3-D feature of self-correlation level contours at 10 10 cm scale in solar wind turbulence
aa r X i v : . [ phy s i c s . s p ace - ph ] A ug Draft version September 4, 2019
Typeset using L A TEX default style in AASTeX62 cm scale in solar wind turbulence Honghong Wu, Chuanyi Tu, Xin Wang, Jiansen He, and Linghua Wang School of Earth and Space Sciences, Peking University, Beijing, People’s Republic of China School of Space and Environment, Beihang University, Beijing, People’s Republic of China
Submitted to ApJABSTRACTThe self-correlation level contours at 10 cm scale reveal a 2-D isotropic feature in both the slow solarwind fluctuations and the fast solar wind fluctuations. However, this 2-D isotropic feature is obtainedbased on the assumption of axisymmetry with respect to the mean magnetic field. Whether the self-correlation level contours are still 3-D isotropic remains unknown. Here we perform for the first time a3-D self-correlation level contours analysis on the solar wind turbulence. We construct a 3-D coordinatesystem based on the mean magnetic field direction and the maximum fluctuation direction identified bythe minimum-variance analysis (MVA) method. We use data with 1-hour intervals observed by WINDspacecraft from 2005 to 2018. We find, on one hand, in the slow solar wind, the self-correlation levelcontour surfaces for both the magnetic field and the velocity field are almost spherical, which indicatesa 3-D isotropic feature. On the other hand, there is a weak elongation in one of the perpendiculardirection in the fast solar wind fluctuations. The 3-D feature of the self-correlation level contourssurfaces cannot be explained by the existed theory. Keywords: solar wind, turbulence, magnetic fields, plasmas INTRODUCTIONThe magnetohydrodynamic (MHD) turbulence exhibits anisotropic features as a result of the preferred directionthat the background magnetic field determines (Shebalin et al. 1983). The solar wind is observed to be in a turbulencestate (Tu & Marsch 1995) and various studies proposed that solar wind turbulence is 2-D anisotropic based on theories(Oughton et al. 1994; Goldreich & Sridhar 1995), simulations (Cho & Vishniac 2000) and observations related to thepower spectral index (Horbury et al. 2008; Podesta 2009; Chen et al. 2010; Wicks et al. 2010; He et al. 2013), structurefunction (Luo & Wu 2010) and correlation function (Matthaeus et al. 1990; Dasso et al. 2005).The Maltese cross is one 2-D pioneering work. It consists of two lobes, one elongated along to the mean fielddirection (slab-like fluctuations), and the other elongated along the perpendicular direction to the mean field direction(2-D fluctuations) and is obtained by a 2-D self-correlation analysis (Matthaeus et al. 1990). Dasso et al. (2005)applied the correlation function method by using two-day-long data from Advanced Composition Explorer (ACE)spacecraft for the slow solar wind and the fast solar wind separately. They find that the fast wind mainly containsslab-like fluctuations and the slow wind 2-D fluctuations. However, Wang et al. (2019) find that the self-correlationfunction level contours of the magnetic field and the velocity field are 2-D isotropic for both the slow solar wind andthe fast solar wind at 10 cm scale using the same method as Dasso et al. (2005) .The 2-D anisotropic studies have been extended to the 3-D scenario which includes not only the mean magnetic fielddirection, but also the perpendicular magnetic field fluctuation direction. Boldyrev (2006) predicted theoretically thatthe solar wind turbulence is 3-D anisotropic with l k > l ⊥ > l ⊥ , where l k , l ⊥ , l ⊥ are correlation lengths in the meanmagnetic field direction, the perpendicular magnetic field fluctuation direction and the direction perpendicular to both,respectively. Chen et al. (2012) used the local structure function method to analyze the 3-D structure of turbulence Corresponding author: Chuanyi [email protected]
