NASA-CR-192930

Oblique MHD Cosmic-Ray Modified Shocks:Two-Fluid Numerical Sinmlations

!/ c

Adam Frankl T. \V. Jones I and Dongsu Ryu 2'3

I Department. of Astronolny, University of Minnesota, Minneapolis, MN 55455

2Princet(m University Observatory, Peyton Hall, Princeton, NJ 08544

:_Del)artment of Astronomy &: Space Science, Chungnaln National University,

Daejeon 305-764, Korea

Abstract

We present the frst results of time dependent two-fluid cosmic-ray (CR) modified MHDsh(_ck simulations. The calculati(ms were carried out with a new numerical code for 1-D

ideal .kIHD. By coupling this code with tlw CR energy transport equation we can simulate

the time-dependent evolution of .k[HD sh(_cks including the acceleration of the CR andtheir fl-edback on the shock structures. \Ve report tests of the combined numerical method

illcluding comparisons with analytical st_'ady state results published earlier by Webb, as

well as internal consistency checks for lnore general MHD CR shock structures after they

appear to have converged to dynamical steady states. We also present results from an

initial time dependent silnulation which extends the parameter space domain of previous

analytical models. These new results support Webl)'s suggestion that equilibrium oblique

shocks are less effective than pax'allel shocks in the acceleration of CR. However, for realistic

models of anisotropic CR diffusi(m, oblique shocks may achieve dvlmmical equilibrium on

shorter timescales than paralM shocks.

(NASA-CR- 192930) O[_L I ;IU£ MHdCOSMIC-RAY MODIFISO SHOCKS:TWO-FLUI5 NUMFRICAL SIMULATIONS

(_innesotd Univ. ) 14 p

N93-25132

tJnclas

,33/_3 0tSg£e0

https://ntrs.nasa.gov/search.jsp?R=19930016943 2020-05-02T10:49:44+00:00Z

1. Introduction

Nonlinear theories of diffusive shock acceleration have demonstrated the importance of

cosmic-ray (CR) feedback on the evolution of shock structures (e.g., Blandford & Eichler

1987). Using two-fluid models it has been shown (e.g. Drury& V/51k 1981, Achterberg

et. al. 1984. I,:ang & Jones 1990) that CR pressures can become large enough to

smooth shocks, eliminating the entrol)y generating gas sub-shock. More generally, the CR

feedback modifies the efficiency of energy transfer from gas to CR in the shock. Recent

numerical sinmlations (e.g., Drmv & Falle 1987. Falle £: Giddings 1989, Jones & I,:ang

1990. Kang £: Jones 1991, I(ang. Jones £: Ryu 1992) have also shown the importance

of time-dependent effects in the determination of CR modified shock properties. For the

most part, pa.st studies of CR modified shocks have focused on pure hydrodynamical,

(or parallel, sonic mode) models of the shock dynamics. Magnetic fields, however, will

generally be dynamically important in many environments where particle acceleration

occurs. It has been suggested that comp(ments of the magnetic field which are aligned

perpendicular to the shock normal (tangent to the shock face) can alter the efficiency

of the acceleration process. Jokipii (1987) has pointed out that. for standard models

of CIR anisotropic dif-i\lsion (see equati()n [4.1] below) perpendicular components of the

field will decrease the shock crossing time fin a CR particle, increasing the rate at which

individual particles gain energy fl(.nn the shock. On the other hand, Baring, Ellison & Jones

(1993) have shown that the efficiency of thermal particle injection into the CR population

can be dramatically decreased by perl)endicular magnetic field components. ;From these

examples it is cleat that to tmders_and the flmdamental nature of shock acceleration in

more r('alistic astrophysical settings, full X[HD calculations are needed. Two-fluid models

_,f CR transport along the lines intr_)duced by Drury _: V&lk ( 1981 ) are an eflCicient means to

l_egin such explorations. Such models enable one to economically calculate the dynamical

features (_f t-l_m,s within the constraints inlp_sed by the need to estimate a priori some

clos_u'e param_'ters for the CIR. Equilil_rimn *IHD CFl-modified shock structures have been

('alc_fl_lted 1)v \\'el3b (1983) using these methods for the case where the gas is cold and its

pressm'e can be ignored. \Vebb's calculations demonstrated that. as for the CR-modified

gas dynamic flows, both sub-shock and sm()othed. CR dominated solutions to the MHD CR

sh_ck equations were possible. However. his solutions suggested that, in the limited range

of conditions he could consider, shock acceleration of cosmic-rays was less effective when

the upstream tangential components of the field were strong. Among other consequences

this appears to expand the portions of the shock parameter space that lead to sub-shock

solutions. That could impact on such issttes as low energy injection processes and themomentum distribution of the CR.

