JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. A2, PAGES 2289-2300, FEBRUARY 1, 1994
MHD simulationsof boundary layer formation along
the daysideVenus ionopausedue to massloading
J. E. McGary
Clarkson Engineering, Sugar Land, Texas
D. H. Pontius, Jr.
Bartol Research Institute, University of Delaware, Newark
Abstract.
A two-dimensional
MHD
si•nulation
of mass-loaded
solar wind flow
around the daysideof Venusis presented. For conditionsappropriateto a lowaltitude ionopausethe simulationsshow that massloading from the pickup of
oxygenions producesa boundary layer of finite thicknessalong the ionopause.
Within this layer the temperaturesexhibit strong gradientsnor•nal to and away
from the ionopause. Furthermore, there is a shear in the bulk flow velocity across
the boundarylayer, suchthat the (predominantlytangential)flow decreases
in speed
as the ionopauseis approachedand remains small along the ionopause,consistent
with Pioneer Venus observations.The total massdensity increasessignificantlyas
the flow approaches
the ionopause,wherethe contributionof 0+ to the total nmnber
density is a few percent. Numerical simulations are carried out for various •nass
additionrates and demonstratethat the boundarylayer developswhen oxygenion
productionexceedsapproximately
2 x ].0Sm
-• s-•. For the upstreamsolarwind
para•neters and mass loading rates chosenfor these simulations, the results are
consistentwith observationsmade on the daysideof Venusfor averageionopause
conditions
near 300 km.
1. Introduction
the solar wind. This complexity has lead researchers
to numerical simulations of the flow to help ascertain
Pioneer Venus provided evidence of oxygen ions in the dominant physical processesresponsiblefor the obthe dayside magnetosheathflowing with a bulk speed served behavior. Key features of interest include the
that decreasedrapidly near the ionopause[Mihalov et structure and location of the bow shock; the bulk flow
al., 1980; Mihalov and Barnes,1981]. The presenceof pattern in the magnetosheath;the magnetic field deplanetary ions within the sheath, where [he postshock scription;and plasmaquantities(e.g., density,tempersolar wind rapidly carries them away, implies the exis- ature, and pressure)in the flow. Severalnumericalmodtence of a persistent sourcemechanismtha.t placesthem els have been recently developed to examine the effect
directly in the flow. One viable source is the ioniza- of massloading on the flow around Venus. These modtion of the hot neutral oxygen corona through charge els differ from one another in a variety of ways, from
exchange,solar irradiation, and electroni•npact [Nagy the mathematical representation of the physics to the
et al., 1981]. This coronaextendspast the ionopa.
use numericalalgorithms usedto solvethe nonlinear partial
and can contribute significantlyto the ion rnassdensity differential equations that describe the flow.
in the magnetosheath,particularly in the region near
The gasdynamicmodels[Belotserkovskii
et al., 1987;
the ionopause where the flow is slowest. These newly Breuset al., 1987;Staharaet al., 1987]assumethat the
created ions are then carried downstream by the solar flow is dominated by gasdynamiceffectsand that the
wind away from their creation point and contribute to magnetic field is carried along passivelywith the flow.
the total mass flux of the flow.
Typically, the flow field v is calculatedby first solving
The interaction
of the solar wind with Venus is a
the gasdynamicequationsfor steadystate flow and then
complicatedsysteminvolving a collisionlessbow shock, determiningthe magneticfield B usingMaxwells equasupersonicand super-Alfvdnic flow, particle-magnetic tions and the convective electric field E - -v x B.
field interactions, and a planetary ionosphere that in- This method, often called the gasdynamic-convected
teracts directly with the flow to present an obstacle to field model, provides a first-order, fluid description of
the interaction without including the magnetic field effectsand givesa reasonableapproximationto the steady
Copyright 1994 by the American Geophysical Union.
state flow field. An advantage of these models is that
Paper number 93JA03166.
the numericalcodesusedcan be (and havebeen)ver-
0148-0227/94/93JA-03166505.00
ified by extensivecomparisonwith wind tunnel exper2289
2290
MCGARY AND PONTIUS: MHD SIMULATIONS
OF MASS LOADING AT VENUS
The MHD formalismadopted here is an intermediate
iments, an important feature giventhe numericalsophistication
requiredto modelsucha cornplicated
phe- step betweenthe gasdynamicand hybrid approaches,
nomenon. The accompanyingdisadvantage,of course, whereinthe magneticforceson the flowareincludedbut
is that the inherently collisionlesssolar wind must be individualparticlemotionsand finite tyroradiuseffects
of the
assumed to act as a classical fluid. However, this are neglected.This approachallowsexamination
practicehas a long and successful
historyin applica- first-ordereffectsof massloadingto determinethe devitions to the terrestrial bow shock and magnetosheath ationsfrom the gasdynamicflow field. The electromagand is generallyacknowledged
to providean excellent netic fieldsare dynamicallycoupledto the plasmaflow,
first-orderrepresentationof the physics.At Ventisthe and duringion pickup,magneticforcesact to transfer
gasdynamic-convected
fieldmodeladequately
describes momentum from the flow to newly loaded ions. Bethe generalbow shockshapeand position,the aver- causeof the aforementioneddifficulty in implementing
agebulkflowin the magnetosheath,
andthe convected a full three-dimensional MHD calculation, the results
vermagneticfield betweenthe ionopause
and shock.The presentedhere are for a reducedtwo-dimensional
obviousnext step in this approachto the problemis sionof the problemwherethe magneticfield is normal
to incorporatethe magneticforcesand allow the field to the plane of calculation. Quite obviously,the solutions of this analogoustwo-dimensionalproblemcannot
to influence the flow dynamics, i.e., to solve the MItD
equations.However,althoughthis measurecan be ex- be immediately acceptedas a proper representationof
pected to improve agreementwith observations,it is the full physicsof the solarwind interactionwith Venus,
difficult to implementpractically becauseincludingthe and care must be taken to interpret and apply the remagneticfield preventsthe useof symmetryarguments sults appropriately.
to reduce the three-dimensional physical problem to an
equivalent two-dimensionalmathematical one.