Wu et al. in the fast solar wind in a new local coordinate system from the outer scale to the proton gyroscale. Verdini et al.(2018) used the same local structure function method to analyze the 3-D structure taking the wind expansion effectinto account.In the present study, we perform the 3-D self-correlation function level contour analysis on the WIND spacecraftmeasurements. We construct the 3-D coordinate system using the mean magnetic field and the maximum variancefluctuation direction L obtained by MVA method. In section 2, we describe the data and methods used in order tostudy the 3-D anisotropy, including the way to construct the 3-D coordinate system and get the 3-D contour surfaces.We show our observational results in section 3. In section 4, we discuss our results and present our conclusions. DATA AND METHODWe use data from the Wind spacecraft during 14 years from 2005 to 2018, when the spacecraft hovers at theLagrangian point L max [ | δB j | ] < max [ | δV j | ] < j indicates x , y , z axis in the geocentric-solar-ecliptic (GSE) coordinate system, and δ means the variation betweenevery 3 s, in order to avoid the influence of shear magnetic field and shear flows.For each interval i , we define the fluctuation as δ ~U = ~U − ¯ U , where ~U is either magnetic field ~B or velocity ~V , and¯ U is obtained by performing a linear fit to ~U . The two-time-point self-correlation function of δ ~U is calculated as R U ( i, τ ) = < δ ~U ( t ) · δ ~U ( t + τ ) >, (1)here, τ = 0 , ∆ , , ..., <> denotes an ensamble time average. In order to easily makecamparison, we use the zero time lag self-correlation R ( i,
0) to normalize the self-correlation function and obtain R uu ( i, τ ) = R U ( i, τ ) /R ( i, R uu ( i, τ ) at τ = 0 is always equal to 1. According to the Taylorhypothesis (Taylor 1938), we transfer the time lag to spatial lag using r = τ V SW , where r is the spatial lag and V SW is the mean flow velocity in the corresponding interval i .Wang et al. (2019) has shown the isotropic feature of the self-correlation level contours in a 2-D coordinate system.We extend this system to 3-D by introducing the maximum variance direction L , which is determined by performingminimum-variance analysis (MVA) method (Sonnerup & Cahill 1967) to the magnetic field data. This 3-D coordinatesystem uses the mean magnetic field ~B and the projection of maximum variance direction L in the plane perpendicularto the mean field as r k and r ⊥ components, respectively. r ⊥ components completes this orthogonal coordinate system.Any angles greater than 90 ◦ are reflected below 90 ◦ . In Figure 1, we show the angle θ VB between the directions of V SW and ~B and the angle φ L between r ⊥ direction and the component of V SW perpendicular to ~B for each interval i . We find 23083 intervals in the slow solar wind ( V SW <
400 km/s ) and 3347 intervals in the fast solar wind ( V SW > θ VB in theleft panel of Figure 2 shows that the magnetic field is more oblique to the solar wind velocity in the slow wind than inthe fast wind, which is consistent with the Parker Spiral theory. In the right panel, we show in the slow wind, thereare more intervals with perpendicular φ L than parallel φ L , while the fast wind group has a roughly even distributionover 0 ◦ and 90 ◦ .For each group, we bin θ VB and φ L into 15 ◦ bins and calculate the average of the normalized spatial self-correlationfunctions as follows: R uu ( θ m VB , φ n L , r ) = 1 n ( θ m VB , φ n L ) X θ m VB − . < = θ VB ( i ) <θ m VB +7 . ,φ n L − . < = φ L ( i ) <φ n L +7 . R uu ( i, r ) (2)where n ( θ m VB , φ n L ) is the number of the intervals in corresponding bin, and, θ m VB = 15 ◦ m + 7 . ◦ ; φ n L = 15 ◦ n + 7 . ◦ ; m, n =0 , , , ..., . We obtain 36 averaged self-correlation functions for 36 ( θ VB , φ L ) = 15 ◦ × ◦ bins. We analyze the contours atlevel R uu ( θ VB , φ L , r ) = 1 /e ≈ . r level value by linear interpolation for each ( θ VB , φ L ). In orderto plot the contour surface in the 3-D coordinate system, we transform ( θ VB , φ L , r level ) into ( r ⊥ , r ⊥ , r k ) by using r ⊥ = r level sin θ VB sin φ L , r ⊥ = r level sin θ VB cos φ L , r k = r level cos θ VB . We reflect the surface in the first octant intothe other seven octants based on the assumption of reflectional symmetry. The result is presented in the next section. -D isotropic feature of self-correlation level contours B L r ∥ r ⊥1 滚滚长江东逝水 r ⊥2 θ VB ϕ L V SW Figure 1. r k corresponds tothe direction of the mean magnetic field ~B , and, the projection of the maximum fluctuation direction L on the perpendicularplane is defined as r ⊥ , and, r ⊥ completes this orthogonal coordinate system. θ VB is the angle between the mean magneticfield and the solar wind velocity, and, φ L is the angle between r ⊥ and the projection of the solar wind velocity on the planeperpendicular to ~B . 3. RESULTSFigure 3 shows the averaged self-correlation functions in r ⊥ , r ⊥ , and r k directions, which correspond to the followingangular bins: r ⊥ → (75 ◦ < = θ VB < = 90 ◦ , ◦ < = φ L < = 90 ◦ ) , (3) r ⊥ → (75 ◦ < = θ VB < = 90 ◦ , ◦ < = φ L < ◦ ) , (4) r k → (0 ◦ < = θ VB < ◦ , ◦ < = φ L < = 90 ◦ ) . (5)In the left panel of Figure 3, we present the averaged magnetic self-correlation functions with standard error bars forboth the slow solar wind (solid lines) and the fast solar wind (dashed lines). It is hard to distinguish the functions ofthe three directions for the slow wind. The phenomenon of the functions almost overlapping with each other means the3-D isotropic feature of the self-correlation level contours in slow solar wind turbulence. For the fast wind, there is aslightly elongation along the r ⊥ direction in the perpendicular plane. Note that the magnetic self-correlation functionof the fast wind is larger than that of the slow wind. When we consider self-correlation function with respect to thetime lag instead of the spatial lag, the magnetic self-correlation functions for both the slow wind and the fast wind arealmost the same (not shown). In the right panel, we show the averaged velocity self-correlation functions. They have Wu et al. θ VB [∘] P D F slowfast ϕ L [∘] P D F slowfast Figure 2.
Probability density function of θ VB (left) and φ L (right). The red and black histograms are for the slow wind andthe fast wind, respectively. almost the same features as the averaged magnetic self-correlation functions except there is no clear elongation alongthe r ⊥ direction. r [10 cm] R bb slow ∥slow ⟂ slow ⟂ fast ∥fast ⟂ fast ⟂ r [10 cm] R ∥∥ slow ∥slow ⟂ slow ⟂ fast ∥fast ⟂ fast ⟂ Figure 3.
Left panel: Averaged normalized self-correlation functions R bb ( r ) of 1-hour-long magnetic field data. The solid anddashed lines are for the slow wind and the fast wind. Red, blue, and yellow colors correspond the r k , r ⊥ , and r ⊥ directions,respectively. The error bar shows the standard error of r level for a given R bb . Right panel: Averaged normalized self-correlationfunctions R vv ( r ) of 1-hour-long velocity data, in the same manner as the left panel. We show the 3-D self-correlation level contour surfaces at level R uu = 0 .
