GEOPHYSICALRES•RCH •S, VOL.18, NO. 5, PAGES801-804, MAY 1991

A TWODIMENSIONAL SHOCK CAPTURING, HYDRODYNAMIC MODEL OF THE VENUS IONOSPHERE A. F. Nagy,A K6r•Ssmezeyl,J.Kim and T. I. Gombosi SpacePhysics Research Laboratory, TheUniversity of Michigan

• A two-dimensional,time-dependent, shock capturing ModelDescription single-sp•ies(O+) hydrodynamicmodel of the Venus ionosphere, Govemin• Eouations whichsolves the coupledcontinuity, momentum and energy v _ _,•ons forthe altitude range of 150- 500km and the solar zenith Thetime-dependent two-dimensional continuity, momentum and anglerange of 0O - 180ø has been developed and is presented inthis energyequations for O+ ions and electronsare solvedself- paper.It was again demonstrated thatthe introduction oftopside consistently.Quasi-neutrality is assumed,i.e. ni = ne, wheren heat inflows leads to calculated dayside electron ,red ion denotesnumber density. Furthermore, the ion and electronsare temperatures,which are consistent with the measured values. In assumedto havethe samevelocity, i.e. Uion= Uelectmn,in other orderto reproducethe measured electron temperatures, which are words the cunvmtis assumedto be zero. Knudsen et al. [1981] have roughlyconstant over all SZA's,the heat inflows had to be reduced shownthat the transterminator flow is not stronglyinfluenced by the significantlyoverthe nightside compared tothe dayside values. The magneticfield; therefore,the use of a hydrodynamicapproach, calculatedtransterminator ion flowsare supersonic and relatively assumedin this model, is sufficient to elucidate the important •ose to the observedaverage values. The modelpredicts a aspectsof thehigh speed ion flow. The bulkmotion of theneutral dec•lerationshock at a SZAof about135 ø, consistent with the ion atmosphereis neglected. This is a reasonableassumption, because, tempe• andvelocity observations. in general,the ion velocities are significantly larger than the neutral ones.The assumptionof only a singleion species,O +, was Introduction necessaryin order to make the numerical solution "feasible". Howeverthis is a reasonablygood assumption,because above Ourunderstanding of the coreroilingphysical and chemical about190 kin, O+ is thedominant ion speciesand the main aim of processesin the ionosphere of Venushas advanced significantly thesecalculations is to establishthe ion flows and their effect, which sincethe launch of the PioneerVenus Orbiter (PVO) [Colin,1980]. becomesignificant only above about 200 kin. The chemical Thediscovery of the transonicflow directedfrom the dayside reactionsincluded in themodel are listedin Table 1. In calculating towardthe nightside of theVenus ionosphere was one of themany thechemical source for O+ fromreaction (R1), the [CO2 +] density -surp•ngfacts that PVO established[Knudsen et al., 1980].The wasapproximated from the photochemical equilibrium relationship: typicalhigh altitude transterminator ion velocitieswere found to be ofthe order of 2-3 km/sand are supersonic across the terminator at P(CO•) highaltitudes. The ion flow is drivenpredominantly by thelarge (k 1+ k2)[O] + k3[ne] lXe•-• gradient,which is presentacross the terminator [Knudsen (1) et al., 198!]. The rapid rise in ion temperaturesand whereP(CO2 +) is the photoionization rateof CO2 +. drop/mdomizationin ion velocities beyond a zenithangle of about Thecontinuity equation is: 150ø, led Knudsen et al., [1980]to suggesta shock deceleration of thesupersonic flow deepon the nightside. 1 J 1 3(AoPu0)=Sc_S1 A numberof modelshave been developed to studyand establish rAo B0 themechanisms responsible for thesesupersonic ion flows,but all ••t+••(Arpur) +----- (2) ofthese previous transport models had some limitations (e.g. use of The ion momentumequation in the radialdirection is: assumedvelocity components, limited spatialcoverage, lack of shock"capturing" ability). The need for a full-scale,two- dimensional,shock capturing model has been obvious for many •t[@Ur]+13 2 •1•[AoPurUo] years.However two dimensionalmodels capableof such calculationsarevery complex and computer intensive and have only = + + + beguntobe employed in the field of spaceplasma physics relatively A r (3) recently[e.g. Gombosi et al., !985]. A two-dimensional,time- Theion momentumequation in theangular direction is: dependent,shock capturing single-species (O+) hydrodynamic modelof the Venusionosphere, which solves the coupled cmtinuity,momentum and energy equations forthe altitude range of I 150- 500 km and the solar zenith angle range of 0 ø - 180ø has been developedandis presentedin this paper. This two-dimensional modelwill help us to evaluate the importance ofsupersonic day-to- = + +uSS- uo% + rA0 + aA__o•0 nightflows and, for the first time address, ina quantitative way, the (4) probabilityofshock formation inthe nightside ionosphere ofVenus. Table !. Ion chenfical reaction rates

