Observations of Short-Period Ion-Scale Current Sheet Flapping
L. Richard, Yu. V. Khotyaintsev, D. B. Graham, M. I. Sitnov, O. Le Contel, P.-A. Lindqvist
mmanuscript submitted to
JGR: Space Physics
Observations of Short-Period Ion-Scale Current SheetFlapping
L. Richard , , Yu. V. Khotyaintsev , D. B. Graham , M. I. Sitnov ,O. Le Contel , P.-A. Lindqvist Swedish Institute of Space Physics, Uppsala, Sweden Space and Plasma Physics, Department of Physics and Astronomy, Uppsala University, Sweden The Johns Hopkins University Applied Physics Laboratory, Laurel, Maryland, USA Laboratoire de Physique des Plasmas, CNRS/Ecole Polytechnique IP Paris/SorbonneUniversit´e/Universit´e Paris Saclay/Observatoire de Paris, Paris, France Space and Plasma Physics, School of Electrical Engineering, KTH Royal Institute of Technology,Stockholm, Sweden
Key Points: • Partially demagnetized ions are observed in flapping current sheet • The wavelength of kink-like current sheet scales with the thickness consistent witha drift-kink instability. • The thickness of thin current sheets can be limited by the lower hybrid drift in-stability
Corresponding author: L. Richard, [email protected] –1– a r X i v : . [ phy s i c s . s p ace - ph ] J a n anuscript submitted to JGR: Space Physics
Abstract
Kink-like flapping motions of current sheets are commonly observed in the magnetotail.Such oscillations have periods of a few minutes down to a few seconds and they prop-agate toward the flanks of the plasma sheet. Here, we report a short-period ( T ≈ –2–anuscript submitted to JGR: Space Physics
The kink-like flapping motion is a global oscillation of the magnetotail current sheet(V. Sergeev et al., 2004; V. A. Sergeev et al., 2006) and is correlated with fast flows (V. A. Sergeevet al., 2006), which contributes to the explosive magnetotail activity (M. Sitnov et al.,2019). The period of these wave-like oscillations varies from couple of minutes (T. L. Zhanget al., 2002; V. Sergeev et al., 2003, 2004; Runov et al., 2005; T. Zhang et al., 2005; Laiti-nen et al., 2007; Shen et al., 2008; X. H. Wei et al., 2015; Rong et al., 2015; Gao et al.,2018; Rong et al., 2018; S. Wang et al., 2019) to a couple of seconds (Y. Y. Wei et al.,2019) and they propagate with a phase velocity of the order of tens to hundreds of km s − in the spacecraft frame. Taking advantage of the abundance of the flapping events in themagnetotail, the statistical surveys carried by Yushkov et al. (2016) and Gao et al. (2018)showed that the wave-like magnetotail current sheet observed in the flanks propagatesaway from the midnight sector. However, Gao et al. (2018) also showed that around mid-night sector ( | Y GSM | < R E ) the flapping motion is dominated by up-down steady flap-ping motion.The up-and-down motion in the midnight sector was proposed by Gao et al. (2018)to be a mechanism of generation of the kink-like flapping of the current on both sidesof the midnight sector. External disturbances such as Alfv´enic fluctuations in the inter-planetary magnetic field (Tsutomu & Teruki, 1976), pressure pulses (Fruit, 2002, 2004)and solar wind directional changes (G. Q. Wang et al., 2019) were proposed as possiblemechanisms to trigger kink-like flapping motions. The magnetohydrodynamics eigenmodesanalysis of the magnetic double gradient mechanism (Erkaev et al., 2008) pointed outthat a gradient of the normal magnetic field induced by a local thinning of the currentsheet, can trigger the flapping instability.The kinetic treatment of the drift-kink and ion-ion kink instability proposed by Lapentaand Brackbill (1997) and Karimabadi (2003) using a Harris model of the current sheet,concluded that the propagation of the wave-like current sheet was in the direction of theion drift which agrees with direction of propagation of the flapping motions observed inthe statistical studies (Yushkov et al., 2016; Gao et al., 2018). Indeed, Kissinger et al.