Transcript What Controls the Rate of Magnetic Reconnection in Astrophysical

Addressing magnetic reconnection on multiple scales:
What controls the reconnection rate in astrophysical plasmas?
John C. Dorelli
University of New Hampshire
Space Science Center
1. Is reconnection a local process?
2. What is the role of boundary conditions?
3. What is the role of the dissipation region?
Axford Conjecture (1984): Reconnection proceeds at a rate
which is completely determined by the externally imposed
electric field.
The closed magnetosphere
Solar wind
Magnetopause surface
Current is distributed on the magnetopause
in such a way that the solar wind field doesn’t
penetrate into the magnetosphere.
(e.g., Stern, D., JGR, 99, 17,169-17,198, 1994.)
Magnetosphere
Boundary conditions chosen so that magnetic field
is tangent to the magnetopause surface.
The perils of living in 2D….
Dungey, J. W., PRL, 6, 47-48, 1961.
Dungey, J. W., in Geophysics: The Earth’s Environment,
eds., C. Dewitt et al., 1963.
Both of these topologies are unstable in 3D!
Observations of magnetospheric reconnection
Evidence that the magnetopause
locally looks like a rotational
discontinuity
Phan et al., GRL, 30, 1509, 2003.
Observations of magnetospheric reconnection
Magnetopause
Bow Shock
Polar VIS UV image of auroral oval (from
http://eiger.physics.uiowa.edu/~vis/examples)
Magnetic Separatrix
Auroral oval marks the boundary between open and closed field lines; the reconnection rate
can be determined from radar observations of ionospheric convection (e.g., de la Beaujardiere
et al., J. Geophys. Res., 96, 13,907-13,912, 1991.).
3D separatrices
(fan)
(spine)

A
B
B
separator
Lau, Y.-T. and J. M. Finn, Three-dimensional kinematic reconnection in the presence of
field nulls and closed field lines, Ap. J., 350, 672, 1990.
Can we apply two-dimensional steady state reconnection
models to the subsolar magnetopause?
Sonnerup, JGR, 79, 1546, 1974.
Gonzalez and Mozer, JGR, 79, 4186, 1974.
Bo
cos 
Bi
reconnection is geometrically
impossible.
Gonzalez, Planet. Space Sci., 38, 627, 1990.
most solar wind-magnetosphere coupling functions can
constructed on the foundation of the Sonnerup-Gonzalez
function.
z

Cowley, S. W. H., JGR, 81, 3455, 1976.
the Sonnerup-Gonzalez function is not valid in 2D reconnection
with a spatially varying guide field (e.g., asymmetric reconnection).
Swisdak and Drake, GRL, 34, L1106, 2007.
x
reconnection is possible for all non-vanishing IMF clock
angles.
Dorelli et al., JGR, 112, A02202, 2007.
reconnection “X line” (separator) is determined by global
considerations.
Determining the X line orientation
Swisdak and Drake, GRL, 14, L11106, 2007.
Reconnection occurs in the plane for which the outflow speed from the X line is maximized
However…in 3D, local magnetic field geometry can differ significantly from global magnetic
field topology!
Dorelli et al., JGR, 112, A02202, 2007.
Vacuum superposition (no dipole tilt and no
magnetic field x component) predicts:
1
tan( c  X )  tan X
2

X

X 

c
2
c
1 

 sin c cos c
2 3
2
2
Sweet-Parker Analysis
y

x

Momentum equation:
Lundquist number:
Flux Pileup Reconnection
Parker, E. N., Comments on the reconnexion rate of magnetic fields, J. Plasma Phys., 9, 49-63, 1973.
2D incompressible MHD equations. Bulk velocity
has the following form:
The upstream magnetic field increases to
compensate for the reduction in resistivity
(and consequent reduction of inflow speed).
Classical 2D Steady State Solutions
Priest, E. R. and T. G. Forbes, New models for fast steady state magnetic reconnection, J. Geophys. Res.,
5579-5588, 1986.
Petschek
Flux pileup

