sebastien leygnac et al- modeling multidimensional effects in the propagation of radiative shocks

8/3/2019 Sebastien Leygnac et al- Modeling multidimensional effects in the propagation of radiative shocks

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arXiv:as

tro-ph/0610876v2

1Dec2006

Physics of Plasmas, 13, 113301 (2006).

Modeling multidimensional effects in the propagation of radiative shocks

Sebastien Leygnac

School of Cosmic Physics, Dublin Institute for Advanced Studies, Ireland and

Laboratoire de lUnivers et ses Theories, Observatoire de Paris, France

Laurent BoireauLaboratoire de lUnivers et ses Theories, Observatoire de Paris, France and

Departement de Physique Theorique et Appliquee, Commissariat a lEnergie Atomique, Bruyeres-Le-Chatel, France

Claire MichautLaboratoire de lUnivers et ses Theories, Observatoire de Paris, France

Thierry LanzDepartment of Astronomy, University of Maryland, USA and

Laboratoire de lUnivers et ses Theories, Observatoire de Paris, France

Chantal StehleLaboratoire de lUnivers et ses Theories, Observatoire de Paris, France

Christine Clique and Serge BouquetDepartement de Physique Theorique et Appliquee, Commissariat a lEnergie Atomique, Bruyeres-Le-Chatel, France

(Dated: July 20, 2007)

Radiative shocks (also called supercritical shocks) are high Mach number shock waves that pho-toionize the medium ahead of the shock front and give rise to a radiative precursor. They aregenerated in the laboratory using high-energy or high-power lasers and are frequently present ina wide range of astronomical objects. Their modelisation in one dimension has been the subjectof numerous studies, but generalization to three dimensions is not straightforward. We calculateanalyticaly the absorption of radiation in a grey uniform cylinder and show how it decreases withR, the product of the opacity and of the cylinder radius R. Simple formulas, whose validityrange increases when R diminishes, are derived for the radiation field on the axis of symmetry.Numerical calculations in three dimensions of the radiative energy density, flux and pressure createdby a stationary shock wave show how the radiation decreases whith R. Finally, the bidimensionalstructures of both the precursor and the radiation field are calculated with time-dependent ra-diation hydrodynamics numerical simulations and the influence of two-dimensional effects on theelectron density, the temperature, the shock velocity and the shock geometry are exhibited. These

simulations show how the radiative precursor shortens, cools and slows down when R is decreased.

Keywords: radiati on hydrodynamics, astrophysics, laser experiments, numerical calculations

I. INTRODUCTION

In many astrophysical systems, the effects of radia-tion on hydrodynamics are strong. This is the casewith fast outflows and shocks, such as those encoun-tered in jets, bow shocks produced by the interaction of

jets with the surrounding interstellar medium,1 or radia-tive shocks arising in the envelope of pulsating evolvedstars.2 Nowadays, these processes can be observed withan increasing level of details. For example, high angu-lar resolution imaging of jets produced by Young Stel-lar Objects1 shows complex structures of bright knotsand shocks inside the jets. The improvement of obser-vation techniques, and more especially the combinationof spectroscopy and high angular resolution techniques,will allow the study of the physical properties of shocks,in addition to the study of their complex shapes. Thiscombination will be achieved by AMBER3 on the VeryLarge Telescope Interferometer for instance.

However, the interpretation of these new data shouldgo together with the improvement and the developmentof new models or theories and new sophisticated nu-merical codes. Since these codes are complex, extensivetesting made through code inter-comparisons and com-parisons with laboratory experiments are very useful.4

Therefore, code benchmarking can be viewed as one mo-tivation for pursuing laboratory high-energy density ex-periments. These experiments can also be considered as

the most relevant approach to check assumptions and toprovide hints and new ideas to address open questions orissues such as radiative shocks in astrophysics.5

High-energy density laboratory astrophysics(HEDLA) experiments are mostly driven on large-scale lasers6,7,8,9,10,11 or on Z-pinches.12,13 In a firstkind of experiment, one measures microscopic quantitiesrequired for the determination of the equation of stateand the opacities of hot and dense matter found ingiant planet interiors or in stellar atmospheres. In
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a second kind of experiment, one seeks to reproduceastrophysical phenomena in the laboratory. The size ofthe experimental targets is of the order of a millime-ter, implying typical durations of a few nanoseconds,whereas the scale of astrophysical phenomena arebetween 15 and 25 orders of magnitude larger. However,scaling laws help to bridge laboratory experiments andastronomical phenomena.14,15 In order to be relevant

for benchmarking, the appropriate energy requirementsneed to be satisfied and, more importantly, a set ofaccurate time and space-resolved diagnostics must beprovided. Many experiments have been devoted to thestudy of typical radiative hydrodynamical situations,such as (i) radiative blast waves in the context ofsupernovae remnants,16,17,18 (ii) radiative precursorshocks waves,9,10,19,20 with applications to the studiesof stellar jets, pulsating stars,21,22 and accretion shockduring star formation,23 and (iii) radiatively collapsing

jets relevant for protostellar outflows.8,13 Reviews canbe found in Refs. 24,25,26.

