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HTLE HYBRID SIMULATION CODES WITH A.PPLTCATION

TO SHOCK: AND UPSTREAM WAVES

AUTHOR(S) D. Winske

SUBMITTEDTO Proceedings

Simulation,

of the Second International School for Space

Kapaa, Kauai, Hawaii, iebr. 3-15, 1985

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rlLJ~8~k~~~ bsAlamos,NewMexico87545

Los Alamos National Laboratory

MSIMUIIWOfMS MWEHT ISUUIIMIIEO

HYBRID SIMULATION CODES WITH APPLICATION TO SHOCKS AND UPSTREAM WAVES

D. WinskeLos Alamos National LaboratoryLos Alamos, New Mexico, 87545us.?

ABSTRACT. Hybrid codes in which part of the plasma 19 represented asparticles and the rest a9 a fluid are discussed. In the past fewyears such codes with particle ions and massless, fluid electronshave been applied to space plasmas, especially to colllsionlessshocks. All of these simulation codes are one-dimensional and 9imi-lar in structure, except for how the field equations are solved. Wedescribe in detail the various approaches that are l!sed(resistiveOhm’s law, predictor-corrector,Hamiltonian) and compare results fromthe various CUUeJ with examples taken from collisionless shocks andlow frequency wave phenomena upstream or shocks.

1, INTRODUCTION

Plasma physics phenomena are characterized by a multitude of lengthand time scales, primarily due to the different responses of thelight electrons and Lhe massiv” ions to the imposed and self-gener-ated electric and magnetic fie.da. Typioally, one is interested inparticular proceasea which occur on some of these scaloa and notInterested in other processes that occur on shorter or longer time ordistance scales. This can be accomplished in numerical simulation bytreating the various plasma sp+cle9 in different waya, for example,as dlacrete partiolea or as fluids. Hybrid oodea are defined asthose numerical algorithms in whloh the varioua plasma species aretreated in a different manner, a~ distinct from particle codee whereall the plasma species are treated as particles or fluid codes whereeach species (or several Qpeoies together) is treated as a fluid.

Various types of hybrid codes are of’course possible, dependingon the problem at hand. One important subclass of hybrid models arethose in which there are two (or more) population~ of one particularoharge Bpeolea, whose prOpOrtiOS on a partloular length or time Scaleare different. For example, oonsider the lnteraotion of a small,cold electron beam with a hot background eleotron population (0’Neil

2

et al., 1971). In this case the unstable waves generated by the——presence of the beam strongly affect it, and thus a particle descrip-tion Is needed to correctly ❑odel the highly nonlinear dynamics ofthe beam electrons. On the other hand, the waves do not affect thebackground population very much. Thus, there is no need to followthe dynamics of each individual background electron; rather, thecontribution of the background component can be included simply as alinear dielectric in Poisson’s Gquation. Similar methods can be usedwhen there are several ion populations. For example, in the study ofthe interaction of an iOn ring velocity distribution with a back-ground ion core, Lee and Birdsall (1979) treated the ring ions asdiscrete particles, with a fluid de~cription fo~-the background ior19(and the electrons).

The most common type of hybrid code, however, occurs when the twospecies involved are the electrons and ions. The simplest kind ofhylvid model of this type is to ignore one species entirely. Forrxample, in the study of high frequency electron behavior it isvery common to eliminate the ions, except as m charge-neutralizingbackground. It 1s also possible to ignore the electrons entirely, ashas been done for tearing mode calculations (Dickman et al. (1969)and later work to be cited in the next section). Another usefulapproximation that is commonly invoked in this type of hybrid modelis qua~ineutrality, which makes use of the fact that the electron(nct and ion (n ) charge dermities are nearly equal.

iIf one is in-

terested in sca e lengths much larger than the Debye length, thecondition ne - nl is imposed; for smaller systems, a Boltzmann rela-tion between the electron charge density and the electrostatic poten-tiai may be used instead, which gives rise tG a nonlinear Poissonequation. (Okudaet al., 1978). Another common approximation in-——volves the electron masa. Depending on tne frequency range of inter-est, the electron mass may be kept (Hewett and Nielson, 1978) or not.

