Electron scale nested quadrupole Hall field in Cluster observations of magnetic reconnection

This Letter presents the first evidence of a new and unique feature of spontaneous reconnection at multiple sites in electron current sheet, viz. nested quadrupole structure of Hall field at electron scales, in Cluster observations. The new nested quadrupole is a consequence of electron scale processes in reconnection. Whistler response of the upstream plasma to the interaction of electron flows from neighboring reconnection sites produces a large scale quadrupole Hall field enclosing the quadrupole fields of the multiple sites, thus forming a nested structure. Electron-magnetohydrodynamic simulations of an electron current sheet yields mechanism of the formation of nested quadrupole.

1 Magnetic reconnection is a fundamental process for the fast release of magnetic energy into kinetic and thermal energy in laboratory, space and astrophysical plasmas. Collisionless reconnection develops in thin current sheets with thicknesses comparable to the electron skin depth d e (= c/ω pe ). The electron current sheet (ECS) with thickness ∼ d e is embedded inside an ion current sheet with thickness ∼ d i (= c/ω pi ). The electron and ion dynamics are decoupled at this scale and the plasma is no longer frozen in the magnetic field, thus enabling reconnection. The Hall current due to the differential flow of ions and electrons in the reconnection region generates an out-of-plane magnetic field with a quadrupolar structure [1,2], which will be referred to as the Hall field. The quadrupole structure of the Hall field is an essential feature of collisionless reconnection and has been detected in space observations [3][4][5], laboratory experiments [6] and simulations [7].
The electron current sheet is susceptible to secondary tearing instabilities which lead to spontaneous reconnection at multiple sites in ECS [8]. The interaction of neighboring sites leads to a new and unique feature, viz. nested quadrupole structure of the Hall field [9], unlike the single quadrupole in the case of reconnection at a single site. This feature arises in electron current sheets with a thickness (∼ few d e ) which is small compared to its extent (∼ few d i ). Such current sheets are unstable to tearing instability, with a growth rate that has a maximum when the perturbation has scale length of a few d e [9,10], thus leading to reconnection at multiple sites. This Letter presents the first evidence of a nested quadrupole structure of Hall field in the Cluster observations of an electron scale current sheet in Earth's magnetotail [3]. The Cluster spacecraft crossed the reconnection region at distances of ∼ 18R E in Earth's magnetotail on 1 October 2001. Among the four spacecraft SC4 was closest to the X-line and crossed the current sheet on the earthward side between 09:46:48 and 09:46:51 UT, and the profiles of electric and magnetic field are shown in Fig.   1 (Fig. 3 in Ref. [3]). The change in sign of the magnetic field components are critical to the structure of the Hall field and the time marks for these are shown by the vertical dashed lines in Fig. 1, viz. L 1 for B z , L 2 for B y , L 3 for B z , and L 4 for B x and B y .
A schematic of the magnetic field structure corresponding to the Cluster observations ( Fig. 1) is shown in Fig. 2, and consists of a primary site, with X-point at P, and a secondary site with X-point at S. In the standard picture of 2-D reconnection with a single reconnection site, i. e., in the absence of the secondary sites, B z should have the same sign on any one side (tail-ward or earthward) of the y − z plane containing the X-point P, and   Fig. 3b. The poles Q S 1 1 and Q S 1 4 of the secondary quadrupole at S 1 penetrates between the poles Q P 2 and Q P 3 of primary quadrupole. At the same time, the poles Q S 1 2 and Q S 1 3 of the secondary quadrupole at S 1 connect to the poles Q P 2 and Q P 3 of the primary quadrupole, respectively, thus increasing the extent of the primary quadrupole Q P .
One of the negative pole (Q P 2 +Q S 1 2 ) of the extended quadrupole is enclosed by a closed loop (red dashed line) in Fig. 3b. The extended quadrupole is nested inside the new quadrupole (Q N ), the poles of which are also marked ('+' and '-') in Fig. 3b.