In this paper, we shall concentrate on the modeling

of radiative-precursor shock wave experiments performedwith high-power lasers. By radiative-precursor shockwaves, we mean highly hypersonic shocks that are drivencontinuously by a piston. Because of the high shock ve-locity, the shocked medium is ionized and emits radia-tion, which in turn ionizes and heats the cold unshockedgas and leads to the apparition of a radiative precur-sor. The structure of the shock depends on the shockvelocity.27 For a given shock velocity, the temperatureof the shocked matter and its mass density are solutionsof generalized Rankine-Hugoniot equations.28 For weaklyhypersonic shocks, the mass density varies continuouslythroughout the shock. At higher shock velocities, it be-comes discontinuous and then again continuous when thevelocity increases because of the contribution of the ra-diation pressure and energy density.27,29 In the so-calledsupercritical regime, the mass density varies discontinu-ously, but the temperature remains constant through theshock30 except in the shock front where the temperaturespikes. The extent of this region is of the order of thephoton mean free path.

The strong coupling between radiation and hydrody-namics in these shocks is difficult to model because of thestrong gradients and different length scales involved forhydrodynamics and radiation transport. Additionally,an accurate set of opacities and equations of state for awide range of plasma conditions is needed. Departuresfrom local thermodynamical equilibrium (LTE) are alsoexpected to be important, especially in the shock front,requiring detailed calculations of the monochromatic ra-diation intensities. A complete and detailed study of theshock structure can therefore be achieved currently onlyin a restricted, 1D geometry. However, the finite radialsize R (in the direction perpendicular to the shock prop-agation) of actual shocks in the cosmos and in laboratoryexperiments may introduce departures from the ideal 1Dbehavior. The purpose of this paper is to examine the

Vshock

R = 0.5 mmt

beamslaser 2 mmz Xe

piston shock front ionisation front

V

target

prec R = 0.25 mm

x

precursor

h

FIG. 1: (Color online) Propagation of the shock (radius R)in the cell (radius Rt) filled with Xe. The laser beams areshown for illustrative purpose: during the experiment, thelaser pulse is ended when the shock propagates in the cell.

importance of these 3D geometrical effects.

In a 1D description, the radial extension R of the re-

gion in the shock that emits radiation is considered tobe infinite. The amount of radiation absorbed at onepoint of the configuration (in the precursor for example)is therefore overestimated by comparison with a descrip-tion where R is finite. This geometrical effect results ina radiative energy loss because a fraction of the energyis radiated radially and does not heat the radiative pre-cursor ahead of the shock. This effect will be dominantwhen the photon mean free path becomes large comparedto R and may have a strong impact on the developmentof the radiative precursor and on the shape of the shock.

Deficiencies in the current modeling of radiative shockexperiments provide the motivation for a detailed investi-

gation of the importance of these lateral radiative losses.For instance, Bouquet et al.10 have recently reported onsupercritical radiative shocks created with a high-powerlaser and the associated modeling work. While hydrody-namical simulations reproduce the main features of theexperiment, questions remain regarding the precise un-derstanding of the formation and of the structure of theradiative precursor.

In this paper, we report a study of the importance ofradiative losses on the structure of shocks produced inlaboratory experiments, aiming at understanding the de-ficiencies of 1D models. Section II briefly recapitulatesthe results of Bouquet et al.10 and some of these deficien-cies. An analytical estimation of the effects of the finitesize on the radiation distribution is given in Sec. IIIA.The numerical calculation of the spatial structure of theradiation field in a stationary case without coupling tothe fluid is described in Sec. IIIB. A full description ofthe radiative shock structure and of the radiation field isstudied using the 2D Lagrangian radiative hydrodynam-ics code FCI,31 with various boundary conditions for theradiation (Sec. IV). We conclude (Sec. V) that it is essen-tial to account for multidimensional effects in modelingradiative shocks with finite radial extension.


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II. RADIATIVE SHOCK EXPERIMENTS

Several shock experiments have been performed withhigh energy density lasers to generate supercriticalshocks.9,10,19,20,32 In such experiments the laser energyis converted into kinetic energy of a piston, which drivesa shock in a millimetric target of radius Rt (see Fig. 1)filled with gas or low-density material.

