Probing the large-amplitude flapping oscillations of current sheet with Cluster V. Sergeev, S. Apatenkov∗,A.Runov, W. Baumjohann, R. Nakamura, T. Zhang †,B.Klecker‡, J.-A. Sauvaud, P.Louarn§
Abstract Comparing a few different methods in case of multiple current sheet (CS) crossings without strong plasma sheet flows we confirmed a possibility to control the time variations of current sheet (CS) orientation and motion. Particularly, methods based on Minimal Variance Analysis show similar values and behavior of estimated CS normal with those based on B-gradient estimation and vertical translational CS velocity estimated from magnetic data of four spacecraft behaves similarly to actual vertical plasma velocity. Flapping oscillations were the kink-like folds with the vertical amplitude about ∼ 1-2 Re, strong tilts in YZ plane and behavior deviating from that of simple harmonic wave.
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Introduction
The current sheets (CS) in the Earth’s magnetosphere, including the tail current sheet, are the only kind of thin boundaries in the space plasma which can be directly probed and systematically studied with in situ spacecraft observations. Large hopes to learn about the tail CS physics are linked to the possibilities of multiple-spacecraft Cluster system which allows to probe gradients and evaluate the current density. To learn about CS physics one needs to know the distribution of plasma parameters across the current sheet, which can only be probed if the (closely-located) spacecraft cross the fast-moving current sheet. This particularly occurs during the so-called flapping motions, which are known for more than 40 years, but whose properties were little explored because of the limitations of single spacecraft techniques. Recent measurements made with Cluster (e.g. [Zhang et al., 2002, Sergeev et al., 2003, 2004] indicated a kink-like character of the flapping oscillations with a number of puzzling properties, like their strong and variable tilts and predominant propagation in the YZ plane. However, these conclusions mostly had a qualitative character since they were obtained using methods based of strong simplifying assumptions, like a plane approximation (which fails in case of the wave-like structure ∗
St.Petersburg State University, Institute of Physics, Petrodvoretz, 198504, Russia, e-mail:
[email protected] † Institut fur Weltraumforschung, A-8042 Graz, Austria ‡ MPE, Garching, Germany § CESR, Toulouse, France
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with comparable CS thickness, wavelength and oscillation amplitude). No methods were yet reported to follow the time variations of the sheet orientation. A similar big problem is to estimate the CS spatial scale (required to obtain the cross-sheet profile of plasma parameters) which can possibly be calculated based on the measured CS velocity. Testing different methods to control the time-varying CS orientation and motion is the main purpose of our paper. The timing of some feature (e.g. the crossing of the neutral sheet, NS) is the most transparent method, But its main assumption of a plane structure moving with constant velocity and constant orientation is questionable in application to wavy phenomena like the flapping is. Nevertheless it is useful for qualitative tests, and on many occasions it showed a good agreement with the MVA estimates (e.g. Zhang et al., 2002, 2003, Sergeev et al.,2003, 2004). One case with a large disagreement was also reported (Zhang et al., 2003). The well-known Minimal Variation Analysis (MVA) [Sonnerup and Schneible, 1998] explores the structure of B-field variance and interprete the least variance direction (having the smallest of three eigenvalues, λ3, being the measure of B-variance in that direction) as the normal to the (supposed) 1d sheet-like structure. (It can perform well also in the case of weakly non-1d structures when the interspacecraft separation is smaller than the wavelength). Its principal difficulty is that this suggestion (1d sheet) can not be justified for any particular part of time seria from the MVA data alone, since 2-3d structures or waves may perturb the variance structure (whose contribution is difficult to separate from contribution of the regular CS). Moreover, in some simple 1d structures the direction of the normal itself may not be well-defined (e.g. in the case of coplanar field-lines, like those in the standard Harris-type model, where both Z- and Y-directions have zero variances). This can be answered only after a cross-comparison with other methods (in our case, based on B-gradient estimations) will be performed. A number of practical questions related to the method performance (determination of the width of sliding window, setting the threshold λ2 /λ3 to control the quality of how MVA works) should also be decided in such comparison. The possibility to evaluate the B-field gradients and estimate the current density j from measurements at four identical closely-spaced spacecraft is a novel feature of the Cluster project. The previous work with ISEE-1,2 spacecraft (e.g. Sanny et al., 1994) showed the way how the time variation (dB/dt) and measured gradient (dB/dn , along the direction of the normal n) can be used to determine the translational CS velocity ( V T = −dB/dt/(dB/dn)) along the normal suggesting small temporal changes in the sheet structure (i.e. dB/dt + V T ∗ dB/dn ≡ dB/dt ∼ 0 ). Continuing along this line we shall do a next step (like Runov et al., 2004PSS) by extending this method to the case of arbitrary and time-varying configuration. Also, a possibility to use the ∇B direction as an estimate of the local sheet normal (discussed first by C. Cully, ’Techniques for monitoring tail current sheet structure and dynamics’, unpublished manuscript, 2003) will also be tried in our study.
