Relativistic MHD simulations of collision-induced magnetic dissipation in Poynting-flux-dominated jets/outflows

5.9 MB·English

by  Wei Deng

Draft version January 30, 2015
PreprinttypesetusingLATEXstyleemulateapjv.05/12/14
RELATIVISTIC MHD SIMULATIONS OF COLLISION-INDUCED MAGNETIC DISSIPATION IN
POYNTING-FLUX-DOMINATED JETS/OUTFLOWS
Wei Deng (邓巍)1,2, Hui Li (李晖)2, Bing Zhang (张冰)1, Shengtai Li (李胜台)2
Draft version January 30, 2015
ABSTRACT
Weperform3DrelativisticidealMHDsimulationstostudythecollisionsbetweenhigh-σ (Poynting-
flux-dominated) blobs which contain both poloidal and toroidal magnetic field components. This is
5 meant to mimic the interactions inside a highly variable Poynting-flux-dominated jet. We discover a
1 significant electromagnetic field (EMF) energy dissipation with an Alfv´enic rate with the efficiency
0 around35%. Detailedanalysesshowthatthisdissipationismostlyfacilitatedbythecollision-induced
2 magnetic reconnection. Additional resolution and parameter studies show a robust result that the
relative EMF energy dissipation efficiency is nearly independent of the numerical resolution or most
n
physicalparametersintherelevantparameterrange. Thereconnectionoutflowsinoursimulationcan
a
J potentiallyformthemulti-orientationrelativisticmini-jetsasneededforseveralanalyticalmodels. We
alsofindalinearrelationshipbetweentheσ valuesbeforeandafterthemajorEMFenergydissipation
9
process. Ourresultsgivesupporttotheproposedastrophysicalmodelsthatinvokesignificantmagnetic
2
energy dissipation in Poynting-flux-dominated jets, such as the internal collision-induced magnetic

] reconnectionandturbulence(ICMART)modelforGRBs, andreconnectiontriggeredmini-jetsmodel
E for AGNs.
H
h. 1. INTRODUCTION Gao & Zhang 2015). Next, strong linear polarization
was discovered during the prompt gamma-ray emission
p The energy composition in the jet/outflow of astro-
- physical systems is an important and fundamental ques- phase for some GRBs (Yonetoku et al. 2011, 2012), and
o during the reverse-shock-dominated early optical after-
tion, since it leads to significant differences for the sub-
r glow emission phase for some others (Steele et al. 2009;
t sequent energy dissipation process, particle acceleration
s Mundelletal.2013),whichhintattheexistenceofglob-
mechanism,radiationspectrumandlightcurve,polariza-
a ally ordered magnetic fields in the jet. Furthermore,
[ tion behavior, neutrino emission luminosity, and so on.
Generally speaking, jets can be separated into two types strong PeV neutrino emission as predicted by the MFD

1 depending on their energy composition: Poynting-flux- models has not been observed from GRBs so far (Ab-
v dominated (PFD) (σ >> 1) and matter-flux-dominated basi et al. 2012), which is consistent with the expecta-
5 tionofthePFDmodels(Zhang&Kumar2013). Finally,
(MFD) (σ <<1), where σ is the magnetization parame-
9 the MFD internal shock (IS) model for GRBs also suf-
terdefinedastheratiobetweentheelectromagneticfield
5 fers some criticisms, such as low energy dissipation effi-
(EMF) energy flux to the plasma matter energy flux.
7 ciency(Panaitescuetal.1999;Kumar1999),electronfast
Many independent observations from Gamma-Ray
0 cooling (Ghisellini et al. 2000), the electron number ex-
Bursts (GRBs), Active Galactic Nuclei (AGNs), micro-
. cess (Bykov & M´esz´aros 1996; Daigne and Mochkovitch
1 quasars, and Crab nebula give strong hints of the PFD
0 outflows at least for some events. Several theoretical 1998; Shen and Zhang 2009), and inconsistency with
5 modelshavebeenproposedwithintheframeworkofPFD some empirical (Amati/Yonetoku) relations (Zhang and
1 jets/outflows to interpret the observations. M´esz´aros 2002; Liang et al. 2010). Zhang & Yan (2011)
v: In the field of GRBs, evidence of PFD jets has been proposed a novel PFD outflow model named as “the
Internal-Collision-induced MAgnetic Reconnection and
i collected independently in several directions. First, a
X prominent thermal emission component as expected in Turbulence (ICMART)”, which can potentially keep the
merits of the IS model but alleviate the criticisms faced
r the fireball-internal-shock model (e.g. M´esz´aros & Rees
a 2000) has been seen only in a small fraction of GRBs by the IS model mentioned above. The main idea of
the ICMART model is that the GRB jets are Poynting-
(e.g. GRB 090902B, Ryde et al. 2010; Zhang et al.
flux dominated. The Poynting flux is catastrophically
2011). The majority of GRBs either show no evidence
discharged at a relatively large distance (e.g. 1015 cm)
of a thermal component or a weak, sub-dominant ther-
from the central engine through collision-induced mag-
mal component (e.g. Abdo et al. 2009; Guiriec et al.
