The Diffusion Region of Asymmetric Magnetic Reconnection

Michael Shay – Univ. of Delaware

Bartol Research Institute

Collaborators

• Paul Cassak

• Our Asymmetric reconnection publications (no guide field):

– General Scaling theory and resistive MHD:• Cassak and Shay, Physics of Plasmas, 14, 102114, 2007.

– Hall MHD simulations• Cassak and Shay, GRL, (In press)

Semantics

• Diffusion region– A non-MHD region where at least one species

is not frozen-in– Not necessarily irreversible dissipation

• Example: Hall region of regular collisionless reconnection.

Review: Reconnection

Magnetic Reconnection

Vin

CA

Process breaking the frozen-in constraint determines the width of

the dissipation region,

Y

XZ

Magnetic Reconnection Simulation

Jz and Magnetic Field Lines

Y

X

QuickTime™ and aBMP decompressor

are needed to see this picture.

• d

Reconnection drives convection in the Earth’sMagnetosphere.

Kivelson et al., 1995

Reconnection in Solar Flares

F. Shu, 1992

• X-class flare: 100 sec.

• B ~ 100 G, n ~ 1010 cm-3 , L ~ 109 cm

• AL/cA ~ 10 sec.

• Reconnection rate Vin

• Conservation of mass: Flow into and out of dissipation region:

Vin ~ (/D) cA

• determined by the process breaking the frozen-in constraint.=> The spatial extent of the dissipation region is of

key importance to determining the reconnection rate.

Calculating Reconnection Rate

Vin

cA

DY

XZ

Two Types of 2D Reconnection

• << D Vin << cA => Slow

• ~ D Vin ~ cA => Fast

D

D

Out of Plane Current

Out of Plane Current

Y

XZ

Kinetic Reconnection (cont.)

• dissipation region in hybrid model ( Shay, et al., 1999)

Je

Ji

Vi

Di

De

cpi

cpe

• Effect of Hall Physics

• Ion dissipation region– Controls R. Rate

Vin ~ (cpi/Di) cA

(cpi/Di) ~ 1/10

No system size Dependence!

• Electron dissipation region– No impact on R. Rate

Vine ~ (cpe/De) cAeY

XZ

Whistler signature

• Magnetic field from particle simulation (Pritchett, UCLA)

•Self generated out-of-plane field is whistler signature•Confirmed with satellite and laboratory measurements.

Overview: Asymmetric Reconnection

• What is Asymmetric Reconnection?

• Diffusion region analysis

• Resistive MHD Simulations– No guide field

• Hall MHD Simulations– No guide field

• Conclusions

Asymmetric Reconnection

• Different B,n on either side of diffusion region.

• Dayside magnetosphere

• Solar reconnection?

• Heliopause reconnection

• d

Intense currents

• MHD not valid

• No frozen-in Kivelson et al., 1995

High nLow B

Low nHigh B

Observation

• Asymmetric– Reconnecting B-field

– Density

– Temperature

Previous Work

• Shock structure– Petschek slow shocks => Intermediate wave+expansion fan (Levy et al., 1964)

– Further work• Petschek and Thorne, 1967; Sonnerup, 1974; Cowley, 1974; Semenov et al., 1983, MHD

(Hoshino and Nishida, 1983; Scholer, 1989; Shi and Lee, 1990; Lin and Lee, 1993; La Belle-Hamer et al., 1995;

Ku and Sibeck, 1997; Ugai, 2000; …), Kinetic - Hybrid: (Lin and Lee, 1993; Lin and Xie, 1997; Omidi et

al, 1998; Krauss-Varban et al., 1999; Nakamura and Scholer, 2000; …), Particle: - Okuda, 1993.