The extension R (shock radius) of the experimentalshock in the direction perpendicular to its propagationalong the x axis (see Fig. 1) is given by the size of thelaser spot focused on the piston. The diameter 2R of thefocal spot is equal to 500 m [full width at half-maximum(FWHM)] with a 250 m flat center where the intensity isconstant. The shock has the same cylindrical symmetrythan the laser focal spot.

The target is impacted by the laser beams on one of itstwo ends consisting in a piston composed of several layers.The first layer is ablated during the 0.8 ns duration ofthe laser pulse. This ablation produces a shock wavethat propagates in the x direction. The second layer is

a radiative shield in titanium that prevents the radiationof the laser or the radiation emitted during the heatingof the piston by the laser from penetrating the xenon gas.The shock finally breaks out in the tube filled with xenongas at 0.1 or 0.2 bar. The shock velocities achieved in thegas are in the range 50 - 100 km s1 and the duration ofthe experiment is 10 ns.

The spatially averaged temperature of the shock ismeasured33 to be 10 - 20 eV and the electron density34

in the precursor is between 1018 and 1020 cm3. Theprecursor propagates ahead of the shock front with a ve-locity that equals up to twice the velocity of the shockfront.

The preparation and the initial analysis of the experi-ments were performed using one dimensional (1D) radi-ation hydrodynamics codes (FCI,31 MULTI35,36). Whilethe experimental shocks deviate from 1D geometry, thesecodes give an overall good agreement with the experi-ments (shock velocity, temperature of the shocked ma-terial). However, they are only moderately successfulregarding the radiative precusor. They fail to reproducethe shape of the electron density profile in the precursoron the whole interval of values accessible to measure-ment (from 1018 to 1020 cm3 in Ref. 10). The differentcodes provide a range of predicted velocities for the pre-cursor. For example, for a particular shot, the velocityat ne = 3 1019 cm3 is 300 km s1 in the 1D version ofFCI and 120 km s1 in MULTI, the last one being inaccordance with the measured velocity. It was reportedby Vinci et al.33 that the time evolution of the radialextension of the shock was well reproduced by the 2Dversion of FCI, therefore showing that 2D effects have tobe considered.

In the work presented here, we estimate the 2D effectsand the consequences of the finite lateral size R of theshock on the radiation field. When investigating the ge-ometry of the problem, one needs actually to consider

r

x

Io

0

h n

min

d

R

R

S = Cte

S = 0

LM(x)

disk

FIG. 2: (Color online) Geometry of the configuration for theanalytical model. The disk is located at x = 0. The limitangle min = cos

1 min is the angle under which the disk isseen from the point M(x).

the optical depth, , which is the geometrical distancenormalized by the photon mean free path. The opac-ity of the medium is therefore an essential parameter ofthe problem. Considering the overall agreement of the

1D codes with the experimental results, we may assumethat the current opacities are a reasonable approxima-tion. As a consequence, we will not examine that specificpoint further.

III. RADIATION FIELD EMITTED BY A

SHOCK

A. Analytical estimation of the multidimensional

effects

We start with an analytical study of the radiation field

generated by a disk in a grey and homogeneous cylinderwith radius R and length L (Fig. 2). We furthermore con-sider that the opacity (cm1, the inverse of the photonmean free path) is uniform (independent of space) andwe neglect scattering for sake of simplicity. The opti-cal depth in the x direction (x) =

x0

dx, is the po-sition along the x axis normalized by the opacity. Thecylinder length L is such that L = 100. The disk radi-ates an isotropic specific intensity Io (J m

2 s1 sr1)and is characterized by its radius R or by its lateraloptical depth R = R. The local source function S(J m2 s1 sr1) is uniform in the cylinder and muchlower than the specific intensity Io. It is zero outside ofthe cylinder.

In this approach, we would like to calculate the fractionof radiation energy emitted by the disk and received atany point M(x) of the cylinder axis x. In addition, it isaimed to examine the way this amount of energy variesby changing the disk radius. A brief presentation of thiscalculation has been presented in Ref. 37 and an extensiveversion can be found in Ref. 20. We will use the threeEddington variables: the mean intensity J, the flux H,and the tensor K analogous to the pressure tensor. Theyare, respectively, related to the radiation energy density


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E, the radiative flux F, and the radiative pressure tensorP through the relations (see Ref. 38): JHK

(r) = 1

4

cEF

cP

(r) = 1

4

1nnn

I(r,n) d,

(1)where c is the speed of light in vacuum. In these expres-

sions, the integration is performed over the solid angle ,and I(r,n) is the specific intensity at position M(r) inthe direction given by the unit direction vector n. Equa-tions (1) are frequency-dependent, but since we assumethat the opacity is grey, we will consider that the radia-tive quantities appearing in Eqs. (1) are integrated overfrequencies. We will refer to J, H and K as the radia-tive moments since they correspond respectively to thezeroth-, first- and second-order moments of the intensityover angles.

