Ann. Geophys., 31, 1535–1541, 2013www.ann-geophys.net/31/1535/2013/doi:10.5194/angeo-31-1535-2013© Author(s) 2013. CC Attribution 3.0 License.

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DiscussionsMagnetic field generation in a jet-sheath plasma via the kineticKelvin-Helmholtz instability

K.-I. Nishikawa1, P. Hardee2, B. Zhang3, I. Dutan4, M. Medvedev5, E. J. Choi6, K. W. Min 6, J. Niemiec7, Y. Mizuno8,A. Nordlund 9, J. T. Frederiksen9, H. Sol10, M. Pohl11, and D. H. Hartmann12

1Center for Space Plasma and Aeronomic Research, University of Alabama in Huntsville, 320 Sparkman Drive,ZP12, Huntsville, AL 35805, USA2Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, AL 35487, USA3Department of Physics, University of Nevada, Las Vegas, NV 89154, USA4Institute of Space Science, Atomistilor 409, Bucharest-Magurele 077125, Romania5Department of Physics and Astronomy, University of Kansas, KS 66045, USA6Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea7Institute of Nuclear Physics PAN, ul. Radzikowskiego 152, 31-342 Krakow, Poland8Institute of Astronomy, National Tsing-Hua University, Hsinchu, Taiwan 30013, Republic of China9Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen Ø, Denmark10LUTH, Observatore de Paris-Meudon, 5 place Jules Jansen, 92195 Meudon Cedex, France11Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25,14476 Potsdam-Golm, Germany12Department of Physics and Astronomy, Clemson University, Clemson, SC 29634, USA

Correspondence to:K.-I. Nishikawa (ken-ichi.nishikawa@nasa.gov)

Received: 18 March 2013 – Revised: 25 July 2013 – Accepted: 26 July 2013 – Published: 6 September 2013

Abstract. We have investigated the generation of magneticfields associated with velocity shear between an unmagne-tized relativistic jet and an unmagnetized sheath plasma. Wehave examined the strong magnetic fields generated by ki-netic shear (Kelvin–Helmholtz) instabilities. Compared tothe previous studies using counter-streaming performed byAlves et al. (2012), the structure of the kinetic Kelvin–Helmholtz instability (KKHI) of our jet-sheath configura-tion is slightly different, even for the global evolution ofthe strong transverse magnetic field. In our simulations themajor components of growing modes are the electric fieldEz, perpendicular to the flow boundary, and the magneticfield By , transverse to the flow direction. After theBy com-ponent is excited, an induced electric fieldEx , parallel tothe flow direction, becomes significant. However, other fieldcomponents remain small. We find that the structure andgrowth rate of KKHI with mass ratiosmi/me = 1836 andmi/me = 20 are similar. In our simulations saturation in thenonlinear stage is not as clear as in counter-streaming cases.

The growth rate for a mildly-relativistic jet case (γj = 1.5) islarger than for a relativistic jet case (γj = 15).

Keywords. Solar physics, astrophysics, and astronomy (En-ergetic particles)

1 Introduction

Recent kinetic simulations have focused on magnetic fieldgeneration via electromagnetic plasma instabilities in un-magnetized flows without velocity shear. Three-dimensional(3-D) particle-in-cell (PIC) simulations of Weibel turbulence(Nishikawa et al., 2005, 2008, 2009a) have demonstratedsub-equipartition magnetic field generation.

Velocity shears also must be considered when studyingparticle acceleration scenarios, since these trigger the ki-netic Kelvin–Helmholtz instability (KKHI). In particular theKKHI has been shown to lead to particle acceleration andmagnetic field amplification in relativistic shear flows (Alveset al., 2012; Liang et al., 2013; Nishikawa et al., 2013).

Published by Copernicus Publications on behalf of the European Geosciences Union.

1536 K.-I. Nishikawa et al.: Magnetic field generation in jet-sheath plasma

Furthermore, a shear flow upstream of a shock can leadto density inhomogeneities via the Magnetohydrodynamics(MHD) Kelvin–Helmholtz instability (KHI), which may pro-vide important scattering sites for particle acceleration.

In our simulations a relativistic jet plasma is surrounded bya sheath plasma (Nishikawa et al., 2013). This setup is similarto the setup of our RMHD simulations (Mizuno et al., 2007).In one of our simulations, the jet core hasvcore= 0.9978c(γj = 15) pointing in the positivex direction in the middle ofthe simulation box as in Alves et al. (2012). Unlike Alves etal. (2012) the upper and lower quarter of the simulation boxcontain a stationary,vsheath= 0, sheath plasma as shown inFig. 1a. Our setup allows for motion of the sheath plasma inthe positivex direction.

