Electron crescent distributions as a manifestation of diamagnetic drift in an electron scale current sheet
A.C. Rager, J.C. Dorelli, D.J. Gershman, V. Uritsky, L.A. Avanov, R.B. Torbert, J.L. Burch, R.E. Ergun, J. Egedal, C. Schiff, J.R. Shuster, B.L. Giles, W.R. Paterson, C.J. Pollock, R.J. Strangeway, C.T. Russell, B. Lavraud, V.N Coffey, Y. Saito
EElectron crescent distributions as a manifestation of diamagneticdrift in an electron scale current sheet
A. C. Rager , , J. C. Dorelli , D. J. Gershman , V. Uritsky , , L. A. Avanov , , R. B.Torbert , , J. L. Burch , R. E. Ergun , J. Egedal , C. Schiff , J. R. Shuster , , B. L. Giles , W. R. Paterson , C. J. Pollock , R. J. Strangeway , C. T. Russell , B. Lavraud , V. NCoffey , Y. Saito , Catholic University of America, Washington DC, USA NASA Goddard Space Flight Center, Greenbelt, MD, USA University of Maryland, College Park, MD, USA University of New Hampshire, Durham, NH, USA Southwest Research Institute, San Antonio, TX, USA University of Colorado Boulder, Boulder, CO, USA University of Wisconsin, Madison, WI, USA Denali Scientific, Healy, AK University of California, Los Angeles, CA, USA Research Institute in Astrophysics and Planetology, Toulouse, France NASA Marshall Space Flight Center, Huntsville AL, USA Institute for Space and Astronautical Science, Sagamihara, Japan
Key Points: • Diamagnetic drift explains out-of-plane current in regions of deviation from frozen-influx • Perpendicular crescents exist in regions where electrons are diamagnetically drifting • New technique for extracting 7.5 ms electron moments produces reliable data
Corresponding author: A.C. Rager, [email protected] –1– a r X i v : . [ phy s i c s . s p ace - ph ] N ov bstract We report Magnetospheric Multiscale observations of electron pressure gradient electricfields near a magnetic reconnection diffusion region using a new technique for extracting 7.5ms electron moments from the Fast Plasma Investigation. We find that the deviation of theperpendicular electron bulk velocity from E × B drift in the interval where the out-of-planecurrent density is increasing can be explained by the diamagnetic drift. In the interval wherethe out-of-plane current is transitioning to in-plane current, the electron momentum equationis not satisfied at 7.5 ms resolution. Magnetic reconnection is often invoked to explain the rapid conversion of magneticenergy into plasma energy in astrophysical and laboratory plasmas. In our solar system,magnetic reconnection is the primary mode by which the solar wind couples electrodynam-ically to magnetized bodies, producing open magnetic topologies and enabling the trans-port of mass, momentum and energy from the solar wind into planetary magnetospheres.While computer simulations have produced a wealth of predictions about the electron scaleproperties of reconnection [
Hesse et al. , 2011;
Chen et al. , 2016;
Shay et al. , 2007;
Jain andSharma , 2009;
Bessho et al. , 2016;
Hesse et al. , 2014], there have been few direct measure-ments to test these predictions.The Magnetospheric Multiscale (MMS) mission was designed to study the basic physicsof magnetic reconnection in Earth’s magnetosphere, resolving both the fields and plasma onelectron time scales for the first time [
Pollock et al. , 2016;
Burch et al. , 2016a]. MMS con-sists of four spacecraft flying in a close tetrahedral formation (nominal spacecraft separationsare ∼
10 km). The close formation and high quality of the MMS tetrahedron permits the ac-curate calculation of sub-ion scale spatial gradients, allowing for the first time a direct test ofexact plasma fluid equations.The MMS Fast Plasma Investigation (FPI) uses a suite of 64 top-hat spectrometers tosample the three-dimensional velocity space every 30 ( ) ms for electrons (ions) [ Pollocket al. , 2016]. The 30 ms resolution electron measurements from the FPI Dual Electron Spec-trometer (DES) have resulted in the first electron scale measurements of a dayside magne-topause current sheet associated with magnetic reconnection [
Burch et al. , 2016b].