Reaction Rate constant .... (cmS,s,e, c'1 ) 1on leave from Central Research Institute forPhysics, Budapest, !525,Hungary (R!) CO2+ + O ..... > O+ + CO2 1.0xl 0'!ø (R2) CO2+ + O ..... > 02 + + CO 1.64xl 0'1ø Copyright1991 by the AmericanGeophysical Union. (R3) CO2+ + e ..... > CO + C 1.!4x10-agrc (R4) O+ + CO2 ..... > 02+ + CO 9.4x104ø Papernumber 9!GL00362 0094-8534/ 91 / 91GL-003 62 $03. O0 (.R5)... O. + + H ..... ,--.:>..H+ + O ...... 2.5X10-!1 Tn 1/2 '

801 8O2 Nagyet al.: A 2D,Shock Capturing, HDModel of the Venus Ionosphere

B9undaryand Initial Conditions Theion energy equation is: The upperboundary is setat 500 km altitude,and the lower boundaryis setat 150kin. Thetwo sideboundaries are at solar zenithangles (SZA) of 0ø and180 ø. The verticaland horizontal resolutionis5 kmand 5 ø SZA, respectively. For ion densities (@), 7t arr + chemicaland diffusiveequilibrium conditions are appliedat the lowerand upper boundaries, respectively. For verticalvelocities (Ur),the conditions Ur=0 and 3u•/3r=-0, are imposed atthe upper boundaryon the day and nightside respectively, but no flows into +-rA0 " A0u0• •i •lpi +Ur• +u0r• thesystem are allowed. Vertical (Ur) and horizontal velocities (u0) arekept equal to zero at the lower boundary. The floating boundary 3 K condition,3u0/3r=-0 is imposedat the upperboundary for the =Qi + •r(F•+F•) +u0(F• +F$) +•3( K•3Ti• W)+ r•( •3T• horizontal velocities (u0). _For ion and electron temperature cri,Te) ' theheat fluxes, • = -KOT/3r,are set at theupper boundary. Ti and Te areset equal to theneutral temperatures (Tn) at thelower u•2 +u 0 + - _ • boundary.The floating boundary conditions (3p/r30--0; 3Ur/rO•-0; +-( 2 o2•i-1)miJkT n•S c (u3 +u•2 +(•i p•l)P[•S1 3Ti/r30=0;3Te/r30=0) are set at theside boundaries (SZA = 0ø, O) 180ø),except that the horizontalvelocities (u0) havereflective boundaryconditions (u0 i,n+ 1 = -u0i n) atSZA= 180 ø. •e electronener• equationis: In orderto establisi•confidence' in our model, a numberof differenttest runswere cardedout. In one casewe droppedall • 1 • 1 • sourceterms and perturbed the density to checkif thepropagation speedis equal to the sonic speed. A testmn with all the source terms ••YPe: i•+••[Arur Ar • L Y•:•Pe•+ r• •[A0u0 •e - t J includedwas alsocarried out for the caseof no horizontalvariatior• of the inputdata (photoionization rates, neutral densities, and photoelectronheating rates) to comparewith previous 6ne. dimensional model results. The results of these pseudoone- Pe dimensionalcalculations areused as the initial conditions forthe =Qe +•• ea% + O K r•) kTn- 1)m•Sc- (7e' 1)pc S1numericaltwo-dimensional method calculations.can be foundin A Kimmore [1991].detailed description ofthe (6) InputParameters. where All inputparameters for the model are for solar cycle maximum r radial component; conditions(F10.7 =200). The neutral densities are from Hedin et 0 angularcomponent; al., [1983]except for the H densities,which were taken from subscripti ions; VIRA(VenusInternational Reference Atmosphere) [Keating etfl., subscripte electrons; 1985]and extrapolated (VIRA only gives densities forF10.7 =15,0 subscriptn neutrals; conditions).The presenceof hot oxygenis neglected. p massdensity (=Pi = Pemi/rne ) Photoionizationrates and electron heating rates are calculated Ar,A0 areafunction in radialand angular every5ø SZA and 5 kmaltitude grids using the two-stream model direction,respectively, Nagyand Banks [1970]. Ion production rates were arbitrarily (Ar= r2; A0= sin0); decreasedfrom 90 ø to 100ø SZA by a factorof 100and remaiv•l Sc,S1 ionproduction andloss