Magnetic Reconnection: Some Answers and Open Questions

Amitava BhattacharjeeSpace Science Center and Center for Magnetic Self-OrganizationUniversity of New Hampshire

NESSC Meeting, Durham, New Hampshire, January 9, 2007

Collaborators

Naoki Bessho, John Dorelli, Terry Forbes, Kai Germaschewski, Chung-Sang Ng, David Pontin, Joachim Raeder, Hong-Ang Yang, UNH

John Greene, GAShuanghui Hu, UC-IrvineZhiwei Ma, Unversity of IowaXiaogang Wang, Beijing University

CMSO : John Finn, Chris Hegna, Hantao Ji, Yi-Min Hwang, Vladimir Mirnov, Stewart Prager, Dalton Schnack, Carl Sovinec, Masaaki Yamada, Ping Zhu, Ellen Zweibel

Outline

• Classical steady-state models of Sweet-Parker and Petschek.

• Impulsive, time-dependent reconnection dynamics: comparative studies of magnetospheric substorms, and solar flares.

• Scaling of the reconnection rate.• 3D reconnection: Separator reconnectionCurrent sheet formation and reconnection in geometries without nulls or closed field lines.

I will not discuss the important subjects of particle acceleration, and turbulent reconnection.

Classical (2D) Steady-State Models of Reconnection

Sweet-Parker [Sweet 1958, Parker 1957]

Geometry of reconnection layer : Y-points [Syrovatskii 1971]

Length of the reconnection layer is of the order of the system size >> width

 Reconnection time scale

Δ

€

τ SP = τ Aτ R( )1/2

= S1/2τ A

Solar flares: ,10~ 12S

€

τA ~ 1s

sSP610~τ⇒ Too long to account for solar flares!

Q. Why is Sweet-Parker reconnection so slow?

Conservation relations of mass, energy, and flux

€

VinL =Voutδ, Aout VV =

€

Vin =δ

LVA,

δ

L= S−1/2

Petschek [1964]

Geometry of reconnection layer: X-point

Length (<< L) is of the order of the width

SAPK lnττ =

Solar flares: τPK ~ 102 s€

Δ

€

δ

A. Geometry

Numerical Simulations of the Petschek Model[Sato and Hayashi 1979, Ugai 1984, Biskamp 1986,

Forbes and Priest 1987, Scholer 1989, Yan, Lee and Priest 1993, Ma et al. 1995, Uzdensky and Kulsrud 2000, Breslau and Jardin 2003, Malyshkin, Linde and Kulsrud 2005] Conclusions• Petschek model is not realizable in high-S plasmas, unless the resistivity is locally strongly enhanced at the X-point. • In the absence of such anomalous enhancement, the reconnection layer evolves dynamically to form Y-points and realize a Sweet-Parker regime.

2D coronal loop : high-Lundquist number resistive MHD simulation

[Ma, Ng, Wang, and Bhattacharjee 1995]

T = 0 T = 30

Biskamp’s Critique of the Petschek Model

Δ increases with increasing Lundquist number; this directly contradicts the Petschek model, which requires Δ δ Flux pile-up: Litvinenko, Forbes and Priest [1996], Craig, Henton, and Rickard[1993], Dorelli and Birn [2001], Simakov, Chacon, and Knoll [2006]

Magnetic energy accumulates upstreamof the current sheet (flux pileup).

Impulsive Reconnection: The Trigger ProblemDynamics exhibits an impulsiveness, that is, a sudden change in the time-derivative of the reconnection rate.The magnetic configuration evolves slowly for a long period of time, only to undergo a sudden dynamical change over a much shorter period of time. Dynamics is characterized by the formation of near-singular current sheets in finite time. ExamplesSawtooth oscillations in tokamaks Magnetospheric substormsImpulsive solar flares

Sawtooth events in MST

[Almagri and the MST group 2003]

Current Disruption in the Near-Earth Magnetotail

[Ohtani, Kokubun, and Russell 1992]

Time Profile of Reconnection Rates for X3 Flare

observed by TRACE

Linear Plot Log Plot

[Saba, Tarbell, and Gaeng 2003]

Hall MHD (or Two-Fluid) Model and the

Generalized Ohm’s LawIn high-S plasmas, when the width of the thin current sheet ( ) satisfies

Δ

c /ω pe<<Δ <<c/ω pi

“collisionless” terms in the generalized Ohm’s law cannot be ignored.