A very commoo type of hybrid code, and the one of interestthroughout the Pest Of this chapter, treats the electrons as amassless, charge-neutralizingfluid. In recent years this type ofmodel has bucome widely used in space physics for the study of phe-nomena at the bow shock (Leroy et al. 1981 and 1982; Leroy andkiinske,1983. Kan and Swift, 19&;~andt and Kan, 1985), upstream ofthe bow shock (Winske and Leroy 1984a; Winske et al. 1984 and 1985;Hada and Kennel,

——1985), the magnetopause (Swift and Lee, 1983), the

magnetotail (Swift, 1983b), and the magnetosphere (Omura~t al. 1985;Tana4a and Goodrioh, 1985).

While all of the calculation cited here are based on hybridmodels with similar properties, thero are differences in the way themodels are implemented numerically, primarily in how the field equa-tions are aolvsd. Three different techniques, referred to hereafteras che resistive (Ohm’s law) method, the predictor-oorreotor method,nnd the Hamlltonlan method, havs been employed. The purpose of thispaper is to expla!n how these various methods work in some detail(Seotion 2) and then compare results of simulations based on eachmethod (Section 3). The examples ohoeen are well $tudied phenomenafrom the earth’s bow shock and the upstream region, and the discus-sion will mphaaize numerics rather than the physics content. Ashort summary 18 given in Section 4.

3

2. DESCRIPTION OF HYBRID CODE MODELS

In this section the numerical schemes used in three hybriC code mod-els are described. In each case particle-jn-ceil techniques areemployed for the ion dynamics, while fluid equations are used for the(massless) electrons. Quasineutrality is assumed and the low fre-quency (~arwin) approximation is used (Nielson and Lewis, 1976). Forthe methods described here the one-dimensional nature of the calcula-tion 1s utilized, but as will be discussed, each of the models hasbeen generalized to two spatial dimensio~s.

2.1 Resistive Ohm’s Law Method

This model was originally devised by Chodura (1975), and furtherdeveloped by Sgro and Nielson (19’76)and Hamasaki et al. (1977), forthe analysis of laboratory magnetic fusion experiments and more re-

——

cently for the study of’the earth’s bow shock by Leroy et al. (1981,1982). The details of the method and references to earlier work aregiven by Mlnske and Leroy (1984b). In this method, as in the two tofollow, a standard leapfrog scheme is used to advance the particlesand fields. The velocities of the particles are known at the halftime steps, while positions of the particles and the fields are knownat the even time steps (time le elJ-1~2to~e~finotfid by superscripts).Thus, at time step N, we know~ ,x, ,~. To advance theparticles, we solve

“)4+1/2 ~ vN-1/2 Atq.- +— m

where~ m vN-1/2+ btq EN ‘-—.

2m

f. 1-l

()

Atq 2 ~N .

7F -

Atq (vN-1/2 . BN)g“g-

and thenxN+l - # + Atv N+l/2

x

W9 then solve Ampere’s law

V2ATN+1 -J!w~TN+l

-—-0

for the transverse components (y

(1)

solving to

(2)

(3)

(4)

or z). The ion Dart of the current,oomeu directly from collecting the ion moments, where we advance (butdo not save) the particles one additional half time step

#+1 . J4+l/2 Atq (EN+ vN+l/2 XBN,O)

‘K(6)

where EN and EN are evaluated at #+1, to obtain the Ion densityN+l

n,N+l and velocities~i . Quasineutrality (in l-D) then gives ne -n; = n and V~x - Vix - Vx.

