JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. E9, PAGES 21,099-21,115, SEPTEMBER 25, 1996
Ejecta entrainment by impact-generated ring vortices:
Theory and experiments
O. S. Barnouin-Jha
and P. H. Schultz
Departmentof GeologicalSciences,Brown University,Providence,RhodeIsland
Abstract. Laboratoryexperiments
indicatethatan advancingejectacurtaindisplacesthe atmosphereandcreatesstrongwindsthatentrainlargeamountsof fine-grainedejecta. A theoretical
modeldescribingthe interactionbetweenan atmosphere
andtheoutwardmovingejectacurtainallows oneto estimatethe magnitudeandvelocityof the windsin the inducedring vortexin orderto
establishthe importanceof suchwindsat planetaryscales.Model estimatesfor the initial magnitudeof the flow within the basalvortexmatchexperimentalresultswithin observationaluncertaintyandrevealthatthe flow in thering vortexis turbulent.Successful
comparisons
of the vortex generation
modelwith experiments
allowpreliminaryapplications
to be madeat muchbroader
scales.Curtaingenerated
windsby a 30-km-diameter
cratershouldentrainejectaparticlediameters
smaller than 5 mm on Mars, 30 cm on Earth, and 8 m on Venus.
Introduction
1982; Ivanov et al., 1994]. Schultz [1992a] suggested that
Ejecta trajectoriesfrom an impact generatea curtain, forming
an inverted hollow cone or frustumthat expandsoutward [Gault
et al., 1968; Oberbeck, 1975]. In an atmosphere,the lower
portion of the ejecta curtain acts as a barrier that forces
atmospherearound it, causingflow separationat its top edge
[Schultz and Gault, 1979, 1982; Schultz, 1992a] in a manner
similar to when flow impingesupon a knife's edge [Prandtl and
Tietjens, 1934]. A ring vortex forms behind this advancing
curtain. Airflow impinging on the curtain and within the vortex decelerates and entrains sufficiently fine grained ejecta
(Figure 1). As the flow decays,winds in the vortex may initially scour ejecta and target surfaces,with the coarsegrained
ejecta depositedin contiguousramparts,and finer fractionsin
flow lobes. Such featuresare strikingly similar to nonballistically emplaced ejecta deposits on Mars, Earth, and Venus.
Evidence consistent with atmosphericeffects on ejecta emplacementincludes(1) absenceof secondariesaroundcraters
with fluidized ejecta on Mars [Schultz and Singer, 1982]; (2)
two stagesof ejecta emplacementwhere the outer lobe of multilobe impact craters on Mars and Venus override the inner
ejecta lobes [Schultzand Gault, 1979; Schultz, 1992b]; (3)order of magnitudeestimatesof the wind intensitycreatedby the
curtain exceed present-daywind strengthsknown to transport
eolian material [Schultz, 1992a]; (4) nonballistic emplacement of ejecta indicated by deflections around obstacles
[Schultz, 1992b]; and (5) consistencyof first-orderrun-out distances with crater size and with regional lithology [Schultz,
1992a].
Other proposedmechanismsdescribe the nonballistic emplacement of ejecta facies on planets. One mechanismhas
been attributed to the presence of water or volatiles which
would reduce the viscosity in initially ballistically emplaced
ejectaand result in flows similar to mud-slide[Carret al., 1977;
Mouginis-Mark, 1979, Gault and Greeley, 1978; Greeley et
al., 1980; Horner and Greeley, 1982; Mutch and Woronov,
Copyright1996 by the AmericanGeophysicalUnion.
Paper number96JEO1949.
0148-0227/96/96JE-01949
$09.00
suchwater-saturatedejecta most likely characterizesthe inner
facies in certain highly fluidized ejecta styles. Alternatively,
volatiles vaporized at impact might entrain some ejecta into a
debris cloud that would flow as a density current and deposit
ejecta[Cart et al., 1977;Wohletzand Sheridan,1983].
Nonballistic ejecta featuresalso could result from secondary
cratering. Secondaryimpactsmight createa flow-like debris
surgeof primary and secondaryejecta [HOrz et al, 1977, 1983;
HOrz, 1982]. An atmosphereentrainedin the falling ejecta
could enhanceturbulence created by the ballistic processgenerating flow-like features[HOrz, 1982; Schaberet al., 1992],
althoughthe specific mechanicsof this processwere not discussed. Volatiles releasedand entrainedduring secondarycratering could enhancerun-out during ejecta emplacementon
Mars [Schultz, 1992a].
Becausean atmospherealso causesthe ejecta curtain to decelerateand steepento near vertical,gravity could causeejecta
small enoughwithin the curtainto collapse[Schultzand Gault,
1979]. Ejecta would then flow downwardand outward,sliding
alongthe surfaceas a sturzstromas describedby Hsii [1975] for
avalanchesto producelobate structures[Schultzand Gault,
1979, 1982; Herrick and Phillips, 1994]. Other than initially
deceleratingthe ejecta, the collapsingejecta model of Herrick
and Phillips [1994], however,ignoresthe action of the ejecta
curtain on the atmosphere.