An alternative modeling approachinvolvesexplicitly
includingthe particle dynamics. This was done in hybrid schemes that treat electrons as a massless fluid and
In section2 we describethe mathematical representation of the physicsand the computationalmethod employed. We adopt parameters appropriate for steady
mass-loaded solar wind flow in the dayside magnetosheath of Venus for ionopausealtitudes of 300 to 400
km. In section3 gasdynamicsolutionswith and without
ions as individual particles [Harried,1982; Brecht and
Thomas,1988; Brecht, 1990; Moore et al., 1991]. The massloadingarepresented
and compared.An enhanced
electric and magnetic fields are coupledto the electron sourcerate is shownto produce a boundary layer near
and ion dynamics and are determined by iterating be- the ionopausebut has little influenceon bow shocklotween the equationsof motion and Maxwells equations. cation. MHD and gasdynamicsolutionsfor the same
In principle, this representsa potentially more accurate
model
of the ion motion
model
the solar wind
and the concomitant
source rate are then contrasted to study the effect of
currents
the magnetic field. In section4 we discussthe relathan either the gasdynamicor the MHD formulations. tion of the presentresults to other simulationsand to
However, the required computer time and numerical observations. The final section is a summary of our
complexity increasesas more features are included to conclusions.
flow.
To date, the gasdynmnicmodels and hybrid codes 2. Model Description
that havebeenusedto model massloadingof solarwind
The mass-loading
model presentedin this paper is
flow around Venushave producedvarying and sometime
dissimilar results. The various numerical approaches
based on the fundamental assumption that the con-
all use the same gasdynamicformulation and include
the massloading effect via a sourceterm in the continuity equation, but different numerical techniques are
adopted. The simulationsof Breus et al. [1987, 1992]
and Belotserkowkiiet al. [1987]predictthat ion pickup
significantlyincreasesthe bow shockdistancefrom the
planet with a negligibleincreasein plasmasheathmass
density½the overall flow field remains essentially unchangedexcept for the shocklocation. In contrast, Stahara et al. [1987]report that massloadingdoesnot significantly changethe shockposition. The hybrid model
of Moore et al. [1991]confirmsthat findingfor comparable levels of mass loading; for enhancedloading they
find that the shockdistanceincreasesonly about 10%.
Furthermore, the hybrid model includesmagneticforces
and finite ion tyroradius effectsto describeion trajectories in the flow and predicts asymmetries in the oxygen ion distribution. These asymmetriesshoulddiminish near the obstacle boundary where the flow speed is
small and the pickup is fluidlike.
tinuum equationsof magnetogasdynamics
adequately
describethe averagepropertiesof the solar wind flow
in the dayside Venus magnetosheath. This assumption is supportedby a comparisonof in situ measurements made by Pioneer Venus with the results from
the gasdynamic-convected
magneticfieldmodelof Spreiter and Stahara[1980]. Their model properlyreproducesthe behaviorand magnitudeof averageobserved
flow field characteristics,e.g., bulk velocity,protondensity, and plasmatemperature,for muchof the magnetosheathregionfrmn the bow shockto positionsupstream of the ionopause. Extending those equations
to includethe magneticfield effectsand massloading
will improvethe descriptionof the averageflow field,
particularly near the ionopausewhere their influence
shouldbe strongest. (Note that in the earlier treatmentthe magneticfieldexertedno forcebut wassimply
conveeredwith the plasma.) With theseassumptions,
numerical solutionsof the time dependent, single-fluid
MHD equations,with mass loading representedby a
MCGARY
AND PONTIUS:
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OF MASS LOADING
AT VENUS
2291
source term in the continuity equation, are computed physicalprocesswhereby oxygen atoms are photoionto determine the steady state flow tield of the mass-' ized and then acceleratedby the convectiveelectricfielet
to the bulk flow speed. In the present treatment, finite
loaded, magnetizedsolar wind.
The MHD conservation equations for mass, momen- gyroradiuseffectsare neglectedand newly createdions
are treated as having zero initial temperature and vetum, and energy are written
locity, as is reflectedby the absenceof additional source
Op
terms in the momentum and energy equations. This is
+ V.pv- $,,,
oqpv
o-q+ v.
0 pe
consistent,with the resultsof Biermann et al. [1967],
-ve +j xB
+ v. b,(p +
J' v,
(3)
where p is the total mass density, e is the total energy per unit mass, v is the bulk flow velocity, P is
the isotropic plasma pressure,and S.,nis the massloading term due to photoioni•,ation. The electric field E
and the current density j are related to the magnetic
field through Arepete'slaw and the solar wind induced
electric
field:
v x
who showedt,hat the dominant effect of mass loading is
through the continuity equation and that contributions
to the momentum and energy density are small. In the
continuity equation, oxygen photoionizationis considered to be the only mass source,and the in situ produc-
tion of hydrogenphotoionsis not includedin the present
treatment.
This
with
observations
show-
ing 0+ is the majorion in the lowerregionof the Venus
exosphere[Nagy and Cravens,1988]. Furthermore,ion
lossmechanisms,charge-exchange,and electron impact
processes
are not included.The productionof 0+ is assumedto be primarily from the hot oxygencoronawith
an altitude distribution appropriate for solar maximum
conditions'
E+vxB=0
(5)
The magneticfield is coupledto the MIlD conservation
q- qoexp
{-(h-h")
}
equationsand solvedsixnultaneously
with equations(1)
- (3) in a time dependentform usingFaradayslaw:
0B
is consistent
-+ v x
= o
(6)
A good discussionof the various approximationsunderlying single-fluidideal MHD is given by Krall and
Trivelpiece[1973,chapter3]. We note that nearthe obstacle both the flow speed and the distance over which
plasmaparametersvary will becomequite small, and an
anonymousreferee has emphatically assertedthat the
failureto includethe Hall currentin equation(5) invalidates tile present results in this region. Brecht and
Ferrante [1991] examined this matter in their hybrid
Venussimula.tion and reported that the length scalefor
spatial variations should be much greater than 14 km
for this term to be safely omitted. The density and
temperaturegradientsproducedalong the boundaryin
the presentmodel will be shownto have gradient length
scalesof about 100 kin, which suggeststhat the ideal
fluid picture may be compromised. However, in their
Ho
(7)
where h is the altitude, % is the production rate at
a referencealtitude ho, and Ho is the scale height of
the hot neutral oxygenpopulation. The massloading
rate S,,• is given by (7) times the massof an oxygen
atom. This production rate is based on Pioneer Venus
observations for solar rnaximum, where the hot neu-
tral oxygencoronahad a scaleheight Ho - 400 km in
the subsolarregion and a referencenumber density of
3.0 x 104cm
-3 at altitude ho = 400 km [Nagy et al.,
1981]. In accordwit,hthesevaluesand solarmaximum
EUV flux measurements
[Tort and Tort, 1985],the oxy-
genphotoionization
rateadoptedhereis 1.0x 10-• s-l,
whichgivesa valuefor the referenceproductionrate of
qo : 3.0 x 104•n-3s-•. For fixed ionopausealtitude,
we have examined the consequences
of massloading at
rates ranging from zero to approximately10%, which
representsthe oxygen ion production rate at approximately 300 kin. The referencealtitude ho is taken to be
the ionopausealtitude, and the neutral oxygenaltitude
distribution
is assumed to be invariant with solar zenith
hybrid model,Mooreet al. [1991]reportedthat pickup angle.