The elec ron part of the current comes from the electron momentumequation with a reslstlve term (i.e. Ohm’s law)

!4+1(alesr’bed shortly), we obtain in the usualAfter solving for & .’-way

ET - ~ x & (Bx - constant)

ET - ‘1 aA.f (8)7X

Then eolving an energy equation for Te,

(9)

we can obtain the last field component, Ex, from the x Cornporlent Of

(7)

Ex = -1 (ye x ~)x - 1 imTe(10)——

7 nq ax

The method used to solve (5) for~ 1s slightly different from8that presented in Winske and Leroy (19 4b) and works better for

oblique shocks. We assume the resistivity tensor to be diagonal (i.e.nil=nl ●n), which has been shown to correctly model turbulent sys-tems, such as the z-pinch, very well (Sgro and Nielson, 1976). Wesolve

~T “ n-’(~ +~ x~/c)T - qn(vi - ve)T (11)

for~T , and substitute into (5), obtaining

;; ~“ +%where

Fy - qn(viy - CEZ - VXBY)

~ Bx

Fz = qn(Viz + CEY - VXBZ)

Bx ~

(13)

‘m(H%9(%x)and no-reference density, u -(4nnoq2/m)1/2

iand all quantities are

at time level N+l. Using ( ? to express ~T’and * in terms of ~To(12) can be written in finite difference form and solved, as de-scribed in Hinske and Leroy (1984b).

In addition to the references cited In klinskeard Leroy (1984b),calculations based on this ❑ethod have been done ta study obliqueshocks (J.eroyand Winske, 1983), the electromagnetic ion beam insta-bility (Winske and Leroy, 1984a), the interaction of heavy ions withthe solar wind (Minske et al. 1984 and 1985) and the steepening ofS1OW waves !Hada a:.~Ke=e~ 1985). The method has been exte,ldedtotwo dimensions (Hewett, 1980) and has been applied to the study ofmagnetic reconnection in laboratory experiments (Hewett, 1984).

2.2 Predictor-Corrector Method

A second method used for hybrid code calculations ~mploys thepredictor-corrector technique. The method is describetiby Byers et~. (19’78)and has been implemented in a one-dimensional code by —Tanaka and Goodrich (198s) and Omura et al. (1985) in the study of——heating of heavy ians by ion cyclotion instabilities. The methodhag been extended to two dimensions by Harried(1982) and used in thestudy of rotational instabilities in laboratory field-reve?sed con-figurations (Harried,1983).

We will describe the method in its simplest form, where the elec-

;;:n$5T92:a:ti:eEAs::p;Nc:::t:::;n‘gain’ ‘e assumeat “me ‘tep N, . In this case the advance of onetime-step involv%s two =teps, a predictor step and a corrector step,each of which involves g 1?

$%throug

P!he particle table.

In the first step, +1 2 and x + are obtained as in the previ-ous case using (1) -}0. lnt~g~rocess,

;fi+~%ticles &~:+’ 2= XN+AtVx

however, we first advance1 2/2 to collect the ion moments,

and ni Me then compute the predictor fields (denotedby subscript p) usi;g Faraday’s law and (7):

BN+l/2 . BN - cAt (VXEN)--

T f14).

~N+l/2 - ,

{

N+l/2w -LN+’’2) xEN+’’TJniN+”2”2

qn!N+l/2 um c }

to obtain

E N+l = -EN ● 2EN+1/2-P - -B N+l - ~N+l/2 - cAt (VXE4N+1)-P - —-.

2

(15)

6

He then go thru the particles again, using the predictor fields,

to calculate v N+3/2 and ~@//2

‘+3’2. again using (1) - (4), in order to

and nl pfl+3’2collect ~i,p ● Then, recalculate BN+3/2 andN+~, B N+@E ‘+3’2 using (14) with ~redictoi’fields% and ion ❑oments

‘R 0b’a’nEi!7 :eY ;#:Y?2–P

+ ~ N+3/21

T--P

(16)#+’ N+l/2 -cAt ~vxEN+l)“g ---

2

Having now advanced Lhe fields, we are thus ready to start the entiresequence again.