Laboratory-scale experiments and simplified calculations
suggestthat curtain-drivenvortex winds may stronglyaffect
ejecta entrainmentand depositionduring planetarycratering
[Schultz, 1992a, b]. However, previous calculations of the
strengthof thesewinds were basedsimplyuponthe advancing
velocity of the ejecta curtain [Schultz, 1992a]. The present
studydevelopsa fluid dynamicalmodelfor the generationand
initial decay of these curtain-drivenwinds, therebyaddressing
quantitatively the physical processesresponsible for the
winds formation and importanceat broad scales. The winds
calculatedby the model are the resultof the ejectacurtainmechanically displacing the surroundingatmosphereat late
stagesin the impactprocessto producea ring vortex. Sucha
ring vortex is not the result of shockand thermal effectsthat
occur early in the cratering process,well before late-stage
21,099
21,100
BARNOUIN-JHA
ANDSCHULTZ:
EJECTA
ENTRAINME• BY RINGVORTICES
Inward folding
ejectscloud
,/%
I
movesfrom left to right, the upper vortex possesses
a clockwise flow while the lower vortex possessesa counter-clockwise flow (Figure2; seeTable 1 for variablesusedin the theoretical models). Laboratory observationsindicate that the
lower or basal vortex controlsthe emplacementof fine grained
ejects, while the upper vortex dissipatesrapidly [Schultz,
1992a, b]. Consequently,this studyfocusesonly on the dynamics of the lower vortex.
clearzone
/
Air flow
Shearstresses
actingat the innersurfaceof the ejectscurtain
due to the upward motion of individualejects in the curtain
probablydo not affectthe formationof the lower vortex. To
demonstratethis, a solid plate analogy for the ejects curtain
indicatesthat the shearstressx acting on the lower vortex by
ejectsparticlemotionin the curtainis givenby [Rosenhead,
Figure 1. Atmosphericresponsefollowingthe passageof 1963; White, 1986]
theejectscurtainat thetimeof craterformation[afterSchultz,
0.33206pu2 (UX/V)
-1/2
1992b]. Note that ejects travelsin two modes:larger ejects
= 0.0135pu2(UJ/V)
-1/7
travel alongballisticpathswhile atmospheric
flow entrains
(la)
(lb)
aerodynamically
decelerated
finer ejecta.
near-rimemplacement
of ejecta[e.g.Artem'evet el., 1994].
Experiments
at theNASAAmesVerticalGunRange(AVGR)
for laminar and turbulentflow respectively;p is atmospheric
density;v is the kinematicviscosity;x is the distancefrom
the top edgeof the curtain;and velocity
u=
test the theoreticalmodel and indicate that it satisfactorilyes-
(2)
timatesthe initial circulationgeneratedin the lowest vortex
createdbehindthe curtain. Suchcomparisons
allow extrapola-
Themagnitude
oftheatmospheric
flowvelocity
vf duetoflow
tionsto planetary
scales
for a varietyof atmospheric
andimpactconditions.
Preliminary
results
indicate
thatthewinds
generated
bythecurtain
onplanets
withatmospheres
aresufficientto entrainsignificant
ejectssizeson Mars,Earth,and
Ring
vortex
creation
• upper
vortex
Venus and shouldsignificantlyaffect the ejectsdeposition
process.
Wedonotaddress
mechanisms
forfinalejecta
depo-
basal
vortex
core
radius,
rcorrma
x
l• _ci
cr
sitionin thisstudy;ratherwefocusonthemechanics
by which
toroid
radius,
R,•: •,••
an ejectscurtaingenerates
strongwinds,therebyentraining vortex
,•t-, ½ •
andcarryingejectspriorto its deposition.
symmetry]
[ • •f•
Theory
In order to determine the variables which control the
strength
andbehavior
of thewindscreated
by an advancing
curtain, we evaluatethe physicalprocessesprimarily control-
ling the complexflow createdby thecurtain.An ejectscurtain
containssufficient debris to create an impermeablebarrier to
the surrounding
atmosphere.Consequently,
the ejectscurtain
can be viewedas an impermeablesolidplate. We first focuson
the variablescontrollingthe ring vortex shedbehindthis advancingbarrier,which laboratoryexperiments
indicateare re-
sponsiblefor the entrainment
and deposition
of fine grained
ejects. Thesevariables
thenpermitcomputing
thestart-upcirculation and core radius of the ring vortex first createdby the
advancing
impermeable
ejectscurtain.Theyalsoallowmodeling the decayof the flow in the vortex,thegrowthof its core
line I
!vo..?x
•
-
crater
Rc
.i-'•, •
-
paratiøn
velocity,
vf
...••••'"" imr•rmea•e
)J /t:' curtainlength,
L
I ¾•'"•:•:•':*'
x,.component
ofejectsparticle
velocity
J•
""I'.....'•¾
parallel
tocurtain
surface,
Up
radius,
i ß""'""•'"':'"•
':i"'--'•":i:•:
' gii...'•i....
i
Ring vortex evolution
I Vortex
core
and
curtain
position
attime
t/T-
/
i "..::,......,,....,,
.....