netic reconnection. The magnetic energy is converted to
2011; Axelsson et al. 2012). These GRBs require that
particleenergyandradiationefficiently,leadingtoavery
the GRB central engine is highly magnetized, and jet is
high radiation efficiency as demanded by the GRB data
still PFD at the emission site (Zhang and Pe’er 2009;
(Panaitescu & Kumar 2002; Zhang et al. 2007). A PFD
jethaslessleptonsthantheMFDmodelsothattheelec-
1DepartmentofPhysicsandAstronomy,UniversityofNevada tron excess problem is avoided. A large emission radius
LasVegas,LasVegas,NV89154,USA;[email protected]; favors a moderately fast cooling, which can account for
[email protected]
2Los Alamos National Laboratory, Los Alamos, NM 87545, the right low-energy spectral index observed in GRBs
USA;[email protected];[email protected] (Uhm & Zhang 2014). It also gives a natural explana-
LA-UR-15-20564 tion of the seconds-duration of “slow variability compo-
2 Deng, Li, Zhang & Li
nent”observed in GRBs (Gao et al. 2012). The rapid emission. Althoughthesemodelsshowgreatpotentialto
“fast variability component” can be interpreted within interprettheobservationsandovercomethecriticismsin
this scenario as mini-jets due to locally Lorentz boosted thetraditionalMFDmodels,someimportantingradients
regions (see also Lyutikov & Blandford 2003; Narayan of the models are still of a speculative nature. Detailed
& Kumar 20094). It is speculated that turbulent recon- numerical simulations are needed to give a solid footing
nection in a moderately high-σ flow can give rise to rel- to these models.
ativistic motion of mini-jets within the bulk relativistic From the morphologic point of view, jets/outflows can
motion of the jets. be categorized into two types: continuous and episodic.
For AGNs, observations show fast variable TeV flares Theoretically, episodic jets can be formed either from
of two blazars (Mrk 501 and PKS 2155-304) (Aharonian a highly variable central engine with variable accretion
et al. 2007; Albert et al. 2007). The light-crossing time rate;ordisruptionofacontinuousjetbyscreworkinkin-
is even shorter than the event horizon size of the black stabilities (Li 2000; Mizuno et al. 2009); or from a MHD
holes, so that emission must come from a small local re- erruption process similar to solar coronal mass ejection
gion. The derived Lorentz factor in the emission region (Yuan et al. 2009; Yuan & Zhang 2012). Observation-
should be larger than 50 (Begelman et al. 2008; Mas- ally, episodic jets or knots in jets have been observed in
tichiadis & Moraitis 2008). This value is much larger manyX-raybinaries(Mirabel&Rodriguez1994;Hjellm-
than the observed Lorentz factor of the bulk motion of ing & Rupen 1995; Fender & Belloni 2004) and AGNs
the global jet, which is generally smaller than 10 (Giro- (Marscher et al. 2002; Chatterjee et al. 2009; Doi et al.
letti et al. 2004; Piner & Edwards 2004). To interpret 2011). Rapid variabilities observed in GRBs also point
these observations, Giannios et al. (2009) proposed a towards highly episodic jets (Rees and M´esz´aros 1994;
“jetsinajet”model,whichconsidersthatsomemini-jets Paczy´nski and Xu 1994). As a result, studying interac-
are generated by local reconnection outflows in a global tions or collisions between magnetic blobs or shells is of
PFD jet. The mini-jets can give extra Lorentz boosting great interest.
and particle acceleration to generate the observed TeV In this paper, we perform detailed numerical simula-
photonsaroundtheselocalreconnectionregionswithfast tions on the global properties of collisions between high-
variability. Even though Giannios et al. (2009) did not σ blobs, as envisaged in the ICMART model of GRBs
specify the mechanism of magnetic dissipation, observa- (Zhang & Yan 2011). In Section §2, we give a brief
tions of AGN jets reveal bright knots that are consistent introduction of our 3D relativistic MHD code and the
withinternalinteractionswithinthejet. WithinthePDF simulation setup. In Section §3, we present an exam-
jet scenario, ICMART processes similar to what are en- ple simulation case to show the key results, and perform
visaged in GRB jets may also play a role. a detailed analysis and resolution study. We then ex-
Another related astrophysical phenomenon is γ-ray pand our simulations on two-blob collisions in Section
flares observed from the Crab nebula. Monte carlo sim- §4 to a large parameter space and discuss how differ-
ulations suggest that the bright γ-ray flares and fluctu- ent parameters affect the simulations results. In Section
ations in longer time scales can be understood within §5, we show preliminary results for multiple collisions
theframeworkthattherearemanymini-jetswithawide among four high-σ blobs. We summarize our results in
distribution of size and Lorentz factor within the PFD Section§6anddiscusstheimplicationsofoursimulation
outflowofthepulsar. Theflarescorrespondtotheepochs results on some high energy astrophysical systems, such
whensomebrightmini-jetsbeamingtowardsearth(Yuan as GRBs and AGNs.
etal.2011). Theobservationssuggestthatsimilarphysi-
calprocessesasthoseoperatinginGRBsandAGNsmay
be playing a role in the Crab nebula.
In another front, recent Partical-In-Cell (PIC) sim-
ulations (Sironi & Spitkovsky 2014; Guo et al. 2014)
show that reconnection under high-σ condition can ef-
ficiently accelerate thermal particles to form a non-
thermalpower-lawpopulationoftheparticles. Thisgives
a good support to the above PFD models from the par-
ticle acceleration point of view.