– Other relevant studies: Ding et al., 1992; Karimabadi et al., 1999; Siscoe et al., 2002; Swisdak et al., 2003; Linton, 2006; many dayside studies

• Scaling studies undertaken only recently– Diamagneticd Stabilization (Swisdak et al., 2003)

– Orientation of X-line, outflow speed (Swisdak and Drake, 2007)

– MHD studies: (Borovsky and Hesse, 2007, Birn et al., 2008)

– Global MHD (Borovosky et al., 2008)

– PIC: (Pritchett, 2008), Tanaka, 2008; Huang et al., 2008

– PIC-Satellite comparisons (Mozer, Pritchett et al., 2008)

Conservation Laws• Write MHD in conservative form ( = mass density, v = flow velocity,

B = magnetic field, P = pressure, E = electric field,

– Integrate over closed surface.

∂∂t

=−r∇⋅

rv( )

∂∂t

rv( ) =−

r∇⋅

rv

rv + P +

B 2

8π

⎛

⎝⎜

⎞

⎠⎟

tI −

rB

rB

4π

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∂E∂t

=r∇⋅ E + P +

B 2

8π

⎛

⎝⎜

⎞

⎠⎟

rv−

rv ⋅

rB( )

4π

rB

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∂rB∂t

=−cr∇×

rE

drS⋅

rv( )∫ =0

drS ⋅

rv

rv + P +

B 2

8π

⎛

⎝⎜

⎞

⎠⎟

tI −

rB

rB

4π

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∫ =0

drS ⋅ E + P +

B 2

8π

⎛

⎝⎜

⎞

⎠⎟

rv−

rv ⋅

rB( )

4π

rB

⎡

⎣⎢⎢

⎤

⎦⎥⎥∫ =0

drS ×∫

rE =0

E=

12v2 +

Pγ −1

+B 2

8π= total energy)

More General Diffusion Region• Steady state diffusion relation• Integrate conservation relations

drS⋅

rv( )∫ =0

drS ⋅

rv

rv + P +

B 2

8π

⎛

⎝⎜

⎞

⎠⎟

tI −

rB

rB

4π

⎡

⎣⎢⎢

⎤

⎦⎥⎥

∫ =0

drS ⋅ E + P +

B 2

8π

⎛

⎝⎜

⎞

⎠⎟

rv−

rv ⋅

rB( )

4π

rB

⎡

⎣⎢⎢

⎤

⎦⎥⎥∫ =0

drS ×∫

rE =0

B1

22L

v1

vout

1

B2 v22

out

Conservation of mass

Conservation of momentum

Conservation of Energy

B/t = 0

More General Diffusion Region• Steady state diffusion relation• Integrate conservation relations

B1

22L

v1

vout

1

B2 v22

out

Conservation of mass

Conservation of momentum

Conservation of Energy

B/t = 0

1v1 + 2v2( )L ~ outvout( )2

PressureBalance

B12

8πv1 +

B22

8πv2

⎛

⎝⎜

⎞

⎠⎟ L ~

12outvout

2⎛

⎝⎜⎞

⎠⎟vout2

v1B1 ~v2B2

Asymmetric Scaling Relations• Solving gives

• Need out

vout2 ~

B1B

2

4πB1 + B2

1B2 + 2B1

E ~1c

out B1B2

1B2 + 2B1

⎛

⎝⎜

⎞

⎠⎟vout

2L

Outflow speed

Reconnection Rate

Outflow Density?

• Assume reconnected flux tubes mix and conserve total volume.

– Each flux tube contains same amount of flux:

• B1A1 ~ B2A2

out ~MV

~1A1L + 2 A2L

A1L + A2L

⇒ out ~1B2 + 2B1

B1 + B2 A2

A1L

L

1

2

v

out2 ~

B1B

2

4πout

and E ~1c

2B1B2

B1 + B2

⎛

⎝⎜

⎞

⎠⎟vout

L

Structure of the Dissipation Region• Since v1 B1 ~ v2 B2, the stronger magnetic field flows in slower

– So it makes sense that the X-line is displaced toward the strong field side ofthe dissipation region.