The techniqueadopted to solvet,he MHD equations
ions produced near the obstacle behave in a fluidlike
fashion,which givesconfidencethat a magnetofluiddescription still capturesthe essentialdynamicsof the solar wind-Venus
interaction.
The reader
is advised
that
potentially important effects related to the Hall term
are absent here, and the interested reader is referred to
(1)- (6) ismosteasilydescribed
bya gasdynamic
formulation. This reducesthe number of terms being manipulated while retaining the fundamental form common
to both set,sof equations. First, the conservationequations for mass, momentum, and energy are cast in or-
Brecht and Ferrante [1991]for a further discussionof
thogonalcurvilinearcoordinates
(t•,r/,() for the simple
tile Hall current, particularly in the context of hybrid
gemhettyof flow around a. blunt obstacle. This representationprovidesa number of analytic expressions
for
models.
The sourceterm &,,. in the continuity equation allows
us to investigatethe effectsof ma.ssloadingon the interaction of the solar wind with Venus. This representsthe
transformations
and reduces stone of the discretization
errors. The equationsare then transformedfrom phys-
ical space(/•,r/,()to computational
space(z,y,z)and
2292
are written
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SIMULATIONS
as
Ut + E:• + F• + G, + S - 0
(8)
--
+
+oe)
+
+ Oz}
1
-
p
pv
(12)
pw
pe
pu
E - h2h3
pu2 .-[-P
puv
(13)
puw
+
pvu
F - hlh3
pv2 -]-P
(14)
pvw
+
pwv
pw2 + P
+
This algorithm is based on discretizing the derivatives
with central differencesand including standard fourthorder, explicit damping terms for stability. Equation
turing, the resultingsolutionsprovideexactjump conditions at the discontinuity. In contrast, shock-capturing
•nethods represent the physical discontinuity in the flow
variables by smearing them across a region of finite
thickness. For external flows with strong shockformation upstream of a blunt body, shock fitting is a reasonable method for determining the flow variables and
the position of the discontinuity. Furthermore, shockfitting methodscan reducethe overall convergencetime
required to obtain a steady st-ate solution. The basic
solution procedure presented here has been used for
the gasdynamic equations to calculate supersonicand
hypersonic flow fields about various three-dimensional
geometries and has been validated against wind tun-
nel data and other numericalmethods[Li, 1982, 1985,
1986, 1987;McGary and Li, 1989].
pw
G - h•h2
AT VENUS
to implement in the solution procedure than shock cap-
p•
U -- hlh2h3
LOADING
(9) is modified for the MHD equationsby including
additional terms for the magnetic field, but the overall solution procedureand numerical algorithms remain
b-asicallyunchanged.
To determine the bow shock boundary upstream of
Venus, the discontinuity is fitted within the flow field
using the Rankine-Hugonoit relations and a local shock
velocity;the shockpositionconvergesas its velocityapproacheszero. Although shock fitting is more difficult
in which
-
OF MASS
(15)
3. Discussion
of Numerical
Results
Severalsimplifyingapproximationshavebeenadopted
in the present model with the objective of focusingatwitere h•, h2, and h3 are the scalefactors for orthogo- tention on the first-order effects produced directly by
nal curvilinear coordinates;the letter subscriptsrefer to massloadingon the flow field. These assumptionsabout
partial derivatives;J is the Jacobtanfor transforming the ionopausegeometry, flow character, and boundary
between the two coordinate systems;and u, v, and w conditionsare rnade to simplify the modeling effort and
are the componentsof velocity in the curvilinear coordi- facilitate identification of the dominant physical pronates. The sourcemat,rix S can include physical source cessesresponsible for the average flow properties obterms for mass,momentum,and energy,althoughonly a servedby Pioneer Venus. The emphasisis on providing
mass source is included in the present case. For Carte- a good qualitative descriptionof the overall flow field
sian coordinatesin the absenceof mass loading, the but not comprehensive detail. The numerical complexscalefactorsareall unityand S - 0, whileequations(8) ity is thereby reduced, as is the required computation
and (9) and the flux vectorsE, F, and G all simplifyto time, and more attention can be devoted to isolating
the commonly representedform of a three-dimensional numerical errors and modelingartifacts from the phys-
Euler gasflow [Andersonet al., 1984].
The sol•tion procedureconsistsof establishingthe
computationalgrid and solvingthe model equationsnumerically with the chosen boundary conditions. The
grid is generated within the cronpurercode as a part
of the solution procedure, and each coordinate is discretized unitbrmly after applying analytic transformations to the nonuniformgrid in the physicaldomain.
This method allows grid points to be clustered in regions of interest, e.g., boundariesnear the shock and
ionopause,and admits conformal mapping transformations to produce smooth contours. The numerical algorithm used to solvethe conservationequations is a modified version of the alternating direction implicit factor-
ical effects.
The two-dimensional ionopauseis modeled as an impenetrable circular boundary of radius Rv along which
the solar wind velocity is strictly tangential, representing a discontinuity betweensolar wind and ionospheric
plamna. Since most of the solar wind flux is deflected
around the ionosphere,a tangential discontinuity is a
reasonable boundary condition to represent the ionospheric deflection of' the solar wind flow. The circular
geometryis usedfor convenienceand shouldbe suitable
for the present model except in the terminator region,
where the modeled shock locations are not expected
to match in situ measurements. In the gasdynarnic-
conveered
field modelsof Spreiterand Stahara[1980],a
ization techniqueof Beam and Warming [1976, ].978]. less idealized ionopausegeometry was used that more
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closely reproduced the observedflow field near the terminator. Specifically, the bow shock in their model is
displacedoutward in the terminator region relative to
that arising frozn a spherical ionopauseboundary. This
result follows directly from the bluntness of the deflecting object they use.
