Serpentine mode oscillation of the current sheet

GEOPHYSICAL RESEARCH LETTERS, VOL., NO., PAGES 1 15, Serpentine mode oscillation of the current sheet M. Volwerk 1,2, K.-H. Glassmeier 3, A. Runov 1, W. Baumjohann, 1, R. Nakamura 1, T.L. Zhang 1, B. Klecker 2, A. Balogh 4 and H. Rème 5 M. Volwerk, IWF, ÖAW, Schmiedlstr. 6, 842 Graz, Austria (martin.volwerk@oeaw.ac.at) 1 IWF, Graz, Austria 2 MPE Garching, Germany 3 TU Braunschweig, Germany 4 Imperial College, London, UK 5 CESR, Toulouse, France

2 VOLWERK ET AL.: SERPENTINE CS OSCILLATION Abstract. Oscillations of the current sheet observed by the Cluster spacecraft are investigated, using the special ability of Cluster to separate between spatial and temporal variations. We find that before substorm onset, as defined by the AE index, a thin current sheet moves with a velocity of 1 km/s mainly in the vertical direction; after substorm onset the current sheet thickens and moves with greater velocity, 25 km/s. Oscillations of the current sheet observed before and after substorm onset can well be interpreted and modelled as a magnetoacoustic eigenmode of the current sheet, with different values for the (Harris) current sheet half thickness and the current sheet velocity. The damping rate of the wave can be shown to be related to the thickening rate of the current sheet.

VOLWERK ET AL.: SERPENTINE CS OSCILLATION 3 1. Introduction Neutral sheet oscillations have been investigated by many experiments [Zhu and Kivelson, 1991; 1994; Chen and Kivelson, 1991; Khurana et al., 1992; Bauer et al., 1995a; 1995b; Sergeev et al., 1998] and in theory [Roberts, 1981a; 1981b; Lee et al., 1988; Smith et al., 1997]. In the Cluster era we have four spacecraft that simultaneously measure the magnetotail properties with an interspacing of approximately 2 km. This gives us the possibility of discerning between temporal and spatial variations in the magnetotail. In this paper we will use this capability of Cluster to investigate a serpentine mode oscillation of the neutral sheet. 2. The Data In Fig. 1 we show the magnetic field [Balogh et al., 21] and plasma flow (CIS CODIF) Fig. 1 [Rème et al., 21] data for the August 22, 21, event in GSM coordinates. The highfrequency oscillations in the data have already been discussed by Volwerk et al. [22], but clearly visible is also a large scale oscillation of the B x component which creates multiple B x = crossings while the spacecraft move from the northern to the southern hemisphere, an indication that the current sheet is flapping (see also Sergeev et al. [1998]). There is an asymmetry in the crossing of the neutral sheet, i.e. in the northern hemisphere of the tail the field strength B x 27 nt, whereas when the spacecraft are in the southern hemisphere the magnetic field remains for an extended period (11-14 UT) at B x 7 nt. Only after half an hour does the magnetic field start to increase in strength towards B x = 27 nt, which is reached at approximately 112 UT. The AE index shows an activation at approximately 955 UT, increasing from 15 to 5 nt, which is the start of a long duration magnetospheric activity. The plasma flows,