\
.-
radius, and the outward motion of the core of the ring vortex
[i!ii!•{iii}i!i!ii}i
positionafterthe ejectscurtainbecomes
permeable.
[5!iiii!iiiii:iii•iii}iii!!!:`.
.:.."":"
'"':'""'"'"""....•:•'
...................
•:•!i:
expansi
in
time
As the curtain movesoutward,flow separationoccursat the
ortex
toroid
radius
EjectsentrainmentEjectstransport
andemplacement
upperendof its lowerthicker,denserportion.The decelerating ejectscurtaincreatestwo ring vorticeswith opposing
flow Figure2. Schematic
of flowseparation
at thetopedgeof an
directions[Schultzand Gault, 1982; Schultz, 1992a] in order outward
movingejectscurtainresulting
in anupperanda basal
to conserve
angularmomentum
in the fluid nearthe tip of the ring vortex(top), and evolutionof the basalring vortex
in thisstudy
impermeable
ejectscurtain. Ring vorticesalsodevelopnear (bottom).The basalringvortexis emphasized
it is responsible
for entraininganddepositing
finethe tip of a decelerating
knife'sedge[Prandtland Tietjens, because
ejects.Variables
usedin thetheoretical
models
arein1934], and this similar responsepromptsviewing the ejects grained
curtainas an impermeablesolid-likebarrier. If the curtain dicated(see Table 1).
BARNOUIN-JHA AND SCHULTZ: EJECTA ENTRAINMENT BY RING VORTICES
Table
1.
Symbols
Symbol
Definitions
F
circulation
generated
byadvancing
curtain
(m2/s)
L
curtain length (m)
n
atmospheric
kinematic
viscosity
(m2/s)
Veff
effective kinematic viscosity accounting for energy
0
angleof curtainwith respectto targetsurface
losses
duetoturbulence
(m2/s)
Rc
crater radius (m)
R
radiusof ring vortex Ctoroid radius")from impactcenter (m)
Ro
radiusof ring vortexfrom impactcenterwhen flow sep-
F
radius of the vortex core (m)
Fc
core radiusof start-upvortex (m)
aration ceases(m)
core radius of vortex where axial flow reaches a maxi-
mum (m)
shearstressalong (ejecta'scurtain)surface(Pa)
t
time sinceimpact (s)
ti
time
for Oseen vortex
to achieve
vortex
core radius
givenby rc(s)
p
time passedfrom impactuntil curtainis permeable(s)
u
velocity of advancingcurtain(m/s)
u
net flow velocity along (ejecta curtain'sinner) surface
u
componentof velocity of ejecta particlesparallel to
t
(m/s)
p
curtain (m/s)
v
horizontalexpansionvelocityof vortex(m/s)
flow velocityresultingfrom flow separation(m/s)
Vo
axial velocity in vortex core (m/s)
x
distancefrom top of ejectacurtain(m)
y
heightfrom targetsurfaceto vortexcorecenter(m)
p
atmospheric
density
(kg/m
3)
,
,
,
Kaden's [1931] assumptionthat the vortex sheet created by
flow separationis a spiral satisfyinga law of similitude. The
solution maintains finite flow at the top of the edge of the
plate (i.e. Kutta-Joukowskicondition) while matchingthe velocity field of a line vortex locatedat the edgeof the plate with
that ahead of the plate. Experimentsshow that such theoretical solutionsadequatelyreproducethe growthof a vortex sheet
formedat the edge of a verticalplate [Rott, 1956; Wedemeyer,
1961].
To determinethe circulation for the start-upvortex sheetof
the ejectacurtain, we considerthe curtaintravelingat an angle
of about45ø with respectto a horizontaltarget surface[Gault
et al., 1968]. Solutions for the circulation and vortex growth
of the flow behind a 90ø wedge or triangularblock provide an
analogyfor the behaviorof the flow due to separationon the
back side of an ejecta curtain. Using Kaden's [1931] and
Antoh'S [1939] approach, Rott [1956] found a first order
asymptoticsolution for such a wedge which differed significantly from experimental results. Possibly the use of the
Kutta-Joukowski condition for a such large wedge angle no
longer applies becausethe upper edge of the wedge is not as
sharpas the edge of a plate [Rott, 1956]. Furthermore,experimental and numerical evidence indicates that viscosity plays
an important role in forming a secondsmall, counterrotating
vortex near the point of flow separation which is not accounted for in the inviscid models [Rott, 1956; Pullin, 1978].