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Data analysis and discussion
For testing purposes we use observations made by the Cluster spacecraft in the dusk sector of the tail plasma sheet (at [-13, 12, -1] Re GSM) between 09 and 11 UT on October 20,
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Z* = Vz dt , km
2001 previously presented by Sergeev et al (2004). During this time 9 large-scale CS crossings were recorded (duration of individual crossing varied between 80 and 400 sec) , with weak plasma flows suiggesting no active processes in the vicinity of the observation point. We used observations at 1s resolution with maximal interspacecraft separation about 2000 km, the estimated CS half-thickness was above 3000 km in all events. 6371 0 -6371
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Figure 1: Summary plot of basic parameters measured and computed based on Cluster observations on October 20, 2001.
Figure 1 shows the measured magnetic field (Bx) and plasma flow (Vy, Vz) variations during the event together with the electric current (rotB) , divB (as estimator of rotB accuracy derivation) based on magnetic field gradients computed as described in Robert et al (1998); the current sheet crossings are labeled by capital (small) letters in case of downward (upward) sheet motions. The points with | divB/rotB |> 0.25 were discarded. The translational velocity VT (see above, with dB/dt estimated at the baricenter) was computed for all points except for those with low gradients (∇Bx ≤ 1nT /1000km). The projection of translational velocity was computed as V T z = V T ∗ dBx/dz/ | ∇Bx | , V T y = V T ∗ dBx/dy/ | ∇Bx |; such approach ignores the CS motion along the sheet which can not be estimated from magnetic measurements. These components given on two bottom plots show similar absolute value and time variations as compared to the corresponding ion flow components (from CODIF instrument). The amount of agreement and difference is further emphasized by Integrated (translational and plasma flow) Vz
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components given on the top plot (z*=- V zdt) which show the variation of spacecraft position with respect to the current sheet. The variations are similar and they are well phased with the changes of Bx component. Amplitudes of vertical flapping-associated motions are about 1-2 Re (somewhat larger amplitude in the integrated plasma flow). The translational Z* shows also a downward trend (corresponding to some negative offset) whose origin is not yet currently understood. (A small instrumental offset in the plasma flow Vz was removed as explained in Sergeev et al., 2003). General agreement of variations indicate the reality of measured Bx gradients and usefulness of the technique based on it. 1
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Figure 2: Comparison between directions of the current sheet normals obtained by different methods A,B (Q = (nA ∗ nB )2 ).
The information on the magnetic field gradients obtained from Cluster was also used to evaluate the sheet orientation. In first method, for each time step we rotated the basis vector system to find the direction in which the gradient is maximal (by rotating the tensor < dBi /dxj > until the gradients in other two directions will be minimal), the corresponding normal is referred later to as RG (”rotated gradient”) method. This method does not rely on any assumptions, but is more time consuming than the second one. In the second case (nj method) we used the 2d reduction and evaluate the CS normal as nj = [n1 xj] where the vector of electric current j(= rotB/µ0 ) is obtained from linear B-gradient estimation (Robert et al, 1998) and n1 is the average direction of maximal variance obtained from the MVA (this direction is often well defined and stable in the long term, this is illustrated later in Figure 3, top). The advantage of nj method is that since the direction of n1 is chosen (e.g. its x-component is positive in the sunward direction) it has no uncertainty in the choice of the direction (under the nominal dawn-dusk current direction the nominal CS normal looks northward). A square of scalar product between two normals, Q = (nRG ∗ nj )2 can serve as a quantitative estimator of their agreement. Comparison in Figure 2d indicate good agreement betwen two methods (except for events C,d), so in the following we shall use the more simple nj- method to
