BURSTY BULK FLOW DRIVEN TURBULENCE IN THE EARTH’SPLASMA SHEET

Z. VOROS1,∗, W. BAUMJOHANN1, R. NAKAMURA1, M. VOLWERK1,2

and A. RUNOV1

1Space Research Institute, Austrian Academy of Sciences, Graz, Austria2Max-Plank-Institut fur extraterrestrische Physik, Garching, Germany

(∗Author for correspondence: E-mail: zoltan.voeroes@oeaw.ac.at)

(Received 8 November 2005; Accepted in final form 25 January 2006)

Abstract. During the last several years significant progress has been made in understanding MHD

turbulence in the Earth’s plasma sheet. Due to the statistically transitory properties of fluctuations,

finite size and boundary effects, however, issues of fundamental importance remain unresolved. Here

we concentrate on such intrinsic features of plasma sheet turbulence as its origin and dynamical

nature. In particular, we investigate bursty bulk flow driven multi-scale transfer of energy towards the

dissipation scale, and provide evidence for the presence of non-linear interactions. We show that, in

contrast with previous results, Alfvenic fluctuations together with 2D eddy interactions may appear

as important constituents of turbulence in the plasma sheet.

Keywords: turbulence, plasma sheet, bursty flows

1. Introduction

The essence of turbulence is its multi-scale non-local behaviour and highly dissi-pative nature (Warhaft, 2002). Non-locality in turbulence is substantiated also bygeneralized thermostatics (Leubner and Voros, 2005). For that reason, turbulencein the Earth’s plasma sheet is envisioned to have large consequences for the transferprocesses of energy, mass and momentum. Among other things, turbulence mightbe able to greatly influence plasma mixing, diffusion, viscosity, particle orbits, mayheat the plasma sheet and produce favourable magnetic field configurations for re-connection and even enhance the reconnection rate. Turbulence in the plasma sheetcan be driven by magnetic reconnection, shear bursty flows, interaction betweencoherent structures or eddies, anisotropic particle distributions, various types ofinstabilities, including specific boundary conditions and solar wind forcing (for acomprehensive review see Borovsky and Funsten (2003) and the references therein).Interestingly, turbulence can be driven by magnetic reconnection or/and the associ-ated bursty bulk flows (BBFs), and at the same time, the presence of turbulence canlocally enhance the probability of reconnection. The possible existence of repeatingcycles, ‘reconnection–BBF–turbulence–reconnection’, initiated theoretical specu-lations about an avalanche like global activity, believed to be also an importantconstituent of substorms (Chapman et al., 1998; Chang, 1999; Klimas et al., 2000).

Space Science Reviews (2006) 122: 301–311

DOI: 10.1007/s11214-006-6987-7 C© Springer 2006

302 Z. VOROS ET AL.

Note, however, that besides indirect indications obtained mainly in the auroral re-gion (Uritsky et al., 2002), there exists no direct in situ evidence yet which couldsupport convincingly this scenario.

The key role of BBFs in the plasma sheet dynamics is well known from previousstudies. The statistical analyses of AMPTE/IRM data in the near-Earth magnetotailhave already shown that there exist rapid plasma flows in the central plasma sheetwhich are preferentially perpendicular to the magnetic field and strongly sunwardoriented (Baumjohann et al., 1989, 1990) Subsequent research indicated that suchflows are preferentially organized into sporadically occurring groups of BBFs,typically lasting for 10 min. Despite their short duration, BBFs are the carriers ofdecisive amounts of mass, momentum and magnetic flux (Angelopoulos et al., 1992,1993; Schodel et al., 2001). The spectral power of compressional waves in the tailcurrent sheet and neutral sheet has been proven to increase with plasma flow velocity(Volwerk et al., 2003, 2004a). Damped eigenoscillations of the magnetotail canalso be initiated by fast flows (Volwerk et al., 2004b). Besides of waves, BBFs candrive fluctuations with characteristic spectral scaling properties. Magnetic field andplasma flow fluctuations in the magnetotail were observed by a number of authors(Coroniti et al., 1980; Borovsky et al., 1997; Neagu et al., 2002). Significantlyincreased magnetic field fluctuation amplitudes in the plasma sheet occur near solarmaximum, but ion velocity and magnetic field fluctuations seem to be independenton the magnetotail distance or on the distance from the neutral sheet (Neagu et al.,2005). Certainly, the results of statistical studies are also dependent on the chosenstatistical measures applied in the analysis of fluctuations. For example, higherorder statistics shows a different level of intermittency in the central plasma sheetthan within the outer plasma sheet (Weygand et al., 2005). On the other hand, theapplication of higher order statistics may be limited due to the shortness of steadydata intervals in the plasma sheet and due to the non-existence of higher orderstatistical moments (Voros et al., 2004a).