(2012) showed that the flows in the tail are diverging toward the flanks which coincidedwith the direction of propagation of the flapping motions. Using a model of an anisotropicthin current sheet, Zelenyi et al. (2009) argued that kink-like flapping waves are low-frequencyeigenmodes of the current sheet, which possess properties of drift waves. Indeed, obser-vations of flapping events by Cluster reported by Zelenyi et al. (2009) showed that kink-like flapping of thin current sheets propagate along the current density at velocities closeto the relative drift velocity of ions and electrons.The full particle simulation carried out by M. I. Sitnov et al. (2006) showed thatthe shape of the flapping wave of thin current sheets differs from the quasi-rectangularshape of thicker sheets. However, considering the normal electric field arising from thedifference of velocities between ions and electrons at such scales, M. I. Sitnov et al. (2006)argued that the thinning of the current sheet is limited.Nevertheless, three questions remain unsolved. First, how do ions behave in an ionscale flapping current sheet? Second, does the usual flankward propagation of the wave-like current sheet differ far from the midnight sector? Finally, does the thickness of thekink-like current sheet scale with its wavelength? And if so, what processes balance thethinning of the current sheet? Is it the charge separation in ion and sub ion-scale cur-rent sheets as suggested, for instance, by (M. I. Sitnov et al., 2006) or flapping motions?In this paper, we use multi-spacecraft analysis on a short period kink-like flapping of athin current sheet. This event was observed in the flank of the Earth by the Magneto-spheric Multiscale (MMS) mission (Burch et al., 2016), preceded by a sunward burstybulk flow of ions and electrons associated with a dipolarization front. –3–anuscript submitted to JGR: Space Physics
We investigate a flapping event observed by the Magnetospheric Multiscale (MMS)mission (Burch et al., 2016) on 14 September 2019 between 07:54:00 UTC and 08:11:00UTC. At this time, the MMS probes were located on the dusk side of Earth ( X GSE = − . R E , Y GSE = 10 . R E , Z GSE = 6 . R E ) with a spacecraft separation of ∼ ∼ . d i ) of the tetrahedron formed by the MMS constellation,and because we will use the curlometer technique (Dunlop et al., 1988) to compute thecurrent density, we present all quantities at the center of mass of the tetrahedron formedby the MMS constellation. An overview of the event is shown in Figure 1. We observe that the magnetic fieldis mainly along the Y axis (GSE) due to the duskside and northward location of MMS,which can be compared with the magnetotail configuration where the magnetic field ismainly in the X direction. The direction of the magnetic field is the same at the begin-ning and the end of the time interval, which indicates that the spacecraft remain on thesame side of the plasma sheet. We observe a typical plasma sheet ion energy spectrumand ion temperature in Figure 1c. Both the magnetic field and the ion energy spectrumindicate that the spacecraft are far from the lobe and inside the plasma sheet. The ionbulk velocity, plotted in Figure 1b, shows 3 jets. While we observe large-amplitude os-cillations in the magnetic field during the two first jets, the third one is associated witha unidirectional northward magnetic field suggesting that it is a dipolarization front. Weobserve in Figure 1d a slow isotropic decrease in the electron temperature behind thedipolarization front. Following the dipolarization front embedded in the third plasmajet, we identify in Figure 1e, large-amplitude oscillations of the three components of themagnetic field whose period varies from 10 s to 100 s, which is consistent with the usualperiod of flapping current sheets in Earth’s magnetotail (Y. Y. Wei et al., 2019). Theobservation of flapping motions in the wake of the dipolarization jet is consistent withthe correlation between flapping motions and bursty bulk flows observed by V. A. Sergeevet al. (2006).
We apply the Minimum Variance Analysis (MVA) (Sonnerup & Scheibe, 1998) tothe magnetic field over the entire time interval of the flapping motion ([07:54:00, 08:11:00]),to estimate the global
LMN coordinates system. Here, L is the direction of the max-imum variance of the magnetic field, N the normal to the mean plane of the oscillatingcurrent sheet, and M = N × L . The coordinates of the three vectors are : L = [ − . , . , . M = [ − . , − . , − .
43] and N = [ − . , − . , .
