Magnetic Reconnection: Recent Developments and Connections with CMSO

Research A. Bhattacharjee, K. Germaschewski, J. Dorelli, J.

Greene, C. S. Ng, S. Hu, and P. Zhu Space Science Center

Institute for the Study of Earth, Oceans, and Space University of New Hampshire

Center for Magnetic Self-Organization, August 4, 2004

Outline• Some recent developments and key questions pertaining to

reconnection in weakly collisional plasmas. Examples: Impulsive reconnection phenomena and its scaling (sawtooth crashes in tokamaks and RFPs, magnetotail substorms, and solar flares)Geometry and dynamics of 3D reconnection (applications to the dayside magnetosphere, solar corona, and laboratory experiments).

• These questions be studied using the experimental (MST, MRX, SSPX), theoretical, and computational (NIMROD, DEBS) resources of the CMSO group. Combined with in situ satellite observations of magnetospheric plasmas (WIND, POLAR, CLUSTER, and MMS), global imaging of solar plasmas (SOHO, TRACE, RHESSI, STEREO) and new simulation codes (Hall MHD/Two-fluid and Particle-In-Cell), there is now a unique opportunity to make progress on important, unresolved questions.

High-Performance Computing Tools

• Magnetic Reconnection Code (MRC), based on extended two-fluid (or Hall MHD) equations, in a parallel AMR framework (Flash, developed at the University of Chicago).

• EPIC, a fully electromagnetic 3D Particle-In-Cell code, with explicit time-stepping.

MRC is our principal workhorse and two-fluid equations capture important collisionless/kinetic effects. Why do we need a PIC code?

• Generalized Ohm’s law

• Thin current sheets, embedded in the reconnection layer, are subject to instabilities that require kinetic theory for a complete description.

€

E + v × B = 1S

J + de2 dJ

dt+ di

nJ × B −∇ •

t p e( )

Adaptive Mesh Refinement

Effectiveness of AMRExample: 2D MHD/Hall MHD

Efficiency of AMR

High effective resolution

Level # grids # grid points0 1 702251 83 1460802 103 2686663 153 5453164 197 10421325 404 19264656 600 1967234

Grid points in adaptive simulation: 6976118Grid points in non-adaptive simulation: 268730449Ratio 0.02

lo g

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 and RFPs Magnetospheric substormsImpulsive solar and stellar flares

Sawtooth crash in tokamaks (Yamada et al., 1994)

Sawtooth events in MST (Almagri et al., 2003)

Magnetospheric Substorms

Current Disruption in the Near-Earth Magnetotail

(Ohtani et al., 1992)

Impulsive solar/stellar flares

Equilibrium:

Model equations:

(Ottaviani/Porcelli 1993)(Grasso/Pegoraro/Porcelli/Califano

1999)

2D Hall MHD: m=1 sawtooth instability

Resolving the current sheet

zoomzoom

Current sheet collapse, s = 0

t

1/current sheet width

current density

t

Island width

magnetic flux function

Island equationddt

˙ λ δLδx

2⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟ ≈

cJ k2λ4

4de2δx

2

˙ δ x ≈−˙ λ δxδL

1+ ρs2

cJ de2

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟ or δx ≈δL exp−λ

δL1+ ρs

2

cJ de2

⎛ ⎝ ⎜ ⎜

⎞ ⎠ ⎟ ⎟

⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥

If x and L attain constant values and are of order of de (or ), the island equation becomes (Ottaviani & Porcelli 1993 for s = 0 ):

de2/3ρs

1/3

d2ˆ λ dˆ t 2

≈14

ˆ λ +cˆ λ 4

with , ˆ t =γLt γL =kde orkde1/3ρs2/3, ˆ λ =λ /δL.

Island equation c.f. simulation

Solid: simulation, dashed: island equation, cJ = 0.025, ρs =0.2,de =0.1,k=0.5,γL =0.0024

Scaling of the reconnection rate: Is it Universal?

Consider scaling of the inflow velocity:

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V in~ fVAd

It has been argued that f ~ 0.1 [Shay et al., 2004], in a universal asymptotic regime, independent of system size and dissipation mechanism.

Using the island equation in the asymptotic regime:

f ~ cJ δL10

kλ3/ 2

de

Note that L is of the order of de (or ).de2/3ρs

1/3

f depends on parameters de, and k. It also depends weakly on time through (t) ~ 1 in the nonlinear phase.