Burchet al. [2016b] reported electron crescent shaped velocity distributions consistent with thoseobserved in two-dimensional particle-in-cell (PIC) simulations near the electron diffusionregion (EDR) (e.g.,
Hesse et al. [2014]), suggesting that the EDR was contained within theMMS tetrahedron.Several explanations of the electron crescent distributions have since appeared in theliterature.
Bessho et al. [2016] modeled electron Speiser orbits [
Speiser , 1965] in a one-dimensional current sheet with a normal electric field, using Liouville’s theorem to showhow crescents can be produced from an assumed isotropic velocity distribution at the mag-netic neutral sheet.
Shay et al. [2016] used a similar model to explain the crescents as conse-quence of cusp-like electron orbits resulting from acceleration by the normal electric field.Both
Bessho et al. [2016] and
Shay et al. [2016] invoke meandering electron orbits toexplain the crescents, suggesting that the observation of crescents can, by comparison withtwo-dimensional PIC simulations, be used to infer proximity to the EDR. In contrast,
Egedalet al. [2016] argue that the electron crescents can be understood by a simple drift-kineticmodel in which the non-gyrotropic electron distributions observed by MMS can be expressedin terms of an equivalent guiding center distribution: f ( x , v , t ) = F g ( X g , v − v g , v (cid:107) , v ⊥ , t ) (1) –2– here x and v are the electron position and velocity, v g is the guiding center drift (includingthe E × B , magnetic gradient and curvature drifts), X g ≡ x − ρ ( x , v , t ) is the electron guidingcenter location, ρ ≡ v × b / Ω e is the electron gyroradius vector, b is the unit vector in thedirection of the magnetic field B , Ω e = qB /( m e c ) is the electron gyrofrequency, and v (cid:107) andv ⊥ are the electron velocity components parallel and perpendicular to the magnetic field.Note that Eqn. (1) makes no assumption about the size of the electron gyroradius rel-ative to the scale over which F g ( X g , v − v g , v (cid:107) , v ⊥ , t ) varies; one only assumes that in theframe of the guiding center drift all of the gyrophase dependence of f ( x , v , t ) can be ex-plained by spatial structure of the gyrotropic guiding center distribution. In particular, thatEqn. (1) allows large deviations from gyrotropy in the electron velocity phase space densitymeasured at a given point and associated perpendicular currents despite the fact that the elec-trons are strongly magnetized.To understand how strongly magnetized electrons can produce a significant perpendic-ular current in the E × B frame, we consider the electron momentum equation, neglecting theinertia terms: ne E + ne V e × B c + ∇ · P e = n is the plasma density (quasineutrality assumed), V e is the electron bulk velocity, and P e is the electron pressure tensor (defined in the electron bulk flow frame). Separating theelectron pressure tensor into its gyrotropic and non-gyrotropic components, P e = P eg + Π e (where P eg = P e (cid:107) bb + P e ⊥ ( I − bb ) , P e ⊥ = [ Tr ( P e ) − P e (cid:107) ]/
2, and Π e is the non-gyrotropiccomponent), the perpendicular component of (2) can be written as follows: ne E ⊥ = − ne V e × B c − ∇ ⊥ P e ⊥ − ( P e (cid:107) − P e ⊥ ) κ − (∇ · Π e ) ⊥ (3)where κ = b · ∇ b is the magnetic curvature. Equation (3) shows that the electron perpen-dicular bulk velocity can differ significantly from c E × B / B (where B is the magnetic fieldmagnitude) even when the divergence of the non-gyrotropic component of the pressure ten-sor vanishes. The essential point is that sub-ion scale electron pressure gradients and associ-ated electron diamagnetic drift, represented by the second term on the right hand side of (3),may produce significant electron current density in the E × B frame even when the electronsare strongly magnetized [ Hoffman and Bracken , 1965]. We emphasize, however, that suchdiamagnetic drift should also be present at an asymmetric current sheet in which electronsexhibit meandering orbits from the high density magnetosheath to the low density magneto-sphere.