76] in GSE coordinates.First we investigate the behavior of the ions during the oscillations of the magneticfield. From Figure 2c we can see that the ion and electron bulk velocity normal to themean current sheet plane, V iN and V eN , are oscillating together with the magnetic fieldbut have a phase shift of 90 ◦ with respect to B L . In other words, the normal bulk ve-locity peaks in the center of the current sheet where B L = 0, decreases as the space-craft moves away from the current sheet, and reverses at the | B L | maxima. We call theregion of maximum | B L | the current sheet edge. The plasma density plotted in Figure –4–anuscript submitted to JGR: Space Physics
2d has a minimum at the current sheet edges and peaks in the current sheet center, whichindicates the presence of density gradients normal to the current sheet.Even though the ion and electron normal bulk velocities, V iN and V eN , have thesame phase, V iN is much lower than V eN so that the ion and electron motions are de-coupled from each other. Since the magnitude of the magnetic field plotted in Figure 2a(green) has non-zero minima ∼ ρ e < B [ n T ] (a) B x B y B z V i [ k m s ] (b) V ix V iy V iz E i [ e V ] (c) T i , T i , E e [ e V ] (d) T e , T e , B [ n T ] (e) Sc position : [-6.1,11.3, 6.4] R E GSE B x B y B z D E F [ k e v / ( c m s s r k e V ) ] D E F [ k e v / ( c m s s r k e V ) ] Figure 1.
Overview of the flapping event observed by MMS on September 14th 2019. (a,e)Magnetic field in GSE coordinates, (b) ion bulk velocity in GSE coordinates, ion energy spectrum(c) and (d) electron energy spectrum. –5–anuscript submitted to
JGR: Space Physics × km) so that the electrons are still magnetized. This difference between the ionsand the electrons bulk velocity suggests that the substantial part of the ion populationis demagnetized, and hence the thickness of the current sheet is comparable to the ionLarmor radius.We also observe in Figure 2b that the M-component of the ion bulk velocity is ratherconstant during the time interval, V iM ∼ −
150 km s − , while V eM oscillates aroundthe ion bulk velocity V iM . This is an indication that the current sheet is a thin currentsheet, embedded in a thicker current sheet. An illustration of the electron motions in theframe of the ions is shown in Figure 3d, where the red dots indicate the electrons. Thedemagnetized ions move mainly in the − M direction,while the electrons remain frozen-in and follow the oscillations of the current sheet in the N direction.Figure 2e shows the current density J computed using the curlometer technique. J N oscillates around zero with the oscillation period similar to the magnetic field. Thisoscillating current density arises from the decoupling between the ions and the electronsseen in Figure 2b and 2c. The M-component of the current density J M oscillates withthe same frequency as J N but with a 90 ◦ phase shift. We observe that J M remains neg-ative during the flapping event, thus there is an overall current flowing in the − M di-rection. J M peaks at the center of the current sheet and goes to zero at the current sheetedges where V iM and V iN reverse sign and the M-components of the flow are equal toeach other, V iM = V eM . To fully characterize the geometry of the oscillating current sheet, we now focuson its thickness. First, we plot the distribution of the magnitude of the current density J MN with respect to the magnetic field B L in Figure 3a. We compare this distributionto the scaling for the Harris model (black solid line): J MN = B µ h H (cid:34) − (cid:18) B L B (cid:19) (cid:35) , (1)where B is the field outside of the current sheet (lobe field) and h H is the current sheetthickness. We see that the Harris scaling with h H = 3 . d i and B = 15 nT fits theobserved distribution rather well.The lobe field B can be alternatively obtained from the pressure balance condi-tion B P BC = ( B + 2 µ P i ) / , which for our case yields B P BC = 45 nT. However,Figure 3a shows that the current density becomes very small for a much smaller valueof B = 15 nT, which indicates that the observed thin and intense current sheet is likelyembedded into a thicker one. This situation resembles the current distributions Figure2 in Runov et al. (2006).The thickness h H obtained above is a global average current sheet thickness. Inorder to access the local scale of the core current sheet, we compute the thickness of thecurrent sheet at every crossing of the current sheet center. We estimate the local thick-ness h of the current sheet as: h = B µ | J | B =0 , (2)where | J | B =0 is the current density estimated by the curlometer technique at the neu-tral line ( B L = 0) and B = 15 nT is the magnetic field at the current sheet edgesobtained earlier. Figures 3b shows the thickness (red vertical lines) computed using equa-tion (2). We see that the thickness of the current sheet