Generalized Ohm’s law (dimensionless form)

E +v×B=1S

J +de2 dJ

dt+

din

J ×B−βe∇pe( )

Electron skin depth Ion skin depth Electron beta

de ≡L−1 c/ω pe( )

di ≡L−1 c/ωpi( )βe

(or

€

βc /ω piif there is a guide field)

Forced Magnetic Reconnection Due to Inward Boundary Flows Magnetic field

Inward flows at the boundaries

Two simulations: Resistive MHD versus Hall MHD [Ma and Bhattacharjee 1996]

€

B = ˆ x BP tanh z /a+ ˆ z BT

€

v = mV0(1+ coskx) ˆ y

€

′ Δ <0,

For another perspective, with similar conclusions, see Horiuchi and Sato [1994] and more recently, Cassak, Shay, and Drake [2005]

€

d lnψ /dt

….. Hall

___ Resistive

Transition from Collisional to Collisionless Regimes in MRX

“Standard Model”

QuickTime™ and aVideo decompressor

are needed to see this picture.

Scaling of Reconnection

• Observations: Yokoyama et al. (2001), Isobe et al. (2002)This Workshop: Lin et al. (2004), Noglik et al, (2004)Reconnection rates : inflow velocity/Alfven velocity~0.001-0.1.

Hall MHD reconnection scaling depends on the ion skin depth, the Lundquist number (weakly), and system size. The range of reconnection rates in Hall MHD and PIC simulations ~ 0.01-0.1.

Observations (Ohtani et al. 1992)

Hall MHD Simulation

[Ma and Bhattacharjee 1998]

Scaling of the collisionless reconnection rate

How does it scale with ion/electron skin depth, resistivity, plasma beta, and system size?

-

+

Δe

Δi

δeδi

-

+

Vin

y

x

out-of-plane magnetic field, Bz

Scaling is a controversial subject

• Is the reconnection rate insensitive to the details of the electron layer (current sheet layer), and controlled by ions?The GEM Challenge Perspective

[Birn et al. 2001; Mandt et al. 1994, Shay and Drake 1998, Hesse et al. 1999, Rogers et al. 2001, Pritchett 2001, Ricci et al. 2002]Reconnection is insensitive to the mechanism that breaks field lines (electron inertia or resistivity).• In the presence of Hall currents, whistler waves mediate reconnection. The characteristic outflow speed is the whistler phase speed (based on the upstream magnetic field).• The inflow velocity where (= system size). This rate is independent of .

€

Vin = Ωeδe2 /Δi

€

Δi ~ k << L

€

me

Scaling is a controversial subject

• The whistler waves generate an out-of plane quadrupolar magnetic field (seen in MRX/PPPL, SSX/Swarthmore as well as

in situ satellite observations in space).• The ratio of the horizontal electron outflow to the horizontal magnetic perturbation scales as k for the dispersive whistler (or kinetic Alfven) wave.

• How does the reconnection rate scale with the system size?

The length of the reconnection layer . Reconnection rate is a “universal constant”, [Shay et al. 2001, 2004, Huba and Rudakov 2004].

€

Vin ≈ 0.1VA

€

Δi ~ 10di

Scaling is a controversial subject

The GEM perspective is not universally accepted. An alternate point of view provides evidence that:

• Reconnection is not a universal constant, and depends on system parameters (such as ion/electron skin depth, plasma beta, boundary conditions)

• Reconnection rate is not independent of the system size, and in fact, often decreases as the system size increases.

[Wang et al. 2001, 2006, Fitzpatrick 2004, Bhattacharjee et al. 2005, Daughton et al. 2006]

Three examples : (1) Forced reconnection without guide field

(2) Undriven reconnection with guide field (3) Undriven reconnection with open boundaries

Scaling of steady-state reconnection rate in adriven Harris sheet (without guide field)

Two-fluid simulations as di is varied [Wang et al., 2006]

Confirms analytical scaling law [Wang, Bhattacharjee, and Ma 2001]

[Daughton, Scudder and Karimabadi 2006]

Effect of Open Boundary Conditions and System Size

Hall currents cancel exactly in electron-positron plasmas: no whistlers

Generalized Ohm’s law does not contain the Hall term.

€

E+v×Bc

=me

2ne2∂J∂t

+∇ ⋅(Jv+vJ)⎡ ⎣ ⎢

⎤ ⎦ ⎥ −

∇ ⋅(t P e−

t P i )

2ne

[Bessho and Bhattacharjee 2006]

Open questions on scaling

• While collisionless reconnection models do provide a clear mechanism for the onset of impulsive and fast reconnection, issues of scaling remain wide open.

• Is reconnection controlled entirely by ions? Or by electrons? Or is it a hybrid of ion and electron parameters, as well as boundary conditions?

• What is the role of periodic and open boundary conditions?

• Is the reconnection rate independent of the system size? If it isn’t, the theory faces great challenges in working for solar coronal plasmas while it may be adequate for magnetotail and fusion plasmas.

• Is the Hall current essential for fast reconnection? • Role of turbulence--the Eyink-Aluie theorem (2006):

“Frozen-in-flux condition in a turbulent quasi-ideal plasma will occur if the sites of current and vortex sheets intersect one another.”