The MHD equations discussedin section 2 are solved
under the assumption that the flow velocity lies in a
two-dimensionalplane perpendicular to the magnetic
field. In other words, the entire configurationis treated
as having translational symmetry along the magnetic
field, and B is everywhereperpendicular [o the [low velocity. Pioneer Venus observationsshow that the average ionopausealtitude is roug]lly 300 km in the subsolar
region and increasesto approximately 900 km near the
terminator with a median dynamic pressure of about
4.5 nPa whichis shownto vary from 3 to 7 nPa [Phillips
et al., 1988]. To be consistentwith theseobservations
the solarwind is assumedto havea large upstreamsonic
Mach number M$o• = 7.3 with other upstream param-
eterstakenas follows:protondensityno• = 10cm-3,
magneticfield magnitude B• - 10 nT, and ratio of the
specificheats?- 5/3. The upstreamsolarwind conditions are the same in all calculationsto allow straightforward comparisonsbetween simulations with different
mass loading.
To vary the mass addition rate, the production rate
of oxygenionsgivenin (7) wasparameterizedby multiplying the referencevalue qo by a factor ranging from 0
to 10. The result was a seriesof simulationsextending
from the nonloadedsolutionto a maximally loadedcase
with a sourcerate 10 times larger than the lower bound
established by Pioneer Venus measurements. All the
figures presentedin this paper represent solutions for
either no massloading or a •nassloadingvalue of 10 q,,.
The ionopausealtitude and the hot oxygencoronascale
height were fixed for all of the flow simulations presented.
Gasdynamic solutions
The steady s•ate gasdynamicmodelswith 0+ production rates less than approximately 6qo indicate a
negligible change from the nonloaded flow field solutions. For values greater than 7qo, however, the calculated flow properties near the ionopause boundary
change dramatically. As shown by the overlay of the
OF MASS LOADING
AT VENUS
•
1.6-
2293
81'K)ck8
2.O
1.4-
P/P
1.2/
,1.5./
•
1.0
//
1.0-
2.4
2.8
.6-
3.0
3.2
3.4
.4-
3.6
3•8
.2-
4.0
-7
0
1.4
1.2
1.0
.8
I
I
I
I
.6
.4
.2
0
X/Rv
Figure 1. Normalizedtotal massdensitiesfor the gasdynamic mass-loadedcalculation(solid lines) and the
gasdynamicnonloadedcase (dashedlines). The subsolar shock position for both models is approximately
1.2 Rw, where Rw is the radius of the ionopauseboundary. Gasdynamic nonloaded contour labels are marked
along the inside ionopauseboundary and the terminato/
with p/p• = 4.4 at the noseand 1.0 at the terminator.
The mass-loaded calculation
shows that the normalized
total massdensity variesfrom 4.5 on the edge of the
boundary layer to 7 at the stagnation point. Near the
circle, gradients are very steep and contourshave been
spacedmore coarselyto improveresolution. Bow shocks
are shown for both cases and marked with dashed and
solid lines. Solar wind conditionsare Mso½ = 7.3,
no• : 10cm-3, and vo• = 600kms-•.
nearthe ionopause
wherethe 0+ creationrate is largest,
producinga layer of enhancedmassdensityalong the
ionopausecomparedwith the NL solution.This finding
is consistentwith a simple scaleanalysisof the continuity equation to determinethe relative importanceof the
sourceterm to the massflux term, as follows. As a parcel of plasma movesa distance L along the ionopause,
two modelsolutionsin Figure 1, the mass-loaded
(ML) its density is enhancedthrough mass loading by the
simulationsshowthat the total massdensity(includ- product S,,.L. For the NL case, the massflux pv near
ing both solarwind H+ and coronalO+} alongthe the ionopauseis approximatelyequalto po•vo•. Taking
ionopauseis significantlygreater than that in the non- that value as an estimate of the ML value for a. parloaded(NL) model. In the stagnationregionthe nor- cel that remains near the ionopauseover a substantial
malized ML densitiesvary from 4.5 to 7, with closely part of the dayside(L • 6000kin), the relativeimpor-
packedisodcnsity
contours(markedwith an opencircle) tance of the terms $,•L/pv is unity for a sourcerate
that are difficult to resolveat this scale;the normalized
density value 7 extendsabout 30o along the ionopause.
As shown in Figure 1, the total mass density at the
stagnation point resulting from ion pickup in the incident flow is p -• 7po,,, almost twice that produced
in the NL model (p • 4p•). Furthermore,the ML
results showa strong inward gradient in massdensity
of about this magnitude. Therefore, massloading can
be expectedto have a considerableeffecton the density
near the ionopause. Furthermore, becausethe production rate decreasesexponentially with height and the
shockedmass flux is large in regionsinside the shock,
the effectsof massloading shouldbe relatively negligible near the bow shock. Indeed, toward the bow shock
2294
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LOADING
AT VENUS
should be interpreted here as the addition of 16 hydrogen ions in place of each oxygenion. Comparisonsof the
gasdynamicsimulationsis that the bow shocklocation temperature plots with observationsat Venus are admovesonly slightly outward from the planet in response mittedly equivocal, but the plots are included because
to mass loading the flow, which will be discussedfur- they dramatically display the boundary layer.