experiences significant variations.The thinnest current sheet is observed during the interval 07:59:30–08:01:15 UT (Fig- –6–anuscript submitted to JGR: Space Physics ure 3c), which corresponds to the interval where the oscillations of B and plasma param-eters have the smallest period (see Figure 2). Note that in Figures 3b and 3c, the ordi-nate is the spatial scale computed using the spatiotemporal derivative technique (see be-low) in the direction of propagation of the wave-like structure. The thinnest current sheet B [ n T ] (a) B L B M B N | B | V M [ k m s ] (b) Ions Electrons4002000200400 V N [ k m s ] (c) Ions Electrons0.100.150.200.250.30 N e [ c m ] (d) J [ n A m ] (e) J M J N B L [ n T ] B L [ n T ] B L [ n T ] B L [ n T ] Figure 2. (a) Magnetic field in the global minimum variance frame LMN, (b) Ion bulk ve-locity (blue) and electron bulk velocity (red) in the M direction, (c) ion bulk velocity (blue) andelectron bulk velocity (red) in the N direction, (d) ion (blue) and electron (red) number density,(e) Current density from curlometer technique in the M (orange) and N (green) directions. Greylines in panels (b-e) show the magnetic field B L .–7–anuscript submitted to JGR: Space Physics we observe has h = 1 . d i , which is nearly three times smaller than the average thick-ness h H . Thus, at this time the current sheet is very thin, comparable to the ion iner-tial length scale, which is consistent with the observed ion demagnetization.In order to estimate the local scale of the oscillating current sheet and its geom-etry, we compute the local velocity of the wave-like structure using the spatio-temporalderivative (STD) method (Shi et al., 2006). Assuming that the structure is quasi-stationary,we compute the velocity of the structure as V str = − d t B [ ∇ B ] T [ S ] − (3)where V str and S are the velocity of the structure and the rotation rate tensor, respec-tively, in the LMN coordinates system. Note that we cannot use the maximum deriva-tive direction frame (Shi et al., 2006) as a coordinate system due to large errors causedby the small spacecraft separation ( ∼
10 km). Indeed, with such small spacecraft sep-aration, the difference of the measured magnetic field between two spacecraft is so smallthat the uncertainty in the magnetic field will be magnified in the gradient estimate mak-ing the eigenvectors of S unstable.In addition to the STD method providing a continuous velocity estimate, we usethe timing method (Vogt et al., 2011) at every crossing of the current sheet ( B L = 0)and project the velocity of the structure onto the global LMN coordinates system. Theresulting velocities from the two methods are shown in Figures 4b and 4c. The two meth-ods provide similar results, which agree particularly well during the short period train07:59:30–08:01:15 UT, where the velocity increases. Besides, we observe that the N-componentof velocity oscillates around zero, while the M-component is always negative, suggest-ing that the M direction corresponds to the direction of propagation of the flapping mo-tion. We also note that the obtained velocity of the flapping structure is close to the ionbulk velocity (blue). Thus we conclude that the flapping structure propagates in the − M direction (the average current direction) and the structure is close to being stationaryin the ion frame of reference.Integrating the velocity of the current sheet given by the STD method, V str , overtime, we obtain the current sheet geometry in the MN plane. Assuming that the space-craft are stationary while the oscillating current sheet moves past the spacecraft, we plotin Figure 3b and 3c the position of the current sheet center with respect to the space-craft (blue). We observe that as the period of the oscillations changes in time from 10s to 1 min, the wavelength of the oscillating current sheet varies from 5 to 10 d i , but re-mains comparable to the peak-to-peak amplitude of the oscillations ∼ d i .The maximum value of the tilt angle α = tan − ( V strN /V strM ) | B =0 , shown in Fig-ure 3d, between the current sheet and the LM plane is ∼ ◦ which gives a wavy shapeto the current sheet. The picture given by the Figure 3d is either compressed or stretchedin the direction of propagation of the wave-like structure as the wavelength becomes shortor long. Hence, for short wavelength the tilt angle α increases and in the special case ofan infinite wavelength we recover the 2D picture of a thin current sheet. To better characterise the force balance in the current sheet, we compute the dif-ferent contributions to the Ohm’s law. Since we observe a current sheet with a thick-ness h ∼ d i , we focus on the ion scale contributions to the Ohm’s law, E + V i × B = J × B ne − ∇ P e ne , (4)where E is the electric field measured by the EDP, V i × B is the ion convection term,( J × B ) /ne is the Hall term and ( ∇ P e ) /ne is the electron pressure gradient term. –8–anuscript submitted to JGR: Space Physics