Some subsequent steady state two-fluid analyses, considering a wider parameter space

support the above impressions (Kennel e¢. al 1985. \Vebl) e* a11986). In this paper we report

the first results of time dependent MHD CR two-fluid sinmlations. We present tests of

a new numerical code against WeblTs analytical models as well as more general internal

checks on the code's ability to evolve shocks to self-consistent steady state MHD CRstructures. Our simulations confirm Webb's calculations. Our numerical models also allow

3

us to extend Welj)'s calculationsby lifting the cold gas. (Pg = 0). restriction to exanainethe effectsof a full rangeof initial lmrameters on CR-modified MHD shock structures.

2. Methods

\Ve soh'e the equations of ideal MHD for one dimensionM flow in Cartesian coordinates

(e.g., Jeffrey 196S). As with two-fluid gas dynamic models the conservation equations are

modified to include momentum and energy source terms fi'om CR feedback (e.g., Drury &

V61k, 1981. Jones & Kang. 1990), The XI'HD equations are written in conservative, vectorf()l'ln as

0q OF

0-7 + 0.:--7 = s. (2.])

wit h

| pu,r

l P"!J (2.2n:/;,\Ej

_t11(1

F

I)ll .F

I,,'i_.+ P* - BI{t)<t't_g - B.rBypu.r,: - B.r B:

B.9,.r- B.ragB: .,r- B.r, z

(E + 1)" )".r - B.r(B.r ,.r + Bgug + Bzu:)

while the CR source terln vector is

(2.3

0

o PH O.,.0000

.,cOPc/O.r + Se

(2.4

The CR are then_seh'es treated as a massless, diffusive fluid through a conservation

equation for the CR energy. Ec. derived from the diffusion-advection equation (Skilling

1975 ); namely.

o.aE_. &,.,.Ec a a-OT,r _ pc_._.r + Se. 2.4)0-7+ o.,. - aL

E = T_p(_,_, + + )+

In these relations

1 2 B 21 _P+ _(B. + B_+ :/,(i,j - I)

2.5)

2.6)

4

Pc- = (3c - 1)Ec. (2.7)

and the magnetic field ccm_ponents are expressed in rationalized units

B

In the expressions presented above the following definitions hold: p is the mass density;

u_:, Bx and u/,,. u:, By, B- are the components of velocity and magnetic field parallel

and perpendicular to shock front: P!j, _9 and Pc, _c are the gas and cosmic-ray pressures

and adiabatic indices, and _ is the energy weighted spatial diffusion coefl:icient for the CR

parallel to the shock normal. The quantity S, is a term that allows direct energy transfer

flOln the gas to the CR. such as through the low energy injection of thermal particles into

the CR popuhttion (e.g.. Jones &: Kang 1990). This is introduced for completeness, but

for the present, we set Se = 0. In this discussion we set 3'q = .5/3, while 7c, which depends

upon the mix of nonrelativistic and relativistic particles in the CR population, will be

treated as an input lmrameter. In _eneral l_oth 3 c and t," are properties of the solution, so

the need tc_ specify their properties a. priori is the lnajor drawback of the two-fluid model.

In tht _ interest of simplicity we will nc,t consider here flows with tangential magnetic

field r()tations, although the mtmerical code is quite capable of handling such features.