rate, respectively; constant after 100ø SZA in orderto removethe effectof Ur,U0 plasmavelocity photoionizationonthe nightside. Onthe nightside, theelectron p pressure; heatingrates are set to thevalues at theterminator inorder to ¾. specificheat ratio; approximatetheheating due to the transported photoelectrons ard Frl,F0i ionforce term precipitatingenergetic electrons. Fre,F0e electronforce term k Boltzmann constant; Results and Discussions m mass; T temperature; Thecomputer intensive nature ofthis model allowed only alimit• K thermalconductivity; numberoftest cases tobe run, therefore theinput parameters and Qi netion heating rate, boundaryconditions could not be "tuned forthe best" results, in Qe netelectron heating rate, termsofan overall agreement withobservations. However thei•Jtial urO,u0o bulkvelocity ofthe newly created species. useand purpose ofthis model isto establish thenature of transterminatorionflow and related energetics ina quantitative a• Sphericalgeometry isused and axial symmetry around the sun- selfconsistent manner, rather than fit the observed values exactly. Venusaxis is assumed.The use of these"five moment" transport Suchstudies will be carried out in thefuture, as further computer equationsclearly limits the accuracy ofthe results obtained fromthis resourcesbecome available. Also some of the effects neglect• model.The use of theseequations implies that the the temperatures thesefirst calculations (e.g.the presence ofhot oxygen atoms) w4I! areassumed to beisotropic, the heat flow tensor is replacedby be includedin the workto be presentedin a planned,more simplethermal conductivities, theflow is inviscid etc; nevertheless comprehensive paper.Inthis short paper wepresent theresults from thismodel still gives a goodfirst order description of the major onlyone test nm, which reproduced theobservations ina reasonable processescontrolling theionospheric dynamics. but not exact fashion. A numberofstudies showed that heating mechanism(s) othertim N•rical Scheme Thecoupled system ofequations (2)-(6) are solved numerically those due to solar EUV radiation must be considered inthe study of forion densities (p), vertical and horizontal velocities (Ur,U0), ion theenergy balance ofthe Venus and Mars ionospheres [e.g.Chen et temperatures(Ti)and electron temperatures (Te).The second order al.,1978; Cravens etal., 1980; Kim et al., 1990]. There have accurateGodunov type scheme, which can handle the shock beensuggestions thatthe required heatinflow onthe nightside simulation,is usedalong with a Crank-Nicholsonscheme for shouldbesmaller than that on the dayside [e.g. Whitten handlingthermal conduction and heat sources in theenergy 1986].For the case presented hereconstant heatfluxes of5x10 •' equationsanda Runge-Kutta method isemployed forhandling other cm-2 s -1 for electrons and2x107 eV cm -2 s -1 for ions are impos• sourceterms. The Crank-Nicholsonscheme is second-order atthe top boundaries overthe dayside andreduced to30% of accurateand the Runge-Kutta scheme used is third-order accurate, valueson the nightside. Themodel provides O+ densities, whichin effectgive us the overall second order accurate solutions. horizontalandvertical velocities andion and electron temperatu• Similarapplication of the Godunov type scheme was used by theseare shown in Figurel(a)-(d). K6r6smezeyand Gombosi [1989] for theirtwo-dimensional The calculated iontemperatures areshown inFigure l(a). Att• cometaryatmosphere model. subsolarpoint,Ti is about 2000øK, decreases withincreasing Nagyet al.: A 2D, ShockCapturing, HD Modelof theVenus Ionosphere 803