Numerically f is seen to be of the order of 0.1, if one uses simulation parameters of the type used by Shay et al.

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s

Role of CMSO experiments in resolving issues of scaling

• CMSO experiments can play an important role in elucidating scaling issues in quasi-steady (MRX, SSPX) as well as time-dependent reconnection (sawteeth in MST, combined with comparative tokamak studies).

• Strategy: vary ion/electron skin depths, plasma beta, guide field, resistivity, and boundary conditions. Theory and PIC simulations predict a range of scaling behavior.

• Identify and measure mechanisms (describable by two-fluid or more complete PIC models) of instability and turbulence of thin current sheets, and challenge simulations to produce them. (This appears to be well under way already.) Can fully kinetic features be incorporated in Hall MHD framework by appropriate closure relations on the anisotropic pressure tensor?

Sawtooth instability in cylinder

• Goal: To investigate the m=1 sawtooth instabilty in a cylinder for parameters very similar to Aydemir’s (1992) four-field simulation which showed near-explosive growth of islands, possibly accounting for the sawtooth trigger.

• This exercise, expected to be a routine benchmark, turns out to be a surprise: no explosive growth is seen. The Hall currents induce poloidally asymmetric shear flows that are strongly stabilizing.

• The runs reported were carried out on IBM Power4 (Cheetah) at CCS, ORNL.

S = 109

Jz

Jz, cut

zoom

Resistive MHDt=200 t=400 t=600

Poloidal velocity streamlines, Vz (color coded)

Hall MHDt=200 t=260 t=320

Two-fluid (or Hall MHD)

Resistive MHD

Growth rate(Time-history)

MRC Cylinder

Magnetospheric Substorms

Observations (Ohtani et al. 1992)

Hall MHD Simulation

Cluster observations: Mouikis et al., 2004

0

0.01

0.02

0 10 20 30 40t/τA

0

1

2

0 10 20 30 40τ/τ

A

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Bx =B0tanh(z/a)vA/c=0.2 a=L z /2 mi /me =25Ti =5Te di /a=0.1Eyb=0.01B0(vA /c)(1+cosπx/Lx)

J y

t=29

Highly CompressibleBallooning Modein Magnetotail(Voigt model)

x = -1 to -16 RE

z = -3 to 3 REky= 25*2e = 126growth rate:0.2

t=5.8

Ux Uz

ky and Dependence

Resistive Tearing Modes in 2D Geometry Equilibrium Magnetic field assumed to be either infinite or periodic along z

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B = ˆ x BP tanh y / a + BT ˆ z

Time Scales

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τA = a /VA = a 4πρ( )1/2 /BP

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τR = 4πa2 /(ηc2)

Lundquist Number

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S = τ R /τ A

Tearing modes

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γ∝ S−3/5 (slower) or S−1/3 (faster)

y

x

(Furth, Killeen and Rosenbluth, 1963)

Neutral line at y=0

Collisionless Tearing Modes at the Magnetopause (Quest and Coroniti, 1981)

• Electron inertia, rather than resistivity, provided the mechanism for breaking field lines, but the magnetic geometry is still 2D.

• Growth rates calculated for “anti-parallel” ( ) and “component” ( ) tearing. It was shown that growth rates for “anti-parallel” merging were significantly higher.

• Model provided theoretical support for global models in which global merging lines were envisioned to be the locus of points where the reconnecting magnetic fields were locally “anti-parallel” (e.g. Crooker, 1979)

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BT = 0

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BT ≠ 0

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

Spine

Fan

(Lau and Finn, 1990)

Towards a fully 3D model of reconnection

• Greene (1988) and Lau and Finn (1990): in 3D, a topological configuration of great interest is one that has magnetic nulls with loops composed of field lines connecting the nulls.

• The null-null lines are called separators, and the “spines” and “fans” associated with them are the global 3D separatrices where reconnection occurs.

• For the magnetosphere, the geometrical content of these ideas were already represented in the vacuum superposition models of Dungey (1963), Cowley (1973), Stern (1973). However, the vacuum model carries no current, and hence has no spontaneous tearing instability. The real magnetopause carries current, and is amenable to the tearing instability.

Equilbrium B-field

Perturbed B-lines

Equilibrium

Perturbed

Field linespenetrating thespherical tearingsurface

Equilbrium Current Density, J

Features of the spherical tearing mode• The mode growth rate is faster than classical 2D tearing modes,

scales as S-1/4 (determined numerically from compressible resistive MHD equations).

