The rotation angle distribution underlying magnetic field fluctuations in the 1/f range of solar wind turbulent spectra
IIL NUOVO CIMENTO
Vol. ?, N. ? ? The rotation angle distribution underlying magnetic field fluctu-ations in the 1/ f range of solar wind turbulent spectra L. Matteini ( )( ) , D. Stansby ( ) , T.S. Horbury ( ) and C.H.K. Chen ( )( ) ( ) LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Univ. ParisDiderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France ( ) Department of Physics, Imperial College London, SW7 2AZ London, UK ( ) School of Physics and Astronomy, Queen Mary University of London, London E1 4NS, UK
Summary. — We discuss properties of large amplitude magnetic field fluctuationsduring fast Alfvénic solar wind streams, focussing on the statistics of the rotationangle between consecutive magnetic field vector measurements for different scalesin the plasma. Since in the fast solar wind fluctuations preserve the modulus ofthe magnetic field to a good approximation, the tip of the magnetic field vector isobserved to move on a sphere of approximately constant radius | B | . We then com-pare statistics of solar wind measurements with that of a simple model of a randomwalk bounded on a spherical surface. The analogy consists in the fact that in bothsystems the geometrical constraint imposes a limiting amplitude at large separa-tions and thus introduces a break scale in the power spectrum of the fluctuations,leading to a shallower slope for scales where the fluctuations amplitude becomesscale-independent. However, while in the case of the random walk the saturationof the fluctuations occurs when the pattern becomes uniform on the sphere (flatdistribution of the cosine of the rotation angle), transitioning then to a white noiseregime, in the solar wind magnetic field fluctuations saturate in amplitude maintain-ing a preferential direction. We suggest that this behaviour is due to the presence ofthe background interplanetary magnetic field, which keeps some long-range memoryin the system also when the fluctuations becomes independent of the scale. Thislong-range correlation is a necessary ingredient in order to produce the 1/ f spectrumobserved at large scales in the solar wind.
1. – Introduction
In a recent paper [1] we have discussed properties of the large scale solar wind magneticspectrum, focussing on the so called 1/ f range. This is the portion of the power spectrumof the fluctuations which corresponds to scales above the turbulent Kolmogorov inertialrange and during fast streams it is characterised by a − spectral slope [2, 3]. The originof this range and its spectral shape are still debated [4, 5, 6, 7]. We briefly summarisehere the main idea behind our model: in magnetised plasmas, if imposing a regime of c (cid:13) Società Italiana di Fisica a r X i v : . [ phy s i c s . s p ace - ph ] O c t L. MATTEINI ETC. low magnetic compressibility (as typically observed in the fast solar wind) we expectthat there is a limiting amplitude for the fluctuations δ B when they reach order of thebackground field B ; indeed, if | δ B | (cid:29) | B | fluctuations in the components induce alsochanges in the magnetic field intensity, so that they become highly compressible. Notethat in the fast solar wind, at large scales | δ B | / | B | ∼ but δ | B | (cid:28) | δ B | , so that thefluctuations are nearly incompressible ( | B | is conserved). In [1] we suggest that in aturbulent plasma, when the amplitude of the fluctuations becomes large so that at somescale l | δ B | ∼ | B | , a saturation occurs to prevent the their further growth at scales l > l , and thus maintaining the plasma at a low level of magnetic compressibility. Thisleads to a break in the spectrum at l and a constant level | δ B | / | B | ∼ for scales l > l .Remarkably, this simple phenomenological scenario applies well to the large amplitudefluctuations of the solar wind; moreover in situ spacecraft measurements show that thescale l at which | δ B | / | B | reaches order unity in the solar wind plasma always identifiesthe spectral break between 1/ f and inertial ranges as observed at various radial distancesfrom the Sun [1].An important comment is in order here. The existence of a flat first order structurefunction over some scales (constant | δ B | ) does not directly imply a unique spectral slope.For example, if the fluctuations of this range are completely uncorrelated (white noise),the associated power spectrum is also flat. On the other hand, when the autocorrelationfunction of the fluctuations is not identically zero (some long range correlation exists inthe system), then the usual connection between second order structure functions at scale l , δB l , and power spectral density (PSD) for k-vectors k = 1 /l , P ( k ) , can be made:(1) δB l = P ( k ) · k so that if the former has slope α , the latter has slope α − [8]. It is then straightforwardto see that our model, based on the saturation of the structure function δB l above a scale l ( α = 0 ), directly predicts a − spectrum. Also note that for eq. 1, − is the steepestpossible slope.However, as mentioned, this is possible only if some long-range correlations are presentin the plasma at large scales [9], otherwise a flat (white noise) spectrum would be ex-pected. The aim of this paper is then to investigate further this aspect in solar windobservations, and compare the result with a simple model of a bounded random walk.