In general, magnetic fluctuations in the near-Earth magnetosphere exhibit multi-scale features (Lui, 2002; Consolini et al., 2005). In this review we emphasize theoccurrence of BBF driven multi-scale magnetic fluctuations (Voros et al., 2003;Volwerk et al., 2004c) which show evidence of non-linear interactions in the plasmasheet turbulence.

2. Turbulence Spectra

Magnetic field measurements on board the Geotail and AMPTE/IRM spacecraftrevealed that the power-law behaviour of the energy density spectra in frequencyspace, P( f ) ∼ f −α depends on the considered frequency range, both in the distanttail (Hoshino et al., 1994), and in the near-Earth plasma sheet (Bauer et al., 1995).This finding was confirmed by Ohtani et al. (1995) using fractal analysis of thefluctuations in the AMPTE/CCE and SCATHA magnetic field measurements. The

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spectral index changes around f1 ∼ 0.01–0.08 Hz (100–13 s). The observed scalingindices are α1 ∼ 0.5–1.5 for f < f1 and α2 ∼ 1.7–3 when f > f1 (Milovanovet al., 2001; Volwerk et al., 2003, 2004a). For frequencies higher than f2 ∼ 1 Hz(time scales smaller than 1 s) dissipation and kinetic effects start to be important. Theestimation of the scaling index α3 for frequencies f > f2, however, is encumberedby noise produced by the magnetometer itself. For a proper estimation of α3 anunbiased wavelet estimator combined with a correction technique was proposed(Voros et al., 2004b).

Several reasons have been proposed to explain the uncertainty of the estimatedvalues of α1 and α2: (a) slightly different frequency ranges studied by differentauthors (Volwerk et al., 2004a); (b) coherent large-amplitude low-frequency fluc-tuations, yielding a spectral index ∼2 (Weygand et al., 2005); (c) 2D turbulence dueto limited excess of the flow channel in GSM z direction, yielding a spectral index∼3 (Volwerk et al., 2003; Voros et al., 2004b); (d) boundary turbulence (Borovskyand Funsten, 2003; Voros et al., 2004a); (e) transitory mechanisms driven by differ-ent physical processes involved in the generation of the observed scalings (Voroset al., 2004b). The latter point (e) led to the pre-mature conclusion that spectraltools are not useful for identifying the presence of turbulence in the plasma sheet.However, using a wavelet-based technique for the estimation of the spectral powerand scaling index, Voros et al. (2004b) have shown that, if the transient character ofthe driver (plasma flow) and of the multi-scale spectral transfer of energy are takeninto account, the spectral index α2 can be robustly estimated. In Figure 1a and dburst mode (67 Hz) magnetic data are compared with the spin resolution (∼4 s)velocity data from Cluster spacecraft s/c 1 and 3. The variation of the scaling indexfor two different ranges of scales and the variation of spectral power, both estimatedwithin sliding windows (window length = 61 s), are depicted in Figure 1b and c.The spectral indices α2 and α3 are estimated over the time scales 0.7–5 s and 0.08–0.33 s. Hereinafter we will refer to those magnetic time scales as large scales andsmall scales, respectively. Note, that the considered scales are roughly around thetime scale 1/ f2 ∼ 1 s. We will argue later, however, that the value of f2 is velocitydependent. The small-scale spectral power, c f , is computed at the scale of 0.33 s.The dashed horizontal lines in Figure 1b and c indicate spectral exponents (∼0.25and ∼0.55) and normalized spectral power (∼1), corresponding to magnetometernoise. Notice that the appearance of bursty perpendicular flows (Figure 1d) are wellcorrelated with the appearance of high frequency fluctuations of BX (Figure 1a)and, as a consequence, also with increased values of small-scale power c f (Figure1c). While the flows persist (e.g. between 0,134 and 0, 202 UT) the large-scalespectral index can be robustly estimated (α2 ∼ 2.6), implying the presence of aflow driven steady process of spectral transfer of energy. Due to multi-scale spec-tral transfer of energy the large-scale energy of the flow reaches the small-scalesduring rapid flows, which is visible from the increased small-scale power c f (Figure1c). When BBFs are absent (e.g. between ∼0,222 and ∼0,236 UT) α2 is not welldetermined and the small-scale power is at the noise level, which means that an

