Grad-Shafranov reconstruction of a bipolar Bz signature in an

earthward jet in the tail

Hiroshi Hasegawa

ISAS/JAXA

@Uppsala (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

2002-08-13 Cluster event(2200-2400 UT)

• Studied by Amm et al. (2006)

• associated with a substorm (onset at ~22:50 UT)

2002-08-13 Cluster event(2312-2318 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

pBj

Grad-Shafranov reconstruction technique (Hau & Sonnerup, 1999)

(A spatial initial value problem)AssumptionsPlasma structures are: • in magnetohydrostatic equilibria (time-independent).

PBJVVt

V

)(× ×

)(002

2

2

2

AjAd

Pd

y

A

x

Az

t

),)(,,( ABxAyAB z

)2( 02 zt BpP

Pt, p, and Bz are functions of A alone (constant on same field lines).

)( zAA

• 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

YReconstruction plane

Lx = VST_X* T (analyzed interval)

X axis: SC trajectory in the x-y plane

VST_X

VST (VHT)(in the x-z plane)

Spatial integration

Spatial initial value problem (Sonnerup & Guo, 1996)

,)0,()0,(00

x

y

xdxxBxd

x

AxA tdxVxd HT ˆ

)( xABy

Grad-Shafranov equation

2

,

2

2

,

)(2

1),(),( y

y

Ay

y

AyxAyyxA

yxyx

yy

AyxBy

y

ByxByyxB

yx

x

yx

xxx

,

2

2

,

),(),(),(

Ad

Pd

x

A

y

A t02

2

2

2

spatial integration in y direction

))(,,( ABxAyAB z

(2nd order Taylor exp.)

(1st order Taylor exp.)

)(002

2

2

2

AjAd

Pd

y

A

x

Az

t

GS eq.

VHT = (237, 27, 23) km/s in GSMi = (-0.999, 0.042, 0.005)j = (-0.022, -0.621, 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 = 0.961

xz

Consistent with multiple X-line models?

2Ry

The plane of the equator

guide-field ：By0

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]

By0=4nT, 2Ry=3.0Re 　

11.25Re

t=135s

Results

Virtual S/C obs. in the MHD run

x

y

37.5Re

11.25Re

-11.25Re

Virtual observation vs real data

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

Virtual spacecraft observations @ (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 2001-08-13 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).