Overall, this structure is similar in spirit, although not inscale, to that proposed for active galactic nuclei (AGN) rela-tivistic jet cores surrounded by a slower moving sheath, andis also relevant to gamma-ray burst (GRB) jets. In particular,we note that this structure is also relevant to the “jet-in-a-jet” or “needles in a jet” scenarios (Giannios et al., 2009,and papers therein), which have been invoked to providesmaller scale high-speed structures within a much larger,more slowly moving AGN jet. Similar smaller scale struc-tures within GRB jets are also conceivable.

This more realistic setup will allow us to compute syn-thetic spectra in the observer frame. As mentioned by Alveset al. (2012), in a non-counter-streaming or unequal den-sity counter-streaming setup the growing KKHI will prop-agate with the flow. For GRB jets, the relativistic jet core isthought to have a much higher density relative to the externalmedium. On the other hand, for an AGN jet the relativisticcore is thought to be less dense than the surrounding sheath.

Recently the KKHI was investigated using a counter-streaming velocity shear setup for subrelativisticγ0 = 1.02and relativisticγ0 = 3 cases (Alves et al., 2012). The shearflow initial condition was set by a velocity field withv0 point-ing in the positivex1 direction, in the upper and lower quar-ters of the simulation box, and a symmetric velocity fieldwith −v0 pointing in the negativex1 direction, in the middle-half of the box. Initially, the systems were charge and currentneutral. In the subrelativistic case the simulation box dimen-sions were 20×20×20(c/ωp)

3, whereωp = (4πne2/me)1/2

is the plasma frequency, and they used 20 cells per electronskin depth (c/ωp). The simulation box dimensions for therelativistic scenario were 250× 80× 80(c/ωp)

3, with a res-olution of 4 cells per (c/ωp). Periodic boundary conditionswere imposed in every direction.

Alves et al. (2012) found that KKHI modulations are lessnoticeable in the relativistic regime because they are maskedby a strong DC magnetic field component (negligible in thesubrelativistic regime) with a magnitude greater than theAC component. As the amplitude of the KKHI modulationsgrows the electrons from one flow cross the shear surfacesand enter the counter-streaming flow. In their simulationsthe protons being heavier (mp/me = 1836) are unperturbed.

DC current sheets which point in the direction of the pro-ton velocity form around the shear surfaces. These DC cur-rent sheets induce a DC component in the magnetic field.The DC magnetic field is dominant in the relativistic sce-nario because a higher DC current is set up by the crossingof electrons with a larger initial flow velocity and also be-cause the growth rate of the AC dynamics is lower byγ

3/20

compared with a subrelativistic case (Alves et al., 2013). Itis very important to note that this DC magnetic field is notcaptured in MHD (e.g., Zhang et al., 2009) or fluid theoriesbecause it results from intrinsically kinetic phenomena. Fur-thermore, since the DC field is stronger than the AC field,a kinetic treatment is clearly required in order to fully cap-ture the field structure generated in unmagnetized or weaklymagnetized relativistic flows with velocity shear. This char-acteristic field structure will also lead to a distinct radiationsignature (Sironi and Spitkovsky, 2009; Martins et al., 2009;Frederiksen et al., 2010; Nishikawa et al., 2009b, 2010, 2011,2012).

We previously reported results from our first simula-tions for a core-sheath case withγj = 15 andmi/me = 20(Nishikawa et al., 2013). Here we report new simulation re-sults using the real mass ratiomi/me = 1836. We find somedifferences from previous counter-streaming cases.

2 Magnetic and electric field generation by KKHI

2.1 Initial core-sheath jet conditions

In this simulation study we investigate velocity shear in acore-sheath jet instead of the counter-streaming jets used inprevious simulations (Alves et al., 2012; Liang et al., 2013).In this RPIC simulation the velocity shear occurs at the edgesof a velocity field withvcore pointing in the positivex di-rection in the middle of the simulation box, with upper andlower quarters of the simulation box containing a velocityfield with vsheathpointing in the positivex direction as indi-cated by the arrows in Fig.1a. Initially, the system is chargeand current neutral.

We have performed a simulation using a system with(Lx,Ly,Lz) = (10051,2051,2051) and with an ion toelectron mass ratio ofmi/me = 1836 (1 is the cell size).The jet and sheath plasma density isnjt = nam = 8. Theelectron skin depth,λs = c/ωpe = 12.21, where ωpe =

(e2na/ε0me)1/2 is the electron plasma frequency and the

electron Debye lengthλD is 1.21. The jet Lorentz fac-tor is γj = 15. The jet-electron thermal velocity isvj,th,e =

0.014 c in the jet reference frame, wherec is the speedof light. The electron/ion thermal velocity in the ambientplasma isva,th,e = 0.03 c. Ion thermal velocities are smallerby (mi/me)

1/2. We use periodic boundary conditions on allboundaries (Buneman, 1993; Nishkawa et al., 2009a, b).