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Figure 1. Two-point (s/c 1,3) estimation of the scaling parameters in comparison with the mag-

netic field and velocity measurements; (a) the BX components of the magnetic field; (b) spectral

indices (α2,3) estimated at large scales ∼(0.7,5) s and at small scales ∼(0.08,0.33) s; (c) spec-

tral power (c f ) estimated at the scale ∼0.33 s; the capital letters A and B show two intervals

further analysed in Figure 3; (d) x-component of perpendicular velocity. (based on Voros et al.,2004b).

observable steady process of spectral transfer over the analysed scales is not present.This comparison clearly shows that spectral tools are useful for turbulence iden-tification in the plasma sheet only when the details of large-scale driving are alsoknown.

One can speculate that during BBF associated intervals the inertial range ofturbulence widens towards small scales. If the inertial range would exist over theanalysed scales, the spectral indices α2 and α3 should be equal. Though α3 ex-ceeds the noise level during BBF associated intervals, its value is different fromα2. However, Voros et al. (2004b) have shown that because of the influence ofmagnetometer noise the value of α3 has to be corrected. After correction, BBFassociated intervals show α2 ∼ α3, while during non-flow intervals both c f andα3 are close to the noise level. It also means that, during BBF associated inter-vals, the spectral break f2 is shifted towards higher frequencies (smaller timescales).

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3. Evidence for Non-linear Interactions in Plasma Sheet Turbulence

In the expanding solar wind, non-linear interactions in turbulence were identifiedon the basis of widening of the inertial range towards large scales with increasingheliocentric distance (Bavassano et al., 1982; Tu et al., 1984). The large scalesbecome part of the Kolmogorov-like turbulence spectrum due to non-linear inter-actions between large-scale eddies which cannot be explained by the theory ofsolely outward propagating Alfven modes (Tu et al., 1984). In the plasma sheet,the large scales of BBF driven turbulence are poorly accessible because of the non-steady and transitory character of plasma flows. In fact, the majority of rapid flowsstays uninterruptedly at high-speed levels for only a few seconds. Neither do thetypically observed groups of high-speed intermittent flows last longer than someminutes (Baumjohann et al., 1990). A thorough statistical analysis of small-scalehigh-frequency fluctuations is easier to accomplish using high-resolution (67 Hz)magnetic data from the Cluster spacecraft. We have already demonstrated in theprevious section that magnetic fluctuations appear over time scales down to ∼0.1 sin association with BBFs. The evidence for non-linear interactions in plasma sheetturbulence comes from statistical analysis of widening of the inertial range withincreasing Reynolds number (Voros et al., 2005a). For large enough Reynolds num-bers the energy containing scales and the dissipation scales are clearly separatedwith a self-similar inertial range in between, where dissipation is supposed to benegligible and energy cascades towards large wavenumbers governed by non-linearinteractions. The width of the inertial subrange increases as the Reynolds numberincreases, which means that the energy containing large scales are separated furtherfrom the small scales (Saddoughi and Veeravalli, 1994). MHD turbulence in theplasma sheet is more complex than the outlined cascade picture would suggest.For example, dissipation can be significant even within the inertial range of scales(Borovsky and Funsten, 2003) or direct interaction between large and small scalesis possible without developing into a full cascade. One might still expect changesin the character of cross-scale coupling, a shift of the dissipation scale or a properlydefined small-scale, when the Reynolds number or the magnetic Reynolds numberincreases. Different types of Reynolds numbers can be derived for the flows in theplasma sheet (Borovsky and Funsten, 2003) corresponding to different aspects ofturbulent flows.

Voros et al. (2005a) have considered the so-called flow Reynolds number whichis a dimensionless number RE = V L/ν, where V is the average flow velocity, Lis the typical large spatial scale of the flow and ν is the kinematic viscosity. It canbe supposed that, in the plasma sheet, within the distances XGSM ∈ (−14, −20)RE ,the viscous forces are not changing substantially. The width of the flow channel,which represents the typical large-scale of the flow in the expression of the Reynoldsnumber, has been estimated on the basis of statistical analysis of Cluster spacecraftmulti-point measurements by Nakamura et al. (2004). It was shown that the fullwidth of the flow channel is 2–3 RE in the “dawn-dusk” direction and 1.5–2 RE in the