A striking feature of spontaneous reconnection at multiple sites is the new quadrupole which, unlike the other three quadrupoles in Fig. 3b, is not directly associated with a reconnection site but arises from their interaction. The physics of the new quadrupole is the whistler response of the upstream plasma to the interaction of inflow to the secondary (weak) sites and outflow from the primary (dominant) site [9]. Because of the magnetic field structure of reconnection, the whistler perturbations are anchored in phase at their origin and propagate away from the reconnection region. The direction of propagation is very well approximated by the wave normal (shown by blue line in Fig. 3b) which is at Storey angle of 19.5 • [11,12] with the background magnetic field along x. Fig. 3c shows the out-of-plane magnetic field B y,W N along the wave normal. The wave propagates away from the reconnection region while its amplitude diminishes. The distance between positive and negative peaks is ≈ 12d e giving a wavenumber kd e ≈ 0.25, as expected for frequency ω = 0.1ω ce [13]. The extension of the primary quadrupole along x, and, the formation of a new quadrupole due to the whistler perturbation at secondary sites in the manner described above make the overall structure a nested structure of quadrupoles. system. Since the simulations are in boundary normal coordinate system, the profile of the electric field in Fig. 4 is obtained by transforming it from boundary normal to the GSE system. The boundary normal vectorn = −0.05x GSE + 0.80ŷ GSE − 0.59ẑ GSE of the highly tilted current sheet in Cluster observations is almost in the y GSE -z GSE plane and shown in the top panel of Fig. 4. Assuming the simulation current sheet to have the same orientation with respect to the GSE coordinate system, the y-component of the electric field in the latter can be obtained from E yGSE = E y sin(α) − E z cos(α), where α is the angle between the normal vector and theŷ GSE , with cos(α) = 0.8.
The electric and magnetic field profiles in the Cluster observation (Fig. 1) and EMHD simulation (Fig. 4) are remarkably similar not only in magnitude but also in the scale and pattern of variation. The current sheet crossing, represented by the change in B x from ≈-10 nT to ≈10 nT in observations (during ∼ 46 : 48 − 46 : 51, Fig. 1) and simulations (Fig. 4), provides more details of the reconnection in the magnetotail. The half thickness of the current sheet in simulations ≈ 7d e compares well with the observed values ∼ 3 − 5d e .  Fig. 4 shows that E yGSE , given by E y sin(α) − E z cos(α), is dominated by the normal component of the electric field, E z cos(α), due to the tilt of the current sheet with respect to the GSE coordinate system. Line L 4 in Fig. 1 and Fig. 4 show that E yGSE crosses zero earlier than both B x and B y , which cross zero simultaneously. The normal component of magnetic field B z remains positive during the current sheet crossing but is negative just before the current sheet crossing (between L 1 and L 3 ). The zero crossing of B z at L 3 coincides with the edge of the current sheet and negative peaks of E yGSE and B y . Both in the manner that results into the nested quadrupole structure. In natural situations, e.g., in the magneto-tail, reconnection at multiple sites is expected, with the one initiated first being dominant over the adjacent sites. Further, in the magnetotail the monotonic decrease of the magnetic field away from Earth (along x) will introduce asymmetry among the multiple reconnection sites, thus leading to the nested Hall field.
In the Cluster observations, the total time of crossing ≈ 6 sec., close to the ion cyclotron period and thus captured the electron dominated physics of reconnection. Since these electron scale observations are by a single spacecraft when the other three spacecrafts were separated by distances much larger than typical electron scales (∼ 20 km), the spatial and time variations are not uniquely distinguished. However, the EMHD simulations show that the electron scale structures form very quickly, in a time of the order of tens of electron cyclotron periods, but evolve very slowly after their formation [9]. Thus the structures observed by Cluster are consistent with spatial variations as described above. The forthcoming multi-spacecraft NASA/MMS mission, designed to resolve the electron scales in the magnetosphere and to distinguish between spatial and time variations, will provide key details of the spatio-temporal structure.
The nested quadrupole structure of Hall magnetic field identified in Cluster observations and the underlying mechanism revealed by EMHD simulations focus only on electron scale processes. Many details of the electron scale physics and the connection to the larger scale ion processes remain unexplored. Such studies will require new studies of electron scale physics in simulations, experiments and satellite observations of magnetic reconnection. In particular, the results presented in this Letter provide a critical step for a deeper understanding of reconnection at electron scales using new kinetic simulations that resolve the electron scales clearly and the data for electron scale physics from the upcoming NASA/MMS mission.