A MULTISCALE CENTRAL DIFFERENCE SCHEME APPLIED TO MAGNETOHYDRODYNAMICSIMULATIONS OF COMETARY ATMOSPHERES

Mehdi Benna, Paul R. Mahaffy, Peter MacNeice, and Kevin Olson

NASA Goddard Space Flight Center, Codes 915 and 931, Greenbelt, MD 20771; mehdi.benna@gsfc.nasa.gov

Receivved 2004 May 24; accepted 2004 August 7

ABSTRACT

The hyperbolic equations describing the interaction between a cometary atmosphere and the solar wind aresolved through a central differencing scheme. This work will enable us to better predict conditions that roboticspacecraft will encounter on comet nucleus flyby or rendezvous missions. We are interested in studying the comacharacteristics from large distances from the nucleus, where the solar wind just begins to be perturbed by thecomet, to very near the nucleus. We employ a multifluid approach in which the equations governing the ions,neutrals, and electrons are coupled. We solve these equations by the use of a total variation diminishing Lax-Friedrichs solver combined with an adaptive mesh refinement technique. Magnetic fields and plasma pressure,velocity, and density profiles are calculated and compared with other models. We show that this approach is wellsuited to provide an accurate solution over a large computational domain, and we present results for a Halley-likecomet.

Subject headinggs: comets: general — methods: numerical — MHD — plasmas — solar wind

Online material: color figure

1. INTRODUCTION

The motivation to understand the physical and chemicalcomposition of the cometary nucleus has led to several recentmissions that followed the groundbreaking in situ investiga-tions of comet Halley in 1986 (Mason 1990). Additional nu-cleus flyby or rendezvous missions are presently being planned.Numerical models of the interaction between the solar wind andthe cometary atmosphere can support ongoing missions andnew missions planned, and also advance the understanding ofthe physics and dynamics of the cometary coma. The large spa-tial domains, combined with the wealth of physical and chem-ical processes that shape the structure and dynamics of thecometary coma, make this a challenging task. Here we describethe results of a new model that will enable us to study the fullspatial domain of interest. We provide a direct comparison ofthe output of our model with previous work.

The numerical magnetohydrodynamic (MHD) simulation ofthe behavior of a cometary coma is a complex problem. Thecomplexity is mainly due to the difference between the twoplasmas involved in shaping this coma. On one hand, the solarwind can be seen as a hypersonic warm plasma composed oflight ions (mainly protons) and electrons embedded in an in-terplanetary magnetic field. On the other hand, the plasma inthe region of the inner nucleus, formed from the cometary gasreleased from the comet nucleus, is much more dense than thesolar wind. It is also composed of different ion and neutralspecies with a wide range of masses. This gas is chemicallyactive, and as soon as it is formed from volatile compoundsat the nucleus, a complex network of chemical reactions pro-duces new chemical species. Because of its low temperature,the cometary gas is very often considered to be completely cold(Tn � 0 K) compared to the warm solar wind plasma (Tsw �10;000 K). The interaction between these two different plas-mas is characterized by a ‘‘mass-loading’’ phenomenon inwhich mass, momentum, and energy are exchanged and shapethe cometary coma. This can lead to a very extended coma

(�107 km) with many sharp density, temperature, and velocitygradients.Past simulations (Schmidt-Voigt 1989; Gombosi et al. 1994)

showed the necessity of simultaneously achieving a high res-olution at transition regions and a large computational domain.One efficient way to satisfy both of these requirements is to usean adaptive grid whose density varies as required without ex-ceeding the available computing capabilities. Another criti-cal issue raised in the study of mass loading in plasmas is theimportance of correctly modeling the major species involved inthe interaction. In our case they are ions, electrons, and neu-trals. The more accurately we model the dynamics of these spe-cies and their interactions with each other, the more realisticwill be the representation of the cometary coma. Several multi-fluid simulations (Wegmann et al. 1987; Schmidt et al. 1988;Sauer et al. 1989, 1994) show the complexity of the interfluidinteractions and point out the need for a multifluid approach forthe accurate modeling of cometary comas.We have developed the Cometary Atmosphere Simulator