2. – A bounded random walk model
The fact that a self-similar process, which is in principle scale invariant, may becharacterised by a transition between different regimes at a particular scale, should not besurprising. Consider in fact random walks on a 2-D surface; the two-point correlation fora random walk ensemble leads to a power law distribution with slope − . This is howevertrue if the walk takes place on an unbounded surface, so that separations at arbitrarylarge scales can be performed. The large scale correlation in this case preserves the sameproperties as the small scale ones, leading to a single power-law spectrum. On the otherhand, consider now the case when the random walk develops on a spherical surface.While on distances (separation times) that are small compared to the curvature radius ofthe surface 2-point correlations still display the same − spectrum as in the unboundedcase, large scale correlations are characterised by a different behaviour. Indeed, if therandom walk is let evolve for sufficiently long time, the pattern starts to cover uniformlythe sphere, and the trajectory will eventually come back to close to the starting point. AGNETIC ROTATIONS IN THE 1/ F SOLAR WIND SPECTRUM Figure 1. – Results from a random walk model bounded on a spherical surface. Left: PDF ofthe cosine of the rotation angle θ between 2 vectors at different time separations encoded bycolours, lag increasing from red to blue. Right: Power spectrum of the resulting time series of therandom walk displacements on the sphere (top) and associated local spectral slopes (bottom).The expected break scale, corresponding to the number of steps needed to cover uniformly thesphere, is indicated by the solid line the in top panel. An example of this dynamics is shown in fig. 1 where we report results from a simplemodel [10] which tracks the motion of a vector whose tip is constrained on a sphere andsubject to a random walk. The model is let evolve for time steps of ∆ θ = 0 . rad,and results are taken as averages of 10 ensembles. The left panel shows the PDFs of therotation angle θ between two distinct vectors for increasing time lags (i.e. number ofsteps N ), from red to dark blue; at sufficiently large time lags, the distribution of cos ( θ ) becomes flat, meaning a uniform coverage of the sphere, as expected, and does not evolvefurther when increasing the time lag. This suggests that, although the underlying processgenerating the pattern is self-similar, the statistics made on time separations that arelarger than the typical time needed to walk around the full sphere is different from thatof smaller scale. Large scales are influenced by the fact that the walk occurs on a closedsurface, and once this uniformly covered, they follow a different law.This behaviour obviously affects the spectral properties of the system. In the rightpanel of fig. 1 we show the power spectrum of the Cartesian components of the resultingseparation time series. We can see that at short time scales a spectral slope of index − is recovered, as expected for a pure random walk; on the contrary, at a scale of N ∼ time steps t , corresponding approximately to the time needed to cover the whole sphere( N ∼ π/ ∆ θ ) and indicated by the vertical solid line, there is a change in the slopeleading to a flat spectrum, as expected for vanishing correlation between points uniformlydistributed (i.e. white noise).The one above is just a simple example of how the presence of a geometrical constraintcan lead to a break in the self-similarity and thus introduce a change in the expectedpower law of the fluctuations. Although it does not apply directly to turbulent interplan-etary magnetic field fluctuations, which cannot be modelled as a simple random walk,this is still an instructive example to compare with the solar wind observations describedin the next section. L. MATTEINI ETC.
Figure 2. – PDF of the rotation angle θ between 2 magnetic field vector measurements sepa-rated by a variable time lag ∆ t in the fast solar wind. Colours encode different scales, fromsmall (red) to large (blue) separations. Left and right panels show the distribution of θ andnormalised θ/ (cid:104) θ (cid:105) , respectively. As a reference, red/green PDFs belong to the turbulence inertialrange ( < ∆ t (s) < ) while almost overlapping blue distributions correspond to the 1/ f range( ∆ t > · s). The PDF approximatively corresponding to the break scale separating the tworanges is highlighted with black diamonds
3. – Solar wind observations
Let us now consider in situ observations from the solar wind. Following [1] we useUlysses measurements at high latitudes, when the spacecraft was continuously embeddedin fast polar wind (i.e. very regular, highly Alfvénic and with little variations of themagnetic and plasma pressures).We focus on the change of orientation between two different magnetic field measure-ments B ( t ) and B ( t + ∆ t ) as a function of the scale ∆ t . We recall that due to the highspeed of the flow, in the solar wind time measurements (frequencies) really correspondto Doppler-shifted spatial scales (k-vectors). Following [11] we calculate the PDFs ofthe angle θ between the two magnetic field vectors for different time lags ∆ t (see also[1] for details). The left panel of fig. 2 shows the distribution of magnetic rotations inthe fast solar wind corresponding to Ulysses measurements at approximately 2 AU, fromday 100 to 250 of year 1995. Colours encode different scales, from small (red/orange) tolarge (blue/purple) scales; the PDF approximatively corresponding to the break betweeninertial and 1/ f ranges ( ∆ t ∼ · s) is highlighted with black diamonds. Due to thesmall power in the fluctuations at small scales the PDF of θ for small time lags peaks atsmall angles. Rotations become significantly larger