306 Z. VOROS ET AL.

“north-south” direction. The largest variations occur in the bulk flow velocity whichcan change from tens of kilometers per second to more than 1000 km/s in the plasmasheet. Therefore, one can consider L/ν ∼ const. and suppose that for increasingbulk velocities a widening of the inertial subrange can be expected. Consequently,instead of analysing the Reynolds number dependence of turbulent small scales,their bulk velocity dependence can be investigated. In principle, the small scalescould be recognized from the occurrence of a spectral break at the dissipation scalewhere kinetic effects and heating of the plasma become important. Then again,magnetometer noise precludes an accurate finding of the frequency range or timescale, over which the dissipation effects strengthen.

In order to estimate the dissipation time scale (τd) in magnetic turbulence Voroset al. (2005a) have introduced a novel method for the classification of velocitydependent patterns of two-point probability density functions’ (PDFs) shapes neartheir maxima. Multi-scale patterns of alterations of PDFs shapes corresponding tomagnetic fluctuations during both low-speed and high-speed flows were identified.It allowed to explore the small-scale behaviour of magnetic fluctuations in responseto the changes in large-scale bulk flows. Figure 2 shows the scatterplot betweenthe large-scale mean flow (quasi-steady bulk velocity during 6 min intervals 〈V 〉)and the dissipation scale Ld estimated as Ld = τd〈V 〉. Rapid flows exhibit largervelocity fluctuations than slow flows, but the slow-flow associated events havelarger uncertainties in τd (not shown). The latter can be the result of an approximatemapping between temporal and spatial structures during slow flows. Notice, that the

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Figure 2. Dissipation scale vs. bulk velocity (based on Voros et al., 2005a).

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dissipation scale (Ld) has a clear tendency to be shifted towards smaller scales whenthe bulk velocity (flow Reynolds number) increases. This is a generic behaviourobserved in many turbulent flows, which also substantiates previous conjecturesthat the observed multi-scale fluctuations in the plasma sheet, over the range fromseconds to minutes, arise due to the non-linear interactions and strongly dissipativemulti-scale turbulence. Notice that a few points exhibit rather large deviationsfrom the general trend in Figure 2. It may happen that the large-scale bulk flowis not the only parameter which controls the characteristics of multi-scale non-linear interactions. We have already mentioned that turbulence statistics may beposition dependent in the plasma sheet (Weygand et al., 2005) and the influence ofchanging boundary conditions and non-homogeneity effects may be significant aswell (Voros et al., 2004a).

The knowledge of the large-scale of the flow channel (L ∼ 2RE , (Nakamuraet al., 2004)) and the dissipation scale (e.g. Ld ∼ 50 km when 〈V 〉 ∼ 300 km/s,Figure 2) allows us to roughly estimate the Reynolds number through R3/4 ∼ L/Ld

(Warhaft, 2002), resulting in R ∼ 1600. This value is significantly lower than R ∼1011, estimated by Borovsky and Funsten (2003) using the value of kinematicviscosity from Coulomb collisions. It also indicates that, because of the substantiallimitations of the transverse section of the flow channel, turbulence might not befully developed in the majority of the cases. Nevertheless, the energy of the large-scale flow transferred through magnetic fluctuations can reach spatial scales smallerthan the ion gyroradius or ion skin depth (a few hundred kilometers).

4. Turbulence Dynamics

Borovsky and Funsten (2003) have argued that, dynamically, the turbulence ofthe plasma sheet appears to be a turbulence of eddies or frozen-in 2D structures,rather than a turbulence of Alfven waves or other MHD modes. They have shownthat when the temporal variations are owed to Alfven waves, propagating at themeasured local Alfven speed, the Alfven wave picture yields spatial sizes that arelarger than the cross-section of the plasma sheet, whereas the frozen-in picturedoes not. Another indication that plasma sheet turbulence might not entirely bethe result of interacting Alfven waves was provided by the presence of scale de-pendent anisotropy of magnetic fluctuations (Voros et al., 2004a,b, 2005b). Scale-dependent anisotropy can be the result of a turbulent cascade in the plane per-pendicular to the local mean magnetic field, strengthening towards small scales.At the same time, the parallel cascade of energy is absent. This scenario leads tothe picture of Alfvenic fluctuations parallel and of dynamically coupled 2D eddyturbulence or coherent structures perpendicular to the mean magnetic field direc-tion (Goldreich and Sridhar, 1995; Chang, 1999). In order to show that Alfvenicfluctuations may occur during BBF associated intervals we have selected two dif-ferent intervals, marked as interval A and B in Figure 1c, during which the average