4 VOLWERK ET AL.: SERPENTINE CS OSCILLATION as measured by Cluster, see Fig. 1, show some bursty features Earthward (94-945 UT), then tailward (945-955 UT). At substorm onset 955 UT, as defined by the increase in AE, and by strong negative bays in the GIMA magnetometers, strong Earthward flow is set up in the period 955-115 UT. Eagle magnetometer (near the footpoint of Cluster) shows a clear substorm onset with a strong negative bay in the H-component. Note, however, that there is a brightening in the FUV at 939 UT and reaches the Cluster footpoints at 941 UT [H. Frey, personnal communication, 22]. This brightening may be related to the strong flow burst observed in the plasma data at 94:3 UT. We have calculated the magnetic, plasma and total pressure over the interval 93-12 UT. In general we find that the total pressure is.4 npa, with magnetic pressure dominating in the outer plasma sheet and the lobes and plasma pressure dominating in the current sheet. However, between 955 and 15 UT, after substorm onset, the total pressure drops to.3 npa and is recovered afterwards. The magnetic field does not reach a value B x 27 nt until 112 UT, until that time cluster stays in a high-β region. Also, after the field has reached -27 nt, just after 1115 UT, there is still significant plasma pressure. This indicates that the current sheet has become thicker, which indeed is the case. Volwerk et al. [22] determined the current sheet thickness using the method of Nakamura et al. [22] and found that between 94 and 1 UT there is a very thin current sheet with half thickness.5 λ.2r E, whereas after 1 UT the current sheet half thickness is.4 λ 2R E (their results are reproduced later in this paper). We low-pass filter the data (Butterworth filter, τ > 5 min.), the result of which can be seen in Fig. 2. First we will look at some of the properties associated with this oscillation, Fig. 2

VOLWERK ET AL.: SERPENTINE CS OSCILLATION 5 such as the velocity and minimum variance direction, in Sect. 3 we will model this flapping motion of the current sheet and in Sect. 4 we will investigate the associated currents. With four spacecraft we can immediately obtain the velocity of a front passing by Cluster, under the assumption of a plane geometry. Whenever we have the four spacecraft crossing the same level of B x we can calculate the time difference between the spacecraft and the difference in location, e.g. for the crossing of B x =, and make a least squares fit to obtain the three dimensional velocity vector. In the low-passed data we identify four regions in which the neutral sheet is alternately moving up and down. For three intervals: A: 941-95 UT; C: 956-15 UT and D: 15-19 UT; all four spacecraft cross B x = B and we can determine the velocity of the neutral sheet from these crossings. The results of the velocity determination in the xzplane can be seen in Fig. 3 middle panel, where we have taken 1 nt steps in B. On average Fig. 3 we find that v A = (4.5, 2.9, 7.7), v C = (2.5, 2.5, 7.1) and v D = ( 7.6, 14.3, 2.2) km/s. Interval B: 95-956 UT however fails to have mutual crossings of all spacecraft and we determine the velocity of the neutral sheet with only C1/2/3 (where there is an implicit assumption that there is no velocity component perpendicular to the plane of the spacecraft), v B = ( 7.9, 2.9, 6.8) km/s. The velocity of the neutral sheet, in intervals A, B and C, varies in and between each interval, usually only a few km/s, and the greatest velocity component is mainly v z. Interval D shows significant changes in the the neutral sheet velocity. A similar sudden change in the neutral sheet velocity (albeit in the other direction, from fast to slow) was observed by Runov et al. [22]. The behaviour of the velocity as we see it in the xz-plane needs to be examined further. One can see that the velocity is mainly in the same direction, only changing sign (including

6 VOLWERK ET AL.: SERPENTINE CS OSCILLATION the y-component, not shown in the figure). This velocity pattern is not consistent with a travelling wave in the current sheet in which the x-component of the velocity would remain either positive or negative. However, it is consistent with a standing wave passing through the tetrahedron, which would not be unusual for an eigenmode. For a serpentine mode oscillation some restrictions are in place on the physical quantaties of the current sheet. The mininum variance direction of the magnetic field for spacecraft at opposite sides of the neutral sheet should be in the same direction. We chose the interval 943:48-944:24 where C3 is in the southern hemisphere and C4 is in the northern hemisphere. Only in z are these two spacecraft significantly separated. For both spacecraft we find a minimum variance direction of N = (.5,.24,.97). Also over the oscillation the magnetic and plasma pressure only vary significantly just after substorm onset. 3. Model for Serpentine Mode Oscillation To describe the large scale oscillation of the current sheet that is observed in the lowpass filtered data (see Fig. 2) we use the numerical model by Smith et al. [1997]. Starting from a simple Harris-model current sheet, they solve the MHD equations (cf. Roberts [1981a; 1981b]), and find for the perturbed magnetic pressure the following equation: iω ρ p 1m = va 2 dv z dz B µρ ( ) db v z. (1) dz Here ω is the frequency, ρ is the density, v A the lobe Alfvén velocity, v z is the current sheet velocity, B is the magnetic field magnitude and µ is the permeability. The perturbed