Thus. withcmt fluther loss (_i'generality we can place the magnetic field in the X-Z plane,

= (B.,.. 0. b':). We will alsc_ restrict oursehes t.o flows with no upstream tangential

v_,h_city. _l.,) = t,: = 0. All of the simulations discussed here are essentially piston driven

shock tubes. \Vc establish the fl(,ws 1)3 projecting magnetized fluid with eml)edded CR

ill ft()lll the right b(nuxdar3", usin_ _tll cq)en ])(mndarv condition and reflecting it off a

wall (pist(,n) at the left boundary. The tangential magnetic field at the left boundary is

"'mirrored" in tim same manner as the gas density. Previous simulations of CR modified

shocks have shown that their time to evoh'e to dynamical equilibrium scales with the so-

called diffusion time. t d. (e.g. Jones & bS,ng 1990), which in the present case is conveniently

,'xpressed ash"

td = --=7. (2.9)

where u., is the sh_,ck speed (see equati,m. 3.2). In our discussion below we will express

_inmlation times in units c_f Id. approl_riate to that simulation. The width of a CR-modified

shock transition scales with the related ditt'usicm length scale..r d,

•%1 -- -- tdU,s. (2.10)

To obtain accurate nume,'ical results with the methods we employ it is important that

c(mlputation zones be small enough to resolve the diffusive shock features on this scale.

For our discussion we define, therefine, the resolution ratio of each simulation to be,

• (2.11)t} r _--- ..__,_ '

Our numerical method soh'es equation [2.1] through the second order finite difference

'Total Variation Diminishing" (T\'D) gas dynamics method of Harten (Harten 1981)

extended to MHD (Ryu & Jones 1993). The pure MHD (S = 0) form of the equationis solve with the aid of an approximate Riemann solver, usedto estimate the fluxes, F.CR sourcecorrections, S. are then added in a manner preservingsecondorder accuracy.Shocksand other discontinuities aregenerally resoh'edwithin a few zones.The CR energyequation is solvedusinga secondorder combinedmonotoneadvectionand Crank-Nicholsonscheme.Further details of the method will be presentedelsewhere(Frank, Jones& Ryu, inpreparation). A ptu'egasdynamical versionof the code was tested against both analytical

steady state solutions and numerical time dependent models calculated with our well-

tested PPM code (Jones & I,:ang 1990). with excellent agreement. The pure MHD code

was tested against a variety of 1-D shock tube problems involving all three families of

MHD waves using a nonlinear MHD Riemann soh'er (R.yu _ Jones 1993).

3. Results: Comparisons with Analytical Models

In order to test the accuracy of our numerical method we first attempted to reproduce

the analytical steady state solutions _f \\'ebb (1983). In that paper _:ebb demonstrated

in addition to discontinuous gas 'sub-shock'" solutions with a smooth CR shock precursor,

that one may obtain completely sm<)oth "'shock" solutions to the MHD CR equations if

the downstream velocity remains super-Alfwhfic and the upstream CR pressure, Pc, is

high. That behavior is analogous t() smooth gas dynamic shock solutions identified earlier

by Drurv a: \'51k (19S1). However. XVebb was able to consider only flows in which the

upstream gas was c_hh i.e.. in which P_I = 0.

In figures 1 _nd 2 xxc present tl_o rim_,-asXml)t_,tic shock structures formed in numerical

simula t ions with upstream ('<mdit i_his c_rresp_,nding t o t hose in Webb's pai)er ( his figure 7

and figure 8). The upst ream conditi(ms for these simulations are given, as models 1 and 2, in

table 1. These simulations were performed with a constant CR diffusion coefficient, _c = .01.

The resolution rati(_s, n ,.. for models 1 and 2 are t_,. = 21 and 32 respectively. The Alfvdnic

Xlach numbers for models 1 and 2 are ]I,, = upv/-fi/Ba. = 1 and 2 respectively where up isthe piston speed. In these models 7 c = 4/3. The simulations where cm'ried out until the

p_stshc_ck state appeared steady: namely, t --- 30t,l. The resultant shock transformations

provide excellent agreement with ore" best estimates of the downstream states found by

\\:ebb. The largest uncertainties in the comparison are. in fact, the determination of the

downstream states from Webb's figures. For model 2 the entropy, s = log(Pg/p'_g), (not

shown in the figure) increases though the shock, demonstrating that the flow contains an

MHD fast mode sub-shock, a.s predicted by \Vebb. The entropy for the flow in figure 1

shows no increase, again as predicted. Recall that these particular calculations were meant

to be carried out under Webb's cold gas (P_ ,_ 0) restriction. For numerical reasons the

upstream gas pressure was set in practice to be a small fraction (10-a) of the CR pressure.

As an additional comparison we have also reproduced the pure perpendicular (B x = 0)

smooth and MHD sub-shock models of \Vehb with the accuracy comparable to that shown

in figures i and 2.