(CASIM), a multifluid code that computes equilibrium solu-tions to simulate the interaction between the solar wind and thecometary atmosphere. This paper will describe this code anddiscuss our early results on a sufficiently large computationaldomain to test the robustness of the approach. This work iscarried out with the future perspective of designing a three-dimensional multifluid simulator that can model the interac-tion of solar wind with a wide range of comet types, not onlyHalley-like comets. In x 2 we present the multifluid governingequations that form the core of the CASIM code. In x 3 wedetail the adaptive grid structure and the refinement techniqueadopted to manage and optimize the computational domain.Section 4 shows some elements of the central difference solverand the time-stepping technique. In x 5 we compare resultsobtained using CASIM with those obtained by Gombosi et al.(1994) and Murawski et al. (1998) in a single-fluid case fora Halley-like comet. This computation was done to validatethe robustness and accuracy of our solver and to enable a

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The Astrophysical Journal, 617:656–666, 2004 December 10

# 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.

comparison with previous models. Finally, in x 6 we presentthe first results of the two-dimensional multifluid version ofCASIM.

2. MULTIFLUID GOVERNING EQUATIONS

The CASIM code employs a three-fluid model. In this model,ions, electrons, and neutrals are considered to be three separatefluids that interact with each other by means of the following:

1. chemical reactions that produce or consume species atspecific rates;

2. mutual elastic collisions that transfer momentum andenergy from one fluid to another; and

3. magnetic field coupling.

As a first step, we assume that the neutral gas released bythe nucleus is composed only of water vapor. We also as-sume that the water group photoionization and recombinationare the only chemical reactions occurring inside the coma.The problem is considerably simplified by assuming the chargeneutrality of the resulting plasma, the zero electric current con-dition, and an ion charge state of Zi ¼ 1. We also assume thatthe ratio of specific heats for all fluids is � ¼ 5=3. The excellentreview of Szego et al. (2000) provides the basic kinetic equa-tions for a multifluid approach. Starting from these equations,we accommodate them to our case to obtain a set of governingequations for the three fluids. The equations for the neutral fluidare

@�n@t

þ:= �nunð Þ ¼ �n � �n; ð1Þ

@�nun@t

þ:= �nunun þ pnIð Þ

¼ �nui � �nun þ �ni ui � unð Þ þ �ne ue � unð Þ; ð2Þ

@

@t

1

2�nu

2n þ

3

2pn

� �þ9

1

2�nu

2nun þ

5

2pnun

� �

¼ 1

2�nu

2i �

1

2�nu

2n þ

3

2�nkBTi�

3

2�nkBTn þ

3�ni�nkB Ti � Tnð Þ2mn

þ 3�ne�nkB Te � Tnð Þmn

þ 1

2�ni�n u2

i � u2n

� �þ �ne�n

mn

meue þ mnunð Þ ue � unð Þ: ð3Þ

Table 1 contains the description of all the parameters and var-iables appearing in these equations. These equations are re-spectively the conservation of mass, momentum, and energy forthe neutral fluid.

In the same way, the ion fluid is described by another set ofgoverning equations that takes into account the magnetic fieldcoupling

@�i@t

þ:= �iuið Þ ¼ �i � �i; ð4Þ

@�iui@t

þ:= �iuiui þ piI þB2

2�0

I � B =B

�0

� �¼ �iun � �iui �:= peIð Þ

þ �in un � uið Þ þ �ie ue � uið Þ � B

�0

:=B; ð5Þ

@

@t

1

2�nu

2i þ

3

2pi þ

B2

2�0

� �

þ91

2�iu

2i ui þ

5

2piui þ

B2

�0

ui �B

�0

B =uið Þ� �

¼ 1

2�iu

2n �

1

2�iu

2i þ

3

2�ikBTn �

3

2�ikBTi � ui = :pe

þ 3�in�ikB Tn � Tið Þ2mi

þ 3�ie�ikB Te � Tið Þmi

þ 1

2�in�i u

2n � u2i

� �þ �ie�i

mi

meue þ miuið Þ ue � uið Þ � 1

�0

: =Bð Þ B =uið Þ:

ð6Þ

The electron fluid equations are simplified by using the chargeneutrality and the zero electric current assumption. The prob-lem reduces to solving only a pressure equation. The electronmass density and velocity are directly derived from the ion fluidvariables

ne ¼ ni; ð7Þ

ue ¼ ui; ð8Þ

@pe@t

þ:= peueð Þ þ 2

3pe:=ue

¼ 2

3He �

�ekBTeme

þ 2�en�ekB Tn � Teð Þmn

þ 2�ei�ekB Ti � Teð Þmi

þ 2

3�en�e un � ueð Þ2þ 2

3�ei�e ui � ueð Þ2: ð9Þ

The production and losses due to photoionization and recom-bination can be expressed by

�n ¼ �Rne�i;

�n ¼�n�;

�i ¼ �n;

�i ¼ �n;

�e ¼me

mi

�i;

�e ¼me

mi

�i; ð10Þ

where � is the mean photoionization lifetime and �R is therecombination rate.

To complete this set of equations we need to add Faraday’slaw governing the magnetic field behavior:

@B

@tþ: = uiB� Buið Þ ¼ � :=Bð Þui: ð11Þ

We will not drop the :=B source term in equations (5), (6),and (11), even if there are no magnetic monopoles. Powellet al. (1999) pointed out that keeping this divergence termmakes the system of conservation laws symmetrizable, whichis indispensable to define the wave speeds during the solverconstruction.

The four sets of governing equations for neutrals, ions,electrons, and magnetic fields presented above can be written

MHD SIMULATION OF COMETARY ATMOSPHERES 657

as three sets of quasi-linear systems in vectors of conservedvariables:

Un ¼ �n; �nun; Enð ÞT ;Ui ¼ �i; �iui; Ei; Bð ÞT

Ue ¼ peð Þ ð12Þ

where the energies En and Ei are

En ¼1

2�nu

2n þ

3

2pn ð13Þ

and

Ei ¼1

2�nu

2i þ

3

2pi þ

B2

2�0

: ð14Þ

Note that we grouped the Faraday equation (eq. [11]) with theset of the ion fluid governing equations (eqs. [4], [5], and [6]).This is done for convenience. In fact, the magnetic field cou-pling appears only between the Faraday equation and the con-servation law for the ion fluid. So it is logical to resolve themtogether. Also, keeping these equations together makes the tran-sition from a multifluid model to a simple one-fluid model easy,since we just have to impose the dynamics of the neutral and theelectron fluids.

The quasi-linear divergence forms of the governing equa-tions reduce to

@Uk

@tþ :=Fkð Þ ¼ Sk ; k ¼ (n; i; e); ð15Þ

where Fk and Sk are respectively the flux vector and thesource term corresponding to the k th fluid. These vectorsare straightforwardly obtained by grouping and manipulating

equations (1), (2), (3), (4), (5), (6), (9), and (11). Equation (15)is the core of the multifluid approach. We obtain the one-fluidideal MHD form by resolving only the divergence form equa-tion for k ¼ i for an imposed neutral and electron dynamics.

3. ADAPTIVE GRID AND COMPUTATIONAL DOMAIN

The comet nucleus is of order 10 km, but the bow shock canbe 5 orders of magnitude larger, and the coma extends down-stream a distance up to 108 km. To resolve the finest of thesescales while including the largest requires the use of a mesh forthe calculation that is adaptive and can adjust dynamically asthe structure in the solution evolves. To achieve this, theCASIM code uses the PARAMESH tool kit (MacNeice et al.2000).PARAMESH implements a subset of the Berger-Oliger type