We present these terms in the directions of propagation M and flapping N of thekink-like structure, as well as the non-ideal contribution | E + V i × B | in Figure 5. Weobserve in Figure 5d and 5h, that there is a non-negligible non-ideal electric field (pink),which is balanced by the Hall term (black). This means that the electron pressure gra-dient term is small and can be neglected. The largest contribution to the non-ideal fieldis coming from the M component (Figure 5b and 5f). The measured electric field (green)is balanced by the Hall term (black), while the ion convection term (blue) is nearly zero.The electric field is spiky with E M peaks reaching E M ≈
10 mV m − . The peaks arelocated away from the center of the current sheet. However, at the center of the currentsheet E M is zero, so that there is a reversal of E M at B L = 0. E M also reverses signat the maxima of B L (current sheet edges). The electric field E M ∼ J N B L /ne ∼ − V eN B L ,where the ion velocity is neglected in comparison with the normal electron velocity (Fig-ure 2c). Thus, E M = 0 where B L = 0 (center of the current sheet) and where V eN =0 (current sheet edges). The Hall electric field comes from the fact that the ion motionis decoupled from the electron motion, which is possible at scales below the gyroradius.Therefore, the presence of the Hall electric field supports the observation of the ion scalecurrent sheet.We observe that during the high-frequency oscillations of B L , the amplitude of thenormal electric field E N presented in Figure 5c and 5h is smaller compared with E M . Figure 3. (a) Histogram of the current density with respect to the magnetic field. The blacksquares represent the median current density and the errorbars the corresponding standarddeviation for 2 nT wide bins of the magnetic field, and the solid black line is the Harris approxi-mation of the current density 1. (b, c) Two-dimensions geometry of the oscillating current sheet(blue solid line) and its thickness (red bars). (d) Interpretation of the electron and current sheetmotion in the ion frame. –9–anuscript submitted to
JGR: Space Physics B [ n T ] (a) B L B M B N V M [ k m s ] (b) Ions STD Timing07:55:00 08:00:00 08:05:00 08:10:00 V N [ k m s ] (c) Electrons STD Timing
Figure 4. (a) Magnetic field in the global LMN coordinates system. (b) Ion velocity (blue)and velocity of the current sheet from spatiotemporal derivative (black) and timing (green) alongthe M direction. (c) Electron velocity (red) and velocity of the current sheet from spatiotemporalderivative (black) and timing (green) along the flapping direction.–10–anuscript submitted to JGR: Space Physics
This normal electric field E N is balanced by the ion convection term, while the Hall termis small. Similarly to E M , the normal electric field E N reverses at the center of the cur-rent sheet. However, unlike E M , the normal electric field does not reverse sign at the cur-rent sheet edges. We expect the Hall electric field to be normal to the current sheet. How-ever, here we observe that the Hall electric field is mostly in the M direction which sug-gests that the current sheet is strongly tilted. We showed earlier that the wave-like structure is propagating along the current di-rection. We now focus on the estimation of the phase velocity of this wave-like motion.To do so, we use the velocity of the structure obtained by the timing method applied tothe crossings of the current sheet center. We compute the wavelength using λ = V str T ,where V str is the magnitude of the velocity given by the timing method along the esti-mated direction of propagation, and T is the period of the oscillations which can be es-timated as the time between two current sheet edges. To plot the result in a more usualway in the Figure 6c, we plot k = 2 π/λ and the angular frequency ω = 2 π/T . In or-der to compare with other studies, we use the data provided by Y. Y. Wei et al. (2019)(red dots). The crossings of the current sheet which are not related to the main phaseof flapping motion are considered as outliers and marked by the black dots. Finally, alinear fit is applied, which yields the phase speed V ph ≈ ±
10 km s − , which is con-sistent with the short period current sheet flapping observed by Y. Y. Wei et al. (2019)We recall that the current density normal to the mean plane J N oscillates aroundzero with the same period as the magnetic field (Figure 2f). In Figure 6a we observe astrong correlation between J N and the time derivative of the magnetic field ∂ t B L . Us-ing the Ampere’s law: µ J N = ( ∇ × B ) N = ( ik L B M − ik M B L ) ≈ − ik M B L Thus, the correlation between J N and ∂ t B L can be written as : µ J N ≈ − ik M B L = − iµ aωB L ⇒ V ph = ωk M ≈ [ aµ ] − (5)The linear fit µ J N = a∂ t B L gives a = 3 . ± .