Fortunately,both of theseissuesare not significantwhen consideringan advancingejectacurtain. First, the back surfaceof
the curtain travels at velocities comparableto the velocities
generatedby flow separationin the fluid behind the curtain,
thereby eliminating any viscousstresses. Second,the KuttaJoukowskicondition must apply becausethe flow impinging
upon the curtain is observedin experimentsto bend sharply
aroundthe impermeableportionof curtain,therebyresembling
the edge of a plate rather than a wedge [Schultz and Gault,
1982; Schultz, 1992a].
Even though the solutionsof flow around a wedge do not
apply strictly to the problem of the ejecta curtain, laboratory
experimentsreveal the theoretically expectedflow behavior.
For example, a robust feature of numerical solutionsthat maintain both inviscid
separationin the basal vortex equalsthe velocity of the advancingcurtainU at the backsurfaceof the curtainto withina
factorof 2. Thecomponent
of thevelocity(up)of theejecta
particlesin the lower portionof the curtaintravelsparallelto
the curtain's inner surface with a similar magnitude U.
Consequently,
equation(2) indicatesthat u ~ 0 for the lower
vortex becauseindividualejecta in the curtainmove parallel to
the flow in the vortex, therebymaintainingx small. Actually,
the flow velocity in the lower vortexcould be increasedif u_is
21,101
flow and the Kutta-Joukowski
is that a wide
wedge createsan elliptical vortex sheet[Pullin, 1978]. Quarter
spaceexperiments reveal that the basal vortex exhibits such a
shapewhen it is first created.
To estimate the circulation or flow strength of the vortex
sheetcreatedbehindthe advancingejectacurtain,we adoptthe
sameassumptions
as for the solutionof flow past a vertical
plate [Anton, 1939; Wedemeyer,1961] but includethe effect
of the curtain slant.
We assume that the relative conditions
necessary
to maintainthe Kutta-Joukowski
conditionfor a vertical plate vary geometricallywith changesin curtainslantas
prescribedby the spiraldeterminedby Kaden [1931], which
A flow separationmodelthereforeshouldallow a reasonable satisfiesthe laws of similitude. Usingthe resultof Wedemeyer
[ 1961] for a verticalplateas a reference,the circulationF genestimatefor the circulationor strengthof the flow generatedin
slightly
greater
than
v?although
this
effect
should
besmall.
the lower ring vortex, sincesheardrag is minimal. Laboratory eratedin the lower vortex sheetis given by
observationsalso indicate that the region of flow separationat
the top of the curtain is small comparedto the impermeable
portionof the curtainwhen the vortexis first created. For such
(3)
0.1
conditions, existing theory allows estimating the circulation
created by a vertical plate advancingat a constantvelocity
from rest into an inviscid and incompressiblefluid [Anton,
where 0 is the curtainangle with respectto the surface,L is the
1939; Wedemeyer,1961]. The solutionfor the flow field that curtain
length,
andU isthecurtain
velocity.
Technically
tpis
allows calculatingthis circulationwas determinedby applying the time from when the impermeableportionof the curtain en-
F= 2•:•
UL,
21,102
BARNOUIN-JHA AND SCHULTZ: EJECTAENTRAINME•
ters the ambientatmosphereto when the curtainbecomespermeable. In this study,we assumethat for both small and large
BY R/NG VORTICES
dicate that the core of the basal ring vortex movesoutwards
following the ballistic ejecta curtain [Schultz and Gault,
scale
impacts,
t•,equals
thetimefromimpact
towhenthecur-
1982]. This extends the toroid radius R and reduces the effect
tain becomespermeable. This solutionfor the circulationincorporatesthe first-order physicsof flow separationof an atmosphereby the advancingejecta curtain.
Both theory and observationsindicate that the vortex is elliptical in shape. Nevertheless,the modified formulation of
Wedemeyer[ 1961] providesan adequatedescriptionof the av-
of theflow withinthevortexcore. Because
R>>rc,we canassumethat the vortexbehavesin a linear and planarfashion.
Hencethe flow in the impact-induced
ring vortexwill be modeled to first orderby a planarline vortexwith someslight
erageradiusrcof thevortexcore(referto Figure2):
modifications(discussedbelow).
