Numerical modeling of relativistic
magnetic reconnection:
Kinetic, two-fluid, and MHD simulations

Seiji ZENITANI
National Astronomical Observatory of Japan
Ackn: M. Hoshino (U. Tokyo), M. Hesse, A. Klimas,
M. Kuznetsova (NASA/GSFC), T. Miyoshi (Hiroshima U)

Outline• 0. Introduction• 1. Kinetic modeling– 2D reconnection• 2. Fluid modeling– RRMHD & two-fluid simulations– New insight: high-speed fluid dynamics• 3. Cross-code comparison• 4. Summary

Magnetic reconnectionSun• Re-configuration of magnetic topology• Release of magnetic energy to plasma energy• Violation of the ideal condition is necessary

Earth’s MagnetosphereJetJet

Magnetic reconnection
in high-energy settings• Magnetically dominatedsettings• Special relativity– + radiation, pair creation …

Gamma-ray flares

Tavani 2011

Jet model

Giannios+ 2009

Striped windCoroniti 1990

Current status of relativisticreconnection research (2015Q2)
MHD theories
Blackman & Field 1994
Lyutikov & Uzdensky 2003
Lyubarsky 2005

• Theories have been discussed over two decades• Many works came out after the Crab flare (2011)
Kinetic (PIC) simulations
MHD
simulations

Comisso & Asenjo 2014

Watanabe &Yokoyama 2006
Zenitani+ 2010, 11
Zanotti & Dumbser
2011
Takahashi+ 2011
Takamoto 2013
(Zenitani+ 2015)• Basic properties areunder debate• Ideal for globalmodeling

Tenbarge+ 2010

Two-ﬂuidsimulations
Zenitani+ 2009

• Meso-scaleevolution

Zenitani & Hoshino
2001-2008
Jaroschek+ 2004-2009
Liu+ 2011
Cerutti+ 2012-2014
Kagan+ 2013
Sironi & Spitkovsky 2014
Melzani+ 2014
Guo+ 2014-2015
Zenitani in prep.
• Fast evolution• Particle acceleration

1. Kinetic modeling2D Particle-In-Cell (PIC) simulation• Fast reconnection and particle acceleration occurs• Online version: http://th.nao.ac.jp/MEMBER/zenitani/files/reconnection.mov

Energy spectra

Zenitani & Hoshino 2001, arXiv:1402.7139

DC particle acceleration
ZEyX

E2-B2>0

Speiser 1965, Michel 1968,
Zenitani & Hoshino 2001, 2007,
Uzdensky+ 2011, Cerutti+ 2012

Cerutti+ 2012--2014

Hard-energy spectra~ -1

Zenitani & Hoshino 2001

Sironi & Spitkovsky 2014

• Power-law index in earlier workss ~ -1(near acc. region), -2~3 (global)• Fermi process? Upstream magnetization
may control the index (Guo+ 2014)

Guo+ 2014

3D reconnection: Onset problem

Drift-kink instability
dominates inanother 2D planeTurbulent
Plasma heatingZenitani & Hoshino 2008

3D large-scale evolution: RX appears to win

Liu+ 2011
Guo+ 2014• Kink-mode often saturates• Reconnection outruns kink, andthen particle acceleration turns on• 3D energy spectra is similar to 2D
Sironi & Spitkovsky 2014

2. Fluid modelingRelativistic Petschek (PK) reconnection• Relativistic resistive MHD (RRMHD) eqs.

(Watanabe & Yokoyama 2006, Komissarov 2007)• Alternative: relativistic two-fluid eqs.
(Zenitani+ 2009a,b, Barkov+ 2014)

-γvx

+γvx

Watanabe & Yokoyama 2006
Zenitani+ 2009a

Relativistic PK reconnection - Alfvénic outflow• Magnetization parameter• Relativistic Alfvén speed

Local maxima

Ux=γVxAlfvénic
relation(adiabatic acc.)

1

Petschekoutflow

1
Nonrelativistic 

10Zenitani+ 2010

 Ultrarelativistic

Relativistic PK reconnection - a paradox• Narrower opening angle(Lyubarsky 2005)• Speed is limited by Alfvén speed• Reconnection rate (fluxtransfer speed) goes highSemi-relativistic (sigma=0.4)

Nonrelativistic  Ultrarelativistic• Heat-dominated flow sustains
fast energy throughput

Relativistic (sigma=4)

Jyγ2

Γ
pv
Γ −1

Relativistic Sweet-Parker reconnection

SPtheorySPry
eothReconnection rate

Takahashi+ 2011

Transition to theplasmoid regime

Lundquist number(Normalized system length)

Takamoto 2013

New insight: High-speed fluid dynamics• Branches of fluid dynamics

SubsonicTransonic• Incompressible fluids

Supersonic• High-speed fluid dynamics• Compressible fluid dynamics• Adiabatic effects• Shocks• Shock=Shock interaction• Relativistic reconnection is usually in the high-speed regime

csc
3cA = c1+c1>
2

High-speed fluid effects (1/3): Adiabatic acceleration• Proposed by Shimizu & Ugai 2000