308 Z. VOROS ET AL.

–1

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in Figure 1c); (a, e) probability density functions estimated from spatial fluctuations δBX S between

s/c 1, 4 and from temporal time-delayed fluctuations, δBXT ; (b–d, f–h) comparison of smoothed and

normalized magnetic and velocity fluctuations.

bulk velocities were 480 and 80 km/s, respectively. Figure 3a–d show fast-flowand Figure 3e–h slow-flow related statistics and a comparison of smoothed mag-netic field and velocity fluctuation components. In the bottom six subplots, thetime resolution of both magnetic field and velocity fluctuations is 4 s and the localmean values were extracted. Additionally, the fluctuations were smoothed using a10-point running mean and normalized to the standard deviation. In the two topsubplots (Figure 3a and e) PDFs of spatial and temporal fluctuations are com-pared. In both cases, high resolution (67 Hz) magnetic measurements were used.The spatial fluctuations, δBX S , were formed from BX , measured at Cluster s/c1and s/c4 separated by a distance of S ∼1,700 km in GSM X direction. The tem-poral fluctuations were computed from time shifted measurement using BX froms/c1 through δBXT = BX (t + τ ) − BX (t), where τ = S/〈VX 〉. We note, that thetypical small scales of turbulence can be shorter that the actual spatial separationbetween the spacecraft. The coincidence between spatial and temporal statisticsat the considered scales is better during the high-speed flows, which means thatTaylor’s frozen-in hypothesis works better than in the case of low-speed flows.At the same time, the high-speed flow interval (Figure 3b–d) shows higher levelof correlation between magnetic field and velocity components fluctuations, that

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is more Alfvenic, than the low-speed flow interval (Figure 3f–h). Nevertheless,the Z components of the magnetic field and velocity fluctuations also tend to beanti-correlated after ∼1.66 UT during the high-speed flow interval (Figure 3d), in-dicating possible interaction of Alfvenic fluctuations with local 2D structures. Wenote that Alfvenic correlations in the fast solar wind are also stronger than in theslow solar wind (Bruno and Carbone, 2005). In relation to Alfvenic fluctuations,Dmitruk et al. (2003) have noticed the crucial importance of different time scalesexisting in a turbulent flow. Large-scale motions define a time scale roughly equalto τdrive ∼ L/〈V 〉, where L is the characteristic length of the flow, in this case notnecessarily the distance across the flow. Another important time scale is that ofthe Alfven waves, τwave, to propagate the length of the flow. Dmitruk et al. (2003)have shown that the slope of the perpendicular energy spectrum depends stronglyon the ratio τwave/τdrive as a consequence of which a wide range of spectral slopescan appear. In the present stage, however, it is not entirely clear if the concept ofparallel Alfvenic and perpendicular 2D fluctuations could be used in the Earth’splasma sheet as successfully as in the solar wind.

5. Conclusions

Hydrodynamic or magnetohydrodynamic turbulent flows are rich of coherent struc-tures and self-similar fluctuations arising as a consequence of the redistribution offlow energy introduced typically at large (system) scales. In this paper, we demon-strated that the multi-scale magnetic fluctuations observed in association with BBFsexhibit typical features of turbulence. Turbulence spectra can be robustly estimatedwhen additional information about large-scale flows and boundaries is available.The widening of the inertial range with increasing bulk velocities (Reynolds num-ber) provides a clear evidence for the presence of non-linear interactions. The lim-ited size of the flow channel prevents fully developed turbulent states, nevertheless,the kinetic energy of the large-scale flow mediated through magnetic fluctuationscan reach spatial scales smaller than the ion gyroradius or ion skin depth. Dynami-cally, both parallel Alfvenic and perpendicular 2D fluctuations can be expected, butwe are still far from fully understanding the emerging phenomena in magnetotailflow driven turbulence.

Acknowledgements

The authors thank A. Balogh, B. Klecker, H. Reme, Y. Asano, T. Takada and T.L. Zhang for helpful discussions, all the Cluster mission and instrument teams forthe high-quality data, H. U. Eichelberger for great help in the data processing. Thework by M.V. was financially supported by the German Bundesministerium fur

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Bildung und Forschung and the Zentrum fur Luft- und Raumfahrt under contract50 OC 0104.

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