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).