VOLWERK ET AL.: SERPENTINE CS OSCILLATION 7 magnetic pressure is defined as p 1m = B b 1 /µ, where b 1 is the perturbation magnetic field. To model the observed oscillation we use the following forms for the used quantaties: ( )) z 2 v z (z) = v (.5 +.5 exp, (2) B(z) = B 1 + (B e B 1 ) tanh a 2 ( z λ), (3) where B e is the lobe field strength and B 1 is the offset. The shape of the velocity v x has been taken from Smith et al. s Fig. 4a, where v in Eq. (2) is taken to be the Alfvén velocity in the lobes. However, as determined in the previous section we find for the B x = B crossings that the current sheet velocity v 1 km/s. As we concentrate mainly on modeling the oscillations during the time interval 93-13 UT, we opted for an offset Harris model, as shown in Eq. (3), where the lobe field changes from 27 nt in the northern to -7 nt in the southern hemisphere. Solving the MHD equations implicates that the perturbed fields have the time dependence exp(iω t). In general one finds that, solving for ω, one obtains complex conjugate pairs of solutions. These correspond to a growing (I(ω ) > ) and damped (I(ω ) < ) solution of the wave equation. We write then that ω = ω ± iγ, where we have the choice of either solution. From the data it is clear that we are dealing with a damped oscillation γ <. Thus we can find the expression for the perturbed magnetic field that has to be added to the Harris current sheet: b 1m = ρ ω [ v 2 A dv z dz B µρ ( ) ] db v z dz cos(ωt π/2 + φ) exp( γt). (4)

8 VOLWERK ET AL.: SERPENTINE CS OSCILLATION Substituting Eqs. (2) and (3) into Eq. (4) we can model the observed current sheet oscillation with the the following parameters before substorm onset: z C2 = 15 km; φ C2 = 14 ; B e = 27 nt; B 1 = 1 nt; ω = 2.5 1 3 s 1 ; γ = 1.25 1 3 s 1 ; ρ = 4 AMU/cm 3 ; λ = 3 km and a = 2λ. The location of the spacecraft, z(t), changes over the time interval and is given by z(t) = z C + v(t)t, where v(t) is taken directly from the Cluster data, but z C z C,GSM due to the offset of the current sheet in z GSM (see e.g. Nakamura et al. [22]). The result for C2 is shown in Fig. 3 top panel. After substorm onset at 955 UT the thickness of the current sheet increases [Volwerk et al., 22] and see Fig. 3. At approximately 1 UT the half thickness λ.4r E. Also, as noticed above, the velocity of the current sheet increases from 1 to 25 km/s. In our modeling of the oscillation we change the half thickness to 18 km and the velocity of the current sheet to 25 km/s. The broad second trough observed in the data at 15 UT is then well modeled, which otherwise would be much narrower in the unmodified numerical model. 4. Magnetospheric Currents Associated With Oscillations Ampère s law can be used to calculate the currents [e.g. Dunlop et al., 22]. The result for the total current density is shown in Fig. 3, bottom panel. The data used have been processed using an intercalibration tool [Kepko et al., 1996; Khurana et al., 1996]. The currents calculated in this paper are the currents flowing inside the Cluster tetrahedron. We therefore have calculated the magnetic field at the center of the tetrahedron, BC as the average between C3 and C4, which are mainly separated in z. BC x is shown in Fig. 3, bottom panel.