Ann. Geophys., 31, 1535–1541, 2013 www.ann-geophys.net/31/1535/2013/

K.-I. Nishikawa et al.: Magnetic field generation in jet-sheath plasma 15372 Nishikawa et al.: Magnetic field generation in jet-sheath plasma

x

z n

n

nsheath

sheath

corev

v

vsheath

core

sheath

(a)

Z/Δ

0

50

100

150

200

0 50 100 150 200

X/Δ

(b)

Figure 1. Figure 1a shows our simulation model where the sheath plasmacan be stationary or moving in the same direction as the jetcore. In this simulation the sheath velocity is zero. Figure1b shows the magnitude ofBy is plotted in thex− z plane (jet flow in the+x-direction indicated by the large arrow) at the center of the simulation box,y= 100∆ at simulation timet = 70ω−1

pe for the case ofγj = 15 andmi/me= 1836. This panel covers one fifth of the whole simulation system. The arrows show the magnetic field in the plane.

0 50 100 150 200

Y/Δ

Z/Δ

0

50

100

150

200(a)

•

°•

°

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Z/Δ

-.04 -.02 0 .02 .04E

(b)

-.02 -.01 0.0 .01 .020

50

100

150

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Z/Δ

B

(c)

Figure 2. Electric and magnetic field generated by a relativistic electron ion jet core withγj = 15 and stationary sheath plasma for the casewith the mass ratiomi/me= 1836. The magnitude ofEz is plotted in they−z plane (jet flow is out of the page) at the center of the simulationbox, atx= 500∆ at t = 30ω−1

pe (panel (a)). Panel (b) showsEz (blue), Ex (black), , andEy (red) atx= 500∆ andy= 100∆ at the same time.Since the magnetic field grows more slowly (see Fig. 4), panel(c) showsBy (red),Bx (black), , andBz (blue) att= 70ω−1

pe.

is also relevant to gamma-ray burst (GRB) jets. In particular,we note that this structure is also relevant to the “jet-in-a-jet” or “needles in a jet” scenarios (Giannios et al., 2009, andpapers therein), which have been invoked to provide smallerscale high speed structures within a much larger more slowlymoving AGN jet. Similar smaller scale structures withinGRB jets are also conceivable.

This more realistic setup will allows us to compute syn-thetic spectra in the observer frame. As mentioned by Alveset al. (2012), in a non-counter-streaming or unequal den-sity counter-streaming setup the growing KKHI will propa-gate with the flow. For GRB jets, the relativistic jet core isthought to have a much higher density relative to the externalmedium. On the other hand, for an AGN jet the relativisticcore is thought to be less dense than the surrounding sheath.

Recently the KKHI was investigated using a counter-streaming velocity shear setup for subrelativisticγ0 = 1.02

and relativisticγ0 = 3 cases, (Alves et al. 2012). The shearflow initial condition was set by a velocity field withv0 point-ing in the positivex1 direction, in the upper and lower quar-ters of the simulation box, and a symmetric velocity fieldwith −v0 pointing in the negativex1 direction, in the middle-half of the box. Initially, the systems were charge and currentneutral. In the subrelativistic case, the simulation box dimen-sions were 20×20×20(c/ωp)3, whereωp= (4πne2/me)1/2 isthe plasma frequency, and they used 20 cells per electron skindepth (c/ωp). The simulation box dimensions for the rela-tivistic scenario were 250×80×80(c/ωp)3, with a resolutionof 4 cells per (c/ωp). Periodic boundary conditions were im-posed in every direction.

Alves et al. (2012) found that KKHI modulations are lessnoticeable in the relativistic regime because they are maskedby a strong DC magnetic field component (negligible in thesubrelativistic regime) with a magnitude greater than the AC

J. Name www.jn.net

Fig. 1. Panel(a) shows our simulation model where the sheath plasma can be stationary or moving in the same direction as the jet core. Inthis simulation the sheath velocity is zero. Panel(b) shows the magnitude ofBy plotted in thex-z plane (jet flow in the+x direction indicated

by the large arrow) at the center of the simulation box,y = 1001 at simulation timet = 70ω−1pe for the case ofγj = 15 andmi/me = 1836.

This panel covers one-fifth of the whole simulation system. The arrows show the magnetic field in the plane.