Because
the flow in the corepushes
the entirering vortex
down towardthe surface,surfaceshearstresses
eventually
shouldreducethe flow velocityin the vortexcore. Suchshear
2/9
r
=
c 0.13•
+0 (2.0•
+0)2/3
2/3
(4)
alsoshouldenhanceany instabilitiesthatmay haveexistedin
the corewhenthe vortexwasfirst created,therebyeventually
causingthe ring vortexstructure
to breakdown
[Magarveyand
MacLatchy,1979]. Whenthe vortexfirst reaches
the target
surface, however, surface shear effects will be minimized be-
This radius defines the position where the flow in the vortex causethe shearstresses
areconfinedto a smallboundary
layer
achievesa maximum. Flow velocity outsidethis core decays at the surface.For thisreasonandbecause
thisstudyfocuses
as 1/r where r is the radius from the center of the vortex core,
and decreaseslinearly to zero at the center of the vortex core.
The core of the ring vortex is small comparedto the region influencedby the vortex flow.
Equations (3) and (4) indicate that the circulation and the
vortex core radius are primarily dependentupon the velocity of
the ejectacurtain and then to a lesserextentuponthe lengthof
the curtain. Thus variationsin the impermeablelength of the
curtain, growing and shrinking while it advancesinto the surrounding atmosphere,should not significantly affect the magnitude of the flow and size of the vortex core. However, varia-
tions in the velocity of the curtain shouldbe very significant.
Once the ejecta curtain becomespermeable,we assumethe
magnitudeof the flow in the vortex is no longercontrolledby
the motion of the ejecta curtain. The flow in the vortex can be
controlled by (1) the flow in the vortex affecting itself, (2)
shear stressesacting on the ring vortex at the target surface,
and (3) atmosphericviscosity. The flow within the vortex
core affects the position in space of the core itself.
Considering the counterclockwise streamlinesof the vortex
core on the right side of the impact-inducedring vortex, one
imagines that these streamlines extend to infinity. These
streamlinespush the core at the left side of the ring vortex
downward. Similarly, the streamlineson the left side of the
ring vortex push the right side of the ring vortex downwards.
Occurring simultaneously, such interactions move the entire
ring vortex down toward the targetsurface(seeFigure 2). Once
at the surface, these interactions keep the ring vortex at the
target surface at least initially [Lamb, 1932; Magarvey and
MacLatchy, 1979; Shariff and Leonard, 1992]. Experiments
and numerical calculationsof smoke rings crashinginto a rigid
boundary show that the center of the core of the ring vortex
will remain at a height approximatelyequal to the radiusof the
vortex core before they disintegrate [Magarvey and
MacLatchy, 1979; Shariff and Leonard, 1992].
The streamlinesnot only move the entire ring vortex, they
also may strain and alter the flow within the vortex core. The
significanceof this effect dependson the velocity of the flow
from one side of the vortex acting on the other. In the caseof
on evaluatingthe initial strengthand behaviorof the winds
createdby the advancing
curtain(ratherthanon its decay),surface shear stresses are not considered.
Atmosphericviscosityplaysa key role in vortexevolution
(growth and velocity) after it separatesfrom the curtainand
causesthe flow in the vortex to decayas a functionof time.
Since a planar geometrycan be assumed,an Oseenor Lamb
line vortex modelis usedto approximatethe effect of atmosphericviscosityon the impact-induced
vortex. Sucha model
successfullydescribesthe flow in thin smokerings (r<<R)
[Sallet and Widmayer, 1974]. When the flow in the vortex is
laminar,the axial velocityin the Oseenline vortexis given
by
_/,2
I•
Vø=• 2gr1-
e
4V(t_ti
)
(5)
wheret is the time sinceimpact;ti is the time for the Oseen
vortexto reachthecoreradiusrcwhenthevortexdetaches
from
the advancingcurtain; and v is the kinematicviscosity. The
radiusof thevortexrma
x wherethe velocityin the vortexcore
reachesa maximum is given by
rma
x = 2.245
)v(t-ti)
(6)
Ring vorticescreatedby an accelerating
pistonare foundto
be turbulent[Sallet and Widmayer,1974]. Sincethe motionof
an ejectacurtain resemblesa deceleratingpiston,we consider
the possibilityof turbulentflow: we replacethe kinematicvis-
cosity
v withaneffective
kinematic
viscosity
Veffthaten-
hancesthe rate of viscousdecayas a functionof the Reynolds
number in the flow of the ring vortex. Here we choosethe
model proposedby Owen [1970] where
Vef
f = 0.6108F
(7)
an impactat latetime,however,theradiusrc of thevortexcore
is small compared to R, the toroid radius of the vortex ring.
Since the velocity in a vortex core decreasesas F/2•r where r
and U/v is the Reynolds number for flow in the vortex. This
model is highly simplified where steep stressgradientssepa-
is the distance from the center of the vortex core, the flow from
rate an inner
one side of the ring vortex has little effect upon the flow on
tional flow. Although Owen's [1970] detailed assumptions
may be heuristic, his model proves to agree adequatelywith
theotherside,wherer ~ 2R>>rc. Furthermore,
experiments
in-
annulus
that is strain-free
from the outer irrota-
BARNOUIN-JHA AND SCHULTZ: EJECTA ENTRAINMENT BY RING VORTICES
experimentsover a wide range of Reynoldsnumbersincluding
for vorticesshedoff commercialjet airfoils [Bradshaw, 1973].