of block-structured or patch-based adaptive mesh refinement(AMR; Berger & Oliger 1984; Berger & Colella 1989). In theseschemes, space is divided into a set of contiguous blocks. Eachblock is a logically Cartesian, uniformly spaced submesh. Eachblock can itself be refined into a set of ‘‘child’’ blocks with halfthe grid spacing of their parent block. In PARAMESH, spaceis recursively divided by bisecting the computational volumein each coordinate direction. In a two-dimensional application,each block is a structured submesh with nx ; ny cells, and eachblock has an identical number of cells. For CASIM, nx ¼ ny ¼4. PARAMESH enforces the condition that jumps in resolutionof more than a factor of 2 are not allowed.Each submesh block is surrounded by guard cells so that

the mesh cells in the submesh that are near to the submeshboundary have enough information about the solution in theneighboring submesh blocks. For CASIM, there are two layersof guard cells at each submesh boundary. PARAMESH fillsthese guard cells by copying data directly from the neighboringblocks if the neighbor is at the same refinement level or, in thecase of a refinement discontinuity at the block boundary, either

TABLE 1

Article Variables and Parameters

Parameter Symbol Parameter Symbol

Neutral mean molecular mass ................. mn Neutral energy.......................................... En

Neutral mass density................................ �n Ion energy ................................................ Ei

Neutral velocity vector ............................ un Electron heating rate ................................ He

Neutral pressure ....................................... pn Magnetic field vector ............................... B

Neutral temperature ................................. Tn Neutral-ion collision frequency ............... �niNeutral number density ........................... nn Ion-neutral collision frequency................ �inMass density production of neutrals ....... �n Neutral-electron collision frequency........ �neMass density loss of neutrals .................. �n Electron-neutral collision frequency........ �enElectron mean molecular mass................ me Electron-ion collision frequency.............. �eiElectron mass density .............................. �e Ion-electron collision frequency .............. �ieElectron velocity vector........................... ue Mean photoionization lifetime................. �Electron pressure...................................... pe Recombination rate .................................. �R

Electron temperature ................................ Te Boltzmann’s constant ............................... kBMass density production of electrons...... �e Specific heat constant .............................. �

Mass density loss of electrons................. �e Permeability of vacuum........................... �o

Electrons number density ........................ ne Comet gas production rate....................... Q

Ion mean molecular mass ........................ mi Comet nucleus radius .............................. rcIon mass density ...................................... �i Ion-neutral collision rate.......................... kinIon velocity vector ................................... ui Solar wind number density...................... nswIon pressure.............................................. pi Solar wind speed ..................................... uswIon temperature ........................................ Ti Solar wind temperature............................ TswMass density production of ions ............. �i Identity matrix.......................................... I

Mass density loss of ions ........................ �iIon number density .................................. ni

BENNA ET AL.658 Vol. 617

by restriction if the neighboring submesh is more refined orby interpolation if it is less refined. PARAMESH allows theuser to specify the restriction and interpolation functions to beused in these situations. For CASIM, restriction is done usinga simple averaging of data at cell center. Interpolation is per-formed using bilinear interpolation with aMUSCL-type limiter.

For a typical simulation, the computational domain extendsfrom �224 to 2 ; 224 km in the direction of the solar wind, andfrom�224 to +224 km in the perpendicular direction. The cometnucleus (with a radius of 6 km in the case of a Halley-typecomet) was located at (0, 0) in this coordinate system. Theinitial mesh is established by first placing an array of 3 ; 2PARAMESH blocks within the computational volume. To re-solve the comet nucleus, these initial, coarse blocks are refinedif they overlap the volume occupied by the comet nucleus. Thisprocess is repeated 25 times, and the refined mesh thus has 25levels of refinement. The smallest computational cell size on therefined mesh is 0.25 km. In addition, after every 1000 iterationsof the fluid solve, the local gradients and divergence of thevelocity are examined, and if necessary, the grid is refined toensure that the velocity is effectively resolved. Figure 1 shows

the final mesh obtained for the simulation described in x 5. Thedifferent panels show the refinement progression starting from alarge scale and zooming into the vicinity of the comet nucleus.For this case, the final mesh contained approximately 96,000cells. To achieve the same resolution with a uniformly refinedmesh would require 4 ; 4 ; 3 ; 2 ; 224 ¼ 1:61 ; 109 cells.

