J. Giacalone and J. R. Jokipii- Magnetic Field Amplification by Shocks in Turbulent Fluids

8/3/2019 J. Giacalone and J. R. Jokipii- Magnetic Field Amplification by Shocks in Turbulent Fluids

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The Astrophysical Journal, 663:L41L44, 2007 July 1 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.

MAGNETIC FIELD AMPLIFICATION BY SHOCKS IN TURBULENT FLUIDS

J. Giacalone and J. R. Jokipii

Department of Planetary Sciences, University of Arizona, Tucson, AZ 85721

Received 2007 March 15; accepted 2007 May 21; published 2007 June 15

ABSTRACTWe consider the effect of preexisting, large-scale, broadband turbulent density fluctuations on propagating hy-

dromagnetic shock waves. We present results from several numerical simulations that solve the two-dimensionalmagnetohydrodynamic equations. In our simulations, a plasma containing large-scale, low-amplitude density andmagnetic field turbulence is forced to flow into a rigid wall, forming a shock wave. We find that the densityfluctuations not only distort the shape of the shock front and lead to a turbulent postshock fluid, but they alsoproduce a number of important changes in the postshock magnetic field. The average downstream magnetic fieldis increased significantly, and large fluctuations in the magnetic vector occur, with the maximum field strengthreaching levels such that magnetic stresses are important in the postshock region. The downstream field enhancementcan be understood in terms of the stretching and forcing together of the magnetic field entrained within the turbulentfluid of the postshock flow. We suggest that these effects of the density fluctuations on the magnetic field areobserved in astrophysical shock waves such as supernova blast waves and the heliospheric termination shock.

Subject headings: magnetic fields shock waves turbulence

1. INTRODUCTION

Collisionless shocks play a large role in many astrophysicalcontexts. The fluids through which they propagate are generallyturbulent, which affects much of the shock physics, includingthe acceleration of charged particles to high energies. The tur-bulence upstream of the shock can either be preexisting or becaused by the shock itself. Thelatter is the case when fastchargedparticles escape upstream and generate instabilities that act toconfine the particles. This process is often taken to be coupledwith the energetic-particle acceleration, although the generationof waves is not necessary for acceleration to occur, since pre-existing turbulence can also efficiently accelerate particles. One

example of an effect of large-scale, preexisting turbulence is therandom walk of the magnetic field, which has been extensivelydiscussed in connection with its role in transport across the mag-netic field and particle acceleration at perpendicular shocks(Jokipii 1966; Giacalone & Jokipii 1999; Giacalone 2005).

Magnetic field observations downstream of the solar-windtermination shock show large-scale fluctuations with an am-plitude comparable to the mean (Burlaga et al. 2006). Also, itis inferred from observations of supernova remnants (Berezhkoet al. 2003) that the magnetic field behind a supernova blastwave is much larger than expected based on the jump condi-tions at the shock. In the present Letter, we conclude fromnumerical simulations of the effects of density fluctuations onshocks that both of these can be a consequence of the interaction

of these shocks with upstream density fluctuations.We note that the enhancement of magnetic fields by a

dynamo-like process in a turbulent fluid has been consideredby many authors (e.g., Kim & Balsara 2006, Balsara & Kim2005, and references therein). However, to our knowledge, theprocess of enhancing the magnetic field downstream of a strongshock that propagates through preexisting turbulence has notbeen discussed explicitly before.

2. TURBULENT FLUCTUATIONS AND FLUID SHOCKS:

THE SIMULATION

We consider the effects of preexisting density fluctuationsin turbulent plasmas on the motion and characteristics of shocks

that propagate through them. Observations of the effects of theinterstellar medium on radio waves reveal the existence oflarge-scale density turbulence with a Kolmogorov power spec-trum (Lee & Jokipii 1976; Armstrong et al. 1981; Rickett 1990).They have also been observed in the solar wind (e.g., Burlaga& Lazarus 2000). In addition to variations in plasma density,the flow also contains a turbulent, frozen-in magnetic field.