• Perturbed configuration has three classes of field lines: closed, external, and open (penetrates into the surface from the outside).

• Tearing eigenfunction has global support along the separatrix surface, not necessarily localized at the nulls.

• The separatrix is global, and connects the cusp regions. Reconnection along the separatrix is spatially inhomogeneous. Provides a new framework for analysis of satellite data at the dayside magnetopause. Possibilities for SSPX, which has observed reconnection involving nulls.

Solar corona

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

Solar corona: heating problemphotosphere corona

Temperature Density Time scaleMagnetic fields (~100G) --- role in heating?

~5×103K ~106K1023m−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 ] ≡ φy

Ax

− φx

Ay

Reduced MHD equationslow limit ofMHD

Magnetostatic equilibrium∂J∂z

+[A, J] = 0 ,or B ⋅∇J = 0

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

(current densityfixed on a field line)

c. f. 2D Euler equation ∂Ω∂t

+[φ,Ω]=0A , J Ω, z t

Existence theorem: If Ω is smooth initially, it is so for allTime. However, Parker problem is not an initial value problem, but a two-point boundary value problem.

Footpoint Mappingx⊥(z)=X[x⊥(0) z] x⊥(L)=X[x⊥(0)L]

wiτh Xz

=∂A∂

(XY z)Yz

=−∂A∂x

(XYz)

For a given smooth footpoint mapping, doesmore than one smooth equilibrium exist?

Identity mapping: x⊥(L)=x⊥(0)e. g. uniform field B=ˆ z or A =const

A proof for RMHD, periodic boundary condition in x

A theorem on Parker's modelFor any given footpoint mapping connected with the identity mapping, there is at most one smooth equilibrium.

(Ng & Bhattacharjee,1998)

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

In 3-D torii (such as stellarators), current singularities are present where field lines close on themselves

• Solutions to the quasineutrality equation,

• General solution has two classes of singular currents at rational surface [i(mn) = n/m] (Cary and Kotschenreuther, 1985; Hegna and Bhattacharjee, 1989).

– Resonant Pfirsch-Schluter current– Current sheet

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∇⋅ J = 0,

€

B ⋅∇λ = −∇⊥⋅

r J ⊥

€

= J ⋅

r B

B2 =Σmn λ mneimθ − inζ ,

Σmn (m − nι )λ mneimθ −inζ = −p'Σmnεmnamne

imθ −inζ

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mn = −p' εmnamn

(m − nι )+ ˆ λ mnδ(ψ −ψ mn )

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=J||

B

Reconnection without nulls or closed field lines

Formation of a true singularity (current sheet) thwarted by the presence of dissipation and reconnection.Classical reconnection geometries involve:(i) closed field lines in toroidal devices(ii) magnetic nulls in 3D

Parker’s model is interesting because it fall underneither class (i) or (ii). Questions:• Where do singularities form? What are the geometricproperties of singularity sites and reconnection? Strategicissue for CMSO research.• Need more analysis and high-resolution simulations.Incompressible spectral element MHD code (underdevelopment by F. Cattaneo) may be very useful.

♦ Start with a uniform B field♦ Apply constant footpoint twisting

φ ( x⊥

, 0 ) = 0 , φ ( x⊥

, L ) = φ0

( x⊥

) with[ φ

0

, ∇⊥

2

φ0

] ≠ 0

♦ Current layers appear after large distortion.♦ Quasi-equilibrium at first, becomes unstable♦ J grows faster in the middle ⇒ non equilibrium

Simulations of Parker's model

Simulations of Parker's model

+-

bottom

topmiddle

Simulations of Parker's model

+-

bottom

topmiddle

tJ

max

Jmax 0

q

2

d

2

qmax

dmax

118.5 9.91 9.86 0.233 0.00028 7.946 0.708

131.6 16.5 10.9 0.930 0.00238 16.04 2.272

145.6 28.3 14.6 4.361 0.1471 32.94 5.997

150.5 35.3 16.4 21.98 13.486 83.39 79.01

with q ≡ ∂ J / ∂ z + [ A J ] ≡ q + ν ∇⊥

2

Ω .

Simulations of Parker's model

More general topologyParker's opticalanalogy (1990)

Main current sheet

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