The models discussed above for different astrophysi-
calsystemssharesomecommonphysicalprocesses, such
as efficient magnetic energy dissipation in the PFD out-
flow/jet, mini-jets generated by the relativistic outflows
due to local reconnections, particle acceleration in the
reconnection region, and production of the non-thermal
2. NUMERICAL METHOD AND PROBLEM SETUP
4 Lyutikov & Blandford (2003) and Narayan & Kumar (2009) 2.1. Code introduction
proposed that GRB variability is a consequence of mini-jets due
to relativistic outflow from reconnection or relativitic turbulence. We use a 3D special relativistic MHD (SRMHD)
There is no simple explanation to the observed slow variability code which solves the conservative form of the ideal
component in these models. Zhang & Yan (2011) attributed the MHD equations using higher-order Godunov-type finite-
twovariabilitycomponents(slowandfast)asduetocentralengine
volume methods. This code is a development version of
activity and mini-jets, respectively. Monte Carlo simulations by
Zhang&Zhang(2014)showedthattheICMARTmodelcanindeed the “LA-COMPASS” MHD code which was first devel-
reproducetheobservedGRBlightcurves. oped by Li & Li (2003) at Los Alamos National Labora-
Collision-induced magnetic reconnections & energy dissipations 3
tory. The equations solved in the code are: respectively. Onecanthencalculatether−andz−com-
ponents of the poloidal field
∂(Γρ)
∂t +∇·(ΓρV)=0, (1) 1∂Φ zr (cid:18) r2+z2(cid:19)
B =− =2B exp − , (7)
∂ Γ2h E×B Γ2h B2+E2 r r ∂z b,0r2 r2
( V + )+∇·[ V ⊗V +(p+ )I 0 0
∂t c2 4πc c2 8π and
E⊗E+B⊗B
− ]=0, (2) 1∂Φ (cid:18) r2(cid:19) (cid:18) r2+z2(cid:19)
4π B = =2B 1− exp − . (8)
z r ∂r b,0 r2 r2
∂ B2+E2 0 0
(Γ2h−p−Γρc2+ )+∇·[(Γ2h−Γρc2)V
∂t 8π The poloidal field is closed and keeps the net global
c poloidal flux as zero. The toroidal field configuration is
+ E×B]=0, (3)
4π motivated by considering the black hole accretion disk
∂B system as a “dynamo”, which shears the poloidal flux to
+c∇×E =0, (4) form the toroidal flux from the rotation. The toroidal
∂t
component of the magnetic field therefore has the form
V
E =− c ×B, (5) αΦ r (cid:18) r2+z2(cid:19)
B = =B α exp − . (9)
whereΓ,ρ,h,P aretheLorentzfactor,restmassdensity, φ r0r b,0 r0 r02
relativistic enthalpy, and gas pressure, respectively, V,
Here the parameter α controls the toroidal-to-poloidal
E, B are the vectors of fluid velocity, electric field, and
flux ratio. Li et al. (2006) showed that when α ∼ 3,
magneticfield,respectively,andthesymbol“⊗”denotes
the two flux components are roughly equal with each
tensor product. We also use the ideal gas equation of
other. We set α = 3 for our example simulation, and
state: p = (γˆ −1)u, where γˆ and u are the adiabatic
explore a larger value of α in Section §4.7. We choose
index and the internal energy density, respectively.
the comoving center-of-mass frame of the blobs as our
We use HLL flux with the piecewise parabolic recon-
simulation frame. The direction of velocity is along Z-
structionmethodtosolvetheRiemannproblem(Colella
axis with a profile
& Woodward 1984), and use the constrained transport
(CT) method (Balsara & Spicer 1999; Guan et al. 2014) V , (r ≤r ),
b,z 0
to ensure ∇·B = 0. We use the Cartesian coordinates 
(x,y,z) in our simulations. Vz = (cid:18) (cid:16) (cid:17)2(cid:19) (10)
Vb,zexp − rr−0/r20 , (r >r0),
2.2. Problem set up
whereV isaconstantvaluewhichcanbeeitherpositive
WeenvisagethatthecentralengineofGRBsorAGNs b,z
or negative corresponding to +Z or −Z direction of the
launch a Poynting-flux-dominated jet/outflow. As dis-
velocity. We also set a uniform gas pressure value (P)
cussed in Sect. 1, episodic jets are preferred from ob-
both inside and outside the blobs. The value of P is
servational data. Even if the jet may be overall continu-
muchsmallerthantheinitialmagneticenergydensityof
ous,itisverylikelynon-uniforminternallyandmayform
the blobs.