01 H − s, which in turns gives V ph =255 ± − consistent with the timing method.Finally, as shown in Figure 3b and 3c, the thickness of the current sheet decreasesas the wavelength decreases. Thus, we now investigate a possible scaling of the thick-ness of the current sheet with the wavelength. Figure 6d shows the thickness of the cur-rent sheet h as a function of the wavenumber k . As in Figure 6c we also provide the val-ues from Y. Y. Wei et al. (2019). Figure 6b presents the distribution of the scales of thethickness with respect to the wavelength of the oscillating current sheet. Using a 1 /k fit(black line), we conclude that the thickness scales with the wavelength as h ≈ . λ .The theoretical values of the wavenumber in terms of current sheet thickness ob-tained in the linear analysis of drift-kink instability (Zelenyi et al., 2009) are shown bythe orange lines in Figure 6d (orange box in Figure 6b). We observe that the theoret- –11–anuscript submitted to JGR: Space Physics ical prediction agrees with the observed scales which suggests that the drift-kink insta-bility may be responsible for the observed wave-like motion of the current sheet. B [ n T ] (a) B L B M B N E M [ m V m ] (b) J × B V i × B J × B V i × B E
EDP E N [ m V m ] (c) J × B V i × B J × B V i × B E
EDP | E | [ m V m ] (d) E + V i × B J × Bne B [ n T ] (e) B L B M B N E M [ m V m ] (f) J × B V i × B J × B V i × B E
EDP E N [ m V m ] (g) J × B V i × B J × B V i × B E
EDP | E | [ m V m ] (h) E + V i × B J × Bne
Figure 5. (a, e) Magnetic field in the global LMN coordinates system, (b, c, f, g) electricfield perpendicular to the maximum variance magnetic field with Hall contribution (black), ionconvection (blue), Hall + ion convection (red) and electric measured by the EDP (green). (d, h)Magnitude of the non-ideal electric field (pink) and the Hall contribution (black)–12–anuscript submitted to
JGR: Space Physics
In addition to the low-frequency oscillations due to the flapping itself, higher fre-quency fluctuations of the electric field are observed at the current sheet edges. Figure 7bshows the electric field fluctuations δ E above 4 Hz, together with the plasma parame-ter β (green). We observe that the amplitude of the high-frequency oscillations of the
10 5 0 5 10 d t B L [nT s ] J N [ n A m ] (a) J N = -3.12*d t B L CI 95%0.1 0.2 0.3 0.4 0.5 0.6 h / (b) 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 k [km ] [ r ad s ] (c) = 262.95* k CI 95%2019-09-14Table Wei20190.0000 0.0005 0.0010 0.0015 0.0020 0.0025 k [km ] h [ k m ] (d) h = *
95% CI2019-09-14Table Wei201910 Figure 6. (a) Normal current density J N with respect to the time derivative of the magneticfield d t B L . The colors indicate the distribution of the values and the black line a y = ax fit (seetext). (b) Distribution of the thickness to wavelength ratio during the event. (c) Frequency ω of the oscillations of the magnetic field B L with respect to the wavenumber k given by the tim-ing method. (d) Current sheet thickness with respect to the wavenumber k given by the timingmethod. The blue dots in panels (c) and (d) are the current sheet crossings of the event reportedin this paper and the red dots are other studies of flapping motions in the tail summarised byY. Y. Wei et al. (2019). The black dots are the crossings far from our event that are consid-ered as outliers. The orange lines in panel (d) and orange bow in panel (b) show the theoreticalprediction of the range of wavenumber in terms of current sheet thickness