Our previous sinmlations of CR-m(_dified ogas shocks required numerical grids that over

resolved the CR shock precursor bv roughly a factor of 10 to assure high accuracy (Jones &

Kang 1990), Ill order to explore the dependenceof the accuracyon numerical resolution fortheseMHD simulations wehaverun a seriesof testswith the upstreamconditions of model

2. varying the nulnerical resolution. We ran sinmlations with n r = 4, 8, 16, 32. In figure 3

we plot a measure of the fl'actional error in the downstream CR pressure, (compared with

the value fi:_r the highest resolution case. I__. = 32),

ec = Pc(_,') - Pc(32)

Pc(32). (3.1)

The Figure shows that the simulated shocks converge quickly once 1_r > 10. As mentioned

above, that state is in good agreement with \Vel)b's analytical result. The converged

nmnerical 1)ostshocl< C12 pressure that is about 4_X. higher than our estimate of Webb's

result, although we attribute much of this error to uncertainties in reading final states from

l)ul)lished figures rather than tables. These results agree well with previously mentioned

gas dynamic behaviors found by Jones X: I,[ang (1990). \Ve believe the limiting factor

t() be th( _ accuracy of the Crank-Nich()Ison method used for updating the diffusive CR

ol2ergy eq_lation. As a f_lrther test of tlle lmmerical method u'e have performed tests of the

self-consistency of more general steady state .MHD CR shock solutions. This was done by

testing the a('cm'acv _f v, ri_ms .'xlHD julnp conditions fi)r apparently steady shocks in the

shocl¢ frame. For example, the various momentunl components of tile full flux vector, F,

in equati(nl [2.3] sh_)uld be the same acr_ss the shock when measured in that frame (with

F_ corrected t(_ include Pc.). The shock velocity for this exercise was determined from

the conservation of mass equation for a steady shock transformed into the piston frame;

namely.

[/'"! {3.2)[/,]"

where [ ] refers to diti'('rcnces acro._s the shock. \\'e find that for the sinmlations with

,,. > 10 the jmn l) c<nl(liti(ms were satisfi_,d to better than one part in 1 × 10 4.

4. Finite Gas Temperature MHD Shocks

Since (mr numerical code solves the full _'xIHD CR two-fluid equations there is no need

t(_ restrict inve._tigations to th_)se cases where P¢I = 0. In figure 4 we present the results

fl'om a simulation of all *IHD CR shock formed from gas of finite upstream pressure, Pg"

The upstream conditions for this simulation, model 4, is presented in table 1. We note

that the sonic Mach nulnber in this case is 3Is = 4. The Alfvbnic Mach number Ma = 12.

Standard weak scattering models of particle diffusion lead to differences in diffusion across

and along field lines. Thus _,', which controls diffusion along the shock normal should

depend on the angle between field and tile shock normal, ¢ = tan-l(_e_.). Thus we adopt

_' of a form, (e.g., \\'ebl) 1983, Jokipii 1987, Zank el. M. 1990),

h = h'll c°s2 o + t,'± sin 2 o, (4.1)

where the directions II and .1_refer t_, tile magnetic field direction. Note, of course, that ¢ is

generally a changing function of space and time. For this initial exploration we arbitrarily

set. _ll = .01 = 10 × ,_±.

7

In general Pc develops more rapidly for a stiffer CR equation of state (Jones & Kang

1990). Thus. in order to keel) the colnl)utational costs low for these frst tests we used

",c = 5/3 in the following model. This wouhl influence the detailed properties of the steady

state flc,w. ])ut will not aher the qualitative character in ways that are important to the

preserlt discussion.

In figure 4 we illustrate the evolution of a time dependent, finite gas pressure simulation.

In this model the upstream magnetic field angle is o0 = 30 °. We present results at three

different times: t = 12, 24. and 36_' d. Since h" is not a constant in space or time in this

simulation, we define t d here in terms of _, = 0.01. Figure 4 shows a strong fast mode

shock driven by the piston. That shock has become ahnost smoothed by the CR pressure.