many knots in the jet, where a much larger amount of
Forthedensityprofile,wefirstdefineaconstantinitial
EMF energy (E ) is concentrated compared with other
em
value of the blob magnetization parameter around the
sparseregionsinthejet. Wecansimplifytheknotsofthe
central region of the blobs:
jet/outflow as many quasi-isolated magnetic blobs with
both poloidal and toroidal field components. Due to the E
intrinsic erratic behavior at the central engine, different σb,i = Γ2emh, (11)
magneticblobsmayhavedifferentvelocitiesattheemis-
sion region, so that multiple collisions are very likely to where h = ρc2+γˆP/(γˆ−1) is the specific enthalpy de-
happen among different blobs. Due to the ultra rela- fined in the fluid’s comoving frame, ρ is the rest mass
tivistic motion of the jet, the relative velocities between density, P is the gas pressure introduced above, γˆ is the
different blobs can easily become relativistic. adiabatic index, Γ is the bulk Lorentz factor calculated
Inoursimulationdomain,weusethemodelfromLiet by the velocity profile introduced above, and E is the
em
al. (2006) to initialize the magnetic field configuration. EMF energy density calculated by E =(B2+E2)/8π
em
The equations are introduced in the cylindrical coordi- from the magnetic field profile introduced above. The
nates(r,φ,z),andwewilltransferthemtotheCartesian density profile is therefore
coordinates (x,y,z) in our simulations. from the center
 (cid:16) (cid:17)
(mre=tri0c)woiftheatchhebplooblosi,dtahleflfiuexldfuinscatsiosunmΦedasto be axisym- ρ= c12 ΓE2σemb,i − γˆγˆ−P1 , (r ≤r0 and ρ>ρbkg),
Φ(r,z)=B r2exp(cid:18)−r2+z2(cid:19), (6) ρbkg, (r >r0 or ρ≤ρbkg),
b,0 r2 (12)
0 whereρ isaconstantparametertocontroltheuniform
bkg
and the relationship between Φ(r,z) and the φ compo- background mass density.
nent of the vector potential A is Φ(r,z) = rA . B We also introduce two position-control parameters z
φ φ b,0 d
and r are the normalization factor for the magnetic and x . For a collision between two blobs, the center of
0 s
strength and characteristic radius of the magnetic blob, the two blobs are located at (x ,y,z ) and (x ,y,z ), so
1 1 2 2
4 Deng, Li, Zhang & Li
z = |z −z | is the initial distance between the center evolution of non-collision case to quantify the EMF en-
d 1 2
of the two blobs in Z direction, and x =|x −x | is the ergy level in the quasi-steady phase. This would serve
s 1 2
initial misalignment between the center of the two blobs as the reference value to be compared with the collision
in X direction due to the possible misalignment of the case in which additional EMF energy drop is expected
blobs. The Y coordinate is the same for both of them. due to additional magnetic dissipation.
In Table 1, we give the normalization relationship be- TheupperpanelofFigure2showstheevolutionofthe
tween the code units and the physical units. There are blob electromagnetic energy E as a function of time
em
only three free parameters, L , B and c to control the (normalized to the initial value E ). The dashed line
0 0 em,0
normalization of the entire system. Defining different shows the evolution in the non-collision case. There is
physical values of L and B , we can normalize the sim- a significant drop of E before t ∼ 6, which is due to
0 0 em
ulation system to different environments and problems. the magnetic field relaxation during the process of es-
In Table 1, we also list two sets of example typical val- tablishingaforcebalancebetweentheoutwardmagnetic
ues to show the way of the normalization. In the rest of pressure force and the inward gas pressure force. After
paper,alltheparametersaregivenusingcodeunits. We the balance is established, E is nearly constant and
em
keep r = 1.0 for all the following simulations. In addi- enters a quasi-steady phase, which can be used as the
0
tion, we use γˆ = 5/3 in most of the simulations, since reference energy level without collision.
mostoftheregimesaremildlyrelativistic. Thismaynot Next, we simulate the collision case between two high-
always be true, especially in the regions of reconnection σ blobs. The initial parameters for these two blobs are
outflows, so in Section §4.8 we also try γˆ = 4/3 to test thesameas the non-collisioncase. The E evolutionof
em
the difference. the two blobs with collision is shown as the solid line in
the upper panel of Figure 2. The efficiency (η) of E
em
3. AN EXAMPLE CASE energy dissipation due to collision-induced process can
In this section, we show a series of detailed analyses be calculated by
based on one example simulation case. We focus on the
E −E
following aspects: the evolution of magnetic energy to η = em,nc em,c, (13)
addresstheefficiencyofmagneticenergydissipation,the Eem,nc
detailsofthecollisionprocess,thepropertiesofmagnetic
reconnection and outflows, and the numerical resolution where Eem,c and Eem,nc are the EMF energy values for
effects. We reveal significant collision-induced reconnec- the collision and non-collision cases, respectively. The
tion events with a remarkable efficiency around 35%, efficiencyoftheexamplecaseisshowninthelowerpanel
which is resolution insensitive. The outflow properties of Figure 2, where we find that the efficiency is about
of reconnection events indicate the potential capability 35% near the end of collision process. This efficiency
to generate super-Alfv´enic relativistic mini-jets. is much higher than the collision-induced kinetic energy
release efficiency in the MFD outflows in the internal
3.1. Initial parameters shock model of GRBs, which is typically a few percent
orless(e.g.Panaitescuetal.1999;Kumar1999;Maxham
The initial parameters of the example run are listed
& Zhang 2009; Gao & M´esz´aros 2014). It is consistent
in Table 2. We consider two identical blobs with initial
withtheanalyticestimateoftheICMARTmodel(Zhang
magnetizationparameterσ =8separatedbyz =4.4,
b,i d & Yan 2011, see more discussion below in §3.5).
with an X-direction offset 1.0. The two blobs move in
Oneimportantquestioniswhatmechanismcausesthis
opposite directions in Z direction with an initial center
efficientmagneticenergydispassion? Fromthemagnetic
speed V = 0.3 c. The background pressure and den-
b,z configuration we can see B and B have opposite di-
sity are P = 10−2 and ρ = 10−1, respectively. In x y
bkg rections around the collision region (see Figure 1). We
order to clearly show the initial magnetic field configu-
suggestthatmostlikelytheadditionalE dissipationis
ration of the blobs, in Figure 1 we show a y = 0 slice em
triggered by strong collision-driven reconnection events.