This model tends to slightly overestimatevortex decay at laboratory scales while accurately estimating decay at larger
21,103
Table 2. Empirical Data
Impactor
Atmospheric
information
Environment
scales [Owen, 1970].
Unlike the experimentsof $allet and Widrnayer[1974], the
curtain-generated basal ring vortex expands outward. The
Oseen line vortex
model
therefore
must be modified
Round Diameter, Mass,g Velocity, Pressure, Gas
cm
931005
931007
931008
931009
931010
931011
931013
931014
decrease
as1/xlRwhere
R isthevortex
toroidradius
(seeFigure
2 and the appendix). The circulationF in (5), (6), and (7) must
thenbereplaced
by F(Ro/R)I/2
where
Roequals
R whenthe
curtain becomespermeable. In order to conserveangular mo-
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
931015 0.635
931016 0.635
mentum,the radiusof the vortexr mustdecrease
as (l/R) TM
(see appendix). This is equivalentto reducingthe time (t-ti) in
placing
(t-ti)by(t-ti)(Ro/R)
1/2increases
thevelocity
in the
931028
931029
931030
931033
931034
vortex core while reducingits radius. The new axial velocity
931035 0.635
equations
(5) , (6) and(7) to (t-ti)(Ro/R)1/2. In a qualitative
sense, such an approachmodels vortex stretching,since re-
will
be
_1. 2
I"
V0 2•r
1-
e
atm
0.376
0.376
0.376
0.376
0.376
0.376
0.376
0.376
1.80
1.80
1.83
1.90
1.82
1.59
1.90
1.77
vacuum
0.25
0.25
0.21
0.21
0.16
0.18
0.16
-air
air
air
air
Ar
Ar
air
0.376
0.376
1.85
1.81
0.14
0.11
CO2
CO2
0.275
CO2
Target
to main-
tain energy and angular momentumconstantfor the ring vortex as the vortex expandsin three dimensionalspace. In order
to conserveenergy, the flow velocity in the vortex core must
in the vortex
m/s
4V(t-ti)(Ro/R)
1/2
(8)
Again,
v andVeff
canbeinterchanged
toexplore
theeffects
of
turbulence.
The outwardmotion of the vortex core away from the impact
center can be estimated using the method of images [e.g.,
LaboratorySetup
0.635
0.635
0.635
0.635
0.635
940601
940602
940610
940902
940920
940921
940922
940923
940927
940928
940929
940930
941117
941118
941119
941120
941121
0.318
0.318
0.318
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.635
0.376
0.376
0.376
0.376
0.376
0.376
0.238
0.238
0.238
0.376
0.376
0.376
0.376
0.376
0.376
0.376
0.376
0.376
0.376
0.376
0.376
0.376
0.376
1.83
1.93
1.66
1.62
1.72
0.35
0.35
0.35
0.35
0.5
1.77
2.01
2.08
1.97
1.98
5.11
4.61
4.91
4.61
2.09
2.04
5.17
4.73
2.17
2.19
2.17
2.17
2.19
0.25
0.5
1.0
0.5
0.24
0.46
0.46
0.69
0.75
1.0
0.12
0.35
0.25
0.33
0.5
0.67
0.75
ß
a•r
air
air
air
air
air
air
Ar
air
Ar
Ar
Ar
Ar
air
air
Ar
Ar
air
air
air
air
air
24
24
24
24
24
24
24
24
sand
sand
sand
sand
sand
sand
sand
sand
24sand
24sand
24
24
24
24
24
sand
sand
sand
sand
sand
24sand
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
24
sand
sand
sand
sand
sand
sand
sand
sand
sand
sand
sand
sand
sand
sand
sand
sand
sand
projectile
path
White, 1986]. Since the ring vortex is not stronglyaffected
by thetargetsurface,inviscidflow theoryallowsusto placean
imageof the ring vortexbelowthe surface,therebyreplacing
it with a plane-surface
streamline.As theflow in the ringvorß:
tex tendsto move down, the image of the ring vortex in the
targetsurface
forcesthevortexto openoutwards
(seeFigure2).
This outwardvelocityof the vortexis givenby
_
F
V = •
1-
2
,,(,Y_,,)
e
(9)
4•zy
where y is the height from the vortex core center with respect
to the target surfacewhen the flow in the vortex is considered
planar and unaffectedby the lateral expansionof the ring vortex. When we conserveenergy and angular momentumfor an
expanding ring vortex, the outward velocity of the vortex is
given by
Figure 3. Schematicof quarter-space
experimentsperformed
at the NASA Ames Vertical Gun Range (AVGR). The front
panel is made of 1.27-cm-thickPlexiglasthat is further stabilized by bars attachedto the AVGR chamberwalls. The highspeedvideo camera (500 to 1000 f/s) capturesthe passageof
the vortex behind the Plexiglas.