Plasma flowing along the stagnation line will have
ther below. It is important to note that while these
results are physically reasonable, care must be taken its flow energy per unit volume converted into interto allow enough time for the computer simulations to nal energy and work at the ionopause. Becausenewly
iterate sufticiently and properly capture the subtle ef- addedionsdo not contributeany energyto the flow,
fects associatedwith this layer in the mass-loadedflow. this statement is true for the ML case as well. The
The NL gasdynamic calculations typically convergein upstream conditions are held fixed throughout, so the
about 500 time steps,whereasdependingon the size of internal energy per unit volume at the stagnation point
the time increment, the ML solutions require between is determined by the compressionof the flow. There2000 and 6000 steps to convergeto the solution where fore, for not-too-dissimilar flow fields at the nose, tile
the boundary layer appears.
stagnation massdensity of the ML flow would be larger
The next point of comparisonbetween the NL and than that of the N L, and the stagnation temperature
ML gasdynamicsimulationsis the temperature distri- of tile ML flow would consequentlybe lessthan that of
bution within the bow shock. Becausethe gasdynamic the NL. Outward frownthis region,where massloading
model treats the flow as a continuous medium, the is small, the ttow behavesas a perfect gasfor both cases;
calculations till now have been insensitive to the conthe plasma.is heated acrossthe shock and the temperstituent particles of the fluid. To examine temperattire, ature increasestoward the planet. The ML flow then
mass density must be converted to number density by coolsas it nears the ionopausewhere ion production bespecifyingthe particle species.However,a single-fluid comessubstantial. Figure 2 comparesthe temperature
model cannot properly treat incident and loaded ions of distributions for the NL and ML gasdynarnicsolutions
different species, and a compromise is necessary. The
and supportsthis reasoning.Mass loadingsignificantly
temperature plots shown are calculated for a gas of pure cools the flow along the ionopauserelative to the nonhydrogen, which means that the specified loading rate loaded flow. Furthermore, the ML flow in this layer
exhibitsa sharp t.emperat•re gradient, with the temperBow shocks
•--.- ... ?
ature along the stagnation line decreasingfrom 19To•
to 13 Toe near the stagnation point over a distance of
approximately 300 kin. Mass loading effectively cools
1,4-T/Too
the flow as zero-temperatureoxygenionsare introduced
into tile liow and thermalized. The result is a boundary
layer with enhanced density and reduced temperature,
adjacent to the ionopause.
the ML mass densities anti behavior approach those of
the N L sixnulation. Another important feature of these
The presstiredistributionin the magnetosheath
(Figure 3) indicatesthat the dynamicsof the flow are not
1.0--
significantlyaffectedby massloading. This result is not
an intuitively obvious one; there is no simple and convincing argument as to what effect massloading should
have on the bow shock position, and the resulting pressure gradients are not easily determined without performing the actual calculation. However, the simulations indicate that the pressuredistribution in the flow
is basically unchangedby mass addition, and the pattern of streamlines is approximately the same as before.
3
14
15
.6-
16
17
.2-
0
1.4
1.2,
1.0
I
I
I
I
I
.8
.6
.4
.2
0
X/Rv
Figure 2. Normalized plasma temperaturesfor the
gasdynamicmass-loaded
(solid lines) and nonloaded
(dashedlines) simulations. The nonloadedsolution
is labeled along the ionopausefrom 20 in the subsolar region to 7 at the terminator. The mass-loaded
solution shows the norma.lized temperature decreasing
from 19 to 13 along the stagnation line toward the
ionopause. The temperaturesfrom both solutionsare
approximately equal in the upper magnetosheathre.
glon.
Simpleenergyconsiderationsthen suggestthat the ML
flow speedshould be decreasedin the massloading region relative to the NL case, becausea larger density
must be transported. This conclusion is borne out by
Figure 4, which compares the flow speedsfor the two
gasdynamic cases. Adjacent to the ionopause, the ML
flow speed is less than that of the NL flow, and a velocity shear in the radial direction is present for the
ML flow that is absent in the NL case. Furthermore,
there is a region near the terminator where the ML
flow speed remains almost constant; the flow appears
to be transonic through most of this region. Pioneer
Venusmeasurements
[Mihalovet al., 1982]indicatethat
magnetosheathflow speedsabove the ionopauseare less
than thosepredicted by the gasdynamic-convected
field
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/
1.0-10
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2295
duced, and the flow speed is even lower. As the MIlD
flow deceleratestoward the stagnation point, flow energy is now converted into both thermal and magnetic
pressure, as well as into work. The fluid elements are
therefore not compressedas much as they would be in
the absenceof the magnetic field, which implies that
the resulting plasma pressure in the MHD case should
be less than that for gasdynamicflow. This analysis
is supported by a comparisonof Figures 3 and 5. The
magneticfield (Figure 9) enhancesthe barrier provided
by the plasma pressure,thus diverting the flow away
from the body at larger distances from the ionopause
and increasingthe divergenceof streamlinesin the nose
region. Note that the MHD plasma pressurecontoursin
Figure 5 exhibit a stronger gradient in the noseregion,
both norma] and tangential to the ionopause,than the
gasdynamicpressures
(Figure 3). The speedof the flow
in the MIlD case(Figure 6) is reducedcomparedwith
the ML gasdynamiccase, and the overall mass density
.8-
.6-
.4-
.2-
(P.'igure
7) is enhanced.Thesetwo findingsare consistent with the continuity equation since the same source
1.4
1.2
1.0
.8
.6
.4
.2
0
st;rengthis usedin both models. At the stagnationpoint
X/Rv
the massdensity is 24 po•, an increaseof roughly a factor of 4 over the NL gas flow results, whereas the velocFigure 3. Normalizedplasmapressurefor the gasdy- ity decreasesto very small values. In tl•e terminator renamicmass-loaded
(solidlines)and nonloaded(dashed
lines) simulations. The pressuredistribution is not gion •he flow velocitiesfrom the simulationsagree wil;h
changedsignificantlyby massloading,and the two cases the observedflow behavior[Mihalovet al., 1982],which
exhibit a velocity shear and an overall flow speed less
are similar.
than NL gasdynamicpredictions. Also, the stagnation
model and decreasenoticeably as the ionopauseis approachedfrom above. Mihatov et al. [1982]report that
the observedflow above the ionopauseappears to be
retarded as though a boundary layer is present, which
1.6-
Bow 8hock8.
/
/
1.4-
(v/voo)x/7M
Soo
is consistentwith the results presented here.