(cut through the blob centers) of the profiles of several
In order to check our conjecture, in the following, we
parameters: projected field line configuration (panel A),
carry out a series of detailed analyses based on our sim-
σ distribution (panel B), B (panel C), and B (panel
x y ulation data.
D).
The E evolution in Figure 2 can be characterized in
For this example run, the 3D box size is chosen as 203 em
fourstages: (1)aninitial“selfadjustment”(steepdecay)
from -10 to +10 in each dimension, which means that
phase before t ∼ 10; (2) a “plateau” phase from about
the position (x,y,z)=(0,0,0) corresponds to the center of
t∼10tot∼38; (3)a“normaldecay”phasefromabout
the box. And the resolution is chosen as 10243. t∼38tot∼120;and(4)afinalquasi-steadyphase. We
analyze these stages in detail below.
3.2. Energy evolution analysis
The major collision starts from the later part of the
Since the initial magnetic configuration is not in com- “self-adjustment” steep decay phase. The collision com-
plete force balance (between the internal magnetic pres- presses the magnetic fields to make the energy level
sure and the background gas pressure), the blobs would higherthannon-collisioncase. PanelAofFigure3shows
quickly expand and evolve into a quasi-steady phase, a series of representative cuts at t = 4. From left to
forming a quasi-force balance between the gas pressure right,thefourimagesdisplaythe3Dcurrentcontourplot
and magnetic pressure. During this process, a fraction viewed from Y-axis, the 3D current contour plot viewed
of EMF energy E is converted to thermal and kinetic from X-axis, the 2D contour cut of the y-component of
em
energy due to magnetic field relaxation. So before per- the outflow velocity (V ) in the YZ-plane (x=0) corre-
y
forming a collision simulation, we first simulate the blob sponding to the current plot, and the 2D contour cut
Collision-induced magnetic reconnections & energy dissipations 5
TABLE 1
The normalization factors between physical units and code units.
Parameters: Length Velocity Time Magneticfield Pressure Density
Codeunits: 1 1 1 1 1 1
Normalizationfactors: L0 c L0/c B0 B02 B02/c2
Typicalvalues1: 1012 cm 3×1010 cm/s 33s 103 G 106 Ba 1.1×10−15 g/cm3
Typicalvalues2: 1013 cm 3×1010 cm/s 333s 10G 102 Ba 1.1×10−19 g/cm3
Fig. 1.—Severalmanifestationsoftheinitialmagneticfieldconfigurationcutintheblob-centerplaneintheexamplesimulation. Panel
√
A:Theinitial3DfieldlineprofileviewedalongtheY direction. ThecolorcontourdenotesthevalueofB/ 4π;PanelB:The2Dcontour
cutoftheinitialσprofileintheXZ plane(y=0);PanelC:The2Dcontourcutofthex-componentoftheinitialmagneticfieldstrength
intheXZ plane(y=0);Panel D:The2Dcontourcutofthey-componentoftheinitialmagneticfieldstrengthintheXZ plane(y=0).
extra 2D contour cut of the x-component of the out-
TABLE 2
The initial parameters for the example simulation. flow velocity (Vx) in the XZ-plane (y=0) corresponding
to the current plots, which presents another important
(cid:12) (cid:12)
σb,i Bb,0 α (cid:12)Vb,z(cid:12) P ρbkg zd xs result that the current layer actually generates multi-
8 √4π 3 0.3c 10−2 10−1 4.4 1.0 orientationoutflowsina3Dstructure. Theseresultssug-
gest that many mini-jets with relativistic speeds can be
potentially generated, if multiple collisions are invoked
of the rest mass density in the YZ-plane (x=0), respec-
in a PFD outflow. Another interesting phenomenon is
tively. From these results we find that a strong current
that although the system undergoes a strong reconnec-
layer and a pair of outflows are forming around the con-
tionprocesswhichinprincipledissipatestheEMFenergy
tact surface, which are consistent with the features of a
significantly, the global E evolution is nearly flat and
em
collision-driven reconnection.
even shows slight increase during this stage. The main
The second stage is the “plateau” phase from about
reason for this feature is that the initial strong recon-
t ∼ 10 to t ∼ 38. Panel B of Figure 3 shows a se-
nection is collision-driven. Besides the strong reconnec-
ries of representative cuts at t = 18. We can see that
tion,collision-inducedstrongcompressionalsoexistsand
the current layer around the contact surface becomes
tends to increase E , which balances and even slightly
em
clearer and more concentrated. The outflows become
surpassesE dissipationduetoreconnection. Theaddi-
em
faster (nearly 0.75c) and are also more concentrated
tional outflow study in the following Section §3.3, which
at the current layer. Besides the four representative
shows that the outflows become super-Alfv´enic at this
cuts shown in all panels, for panel B, we also add one
6 Deng, Li, Zhang & Li
mode magnetosonic velocity
(cid:113)
V = V2+C2(1−V2/c2), (15)
ms A s A
where h(cid:48) and B(cid:48) are the specific enthalpy and magnetic
strengthinthelocalcomovingframeofthefluid,andC
s
is the relativistic sound speed calculated by
(cid:112)
C =c γˆP/h(cid:48). (16)
s
In order to investigate whether the fluid velocities ex-
ceed the two characteristic velocities, we define
Γ
R ≡ out, (17)
A Γ
A
Fig. 2.— Upper panel: The Poynting flux energy (Eem) evolu- R ≡Γout. (18)
tionoftheexamplesimulationcase. Dashedlinedenotesthenon- ms Γ
collisioncase,whichservesasthereferenceforadditionalmagnetic ms
dissipation. Solid line denotes the case of collision between two Figure 4 shows the selected 2D contour cuts of R .