The formal solution38 of the radiation transfer equa-tion leads us to decompose the radiation field at positionM(r) in two components: the contribution Jo, Ho andKo coming from the disk, and the emission JS, HS andKS by the uniform medium. The total radiation field is

JHK

(r) =

JoHoKo

+

JSHSKS

. (2)

In this simple model, it is straightforward to calculatethe radiation field existing at point M() on the x axisof the cylinder (r = 0). The contributions of the sourcefunction, JS , HxS and KxxS , are calculated by summingthe emission coming from all the points of the cylinder.The contribution of the disk is given by

JoHo xKo xx

(, r = 0) = Io2

1min

12

e/d, (3)

where we have considered only the x component of theflux and the xx component of the tensor Ko. The quan-tity is defined by = cos , where is the angle be-tween the direction n and the x axis. The integrals givenby Eqs. (3) are exponential integrals in the case min = 0corresponding to a disk with a radius R going to infinity.For a finite radius of the disk, the specific intensity is in-tegrated over the solid angle under which we see thedisk from point M() (see Fig. 2).

The total value J() = Jo() + JS() (and similarlyfor H and K) is plotted in Fig. 3. Because the generalbehavior of J, Hx and Kxx is similar, we can restrictour description mostly to the mean intensity J. Thedecrease of the radiative moments with R enables us tounderstand the effect of the finite radial extension R ofthe medium on the structure of the radiation field.

Far from the disk, the radiation emitted by the diskhas been considerably absorbed, and the moments de-pend only on the local source function S. In the calcula-tions shown in Fig. 3, the ratio S/Io = 10

4. This value

1

101

102

103

104

105

106

J(,r=0)/Io

1

1/10

R = 1/100

eq(1)eq(1/100)

(a)

1

101

102

103

104

105

106

Hx

(,r=

0)/Io

1

1/10

R = 1/100

(b)

1

101

102

103

104

105

106

107

0.001 0.01 0.1 1 10

Kxx

(,r

=0)/Io

1

1/10

R = 1/100

(c)

FIG. 3: (Color online) Radiative moments (a) J/Io, (b) Hx/Ioand (c) Kxx/Io produced by a disk in a grey and homogeneousmedium with S = 104Io. The exact solution is shown for

the infinite case (plain line with crosses) and for R

= 1, 1/10and 1/100 (stars, triangles and big dots). Also shown arethe approximate solutions given by Eqs. (5) (dashed lines)and Eqs. (5) times min (plain curves), which are given byEqs. (6) for J and Hx. Positions of eq(R) are shown onframe (a) for R = 1 and 1/100.

corresponds to a system in which the radiation is dueto Planck emission and the temperature of the mediumis 10 times lower than the disk temperature. The flux


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Hx() goes to zero when is greater than 1. An orderof magnitude of JS can be estimated by assuming themedium is a sphere of radius R,

JS( 1) 1

2

11

d

R0

d S e S

1 eR

.

(4)Though a rough estimate, it is quite correct in the cases

considered here. For small R, the expansion of eR

gives JS( 1) S R, an evaluation quite close to theexact value [see in Fig. 3 the value of J( > 10) JS SR].

An interesting feature of the decrease of J with R canbe illustrated with Fig. 3: for R smaller than one, ittakes a distance of a few R for J to be absorbed by afactor of 10, whereas, in the infinite case, where R isgreater than one, the radiation is absorbed tenfold aftera propagation of a few mean free paths of photons, i.e.at 1 2. The absorption of the radiation field istherefore determined by the smallest of the characteristicscales. For the system studied here, the smallest scale iseither the mean free path of photons 1/ or R.

In the case R = 1 for which the two scales are thesame, Jo is not far from the mean intensity J0 calcu-lated in the infinite case. At position = 2, Jo is only afactor of four lower than J0. Therefore, a characteristiclateral optical depth R of a few is sufficient to considerthe medium as optically thick and consequently nearlyidentical to a medium with an infinite lateral dimension.

The other two moments for the disk radiation H0 xand Koxx have similar profiles, and the three momentsJo, H0x and Ko xx have nearly the same value sufficientlyfar away from the disk, where it is possible to have anapproximate solution of Eqs. (3). At a distance from thedisk of a few times R, the solid angle is small and

min 1 2/2 1 (R/)2/2. The limiting valuemin is close to unity as soon as is larger than 3 R or4 R and asymptotically goes to unity.

The limits of the integrals in Eqs. (3) are JoHo x

Ko xx

(, r = 0)

min1

I04

R

2e/min

1min

2min

.