Comparisons of this model with an analogous parallel field (O = 0 o) simulations show a

number important differences. First. the parallel shock model reaches a dynamical steady

state more slowh" than the oblique she,ok case. That is simply because, according to

equaticm [4.1] the diffusion cc)cfficient. _. in the parallel case is greater, so that energy

gain 1)v the CR is d,mcd ("-v,.- .h,kipii 19ST). ()n the other hand. while the dynamical

steady state may lw rcm'hed m_,r_' quickly in _,blique shocks than in parallel shocks the

eflbctivencss (,f the accelm'aticm in th,' (,1)lique ('as(: is reduced. The downstream value of

P_. in the cA)lique sh_ck case is decreas('d 1)v 8t7_.fr_)m what is obtained in the parallel shock

model. It is reasonal)le to ,expect that in the oblique models the upstreanl momentum flux

which would have gone into accelerating CR is being used to do work on the tangential

magnetic field. We note that behind the fast mode shock a weaker slow mode shock

co,represses the CR driven transiticm density spike (see Jones S,: 1,2ang 1990 for a discussion

_,f thi_ feature). Close examinati<m also, shows that CR particle acceleration is taking place

as that sh,w re(Me sh(,('l_ devdol,._. Xlo(lificati(,n (,f tit(" density spike, which can only be

seen in time del)Vn(l_-'nt models, is mt example the additional complications which arise due

t_, the multiple wave families present in MHD.

•_5. Conclusions

1) The lmlnerical code we have developed accurately computes two-fluid models ofCR-modified XIHD shocks. If the resc_luticm ratio defined in the text, nr > 10, then the

tilne asymptotic properties of the simulations appear to converge to analytically predicted

steady states. The time asymlm)tic lmmcrical sh(,cks are also internally consistent in terms

c,f c_mservation laws expected t_ I_e satisfi_,d acr_,ss steady shocks to at least one part in

i04.

2) Because of the work done on tangential fields within the shocks, time asymptotic

particle acceleration will tend to be more efficient in parallel shocks than in oblique

shocks. H_,wever, the oblique shocks reach dynamical steady states more quickly for a

given diffusion coefficient parallel to the magnetic field.

3) Tl'ansient features deveh,p flom the MHD fast mode CR shocks similar to those seen

in pure CR hydrodynamical shocks. Hcm'ever. these transients are modified and made more

Colnplex by the developllxent of MHD shin' m(_de shocks.

Acknowledgments

This w<_rkwas supported in part by NASA through grant NAGW-2548, the NSFthr()u_h ,arantAST-9100486and l_vtile University of MinnesotaSupercomputerInstitute.

References

Achterberg. A., Blandfl:_rd,R.D. & Periwal, V. 1984,A&A, 98, 195.

Baring. M. G.. Ellison. D. C.. & .h_ne.-s. F. C. 1993. ApJ, in press.

Blandford. R.D. & Eichler. D. 19S7. Phys Rept, 154, 1.

Drury. L. O'C.. & V61k. H. J. 19S1. ApJ. 24S. 344.

Drury, L. O'C.. & Falle, S. A. E. G. 19S6..XINRAS. 223, 3,53.

Falle. S. A. E. G.. & Giddings .1. i2. 19S7..XINRAS. 22.5,399.

Hat'ten, A. 19S3. J. Comp. Phy>.. 49.3.57.

.Ieffrev, A. 1966. Map,,et, oh.!ldr,_d_/7_,m_,_ (L_>ndon: Oliver and Boyd).

J_kipii. 3. 12. 19S7. Ap]. 313. 546.

.]()IDES. r. \\. _s [,_.allg. H. 1990. AI).J. 363. 499.

1,2ang. H.. & .J<)ne._. T. \V. 1990. Ap.J. 3.53. 149.

I;ang. H.. & .l_>IleS. T. \V. 1991..XI.XRAS. 249. 439.

1,2ang, H...]ones. T. \V.. & Rvu. D. 1992, ApJ. 3S.5, 193.

Kennel. C. F.. Edmiston..I.P. _: Hada, T. 19S.5. in Colli,_ionles_ Shocks in the IIelio_phere:

A T_Ltorial Review. ed. R,_l)ert G. St_nte & Bruce Tsurutani (American Geophysical

Union: \\'ashington. D. C.). p.1.

R)tt. D.. & Jones. T. \\'. 1993. ApJ. to l)e submitted.

Skilling, J. 197.5. _X[NRAS. 172..557.