2 Nishikawa et al.: Magnetic field generation in jet-sheath plasma

x

z n

n

nsheath

sheath

corev

v

vsheath

core

sheath

(a)

Z/Δ

0

50

100

150

200

0 50 100 150 200

X/Δ

(b)

Figure 1. Figure 1a shows our simulation model where the sheath plasmacan be stationary or moving in the same direction as the jetcore. In this simulation the sheath velocity is zero. Figure1b shows the magnitude ofBy is plotted in thex− z plane (jet flow in the+x-direction indicated by the large arrow) at the center of the simulation box,y= 100∆ at simulation timet = 70ω−1

pe for the case ofγj = 15 andmi/me= 1836. This panel covers one fifth of the whole simulation system. The arrows show the magnetic field in the plane.

0 50 100 150 200

Y/Δ

Z/Δ

0

50

100

150

200(a)

•

°•

°

0

50

100

150

200

Z/Δ

-.04 -.02 0 .02 .04E

(b)

-.02 -.01 0.0 .01 .020

50

100

150

200

Z/Δ

B

(c)

Figure 2. Electric and magnetic field generated by a relativistic electron ion jet core withγj = 15 and stationary sheath plasma for the casewith the mass ratiomi/me= 1836. The magnitude ofEz is plotted in they−z plane (jet flow is out of the page) at the center of the simulationbox, atx= 500∆ at t = 30ω−1

pe (panel (a)). Panel (b) showsEz (blue), Ex (black), , andEy (red) atx= 500∆ andy= 100∆ at the same time.Since the magnetic field grows more slowly (see Fig. 4), panel(c) showsBy (red),Bx (black), , andBz (blue) att= 70ω−1

pe.

is also relevant to gamma-ray burst (GRB) jets. In particular,we note that this structure is also relevant to the “jet-in-a-jet” or “needles in a jet” scenarios (Giannios et al., 2009, andpapers therein), which have been invoked to provide smallerscale high speed structures within a much larger more slowlymoving AGN jet. Similar smaller scale structures withinGRB jets are also conceivable.

This more realistic setup will allows us to compute syn-thetic spectra in the observer frame. As mentioned by Alveset al. (2012), in a non-counter-streaming or unequal den-sity counter-streaming setup the growing KKHI will propa-gate with the flow. For GRB jets, the relativistic jet core isthought to have a much higher density relative to the externalmedium. On the other hand, for an AGN jet the relativisticcore is thought to be less dense than the surrounding sheath.

Recently the KKHI was investigated using a counter-streaming velocity shear setup for subrelativisticγ0 = 1.02

and relativisticγ0 = 3 cases, (Alves et al. 2012). The shearflow initial condition was set by a velocity field withv0 point-ing in the positivex1 direction, in the upper and lower quar-ters of the simulation box, and a symmetric velocity fieldwith −v0 pointing in the negativex1 direction, in the middle-half of the box. Initially, the systems were charge and currentneutral. In the subrelativistic case, the simulation box dimen-sions were 20×20×20(c/ωp)3, whereωp= (4πne2/me)1/2 isthe plasma frequency, and they used 20 cells per electron skindepth (c/ωp). The simulation box dimensions for the rela-tivistic scenario were 250×80×80(c/ωp)3, with a resolutionof 4 cells per (c/ωp). Periodic boundary conditions were im-posed in every direction.

Alves et al. (2012) found that KKHI modulations are lessnoticeable in the relativistic regime because they are maskedby a strong DC magnetic field component (negligible in thesubrelativistic regime) with a magnitude greater than the AC

J. Name www.jn.net

Fig. 2.Electric and magnetic field generated by a relativistic electron ion jet core withγj = 15 and stationary sheath plasma for the case withthe mass ratiomi/me= 1836. The magnitude ofEz is plotted in they-z plane (jet flow is out of the page) at the center of the simulation box,atx = 5001 at t = 30ω−1

pe (a). Panel(b) showsEz (blue),Ex (black), andEy (red) atx = 5001 andy = 1001 at the same time. Since the

magnetic field grows more slowly (see Fig.4), panel(c) showsBy (red),Bx (black), andBz (blue) att = 70ω−1pe .

2.2 Core-sheath jet KKHI results

Our new simulation results using the real mass ratio areshown to compare with our previous results (Nishikawa etal., 2013).

Figure1b shows the magnitude ofBy plotted in thex-zplane (jet flow in the+x direction indicated by the large ar-row) at the center of the simulation box,y = 1001 at simu-lation timet = 70ω−1

pe for the case ofγj = 15 andmi/me =

1836.