Torbert et al. [2016] examined the terms in the generalized Ohm’s law for the
Burchet al. [2016b] EDR event and found significant deviations of the perpendicular electron bulkvelocity from c E × B / B that were associated with the divergence of the full electron pressuretensor; however, they did not further separate the electron pressure tensor into its gyrotropicand non-gyrotropic components, so that an important question remains unanswered: Are thenon-gyrotropic electron crescent distributions observed by Burch et al. [2016b] a manifesta-tion of the electron diamagnetic drift of strongly magnetized electrons in a thin sub-ion scalecurrent sheet? In what follows, we address this question using a new method we have devel-oped to extract 7 . . . –3– Data and Results
In Figure 1 (a)-(d) we show the four 7 . Figure 1. (a)-(d) Four intermediate electron velocity distributions that are used to generate (e) the 30 msFPI electron velocity distribution. (f) The simple image composite of distributions (a)-(d) for comparison tothe 30 ms distribution.
Burch et al. [2016b] demonstrated that MMS encountered an electron diffusion re-gion (EDR) on 2015-10-16. They identified the EDR based on several criteria: 1) a bipolarexhaust signature in the L component (in boundary normal coordinates) of the ion bulk ve-locity, 2) a strong perpendicular current in which the electron perpendicular bulk velocitydiffers significantly from E × B drift, 3) a strong depression in the magnetic field magnitude(suggesting a magnetic null in the reconnection plane), 4) strong parallel electron heating,5) a strong electric field pointing outward along the current sheet normal, 6) crescent shapedelectron velocity distributions, and 7) a strong J ⊥ · E ⊥ signature in the electron rest frame.The presence of the electron crescent velocity distributions supporting J ⊥ · E ⊥ > Hesse et al. [2014]) as a feature of the flow stagnation region. How-ever, it is interesting to note that the onset of perpendicular crescents observed by MMS2(at about 13:07:02.16 UT) is during the rising edge of J M and earlier than the onset of large –4– mplitude electric field fluctuations (at about 13:07:02.2 UT). Recently, Burch et al. [2017]interpreted the large amplitude electric field fluctuations as nonlinear electrostatic whistlerstructures that can produce intense localized magnetic dissipation that drives magnetic recon-nection at the boundary between closed and open magnetic field lines, where the perpendicu-lar crescents are just beginning to transition to parallel crescents.Figure 3 shows the electric field and electron pressure at 30 ms and 7 . E × B drifting (frozen-in).The perpendicular components of the electric field are shown in red (FPI at 30 ms resolu-tion), green (FPI at 7 . . . E × B drift. This improved agreement between E ⊥ and − V e × B / c serves asvalidation of our 7 . . E × B drift coincides with the onset of electron diamagnetic drift. That is, in the region of non-gyrotropic perpendicular electron crescents prior to region of large electric field fluctuations,the perpendicular gradient of the perpendicular electron pressure is making the dominantcontribution to E ⊥ + V e × B / c , as shown in Figure 4 panels (b)-(d) and (f)-(h). This resultsuggests that the strong non-gyrotropy of the electron velocity distributions in this interval isa manifestation of the energy-dependent magnetic gradient drifts that – when integrated overvelocity space – produce an electron pressure gradient contribution in equation (3).Prior to the onset of J M , there is an electron pressure gradient signal but no corre-sponding current density. This displacement in time between the pressure gradient signaland the J M signal is a result of averaging the J M measurements from the four observatoriesas they cross into the current sheet. The finite difference pressure gradient signal does notproduce such a delay.During the interval of large electric field fluctuations beginning at 13:07:02.2 UT, theelectron momentum equation does not appear to be satisfied at FPI 7 . In summary, we have shown that the deviation of the electron bulk velocity from E × B drift observed between 13:07:01.199 UT and 13:07:02.180 UT can be explained by elec-tron diamagnetic drift in an electron scale current sheet. In the region where the out-of-planecurrent is transitioning to in-plane current, the electron momentum equation is not satisfiedat 7 . J · E (cid:48) ≈ Burch et al. [2016b]. The observation of perpen-dicular crescents preceding the region of J · E (cid:48) > Burch et al. [2016b]) or anomalous resistivity (assuggested by