V-
1- e
(10)
4•zy
Here,too,v andreftCanbeinterchanged
toexplore
theeffects
of turbulence.
21,104
BARNOUIN-JHAAND SCHULTZ:EJECTAENTRAINMENTBY RINGVORTICES
Table 3. Ejecta Properties
EjectaType
GrainSize,!xm
24 sand
Micro spheres
Quarter-spaceexperiments(Figure 3; Table 2) at the NASA
AVGR allow one to measure the variables necessary to
computeboth the circulation and the radius of the ring vortex
just beforethe curtainbecomespermeable,as well as the radius
and positionof the ring vortex thereafter. A coarsesandtarget
(24 sand in Table 3) createsa sufficiently impermeableejecta
curtain to induce flow separation and a ring vortex. The
magnitude of the flow, however, is insufficient to entrain the
Density,
kg/m3
457
1700
100
500
Experiments
sand [Schultz, 1992a]. In order to visualize the flow without
The above theory can be testedexperimentallyby first measuring the parameters necessaryto compute both the circulation and the radius of the vortex sheetcreatedby an advancing
ejecta curtain. Once the circulationF and the radiusof the core
of the start-up ring vortex are estimated,computationscan be
comparedwith measurementsof the vortex size during formation and decay, as well as with the position of the vortex core
as a function of time. Becausethe theory for the flow velocity
within the vortex and the positionof the vortex core as a function of time applies to a single phase fluid, our experiments
are designedto minimize ejecta entrainment.
significantly altering the kinematic viscosity and density of
the flow, small microspheres(Table 3) are placedon the target.
A high-speed video camera (500 and 1000 f/s) permits
measuring all the time-evolving variables necessaryfor the
comparison between theory and experiment: the length L of
the impermeableejecta curtain; the positionof the curtain;the
radiusrc (or rmax after curtainbecomespermeable)of the coreof
the ring vortex; and the height and outwardmotion of the core
of the ring vortex. Careful analysisof the video also allows
oneto determine
the approximate
time tlowhenthecurtain
becomespermeableand the time of craterformation(unaffected
2.0
Ar
P = 1.0 atms
r = 0.238 cm
V = 1.97 km/s
co•
P =0.14
atms
r=0.3175 cm
l'
I'
V = 1.85 km/s
.t
0.0
,
ß
012
0.4
016 ß
'
,
o.o
oi,,
TIME (tin
2.0
o16
0.8
1.0
TIME(t/T)
•' • '
D
r,.) 1.0
O
•
Ar
0.5
AIR
P = 0.46 atm
r=0.3175
cm
P = 0.67 atms
ß
r = 0.238 cm
V = 4.91 km/s
0.0
0.0
ß
0'.2 '
014 '
016 '
018 '
V=2.17
.0
TIME(t/T)
0.0
0.2
0.4
0.6
km/s
0.8
TIME(t/T)
Figure4. Measured
(solidsquares)
andcalculated
(opendiamonds)
radius
of initial(start-up)
vortexcoreasa
function
of time(t) normalized
to thetimeof craterformation
(T) for different
experiments.
Theclosematch
betweentheoryandobservations
indicates
thatthestart-up
modelof equation
(3) estimates
well thecirculation
inthevortex.
Theshaded
area
shows
theexperimental
uncertainties
duethemeasurements
ofL,Uandtpused
to thecalculate
theoretical
vortexradius.Because
thevortexcorepossesses
anellipticalshape,
theerrorbars
indicatethe size rangeof its semimajorand semiminoraxes.
1.0
BARNOUIN-JHA AND SCHULTZ: EJECTA ENTRAINMENT BY RING VORTICES
Ar
CO2
A
P=0.14arms
B
P= 1.0 auns
turbulent
r = 0.3175 cm
0.8
ß
r = 0.238 cm
V = 1.82
kn•s
t
,
V = 1.97 kn•s
RTM_ ß•
I
21,105
ßß
•
RTM
turbulent
ß
;>
0.2
I ß
I
lamSnar(•1 LMs)
lan•nar (a• LMs)
o.o
1.0
o.o
2.0
3.0
4.0
5.0
6.0
0.0
1.0
2.0
3.0
True (t/T)
ß
i
ß
i
ß
i
•l
ß
r = 0.238 cm
V=2.17
kags rrM
V=4.91
kags l•
turbulent
turbulent
S
• 0A
'-,'
ß
"
ßll•
0.0
1'.0
2•
7.0
P = 0.67 atms
r = 0.3175 cm
ßß
6.0
Air
D
P= 0,46atms
0.8
5.0
ß
Ar
C
4.0
TIME (m3
' '
4.0
0.0
i1
1.0
2.0
TnVm
(t/T)
•
-
ß
lanfinar
(a•
LMs)
t
3•
•
3.0
lan•nar
(a•
LMs)
4.0
5.0
6.0
TnVm
(t/T)
Figure 5. Measured(solid squares)and leastsquaresfit (curves)of verticalheight(y) of the coreof the ring
vortex scaled to the crater radius (R c) as a function of dimensionlesstime (time, t normalized to the time of
crater formation, T). The turbulent models depart from the least squaresfit of the observeddata becausethe
theoreticalestimateof the diameter of theseturbulent ring vorticesexceedstheir observedheight. The solid
curve excludesany effects due to the lateral growth of the ring vortex (ideal turbulent model, or ITM). The
dashedcurve considersonly the conservationof energy (energy conservationturbulentmodel, or ETM). The
dotted curve conservesboth energy and angular momentum while the ring vortex expands horizontally
(realistic turbulent model, or RTM).