A
blobs. lower panel: Ratiocalculatedby(Eem,nc−Eem,c)/Eem,nc The three panels in the upper row correspond to the
tcoollsihsioown-tinhdeuacdedditpioroncaelsEseesm. dissipationefficiencytriggeredbythe starting time when RA > 1 is reached (t = 4), the time
when R is the largest (t = 18), and the ending time
A
for the condition of R >1 (t=38), respectively. After
A
stage, also supports the above analysis. t ∼ 38, the Γout starts to become slightly smaller but
Thenextstageisthe“normaldecay”phase. Wechoose still close to ΓA (see the three panels in the lower row of
two series of representative cuts at t=58 (Panel C) and Figure 4). These results are consistent with the energy
t = 94 (Panel D), respectively. The current strength evolution analysis presented above in Section §3.2. The
and outflow velocity are similar between panels C and duration when RA > 1 is satisfied is just the “plateau”
D,whiletheyaresystematicallyweakerandslowercom- phase of energy evolution, in which strong compression
pared with the “plateau” phase (panel B). This means exists and drives the outflows to become super-Alfv´enic.
that the initial collision-driven effect becomes weaker After t ∼ 38 the energy evolution enters the “normal
and the reconnection-facilitated dissipation enters a rel- decay” phase, which corresponds to the phase of rela-
atively steady phase. In the mean time, compression tivelysteadyreconnection-facilitateddissipationwithout
becomes sub-dominant, so that globally E dissipates strong compression, so that the outflow velocity is close
em
with a relatively steady rate, which roughly equals to to the theoretical Alfv´enic velocity.
0.1Eem,0 = c·Eem,0 in the center-of-mass frame of the Figure 5 show the contour cuts of Rms. Since Vms is
40t0 400L0 the maximum wave propagation speed in a MHD sys-
blobs (L is the length normalization factor introduced
0 tem, if R > 1, a local shock in the front of the out-
in Table 1). The additional outflow study in the follow- ms
flow would potentially be generated. The three epochs
ing Section §3.3, which shows that the outflow velocity
shown in Figure 5 correspond to the starting time when
keepsbeingaroundtheAlfv´envelocityatthisstage,also
R > 1 is satisfied (t = 6), the time when R is the
supports this conclusion. ms ms
largest(t=18),andtheendingtimefortheconditionof
Finally, after t ∼ 120, the reconnection-dissipation
R > 1 to be satisfied (t = 20), respectively. These re-
gradually becomes weaker, and the system enters the ms
sultsindeedshowaperiodofabout15timeunitsduring
quasi-steadyphasewithoutobviousE dissipation. The
em whichR >1issatisfied. Thisdurationisshorterthan
E evolution becomes nearly parallel with the non- ms
em the duration when R > 1 is satisfied. For this case,
collision case in Figure 2. A
thelargestvalueofR isabout1.13. SincetheΓ de-
Fromtheseanalyses,weconcludethatthecollisionbe- ms out
pendsonnumericalresolution(seeSection§3.4belowfor
tween two high-σ blobs can indeed trigger strong mag-
details) and other physical parameters, it is worthwhile
netic reconnections and dissipate a significant fraction
to perform a more detailed study for this feature in the
of EMF energy due to the reconnection-facilitated pro-
future. In this study, since V is only slightly larger
cesses. out
than V in a small local region and for a short dura-
ms
tion, we do not resolve an obvious shock feature from
3.3. Additional outflow study
the numerical data.
Following the above analyses, in this subsection, we
carry out an additional study on the outflow velocity. 3.4. Resolution study
WecomparethelocalLorentzfactoroftheoutflow(Γ )
out We now discuss the effects of numerical resolution on
with the critical Lorentz factor Γ calculated from the
A our results. Although the ideal MHD code that we use
local relativistic Alfv´en velocity
does not have explicit resistivity, it still has numerical
c resistivityfromthenumericalscheme, whichdependson
V = , (14)
A (cid:112)4πh(cid:48)/B(cid:48)2+1 the resolution of the simulation. This may affect the
reconnectionrateandenergydissipationrateinthesim-
and the critical Lorentz factor Γ calculated from the ulations. Toaddressthisuncertainty,weperformareso-
ms
maximum possible value of the local relativistic fast lutiontestbasedontheaboveexamplecase. Wekeepthe
Collision-induced magnetic reconnections & energy dissipations 7
Fig. 3.— The representative cuts of current, velocity and density for the different evolution stages corresponding to Figure 2. Panel A
correspondstotheinitial“selfadjustment”phase;PanelBcorrespondstothefollowing“plateau”phase;PanelC&Dcorrespondtothe
“normaldecay”phase. Thelastquasi-steadyphasehasnoobviousfeature,sowedonotdrawcutsforthatstage. Foreachpanel,thecuts
fromlefttorightarethe3DcurrentcontourplotviewedfromY-axis,the3DcurrentcontourplotviewedfromX-axis,the2Dcontourcut
of the y-component of the outflow velocity (Vy) in the YZ-plane (x=0) corresponding to the current plot, and the 2D contour cut of the
rest mass density in the YZ-plane (x=0), respectively. In Panel B, we add an additional 2D contour cut of the x-component of outflow
velocity(Vx)intheXZ-plane(y=0)toshowtheexistenceofmultipledirectionsoftheoutflows.