368 in Figure 4. In Figure 4(a), the slowwind magnetic field self-correlation function contour surface is almost a spherical surface. The projection closed curves -D isotropic feature of self-correlation level contours r level in the r ⊥ direction is slightly longer. The projection closed curveson three planes are not round and have different sizes between each other. In Figure 4(c), the slow wind velocityfield self-correlation level contour surface is almost spherical and the projection closed curves on the 2-D planes is alsoround and identical to each other, which shows a clear isotropic feature as the magnetic field. In Figure 4(d), the fastwind velocity field self-correlation level contour surface has a similar shape with that of magnetic field. The similaritybetween the magnetic field and velocity field contour shape supports the applicability of the data analysis techniquehere. We should also note that, the r level is shorter for the slow wind than for the fast wind and shorter for the velocityfield than for the magnetic field.In order to evaluate the unevenness shown in Figure 4, we reduce the 3-D surface into the line trend with θ VB and φ L , as shown in Figure 5. We calculate the averaged r level in 6 θ VB bins from 0 ◦ < = φ L < = 90 ◦ with a weight ofinterval number in each φ L bin. The result is shown in the left panel. We can clearly see that the variation with θ VB is rather small for both the slow wind and the fast wind and for both the magnetic field and the velocity field. Wecalculate the average r level in 6 φ L bins from 60 ◦ < = θ VB < = 90 ◦ with a weight of interval number in the two θ VB bins. The result is shown in the right panel. For the slow wind, the variation with φ L is very small; while for the fastwind, there is a weak elongation along r ⊥ . Again, it is easily seen that r level is shorter for the slow wind and thevelocity field.In Figure 6, we show the variations with θ VB and φ L for the fast solar wind. The black solid and black dashed linesare the same as in Figure 5. We check the data intervals in the fast wind (group A) and further rule out the intervalswith large gradient by visual inspection. We reserve 2272 cases (group B). The variations with θ VB and φ L for thisnew fast wind group B are calculated and shown in blue lines in Figure 6. The anisotropy for the fast group becomesweaker after we remove the structures more strictly. DISCUSSION AND CONCLUSIONSWe present for the first time the 3-D self-correlation level contours of the magnetic field and the velocity field at10 cm scale based on WIND spacecraft measurements from 2005 to 2018. We construct a 3-D coordinate systemaccording to the mean magnetic field direction and the maximum variance direction L of the magnetic field. Theself-correlation contour surfaces at level R uu ≈ /e in the slow solar wind are 3-D isotropic for both the magneticfield and the velocity field. The self-correlation contour surfaces at level R uu ≈ /e in the fast solar wind show weakanisotropic feature in the perpendicular plane with an elongation along r ⊥ . However, the anisotropy becomes weakerwhen we exclude the intervals with structures more strictly.The 3-D coordinate system constructed here is consistent with the 3-D coordinate system presented by (Chen et al.2012) if we consider the maximum variance direction L as the ( ~B − ~B ) in their work. Chen et al. (2012) presenta structure function analysis in a scale-dependent 3-D coordinate system defined as follows: for each pair of points,the local mean field B local = ( ~B + ~B ) / B local × [( ~B − ~B ) × B local ] as another axis. Verdini et al. (2018) perform a structure function analysis in the same3-D coordinate system. However, our maximum variance direction L is based on the whole interval, while their localperpendicular fluctuation direction is adjusted for every two time instances.The surfaces of the fast wind have an approximately 1 . . cm corresponds more or less to the low-frequency break scale universally observed in the fast solarwind turbulence. The 3D quasi-isotropic feature of the self-correlation level contours in the fast solar wind is consistentwith Figure 1 by Wicks et al. (2010) that the power spectrum is isotropic at the low-frequency break. Whether theanisotropy of the 3-D self-correlation level contours increase at smaller and smaller scales needs further study. Thatthe contour of the velocity field is similar to the contour of the magnetic field is reasonable since magnetic and velocityfluctuations are coupled. Currently, we have no idea why the size of the magnetic contour is larger. Wu et al. r ⟂ −1.4−0.70.00.71.4 r ⟂ −1.4 −0.7 0.0 0.7 1.4 r ∥ −1.4−0.70.00.71.4 ∥a) ----0.1----0.5R bb (slow wind) r ⟂ −1.4−0.70.00.71.4 r ⟂ −1.4 −0.7 0.0 0.7 1.4 r ∥ −1.4−0.70.00.71.4 ∥b) ----0.1----0.8R bb (fast wind) r ⟂ −1.4−0.70.00.71.4 r ⟂ −1.4 −0.7 0.0 0.7 1.4 r ∥ −1.4−0.70.00.71.4 ∥c) ----0.1----0.4R vv (slow wind) r ⟂ −1.4−0.70.00.71.4 r ⟂ −1.4 −0.7 0.0 0.7 1.4 r ∥ −1.4−0.70.00.71.4 ∥d) ----0.1----0.7R vv (fast wind) Figure 4. R uu = 0 .