(5)This asymptotic expression (represented in Fig. 3) is

useful to approximate the exact Eqs. (3) over a range[0, eq] and to describe more simply the absorption ofthe radiative moments. Let us first estimate the lowervalue 0. At position = 5R, the error on Jo definedas the relative difference (Japproxo Jexacto )/Jexacto is, re-spectively, 6%, 0.6% and 0.06% for, respectively, R = 1,0.1 and 0.01. We see on Fig. 3 that 0 is of the orderof a few R. Therefore, the approximation stands onlyfar enough from the disk, where the mean intensity hasbeen greatly reduced by the absorption. For example,Jo( = 5R)/Jo(0) 1/100 and 1/50 for respectivelyR = 0.1 and 0.01.

Let us define eq(R, Io/S) as the optical depth forwhich the disk radiation is equal to the local radiation.

This is a straightforward way to separate the medium in aregion < eq where the radiation field coming from thedisk dominates and the region > eq where the diskdoes not have a direct influence on the radiation field.The transition between the two regions is sharp becausethe radiation field emitted by the disk decreases expo-nentially around eq. This eq can be easily calculatednumerically. Its approximate position is shown Fig. 3.a

for R = 1 and 1/100. We find that eq decreases slowerthan R since the local radiation JS decreases like Rwhen R is small [see Eq. (4)], whereas the disk radi-ation decreases like (R/)

2 exp(). We see in Fig. 3that eq 5 for R = 1 and eq 2 for R = 1/100.Therefore, eq decreases by less than a factor 5 when Rdecreases by a factor 100. Then, in this particular cylin-drical configuration, the range [0, eq] over which theapproximation (5) is good increases as R decreases since0 decreases proportionally to R and eq does not varya lot.

Equations (5) show that the radiative moments on thex axis are absorbed by a factor e/min and vary like(

R/)2 which, for a uniform opacity, is the solid angle

(R/x)2 under which the disk is seen from position .Furthermore, we can calculate from Eqs. (5) the Ed-

dington factor f = Ko xx/Jo 2min and the anisotropy

factor Ho x/Jo min. We conclude that the radiationfield has a high degree of anisotropy at distances wherethe approximation is acceptable.

The approximation (5) can be improved for Jo andHo x (but not for Koxx) if they are multiplied by min,

Jo

Ho x

(, r = 0)

I04

R

2e/min

min2min

.

(6)The error for Jo is then decreased by a factor of 5 to10, and this approximation is valid for a larger rangethan approximation (5) (see Fig. 3), namely for valuesof greater than approximately R/2. For Ho x, theimprovement is very good since the error is then lowerthan R for all values of . It is therefore very useful forR < 0.1, since the accuracy is then better than 10%.With this new approximation, the Eddington factor be-comes f = Ko xx/Jo min and, when compared withthe exact value, proves to be more accurate than 2min.

B. Numerical calculation of the radiation fieldgenerated by a planar shock

1. Multidimensional radiative transfer with the short

characteristics method

The above analytical calculation is limited to the val-ues of the radiation field on the x axis in a uniformmedium with a grey opacity. We now perform a multi-dimensional numerical computation of the radiation fieldgenerated at a given time by a planar shock. We varythe lateral size of the shock and study how the radiation


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107

108

109

0.1 10

10

20

30

40

50

60

Flux

Hx

(x)(Jcm

2s

1)

T(eV)

x (mm)

T

Hx 1Dyz, R=500 myz, R=250 myz, R=150 m

Hx (x,r=0)

FIG. 4: (Color online) 3D and 1D numerical computationof the flux Hx (from the model presented in Sec. IIIB3) forseveral values of R. The lines with symbols are the meanvalues, the dashed lines are the values at position r = 0, andthe error bars are the mean deviations. The temperature T(x)of the shock is plotted.

field is modified while all the other quantities are kept in-variant. The coupling of the radiation and the fluid willbe studied in Sec. IV. For the moment, we are only in-terested in the shape and the value of the radiation fieldin a nongrey description.

The structure of the shock (temperature and density)along x is given by a one-dimensional radiation hydrody-namics code developed by J. P. Chieze and that has alsobeen used in Ref. 10. It is chosen to be as close as possi-ble to the laboratory shocks we have observed10 in xenon(velocity 60 km s1, P = 0.2 bar). The temperature

is shown in Fig. 4. This 1D shock structure is put in athree-dimensional (3D) grid by assuming a plane parallelshock structure.

In the 3D radiative transfer calculation, the mediumis assumed to be in LTE. The source function is there-fore assigned to be the Planck function. The opacity isapproximated by a screened hydrogenic model, and cal-culated at n = 200 frequencies.