Outflow 4-velocity
iγVjet

caLo

lx
ma

• Super-Alfvénic flow was also reportedby Zanotti & Dumbser 2011

maux

ekhtscPe flowtou

1

1

10

Magnetic island(“plasmoid”)

32Ux

Sound 4-velocity

10Laval nozzle

020

40

60

80
X

100120High-speed fluid effects (2/3): normal shocks
ux

Postplasmoid
slow shocks

Downstream

Upstream

Forwardslow shocks

Normal shock (Recompression shock)• Magnetic island = Airfoil

Subsonic(V << cs)

Transonic(0.8cs <V)(b) t = 250zSupersonic(cs < V)vx

x

Compressible effects (3/3): Shock diamondux
Outflow jet

Lab frame

Plasma sheet

Plasmoid frame
PlasmasheetuxShock diamond

Plasmoid frame

Plasmasheet• (a) Over-expanded flow

cs < Vjet

pjet < pambientExpansionwaveShock• (b) Under-expanded flow

cs < Vjet

pjet > pambient

Expansion
wave

Shapiro 1953

Shock diamonds in an extragalactic jet
2x106 light years

PKS 0637-752

Godfrey+ 2012

Shock diamonds in aeronautics

BBC online

http://www.bbc.com/future/story/20130701-flying-the-worlds-fastest-planeShock diamonds in video game

Microsoft Flight Simulator X

https://www.youtube.com/watch?v=S8QGaiE4yWc

Some more shock-diamondsux
1. Over-expanded
shock diamonds

Normal shocks

2. Under-expanded
shock diamonds

±1.5%
uz

3. Mach disk
This may evolve toover-expanded
shock diamonds

Structure of a reconnection-plasmoid system• High-speed effects
were also missing in
nonrelativisticreconnection industry• They result in complex
shock structure in a
low-β plasma=2cscA

2Zenitani & Miyoshi 2011
Zenitani 2015aprelimina

3. Cross-code comparison

ZryRelativistic kinetic Ohm’s law• MHD simulation relines on pre-defined Ohm’s law• What about self-consistent PIC results?
• Stress-energy tensorW=f (u)u u

d3 u• Standard decomposition• Ohm’s law (∂t=0)

Bulk term(incl. relativistic pressure)

Heat flow term
(Only in relativistic kinetic plasma)

Local momentum transport(Incl. kinetic div.P term)

Kinetic Ohm’s law (cont.)• Normalized energy dissipation (~ j.E/n ~ ηj2/n)
De/nezx• Ohm’s law: relativistic heat flow term appears
0.2
0.15

Ey
B00.1
0.05

1nq

1
nqi j· wu u

0· Qij1nq-0.05
2040z

60· P ij80

Zenitani in prep.

PIC vs Two-fluid vs RRMHD - energy throughput
RRMHD

Two-ﬂuidPIC2222 2nmc v)↔
(γnmcv
Matternmcnmcvv) v)(↔
Matter
↔
nmc
γnmc
nmc
vvv flow((↔
MatterMatterflow
flow flow
γγnmc
Matterflow
1& & 2 &2 1 1#
###
122
↔
2%nmv−12v)vnmc
v nmv
2 2nmv v((nmcBulk kinetic
Bulk
kinetic↔
nmv%↔
γ (γγ (γ−1γ−1−1v %%↔
)γnmc)γnmc$ v( v( '2 22Bulk
Bulk
kinetic
kinetic
Bulk
kinetic

RRMHD

Two-ﬂuid()€$$ $ 22 2

2' '# 5 &€2€€Γ flux)
Γ#ΓΓ##&%↔& pv(5γpv(Enthalpy552€ γflux
2 2Enthalpy
↔
pv
γpv%
↔
↔
pv
γpvpv
€%%
(€ flux
Γ −1 $$ $ 22 2 $'pv(' 2 'Enthalpy
flux
Enthalpy
Enthalpy
flux
ΓΓ−1
Γ−1−1
€ 0i
€
Heat€Q€€c flow:€cE×B()€ €Poyntingflux
c c (E ×4B
Poyntingflux)€×EB×π) B)(Poynting
Poynting
flux
flux 4π(E€Zenitani+ 2009b€ € flux
€ 4 π4 πPoynting
Kinetic€€€€ to be important• Kinetic term (heat flow ) appears€€€

Summary• Kinetic modeling– Reconnection features DC particle acceleration– 3D evolution - a kink instability– Now it is a field of active research!!

• RRMHD, Two-fluid modeling– Straightforward extension of nonrelativistic MHD reconnection– Narrower exhaust, faster reconnection rate, heat-dominated flow
– High-speed fluid effects - adiabatic acc., shocks, shock diamonds• Cross-code comparison– Heat flow appears in the Ohm’s law and dominates in the energy
budget (due to accelerated particles)