VOLWERK ET AL.: SERPENTINE CS OSCILLATION 9 The currents in the neutral sheet start as soon as the BC x component starts to decrease, and are maintained until the magnetic field has reached its temporary field strength of -7 nt in the southern hemisphere. As is to be expected, the strong B x variation observed in the data is mostly reflected in the J y component. Panel III in Fig. 3 shows that a maximum in J is reached at 944 UT, where the magnetic field strength BC x 12 nt. This means, that unlike in the Harris current sheet, the maximum current is not at BC x = nt. This could be an indication of a bifurcated current sheet [see e.g. Runov et al., 22] or a temporal variation. However, we lack sufficient evidence for the former from the other side of the neutral sheet, where we find a flat distribution of the current over the sheet. However, the off-centered current would be in agreement with the offset Harris sheet that we have used to model the current sheet oscillation. We would then expect to find current maximum at B x 1 nt. In this case a temporal variation of the current sheet would be the preferred model. Panel IV of Fig. 3 we show the half thickness of the current sheet over time on a logarithmic scale for the times that a good fit to a Harris sheet was possible. These results are reproduced and expanded from Volwerk et al. [22]. The goodness of fit is determined by the spacecraft that is not used in determining the current sheet thickness, for which one only needs three spacecraft [see Nakamura et al., 22]). The thickness is calculated for C1/3/4 (circles) and C2/3/4 (triangles). The current sheet remains thin (λ 1R E ) within the interval 937-955 UT, after which the current sheet starts to expand. One sees changes in the thickness of the current sheet. One finds that the current sheet thickens exponentially at a rate of 1 R E in 13 minutes (.15 R E at 938 UT to 2 R E at 14 UT) indicated by the thick solid line.

1 VOLWERK ET AL.: SERPENTINE CS OSCILLATION 5. Discussion We have studied a neutral sheet oscillation with the four cluster spacecraft during an interval around substorm onset. The half thickness of the current sheet before substorm onset was small, λ.5.2r E, whereas after substorm onset it increases to λ.4r E. Also the velocity of the neutral sheet changes from approximately 1 km/s before to approximately 25 km/s after substorm onset. The onset of the decrease in B x coincides with a strong flow burst in v x and v y seen in the CIS CODIF data at 94:3 UT. This means that the magnetoacoustic eigenmode, the serpentine oscillation discussed in this paper, finds its onset in this flow burst and related to an auroral brightening in the FUV data starting at 939 UT and which reaches the Cluster footpoints at 941 UT. The serpentine mode is a standing wave of the current sheet, shown by the sign changing of both the v z and v x components and is an oscillation of the whole current sheet, shown by the minimum variance direction that is the same for the northern and the southern hemisphere of the current sheet. Note that the average velocity vector in 943:48-944:24 UT (during interval A) v = (4.5, 2.9, 7.7) km/s and the minimum variance direction obtained from both C3 and C4 is N = (.5,.24,.97). This means that in GSM coordinates the the magnetic field lines are not much bent yet, which would be expected at the onset of the oscillation. In a later stage 956:24-1 UT (during interval C) we find that the minimum variance direction N = (.2,.5,.8) and the average velocity over this interval is v = (2.5, 2.6, 7.1) km/s. This means that the velocity does not change much in similar parts of the oscillation, but now we see that the normal of the neutral sheet is slightly more tilted. The spacecraft are mainly traversing the current sheet in