Intuitively, we expect that regions of enhanced density, asthey encounter the shock, will push in the shock front and notbe slowed as much as less dense regions. This will cause arippling of the surface that will lead to significant, randomtransverse flows behind the shock. These will, in turn, lead to

rotation and vorticity in the downstream fluid. The flow patternis reminiscent of the Richtmyer-Meshkov instability (e.g.,Brouillette 2002) as a sudden density jump encounters a shock.However, our system starts out with preexisting, large-scaleand gradual fluctuations having a broad range of scales. Ourresulting flow is initially highly nonlinear, so a linear instabilityanalysis is not useful. The downstream cells of rotation andvorticity will drag the frozen-in magnetic field and result in itsdistortion and amplification. In order to follow this quantita-tively, a simulation is necessary.

The physical problem of a shock propagating in a turbulentfluid has been discussed previously by a few authors. Lee etal. (1993) discuss a direct numerical simulation of isotropicturbulence interacting with a weak shock wave. The discussion

of Zank et al. (2002) is based on Burgers equation and usesapproximations to make the resulting analysis tractable ana-lytically. We present here the results of large-scale magneto-hydrodynamic simulations in which a planar shock propagatesthrough a plasma containing preexisting density and magneticfield fluctuations.

In order to make the problem manageable, we consider atwo-dimensional system. Many coherence scales of the im-posed turbulence are contained within the computational do-main, and the simulation proceeds for many coherence times.Results from fully three-dimensional simulations should differsomewhat from those presented here; however, we expect thatthe basic conclusions will remain the same. This will be thetopic of a future study.


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TABLE 1

Summary of Simulation Parameters and Results

Run Ms0 MA0 vBn t /(L/u )max 0 x /Lmax AB S/B2 0 B /B2 , max 0

1 . . . . . . 10 10 90 10 8.25 3.9 8.32 . . . . . . 31.6 90 10 8.25 6.9 24.73 . . . . . . 100 90 10 8.25 8.6 56.64 . . . . . . 316.2 90 10 8.25 12.1 214.55 . . . . . . 1000 90 10 8.25 13.3 422.6

6 . . . . . . 100 0 10 8.25 4.8 66.47 . . . . . . 1000 500 90 20 14.75 17.4 268.08 . . . . . . 5 15 90 10 9.25 3.9 10.5

Note.See the text for descriptions of these parameters.

We solve the MHD equations for Cartesian geometry (x-yplane) over a region with dimensions . At the startx # ymax maxof the simulation, our fluid consists of a uniform flow in thepositive x-direction with a speed and a turbulent densityu 0having a mean of and a fluctuation variance of 2n ADn Sp0

. The density fluctuations are generated so that they ini-20.2n 0tially have a two-dimensional Kolmogorov-like power-lawspectrum of the form

1P(k) , (1)

8/31 (kL)

where k is the magnitude of the wavevector and L is the tur-bulence coherence length.

The turbulence is generated by summing over a large numberof discrete wave modes. We choose the turbulence such thatthe maximum wavelength is (which is one-half ofl p 5Lmaxthe box dimension in the y-direction) and the minimum wave-length is . The grid-cell size is of the smallest1l p 0.005Lmin 2wavelength of the system (i.e., ). The co-Dxp Dyp 0.0025Lherence scale of the turbulent fluctuations, L, determines thescale of our simulations since MHD has no intrinsic scale.

We create a realization of the spectrum by summing over alarge number of discrete wave modes with random phases, toobtain a function , which has zero mean, given bydf

Nm

df(x, y)p C2pk Dk P(k ) n n nnp1

# exp i(k cos v x k sin v y f ) , (2)[ ]n n n n n

where C is a normalization constant, is the direction of thevnwavevector with magnitude , and is the phase. To obtaink fn nany given random realization, for each n we randomly selecta phase and a direction from the ranges and0 ! f ! 2p 0 !n

, respectively. The total number of modes isv ! 2p N pn m. This method for generating a random function, producing903

a Kolmogorov-like turbulence spectrum, has also been dis-cussed in our previous papers (e.g., Giacalone & Jokipii 1999).

