldreich & Sridhar 1995) for MHD turbulence, seems to describe well the observed anisotropy ofsolar wind turbulence. The role of k , which was once neglected in previous observational studies,in shaping the anisotropy shall be emphasized here. It may be the reduction in k which causesthe development of the spectral anisotropy (increasing inclination toward k ⊥ at the same | k | ) ininterplanetary space as heliographic distance increases.Solar wind heating mechanism may be inferred from the radial evolution of the ridge trend.According to linear Vlasov theory, Alfv´en waves with plasma β p ∈ [0 . , .
0] usually become dis-sipated due to proton cyclotron resonance when they have k k c/ω p ∈ [0 . , . c/ω p isthe proton inertial length (Gary & Nishimura 2004). On the other hand, Landau resonance be-comes more and more prominent as plasma β p rises (Gary & Nishimura 2004) and k ⊥ ρ g increases(Howes et al. 2006), where ρ g is the proton gyroradius. Howes (2011) pointed out that, Landaudamping calculated in the gyro-kinetic limit is not sufficient for the empirically estimated protonheating (Cranmer et al. 2009) at small heliocentric distances ( R < . k k = α · k / · k / ⊥ with α ∈ [3 , k k c/ω p > .
5) ahead of Landau resonance (marked by k ⊥ ρ g > . ≥ k , the ridge profilewould first exceed the threshold k ⊥ ρ g = 1 before approaching to k k c/ω p = 0 .
5, which implies a 10 –dominance of Landau damping over cyclotron damping at larger distances. According to critical-balance theory in MHD and kinetic regimes(Schekochihin et al. 2009), the extension of ridge in theMHD inertial range may be still related to Equation 10, while the extension part in the kinetic(dissipation) range may deviate from Equation 10 with more inclination towards k ⊥ . For sake ofsimplicity, we neglect such deviation of extended ridge in the kinetic range from that in the inertialrange.The value of coefficient α is also worth emphasizing here. If α were one third of the ap-proximated value (3 . / . k ⊥ ρ g = 1 without approaching to k k c/ω p = 0 .
5, leading to insufficient heating rate by Landau res-onance within 1 AU according to the gyro-kinetic prescription by Howes (2011). On the other side,if α were too large (saying 3 . × k ⊥ ρ g = 1, implying the absence of transition from cyclotron damping to Landau dampingaround 1 AU (inconsistent with the conclusion by Howes (2011)). Therefore, the α value ( ∈ [3 , k is another important parameter for grasping the essence of solar windheating mechanism. α may be expressed as the ratio of ε to V k with ε being the energy cascaderate if k k = ( ε/V ) / k / ⊥ , which is usually assumed in critical-balance theory (Goldreich & Sridhar1995; Schekochihin et al. 2009).
4. Summary and discussion
We have made the first successful attempt to reconstruct, on the basis of single spacecraftmeasurements, the 2D spectral density PSD (k k , k ⊥ ) for solar wind MHD turbulence. We esti-mate the angular distribution of the second-order structure function SF( τ, θ RB ), and derive thecorresponding correlation function CF (r k , r ⊥ ), which in principle is an inverse 2D Fourier trans-form of PSD (k k , k ⊥ ). The transformation from time scale τ to spatial scale r , when buildingup CF (r k , r ⊥ ), is based on Taylor’s hypothesis that solar wind fluctuations are quasi-stationarywithin the flow transit time scale, as the solar wind passes by the spacecraft. The 2D direct Fouriertransform of CF (r k , r ⊥ ) fails to guarantee the required positivity of PSD (k k , k ⊥ ). Alterna-tively, we employ for the first time a method based on the projection-slice theorem, which connectsthe integrated CF (r , θ ′ ) with the corresponding slice PSD (k , θ ′ ) of the PSD via a 1D Fouriertransform, to fulfill that task. Before the 1D Fourier transformation, CF (r , θ ′ ) is fitted smoothlyto guarantee the positivity of the transformed PSD (k , θ ′ ).As a result, SF( τ, θ RB ) shows a non-uniform angular distribution with more power being locatedin the perpendicular region ( θ RB ∼ ◦ ) than in the parallel region ( θ RB ∼ ◦ ) of wave-vector space.Moreover, there is angular dependence of the scaling law for SF( τ, θ RB ) at short scales, whereby thescaling index p drops from ∼ . θ RB = 0 ◦ to ∼ . θ RB = 90 ◦ . We find that SF( τ, θ RB ) haveat all three positions (0.29, 0.65, and 0.87 AU) the two above properties, indicating the prevalenceof anisotropy in the turbulence throughout the inner heliosphere. This result obtained within 1 AU 11 –is similar to that found for the SF anisotropy beyond 1 AU (Luo & Wu 2010). The correspondingcorrelation functions CF (r k , r ⊥ ) clearly show that magnetic fluctuations are correlated at longer(shorter) length along (across) the background magnetic field.The corresponding PSD (k k , k ⊥ ) at the positions within 1 AU is revealed to have a ridge dis-tribution with a bias towards k ⊥ as compared to k k , suggesting a preferential cascading along k ⊥ .This kind of ridge distribution has never been reported in previous studies at 1 AU, e.g. those basedon the wave-telescope (k-filtering) method (Narita et al. 2010; Sahraoui et al. 2010). Furthermore,this ridge distribution is found to become ever more inclined toward the k ⊥ axis with increasingheliographic distance, thus indicating a radial development of the wave-vector anisotropy. The ob-served radial evolution of the ridge casts new light on the scaling relation between k k and k ⊥ , whichmay empirically be approximated by k k ≃ αk / k / ⊥ , with α ∈ [3 ,
4] and k being the wave-numberof the outer scale. This approximation for the wave-vector anisotropy seems to indicate critical-balance-type cascading (Goldreich & Sridhar 1995) of solar wind turbulence. A possible influenceof k on the anisotropy development, which was neglected in previous observational analyses, isalso found.However, the evolution of the ridge distribution cannot represent the whole story about wave-vector anisotropy of solar wind turbulence. There seems to be a minor population located near k k , which is beyond the scope of critical-balance turbulence theory. The apparent two-componentdistribution of PSD (k k , k ⊥ ) seems to be connected with previous two-component models, e.g.,models with “slab”+“2D” (Matthaeus et al. 1990), models composed of Alfv´en waves and convectedstructures (Tu & Marsch 1993), and conjectures with critical-balanced component plus slab com-ponent (Forman et al. 2011; He et al. 2012b). We find that the minor population seems to weakenfurther with increasing heliographic distance, leaving more energy distributed in the region closeto the k ⊥ axis. This gradual migration of energy towards k ⊥ might indicate a relative enrichmentof turbulence energy carried by convective structures and explain the observed associated shortageof Alfv´enicity, which was already discussed in the previous two-component model by Tu & Marsch(1993). The observed “slab”-like minor component is crucial for scattering of energetic particles inthe interplanetary space (Bieber et al. 1996; Chandran 2000; Qin et al. 2002). The radial evolutionof anisotropic turbulence may be quantified in the future and incorporated into the transport modelof energetic particles.The estimated PSD (k k , k ⊥ ) is believed to impose valuable observational constraints on thetheoretical models of solar wind turbulence. Recently, Cranmer & van Ballegooijen (2012) modelledPSD (k k , k ⊥ ) at different heliocentric distances by solving a set of 2D cascade-advection-diffusionequations, with the total power pre-determined by the damped wave-action conservation equa-tion and the reduced PSD (k ⊥ ) pre-set by the 1D advection-diffusion equation. Their modelledPSD (k k , k ⊥ ) looks partly similar to our observational spectrum, in the sense of where the majorpower is located. However, the differences in distribution pattern and radial evolution betweenobservational and modelled spectra call for a substantial improvement of the models for solar windturbulence. 12 –The approximated ridge profile as extended to large k k and k ⊥ may give a hint about theresonance type responsible for solar wind heating at different radial distances. The extended ridgeprofile at small distances ( R < . k k where proton cyclotron reso-nance acts before Landau damping sets in. As the distance increases, the extended ridge profile,which is inclined more towards k ⊥ due to the reduction of k , tends to arrive at Landau resonancebefore cyclotron resonance. Such performance of the approximated ridge profile confirms obser-vationally previous conjecture about the transition from cyclotron resonance to Landau resonancewith increasing heliographic distance (Howes 2011).Our results are just limited to the MHD inertial range, but the analysis should be extendedto kinetic scales where several typical properties have been revealed: steeper power-law mag-netic spectrum (Sahraoui et al. 2009; Alexandrova et al. 2009), enhanced electric-field spectrum(Bale et al. 2005), enhanced magnetic compressibility (Smith et al. 2006; Hamilton et al. 2008;Salem et al. 2012; He et al. 2012a), and two-component pattern in the magnetic helicity (He et al.2011; Podesta & Gary 2011; He et al. 2012a,b). These observations seem to be in favour of theoblique Alfv´en waves or kinetic Alfv´en waves (KAW) as the candidate for explaining the dominantfluctuations in ion-scale turbulence. The oblique Alfv´en/ion-cyclotron waves may be via resonancediffusion (Marsch & Bourouaine 2011) responsible for the formation of the observed wide protonbeam. The theory of