The radiative transfer code calculates the specific in-tensity I(r,n) at each grid point of a 3D Cartesian grid(number of grid points: nx = 200, ny = 40, nz = 40) withthe short characteristics method.39 We use the cartesiancoordinates because a solution of the radiation transportin these coordinates is easier to express and faster tocompute than in the equivalent natural cylindrical sys-tem (see also Ref. 40). The smallest radial size of theshock for which the calculation has been performed isY = Z = 2R = 300 m, i.e., the size of the ex-perimental shock. For this value of R, and for a celllength x = 2 mm, the number of points for the anglesis n = 200, n = 64 where the range of is [1, 1] andthe range of is [0, 2]. Solving the radiation transportin 3D is much more demanding than in 1D not only be-cause of the larger number of grid points (nx, ny and

108

109

1010

x (mm)

r(m)

J(x,r)

(J cm-2

s-1

)

0.2 0.4 0.6 0.8 1 1.2

0

50

100

150

200

250

FIG. 5: (Color online) 2D plot of the mean in-tensity J(x, r) (J cm2 s1) for a planar shock of width R = 250m. Isocontours are drawn forJ = 2.108, 4.108, 6.108, 8.108, 109, 2.109, 4.109 and 6.109

J cm2 s1.

nz) but also because of the required angular sampling(n and n) which must be more refined as R decreases.Generally, this implies that some trade-off such as lim-iting the number of frequencies describing the opacityis necessary. In the 1D case, 105 to 106 frequencies canbe easily managed, whereas this number is drastically re-duced from 10 to 103 in 3D calculations (here, n = 200).In this calculation, the total number of mesh points isnx ny nz n n n 1012.

2. Curvature of the radiation field

The curvature of the radiation field is evidenced inFig. 5. This curvature is due only to the finite lateralsize of the medium since the source function is kept pla-nar, that is S(x, r) = B(T(x)), with B(T) the Planckfunction. The profiles of Hx and Kxx are similar to theprofile ofJ. At position x = 500 m in the radiative pre-cursor, at mid distance between the shock front and theionization front, the value of the mean intensity on theboundary J(x, R) is roughly a factor of two lower thanJ(x, r = 0), the value on the axis. The medium is thenin an intermediate state between optically thick and op-tically thin. At places x 200 m where the medium isoptically thick, for example in the shock front, J is nearlyconstant, which makes it identical to the 1D case. Themedium becomes optically thinner as we go farther awayfrom the shock front. The radiation field deviates fromthe planar geometry and less radiation is available to ion-ize the medium on the outer border of the ionization front(r R) than on the axis. This leads the ionization frontto also depart from a plane. This result is confirmed bythe subsequent calculations of Sec. IV.


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3. Comparison between 1D and 3D

A comparison between the radiation field in 3D and in1D can be done by computing its value averaged on thesurface YZ at each position x:

yz=Y/2

Y/2dy

Z/2

Z/2dz J(x, y, z)

YZ

where J is either the mean intensity J, a component ofthe flux Hi or a component of the tensor Kij , where i, j =x, y, z.

The flux Hx calculated with the 3D model for R = 150,250 and 500 m and in 1D for R = is shown Fig. 4.The error bars are standard deviations on the lateral sur-face and the dashed lines give the value at r = 0 foreach value ofR. The magnitude of the radiation field de-creases with the lateral size of the shock, as was shown byour analytical approach. Although the radiation field isnearly constant in the 1D infinite calculation, it decreases

by one order of magnitude between the shock front (thespike of temperature) and the ionization front (where thetemperature goes to zero) for R = 500 m and by twoorders of magnitude for R = 150 m. We should note,however, that these calculations most likely overestimatethe radiation field since the temperature profile is keptthe same for all the values of R. But the decrease of theradiation field should result in a cooler and shorter pre-cursor, which will therefore emit even less radiation thanthe 1D infinite-R profile. We are going to see in Sec. IVthat when R decreases, the temperature in the precursordecreases too.

IV. BIDIMENSIONAL TIME-DEPENDANT

NUMERICAL SIMULATIONS

In the general case, the determination of the radiationfield [for example by solving Eqs. (3)] is difficult becausethe value of the optical depth (r), which depends on theopacity (r), must be known. But the opacity dependson the local density, temperature and the ionization andexcitation state of the medium, which can be modifiedby the radiation field.

In the previous sections, we studied the consequenceson the radiation field of the variation of the lateral size ofthe medium, without calculating the effect on the struc-ture of the medium. We now focus on the consequencesof the bidimensional shape of the shock on the radia-tive precursor structure. We first compare 1D and 2Dnumerical simulations with various boundary conditionswith respect to radiation, and investigate the transitionbetween these two kinds of calculations, stressing the roleof the radial (transverse) radiative flux. In a second part,we study the 2D structure of the radiation field.