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evaluate the performance of MVA in different modifications. As concerns the MVA, we tried sliding window approach with different window lengths (ranging from 10s till 30s) and using data from either a single spacecraft or all spacecraft together. Only results with large enough contrast between medium and lower eigenvalues are shown (with λ2 /λ3 ≥ 3). The amount of data points in the window was always taken to be > 20. The uncertainty in the choice of the sign of eigenvector n3MVA is solved by requirement that (n3MVA ∗ n3j ) > 0. Since we are mostly interesting in the CS properties during the fast crossings, the time intervals between them are not shown. With reference to the Figures 2 and 3 the main results can be briefly summarized as follows. (a) We found that using the data of 4 SC together provides MVA estimations similar to those obtained from a single spacecraft data set (with similar amount of data points, e.g. panel 2c), this however allows to reduce the window down to 10 sec and, often, to reach better agreement with ∇B-based estimates; (b) The MVA obtained with a short window show more structure although it often display the larger λ2 /λ3 ratio than those obtained with a longer window (panel 2b); (c) The agreement between MVA-based and ∇B-based normals is fairly good (angular deviation smaller than 30deg for most of points, see panel 2e). (d) The crossings are individual: the crossings C,c are more influenced by the noise (see fig.2) as compared to the well-behaved properties during crossings a,B,b,D,E,e. 1
Compared: MVA 10sec 4sc (
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Figure 3: Comparison of variations of the components of minimal variance unit vector (N3 ) and x-components of max variance unit vector (N1X ) obtained by MVA-based and ∇B-based methods.
As concerns the physical properties of the flapping current sheet, the major result is that good consistency between MVA and gradB-based methods gives more credit to the
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identification of kink-character of the current sheet motions. This is clearly displayed by alternating signs of (dominant) N3Y component of the normal during upward/downward crossings in Fig.3. Large tilts toward Y axis shown previously in Sergeev et al. (2004) are clearly confirmed. Stability of the orientation during the crossings a,B,b,E explains why the plane sheet-based methods often give good estimates of the sheet orientation and motion. The change of the CS orientation between crossings E and e looks more like a discontinuity than like a smooth harmonic oscillation. In conclusion, we compared different methods and confirmed a possibility to control the time variations of current sheet (CS) orientation and motion in the case of multiple current sheet (CS) crossings: the methods based on Minimal Variance Analysis provide similar values and behavior of estimated CS normal as those obtained from B-gradient, and translational CS velocity estimated from magnetic data of four spacecraft appeared to behave similarly to actual vertical plasma velocity. Flapping oscillations were confirmed to be the due to kink-like folds with the vertical amplitude about 1-2 Re, strong tilts in YZ plane and behavior deviating from the behavior of a simple harmonic wave. Acknowledgement. This work was supported by INTAS grant 03-51-3738 , RFBR grants 03-02-17533 and 03-05-20013 -BNTS. The work by S.A.was also supported by grant
References [1] Robert, P., M.W. Dunlop, A.Roux, G.Chanteur, Accuracy of current density determinations, In: Analysis Methods for Multi-Spacecraft Data, G. Paschmann and P. Daly (Eds.), ISSI Scientific Report SR-001, ISSI/ESA, 398, 1998. [2] Runov,A.,V.Sergeev, R.Nakamura, W.Baumjohann, T.Zhang et al., Reconstruction of the magnetotail current sheet structure using multi-point Cluster measurements, Planet.Space Sci.,in press, 2004 [3] Sanny, J., R.L.McPherron, C.T.Russell, D.N.Baker, T.I.Pulkkinen, A.Nishida, Growth-phase thinning of the near-Earth current sheet during CDAW-6 substorm, J.Geophys.Res., 99, 5805, 1994 [4] Sergeev, V., et al., Current sheet flapping motion and structure observed by Cluster Geophys. Res. Lett., 30, N 6, 10.1029/2002GL016500 , 2003 [5] Sergeev, V. et al., Current sheet flapping motion and structure observed by Cluster, Geophys. Res. Lett., 30, N 6, 10.1029/2002GL016500 , 2004 ¨ and M. Schneible, Minimum and maximum variance analysis, In: [6] Sonnerup, B.U.O., Analysis Methods for Multi-Spacecraft Data, G. Paschmann and P. Daly (Eds.), ISSI Scientific Report SR-001, ISSI/ESA, 185 - 220, 1998. [7] Zhang,T.I. et al, A wavy twisted neutral sheet observed by Cluster, Geophys. Res.Lett.,29, 1899, 2002 [8] Zhang,T.I. et al, Neutral sheet normal direction determination, Adv.Space Res., in press, 2003