The normalization constant in the equation above is givenby

N 1m

2Cp Adf S 2pk Dk P(k ) , (3)( )n n nnp1

where is the variance of , which is related to both2Adf S df n 0and , as we discuss below.2ADn S

The resulting fluctuating density is then computed from theequation

n(x, y, tp 0)p n exp (f df), (4)0 0

where is a constant that is chosen to give the proper valuesf0of and . Note that the relationships between and2n ADn S f0 0

to and can be obtained in a straightforward way by2df n ADn S0linearizing equation (4) and assuming small-amplitude fluctua-tions, as we do. We note that this description of the turbulencegives a lognormal probability distribution of the fluctuating den-sity, as is observed in the solar wind (Burlaga et al. 2006).

The upstream plasma carries with it a magnetic field. Thisfield has a mean component with a direction that is prespecifiedat the start (at an angle of with respect to the incident flow)vBnand an irregular component with a Kolmogorov spectrum thatis generated in a manner similar to that which we have de-scribed elsewhere (e.g., Giacalone & Jokipii 1999). Thestrength of the upstream field and the pressure of the upstreamplasma are determined from the Alfven and sonic Mach num-bers, which are input variables. It is understood that the AlfvenMach number is defined using the mean magnetic field strength

( ) and the mean plasma density ( ). Note that these fluc-B n0 0tuations are also continuously created at , as new fluidxp 0flows in from that boundary, and generated as above, exceptthat they are evaluated at the position .xpu t0

In order to create a shock wave, a rigid reflecting boundaryis placed at the plane . The fluid, which is movingxp xmaxinitially in the positive x-direction, reflects off of this boundary,where the flow velocity normal to it is set to zero. As thedensity builds up at , a shock forms and propagatesxp xmaxin thex-direction. In the frame of reference of the simulation,the plasma speed downstream of the shock is zero, on average.In the shock frame of reference, the upstream flow is u p0

, where is the speed of the shock in the sim-u u u0 shock shock ulation frame and is also the downstream flow speed in the

shock frame. Thus, the upstream plasma speed in the simulationframe is related to that in the shock frame by u p u 0 0

, where r is the average density compres-u p u (1 1/r)shock 0sion across the shock (4 for a strong shock).

The MHD equations are solved by using a fourth-order-accurate (in time) predictor-corrector scheme. All quantities areperiodic in the y-direction. Spatial derivatives are determinedusing fourth-order-accurate centered differencing in the y-direction and sixth-order-accurate spline derivatives in the x-direction. The shock thickness is of the order of a few gridcells (due to a small amount of artificial viscosity included inthe calculation), which is less than 0.01L. Our simulations havebeen tested for compatibility with the jump conditions for thecase of initially quiet upstream plasma, and they are satisfiedto very high accuracy. Moreover, we have found that the di-vergence of the magnetic field is less than throughout.610 B /L0

3. SIMULATION RESULTS

The results from the simulations are summarized in the righttwo columns of Table 1 and are shown in Figures 1, 2, and 3.In Table 1, and are the sonic and Alfven Mach numbersM Ms0 A0in the shock frame of reference, respectively, and is thevBnangle between the upstream magnetic field and shock normaldirection; tmax and xmax are the maximum time and spatial di-mension along the shock-normal direction, respectively (for allsimulations, we take ). is the magnetic fieldy p 10L AB Smax 2


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No. 1, 2007 DENSITY FLUCTUATIONS AND SHOCK WAVES L43

Fig. 1.Results from run 7. Shown are cross section profiles of the plasmadensity, the flow speed transverse to the average shock propagation direction,

and the magnetic field as a function of x. Only a portion of the entire simulationdomain is shown.