FIG. 6: (Color online) Transition from the FCI-1D referencesimulation to the FCI-2D reference case, using decreasing tar-get radius Rt (in m) and reflecting wall cells. Axis profilesof (a) electron temperature (eV), (b) electron density (cm3)and (c) radiation energy density (J.cm3) are plotted at t =3 ns.


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FIG. 7: (Color online) 2D plot of Rosseland mean free pathin mm, in the reference 2D case, at t = 5 ns.

A. Transition from 1D to 2D

We perform time-dependant 2D radiation hydrody-namics simulations in cylindrical geometry with the codeFCI.31 This code, devoted to the modeling of inertialconfinement fusion, is therefore well suited to simulatethe radiative shock experiments described in Sec. II fromthe laser pulse to the shock propagation in the Xe gas.The radial and temporal laser profiles can be well ap-proximated by constant functions, with Gaussian wingsand 500 m FWHM. The value of the laser flux (3.1013

W/cm2 at peak power in time) is chosen so that thevelocity of the produced shock (55 km/s) matches the

experiment shock velocity.We define a reference 2D simulation similar to a typical

shot performed in the 2002 campaign.10,32 The targetdiameter is 1 mm and the walls of the cell, which weremade of quartz, are transparent to radiation.

The transition between 2D and 1D has been investi-gated by performing 2D simulations with walls reflectingradiation and decreasing the target radius Rt. However,the laser profile and the shock radius R are the same inall simulations. Reflective walls prevent lateral radiativelosses at the boundary r = Rt. A 2D simulation withreflective walls and Rt equal to the laser spot radius Ris nearly equivalent to a 1D simulation. On the con-

trary, a calculation with Rt much larger than R clearlyexhibits the 2D aspects of the problem. Increasing Rtin these simulations is then comparable, although notstrictly equivalent, to decreasing the shock radius in themodel of Sec. III. As stated earlier, the geometrical effectof decreasing R can be described as a radiative loss.

This procedure emphasizes the importance of radia-tive losses both in simulations of radiative shocks andin related experiments by varying only one geometricalparameter.

FIG. 8: (Color online) Comparison of axis longitudinal radia-tive flux (J cm2 s1) with various boundary conditions, at t= 3 ns.

In Fig. 6.a, we compare the axial temperature profilesat t=3 ns in the reference 2D simulation (lower curve)and the reference 1D simulation (upper curve) computedwith the 1D version of the code.31 Intermediate plots giveelectron temperature profiles for 2D simulations using re-flective walls and decreasing target radius Rt = 500, 450,350, 300, and 250 m. Reflecting boundaries suppresseradiation losses and give higher radiative flux and energyin the precursor (Fig. 6.c), which then produce highertemperatures (Fig. 6.a) and electron densities (Fig. 6.b).As the radius of the reflecting wall simulations decreases,the resulting temperature profile gets closer and closer

to the reference 1D profile. With reflecting walls and a350 m target radius, the axial 2D electron density inthe precursor is about half that of the 1D value. Withreflecting walls and a 250 m radius, the 2D simulationyields an axial profile of the electron density quite simi-lar to the 1D simulation, as expected. Radial flux lossesthus seem to be the main source of differences between1D and 2D simulation results.

One can also notice that the difference between the ref-erence 2D profiles and the others increases farther awayfrom the shock front. This can be explained by the cumu-lative effect of radial losses on the radiative flux emittedby the shock front.

A cartography of the Rosseland photon mean free pathin xenon is shown Fig. 7 for the reference 2D simulationat t=5 ns, when the shock position is x 250 m. Onlythe xenon gas has been represented in the figures. Theregion in white color corresponds to the xenon gas, whichhas been swept up by the shock initially produced atx = 0. The radius of this region is about 300 m at theorigin x = 0 whereas the radius of the laser focal spot isabout 250 m. We conclude, therefore, that the radius ofthe shock, R, increases with time while it is propagatingto the right in the xenon still at rest. In addition, the


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FIG. 9: (Color online) 2D profile of the electron temperaturein the gas in the reference 2D case, at t = 5 ns.

curvature of the shock front is clearly evidenced.The values of the Rosseland mean free path 1/ inthe precursor (400 m x 800 m) range typicallybetween 200 m and 1 mm (or less), which correspond,respectively, to R = R 1.25 and 0.25, whereas theystrongly decrease in the shocked xenon and the shockfront (250 m x 400 m), where R 2.5 for 0.1mm. With these characteristic values of R, the shockmight be considered as nearly 1D in the shock front and3D in the precursor.