3D Reconnection:Geometry and Dynamics

Resistive Tearing Modes in 2D/2.5D

Geometry

€

B = ˆ x BP tanh y / a + BT ˆ z

Time Scales

€

τA = a /VA = a 4πρ( )1/2 /BP

€

τR = 4πa2 /(ηc2)

Lundquist Number

€

S = τ R /τ A

Tearing modes

€

γ∝ S−3/5 (slower) or S−1/3 (faster)

y

x

[Furth, Killeen and Rosenbluth 1963, Coppi, Galvao, Pellat,Rosenbluth, and Rutherford 1976]

Neutral line at y=0

Magnetic nulls in 3D play the role of X-points in 2D

Spine

Fan

[Greene 1988, Lau and Finn 1990]

Dungey’s Model for Southward and Northward IMF

“Magnetopause phenomena are more complicated as a result of merging. This is why I no longer work on the magnetopause.” -- J. W. Dungey

[Dungey 1961, 1963]

Observation of magnetic null in space [Xiao, Pu, Wang et al., Nature, 2006]

Equilbrium Current Density, J

[Greene and Miller 1995]

Equilbrium B-field

B-lines in a spherically tearing state

[Hu, Bhattacharjee, Dorelli, and Greene 2004]

Equilibrium

Perturbed

Field linespenetrating thespherical tearingsurface

Magnetopause reconnection topology in global MHD

Greene, J., Locating three-dimensional roots by a bisection method, JCP, 1992.

[Dorelli, Bhattacharjee, and Raeder 2006]

Magnetopause reconnection geometry

Current density is concentrated at the intersection of the two separatrix surfaces; the X line has collapsed into a double Y line topology.

[Dorelli,Bhattacharjee, and Raeder, 2006]

Such ribbons were discussed by Longcope and Cowley [1996], but for force-free fields.

Solar corona

astron.berkeley.edu/~jrg/ ay202/img1731.gifwww.geophys.washington.edu/ Space/gifs/yokohflscl.gif

Solar corona: heating problem

photosphere corona

Temperature

Density

Time scale

Magnetic fields (~100G) --- role in heating?

~5×103K ~106K

1023m−3 1012m−3

~104s ~20s~ ~

Alfvén wave

current sheets

Parker's Model (1972)Straighten acurved magnetic loop

Photosphere

∂ Ω

∂ t

+ [ φ , Ω ] =

∂ J

∂ z

+ [ A , J ] + ν ∇⊥

2

Ω

∂ A

∂ t

+ [ φ , A ] =

∂ φ

∂ z

+ η ∇⊥

2

A

B = ˆ z + B

⊥= ˆ z + ∇

⊥A × ˆ z --- magnetic field,

v = ∇⊥

φ × ˆ z --- fluid velocity, Ω = − ∇

⊥

2

φ --- vorticity, J = − ∇

⊥

2

A --- current density ,

η --- resistivity, ν --- viscosity, [ φ , A ] ≡ φ

yA

x− φ

xA

y

Reduced MHD equations

low βlimit ofMHD

Magnetostatic equilibrium∂J∂z

+[A, J] = 0 ,or B ⋅∇J = 0

with φ = η = 0. Field-lines are tied at z = 0, L .

Key Question: What is the nature of the solutions, givena sufficiently complicated footpoint mapping?

Objections to Parker’s claim:van Ballegooijen [1985], Longcope and Strauss [1994].Cowley et al. [1997] and others.

Caveat: A proof based on reduced MHD equations, periodic boundary condition in x

A theorem on Parker's model

For any given footpoint mapping connected with the identity mapping, there is at most one smooth equilibrium.

[Related results by Aly 2005 and Low 2006]

ImplicationAn unstable but smooth equilibrium cannot relax to a second smooth equilibrium, hence must have current sheets.

Possible current sheet topology in line-tied geometry at Quasi-Separatrix Layers (QSL)

Summary• Collisionless magnetic reconnection, governed by a generalized

Ohm’s law holds the promise to resolve a number of outstanding questions pertaining to impulsive reconnection dynamics in laboratory and astrophysical plasmas, such as sawtooth oscillations in tokamaks and reversed-field pinches, magnetotail subtorms, or solar flares. We have elucidated the role of two-fluid effects in triggering fast reconnection.

• The question of scaling of reconnection rates in driven as well as undriven 2D/2.5D systems remains open. The standard picture suggests that there is a “universal” fast reconnection rate, but there is now a strong body of evidence that suggest that reconnection rates depend on plasma parameters (ion/electron skin depths, guide field, plasma beta) as well as the system size.

• 3D reconnection calls for a new topological framework. We have discussed the role of magnetic nulls and null-null lines in defining magnetic skeletons. Current sheets can form in 3D in ideal quasi-separatrix layers, but the reconnection rate in the presence of such singularities remains an open question.

• Confluence of experimental and theoretical results in laboratory, magnetospheric, and solar physics is key to answering fundamental questions.