Grad-Shafranov reconstruction of a bipolar Bz signature in an earthward jet in the tail Hiroshi Hasegawa (2007/02/14)

Observation of bipolar Bz ＋ to － Bz （ GSM ） : in the mid- to distant- tail along with tailward flows studied in association with substorms (Ieda et al., 1998, etc.)

Earthward moving flux rope? － to ＋ Bz often seen in the near-tail (from Geotail and Cluster observations). Slavin et al. (2003)

Superposed epoch analysis Core By field Observed along with earthward flows (BBFs) Slavin et al. (2003)

Multiple X-line reconnection (forming magnetic flux ropes) (e.g., Slavin et al., 2003) Transient reconnection (e.g., Sergeev et al., 1992) Localized reconnection under guide-field By (Shirataka et al., 2006) Models for bipolar Bz in earthward flows Vz

Cluster event ( UT) Studied by Amm et al. (2006) associated with a substorm (onset at ~22:50 UT)

Cluster event ( UT) － / ＋ Bz embedded in an earthward flow C3 exactly at the center of the current sheet C1, 2, 4 on the northern side Separation ~ 4000 km Bz Vx Bx

Grad-Shafranov reconstruction technique (Hau & Sonnerup, 1999) (A spatial initial value problem) Assumptions Plasma structures are: in magnetohydrostatic equilibria (time-independent). ×× P t, p, and B z are functions of A alone (constant on same field lines). 2-D (no spatial gradient in the z direction) Grad-Shafranov (GS) equation (e.g., Sturrock, 1994) Magnetic field tension balances with force from the gradient of total (magnetic + plasma) pressure.

X A 2D structure X Y Z (invariant axis) Reconstruction procedure Y Reconstruction plane Lx = V ST_X * T (analyzed interval) X axis: SC trajectory in the x-y plane V ST_X V ST (V HT ) (in the x-z plane) Spatial integration

Spatial initial value problem (Sonnerup & Guo, 1996) Grad-Shafranov equation spatial integration in  y direction (2nd order Taylor exp.) (1st order Taylor exp.) GS eq.

V HT = (237, 27, 23) km/s in GSM i = (-0.999, 0.042, 0.005) j = (-0.022, , 0.784) k = (0.036, 0.783, 0.621) Roughly circular flux rope Flux rope with half width of ~1 Re Strong core field (mostly By) cc = x z Consistent with multiple X- line models?

2Ry The plane of the equator guide-field ： By 0 The Northern hemisphere The Southern hemisphere N S 3D-MHD simulation of localized reconnection with guide-field (Shirataka et al., 2006) [Slavin et al. 2003] N E W S Y X Z 2Ry = 3 Re

N S E W Results Reproducing the southward magnetic field Shirataka et al. (2006)

Bz [z=0] By 0 =4nT, 2Ry=3.0Re 11.25Re t=135s Results Virtual S/C obs. in the MHD run x y 37.5Re 11.25Re Re

Virtual observation vs real data What will be reconstructed, when applied to the simulation data in which no flux rope is created?

Virtual spacecraft (x,y,z) = (11.25, 0, 0), (11.25, 0, 1), (11.25, 1, 0), (11.25, 2, 0) Re Applied to the interval T = 105 – 195 s (A suitable model may be determined if the separation is ~2Re. )

A flux rope, which does not really exist in the simulation, is reconstructed erroneously. Map recovered from data sampled at (x,y,z) = (11.25, 0, 0) Re Z(GS) = (0.000, 0.996, 0.087) GS map recovered from virtual observation

Map recovered from data sampled at (x,y,z) = (11.25, 1, 0) Re The presence of a flux rope-like structure in GS maps does not necessarily mean that it exists in reality. But, are GS results totally meaningless?

Map recovered from data sampled at (x,y,z) = (11.25, 0, 0) Re Simulation result at the time when Bz reversal is at x=11.25 Re (in the same plane)

Map recovered from data sampled at (x,y,z) = (11.25, 1, 0) Re Simulation result at the time when Bz reversal is at x=11.25 Re (in the same plane)

Which model is more reasonable (for the Cluster event)? CL event: Roughly circular Pressure minimum at the core Simulation result: Elongated in the x direction Enhanced P at the front

Summary The GS method cannot accurately recover the magnetic topology. One must be cautious about interpretation of model-based (force-free, or GS model) results. It seems possible to get some information on the basic structure (shape, pressure distribution, etc.) in the reconstruction plane. The Cluster bipolar Bz event on is most likely explained by a flux rope (multiple X-line reconnection). A suitable separation distance for discriminating models is a few Re (comparable to the jet width).