4. CENTRAL DIFFERENCE SOLVERFOR CONSERVED LAWS

4.1. Finite-Volume Discretization and Time Steppingg

To solve the set of divergence form equations (eq. [15]),we have developed a multistage finite-volume explicit centralscheme. In a finite-volume approach over a Cartesian-structuredgrid, equation (15) can be written locally as

@Uk; i

@tþ 1

Vi

Xf : cell faces

Ff (k; i) =nf ds ¼ Sk; i; k ¼ (n; i; e); ð16Þ

where Uk; i, Ff (k; i), and Sk; i are respectively the cell-averagedconserved variables vector, face flux, and cell-averaged source

Fig. 1.—Mesh obtained at the end of the one-fluid simulation for a Halley-type comet (see x 5). (a–d ) Progression of the refinement from the coarsest to the finestcells. The nucleus can be seen in the center of (d ). Note that (a) shows only a part of the more extended computational domain.

MHD SIMULATION OF COMETARY ATMOSPHERES 659No. 1, 2004

term relative to the k th fluid; nf is the normal to the cell facef (facing out). In a three-dimensional case, Vi is the cell volumeand ds is the cell face area. In a two-dimensional model, Vi

would be the cell area and ds is the edge length. In the lattercase, one should sum the fluxes along the cell edges insteadof the cell faces. We are interested in obtaining steady-statesolutions for our model problem. To achieve this, we use asecond-order Runge-Kutta time integration scheme to advanceequation (16) in time. The time step is set locally within eachcomputational cell according to the CFL condition (Courantet al. 1967). The solution in each cell is then advanced in timeaccording to its own local time step. This scheme is applied toeach cell in the mesh until we reach a prescribed number ofiterations. The final number of iterations is chosen high enoughto ensure that at the end the scheme converged globally to thefinal steady-state solution.

4.2. The Total Variation Diminishingg Lax-Friedrichs Solvver

The total variation diminishing Lax-Friedrichs (TVDLF)solver is one of the high-resolution schemes designed to solvehyperbolic conservation laws and capture complex shocks with-out oscillations. Yee (1989) provides a good review of this classof schemes. The TVDLF method is known to be robust, and inmost cases there are no spurious oscillations, but the TVDLFis more diffusive than other TVD methods. In their one-fluidmodel, Gombosi et al. (1994) used a more accurate upwindsolver (Roe-type solver) based on the local resolution of theapproximate Riemann problem. That solver, well known inpure hydrodynamic problems for efficiently capturing shocksand sharp transitions, was successfully applied to complexmultidimensional MHD problems by Powell (1994). Our prob-lem involves solving the multifluid governing equations for alarge number (up to a million) of cells. Applying the Roe-typesolver for our case would be computationally expensive. Infact, applying this process would involve solving the localapproximate Riemann problem for every fluid and wouldrequire computing many eigenvalues and eigenvectors. InCASIM we chose to use the TVDLF, since it does not use aRiemann solver and is usually much faster than the classic Roe-type solver.

In the TVDLF approach the average face flux vector for thekth fluid is given by (we drop the subscripts k and i for clarity)

Ff =nf ¼1

2FfR = nfR þ FfL = nfL� �

� kf2

UfR � UfL

� �; ð17Þ

where UfR and UfL are the reconstructed vectors of the con-served variables at the midpoints of the left and right sides ofthe considered face f; FfR and FfL are the flux vectors computedusing UfR and UfL , and kf is the coefficient of artificial viscosityevaluated for the average primitive variables and given by

kf ¼j u j þ a; ð18Þ

where u ¼ u = nf� �

with u as the average velocity vector at theface f. The speed a for the various fluids is as follows:

1. the speed of sound for the neutral fluid:

a ¼ffiffiffiffiffiffiffiffi�pn�n

r; ð19Þ

2. the fast magnetoacoustic wave for the ion fluid:

a ¼

1

2

(�pi þ B = Bð Þ=�0

�i

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�pi þ B = Bð Þ=�0

�i

� �2� 4

�pi B =nf� �2�0�

2i

s )!1=2

; ð20Þ

3. and zero for the electrons.