Figure 2a shows the magnitude ofEz plotted in they-zplane (jet flow is out of the page) at the center of the sim-ulation box,x = 5001 at t = 30ω−1

pe . Figure2b showsEz

(blue),Ex (black), andEy (red) electric field components atx = 5001 andy = 1001 at time t = 30ω−1

pe . As shown inFig. 4, thez component of electric field,Ez grows quickly,this DC electric field is shown in Figs.2a and b. As ex-pected, they component of magnetic field,By is very smallat this earlier time. The magnetic field grows later. Figure2cshowsBy (red), Bx (black), andBz (blue) magnetic fieldcomponents atx = 5001 andy = 1001 t = 70ω−1

pe .

www.ann-geophys.net/31/1535/2013/ Ann. Geophys., 31, 1535–1541, 2013

1538 K.-I. Nishikawa et al.: Magnetic field generation in jet-sheath plasmaNishikawa et al.: Magnetic field generation in jet-sheath plasma 3

60 70 80 90 100 Y/∆

40

50

60

70

30

Z/∆

Tω = 30pe (a)

°·

60 70 80 90 100 Y/∆

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Tω = 40pe (b)

°•

60 70 80 90 100 Y/∆

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Tω = 50pe (c)

°•

60 70 80 90 100 Y/∆

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Tω = 30pe (d)

°•

60 70 80 90 100 Y/∆

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Z/∆

Tω = 40pe (e)

°•

60 70 80 90 100 Y/∆

40

50

60

70

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Z/∆

Tω = 50pe (f )

°•

Figure 3. The time evolution of current filaments (Jx) and electric field (Ez) in the area denoted by the small box in Fig. 2a. KKHI starts togrow at (a)t= 30ω−1

pe and current filaments become large but have not yet merged by (c) t= 50ω−1pe. The maximum current density (simulation

units) is (a)±0.108 att= 30ω−1pe, (b)±0.166 att= 40ω−1

pe, and (c)±0.296 att= 50ω−1pe. The arrows show the magnetic field (By,Bz) (the length

of the arrows are not scaled to the strength of the magnetic fields). Panels (d), (e), and (f) show the time evolution of the electric field (Ez)in the same area as the current filaments. The rather uniform electric field grows even att= 30ω−1

pe. The maximum values ofEz change withtime (d): ±0.032, (e):±0.040, and (f):±0.040. As seen from green lines in Fig. 4a, electric fields achieve a maximum att = 45ω−1

pe andsubsequently decreases.

component. As the amplitude of the KKHI modulationsgrows the electrons from one flow cross the shear-surfacesand enter the counter-streaming flow. In their simulationsthe protons being heavier (mp/me= 1836) are unperturbed.DC current sheets which point in the direction of the pro-ton velocity form around the shear-surfaces. These DC cur-rent sheets induce a DC component in the magnetic field.The DC magnetic field is dominant in the relativistic sce-nario because a higher DC current is set up by the crossingof electrons with a larger initial flow velocity and also be-cause the growth rate of the AC dynamics is lower byγ3/2

0compared with a subrelativistic case (Alves et al. 2013). Itis very important to note that this DC magnetic field is notcaptured in MHD (e.g., Zhang et al. 2009) or fluid theoriesbecause it results from intrinsically kinetic phenomena. Fur-thermore, since the DC field is stronger than the AC field,a kinetic treatment is clearly required in order to fully cap-ture the field structure generated in unmagnetized or weaklymagnetized relativistic flows with velocity shear. This char-acteristic field structure will also lead to a distinct radiationsignature (Sironi & Spitkovsky 2009; Martins et al. 2009;Frederiksen et al. 2010; Nishikawa et al. 2009b, 2010, 2011,

2012).We previously reported results from our first simula-

tions for a core-sheath case withγj = 15 andmi/me = 20(Nishikawa et al. 2013). Here we report new simulation re-sults using the real mass ratiomi/me= 1836. We find somedifferences from previous counter-streaming cases.

2 Magnetic and Electric Field Generation by KKHI

2.1 Initial Core-Sheath Jet Conditions

In this simulation study we investigate velocity shear in acore-sheath jet instead of the counter-streaming jets usedinprevious simulations (Alves et al. 2012; Liang et al. 2013).In this RPIC simulation the velocity shear occurs at the edgesof a velocity field withvcore pointing in the positivex di-rection in the middle of the simulation box, with upper andlower quarters of the simulation box containing a velocityfield with vsheath pointing in the positivex direction as in-dicated by the arrows in Figure 1a. Initially, the system ischarge and current neutral.