same box size and the parameters in Table 2, and only to the results with numerical resolution 1283, 2563, 5123
change the resolution. Figure 6 shows the results. The and 10243, respectively. When the resolution decreases,
magenta, red, green and blue groups of lines correspond we find that the level of E evolution is systematically
em
8 Deng, Li, Zhang & Li
Fig. 4.— The selected 2D contour cuts of RA for different stages. The three panels in the upper row correspond to the starting time
when RA >1, the time when RA is the largest, and the ending time for the condition of RA >1, respectively. These correspond to the
plateau stage. The three panels in the lower row correspond to three epochs during the normal decay phase, during which Γout becomes
relativelysteadyandclosetoΓA.
Fig. 5.— the selected 2D contour cuts from the results of Rms. The three panels correspond to the starting time when Rms > 1 is
satisfied,thetimewhenRms isthelargest,andtheendingtimewhentheconditionofRms>1issatisfied,respectively.
lower and the efficiency also slightly decreases. On the responding to the maximum y-component of the outflow
other hand, the change of efficiency is only several per- velocity (V ) in the YZ-plane (x=0) for different resolu-
y
centagefromthehighesttothelowestresolutions,which tions. ThemaximumvaluesofV areabout0.45c,0.55c,
y
means that the Eem dissipation efficiency is insensitive and0.75cforresolutionof2563,5123,and10243,respec-
to numerical resolution. In addition, the Eem level and tively. The reason is probably that the higher resolution
the efficiency in the final quasi-steady phase also show a decreasestheeffectivenumericalresistivityanddecreases
trend of convergence when the resolution increases. the aspect ratio between the thickness and the length
Another important result from the resolution study is of the reconnection layer, so that the outflow speed is
that the maximum outflow velocity increases when the forced to reach a higher value in order to balance the
resolutionincreases. Figure7showsthecontourcutscor- similar compression forced inflow. This analysis is also
Collision-induced magnetic reconnections & energy dissipations 9
1 Model: example--collision--res 1283 back and reorganizes the field configuration to make the
0.8 MMMooodddeeelll::: eeexxxaaammmpppllleee------ncnooolnnli--scciooonllllii-ss-riiooenns-- 2--rr5ee6ss3 12258633 two blobs merge into one larger blob with a new field
Model: example--collision--res 5123 configurationwitha“∞”shapeatthefinalquasi-steady
Model: example--non-collision--res 5123
E/Eemem,0 00..46 MMooddeell:: eexxaammppllee----cnoolnli-scioonlli-s-rioens- 1-r0e2s4 130243 setnaDtgitueye. atTfothetrihstehsuemgcigose-lsalitlssiigontnhm.aetntthientXwodbirloebctsiomneorgfethteo townoe
0.2 blobs, the collision would induce rotation during the
merging process. This would render the collision pro-
Efficiency 0000 ....02468 MMMMooooddddeeeellll:::: eeeexxxxaaaammmmpppplllleeee--------rrrreeeessss====25115102622833433 citemwenseepsreognnryottat(hnEcetomtrho)pet.alreTtotieothalneytireioonnnteealeratgffisyoetncic(t.EeinsIr,noetwr)ogeryadcneoadrflcttutohhlaeeintteivwnetioshttieibagrllaoaktbteiisnohecbotaeiwnc-
-0.2 k,i
0 20 40 60 80 100 120 140 160 beestimatedasE =2×(1Iω2), wherethemomentof
rot 2
Time inertiaofoneblobcanbeestimatedasI = 2mr2+mr2,
Fig. 6.— A numerical resolution study based on the above ex- 5
where the first term denotes the moment of inertia of
ample case in Section §3.2. The magenta, red, green and blue
groups of lines correspond to the resolutions of 1283, 2563, 5123 anideasphere,andthesecondtermdenotethedisplace-
and10243,respectively. TheEemdissipationefficiencyatthefinial mentfromtherotationaxis. Sincetheblobsexpandwith
quasi-steadyphaseisnearlythesameinallcases. time, the size of the blob and its displacement increase
withtime. Weestimatethatafterthemergingprocess,r
supported by Figure 7, which shows that with an in-
is about three times of r =L =1. We therefore derive
0 0
creasing resolution, the length of the reconnection layer I ∼ 63mL2 = 63m. For the angular velocity ω, we can
is similar, but the thickness becomes thinner. 5 0 5
estimate it from Figure 8, which shows a roughly π/4
angular change within ∆t = 90L /c = 90. As a result,
3.5. Physical analyses 0
onecanestimateω ∼π/360,sothatE =10−3m. The
rot
In this subsection we carry out some physical analy- ratio between E and E is therefore
rot k,i
ses to understand the ∼ 35% E dissipation efficiency
em
obtained from our numerical simulations. E 10−3m
rot = ≈10−2. (21)
Assuming a complete inelastic collision between two E 2× 1mV2
high-σblobs,Zhang&Yan(2011)analyticallyestimated k,i 2 b,z
the total efficiency of the collision-induced E dissipa- So the rotation energy is only a small fraction of the
em