Ion temperature(øK) for SZA<100ø and increaseswith SZA in the nightside,up to 4000øKpartially due to theconvergence of thetransonic horizontal flows.There is a dropin the calculatedion temperaturesin the regionbetween 90-100øSZA as shownin Figurel(a) dueto the adiabaticexpansion of thehorizontal flow in theregion [Knudsen et al., 1980]. Miller et al.[1980] also found someevidence of a slight 3500 3000 coolingin the regionimmediately antisunward of the terminator. , Thereis alsoa smalldrop in the ion temperaturespast the shock, 2000 whichis causedby a smallincrease in themodel neutral densities 1500 nearthe antisolarpoint and adiabaticcooling resulting from the 1000 radiallyoutward flow. SO0 The calculatedelectron temperatures shown in Figure1 (b) are 25000 roughlycomparable to the measuredvalues, but are somewhat higheron thenightside because the assumedheat inflows are too largefor thelow densityplasma on thenightside. The variationof theelectron temperatures are within about3000 øK acrossall SZA range.The medianof theobserved Te showlittle variationover all SZAs [Miller et al., 1980]. In the daysideVenus ionosphere, the two heat sourcesconsidered are the topside heat flow and the photoelectronheating. The latterdecreases as SZA increaseson the Electron temperature (øK) daysideand drops sharply after the terminator,unless the effect of energeticelectron precipitation is included.Thus the Te would decreasemonotonically from the subsolarto antisolarpoint without an assumedtopside heat flow. The calculatedvelocity fields shown in Figure l(c) reach supersonicvalues up to 3.5km sec-1, whichare in reasonably 60OO goodagreement with the measurements [Knudsen et al., 1982].The 5OOO horizontalvelocities at 397.5 km are shownin Figure 2 together 4OOO with the observed values at 400 km taken from Whitten et al., 3OOO [1984]. The flow has to be slowed down before it reachesthe 200O tooo antisolarregion. A shockwave is formed at around 135ø SZA, o whichcan be seenin the vectorplot shownin Figure 1(c). Beyond the shockwave, the ion densitybuilds up only slightly [seeFigure l(d)] because horizontally transported ions are lost via recombination. There are strong downward motions on the nightside,with a maximumdownward speed of about 600 m sec -1.

• ---- I ' t i I i Velocity 5..•5 _ Horizontalvelocity(km/s) !----I 4 km/sec

-1 I I I I I D 3• 6•1 IlO 120 150 180 Ion densities(cm a) Solar Zenith Angle (øK} Fig. 2 Calculatedhorizontal velocities (solid) with sonicspeed (dotted) at the altitude of 397.5 kin. Squares are the observed 4.$ horizontalvelocities at 400 km taken from Whirten et al. [1984]. 4A 4.1 I 4.O