We have performed a simulation using a system with(Lx,Ly,Lz) = (1005∆,205∆,205∆) and with an ion to elec-

www.jn.net J. Name

Fig. 3.The time evolution of current filaments (Jx ) and electric field (Ez) in the area denoted by the small box in Fig.2a. KKHI starts to growat (a) t = 30ω−1

pe and current filaments become large but have not yet merged by(c) t = 50ω−1pe . The maximum current density (simulation

units) is(a) ±0.108 att = 30ω−1pe , (b) ±0.166 att = 40ω−1

pe , and(c) ±0.296 att = 50ω−1pe . The arrows show the magnetic field (By ,Bz) (the

length of the arrows are not scaled to the strength of the magnetic fields). Panels(d), (e), and(f) show the time evolution of the electric field(Ez) in the same area as the current filaments. The rather uniform electric field grows even att = 30ω−1

pe . The maximum values ofEz change

with time (d) ±0.032,(e)±0.040, and(f) ±0.040. As seen from green lines in Fig.4a, electric fields achieve a maximum att = 45ω−1pe and

subsequently decrease.

Figure 3 shows the evolution of current filaments alongthe jet (Jx) and thez component of the electric field (Ez)at an earlier time. Jet flow out of the page is indicated by�. Current filaments inside the jet (z/1 > 50) have posi-tive Jx and outside the current filaments have negativeJx .The width of both positive and negative current filaments isabout 2–3 electron Debye lengths. This suggests that the jetelectrons near the velocity shear move out of the jet slightlyand/or the sheath electrons are dragged by the jet. The steadycurrent filaments at the velocity shear create a steadyEz

component. This electric field component decreases aftert = 50ω−1

pe . These current filaments create aBy componentwhich has been reported on in the previous work (Nishikawaet al., 2013).

We have compared the differences between cases withmass ratiosmi/me = 20 and 1836 for the relativistic jet withγj = 15. We find that the structure and growth rate of kineticKHI is very similar (see Fig.4). The heavier ions in the real

mass ratio case keep the system thermal fluctuations smaller,but the kinetic KHI grows similarly. The magnetic field en-ergy becomes larger than the electric field energy at a simi-lar time in both cases aroundt = 87ω−1

pe . We also performeda simulation withγj = 1.5 andmi/me = 20. For this non-relativistic case the magnetic field grows earlier and over-takes the electric field energy att = 46ω−1

pe .Figure5 shows the evolution of electric and magnetic field

energy for the core jet withγj = 15 and stationary sheathplasma for mass ratiomi/me= 1836. Figure5a shows theevolution of the total electric field energy (black),Ey (green),Ex (red), andEx (blue). Thez component grows first asshown in Fig.3 and they component of the magnetic fieldgrows as shown in Fig.5b. After By grows an induced elec-tric field Ex grows as indicated by the red line in Fig.5a. Itshould be noted that the magnetic field grows at a slower ratethan a counter-streaming case withγ0 = 3.0. At t = 170ω−1

pe ,saturation has not yet occurred. This may be due our highly

Ann. Geophys., 31, 1535–1541, 2013 www.ann-geophys.net/31/1535/2013/

K.-I. Nishikawa et al.: Magnetic field generation in jet-sheath plasma 1539

10 20 30 40 50 60 70 80 90t/ωpe

log E10

Fig. 4. The panel shows the time evolution of electric field energy(red) and magnetic field energy (black) forγj = 15 andmi/me = 20(Nishikawa et al., 2013). For the case with the real mass ratio thegreen and blue lines shows the time evolution of electric field en-ergy and magnetic field energy, respectively. The dashed red andblack lines show the time evolution of electric field energy andmagnetic field energy, respectively for the case withγj = 1.5 andmi/me = 20. For all cases the kinetic KHI grows quickly, the mag-netic field energy plateaus, and later gradually grows again. In thenon-relativistic case the magnetic field energy becomes larger thanthe electric field energy at an earlier time.

relativistic jet core resulting in a larger Lorentz factor inthe “electrostatic” relativistic jet plasma frequency and largerelectron inertia in the simulation frame.

3 Core-sheath dispersion relation

We have extended the analysis presented in Gruzinov (2008)to core-sheath electron-proton plasma flows, allowing fordifferent core and sheath electron densitiesnjt andnam, re-spectively, and core and sheath electron velocitiesvjt andvam, respectively. In this analysis the protons are consideredto be free-streaming, whereas the electron fluid quantitiesand fields are linearly perturbed. We consider electrostaticmodes along the jet. The dispersion relation becomes

(k2c2+ γ 2

amω2p,am− ω2)1/2(ω − kVam)2

×[(ω − kVjt)2− ω2

p,jt]

+(k2c2+ γ 2

jt ω2p,jt − ω2)1/2(ω − kVjt)

2

×[(ω − kVam)2− ω2

p,am] = 0, (1)

where ωp,jt and ωp,am are the plasma frequencies (ω2p ≡

4πne2/γ 3me) of jet and ambient electrons, respectively,k =

kx is the wave number parallel to the jet flow, andγjt andγamare Lorentz factors of jet and ambient electrons, respectively.

