Addressing magnetic reconnection on multiple scales:What controls the reconnection rate in astrophysical plasmas?

John C. DorelliUniversity 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

Magnetopause surface

Magnetosphere

Solar wind

Current is distributed on the magnetopausein such a way that the solar wind field doesn’tpenetrate into the magnetosphere.

(e.g., Stern, D., JGR, 99, 17,169-17,198, 1994.)

Boundary conditions chosen so that magnetic fieldis 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 reconnectionEvidence that the magnetopauselocally looks like a rotationaldiscontinuity

Phan et al., GRL, 30, 1509, 2003.

Observations of magnetospheric reconnection

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.).

Polar VIS UV image of auroral oval (fromhttp://eiger.physics.uiowa.edu/~vis/examples)

Bow Shock

Magnetic Separatrix

Magnetopause

3D separatrices

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.

(fan)

(spine)

Can we apply two-dimensional steady state reconnectionmodels to the subsolar magnetopause?

x

z

Sonnerup, JGR, 79, 1546, 1974.Gonzalez and Mozer, JGR, 79, 4186, 1974.

€

cosθ >Bo

Bi

reconnection is geometricallyimpossible.

Gonzalez, Planet. Space Sci., 38, 627, 1990.

most solar wind-magnetosphere coupling functions can constructed on the foundation of the Sonnerup-Gonzalezfunction.

Cowley, S. W. H., JGR, 81, 3455, 1976.

the Sonnerup-Gonzalez function is not valid in 2D reconnectionwith a spatially varying guide field (e.g., asymmetric reconnection).

Swisdak and Drake, GRL, 34, L1106, 2007.

reconnection is possible for all non-vanishing IMF clockangles.

Dorelli et al., JGR, 112, A02202, 2007.

reconnection “X line” (separator) is determined by globalconsiderations.

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 magneticfield topology!

Dorelli et al., JGR, 112, A02202, 2007.Vacuum superposition (no dipole tilt and nomagnetic field x component) predicts:

€

tan(θc −θX ) =1

2tanθX

€

θX ≈θc

2+

1

3sin

θc

2cos

θc

2

€

θX

€

θc

2

Sweet-Parker Analysis

x

y

Momentum equation:

Lundquist number:

Flux Pileup ReconnectionParker, E. N., Comments on the reconnexion rate of magnetic fields, J. Plasma Phys., 9, 49-63, 1973.

2D incompressible MHD equations. Bulk velocityhas 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 SolutionsPriest, 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 SaturationSonnerup, 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?

steady magnetopause reconnection occurs viathe flux pileup mechanism -- local conditions adjustto accommodate (but not necessarily match!)the solar wind electric field

Dorelli et al., JGR, 109, A12216, 2004.

Borovsky et al., JGR, in press, 2008.

magnetopause reconnection is controlled by localplasma parameters (local magnetic fields and densitiesupstream of the diffusion region).

1. reconnection rate doesn’t match the solar wind electric field.

2. pileup is not observed to depend on the IMF clock angle

3. a “plasmasphere effect” was observed, consistent with a local Cassak-Shay electric field.

we don’t expect the reconnection rate to matchthe solar wind electric field in 3D; instead, the localparameters adjust themselves so that

€

∇×E = 0

in 3D flux pileup reconnection, the degree of pileupis independent of the IMF clock angle; nevertheless,the degree of pileup increases with decreasingresistivity and increasing solar wind speed.

Note: Cassak-Shay assumes E constant -- “driven”in the Borovsky et al. [2007] sense!

€

∇×E = 0

Sonnerup and Priest (1975)

Calculating the parallel electric field at the subsolar X lineAssumptions:

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.

€

Ex = −UyBz

c+

UzBy

c+

ηc

4π

∂Bz

∂y−

∂By

∂z

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Ey = −UzBx

c+

UxBz

c+

ηc

4π

∂Bx

∂z−

∂Bz

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟

€

E z = −UxBy

c+

UyBx

c+

ηc

4π

∂By

∂x−

∂Bx

∂y

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Ux = −U1

Lx

€

Uy =U1

2Ly

€

Uz =U1

2Lz€

Ey (0) = E(L) −∂Ex

∂y0

L

∫ dx

€

∇×E = 0

Local conditions adjust themselves so that this equation is satisfied.

Asymptotic matching

X

f

€

εf ' '+ξf '+1

2f = 0

€

δ ~ ε1/ 2

Ideal MHD

€

ε = ηc 2

4πLU1

€

f =By

B1

€

ξ =x

L

€

L −δ

€

ξf '+1

2f = 0

€

f (0) = f0

€

f (1) = sinθc

€

fout ~sinθc

ξ 1/ 2

€

εf ' '+ξf '= 0

€

f in ~2

3

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2sinθc

ε 3 / 4−

f0

ε1/ 2

⎛

⎝ ⎜

⎞

⎠ ⎟ exp −

u2

2ε

⎛

⎝ ⎜

⎞

⎠ ⎟

0

ξ

∫ du + f0

€

∇×E = 0Sonnerup and Priest (1975)

Diffusionregion

A global topological constraint

X

(f, g)

€

δ ~ ε1/ 2

€

f =By

B1

€

L −δ

€

f (0) = f0

€

fout ~sinθc

ξ 1/ 2

€

f in ~2

3

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2sinθc

ε 3 / 4−

f0

ε1/ 2

⎛

⎝ ⎜

⎞

⎠ ⎟ exp −

u2

2ε

⎛

⎝ ⎜

⎞

⎠ ⎟

0

ξ

∫ du + f0

€

g =Bz

B1

€

g(0) = g0

€

f0

g0

= tanθX

Current density maximum occurs at the magnetic separator.

Subsolar magnetopause reconnection rate

€

E || =η1/ 4B1U1

3 / 4

L1/ 4

4π

c 2

⎛

⎝ ⎜

⎞

⎠ ⎟3 / 4

2

3

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

sinθc

2

⎛

⎝ ⎜

⎞

⎠ ⎟ 1−

1

3cos2 θc

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

sinθc

2

⎛

⎝ ⎜

⎞

⎠ ⎟ 1−

1

3cos2 θc

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

sin8 / 3 θc

2

Half-wave rectifier

Cassak-Shay predicts that the reconnection rate scales likethe 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’sDayside Magnetopause?

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!)

Cassak and Shay, Phys. Plasmas, 14, 102114, 2007.

€

E =B1B2

B1 + B2

⎛

⎝ ⎜

⎞

⎠ ⎟Vout

c

2δ

L

€

Vout2 =

B1B2

4π

B1 + B2

ρ1B2 + ρ 2B1