But this is incorrect!

Weak field B1

Strong field B2

Weak field B1

Strong field B2

X2

X1X

B22

8πv2 ~

B1B2

8πv1 ~

B2

B1

⎛

⎝⎜

⎞

⎠⎟

B12

8πv1

• The X-line is actually shifted toward the weak field side!– Why? While the flow coming in the strong field side is slower, the flux of energy is larger.

Calculation of Location of X-line• Evaluate conservation of energy for volume from edge to X-line

B1

X22L

v1

vout

1

B2 v2 2

out X1

B12

8πv1

⎛

⎝⎜

⎞

⎠⎟ L ~

12outvout

2⎛

⎝⎜⎞

⎠⎟vout X 1

B22

8πv2

⎛

⎝⎜

⎞

⎠⎟ L ~

12outvout

2⎛

⎝⎜⎞

⎠⎟vout X 2

X

X 2

X 1

~B2

B1

Their ratio gives:

Location of the Stagnation Point• Similar argument for mass flux

• Stagnation point offset toward side with smaller B/.

B2

B1, 1

B2, 2

S2

S1

v2

v1

S

2v2 ~2v1B1

B2

~2B1

1B2

⎛

⎝⎜

⎞

⎠⎟1v1

S2

S1

~2B1

1B2

X-line and Stagnation Point are not colocated!

• There is a flow across the X-line – Generic to asymmetric reconnection!– Previous magnetopause simulations (Siscoe, 2002; Dorelli et al., 2007, …) – Quantitative predictions of the location of X-line and stagnation point (Cassak and

Shay, 2007) have been questioned (Birn et al., 2008)

Which plasma flows across the X-line?

• Inflow Alfven speeds control flow across X-line

S1 >X 1

B22

4 π2

>B1

2

4 π1

Since cAsp > cAsh there is a flow of magnetosheath plasma in to magnetosphere. (Matches observations.)

Results are General

• These relations give E and vout in terms of upstream parameters.

– No specificaion of diffusion mechanism or Hall term.– General applicability

• Require diffusion (non-MHD) mechanism to determine absolute values:

– Sets diffusion region widths 12 and L

– Determines actual reconnection rate

Resistive MHD

• To find an absolute reconnection rate, we need to specify a dissipation mechanism. For asymmetric Sweet-Parker,

• Uniform resistivity Sweet-Parker reconnection

v1

c×B1 : ηJ 1

v1

1~

ηc2

4π12

E ~

1c

B1B

2v

out

ηc2

4πvout L

Fluid Simulations

• Double current sheet configuration

• x = outflow y = out-of-plane z = inflow

• B, T tanh functions

• n balances B2

Resistive MHD Simulations

• V normalized to cA, Length normalized to L0

• Size: 409.6 X 204.8, 4096 X 2048 grids η0.05• (Lundquist number = 8,192-40,960) min = 1 initially

• n1 = n2

• [B1,B2] = [1,1], [2,1], [3,1], [4,1], [5,1], [4,2]

Resistive MHD Simulations

• [B1,B2] = [1,3]

• [n1,n2] = [1,1]

• x = outflow y = out-of-plane z = inflow

MHD Results

Out-of-plane current density J

Cut across X-linealong inflow

Cut across X-linealong inflow

X S

Decoupling of X-line and

stagnation point borne out in MHD

simulations.