MHD
solutions
The single-fluidMHD equationswere initially solved
without a sourceterm. For upstream magnetic fields
ranging from 0 to 10 nT, the nonloadedMHD flow field
was al•nost identical to the N L gasdynamiccase, exhibitingonly minor quantitativechanges.The contours
of the flowvariables(not shown)are qualitativelyidentical, with only small perturbationsdue to the magnetic pressureterm. These differencesare quite unlike
the severedistortionsof density,temperature, and flow
speedthat arose owing to mass loading in the gasdynamiccase. For the maximumsolarwind magneticfield
strength of 10 nT and identical upstream flow parameters, the bow shock location was essentiallyunchanged
from the value 1.2 Rv found in the nonloadedgasdynamic calculations.
--
7
1.2-
We therefore
turn to the effects of
the magnetic field on the mass-loadedsolutions.
The soh•tionsfor the MIID simulationsshownin Figures5- 9 can be comparedwith the previouslydiscussed
gasdynanficresultsto elucidatethe effectsof the magnet.ic field on the mass-loadedflow. The principal effects are that the bow shock is displacedfarther from
the planet and t,he characteristicsof the boundary layer
are accentuated, i.e., the thickness increases,the mass
densityis enhancedwhile the plasmatemperature is re-
6
1.0-
8
7
.6-
.4-
.2-
0
1.4
I
1.2
1.0
.8
I
I
I
I
.6
.4
.2
0
X/Rv
Figure 4. Normalizedflow speeds,where 3' - 5/3
and v/? Msoo - 9.4. The mass-loadedsimulation
(solidlines) showsthat a velocityshearexistsacross
the boundary layer with decreasingspeedtoward the
ionopause. In general, the mass-loadedflow speeds
along the ionopauseare lower than in the nonloaded
case (dashedlines), remainingalmost constantover
roughly the last half of the daysideionopause.
2296
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4. Comparison with observations and
1.6-
other
1.4--
LOADING
P/Poo
models
The two-dimensionalMIlD simulationspredict a subsolar shock location between 1.2 and 1.3 Ru , depending on the massloading rate, in contrast to the Pioneer
1.2--
VenusOrbiter (PVO) observationsthat showthe averagesubsolarshockdistanceto be about 1.5R7 [Zhang
et fl., 1•0]. In addition,the calculatedmaximumvalue
1.0-
of the magneticfield is lessthan the averagevalue taken
from measurementsin the magneticbarrier region;however, the magnetic peak value relative to the ionopause
seemsto fit the description of the magnetic barrier dis-
.6-
cussedby Zhanget fl. [19•1]. The bow shocklocation
and magnetic field magnitude predicted by the models
.4-
of Brechto,nd Fewante[19,91]and Mooreet fl. [1•1] are
in better agreement with observationaldata than the
.2-
two-dimensional MHD model presented here, though
all the models are subject to various nontrivial scaling
considerations. In particular, each method represents
the ionopause as an obstacle with fixed location and
properties, and the identification of that barrier with a
particular physical position determines where features
0
1.4
I
1.2
1.0
.8
I
I
I
I
.6
.4
.2
0
X/Rv
in the solution
Figure 5. Normalized plasma pressurefor the MHD
mass-loadedsimulation, where the boundary conditions
are the same as in the gasdynamic simulations. The
pressuresare generally lower than those of the gasdynamic
solutions.
temperatureis decreasedby almosta factorof 4 (Figure
8), with T = 3To,, a,nd the flow reinfins cool toward
are located
relative
to Venus.
Mooreet fl. [1•1] scaledthe gasdynamicsolutionof
Stahavaet fl. [1•87] to match the subsolarshockposition to that in their hybrid simulation. They found that
the scaledgasdynamicobstaclefell within the magnetic
barrier identified in the hybrid results and suggested
that the magnetic barrier should be consideredas the
appropriate flow obstacle in a gasdynamic approach.
1.6-
the terminator.
The calculated bow shockmovesslightly outward in
responseto adding the magnetic terms to the massloaded gasdynamic equations, and the magnitude of
the displacement seems to correlate roughly with the
boundary layer thickness for both cases. A suitable
description of the boundary layer is needed to explore
properly this correlation between the boundary layer
thickness and the shock distance. ttowever, choosing
a characteristicscale length to representthe boundary
layer thicknessover the entire region is not stra.ightforward, and should probably considerother flow field features should probably be consideredas well. In the subsolar regionthe boundary layer is easily identifiedby the
rapid decreasein temperature normal to the ionopause,
eve• though the massdensity and velocity changeacross
this region. Near the terminator, however,the temperature gradient differs little from that of a gasdynamic
solution and is probably not a. good characteristicindicator. In that region the velocity shear is a better indicator of the boundary layer. For the moment, defining
• to be the boundary layer thickness associated with
the temperature gradient along the stagnation line, the
present simulations indicate that the bow shock is dis-
placedby massloadinga distanceof approximately•/2.
1.4-
(v/vo)7M
1.2-
1.0-
.6-
.4-
.2-
0
1.4
1.2
1.0
I
I
I
I
I
.8
.6
.4
.2
0
X/Rv
Figure 6. Normalized flow speedsfor the MHD mass-
loadedsimulation,where? = 5/3 and v/? Ms• = 9.4.
The flow speedsare lower than calculatedfor the gasdynamic caseand extend over a larger region. The velocity
sizeof the obstacleby about •/2, thus [orminga shock shear has increasedfrom the additional magneticforces
on the flow.
at a slightly larger distance than the nonloadedcase.
This suggeststhat massloading effectivelyincreasesthe
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2297
cant effect and predicts a similar expansion of the bow
shock when the source rate was increased by a factor of
5. The hybrid simulationof Brechtand Ferrante[1991]
P/Poo
1.4
1.2
2.0
1.0
4.0
did not include mass loading; however,they varied solar wind conditions and the obstacle radius, as well as
numerical features suchas grid cell size and simulation
domain, to explore their influence on bow shock location. Depending on these input parameters, the shock
distancevaried betweenroughly 1.3 and 1.4 Rv, where
the best results were obtained
.6
.4
.2
1.4
6- 24
1.2
1.0
.8
.6
.4
.2
0
X/Rv
Figure 7. Normalized total mass densities for the
MHD mass-loaded calculation, showing a larger increase in the total mass density due to mass loading
in comparisonwith that for the gasdynamicsimulation.
Along the stagnation line, the density falls from 24 p•
at,the stagnation point to 6 p• at the edgeof the boundary layer. The circle representssteep gradients in density toward the boundary, which are difficult to resolve.