in the inertial range of the turbulence(green curves), starting to reach the extreme boundary of 180 degrees. At larger scales,corresponding to the 1/ f range (blue/purple), the PDF approaches a quasi-symmetricshape between 0 and 180 degrees; moreover, PDFs do not evolve further, as expectedfor scale-independent fluctuations (consistent with that, the corresponding PDFs of thefluctuations δB also do not change inside the 1/ f range and saturate to a constant level | δ B | / | B | ∼ , see [1]).The right panel of fig. 2 shows the same distributions each normalised to its meanvalue (cid:104) θ (cid:105) . As discussed in [11], at small scales the normalised PDFs can be reasonablywell described by a lognormal distribution; at these scales the PDFs of θ (left panel) aresufficiently far from the 180 degrees boundary, so that a high-tail can be sustained. How- AGNETIC ROTATIONS IN THE 1/ F SOLAR WIND SPECTRUM Figure 3. – Left: PDF of the cosine of the rotation angle θ between 2 magnetic field vectors inthe fast solar wind at 2 AU (Ulysses); colors encode different scales as in previous figure. Atthe largest scales (1/ f range, blue) the distribution tends to become shallower, with a roughlyconstant slope, although not entirely flat; this implies that the distribution of the underlyingmagnetic field vector tends to spread over the full sphere of radius | B | , but not yet uniformly.Right: same analysis for the fast wind at 0.3 AU (Helios measurements). ever, moving to larger scales and approaching the 1/ f range, the effect of the boundaryon the lognormal tail is apparent. As a consequence, the distribution of θ/ (cid:104) θ (cid:105) in the 1/ f has a more Gaussian-like shape, although some skewness is visible.In order to better appreciate the spread of rotation angles, the left panel of fig. 3 showsthe distribution of cos θ for all scales, with the same colour code as fig. 2. This goes froma strongly peaked shape at small scales -where angles are almost always smaller thana few degrees- to shallower distributions at larger scales. Once more, note that curvescorresponding to the 1/ f range all show the same shape and fall on top of each other.However, unlike fig. 1, the PDFs never reach a flat shape, indicative of a uniform distri-bution over the sphere of constant | B | ; indeed, a clear asymmetry between positive andnegative values of cos θ is maintained at all scales, including the 1/ f range. As a conse-quence, rotations are preferentially distributed around the direction of the backgroundfield (approximately the Parker spiral). This implies that even at the largest scales (inthe 1/ f range, where | δ B | / | B | ∼ ) the fluctuations are not totally uncorrelated and havesome long-range correlation, the memory of the background field direction.In [1] we have shown that the saturated shape observed for the 1/ f range, correspondsto an approximatively exponential relation:(2) P DF (cos θ ) ∼ exp ( α cos θ ) where α is an empirical constant, whose value can be obtained from the observations.Relation (2) is shown in fig. 3 as a dashed red line; it seems to describe reasonably wellthe main part of the PDF in the 1/ f range. For the Ulysses observations, we get α ∼ . .As a comparison, we show in the right panel of the same figure the same analysis fora fast solar wind stream observed by the Helios spacecraft at 0.3 AU (see [1] for moredetails). Also in this case we recover the same qualitative picture as for Ulysses (takenat ∼ AU); for Helios however, the PDFs in the 1/ f are less shallow, and α ∼ . . Thissuggests a possible evolution of the PDF saturated shape with increasing radial distance. L. MATTEINI ETC.
4. – Discussion and conclusion
To summarise, we have discussed properties of large scales fluctuations in the solarwind, when | δ B | / | B | ∼ and the tip of magnetic field vector approximatively moves ona sphere of constant radius | B | [12, 13], and compared them to a simple model basedon a random walk bounded on a sphere. In particular we have focussed on the rotationangle θ between two different vectors separated by a variable time lag, to probe the effectof the geometrical boundary on the statistics of different scales.We have found that the difference in the two cases lies in the condition that char-acterises and constrains the asymptotic state. In the simple random walk model, theself-similarity is broken by a purely geometrical condition, i.e. the pattern of the walkbecomes uniform over the sphere. The appearance of a spectral break is then caused bythe lack of two-point correlations above a certain scale (the size of the sphere) and as aconsequence the associated spectrum becomes flat, i.e. white noise.On the other hand, in the solar wind case some physical mechanism must be respon-sible for the saturation of the amplitude of the fluctuations typically observed at largescales [14]; in [1] we propose that this is related to a constraint of low magnetic compress-ibility in the plasma. As for the bounded random walk, the saturation of the amplitudeof the fluctuations naturally introduces a break in the solar wind magnetic spectrum.However, unlike fig. 1, the distribution of cos θ in the fast solar wind never reaches aflat shape (fig. 3), suggesting that a preferential direction common to all fluctuationscontinue to exist: this is the interplanetary background magnetic field. This maintainssome level of long-range correlation between the fluctuations, as required to build a 1/ f spectrum like that observed in the solar wind at large scales. ∗ ∗ ∗ This work was supported by national program PNST of CNRS/INSU co-funded byCNES. DS and TH are supported by STFC grants ST/K001051/1 and ST/N000692/1,respectively. CHKC is supported by STFC Ernest Rutherford Fellowship ST/N003748/2.
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