Fig. 2.More results from run 7. Shown is the gray-scale representationof the magnitude of the magnetic field over the entire simulation domain.White represents values below 3.16 , black represents values larger thanB0316 , and the shades between these two extremes are equally spaced in aB0logarithm of .B/B0

Fig. 3.Results from runs 3 and 6. Plotted is the magnetic field, averagedover the entire downstream region, which increases as the shock moves awayfrom the wall, as a function of time. The only difference between these twosimulations is the angle between the upstream mean magnetic field and theplasma flow velocity, as indicated.

strength averaged over the entire downstream region, andis the maximum.B2, max

The most important new result of our analysis is that the down-stream magnetic field is enhanced by the stretching and forcingtogether of magnetic fields that are entrained within the turbulentflow. The right two columns of Table 1 indicate that both the meandownstream magnetic field, (which is an average of the mag-AB S2netic field strength over all y and over all x from just behind theshock to ), as well as the maximum magnetic field canxpxmaxbe significantly larger than the upstream field. Both the mean andmaximum fields increase with increasing Alfven Mach number.

As the Alfven Mach number is increased, the initial magnetic fieldis very weak and is carried as a passive additive with the flow.The downstream field is enhanced up to the point where magneticstresses are large enough to influence the plasma motions. Forhigher Alfven-Mach-number shocks, the field can be enhancedgreatly over the initial field.

Note that the last two simulation runs shown in Table 1 useinput quantities that are representative of a typical supernovablast wave (run 7) and the termination shock of the solar wind(run 8). For the case of the termination shock, reachedAB S2the equilibrium value shown in the table, whereas for run 7,

was still increasing slightly at the end of the simulation,AB S2despite the fact that it was run twice as long.

Figure 1 shows a cross section of the plasma density, the y-

component of the flow speed, and the magnitude of the magneticfield as a function of x. The plasma density jumps by a factorof about 4, which is the strong shock limit of the jump conditions.Note that the transverse flow jumps at the shock to a fractionof the incident flow speed. This is because of the distortion ofthe shock front, which depends on the location along the shock.Also note that the magnetic field is compressed by a factor ofabout 4 right at the shock, but farther downstream there areregions of both a greatly enhanced field, reaching values of 100times the initial field or more, and a relatively weak field.

Figure 2 shows the two-dimensional gray-scale representa-tion of the magnitude of the magnetic field over the entiresimulation domain. Note that the different gray-scale shades

are separated equally in a logarithm of field strength, with theminimum value set to and the maximum value0.510 p 3.16set to . We first note that the turbulent density2.510 p 316fluctuations cause the shock wave to become rippled and un-even, as regions of higher or lower density, and hence higheror lower inertia, encounter the shock. These ripples in the shockfront then introduce vorticity into the flow. This downstreamvorticity is readily apparent in the magnetic field gray-scaleplot shown in Figure 2. The frozen-in magnetic field is draggedby the vortices, and the magnetic field lines become stretchedand deformed, creating regions of larger magnetic field inten-sity. Both the maximum and the spatially averaged magneticfield magnitude increase rapidly behind the shock, as illustratedin both Figures 1 and 2, as well as in Table 1, and are limited

only by the dynamical effects of the increasing magnetic field.The resulting amplification of the magnetic field can be quitelarge for strong shocks having a high Alfvenic Mach number.


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Figure 3 shows the mean magnetic field, averaged over theentire downstream region (whose size increases with time as theshock propagates away from the wall), as a function of time, fora parallel shock ( ) and for a perpendicular shock (v p 0 v p Bn Bn

). Note that the mean field was still increasing when the sim-90ulation was stopped. While the mean magnetic field apparentlytakes some time to saturate, the maximum field is obtained veryquickly. We have found that very strong fields are obtained within

a coherence scale of the shock front (see Fig. 2). The mean down-stream field for runs 1 and 2 did saturate at the values indicatedin Table 1 within the chosen simulation time because the AlfvenMach numbers are much lower. Figure 3 shows that the meandownstream field is stronger at the perpendicular shock comparedto the parallel shock, whereas the maximum compression is com-parable at the two (see Table 1).