In our method, the extrapolation of the cell-centered values toface midpoints for the construction of the fluxes in equation (17)is achieved by the used of a second-order reconstruction scheme.We also limit the slopes of the reconstructed solution so as toavoid overshoots. Different limiters were tried, but we found thatthe MINMOD-type limiter described by DeZeeuw (1993) givesthe best results.The additional artificial viscosity in the TVDLF scheme

damps oscillations near shocks and makes the scheme stable.We show in x 5 that even with an excessive artificial viscosity,we still capture correctly the shocks and all the sharp featuresin the cometary coma. The advance from previous methods isthe computational efficiency that enables us to simulate a three-fluid cometary coma and to handle a large number of cells thatstart from the nucleus and extend over more than 8 orders ofmagnitude.

5. ONE-FLUID SIMULATIONOF A HALLEY-LIKE COMET

To evaluate the performance of CASIM we carried out aone-fluid simulation of the interaction of the solar wind with aHalley-like comet. In this one-fluid version we only solve theion fluid equation assuming no recombination reactions. Theion-electron friction was also neglected, making any assump-tion about the electrons temperature profile unnecessary. Tocompute the production rate and the ion-neutral friction, weassumed a radially expanding neutral gas with

�n ¼mnQ

4�unr 2exp � r � rc

un�

� �;

un ¼ constant;

Tn ¼ constant;

�in ¼ kin nn; ð21Þ

where r is the cometocentric distance. The neutral gas param-eters chosen for this simulation are Q ¼ 1030 molecules s�1,rc ¼ 6 km, un ¼ 1 km s�1, Tn ¼ 10 K, kin ¼ 1:1 ; 10�15 m3

s�1, and � ¼ 106 s. The unperturbed solar wind is chosen tohave a velocity usw ¼ 400 km s�1, a number density nsw ¼10 ions cm�3, and a temperature Tsw ¼ 10;000 K. The mag-netic field value is Bsw ¼ 5 nT. These parameters correspondglobally to those used by Gombosi et al. (1994) and Murawskiet al. (1998) in their simulations and provide us a direct com-parison with their results.In a first simulation we assumed no ion-neutral friction. Fig-

ure 2 shows the large-scale plasma pressure. This pressure profileis globally consistent with the ones obtained by Gombosi et al.(1994) and Murawski et al. (1998). Gombosi and Murawski re-ported a bow shock position at x � 0:45k, where k ¼ 106 km isthe ionization scale length. In his paper, Murawski noted that thisposition value is smaller than the value predicted by Biermann

BENNA ET AL.660 Vol. 617

et al. (1967) for the same set of parameters. In our case we founda bow shock position at x ¼ 0:624k, which is the value predictedby Biermann et al. We believe that the lower value of Gombosiand Murawski is mainly due to the value approximation made intheir source term.We note also that our plasma pressure profile iscloser to the one obtained by Gombosi than to the Murawskiprofile. In fact, the Murawski pressure profile shows a relativelylarge transition region on the bow shock and a local pressuremaximum near the stagnation region. In our case we found, asdid Gombosi, a very sharp and narrow transition in the bowshock region and a flat pressure plateau between the bow shockregion and the stagnation region.

In a second simulation, we included the ion-neutral frictionand focused on analyzing the inner nucleus region. Figure 3shows the plasma velocity profile in the inner nucleus region.In this figure we clearly identify the contact surface positionsituated at x ¼ 2:2 ; 10�2k in the subsolar direction. The in-ner shock corresponding to the deceleration of the cometaryplasma from supersonic to subsonic speeds is located at x ¼1:95 ; 10�2k on the dayside and x ¼ 2:8 ; 10�2k on the night-side. These values are in complete agreement with those foundby Murawski.