which maximize thegrowth rate of the drift-kink instability. –13–anuscript submitted to JGR: Space Physics electric field are almost zero at the local maxima of β (current sheet center), while theamplitude of the high-frequency electric field peaks at the minima of β (current sheetedges). Thus, it suggests that the high-frequency electric field waves grow at the cur-rent sheet edges and are suppressed within the current sheet which is consistent with Yoonet al. (2002) and Daughton (2003).Furthermore, the electric field spectrum presented in Figure 7c and the magneticfield spectrum presented in Figure 7d show that these fluctuations are essentially elec-trostatic waves with a frequency close to the lower hybrid frequency (black). Thus, weidentify these waves as lower hybrid drift waves (LHDW). It is important to note thatthese fluctuations are seen only before 08:01:10, in the region where the current sheetthickness remains small ( h ∼ . d i ) and the density gradients observed in Figure 2dare large. At later times, as the current sheet becomes thicker ( h > d i ) and the den-sity gradients become smaller, the LHDW vanish. This suggests that the LHDW are drivenby the density gradients arising from the thinning of the current sheet. We observed that the wave-like current sheet propagates along − M = [0 . , . , . − M = [0 . , . , .
43] GSE,which is consistent with the usual direction of propagation of wave-like current sheetsobserved in the tail, and suggests that this property can be extended to the flanks. Theobserved propagation velocity is ∼
200 km s − in the spacecraft frame is close to the ionbulk flow velocity, and thus the structure is nearly stationary in the ion frame of refer-ence. Similar slow propagation speeds for flapping structures were observed in PIC sim-ulations (M. I. Sitnov et al., 2014).The observed wavelength of the flapping waves is ranging between 2 . × and3 . × km (5.5 and 56 d i ). This is clearly much longer than the prediction given byDaughton (2003) and elaborated by M. I. Sitnov et al. (2014) for a long-wavelength ex-tension of the lower hybrid drift instability (LHDI) with λ ∼ π √ ρ i ρ e ∼ . d i . Theshortest wavelength we observe is λ ∼ . d i which is close to the flapping waves behindthe dipolarization front in 3D PIC simulations (M. I. Sitnov et al., 2014).The average thickness of the observed current sheet is 3 . d i . The thickness variesduring the event and reaches a minimum thickness h = 1 . d i = 0 . ρ i . The observedgeometry of the current sheet presents a wavy shape which resembles the one seen in sim-ulations of thin h ≈ ρ i current sheets (Lapenta & Brackbill, 2002; M. I. Sitnov et al.,2006). The wavelength λ of the observed wave-like current sheet scales with the thick-ness h as h ∼ . λ , or kh ∼ .
69, where k = 2 π/λ is the wavenumber. This observedscaling of the wavelength of the wave-like current sheet with its thickness can be usedin further development of the theory of the flapping motions.This observed scaling can be used to characterize the instability which causes theobserved wave-like structures. Linear stability analysis of an ion-scale Harris sheet, h = ρ i , showed that the maximum growth rate of the kink mode corresponds to kh ∼ m p /m e = 64 (Daughton, 1999). Also, using a thin anisotropic current sheet model, Zelenyiet al. (2009) showed that the maximum growth rate of the drift-kink instability belongsto the range kh ∈ [0 . , .
8] depending on the propagation angle. The observed scal-ing kh = 1 .