368 of (a) magnetic field in the slow wind; (b) magneticfield in the fast wind; (c) velocity field in the slow wind; (d) velocity field in the fast wind. The color represents r level [10 cm],which is the distances from the origin. The dashed red and blue lines in r ⊥ = − . r ⊥ = A r ⊥ = A
2, respectively, where A A r ⊥ = − . r ⊥ = A r ⊥ = A
2, respectively; the dashed red and blue lines in r k = − . r k = A r k = A
2, respectively.
Recently, Bruno et al. (2019) find that the low-frequency break is also present in the slow solar wind magneticspectra. They show a case with the break located around 10 − Hz and an average velocity of 316 km / s. Our scale inthe slow solar wind is smaller than the approximate break scale 3 × cm and is in the typical quasi-Kolmogorovrange, as in Bruno et al. (2019). Our 3-D self-correlation level contour analysis of both the magnetic and velocityfield shows a 3-D isotropic feature. We consider the self-correlation level contours represent the angular feature of theturbulence eddies. These results are not consistent with the predictions of the existed MHD theories. The 3-D isotropicfeature of self-correlation level contour supports the Kolmogorov’s theory (Kolmogorov 1941). How to interpret thisresult in the slow solar wind requires further investigation. -D isotropic feature of self-correlation level contours θ VB [∘] r [ c m ] R bb − slowR vv − slow R bb − fastR vv − fastR bb − fastR vv − fast ϕ L [∘] r [ c m ] R bb − slowR vv − slow R bb − fastR vv − fastR bb − fastR vv − fast Figure 5.
Left panel: averaged r level in each θ VB bin. The solid and dashed lines are for the magnetic field and the velocityfield. And, the red and black lines indicate the slow wind and the fast wind, respectively. The error bars show the standarderrors of the averaged r level . Right panel: averaged r level in each φ L bin, in the same manner as the left panel. θ VB [∘] r [ c m ] R bb − fast∘A)R vv − fast∘A) R bb − fast∘B)R vv − fast∘B)R bb − fast∘B)R vv − fast∘B) ϕ L [∘] r [ c m ] R bb − fast∘A)R vv − fast∘A) R bb − fast∘B)R vv − fast∘B)R bb − fast∘B)R vv − fast∘B) Figure 6.
Left panel: averaged r level in each θ VB bin. The solid and dashed lines are for the magnetic field data and velocitydata. And, the black and blue lines indicate the fast wind group A and the fast wind group B, respectively. The error bars showthe standard errors of the averaged r level . Right panel: averaged r level in each φ L bin, in the same manner as the left panel. We thank the CDAWEB for access to the Wind data, Dr. Liping Yang and Dr. Junxiang Hu for helpful discussions.This work at Peking University and Beihang University is supported by the National Natural Science Foundation ofChina under contract Nos. 41474147, 41504130, 41874199, and 41674171.REFERENCES
Boldyrev, S. 2006, Physical Review Letters, 96, 115002,doi: 10.1103/PhysRevLett.96.115002 Bruno, R., Telloni, D., Sorriso-Valvo, L., et al. 2019, A&A,627, A96, doi: 10.1051/0004-6361/201935841