Analytic solutions to Eq. (1) are not available except in thelow (ω � ωp andkc � ωp) and high (ω � ωp andkc � ωp)frequency and wavenumber limits. In the high wavenumberlimit with Vam = 0, the real frequency becomesω ' kVjt ±

ωp,jt/√

2. In the low wavenumber limit an analytic solution

to the dispersion relation is expressed as follows:

ω '(γamωp,jtkVam+ γjtωp,amkVjt)

(γamωp,jt + γjtωp,am)

± i(γamωp,jtγjtωp,am)1/2

(γamωp,jt + γjtωp,am)k(Vjt − Vam). (2)

Here the real part gives the phase velocity and the imagi-nary part gives the temporal growth rate and directly showsthe dependence of the growth rate on the velocity differenceacross the shear surface. Equation (2) shows that the wavespeed increases and the temporal growth rate decreases asγjtωp,am/ωp,jt = γ

5/2jt (nam/njt)

1/2 increases.We show exact solutions to Eq. (2) in Fig. (6) for a test

case using the parameters,njt = nam = 0.1 cm−3, Vjt/c =

0.979796 withγjt = 5.0, andVam = 0. Note that the num-ber densities are determined in their proper frames. In theambient observer frame the jet’s number density would ap-pear to ben′

jt = γjtnjt = 0.5 cm−3. With the number densi-ties determined in the proper frames the plasma frequen-cies areωp,jt = (4πnjte

2/γ 3jt me)

1/2= 1.60×103 rad s−1 and

ωp,am = (4πname2/γ 3amme)

1/2= 1.78×104 rads−1. At small

and large wavenumbers the numerical solutions follow theanalytic low and high wavenumber solutions almost ex-actly. The maximum temporal growth rate and associatedwavenumber found from the numerical solutions shown inFig. 6 areωmax

I ' 0.93ωp,jt at kmaxc ' 7.0ωp,jt or kmaxVjt '

6.9ωp,jt .If we assume thatωmax

I ∼ ωp,jt then we can use the imag-inary part of Eq. (2) to estimate the wavenumber at whichtemporal growth is a maximum with result that we esti-matekmaxVjt ∼ 7.6ωp,jt and these estimates are within 10 %of the exact numerical solution. The low wavenumber solu-tion, Eq. (2), provides excellent estimates of the complex fre-quency to wavenumbers within a factor of 2 of the maximallygrowing wavenumber.

4 Summary and discussion

We have investigated generation of magnetic fields associ-ated with velocity shear between an unmagnetized relativis-tic jet and an unmagnetized sheath plasma (core jet-sheathconfiguration). We have examined the evolution of elec-tric and magnetic fields generated by kinetic shear (Kelvin-Helmholtz) instabilities. Compared to the previous studiesusing counter-streaming performed by Alves et al. (2012),the structure of KKHI of our jet-sheath configuration isslightly different even for the global evolution of the strongtransverse magnetic field. We find that the major componentsof growing modes areEz andBy . After theBy component isexcited, the induced electric fieldEx becomes larger. How-ever, other components are very small. We find that the struc-ture and growth rate withmi/me = 1836 andmi/me = 20are similar. In our core-sheath simulations no saturation at

www.ann-geophys.net/31/1535/2013/ Ann. Geophys., 31, 1535–1541, 2013

1540 K.-I. Nishikawa et al.: Magnetic field generation in jet-sheath plasma

4 Nishikawa et al.: Magnetic field generation in jet-sheath plasma

10 20 30 40 50 60 70 80 90t/ωpe

log E10

Figure 4. The panel show time evolution of electric field energy (red) and magnetic field energy (black) forγj = 15 andmi/me = 20(Nishikawa et al. 2013). For the case with the real mass ratiothe green and blue lines shows the time evolution of electricfield energy andmagnetic field energy, respectively. The dashed red and black lines show the time evolution of electric field energy and magnetic field energy,respectively for the case withγj = 1.5 andmi/me= 20. For all cases the kinetic KHI grows quickly, the magneticfield energy plateaus, andlater gradually grows again. In the non-relativistic case the magnetic field energy becomes larger than the electric field energy at an earliertime.

