Reconstruction of Reconnection Configurations From Spacecraft Data

Bengt Sonnerup and Wai-Leong TehDartmouth College, Hanover, NH, USA

Hiroshi Hasegawa ISAS/JAXA, Japan

• Magnetohydrostatic Grad-Shafranov (GS) Reconstruction of FTEs Seen by Cluster• GS reconstruction with Field-Aligned flow: Onset of Reconnection Seen by Cluster• Prospects for General MHD Reconstruction of 2D Time Independent Structures

Magneto-hydrostatic GS Reconstruction

• and 2D, steady geometry

• GS equation:

∇p = j × B

∂2 A / ∂x2 + ∂2 A / ∂y2 = −μ 0 jz = −μ 0dPt / dA

B(x, y)=[∂A / ∂y,−∂A / ∂x,Bz(x,y)]

Pt (A)=p(A)+Bz2 (A) / 2μ0

• Get potential A on the x-axis (i.e., along the spacecraft trajectory):

• Can then determine .

• Step away from the x-axis:

A(x,0)=− By(x,0)∫ dx=VHTx By(t)dt∫Pt (A)

A(x,Δy) =A(x,0)+ Δy∂A / ∂y+12(Δy)2∂2A / ∂y2

Reconnection Related Results from the FTE study

1. High correlation between GS map and actually measured B fields at the four SC indicates the GS model assumptions are well satisfied.

2. Consistent results for invariant axis indicates that it is well defined; it is tangent to the magnetopause and embedded in it: length >2

3. The FTEs studied are flux ropes with strong core (axial) fields.

4. They must have been produced by intermittent component reconnection at site sunward of Cluster. They convect tailward with the magnetosheath speed

5. When they reach Cluster, reconnection has ceased and the flux rope evolution is slow (relaxation toward minimum-energy state)

6. Pressure minimum in the FTE center may indicate a connection to the magnetosphere at one end (or possibly both ends)

7. Circumferential flux is generated in less than 5 minutes: reconnection electric field > 0.17 mV/m for FTE 2 (reconnection rate > 0.058)

RE

E0

∇• [(1−MA2 )∇A] =RHS

=μ0ρ[T(dS dA)−(dH dA)] −Bz(dCz dA)−(B2 / 2μ0ρ)(dG2 / dA)

S =cv ln(T / ργ−1) H =cpT +v2 / 2

Cz =(1−MA2 )Bz G =μ0ρv/ B=MA μ0ρ

Structures with field-aligned flow[Sonnerup et al., JGR, September, 2006]

Field-line invariants:

M A2 =ν 2μ0ρ / B

2Alfvén - Mach number

M XT =Y T

X =[(∂T / ∂y),(∂ρ / ∂y),(∂v2 / ∂y),(∂MA2 / ∂y),(∂Bz / ∂y),(∂B

2 / ∂y),(∂2A / ∂y2 )]Y =[(BxdS /dA),(BxdH /dA),(BxdG2 /dA),(BxdCz /dA),0,(By∂Bx / ∂x),Q]

Q =RHS−(∂ / ∂x)(1−MA2 )∂A / ∂x

|M |=0

M A2 =1

M 2 + MA

2 −1=M 2MA2 (By / B)

2

Find derivatives from 7 linear equations:

where Note that

for and for

A=ρ =Bz2 =e−r2 ,MA

2 =MA02 re−r2

M A2

Benchmarking by use of exact solution

Black = error in potential A (0.1% levels)

Red = error in (2% levels)

Cluster Magnetopause Crossing

Crossing order: C4 (blue); C1 (black); C2 (red); C3 (green)

+ x component (GSE)□ y component (GSE)○ z component (GSE)

Walén Test

HT FrameQuality

VΗΤ =(−243,−85,+162)kms

(C1)

C1

C3

Εc =−ν ×Β

ΕΗΤ =−νΗΤ ×Β

Reconstruction from Cluster 1

Reconstruction from Cluster 3

Supersonic Field-Aligned Flow On Reconnected Field Line

Distance along the field line

Onset of local reconnection : Results 1. Strong field-aligned flow seen in the HT frame.

Its dynamic effects are included

2. Wedge of reconnected field lines is created in less than 30 s

3. Flux content in the wedge indicates 0.47 mV/m

(reconnection rate 0.025)

4. Supersonic ( 2.1 – 2.4) field-aligned flow in the wedge

5. Axial extent > 4

6. Guide field changes strength across the magnetopause

E0 ≥≥

M =

RE

YT =

-∂[ p+ (By2 +Bz

2 ) / 2μ0 ] / ∂x−ρ∂(vx2 / 2) / ∂x

−ρvx∂vy / ∂x+vy∂ρvx / ∂x+ (Bx / μ0 )∂By / ∂x−ρvx∂vz / ∂x+ (Bx / μ0 )∂Bz / ∂x

−Bx∂vz / ∂x+vx∂Bz / ∂x−(vxBz / ρ)∂ρ / ∂xρvx∂Bx / ∂x−Bx∂ρvx / ∂x

ρvxdS(ψ ) /dψ

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

X =[(∂ρ / ∂y),(∂p/ ∂y),(∂vz / ∂y),(∂Bz / ∂y),(∂2ψ / ∂y2 ),(∂2A / ∂y2 )]

• Velocit y field: v=(1/ρ)∇ψ(x,y)×̂z+vz(x,y)̂z • Magnetic field: B=∇A(x,y)×̂z+Bz(x,y)̂z • 6×6 matrix equation: MXT=YT

|M |= cv (ρ2 / p)(VAy2 −vy

2 )[−vy4 +vy

2 (VA2 +c2 )−c2VAy

2 ]

General MHD Reconstruction of 2D, Time-Independent Structures

Will ultimately include the Hall effect, the electron pressure gradient, and a simple resistive term in Ohm’s law. This will require an iterative procedure but will allow reconstruction near a reconnection site.

Test case: Ideal radial flow with circumferential fieldSub-Fast results

Error contour line: 0.01% ([black, blue]), 0.1% ([red]).

Comparison between exact solution and reconstruction. Solid lines are field lines; dashed lines are streamlines; color indicates normalized density. Solution has in lower left corner and in upper right corner.

M 2 =1;MA2 =0.5

M 2 =4;MA2 =0.9

MHD reconstruction from C3 of magnetopause crossing at 06:23 UT, July 5, 2001