2.3 Hamiltonian Method

This method employs the canonicalof the velocities. A descriptionpie, by Morse and Nielson (1971),Hamiltonian and Lagragian methods

❑omenta of’the particles P in placeof the method is given, f~r exam-and a comparison between the1s discussed by Nielson and Lewis

(1976). The Hamilton~an method has been used by-Swif’.and Lee (1983)to study tha rotational discontinuity at the magnetopause, Swift(lf183b)to examine magnetic slow shocks, Kan and Swift (1983) andMa’ldtand Kan (1985) to simulate nearly parallel shocks. The methodhas been generalized to tw spatial dimensions in the limited senseof either completely ignoring the electrons (Dickman et al.,1969;Ambrosiano et al.;

.—lg83; Teresawa, 1981) for the study of tearing——

nlcdesand ion fusion physics (Friedman et al., 1977; Mankofsky eta_l.., 1981; tlankofskyand Sudan, 1984) or in restricting the zyp= ofperturbations allowed (Swift, 1983a).

In this case, at t!me step N, the canonical momenta of the parti-cles

ET = ❑v-T + q~T/c(17)

are known at the half time step N-1/2. The particle equation ofmotion is

(18)

Note that In the case BX-Q the method can be very z~tractive since ET

is a conserved quantity. Using (17), (18) can be written as

Another nloe featuro 1s that using a complex representation (P - Py +

tPz,‘- Ai: ‘*Z’

11- qBx/mc) the equations of mction can be writtenoompaotly

dPN = -ifJ(PN- c@)-n c

(20)

or~-im ~ #eiSit . lRNN

ZE c

7

(21)

Thus, in difference form (Swift and Lee, 1983)

#+1/2 - ~N-1/2e-ifiAt N ‘iCIAt/2+iA~Ae (22)c

ThenN. I (~TN+l/2 + p N-1/2,~T — –T -d

2m mc (23),

is known and Vx can be advanced

as can the particle positions, #’l using (4). (I have used v herefor simplicity. It is possible to &ite Px = mvx (AX-O, since ~“A =O) and use P’s throughout.)

--

Again, we need to solve Ampere’s law (5). There are several

different ways to difference the equationg. One method is to useN+l/2

(22) to advance PTN+l/2 to pTN+3/2 and express It in terms Of PT

and A ‘+l. Another (Swift and Lee, 1983) is to solveN~l/2 4mJ N+l/2/c, expressinE~T N+l/2

‘2~T ~+1-- -T as an average between ~TN

and ~T . In either case the resulting equation is implicit in that

it involvesA.T‘+1 on the right hand side. This adds a slight compli-cation, known as double area weighting, in gathering the moments, agdescribed by Forslun

i+t al. (1972), Forslund (1974) and Mankofsky et

Q. (1981). Once& + l~known, E and B can be obtained u9ing (8)~An equation for T , such as (9), c=n be=olved !Kan and Swift (1983)use at,equation of state instead), and then Ex can be obtained agbefore, from (10).

Finally, it should be noted that there is an extra complic&tionlf there 1s an external B or B in the system. A constant B can beadded in through an addlt~onal term A This contribution toP of the particles must be added or ~~t%~~ed if the boundary con-d~tions are such that particles exiting at one end of the systemreenter at the other end.

3. NUMERICAL COMPARISON OF THE MODELS

In this section we compare results of simulations based on the threehybrid models discussed previously for two test problems. The firstis an electromagnetic ion beam lnatability, the second is aquasiperpendicular colllsionless shock. Both problems have been wellstudied, and the reader 1s referred to the literature for details,

Tho first problem involves the inn beam instability, which 18tt:oushtto be the mechanism which produce~ low freq’iencyhydromagnetic wavee and diffutieion populatlwv! upstream of theearth’s bow shock (e.g., see Car] ot al. (19L1) and Wlnske and Leroy(1984a)).

---—The instability results from the interaction of a weak

8

beam of ions backstreaming from the shock and the incident solarwind. The situation to be simulated consists of an ion beam driftingreflatlveto an ambient plasma along a uniform magnetic field Bo. Theparameters are choseilto ❑atch tnose in Uinske and Leroy (1984a) forthe resonant instability: the beam is weak, 1.5$ of the total iondensity, with a drift speed of 10 A relative to the background i0n9.

