NASA-CR-192930 Modified Shocks: T. Abstract€¦ · steady state solutions and numerical time...
Transcript of NASA-CR-192930 Modified Shocks: T. Abstract€¦ · steady state solutions and numerical time...
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.
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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.
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9
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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
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