tion efficiency based on energy and momentum conser- initial kinetic energy, which means that the collision is
vation laws. Their Equation (51) can be written as very close to completely inelastic collision for this ex-
ample case with x = 0.5. While even if E becomes
1 Γ (m +m ) s rot
η = − m 1 2 , (19) a larger fraction of Ek,i when the misalignment xs in-
1+σb,f (Γ1m1+Γ2m2)(1+σb,i) creases, it would only reduce the kinetic energy dissi-
pation efficiency, but would have little direct effect on
whereσ istheinitialσ valueofthetwocollidingblobs,
b,i the E dissipation efficiency that is our primary con-
σ is the final σ value after the inelastic collision is em
b,f cern6. Due to the initial high-σ property of the blobs,
over, Γ , Γ , and Γ are the Lorentz factors of the two
1 2 m the contribution from the E dissipation to the total
colliding blobs and the merged blob, respectively, and k,i
dissipation efficiency is only a minor fraction when E
m , m are the masses of the two colliding blobs. In our em
1 2 has significant dissipation, as we have found above.
simulations,thetwoblobsareidenticalsothatm =m .
1 2 With the above preparation, we can achieve a phys-
SinceweareobservinginthemergedframesothatΓ =
m ical understanding of the high efficiency obtained from
1, Γ =Γ =Γ, the final expression of the efficiency can
1 2 our simulation. Based on Eq.(20), we can derive the ex-
be reduced to5
pected efficiency. From the initial condition, we derive
1 1 Γ = 1.05. From simulation results, we can also calcu-
η = − . (20)
1+σb,f Γ(1+σb,i) late σb,f. Since σb,f has a complex spatial distribution,
we perform a spatial average for all the positions with
In order to connect this analytical equation with our σ >1 and also perform a time average from t=90 to
b,f
simulatedresults, wefirstcarryoutsomeanalysestosee t = 120 to get the σ ≈ 1.16. As a result, we derive
b,f
if the condition of complete inelastic collision is satis- η ≈35.7% based on the analytical calculation (Eq.(20)).
fied. For ideal MHD simulations, fluid elements are at- ThisiswellconsistentwiththeE dissipationefficiency
em
tached to the field lines. Tracking the evolution of mag- calculated directly from the energy evolution of the sim-
netic field configuration is therefore a convenient way to ulations using Eq.(13), as shown in Figure 2.
study whether collision is inelastic. Figure 8 shows sev-
eral contour cuts of the 3D field line evolution. Initially 3.6. Summary for this section
the fields are compressed around t=6, and then bounce
In this section, we revealed a collision-induced strong
back around t=12. Later strong collision-driven recon-
reconnectionprocesswiththeEMFenergydissipationef-
nections on the contact surface efficiently dissipate the
ficiency about 35%, which is resolution insensitive. The
compressed magnetic energy and reduce the magnetic
pressure in the center. This prevents further bouncing
6 However, the misalignment xs itself does have a direct effect
on the Eem dissipation efficiency due to the different field config-
5 This can be also derived directly by writing energy and mo- urations around the initial contact surface. See details in Section
mentumconservationsincenter-of-massrestframe. 4.2.
10 Deng, Li, Zhang & Li
Fig. 7.—Thecontourcutscorrespondingtothemaximumy-componentoftheoutflowvelocity(Vy)intheYZ-plane(x=0)fordifferent
resolutions. ThemaximumvaluesofVy areabout0.45c,0.55c,0.75cfortheresolutionsof2563,5123,and10243,respectively. Theaspect
ratiobecomessmallerforahigherresolution.
Fig. 8.—Theevolutionofthefieldlinesduringthecollisionprocess. Thetwoblobsmergeintoonelargerblob,forminga“∞”-shaped
fieldlineconfigurationatthefinalquasi-steadystageoftheevolution.
outflow can locally become super-Alfv´enic during the σ evolution controls the E dissipation efficiency. The
b em
initial strong compression stage. The outflow velocity simulation results may then depend on the initial value
can potentially become relativistic in higher resolution σ . Second, the initial misalignment x gives different
b,i s
simulationsandgeneratemulti-orientationmini-jetsina magnetic field configurations around the contact surface
global PFD jet. which may control the fraction of the free energy that
can be released due to the reconnection processes. Next,
4. EXTENDED PARAMETER SPACE STUDIES different initial relative speed (kinetic energy) between
In Section §3, we find significant EMF energy dissipa- the two blobs define the strengths of the initial collision-
tion(about35%)facilitatedbycollision-drivenmagnetic driven effect, so that it may be another factor to effect
reconnection. Based on the above analyses, we expect the conclusion. In addition, the initial displacement zd
that some parameters may affect the results. First, the controls the delay of the collision. It is also interesting