VOLWERK ET AL.: SERPENTINE CS OSCILLATION 11 z GSM and therefore one should keep in mind that the oscillation seen in Fig. 3 does not show an xz-view of the current sheet. We have succesfully attempted to model the neutral sheet oscillation by a serpentine mode oscillation as described by Smith et al. [1997]. Superposing the model onto an offset Harris sheet describes quite well the oscillation of the neutral sheet observed by Cluster, when we take into account the changing of the thickness and the velocity of the current sheet after substorm onse. Using this simple 1D model to describe one component of the 3D oscillation of the neutral sheet works very well for the middle spacecraft C1 and C2. A goodness of fit can be determined by the RMS error with respect to the low-pass filtered data over the interval 93-112 UT. The good fit for C2 as shown in Fig. 3 has RMS 2 = 4, for the other data we find RMS 1 = 5, RMS 3 = 9 and RMS 4 = 9. The parameters of the numerical model show that the damping of the magnetoaccoustic eigenmode is over a timescale of the observed oscillation itself. The mean period of the oscillation is close to 8 seconds, whereas in the model we have used γ = 1/8 s 1. Dissipation on a timescale of several hundreds of seconds is slow, however when we look at the development of the current sheet half thickness λ we find that this is similar to the time over which the current sheet changes its characteristics. It has a growth rate of approximately 1 R E in 13 minutes (78 sec.). Most probably the thin current sheet is set off to oscillate in the serpentine mode by a flow burst. The current sheet then evolves, growing in thickness, increasingly so after 1 UT, on a timescale that is comparable with the oscillation time scale. Acknowledgments. The authors would like to thank R. Treumann and V. Sergeev for very useful discussions, H.-U. Eichelberger and Y. Bogdanova for preparing the Cluster

12 VOLWERK ET AL.: SERPENTINE CS OSCILLATION MAG and CIS data. We acknowledge the help from the UCLA FGM Co-Is for providing intercalibrated data for calculating the currents. We thank H. Frey for reprocessing and making available the Image FUV data. Furthermore, we would like to thank Dr. J. Olson at the UAF Geophysical Institute for making the GIMA magnetometer data available on the web. The AE index was obtained from the World Data Center for Geomagnetism website in Kyoto. The work by KHG was financially supported by the German Bundesministerium für Bildung und Forschung and the Zentrum für Luft- und Raumfahrt under contract 5 OC 13. References Balogh, A., C.M. Carr, M.H. Acuña, M.W. Dunlop, T.J. Beek, P. Brown, K.-H. Fornacon, E. Georgescu, K.-H. Glassmeier, J. Harris, G. Musmann, T. Oddy and K. Schwingenschuh, The Cluster magnetic field investigation: overview of in-flight performance and initial results, Ann. Geophys., 19, 127-1217, 21. Bauer, T.M., W. Baumjohann, R.A. Treumann, N. Sckopke and H. Lühr, Low-frequency waves in the near-earth plasma sheet, J. Geophys. Res., 1, 965-9617, 1995a. Bauer, T.M., W. Baumjohann and R.A. Treumann, Neutral sheet oscillations at substorm onset, J. Geophys. Res., 1, 23,737-23,743, 1995b. Chen, S.-H., and M.G. Kivelson, On ultralow frequency waves in the lobes of the Earth s magnetotail, J. Geophys. Res., 96, 15,711-15,723, 1991. Dunlop, M.W., A. Balogh, K.-H. Glassmeier and P. Roberts, Four-point Cluster application of magnetic field analysis tools: the Curlometer, J. Geophys. Res., in press, 22.

VOLWERK ET AL.: SERPENTINE CS OSCILLATION 13 Harris, E.G., On a plasma sheet separating regions of oppositly directed magnetic field, Nuovo Cimento, 23, 115-121, 1962. Kepko, E.L., K. Khurana, M.G. Kivelson, R.C. Elphic and C.T. Russell, Accurate determination of magnetic field gradients from four point vector measurements Part I: Use of natural constraints on vector data obtained from a single spinning spacecraft, IEEE Transactions on Magnetics, 32, 377-385, 1996. Khurana, K.K., S.-H. Chen, C.M. Hammond and M.G. Kivelson, Ultralow frequency waves in the magnetotail of the Earth and the outer planets, Adv. Space Res., 12, 57-63, 1992. Khurana, K.K., E.L. Kepko, M.G. Kivelson and R.C. Elphic, Accurate determination of magnetic field gradients from four point vector measurements II: Use of natural constraints on vector data obtained from four spinning spacecraft, IEEE Transactions on Magnetics, 32, 5193-525, 1996. Nakamura, R., W. Baumjohann, H. Noda, G. Paschmann, B. Klecker, P. Puhl-Quinn, J. Quinn, R. Torbert, A. Balogh, H. Rème, H.U. Frey, C. J. Owen, A.N. Fazakerley, J P. Dewhurst, Substorm expansion onsets observed by Cluster, Proc. 6th International Conference on Substorms, Seattle, USA, in press, 22. Roberts, B., Wave propagation in a magnetically structured atmosphere I: Surface waves at a magnetic interface, Solar Phys., 69, 27-38, 1981a. Roberts, B., Wave propagation in a magnetically structured atmosphere II: Waves in a magnetic slab, Solar Phys., 69, 39-56, 1981b. Runov, A., R. Nakamura, W. Baumjohann, T.L. Zhang, M. Volwerk, H.-U. Eichelberger and A. Balogh, Cluster observations of a bifurcated current sheet, sumbitted to Geophys.