The rapid increase in the downstream magnetic field maybe shown to be caused by the vorticity or twisting induced inthe downstream flow by the rippled shock front, which in turnis caused by the density fluctuations being convected on to theshock front. Such twisting of the magnetic field may be shownto increase the magnetic field strength by considering two sim-ple calculations. Consider first a magnetic field Bp B(x, t)y

having the initial value , which is frozen into the time-in-B0dependent flow . ClearlyUp [U sin (kx), U ky cos (kx )]0 0

. In this simple case, an analytic solution for the Up 0evolution of B may be found. In this, the mean of B remainsconstant, but at the intensity increases asxp p/2 B(xp

. A somewhat more complicated case,p/2, t)p B exp (kU t)0 0, which alsoUp [U sin (kx ) cos (ky), U cos (kx ) sin (ky)]0 0

has zero divergence, and with , Bp B (x, y, t)xB (x, y, t)yx ycan be shown with a simple numerical integration to increaseboth the average and the maximum of with a comparableFBFtimescale to that for the simpler case (see also Parker 1974).From these simple cases, we conclude that the magnetic fieldamplification seen in the shock simulations is reasonable andis caused by the vorticity induced by the density fluctuations.

4. DISCUSSION AND CONCLUSIONS

We have shown that including large-scale, broadband, pre-existing, upstream turbulent density fluctuations in a hydro-magnetic fluid has significant effects on the downstream mag-netic field. In particular, a weak magnetic field in a strong shockcan be amplified by a large factor. This appears to be a conse-quence of the density fluctuations causing a rippling of the shocksurface that then introduces vorticity and swirling in the flowthat then stretches and folds the entrained magnetic field. In ouridealized hydromagnetic system, the magnetic field amplificationis limited by the increasing magnetic stresses as the field mag-nitude grows. This implies that the increase in magnetic field

can be larger for stronger shocks. We note that the process dis-

cussed here has some similarities with the Richtmyer-Meshkovinstability (see, e.g., Brouillette 2002). However, in our case, thefluid variations are gradual, and the fluctuations do not growfrom small perturbations. Although our simulations are for planarshocks, the general nature of the conclusions should apply tosupernova blast waves. Dickel et al. (1989) discussed the con-sequences of an inhomogeneous interstellar medium on super-nova remnants using one-dimensional simulations, and they sug-

gested that a turbulent dynamo might amplify the magnetic field.This idea is similar to that put forth here. Balsara et al. (2001)presented results from a three-dimensional simulation of an ad-iabatic supernova blast wave propagating into a turbulent mag-netized medium. Results similar to ours were seen, but the gen-eral magnetic field amplification was not mentioned, althoughtransient magnetic hot spots were noted. Perhaps the differencebetween our results and those of Balsara et al. (2001) is due tothe quite different boundary and initial conditions or the three-dimensional geometry.

Supernova blast waves are very strong, and the magneticfield amplification implied by our simulations should be large,exceeding factors of 100. Recent analyses of X-ray emissionsfrom supernova remnants have concluded that the magnetic

fields required to explain the observations are approximately100 mG, much larger than expected from the Rankine-Hugoniotconditions on the shock wave (Berezhko et al. 2003; Volk etal. 2005). Bell & Lucek (2001) and Bell (2004) have proposedthat the interaction of cosmic rays with the upstream fluid andmagnetic field naturally causes an instability that producesstrong magnetic field amplification. The present analysis sug-gests the likelihood of an alternate scenario whereby preexistinglarge-scale upstream turbulence causes the magnetic field am-plification. Perhaps both scenarios are applicable. It should alsobe noted that the effects of density fluctuations will leave be-hind a very turbulent medium and magnetic field, with velocityfluctuations of the order of the sound speed. This will contributeto the general turbulence in the interstellar medium.

In the heliosphere, the solar-wind termination shock is muchweaker (and has a smaller Alfven Mach number) than a su-pernova blast wave. Observations (Burlaga et al. 2006) showthat the magnetic field downstream of the shock is very dif-ferent from that observed upstream. In particular, it exhibitslarge fluctuations in magnitude, some with magnitudes muchlarger than expected from the normal jump conditions at theshock. We suggest that the effects of density fluctuations foundin our simulations may play a large role at the terminationshock and heliosheath as well.

This work was supported in part by NASA under grantsNNG05GE83G and NNG04GG20G and by the NSF under

grant ATM0447354.

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J. Giacalone and J. R. Jokipii- Magnetic Field Amplification by Shocks in Turbulent Fluids