Torbert et al. [2016]) since the associated features in the electron velocity dis-tribution may be very difficult to measure.Our results raise important questions about the nature of magnetic energy dissipa-tion at the magnetopause. In steady laminar reconnection, with a reconnection rate of about0 . V A B / c (where V A is the Alfvén speed), the corresponding reconnection electric field is on –5– he order of 0 . − / m. The observed electric field fluctuations in the interval where theelectron momentum equation is not satisfied, however, are much larger than that, approaching50 −
100 mV / m and varying over the 7.5 ms time scale of the FPI energy sweep. What roledo these fluctuations play in changing the magnetic field topology and dissipating magneticenergy? What is their contribution to the global integrated reconnection rate? Burch et al. [2017] has recently suggested that large amplitude electric field fluctua-tions over a region much more localized than that of the out-of-plane current density directlydrive reconnection by producing localized J · E (cid:48) at the boundary between open and closedmagnetic field lines. Our results, demonstrating that the electrons are diamagnetically drift-ing in the region of perpendicular crescents prior to the onset of large amplitude electric fieldfluctuations, are consistent with this suggestion.However, it is also possible that the global reconnection rate is supported by electronmeandering orbits interacting with a much smaller electric field on the order of 0 . V A B / c (the global integral of which gives the reconnection rate), as shown in two-dimensional PICsimulations (e.g., Hesse et al. [2014]). Although the observation of crescents by themselvesdoes not imply meandering orbits, crescent distributions observed in such close proximity toan electron-scale magnetic field reversal supports the idea of meandering orbits [
Burch et al. ,2016b;
Egedal et al. , 2016].A third possibility is that turbulent fluctuations facilitate anomalous transport at themagnetopause. For example, there is evidence from three-dimensional PIC simulations ofthe
Burch et al. [2016b] event that lower hybrid turbulence (driven by the diamagnetic drift)can lead to anomalous heating and transport of plasma from the sheath onto closed magneticfield lines [
Le et al. , 2017].
Torbert et al. [2016] has suggested that violation of the Gener-alized Ohm’s Law at 30 ms resolution might be evidence of such anomalous resistivity, andour 7.5 ms results have not eliminated this possibility.Further progress will require the development of new techniques that move beyond thecalculation of velocity moments and extract information about phase space density and itsvelocity space gradients on time scales shorter than the FPI 7.5 ms energy sweep.
Acknowledgments
This research was supported by the NASA Magnetospheric Multiscale Mission in asso-ciation with NASA contract NNG04EB99C. IRAP contributions to MMS FPI were sup-ported by CNES and CNRS. We thank the entire MMS team and instrument leads for dataaccess and support. The L2 data of MMS can be accessed from MMS Science Data Center(https://lasp.colorado.edu/mms/sdc/public/).
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MMS2 overview of the 5 s surrounding the EDR event on 16 October 2015. (a) Parallel (0-30 de-gree) electron energy-time spectrogram, (b) anti-parallel (150-180 degree) electron energy-time spectrogram,(c) omni-directional ion energy-time spectrogram, (d) magnetic field, (e) current from FPI, (f) electric field,(g) parallel and perpendicular electron temperature. –8– ) b) c) d) e)
Figure 3.
The 7 . E × B drift in theinterval from 13:06:57 to 13:06:59 where we expect the electrons to be frozen-in and strongly magnetized.Panels (a)-(c) show the 8 kHz EDP electric field at MMS2 averaged to 7 . − V e × B / c at 30 ms (red) and 7 . . − E ⊥ , L m V / m − − − − E ⊥ , M m V / m − E ⊥ , N m V / m − B L nT − − − J u A / ( m ) − − E ⊥ , L m V / m − − − − E ⊥ , M m V / m − E ⊥ , N m V / m − − E ′ + ∇ ⊥ P e , ⊥ m V / m − − R m V / m − − − J u A / ( m ) ( − v e xB) L − (v e xB+(ne) − ∇ ⊥ P e, ⊥ ) L ( − v e xB) M − (v e xB+(ne) − ∇ ⊥ P e, ⊥ ) M ( − v e xB) N − (v e xB+(ne) − ∇ ⊥ P e, ⊥ ) N LMNMMS1MMS2MMS3MMS4( − v e xB) L − (v e xB+(ne) − ∇ ⊥ P e, ⊥ ) L inertia( − v e xB) M − (v e xB+(ne) − ∇ ⊥ P e, ⊥ ) M inertia( − v e xB) N − (v e xB+(ne) − ∇ ⊥ P e, ⊥ ) N inertiaLMNLMNLMNMMS1MMS2MMS3MMS4 Seconds16 − Oct − ne − ( ∇ ⋅ P e − ∇ ⊥ P e, ⊥ ) a) b) c) d) e) f) g) h) i) j) k) perpendicular crescents perp -> para crescents Figure 4.