KAW itself and its role in kinetic turbulence have been studied intensively(Hollweg 1999; Wu & Chao 2004; Howes et al. 2008; Zhao et al. 2011; Voitenko & de Keyser 2011;Howes et al. 2011). Ion cyclotron waves, which are considered responsible for the ion perpendicularheating (Bourouaine et al. 2010), were also identified (Jian et al. 2009; He et al. 2011). There areother possible wave modes, e.g. fast whistler waves, ion Bernstein waves, and fast-cyclotron waves,which may exist in kinetic turbulence (Gary et al. 2012; TenBarge et al. 2012; Xiong & Li 2012).Spectral break at the ion-kinetic scale seems to be almost constant (about 0.5 Hz in the space-craft frame) with radial distance (Perri et al. 2010; Bourouaine et al. 2012). The spectral breakfrequency might corresponds to the proton inertial length in quasi-2D turbulence when consideringa large-scale background magnetic field B (which is obtained through averaging over a time periodhigher than 1 hour) (Bourouaine et al. 2012). However, not much is presently known about theradial evolution of solar wind turbulence at ion-kinetic scales.In the future, with the help of high-time-resolution measurements to be made by the waveand particle instruments flown on such mission like Solar Orbiter and Solar Probe Plus, the radialevolution of the wave-vector anisotropy at kinetic scale may be studied, and more new results willbe obtained on the spectrum anisotropy in the inertial range that was analysed here. Acknowledgements:
This work was supported by the National Natural Science Foundationof China under Contract Nos. 41174148, 41222032, 40890162, 40931055, and 41231069. JS Heappreciates helpful discussions with J.-S. Zhao, R. Wicks, and Y. Voitenko. 13 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
17 – r θ k θ r ^ r P k P k ^ D i n t eg r a t i on p i e c e t o g e t h e r k θ r θ k θ r ^ r P k P k ^ D i n t e g r a t i o n p i e c e t oge t he r r θ ab Fig. 1.— Sketch of relation between 2D-PSD and 2D-CF based on the projection-slice theorem.(Top) Two approaches to derive PSD (k k , k ⊥ ) from CF (r k , r ⊥ ): direct 2D Fourier transformand indirect Fourier transform of the projected CF (r , θ ). (Bottom) Vice versa for the derivationof CF (r k , r ⊥ ) from PSD (k k , k ⊥ ) 18 –Fig. 2.— (Top) Time averaged second-order structure functions based on Equation 1 (red) at threepositions (0.29, 0.65, and 0.87 AU) and their corresponding fitting results according to Equation 9(blue). The sets of the three fitting parameters ( R [nT ], t c [ s ], and p ) are (827, 116, 0.61), (53,465, 0.61), and (25, 857, 0.67) at 0.29, 0.65, and 0.87 AU. (Bottom) Corresponding correlationfunctions based on Equation 3 (estimations in red, fitting results in blue). 19 –Fig. 3.— (Top) Angular distribution of second-order structure functions SF( τ, θ RB ) estimated onthe basis of Equation 2. (Middle) Fitting results for SF( τ, θ RB ) with SF( τ ) at every θ RB beingfitted according to Equation 9, whereby p is fitted in angular dependence. (Bottom) Fit parameter p as a function of θ RB , revealing the scaling anisotropy of the structure function. The error-barsdenote the fitting errors of the parameter p . 20 –Fig. 4.— Angular distribution of CF as derived from SF( τ, θ RB ) according to Equation 4 anddisplayed in ( r k , r ⊥ ) space under the Taylor hypothesis of near time-stationarity. An elongationof CF along r k , implying the location of most turbulent energy close to k ⊥ , is revealed at allpositions. The proton gyroradius ρ g is 17, 48, and 70 km at 0.29, 0.65, and 0.87 AU. It is used fornormalization of the spatial coordinates. 21 –Fig. 5.— PSD (k k , k ⊥ ) at three positions (0.29, 0.65, and 0.87 AU) as derived from CF (r k , r ⊥ )according to Equation 7 following the projection-slice theorem. The major components (ridgedistribution with its centroid position aligned as black dashed line) may be roughly described byEquation 10 shown as red solid line, which means the wave-vector anisotropy develops as the outer-scale wave-number ( k ) becomes smaller with increasing heliocentric distance. A weakening trendof the minor component that is inclined to k k becomes also visible. 22 –Fig. 6.— Implication of solar wind heating mechanism from the extension of ridge profile. Redsolid lines denote the extended ridge profiles ( k k = α · k / · k / ⊥ with α = 3 . , . , . (k k , k ⊥ ) in larger wave-vector space. Large k k with k k c/ω p > . k ⊥ ρ g > k k c/ω p ≪ .
0. Radialevolution of the approximated ridge profile and its intersections with the threshold lines ( k k c/ω p =0 . k ⊥ ρ g = 1) indicate the transition of cascade termination from cyclotron resonance to Landauresonance as the solar wind flows further away. Ridge profiles with larger (smaller) α (= 10(1 . ρ g ( c/ω pp