Current Sheet and Vortex Singularities: Drivers of Impulsive Reconnection

A. Bhattacharjee, N. Bessho, K. Germaschewski, J. C. S. Ng, and P. Zhu Space Science Center

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

Isaac Newton Institute, Cambridge University, August 9, 2004

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 =1

SJ + de

2 dJ

dt+

din

J × 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 Problem

Dynamics 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:

Two-Field Reduced Model for large guide field and low plasma beta

(Schep/Pegoraro/Kuvshinov 1994)(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 equation

ddt

˙ λ

δLδx2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ≈

cJ k2λ4

4de2δx

2

˙ δ x ≈−˙ λ δxδL

1+ρs

2

cJ de2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ or δx ≈δL exp−

λδL

1+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 orkde1/3ρs

2/3, 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:

€

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 δL

10kλ3/ 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 for certain popular cloices of simulation parameters, but this is not universal.

€

s

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 40/t τ

A

€

Bx =B0tanh(z/a)

vA/c=0.2 a=L z /2 mi /me =25

Ti =5Te di /a=0.1

Eyb=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 RE

ky= 25*2e = 126growth rate:0.2

t=5.8

Ux Uz

ky and Dependence

Euler equation for incompressible flow:

∂v∂t

+v⋅∇v=−∇p

The incompressibility condition produces a’ Poisson s equation for pressure

- Integro differential equation for pressure

p(x,t) =14

∇⋅(v⋅∇v)x− ′ x

d ′ x∫ + b. .c

Solutions to Euler are essentiallynonlocal.

∇⋅v=0

∇2p=−∇⋅(v⋅∇v)

High-symmetry flow (Pelz 1997)

t=0

t=.49

t=.33

Vorticity in the high-symmetry flow

Vorticity2D and 1D cuts

Growth of vorticity

Distortion ofvortices

vorticity pressure

Sufficient condition for finite-timesingularity

Consider flow along x -axis ( y = z = 0 ). ByKi dasymmet ry v

y= v

z= 0 , ω = 0

Eule r equat ion ˙ vx

= − px

Defi neα = ∂x

vx

, = ∂y

vy

, γ = ∂z

vz

( α + + γ = 0 )

Differentiati ng Euler, obtai n exa ctequation

˙ α + α

2

= − pxx

I f pxx

> 0 followi ng aLagr angianelement,t henα wi ll besingular i n finit eti .me

Sufficient condition (Eulerian)

Origin is a

• velocity null point

• point of intersection of vorticity nulllines

Kida symmetries ensure

px

( 0 ) = pxx

( 0 ) = pxxx

( 0 ) = 0

Fro mth eexac tequati (on al ongy = z = 0 ), by Taylor expansion

˙ α + α

2

= − pxx

( x ) = −

pxxxx

( 0 )

2

x

2

+ O ( x

4

) ,

whe reα ≡ ∂x

vx.

I f the reex ists ar ange0 < x < X ( t ) i n whichp

xxxx is positiv e andincreasi ngrapidly

, enoughther e is a finit -e tim esingularit .y

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

€

τA = a /VA = a 4πρ( )1/2 /BP

€

τR = 4πa2 /(ηc2)

Lundquist Number

€

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)

€

BT = 0

€

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 SSX, 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 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

y

Ax

− φx

Ay

Reduced MHD equations

low 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]=0

A , 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] ,

withdXdz

=∂A∂y

(X,Y,z) ,dYdz

=−∂A∂x

(X,Y,z)

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 [mn) = n/m] (Cary and Kotschenreuther, 1985; Hegna and Bhattacharjee, 1989).

– Resonant Pfirsch-Schluter current– Current sheet

€

∇⋅ r

J = 0,

€

rB ⋅∇λ = −∇⊥⋅

r J ⊥

€

= r

J ⋅r B

B2=Σmn λ mne

imθ −inζ ,

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

imθ −inζ

€

mn = −p'εmnamn

(m − nι )+ ˆ λ mnδ(ψ −ψ mn )

€

=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

top

middle

Simulations of Parker's model

+-

bottom

top

middle

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 ] , d ≡ q + ν ∇⊥

2

Ω .

Simulations of Parker's model

More general topologyParker's opticalanalogy (1990)

Main current sheet

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