Incompressible MHD equations are solved in the “outer region” (outside the field reversal region).
 is determined from a Sweet-Parker analysis of the diffusion rectangle:
Flux Pileup Saturation
Sonnerup, B. U. Ö., and E. R. Priest, Resistive MHD stagnation-point flows at a current sheet, J. Plasma Phys.,
14, 283-294, 1975.
Biskamp, D. and H. Welter, Coalescence of magnetic islands, Phys. Rev. Lett., 44, 1069-1072, 1980.
Litvinenko, Y. E., T. G. Forbes and E. R. Priest, A strong limitation on the rapidity of flux pileup
reconnection, Solar Physics, 167, 445-448, 1996.
Craig, I. J. D., S. M. Henton and G. J. Rickard, The saturation of fast dynamic reconnection, Astron.
Astrophys., 267, L39-L41, 1993.
Is magnetopause reconnection “driven” by the solar wind?
Dorelli et al., JGR, 109, A12216, 2004.
steady magnetopause reconnection occurs via
the flux pileup mechanism -- local conditions adjust
to accommodate (but not necessarily match!)
the solar wind electric field
E 0
Sonnerup and Priest (1975)
Borovsky et al., JGR, in press, 2008.
magnetopause reconnection is controlled by local
plasma parameters (local magnetic fields and densities
upstream of the diffusion region).
1. reconnection rate doesn’t match the solar wind
electric field.
we don’t expect the reconnection rate to match
the solar wind electric field in 3D; instead, the local
parameters adjust themselves so that   E  0
2. pileup is not observed to depend on the IMF clock angle
in 3D flux pileup reconnection, the degree of pileup
is independent of the IMF 
clock angle; nevertheless,
the degree of pileup increases with decreasing
resistivity and increasing solar wind speed.
3. a “plasmasphere effect” was observed, consistent with a
local Cassak-Shay electric field.
Note: Cassak-Shay assumes E constant -- “driven”
in the Borovsky et al. [2007] sense!
Calculating the parallel electric field at the subsolar X line
Assumptions:

U
1.
Magnetosheath flow is nearly
incompressible and symmetric about the
Sun-Earth line.
2.
Field line curvature near Sun-Earth line is
negligible.
3.
Resistive MHD is valid along the Sun-Earth
line.
E 0
U y Bz U z By c Bz By 
Ex  





c
c
4  y
z 
U B U B c Bx Bz 
Ey   z x  x z 



c
c
4  z x  E y (0)  E(L) 
U x By U y Bx c By Bx 
Ez  





c
c
4   x
y 
U1
1
Ux   x Uy 
y
L
2L


U1 
Uz 
z
2L
E x
 y dx
0
L
Local conditions adjust
themselves so that this
equation is satisfied.
Asymptotic matching
c 2

4 LU1
E 0
Sonnerup and Priest (1975)

f

Diffusion
region

 ~ 1/ 2
f (0)  f 0
1
f ''f ' f  0 f (1)  sin 
By
c
f 
2


Ideal MHD

L 
f ''f ' 0
 f   u2 
2 1/ 2sin c
f in ~    3 / 4  1/02  exp du  f 0
3   
 0
 2 

X

1
f ' f  0
2
sin  c
f out ~ 1/ 2

B1
x

L

A global topological constraint
By
f (0)  f 0
f 
B1
Bz
g(0)  g0
g
B1


(f, g)

 ~ 1/ 2
L 

 u 2 
2 1/ 2sin c
f 0 
f in ~    3 / 4  1/ 2  exp du  f 0
3   
 0
 2 


X
f out ~
f0
 tan X
g0
sin  c
1/ 2


Current density maximum occurs at the magnetic separator.

Subsolar magnetopause reconnection rate
E || 
 B1U
3/4
1
1/ 4
L1/ 4
4  3 / 4 2 1/ 2  c  1
2  c 
 2    sin  1 cos

 c  3 
 2  3
2 
  1
 
sin  c 1 cos2 c 
 2  3
2 


sin 8 / 3
c
2

Half-wave rectifier
Cassak-Shay predicts that the reconnection rate scales like
the square root of the resistivity in the resistive MHD case.
Conclusions
1.
Magnetic reconnection is a global process: 1) The geometries of X
lines are determined by global considerations (e.g., the locations of
magnetic separators), 2) The reconnection rate is computed by
evaluating line integrals (along magnetic separators) of the electric field
(i.e., by computing the rates of change of magnetic flux within distinct
flux domains).
2.
Ultimately, all reconnection is “driven” in the sense that large scale
plasma flows impose constraints which dissipation regions must
somehow accommodate. Nevertheless, if the system is either 3D or
time-varying (i.e., “real”), the dissipation region will also have
something to say (via “asymptotic matching”).
3.
The Axford Conjecture does not apply to magnetic reconnection at
Earth’s dayside magnetopause; magnetic flux pileup renders the
reconnection rate sensitive to the Lundquist number. Thus, resistive
MHD simulations will never accurately model the magnetosphere in the
high Lundquist number limit.
Can we apply the Cassak-Shay formula to Earth’s
Dayside Magnetopause?
Cassak and Shay, Phys. Plasmas, 14, 102114, 2007.
 B1B2 Vout 2
E  

B1  B2  c L
2
out
V
B1B2 B1  B2

4 1B2  2 B1
1. It’s a 2D steady state argument, which means that the
electric field is a constant in space.

2. There’s no way to determine the orientation of the X
line from local conditions.
3. The resistive MHD version predicts the wrong scaling
of the reconnection rate with resistivity (the 3D nature
of the magnetopause flow is essential!)