Relativistic Reconnection: From Flat to Curved Spacetime · From Flat to Curved Spacetime Luca Comisso In collaboration with F.A. Asenjo (Universidad Adolfo Ib anez)~ ... space and

Relativistic Reconnection:From Flat to Curved Spacetime

Luca Comisso

In collaboration with F.A. Asenjo (Universidad Adolfo Ibanez)

Department of Astrophysical Sciences, Princeton Universityand

Princeton Plasma Physics Laboratory

Astroplasmas Seminar - April 28, 2017

Luca Comisso Astroplasmas Seminar

Outline

I BackgroundI Motivations to work on this topicI Magnetic connections and reconnectionI Some classical reconnection models

I Special relativistic generalizations of Sweet-Parker andPetschek models

I Relativistic resistive MHD (Lyubarsky 2005)I Relativistic pair plasmas (Comisso & Asenjo 2014)

I Relativistic Sweet-Parker extension in Kerr spacetimeI Equatorial-azimuthal and equatorial-radial current sheets

(Asenjo & Comisso 2017)

I Questions and perspectives


Why are we interested in magnetic reconnection?

Magnetic reconnection is thought to play a key role in manyphenomena in laboratory, space and astrophysical plasmas.

Solar flares and CMEs

Multiwavelength image of the Sun

taken by Solar Dynamics Observatory

Magnetospheric substorms

Aurora Australis recorded from the

International Space Station



Sawtooth crashes

Interior of a tokamak device

with a hot plasma

Gamma-ray flares

X-ray image of the Crab Nebula

taken by Chandra X-ray Observatory

Many other systems where magnetic reconnection is important!



It could be important also in the vicinity of black holes...

Artist’s conception of a supermassive black hole and accretion disk


Magnetic connections

Magnetic connections: In an ideal plasma, two plasmaelements connected by a magnetic field line at a given time willremain connected by a magnetic field line at any subsequenttime (Newcomb, 1958).

⇒ If dl×B = 0 at t = 0, then it remains 0 for all times.



Given that dl×B = 0 at t, magnetic field line conservationrequires

d

dt(dl×B) = 0

UsingdB

dt= (v · ∇)B +∇× (v ×B) and

dl

dt= (dl · ∇)v

⇒ d

dt(dl×B) = −(dl×B)(∇ · v)− [(dl×B)×∇]× v



It is possible to generalize this theorem in a covariant way(Pegoraro, EPL 2012; Asenjo & Comisso, PRD (in preparation)):

I Define the infinitesimal spacelike 4-vector dlµ (distancebetween two close events) such that

dlµFµν = 0

I It can be shown that if UνFµν = 0, then

d

dτ(dlµF

µν) = − (∂νUβ)(dlαF

αβ)

I In the frame where dl0 = 0, the condition dlµFµν = 0 is

equivalent to dl×B = 0 and includes dl ·E = 0

I If dl0 6= 0, simultaneity can be restored resetting the timeby changing dlµ → dl′µ = dlµ + Uµdλ such that dl′0 = 0.


Magnetic reconnection

=t1t =t2t

A

C

D

B

A

C

B

D

Magnetic reconnection: process whereby the connectivity ofthe magnetic field lines is modified due to the presence of alocalized diffusion region.

I It results in conversion of magnetic energy into bulk kineticenergy, thermal energy and super-thermal particle energy.

I It causes a topological change of the macroscopic magneticfield configuration.


Sweet-Parker model

2L

2avout

vin B

I The Sweet-Parker model assumes a resistive current sheet,2-D, ∂/∂t ≈ 0 and ∇ · v ≈ 0.

I It gives some important quantities as a function of theLundquist number S = LvA/η :

a ≈ L/√S , vout ≈ vA , vin ≈ vA/

√S .

I For solar corona parameters S ∼ 1014, τA = L/vA ∼ 1 s⇒ τSP = L/vin ∼ 107 s (solar flares last 102 − 103 s)


Several other models

Other models have been proposed to explain “fast” reconnection

Petschek model(also Priest and collaborators)

2L

2a

2L*

vout

vin B

Turbulent models (e.g. LV99)

Collisionless models(many contributions...)


Sweet-Parker at large S is too slow to be true...

I Sweet-Parker scalings do not hold for S = LvA/η > Scriticalbecause of the plasmoid instability !