The effects of boundary conditions on the radiation be-come less important as the radius of the target increases,as seen in Fig. 8. The longitudinal flux calculated fortransparent walls increases as the radius of the targetincreases, and the volume of gas ionized by radiation in-creases too. The Rosseland photon mean free path (seeFig. 7) lies typically between 200 m and 1 mm or less inthe precursor. Therefore, most of the radiation emittedby the shock has been absorbed by the gas before reach-ing the walls of the cell. The radiation emitted by allthe ionized gas contributes to the longitudinal flux whichincreases until the maximum volume of ionized gas is ob-tained in the precursor, ahead of the shock.

On the other hand, the longitudinal flux calculatedwith reflective boundaries decreases as Rt increases be-cause the radiation that reaches the walls when the radiusis small enough is reflected and then contributes again to

the flux and to the ionization of the gas. The limit sit-uation is reached when the boundary is sufficiently faraway so that most photons are absorbed in the gas.

Another consequence of the smaller amount of radia-tion emitted in 2D than in 1D is the smaller precursorvelocity. The precursor velocity, defined as the speed ofa chosen isocontour value of electron density, decreasesfrom 350 km s1 in the 1D simulation to 100 km s1

in the 2D reference case, for ne = 1019 cm3. The 1D

simulations thus also overestimate precursor velocities ina systematic way.

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r(mm)

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Log 10 [Radial Flux (10 J/cm .s)]

Log 10 [Longitudinal Flux (10 J/cm .s)]

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FIG. 10: (Color online) 2D plot of the (a) longitudinal and(b) radial fluxes (in J cm2 s1), and (c) their ratio in thereference 2D case, at t = 5 ns.

B. Bidimensional structure of the radiation field

Figure 9 shows that the electron temperature is muchhigher in the shock front than in the precursor: 25 eVversus 5 eV and less. Assuming that emissivity scalesas T4 like a black body, this makes precursor emissivitymostly negligible compared to the emissivity of the shock


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front. Since in addition the average mean free path inthe precursor is similar to the cell and precursor lengthscales, we can expect the radiation field to be effectivelydominated by the radiation coming from the shock front.This will not be true anymore far enough from the shockfront. The analysis of Fig. 8 done in Sec. IV A showsthat, for Rt 500 m, the radiation from the volume ofthe gas (the precursor) becomes important at distances

from the shock front greater than about 1 mm.Let us consider the radial and longitudinal componentsof the radiative flux, plotted in Fig. 10.a and Fig. 10.b.On the x axis, the longitudinal flux is maximum, whereasthe radial flux is zero because of the cylindrical symme-try. The radial flux is most important near the shockfront. On Fig. 10.c showing the ratio between both fluxes,we see that the medium can be separated in two regionswith different regimes. On the dashed line, the two com-ponents are roughly equal, while the radial flux domi-nates above this line and the longitudinal flux dominatescloser to the axis, below this separation.

The radiation flux field appears, therefore, veryanisotropic near the axis. It is strongly oriented in theshock propagation direction. This structure of the radi-ation field is very similar to that obtained in the modelpresented in Sec. III, which had a planar shock struc-ture and transparent walls. This suggests that the 2Dradiation at r < 250 m could be well modelled by aplanar uniform source with finite radius R (representingthe shock front) and transparent walls.

V. CONCLUSION

Geometrical effects on the radiation field are impor-tant when the radial optical depth R = R is lower

than unity. Analytical arguments explain the decrease

of the energy density, flux, and pressure when the radiusR of the emitting surface diminishes. They show that,when R is lower than the photon mean free path 1/,the radiation is absorbed roughly by a factor of ten ata distance x from the emitting surface of a few R. Thedimension of the heated region therefore scales like R inthis simple model. Approximate solutions of the radia-tion energy density, flux, and pressure are found in the

asymptotic limit x R. Further studies are needed toestimate their relevance in more complexe models. Veri-fications of the radiative transfer code can also be madeby comparison with the analytical solution.

Numerical calculations of radiative shocks confirm thatthe radius of the shock must be taken into account whenR = R < 1 even though is a mean opacity (theRosseland mean in the present case), which is a veryrough representation of the actual frequency dependencein the radiative transfer calculations. One-dimensionalcalculations for such a complex system overestimate theamount of energy radiated and transfered ahead of the

shock. This results in overestimating the energy depo-sition in the precursor. Therefore, all the properties ofthe radiative precursor are overestimated: its velocity bya factor 3, the temperature and electron density by oneorder of magnitude, and the extension by a factor thatdepends on the electron density, but can be up to anorder of magnitude. These values are for the propertiesof the precursor far enough from the shock front, say atmore than a photon mean free path. Moreover, the shapeof the radiation field in the precursor and the structureof the temperature and electron density in the precur-sor depart from a plane parallel geometry. Our work,therefore, emphasizes the need of considering 3D or 2D-cylindrical numerical simulations for modeling radiative

shocks in the laboratory and in cosmic settings.

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