!Both ion components have B=8nnT/Bo -1.0, with a cold (Be=O.O1) cur-rent-neutralizing electron back~round. The instability leads to thegeneration of low frequency wave9 of well defined wavelength thatscatter the beam. In the nonlinear regime these regular waves breakup intO very nOnlinear waveforms, producing a diffuse ion distribu-tion In the process.

In Figure 1 we compare results of three simulation. The toppa!els correspond to the resistive code with rI-O;the middle panelsare from the predictor-corrector code; the bottom panels are fram acode based on the Hamiltonian method. In each cage we have used10,000 partic es (half to represent the beam, half for the backgroundions) on a s of 256 cells wit~ cell size Ax=c/u .

1In each case we

use tilesame random numbers to initialize the part cle velocities.The left side of the figure show9 one component of the magneticfield, Bz, normalized in terms of the ambier,tfield BO at about thetime when the waves have achieved their largest amptitude, fiit.38.11(0 -eB /mic).If

The results from the three cases are very similar,di fer n~ only In the amount of 10U amptltude, short Wavelength noisethat is superimposed on the dominant structures. The right side ofthe figure shows the tim~ histories of the fluctuating magnetic fieldenergy dens’ty, Ub-SdxB /~dxBo2.

TAgain, the overal,lresults in each

case agree very well. he peak field energy density achieved isslightly (-3%) larger in the predictor-corrector case, which may be areflection of the oetter energy conservation in thepredictor-corrector code (AE/E -0.03%), compared with the resistivecode (AE/Eo-1.2$) and the Hami?tOnian code (AE/Eo-5.4%). The poorerenergy conservation in the Hamiltonian code suggests that thedifferencing scheme used here for the test problem could be improvedand should not be taken as an indication that this method is inher-ently inferior.

The sncond test problem involves a quasiperpendicularcollisionless shock. Again, in this case the physics ha9 been inves-tigated in detail (Leroy et al, 1981 and 1982; Leroy and Winske,——1983; Forslund et alfl,1984). The simulation is iniciallzed withuniform upstream and downstream states related by Rankine-Hugoniotconditions and then Is allowed to evolve In time. Trleparameterschosen for the test case are upstream Mach number MA-Vi/VA-8, up-stream shock normal angle 0Bn-600, and UPStreaIII Be-Bi~0.5. Again, weshow (Figure 2) the results of three simulations: top panels corre-spond to the resistive code with resistive length L -(nui/4m)

!(O/V1)(C/UJ) equal to the P>ll size Ax-o.3 C/LIJi,rridlc panels corre-kspend to t e results of the resistive code with LR-O.OIAX, and bottom

panels correspond to the reslstanoe-free predictor-corrector code.In eaoh case 10,000 particles on a grid of 200 cells with a time stepfJ~At-0.0125 (where the upstream magnetic field B, 1s used to computea~).

9

The left side of Figure 2 shows one component of the transverse

::Ke::cL::l;:t:::tt”’ 2”’”The upstream magnetic field 1s to the

he shock occurs in the center uhere the mag_netic field rises rapidly. The peak value of the magnetic fieldcomponent B IS somewhat larger than the downstream value B ~ (cowputed from ~he Rankine-Hugoniot relations), which lsheldf?xedatthe right hand boundary. This peak value, referred to as theovershoot, is followed by an undershoot and then several moreoscillations. The middle panel shows the same snapshot of the ❑ag-netic field in the run where the resie?ivity is lowered. The sameoverall stucture is observed, except that the overshoot is larger andthe oscillations behind the overshoot are somewhat less regular. Inaddition, there are small oscillations on the ❑agnetic field in theupstream region and ti,esharp rise of the ❑agnetic field is betterresolved, consisting of a precursory smaller increase (called thefoot) and a very steep ramp. The field profile obtained with thepredictor-corrector code (bottom panel) shows the same general fea-tures as In the second case, except th~t the oscillations behind theshock are somewhat larger in amplitude and have a shorter wavelength.