104 105 106 107100

101

102

St re

c

=1e−3 =1e−4 =1e−5

S1/2

Bhattacharjee et al., PoP 2009

Huang & Bhattacharjee, PoP 2010

(also Uzdensky et al., PRL 2010)


Old classics revisited for σ � 1

I What about magnetically dominated environments?

σ =B2

4πh� 1 (vA ≈ c)

where h = nmc2f

I Different relativistic generalizations of Sweet-Parker andPetschek models:

Blackman & Field, PRL 1994Lyutikov & Uzdensky, ApJ 2003Lyubarsky, MNRAS 2005

collisional

Comisso & Asenjo, PRL 2014}

collisional + collisionless

I Resistive RMHD simulations are more consistent withLyubarsky 2005 and Comisso & Asenjo 2014 analyses


Old classics revisited for σ � 1

I Lyubarsky (2005) showed that the Sweet-Parker andPetschek models can be generalized straightforwardly bysubstituting vA with c in the classical formulae:

vout ≈ c , γout ≈ 1

vinc≈ 1√

S

γout ≈√σ , θ ≈ 1/σ

vinc

∣∣∣max≈ π

4 lnS

where S = 4πL/ηc is the relativistic Lundquist number.


Numerical verification

Takahashi et al., ApJL 2011 Zenitani et al., ApJL 2011


But what about collisionless effects?

I In recent years, a lot of work has been devoted torelativistic pair plasmas.

I This was mostly driven by particle acceleration studies.

Sironi & Spitkovsky, ApJL 2014 Guo et al., PRL 2014

Also Melzani et al. A&A 2014, Werner et al. ApJL 2016, ....

I Is it possible to include collisionless effects in therelativistic Sweet-Parker and Petschek models?


Pair plasma relativistic Sweet-Parker

I It is possible to perform a more general analysis (Comisso& Asenjo, PRL 2014) starting from pair plasma equationswith two-fluid/collisionless effects (see Koide ApJ 2009):

Continuity equation

∂µ (nUµ) = 0

Generalized momentum equation

∂ν

(hUνUµ +

h

4n2e2JνJµ

)= −∂µp+ JνF

µν

Generalized Ohm’s law

1

4ne∂ν

[h

ne(UµJν + JµUν)

]= UνF

µν − η c [Jµ + UαJαUµ(1 + Θ)]


Pair plasma relativistic Sweet-Parker

I Assuming pressure balance across the layer and fluxconservation we obtain

vout ≈ c , γout ≈ 1

I Matching the reconnection electric field inside and outsidethe current sheet we find

δ =√δ2η + δ2ti , with δη ≈ S−1/2L , δti ≈

√fλe2

vinc≈

√1

S+f

4

(λeL

)2

, with f =K3

(mc2/kBT

)K2 (mc2/kBT )


Pair plasma relativistic Petschek

I In a Petschek-like scenario we need to formulate the jumprelations at the shocks connected to the diffusion region:

ρ1γ21

v1c

+1

4πBt1Et = h2γ

22

vn2c

+1

4πBt2Et

ρ1γ21

v21c2

+ρ1

4n2e2J21+

1

8πB2t1 = h2γ

22

v2n2c2

+h2

4n2e2J2n2+p2+

1

8πB2t2

− 1

4πBnBt1 = h2γ

22

vn2vt2c2

+h2

4n2e2Jn2Jt2 −

1

4πBnBt2

- subscripts 1 and 2 refer to upstream and downstream

- subscripts n and t refer to normal and tangential components

I Assuming switch-off type shocks (Bt2 = 0) we obtain

γout ≈√σ , θ ≈ 1/σ


Pair plasma relativistic Petschek

I Assume that the magnetic field in the inflow region is asmall perturbation to a uniform magnetic field B0

I Assume that the magnetic field changes mainly within thediffusion region, whereas outside is irrotational

I Then: L∗ ≈ ηc+ βc

4π

(c

vin

)2

vinc

∣∣∣max≈ π

8

[ln

(4πL

ηc+ βc

)]−1, with β =

πfλ2eLc


Importance of the thermal-inertial effects

⇒ L. Comisso and F.A. Asenjo, Phys. Rev. Lett. 113, 045001 (2014)

I In both Sweet-Parker and Petschek relativistic scenariosthere is an increase of the reconnection rate owing to thethermal-inertial effects

I If the thermal-inertial layer width δti ≈√fλe/2 exceeds

the resistive layer width δη ≈ S−1/2L, the reconnectionprocess enters into the collisionless regime

I This occurs when πfλ2e � ηLc (⇒ the thermal-inertialeffects have a fundamental role in hot tenuous plasmas)

I β plays the role of a “thermal-inertial resistivity” thatlimits the response of the electrons and positrons to thereconnection electric field


Can we extend this analysis to curved spacetime?