\\el_b. G..",I. 19S3, A&A. 121.9?.

Webb. G. _I.. Drury. L. O'C. & \'i;lk H. J. 19S6. A&A. 160, 335.

9

Zank, G.P., Axford, \V. I., & ._IcI(enzic,J. F. 1990,A_A, 233,275.

Figure Captions

Fig. 1.-- Model 1 in table 1..NIHD CII shock transition region for a piston driven shockwith upstream condition._taken t'r<mlfigure 7 of \Vel)13(19S3). Shownare the density,p. n()rnial c()nll)Oiicnt (if veh_city, u,.. tangential velocity, u z, tangential component of

niagrietic field. B:. nmgnetic fichl _wientation angle, O = tal_-l(_), and cosmic-ray

pressure. Pc. S('(' tabh' 1 for upstreani ttc)w conditions. The abscissa is given in units of

ditt\tsion length .r d = h/u.,. The dashed lines are post-shock values taken from Vv'ebb's

figlu'e 7. except in the ph)t <_f ,-: where the value was calculated using equation [3.2]

and th( • MHD steady state junll) conditions.

Fig. "2. .XIodel 2 from table 1. .N[HD CR Shock transition region for a piston driven

sli()ck with Ul).strealn conditi(ms tld_eu fr<nn \Vcl)l)'s figure 8.

Fig. 3. - Fracti_mal ovr_,r (eqm_ti_m [3.1] in the c()mputed value of the post-shock CtR

1)res_lu'cL Pc. ill m()del 1 as a fum'ti(m ()f resolution, _ _. = ,rd/_,v.

Fig. 4.- Model 3 from table 1. This mc)dcl has an upstream field orientation, _ o = 30 °.

1Results are sh()wn at t = 12, 24 and 36t d.

Table 1

Upstream Conditions for Models

Model -9-- Up Bx__ B___z__ Eg_ P_

1 1 .2 .32 .45 0.0 .25

2 1 .2 .32 .32 0.0 .075

3 1 .2 .05 .03 1.49x10 -3 1.49x10 -3

__K

.01

.01

_c(B)

Transition

smooth

sub-shock

smooth

...... " - " - • " " - • - - - I' " " "

0d

0

0 x

u')I

0

0 c_ _ cO CO 0 10 0 0 0 0 ,-

o o d d d dt I I I I I

I n

, - . _ . . .

d d d d d d

0

x

c_

_d

, I

d dI

I n

0

0 X

I

0

dI

0

I' .... " - " '

li.... ||" J,,,,I

0 Lr)

0

0 x

I

G

0

• • • . I i i i I . . .

d

0st..-.

0 X

I

0

00 1

0

0

u_

- 0 x

I

, i11 ........ 0

0 0 0 0 0 I

o d o d d

'8

I

I

I

I

I

10 _ 0 Ln 0 I0 0 -- -- C_l

o c5 d 0 dI I I t

0 X

I n

0

d

1' T . " ' - - " " " • " / 1 _

JI

I

I

I

II. , ........ , C

d d o o o

I

"d

0

0 X

I

0

0 c_l _ _i0 cO I

o o d d oI I I I

I n

0 0

0

0 X

o, , | , .... A .....

-| - . ,

u_

c_

0

0..1 '-||,1,,,,

d

°xII

0_i_ Ill|l,,

o d d

'8

0

i°I0

c_ I

0

0.2

0.0

-0.8 , , , I , , ,

0 20 _,0

n r

r7_

.... , .... , .... , ........

0

0

0

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_ .5 un u5 .5 _I I I I I I

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0

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d

o00

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0

0

X

........ t .... , .... , .... , ....

0

0

0

... 0

_0_0_0_ 0000_ _0000000

ooodoooI KIil

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C"4.... , .... , .... , .... , ....

0

oO

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X

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0

c'q

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u") 0 urb 0 u_ 0 00 0 0 _ _ oq

d c5 c5 c5 c5 oI I I I

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0

0

oq

0

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0 0 0 0 0 0 0,_ t,O o,,I _ 0

0 0 0 0 0 0

c5 c5 o c5 do

_d

0

0

_ o

0

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0

O0 0 0 0 O0 0

I I I I I I I

6c-i

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jl

0 0co co

0

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LD

0

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L"'q

0

O0

0

X

X

X

,3"--