The flow at the top boundarieson the nightsidewas not prescribed, •.o but was free to float, while the radial flow on the daysidewas kept at zero. This top boundarycondition for the radial flow, together with the decreasingion-neutral friction with SZA and masssink, acceleratethe alreadyexpanding flow further.Downward motions aredominant in the regionbetween 90 ø and 150ø SZA, whichcan be explainedwith the stronger O+ lossat lower altitudes compared to at 1.• higheraltitudes, thus forming a relativesink of O+ ionsat the lower altitudeson nightside.The shock wave front is tilted becausethe flow stopsat loweraltitudes earlier than at higheraltitudes due to the Ion temperature(øK) frictionprovided by thedenser neutral atmosphere at loweraltitudes, especiallythe hydrogenwhich has a nightsidebulge. In a test run Fig. 1 (a) Color-shadedcontour plot of calculatedion without the hydrogencomponent, the shock wave front was not temperatures,(b)plot of calculatedelectron temperatures, (c)vector tiltedand was located further away at around160 ø SZA. Beyondthe plotof calculatedvelocity fields, and (d) plot of logarithmof shockwave, there are relatively weaker (-200 m sec-1) upward calculatedion densities. motionat all altitudes,allowing the plasmato escapeinto the tail. 804 Nagyet al.: A 2D, ShockCaptunng, HD Modelof theVenus Ionosphere

The calculatedion densitiesshown in Figurel(d) decreasewith Cravens,T.E., Gombosi,T.I., Kozyra,J.U., and Nagy, A.F., increasingSZA up to about150 ø SZA andincrease slightly with Modelcalculations of thedayside ionosphere of Venus: respectto SZA beyondabout 150 ø SZA. The calculatedion densities Energetics,J. Geophys.Res., 85, 7778, 1980. above200 km dr9Pby abouta factorof 600from 0 ø to 150ø SZA. Cravens,T. E. S.L. Crawford,A. F. Nagyand T. !. Gombosi,A Boththe peak O* densityand the altitudesof thepeak ion density two-dimensionalmodel of the ionosphere ofVenus, J.G• tend to decreasewith increasingSZA due to the increasing •, 88, 5595, 1983. downwardmotion from the dayto nightside.The photoionization Gombosi,T.I., Cravens,T.E., andNagy, A.F., A time~dependem ratesof O+ andCO2 + remainalmost constant on daysideand start theoreticalmodel of thepolar wind: Preliminaryresults, to drop sharplyafter 80ø SZA. The ion productionrate on the Geophys.Res. Lett., 1_22,167, 1985. nightsideis practically zero. Thus in this model calculation, Hedin,A.E., Niemann,H.B., Kasprzak,W.T., andSeiff, A., transportfrom dayside across the terminator maintains the nightside Globalempirical model of the Venus thermosphere, ionosphereshown in Figure1 (d). Ourmodel includes the loss O + Res., 88, 73, 1983. by chargeexchange to H, but not thereverse source reaction. A test Keating,G.M., Bertaux,J.L., Bougher,S.W., Cravens,T.E., nm which eliminatedthis lossterm, resultedin a fourfoldincrease in Dickinson,R.E., Hedin, A.E., Krasnapolsky, V.A., Nagy, the nightsideion densities.Furthermore we usean upperboundary Nicholson,J.Y., Paxton,L.J., andZahn, U.v., Modelsof Venus of 500 kin; earliercalculations [e.g. Cravens et al., 1983]showed neutralupper atmosphere: Structure and composition, that increasingthe effective ionopausealtitude leads to higher Res., 5, 117, 1985. transtenuinatorfluxes and nighttime densities. Finally it wasshown Kim, J., A. F. Nagy, T. E. Cravensand H. Shinagawa, by McCormicket al., [1987] that the nightsidedensities are very Temperatureof individual ion species and heating due to charge sensitiveto theneutral density model used; they could not reproduce exchangein the ionosphereof Venus,J. Geophys.Rest., 95, the observeddensities with the Hedin et al. [1983] model, which 6569, 1990. was used in this work. Kim, J., Model studiesof the ionosphereof Venus:Ion composition,energetics and dynamics, Ph.D. Thesis,U. of evaluatedThetotal and height found integratedto be aboutplasmalx! OZO•u,,xparticlesacross