I General Relativistic MHD simulations commonly show theformation of reconnecting current sheets

McKinney et al., MNRAS 2012 Lyutikov & McKinney, PRD 2011

I Typical locations of current sheet formation are in theequatorial plane (they occur in other locations as well)


Some preliminaries... (Asenjo & Comisso, PRL 2017)

I An effective representation of the GRMHD Eqs. can beobtained by writing them in the 3 + 1 formalism.

I Decomposing the spacetime into “space”+“time” allows todisplay curvature effects explicitly in a set of vectorial Eqs.

I It is convenient to introduce a locally nonrotating frame,the ZAMO (“zero-angular-momentum-observer”) frame.

I For observers in the ZAMO frame, the spacetime is locallyMinkowskian


Some other preliminaries...

I We consider the spacetime given by the Kerr metric inBoyer-Lindquist coordinates (t, r, θ, φ), for which

h0 = (1− 2rgr/Σ)1/2 , h1 = (Σ/∆)1/2 ,

h2 = Σ1/2 , h3 = (A/Σ)1/2 sin θ ,

ω1 = ω2 = 0 , ω3 = 2r2gar/Σ .

I rg = GM is the gravitational radiusI a = J/Jmax ≤ 1 is the rotation parameter (Jmax = GM2)I Σ = r2 + (arg)

2cos2θ

I ∆ = r2 − 2rgr + (arg)2

I A =[r2 + (arg)

2]2 −∆(arg)2sin2θ

I Recall thatα = (∆Σ/A)1/2 and βj = βφδjφ

I βφ = h3ω3/α measures the rotation of this spacetime


Azimuthal reconnection layer

I Assume pressure balanceacross the layer

I Assume magnetic fluxconservation

I Match the reconnectionelectric field inside andoutside the current sheet(Eθ is not uniform, butEθ|i ≈ Eθ|X if δ is small)

I In this way we arrive at: vout ≈ c , γout ≈ 1

δ

L≈ 1√

S

√r

h21h3

∣∣∣∣X

vinc≈ 1√

S

√r

h3

∣∣∣∣X

I The spacetime curvature gives r/h3|X < 1


Radial reconnection layer

I What if the current sheet is oriented along the radialdirection? In this case:

I γout ≈(

1 + L∂ lnα

∂r

∣∣∣∣o

)−1/2

Ivoutc≈(

1

2− L

2

∂ lnα

∂r

∣∣∣∣o

)1/2

Iδ

L≈ h1|o

vinγoutvout

vinc≈ 1√

Sh1|o

[1− L∂r lnα|o1 + L∂r lnα|o

]1/4

I In the flat spacetime limit α→ 1 and h1|o → 1

I Sufficiently far (rg � ro):vinc≈ 1√

S

[1− rg

2ro+

(2a2−3)r2g8r2o

]Luca Comisso Astroplasmas Seminar

Effects of the spacetime curvature

⇒ F.A. Asenjo and L. Comisso, Phys. Rev. Lett. 118, 055101 (2017)

Azimuthal reconnection layer:

I The reconnection rate decreases due to the black holerotation

I The spacetime curvature created by the black hole massitself does not play an important role

Radial reconnection layer:

I The black hole rotation acts to increase the reconnectionrate

I The black hole mass acts to decrease the reconnection rate

I The reconnection rate decreases because of the black holemass that has the dominant contribution


Questions and perspectives

BlackHole

Br

X

2δ

2L

Is it possible to include also collisionless effects? Sure!

Stay tuned! Comisso & Asenjo (in preparation)


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Relativistic Reconnection: From Flat to Curved Spacetime · From Flat to Curved Spacetime Luca Comisso In collaboration with F.A. Asenjo (Universidad Adolfo Ib anez)~ ... space and