14 VOLWERK ET AL.: SERPENTINE CS OSCILLATION Res. Lett., 22. Sergeev, V., V. Angelopoulos, C. Carlson and P. Sutcliffe, Current sheet measurements witin a flapping plasma sheet, J. Geophys. Res., 13, 9177-9187, 1998. Smith, J.M., B. Roberts and R. Oliver, Magnetoacoustic wave propagation in current sheets, Astron. Astrophys., 327, 377-387, 1997. Volwerk, M., W. Baumjohann, K.-H. Glassmeier, R. Nakamura, T.L. Zhang, A. Runov, B. Klecker, R. Treumann, Y. Bogdanova, H.-U. Eichelberger, A. Balogh and H. Rème, Compressional waves in the neutral sheet, submitted to Annales Geophys., 22. Zhu, X. and M.G. Kivelson, Compressional ULF waves in the outer magnetosphere 1. Statistical study, J. Geophys. Res., 96, 19,451-19,467, 1991. Zhu, X. and M.G. Kivelson, Compressional ULF waves in the outer magnetosphere 2. A case study of Pc 5 type wave activity, J. Geophys. Res., 99, 241-252, 1994.

VOLWERK ET AL.: SERPENTINE CS OSCILLATION 15 Figure 1. The August 22, 21. Top 3 panels: FGM data in GSM coordinates, in 22 Hz resolution. Bottom 3 panels: CIS CODIF proton flow data, in 4 second resolution. The vertical bar in the panels shows the time of substorm onset for this event as determined from the AE index. Figure 2. Low pass filtering the data with a Butterworth filter (τ > 5 min.). Top panel: The B x component of the magnetic field for all spacecraft. The panels below show the low-passed filtered data for the x, y and z component of the magnetic field, respectively. Figure 3. I: Modeling the current sheet oscillation. The four second resolution (thin line) and the low-pass filtered data (thick dashed line) for C2 and the numerical model (thick line). II: The associated neutral sheet velocities. Shown as vectors are the v x (along time axis) and v z (in vertical direction) components. III: The total current density in the Cluster tetrahedron as determined from the magnetic field in na/m 2 and the x- component of the magnetic field at the center of mass of the Cluster tetrahedron. The dashed vertical line in all panels shows the time of substrom onset. IV: The half thickness of the current sheet (on a log scale) calculated for C1/3/4 (circles) and C2/3/4 (triangles). The thick solid line displays a growth trend of the current sheet thickness.

v z v y v x B z B y B x 2 2 2 2 2 2 75 75 75 75 75 75 9 9.5 1 1.5 11 11.5 12 UT

2 B x (nt) 2 B x,lp (nt) 2 2 B y,lp (nt) 2 2 B z,lp (nt) 2 2 93 945 1 115 115 UT

2 I B x (nt) 2 A B C D II v z v ns 1 km/s v x 2 III BC x, J t 2 3 λ (R E ) 2 1 IV 93 945 1 115 13 UT