thesec- te•ninator . This valuewas Michigan,!99!. is very nearlyequal to the totalloss rate integrated over the night Knudsen,W. C., K. Spenner,R. C. Whirtenand K. L. Miller,Ioa side' bec• use the •'megrated escape flux ( hich occursbetween 1 energeticsin theVenus nightside ionosphere, Geophys. Res._ about150and 180 SZA) isonly about lxlg 3' particles sec-. Lea,, 7, 1045, !980. Knudsen,W.C., Spenner,K., Miller, K.L., andNovak, V., Conclusions Transportof ionosphericO+ ionsacross the Venus terminator and implications.,J. Geophys.Res., 85, 7803, 1980. The first full-scale, self-consistenttwo-dimensional calculations Knudsen,W.C., Spenner,K., andMiller, K.L., Anti-solar capableof handlingshock waves were carriedout for the Venus accelerationof ionosphericplasma across the Venusterminator, ionosphere,corresponding to solarcycle maximum conditions. This Geophys.Res. Lett..,8_8_, 241, 1981. modelsuccessfully simulates the plasmaflow overthe 0-180 o SZA Knudsen,W. C., P.M. Banksand K. L. Miller, A newconcept of range,where traditional numerical scheme could not properly handle plasmamotion and planetary magnetic field for Venus,Geophys, theregion beyond 150 o SZA. Res. Lett., 9, 765, 1982. It was againdemonstrated that the introductionof topsideheat K6r6smezey,A., andGombosi, T.I., A time dependentdusty-gas inflowsleads to calculateddayside electron and ion temperatures, dynamicmodel of axisymmetriccometary jets, Icarus, 84, 118, which are consistent with the measured values. In order to 1989. reproducethe measuredelectron temperatures, which is roughly McCormick,P. T., R. C. Whittenand W. C. Knudsen.Dynamics ] constant over all SZA's, the heat inflows had to be reduced of the Venusionosphere revisited, Icarus, 70, 469, 1987. significantlyover the nightside compared to the daysidevalues. Miller, K.L.., Knudsen,W.C., Spenner,K., Whitten,R.C., and The modelpredicts lower ion densitieson the nightsidethan the Novak,V., Solarzenith angie dependence of ionosphericion and measuredones; this is believedto be due to 1) the inclusionof the electrontemperatures and density on Venus,J. Geophys.Res., chargeexchange loss but not the sourcereaction, 2) the use of an 85, 7759, 1980. effectiveionopause which is too low and3) thechoice of theneutral Nagy, A.F., and Banks,P.M., Photoelectronfluxes in the atmospheremodel. These issues will be addressedin the planned ionosphere,J. Geophys.Res., 75, 6260, 1970. future work, mentioned earlien Whirten, R.C., McCormick, P.T., Merritt, D., Thompson,K.W., The calculatedtransterminator ion flows are supersonicand Brynsvold,R.R., Eich, C.J., Knudsen,W.C., andMiller, K.L., relativelyclose to the observedaverage va! es The modelpredicts a Dynamicsof theVenus ionosphere: A two-dimensionalmodel decelerauon shock at a SZA of about 135, consistentwith the ion study,Icaru_.__2ss, 613,317, 1984. temperatureand velocity observations. Whitten,R.C., Singhal,R.P., andKnudsen, W.C., Thermal structureof theVenus ionosphere: A two dimensionalmodel Acknowledgements study,Geophys. Res. Lett., 13, 10, 1986.

The work presentedin thispaper was supported by NASA Grants NAG2-491 and NAGW-1619. Acknowledgementis made to the A. F. Nagy, $. Kim,T. I. Gombosi,Space Physics Research NationalCenter for AtmosphericResearch, which is sponsoredby Laboratory,Department of Atmospheric,Oceanic and Space theNational Science Foundation, and the San Diego Supercomputer Sciences,The Universityof Michigan,Ann Arbor,MI 48109- Centerfor computationresources. 2!43. A. K6r6smezey,Central Research Institute for Physics, Budapes• References 1525, Hungary.

Chen,R.H., Cravens,T.E., and Nagy, A.F., The Martian ionospherein light of theViking observations,J. Geophys,Res,, • 387!, 1978. Colin, L., The PioneerVenus program, J. Geophys.Res., 85. (ReceivedDecember 28, 1990; 7575, 1980. acceptedJanuary 24, 1991.)