MHD Results

• Color = out-of-plane currentWhite = magnetic field lines

• Initial field asymmetry = 3,no density asymmetry

• Signatures– Typical “bulge” into

low-field region

– Particles flow acrossX-line

Energy and Mass Flux Check

• Determined geometry of diffusion region from simulations.– Non-trivial

• Energy and Mass Flux balances in each sub-region

Flux in

Flux out

Verification of Scaling• Scaling laws for outflow speed vout and reconnection rate E in terms of geometry and upstream

parameters tested

• Very good agreement• Other studies find agreement:

– Borovsky and Hesse, 2007 (anomalous resistivity MHD)– Birn et al., 2008 (Anomalous resistivity MHD)– Borovsky et al., 2008 (Global MHD)– Pritchett, 2008 (Kinetic PIC)

vout E

B

1B

2/ 4π

2L

vout

cB1B2

B1 + B2

ηc2

4πLvout B1B2

E

Hall MHD Simulations

• Two dimensional Hall-MHD simulations

• Anti-parallel magnetic fields

• Three sets of runs– Asymmetric fields [B01,B02] = [1,1], [2,1], [3,1], [0.5,1], Symmetric density

– Asymmetric density [n01,n02] = [1,1], [2,1], [3,1], [0.5,1], Symmetric field

– Asymmetric density and field [B01(n01),B02(n02)] = [2(1),1(2)], [1(1),0.5(4)]

• Asymmetric initial temperature to balance pressure

• Box size = 204.8 x 102.4 c / pi

• Grid scale = 0.05 c / pi

• me = mi / 25 (density asymmetry not included in electron inertia term)

min = 4 initially

Hall MHD Simulations

• [B1,B2] = [1,2]

• [n1,n2] = [2,1]

• x = outflow y = out-of-plane z = inflow

Hall-MHD Results

Cuts across x-line along inflow

Top - field lines (white) and out-of-plane magnetic field (color)Bottom - electron (black) and ion (white) flow lines and out-of-plane current (color)

Initial field asymmetry = 3

• Electron and ion stagnation points different!

Verification of Scaling• Generalized Sweet-Parker like scaling is satisfied for both electrons and

ions.

B1B

2(B

1+ B2 )

4π( 2B1 + 1B2 )

B1B

2(B

1+ B2 )

4π( 2B1 + 1B2 )

vout (ions) vout (electrons)

theory

E

The Big Picture

Magnetospheric Applications?• Agreement of the scaling of E for Hall reconnection

/ L ~ 0.1 is independent of the asymmetry in B and

• Are the results applicable to dayside magnetopause reconnection?– Yes (Borovsky)

• In global MHD simulations, the reconnection rate at the nose of the magnetopause agreed with E based on local parameters rather than the solar wind electric field (Borovsky et al., 2008).

– No (Dorelli)• The analysis is manifestly two-dimensional, whereas 3D effects

(such as flows) are important at the magnetopause.• The orientation of the X-line between arbitrary fields not predicted.

– Critical Question:• Can significant portions of dayside reconnection be characterized as quasi-2D?• Does a fluid element traversing the diffusion region see 3D effects?

Solar Wind-Magnetospheric Coupling Models

• Newell et al., 2007– Best model to date,

but it uses ad hoc fitting to achieve performance

• Borovsky (2008) usedour scaling result to derivea coupling function fromfirst principles– It performed as well as

Newell’s

Scaling of reconnection is a potential starting point for a quantitative understanding of

solar wind-magnetospheric coupling

Conclusion• We have derived the scaling of the reconnection rate and outflow speed

with upstream parameters during asymmetric reconnection [Cassak and Shay, Phys. Plasmas 14, 102114 (2007)]– Numerical simulations agree with the theory for

collisional and collisionless (Hall) reconnection

• Signatures of Asymmetric Reconnection– X-line and stagnation point not coincident for asymmetric B field– There is a bulk flow across the X-line

• Potential applications to the dayside magnetosphere (Borovsky, 2008; Turner et al., in prep), though future work is needed

Future Directions• Much work to be done• Effect of guide field

– Diamagnetic stabilization (Swisdak et al., 2003)– Orientation of X-line (Swisdak et al, 2007)

• More realistic two-scale diffusion region– Requires Kinetic PIC– Pritchett, 2007

• Separatrix structures– Mozer et al., 2007

• Linking separatrix structures with diffusion region structure.