The pressure profile (Fig. 4) confirms the position of theinner shock. Figure 5 shows the magnetic field structure andreveals the shape of the diamagnetic cavity. We found that ourmagnetic field structure is similar to the one presented byMurawski, even though we observe a smaller magnetic fieldenhancement on the nightside. The magnetic field reported byGombosi et al. (1994) is not consistent with these results, as itshows a null field on the tail. In this respect our results arecloser to those of Murawski et al. (1998).

The initial comparison of the plasma density profile with thatgiven by Murawski et al. (1998) showed substantial differ-ences. With CASIM the sharp density evolution near the nu-cleus was smoothed by the artificial viscosity included in thealgorithm, leading to a higher plasma density. We carried outseveral simulations to estimate the magnitude of artificialviscosity error. Our approach was to compare the deviationbetween the numerical solution and the analytical solution of asimple unperturbed, radially expanding outgassing. No solarwind was considered in these test cases. We found that the

excess viscosity superposed a multiplicative logarithmic errorto the analytical solution. This multiplicative error appears tobe of the form a log (r)þ b, where r is the cometocentric dis-tance and a and b are constants. We realized many tests withdifferent outgassing parameters in order to estimate the a and bparameters for the grid geometry we are using, and to derivethe correct analytical correction to use in our specific case. Thiscorrection removes the effect of viscosity excess. Figure 6shows the corrected density profile, which is very comparableto the one obtained by Murawski. In fact, we report the samedensity enhancement of a factor of 3 over the backgrounddensity on the subsolar direction and of a factor of 2 in the tail.This enhancement is well located at the contact surface.

In general, our results are in good agreement with thoseobtained by Gombosi et al. (1994) and Murawski et al. (1998).Our results are more consistent with Gombosi’s results at largecometocentric distances, butmore in agreement withMurawski’sresults in the inner nucleus region. The artificial viscosity in-cluded in our algorithm did not introduce any artifact or incon-sistency in the plasma’s variable profiles at large cometocentricdistance. The only effect of the excess viscosity was in the plasmadensity in the inner cometary region. This effect was easily re-moved by the application of a correction factor.

6. MULTIFLUID SIMULATIONOF A HALLEY-LIKE COMET

In this section we present the first results of a multifluidsimulation of the solar wind–comet atmosphere interaction fora Halley-type comet. Both photoionization and recombinationreactions are considered, and all the mutual friction terms be-tween ions, neutrals, and electrons are included. We use thesame solar wind parameters described in x 5. The neutralsreleased by the nucleus surface are assumed to have an initialvelocity un ¼ 1 km s�1 and a temperature Tn ¼ 200 K. Thesolar wind is assumed to be composed of ions and electronswith temperatures Ti ¼ Te ¼ 10;000 K. This three-fluid sim-ulation was carried out on a computational domain of (3:3 ;107 km) ; (5:0 ; 107 km) using about 100,000 computationalcells. The finest cell resolution is 250 m near the comet nucleussurface.

One of the interesting results is the neutral density distri-bution. In one-fluid models we always suppose that the neutralgas is expanding symmetrically. Figure 7 shows that the neu-trals are pushed in the downstream direction by the ions and theelectrons. The intensity of this phenomena is a direct functionof the neutral-ion and neutral-electron drag forces. We foundthat the neutral density profile is also axisymmetric in thecometopause and the inner nucleus region. Figure 8 shows theion and electron temperatures. These temperature profiles arein good agreement with those presented by Wegmann et al.(1987) showing an electron temperature enhancement in thecometopause. This electron temperature profile is important foran estimation of the recombination rate. Also, this figure showsclearly the cooling effect of electrons by the water molecule.We are initiating additional studies to more fully understandthese profiles and to include other effects like the charge ex-change between the cometary neutrals and the fast solar windprotons. The results of these additional multifluid simulationswill be presented in future papers.