However, comparisonsbetween nonequivalenttypes of
numerical
simulations
in simulations
that
used
a realistic barrier altitude and a physical domain that
extendedto solar zenith angle 900. Varying the cell size
was shownto changethe standoffdistancein their simulations, and they concluded that resolving gradients
was important in determining the shock location.
Another differencebetween the present solutionsand
the mass-loaded hybrid simulations is that the hybrid
codespredict a density depletion near the obstaclesurface where our results show an enhancement. However,
this disagreementis not as seriousas it might seem because the depletion in the hybrid calculations occurs
only in the solar wind ions. Single-fluid models do not
distinguishincidenttt + ionsfrom planetary0+ ions,
so a comparisonwith hybrid results requires combining
speciesand consideringthe total mass density. Moore
et al. [1991]reporta significant
densityof planetary0+
ions near the obstacle,but they did not provide specific
numbers. In either approach, planetary ions are likely
to be concentrated where the loading rate is greatest,
1.6
T/Too
1.4
should be made with caution ow-
ing to the difficulty of isolating particular physicalef-
1.2
fects from those inherent in the different approaches.
The influenceof the magneticfield can be studiedmore
directly in a fluid model where its strength can be set
to zero. Coatparing runs of the presentmodel with the
strength of the magnetic field varied does not support
this hypothesisof Moore et al. [1991]. Rather, our results indicate that the magnetic barrier occursas a feature within Lhe post-shockflow field and has a relatively
12
1.0
.6
small effect on bow shock location.
A more meaningfulcomparisonamong different simulations can be made regardingthe effect of massaddition on bow shocklocation. Including massaddition
in the MHD model dramatically changesthe flow field
field character but only increases the bow shock dista.nces by approximately 10% for the maximum source
rate consideredhere. This agreeswith the gasdynamic-
convectedfield model of Stahara et al. [1987], which
.4
.2
o
1.4
I
1.2
1.0
I
.8
.6
I
.4
I
!
.2
0
X/Rv
showedan insignificant change in the bow shock position due to tnassloading. In contrast, the gasdynamic Figure 8. Normalized plasma temperatures for the
MHD mass-loaded simulation. The temperature desimulationof Belotserkovskiiet al. [1987] predictsa creasesfrom 20 Tc• at the edge of the boundary layer
much larger increasein the bow shockdistance, though to 3 T• at the stagnation point. The temperature gracuriously, the presenceof mass loading did not alter dient is enhancedby including the magnetic field in the
the density at the subsolarpoint. The hybrid model of simulations, and the overall temperature is significantly
Moore et al. [1991]concurswith the presentresultsthat lowerthan that calculatedfrom the gasdynamicapproxan ordinary intensity of mass loading has an insignifi- i•nations.
2298
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1.6-
2.2
2.0
1.4-
1.8
B/B
1.6
1.4
1.2-
1.2
1.0
.8
2.4
2.6
.6-
3.6
.23.8
1.2
1.0
LOADING
AT VENUS
we can speculate on how the mechanism may affect a
hill three-dimensionalMHD mass-loadingmodel. The
population of solar wind ions at •he boundary should
be reduced from the gasdynamic result, but the influence of planetary ions on its effectiveness is not clear.
The depletion of solar wind ions is most pronouncedon
streamlines close to the ionopause, becausethat flow
passesnear the stagnat,ion point where the flow speed
is slowest,thus giving the Zwan-Wolf effect rnore time
to operate. Of course, it is in this region that mass
loading is greatest. Becausethere is no magnetic force
along B, momentum along the field can be acquired
only from the pressuregradient in the particles. However, the field-aligned gradient is severely reduced as
t,he flux tube convectsand is depleted(cf. Zwan and
Wolf [1976, Figure 6], using the adiabatic relation to
convertparticle densityto pressure),whichmeansthat
.4-
1.4
OF MASS
.8
.6
.4
.2
0
X/Rv
Figure 9. Normalizedmagneticfield magnitudefor the
MHD mass-loadedsimulation, where Boo - 10 nT. The
magnetic field magnitude is labeled along the ionopa,use
a.nd the terminator. The nearly constantmagneticpressure indicates that most of the gradient in the total
pressureoccursthrough the thermal component.
and their density will depend inversely on the rate at
which they are removedby the flow. Becausethe mag-
neticfieldslowsthe flowin the subsolarregion(compare
Figures4 and 6), it is to be expectedthat the subsolar massdensity is enhancedcompared with that in the
gasdynamic
simulations(compareFigures1 and 7).
The depletion of solar wind ions in the hybrid codes
appearssimilar to that predicted on theoreticalgrounds
from single-fluid MHD arguments. Zwan and Wolf
[1976]considered
the flowof a magnetizedplasmaabout
a compact obstaclewith the undisturbed magneticfield
oriented normal t,o the incident flow. They argued that
a postshockflux tube will be preferentially compressed
where it is closest to t.he obsta,cle, t,h•tslea•ling t,o a
field-alignedpressuregradient. The resulting force acceleratesflow along tile magnetic field away from that
point and reduces the plasma density there. The depletion continuesa.sthe flux tube is convectedthough
the magnetosheath, and the incoming flow is increasingly deflected by magnetic rather than particle pressure. Zwan and Wolf predicted that the number density could be depleted substantially relative to gasdynamic predictions, and this hypothesishas receivedconfinnation from observationsnear the terrestrial magne-
it will become increasingly difficult to accelerate tile
newly added mass along B. If changesin the pressure
distribution causedby mass addition are relatively minor, as the present study suggests,then the available
force must accelerate a much larger total mass density,
thus reducing the speed of the flow along the field and
increasingthe time for planetary ions to accumulate.
These plausibility arguments suggestthat the ZwanWolf effect may be q,it.e different at Venus, but a full
examinat,ion of this question must await a full threedimensional
MHD
simulation.
rrhe
abundance
of in-
jected oxygen ions relative to the shocked solar wind
protons at the stagnation point will almost surely be
different, but the total mass density near the boundary
layer should be larger than the two-dimensional nonloaded case,even if flow parallel to B decreasesthe stagnation proton density substantia.lly. We predict that
a. boundary layer will still be present, in a fitll threedimensionalmass-loadedsolution and that the magnitude of the magnetic field will be an important factor
determining its extent.