20 40 60 80 100 120 140 160 Tωpe

log E10

–12

–11

–10

–9

–8

(a)

20 40 60 80 100 120 140 160 Tωpe

log E10

–12

–11

–10

–9

–8

(b)

Figure 5. Evolution of electric (a) and magnetic (b) field energies forthe core jet with withγj = 15 and stationary sheath plasma for thecase with the mass rationmi/me= 1836. Panel (a) shows the evolution of the total electric field energy (black),Ez (green),Ex (red), andEy

(blue). The evolution of the total magnetic field energy (black), By (blue),Bz (green), andBx (red) is plotted in panel (b).

tron mass ratio ofmi/me= 1836 (∆ is the cell size). The jetand sheath plasma density isnjt = nam= 8. The electron skindepth,λs= c/ωpe= 12.2∆, whereωpe= (e2na/ǫ0me)1/2 is theelectron plasma frequency and the electron Debye lengthλD

is 1.2∆. The jet Lorentz factor isγj = 15. The jet-electronthermal velocity isvj,th,e= 0.014c in the jet reference frame,wherec is the speed of light. The electron/ion thermal ve-locity in the ambient plasma isva,th,e= 0.03 c. Ion thermalvelocities are smaller by (mi/me)1/2. We use periodic bound-ary conditions on all boundaries (Buneman 1993; Nishkawaet al. 2009).

2.2 Core-Sheath Jet KKHI Results

Our new simulation results using the real mass ratio areshown to compare with our previous results (Nishikawa etal. 2013).

Figure 1b shows the magnitude ofBy plotted in thex− zplane (jet flow in the+x-direction indicated by the large ar-row) at the center of the simulation box,y= 100∆ at simula-tion timet = 70ω−1

pe for the case ofγj = 15 andmi/me= 1836.Figure 2a shows the magnitude ofEz plotted in they− z

plane (jet flow is out of the page) at the center of the sim-ulation box, x = 500∆ at t = 30ω−1

pe. Figure 2b showsEz

(blue),Ex (black), andEy (red) electric field components atx= 500∆ andy= 100∆at timet= 30ω−1

pe. As shown in Fig. 4,thez component of electric field,Ez grows quickly, this DCelectric field is shown in Figs. 2a and 2b. As expected, theycomponent of magnetic field ,By is very small at this earliertime. The magnetic field grows later. Figure 2c showsBy

(red),Bx (black), andBz (blue) magnetic field components atx= 500∆ andy= 100∆ t= 70ω−1

pe.Figure 3 shows the evolution of current filaments along

the jet (Jx) and thez component of the electric field (Ez)

J. Name www.jn.net

Fig. 5. Evolution of electric(a) and magnetic(b) field energies for the core jet with withγj = 15 and stationary sheath plasma for the casewith the mass rationmi/me= 1836. Panel(a) shows the evolution of the total electric field energy (black),Ez (green),Ex (red), andEy

(blue). The evolution of the total magnetic field energy (black),By (blue),Bz (green), andBx (red) is plotted in panel(b).

Fig. 6. The upper panel shows electrostatic mode solutions as afunction of wavenumber,k, parallel to the flow direction. The realpart,ωR, and imaginary part,ωI , of the frequency are indicated bythe solid and dashed lines, respectively. The lower panel shows thephase velocity,γwβw, whereβw = ωR/kc.

the later time is seen as in the counter-streaming cases. Thisdifference seems come from that fact that the jet is highlyrelativistic and our simulation is done in a core-sheath con-figuration. The growth rate for the mildly relativistic jet case(γj = 1.5) is larger than the relativistic jet case (γj = 15).

We have calculated, self-consistently, the radiation fromelectrons accelerated in the turbulent magnetic fields in therelativistic shocks. We found that the synthetic spectra de-pend on the Lorentz factor of the jet, the jet’s thermal tem-perature, and the strength of the generated magnetic fields(Nishikawa et al., 2011, 2012). In forthcoming work we willobtain synthetic spectra from particles accelerated by KKHIas we have done for shock simulations (Nishikawa et al.,2011, 2012).

Acknowledgements.This work is supported by NSF AST-0908010,and AST-0908040, NASA-NNG05GK73G, NNX07AJ88G,NNX08AG83G, NNX08 AL39G, NNX09AD16G, andNNX12AH06G. J. Niemiec is supported by NCN through grantDEC-2011/01/B/ST9/03183 and DEC-2012/04/A/ST9/00083.Y. Mizuno is supported by the Taiwan National Science Councilunder the grant NSC 100-2112-M-007-022-MY3. Simulationswere performed at the Columbia and Pleiades facilities at theNASA Advanced Supercomputing (NAS), Kraken and Nautilusat The National Institute for Computational Sciences (NICS), andStampede at The Texas Advanced Computing Center (TACC)which are supported by the NSF. This research was startedduring the program “Chirps, Mergers and Explosions: The FinalMoments of Coalescing Compact Binaries” at the Kavli Institutefor Theoretical Physics which is supported by the National ScienceFoundation under Grant No. PHY05-51164.

Guest Editor M. Balikhin thanks two anonymous referees fortheir help in evaluating this paper.

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