7. CONCLUSION

We demonstrated in this paper the capability of the CASIMcode as a multifluid simulator. We have successfully reproduced

Fig. 2.—Cross section of the plasma pressure profile along the Sun-cometline ( y ¼ 0) in the absence of an ion-neutral drag force; the pressure is in unitsof the unperturbed solar wind pressure. This profile is to be compared with theones given in Fig. 1c by Murawski et al. (1998) and in Fig. 3a by Gombosiet al. (1994). The solar wind direction in all the figures is from left to right.

MHD SIMULATION OF COMETARY ATMOSPHERES 661No. 1, 2004

Fig. 3.—Profile of the plasma velocity in the inner coma region. (a) Two-dimensional profile on the half x-y plane ( y � 0). (b) Velocity profile along the Sun-comet line, where dashed lines indicate the position of the inner shock. The velocity is in units of the unperturbed solar wind Alfven speed. This profile is to becompared with the one given in Fig. 2 by Murawski et al. (1998).

662

Fig. 4.—Plasma pressure profile in the inner region of the cometary coma. (a) Two-dimensional profile on the half x-y plane ( y � 0). (b) Pressure profile along theSun-comet line, where dashed lines indicate the position of the inner shock. The pressure is in units of the unperturbed solar wind pressure. This profile is to becompared with the ones given in Fig. 3 by Murawski et al. (1998).

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Fig. 5.—Magnetic field in the inner region of the cometary coma. (a) Two-dimensional profile on the half x-y plane ( y � 0). (b) Magnetic field profile along theSun-comet line, where dashed lines indicate the position of the inner shock. The magnetic field value is in units of the unperturbed solar wind magnetic field. Thisprofile is to be compared with the ones given in Fig. 4 by Murawski et al. (1998).

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Fig. 6.—Plasma mass density profile in the inner region of the cometary coma. (a) Two-dimensional profile on the half x-y plane ( y � 0). (b) Density profilealong the Sun-comet line, where dashed lines indicate the position of the inner shock. The plasma mass density is in units of the unperturbed solar wind massdensity. This profile is to be compared with the ones given in Fig. 5 by Murawski et al. (1998).

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the one-fluid simulations presented by Gombosi et al. (1994)and Murawski et al. (1998) for a Halley-like comet. The excessof the artificial viscosity introduced by the TVDLF method doesnot substantially impact the quality of the simulation, since itsonly effects are seen in the sharp plasma density profiles in theinner coma and can easily be corrected. We also presented theearly results of a multifluid simulation for a Halley-like comet.

This result shows the effect of the ion and electron frictionson the neutral gas. The solar wind pushes back the expand-ing neutral coma to form a tail-like structure. Our code also re-produces the expected electron temperature profile and showedthe expected electron cooling effect by water molecules in theinner coma. In future work we will focus on analyzing these re-sults in more detail, and we will carry out similar simulationswith different parameters to try to explain the features seen inthe neutral profiles.

REFERENCES

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MacNeice, P., Olson, K. M., Mobarry, C., Fainchtein, R. D., & Packer, C. 2000,Comput. Phys. Commun., 126, 330

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Sauer, K., Motschmann, U., & Baumagartel, K. 1989, Adv. Space Res., 9,309

Schmidt, H. U., Wegmann, R., Huebner, W. F., & Boice, D. C. 1988, Comput.Phys. Commun., 49, 17

Schmidt-Voigt, M. 1989, A&A, 210, 433Szego, K., et al. 2000, Space Sci. Rev., 94, 429Wegmann, R., Schmidt, H. U., Huebner, W. F., & Boice, D. C. 1987, A&A,187, 339

Yee, H. C. 1989, A Class of High-Resolution Explicit and Implicit ShockCapturing Methods (NASA TM-101088; Washington: NASA)

Fig. 8.—Ion (solid line) and electron (dashed line) temperature in degrees K.

Fig. 7.—Neutral mass density for a Halley-like comet. The mass density isgiven in units of mass density of the unperturbed solar wind. [See the elec-tronic edition of the Journal for a color version of this figure.]

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