At altitudes higher than 400 krn the total oxygen
population is dominated by hot neutrals forming an
oxygen corona that remains almost constant with so-
la.r zenith angle [Nagy et al., 1981; Nagy and Cravens,
1988]. The oxygendensityat 400 k•n deducedfrom the
PVO ultraviolet spectrometermeasurementsis roughly
3 to 6 x 101øm
-3, implyinga productionrate of 3 to
6 x 104m-3s-•. The presentsimula.•ions
considered
oxygenion massloadingrates between4 x 104 and
4 x 105m-3s-•. The production
rate valuesrequired
to generate distinctive boundary layers were found to
be of the order2 x 10•m-3 s-• whichis approximately
the 0+ productionrate due to the coldambientneutral
oxygenatomsat loweraltitudesnear 300 km [Nagyet
al., 1980]. Small changesin ionopauseradius,from 400
to 300 kin, do not alter the general character of the cal-
topause.(Note that in hybridsimulations
similareffects culated flow field; the adopted value of qo is therefore
may arise owing to violations of the frozen field condition and to the assumption that ions are absorbed at
•he boundaryand removedfrom the system.)
The two-dimensional aspect of the present work prevents the Zwan-Wolf effect from occurring. However,
applicable for an averageionopausealtitude of 300 km
for the given solar wind parameters. The ion massloading rate required to generatethe boundary layersin the
•nodel may be due to the cold oxygenatom component;
however,equation(7) wouldrequirean additionalterm
MCGARY
AND
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SIMULATIONS
with a correspondingscale height of the thermospheric
oxygen distribution to properly model this effect.
In the present model the mass addition rate and
ionopausealtitude are externally choseninput parameters, but somemethod of consistentlydetermining those
values is necessary for a complete model of the solar
wind interaction with the Venus ionosphere. A simple
relation is not currently available to describe how the
upstreamsolar wind conditions(po•,Po•,vo•, Bo•) de-
OF MASS
LOADING
AT VENUS
2299
Comparison of the two mass-loaded calculations, the
gasdynamicand the MHD, indicates that the magnetic
field enhances the barrier to the flow, increases the
boundary layer thickness5, and displacesthe shockoutward from the planet a distance proportional to 5. The
change in bow shock location compared with the nonloaded case is fairly small in both these simulations,
in agreementwith the findingsof Staharaet al. [1987]
and Mooreet al. [1991].Increasingthe solarwind mag-
termine the position and nature of the ionopauseand
the boundary layer. Examination of the data shows
t,hat the ionopausealtitude varies with solar wind conditions in a way not completelydeterminedby either the
netic field strength while keeping the other upstream
parameters fixed increasesthe boundary layer thickness
and expands the overall flow behavior in the boundary layer. Mass loading dramatically affects the region
ram pressureor magneticpressure[Braceet al., 1980]. where ion pickup is large and leavesthe remaining magObservationsshow that, in general, the ionopausealti- netosheathflow field relatively unchanged. Therefore,
tude correlateswith the total pressure[Phillips et al., the flow fields between the shockand the boundary layer
1988]. However, even though the averageupstream
are similar
ram pressureis larger than either the average thermal
or magnetic pressure by at least an order of magni-
These simulations model boundary layers for average
ionopauseconditions on the dayside Venus ionosphere;
knowing or understandingthe extent of the boundary
la.yer flow field at higher altitudes may not be easily
extrapolated froin the solutions modeled at lower altitudes without performing the full calculations. It is not
yet known what solar wind conditions and ion source
rates are necessary to produce the boundary layer for
a higher ionopause,and this question needsto be addressed in the future. Because the boundary layer is
associatedwith the ionopause,ascertainingthe conditions required for its formation will provide more information on the ionopauseand the physical processesdetermining its structure. Finally, the simulations show
that the boundary layer is associated with a velocity
shear extending over a range of 100 to 300 km adjacent
to the obstacle surface and is likely to produce turbulence along this boundary and further decreasethe flow
speed. Further simulations such as those describedhere
will provide greater insight into the nature of the transition between the ionosphereand magnetosheath.
tude (pv2 • 1 x 10-8, while P2• ----6 x 10-•ø and
Pt• • 4 x 10-•{•dyncm-•),thereis no clearrelation
between the ionopausealtitude and the total pressure.
Another potentially important factor determining the
ionopausealtitude may be massloading. Becausemass
loading ca.n dramatically alter the flow immediately
above the ionopause,its position may be determined by
the regionwheresourcesbecomeimportant and significa,ntly slow the flow. Additional si•nula.•ionsare needed
to establish a correlation between the boundary layer
thickness and upstream solar wind conditions and to
evaluate the relative importance of the various con•rol-
ling parameters,e.g., the massflux, ram pressure,and
the thermal and magnetic pressure. A first-order analysisusingthe continuityequationsuggeststhat the upstream mass flux is the controlling parameter for the
gasdynamicmodel, but the problem is coupledto the
Mach nu•nber and shock distance, and a more involved
treatment may not support this simple relation.
5. Summary
for the nonloaded
and mass-loaded
flows.
and Conclusions
Acknowledgments.
We thank J. E. Bishop, P. A.
Cloutlet, P. C. Gray, T. W. Hill, J. G. Luhmann, S.S. Stahara, and R. A. Wolf for useful and stimulating conversaWe havemodeledmassloadingof the solarwind flow tions. Computer resources for nmnerical calculations were
around the daysideVenusionosphereusingthe single- provided by Lockheed in 1989. The manuscript was writ-
fluid gasdynamicand MIlD equationswith ion pickup ten while JM was a visitor at the Bartol Research Institute
representedby a sourceterm in the continuityequation. in 1992, and he wishes to extend his thanks for hospitality.
Using0+ as the pickupion and a para,
meterizedsource Additional support provided by N.J. Logsdon and NASA
rate, the simulations indicate the existence of a boundary layer along the ionopauseconsistentwith the average ionopauseconditions observedfor altitudes near 300
grants NASW-4805 and NAG 5-1573.
The editor
thanks
J. G. Luhmann
and another
referee for
their assistancein evaluating this paper.
kin. The characteristicfeaturesof the boundary layer
are decreasingtemperaturestoward the ionopausewith References
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(ReceivedAugust 21, 1992; revisedOctober 29, 1993;
acceptedNovember4, 1993.)