69 is close to the expected scaling for the drift-kink instability, which pointsat this instability as a likely cause of the observed wave-like structures.We have shown that in the observed thin current sheet, the ions are partially de-magnetized. We also observe a Hall electric field consistent with the demagnetization –14–anuscript submitted to
JGR: Space Physics of the ions. In the PIC simulation of super thin current sheet ( h < ρ i ) an electric fieldnormal to the current sheet was observed and it was suggested to be important for con-trolling the thickness of the thin current sheet (M. I. Sitnov et al., 2006). The observedHall electric field has a large component in the direction of propagation of the wave-likestructure; we expect the Hall field to be locally normal to the current sheet, so this means B [ n T ] (a) B L B M B N E [ m V m ] f > E E E f [ H z ] (c) f lh f ce f pi f [ H z ] (d) f lh f ce f pi E [ m V m H z ] B [ n T H z ] Figure 7. (a) Magnetic field in the global LMN coordinates system. (b) Electric field fluctua-tion above 4 Hz in field aligned coordinates together with the plasma β (green). (c) Electric fieldspectrum and (d) magnetic field spectrum. The black, white and red lines in panels (c) and (d)show the lower hybrid frequency, the electron cyclotron frequency and the ion plasma frequency.–15–anuscript submitted to JGR: Space Physics that the observed current sheet is strongly tilted (tilt angle α > ◦ , see Figure 3) andthe local normal to the sheet is very different from the global N direction, i.e. the lo-cal normal is close to the global M direction (the direction of propagation of the wave-like structure). This resembles the strongly-tilted configuration ( α ∼ ◦ ) in Figure 16of M. I. Sitnov et al. (2006).One possible mechanism to limit the thickness of the current sheet is through lowerhybrid drift instability (LHDI). We observed such waves associated with strong gradi-ents of density at the thinnest current sheet crossings. Similar strong gradients are foundin the PIC simulation of thin current sheet (M. I. Sitnov et al., 2006). The non-linearLHDI could modify the initial current sheet equilibrium and enhance the growth rateof the kink instability (Daughton, 2003). The amplitude of the observed LHDW changeswith the plasma parameter β , consistent with damping in the high-beta region in thecurrent sheet center (Bale et al., 2002). The LHDI can be excited by the strong gradi-ents, and will broaden these gradients (Winske & Liewer, 1978). Thus, our observationssuggest that the LHDI can be important for limiting the thinning of the current sheet. We have presented observations of a short-period (wave period T ≈
25 s = 35 f − ci )kink-like flapping of a thin current sheet in the flank of the Earth’s magnetotail. Thewave-like flapping structure propagates along the current direction.We find that the observed current sheet thickness varies during the time we observethe flapping, and the minimum thickness is comparable to the ion inertial length. Thewavenumber of the flapping oscillations scales with the current sheet thickness as kh =1 .
69. Such a value of the wavenumber in terms of current sheet thickness belongs to thedomain kh ∈ [0 . , .
8] of predicted maximum growth rate of the drift-kink instability,suggesting this instability can be responsible for the observed flapping.The observed small value of the current sheet thickness is consistent with the ob-servation of a decoupling between ions and electrons, which is related to the fact thatthe electrons are well magnetized but the ions are partially demagnetized. A large Hallelectric field arises because of the small thickness of the current sheet. We observe thatthis Hall electric field is directed mainly in the direction of propagation of the wave-likestructure which indicates that the current sheet is locally strongly tilted.At the times when the current sheet thickness approaches the smallest ion-inertial-length scale we observed strong density gradients and associated lower hybrid drift waves(LHDW) at the edges of the current sheet, which suggest that the LHDW are generatedin response to the thinning of the current sheet. Thus, the LHDW can potentially pro-vide a mechanism limiting the further thinning of the current sheet.
Acknowledgments
We thank the entire MMS team and instrument PIs for data access and support. All ofthe data used in this paper are publicly available from the MMS Science Data Center https://lasp.colorado.edu/mms/sdc/ . We also wish to thank Victor Sergeev for valu-able discussions. Data analysis was performed using the pyrfu analysis package availableat https://github.com/louis-richard/irfu-python . The codes to reproduce the fig-ures in this paper are available at https://github.com/louis-richard/flapping . Thiswork is supported by the Swedish National Space Agency grant 139/18.
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