The right hand panels of Figure 2 show the time histories of then~agneticovershoot for the three runs. In the case with the largerresistivlty, the overshoot has a nearly constant value. For smallresistivity, the overshoot oscillates in time and the average valueis about 20% larger. The resistance-free predictor-corrector codegives a slightly larger average overghoot with larger, more frequentoscillations.

We corrlude from these results that the three shock simulationsgive overall similar results, but the effect of resistivity is todamp out some of the oscillations. This naturally leads to the ques-tion of how much resistivity (if any) should be included. It shouldbe kept in mind that the resistivlty is ad~ed in to compensate forthe fact that the simulations are one-dimensional and therefore donot include microinstabilities due to the cross-field current thatare seen, for exam~le, in 2-D particle simulations of shocks(Forslund et al., 1984). In the present simulations, as in Leroy ~——~1-. (1982), the reslstivity is taken as r constant, although morerealistic forms for the resistivity, either based on phenomenologicalexpressions (Chodura, 1975) or microphysics (Hamasaki et al., 1977)are possi’ble. In this regard, one has to be guided by~p~e observa-tions or laboratory data (which indeed show rather steady structures)to infer the proper amount of resistlvlty that should be used.

As a final note, the question of resistivity becomes increasinglycomplex as the Mach number is raised. Quest (1985) has recentlycarried out resistive hybrid simulations of perpendicular shocks withMach numbers greater than 20. Ee finds that a shock which is fairlysteady in time can be produced with reslstivity such that LR=Ax, butthat with weaker reslstlvlty the magnitude of the oscillations of theovershoot is comparable to its average value. In this case the shocksteepens to a very narrow (-Ax) width, then collapses. Because theamount of resistivlty that would be needed to maintain the shocksteady is unphyslcally large at high Mach numbers, it is suggestedthat very high Mach number shocks indeed are oscillatory in character(Quest, 1985).

10

4. SUMMARY

We have described hybrid codes in which the various plasma speciesare given different representations. Specializing to the importantcase where the ions are treated as partiCleS and the electrons are amassless, charge-neutralizingfluid, we have discussed in some detailthree ways for solving the field equation~, referred to here as theresistive, predictor-corrector,and Hamlltonian ❑ethods. Using simu-lation codes based on each of these techniques, we have comparedresults for two problems of current interest in space physics: 1Owfrequency waves driven by an ion beam and quasiperpendicularcollislonless shocks. For the lon beam problem all three methodsgive essentially the same results, with the predictor-corrector❑ethod giving better overall energy conservation. In the case of thequasiperpendicular shock, the effect of the resistivity on producingtime steady solutions has been emphasized. Mhile the use of thesecodes in one spatial dimension has been stressed throughout thisarticle, all three methods discussed work in two dimensions, and willundoubtedly become widely applied in space physics, as they havealready in magnetic fusion problems.

ACKNOWLEDGEMENTS. This work was supported by the NASA Solar-Terres-trial Theory Program and the Department of Energy.

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—

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—

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—

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—

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11

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—

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—

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—

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—

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13

21 II

I i1

‘W4

‘Wti-2 ~

o x(c/fAJ$ 25e

0.52 I

‘b

0.0

0.52

‘b

0.0

0.52 ~

‘b

0.00 i-lit 102.4

Figure 1. Results of three different hybrid simulations of theelectromagnetic ion beam instability (top: resistive model; middle:predictor-corrector model; bottom: Hamiltonian model) showing: (left) onecomponent of the iluctuatlng magnetic field at flit■ 38.4, (right) timehiutory of the fluctuating magnetic field energy denaik.y.

14

8 i I I

&01

0 J 1 I I

8 I I 1

~

B1

o 1 1 1

8

+1

00 X(clwl) 60

1- -i

?---l

oL_L_J-.d

2.8

B zm

BZ2

o I I 1

0 flit 12.6

Figure 2. Re3ulte of three hybrid simulations for a q~aslperpendicularshock (top: resistive code with large reaifltivlty;middle: resistive codewith smaller reSiStivity; bottan: predictor-corrector code) showing:(left) one oomponent of the magnetic field at flit - 12.5, (rj8ht) timehistory of the magnet~c field overshoot.