by the presenceof the atmospherefor the target used). By stresses when the vortex first reaches the surface. The above
fitting a power law to the positionof the curtainas a function variables also allow one to compute the expected vortex flow
time (basically a Z-model [Maxwell, 1977]) we evaluate the velocity at the target surface. In order to test the theory further, these models' flow velocities are compared to the velocvelocity U of the advancingcurtainin time.
With measurements
of L, U andtp,equation
(4) allowsone ity of small Styrofoam balls placed on the target which the
to estimatetheoretically the radius r c of the core of the ring vortex pushesoutward after the passageof the curtain (Figure
vortex, which then can be compareddirectly to measuredval- 9).
Uncertainties in the theoretically derived values are due to
uesof rc (Figure 4). In addition,measurements
of the vortex
heightallow computingthe horizontalpositionof the core of uncertainties
in themeasured
valuesof L, U andtp(seeFigures
the ring vortex as a functionof time. When the vortexradius 7-9). Most of the uncertainty arise from measuresof the currmax exceedsthis height,it is maintainedat a heightequal to tain length and the time when the curtain becomesimpermermax,avoidingthe unreasonable
situationwherethe vortex able. To determine the length of the curtain, we establishas
core theoreticallyproceedsunderthe target surface(Figure 5). accuratelyas possiblethe position where the flow rushesover
Such an assumptionis consistentwith numerical calculations the top of the curtain. This position is indicatedby the fine
and laboratoryexperiments[Magarveyand MacLatchy, 1979, microspheresin the ejecta curtain which are strippedaway by
Shariff and Leonard, 1992]. The theoreticalresultsare com- the flow traversing through the top permeable part of the
paredwith the outwardmeasuredmotionandradiusof the vor- ejecta curtain. Repeatedmeasurementssuggestthat this length
tex obtained from the video after the curtain becomes perme-
can vary by about 1 cm.
The ejecta curtaindoesnot becomepermeablesuddenly,but
able (Figures6, 7 and 8). For all calculationsafter the curtain
becomespermeable, we considerboth turbulent and laminar over a small time period as the slow ejecta in the curtain begin
flow in the vortex, but neglect the effect of surface shear to deposit ballistically and the fast ejecta separatefrom each
21,106
BARNOUIN-JHA AND SCHULTZ: EJECTA ENTRAINMENT BY RING VORTICES
6.0
Ar
5.0
P = 1.0 atms
r = 0.238 cm
4.0
V = 1.97 km/s
3.0.
turbulent ......
,,:!:.::;i:::•::.::
:...:•:•....:•:•i•:•:•.':.-":../•::•i::•i•:•5•;•.-'.'•i"--":.•...-'•...
'•.
,•:.-'.:.:
;";;:;•;
..::"'::'...::.-':i..-'.("""'"'""•'
..-•..;...:..•.:..-....-:•:....•-....., ., ,,*:•
o
1.0
............................
•..M........•.d...:,..E._..•
larninar
0.0
....
0.0
1'.0
2:0
3'.0
4'.0
5.0
0.0
0.0
1.0
TIME (t/T)
6.0.
5.0
,
•
,
•
,
2.0
3.0
4.0
5.0
RLM
6.0
7.0
TIME (t/T)
•
.... .,.:::•,!:•:•:•:•,....,•
....••.
...........
•::::....•
6.0 ' • ' • ' • ' • " • ' • :'.•:::-'.:•.:½•
/ ß turbulent
....
•:½i•:;::.:.'•i•:.-.:;•i•i•::•i•::
...:....:;•?•:!•?•ii::ii:i:•.::!:i:::?ii•!?
...-:::•:•i•.
.:..-:....--•i??i:::i':
...........
>•i•;i?•::•:.•..'i•:•i?
:..•
.......
•--0
67
atms
.'"""••i
/,tr
....
...:.::;:::-.:'iiiiiiii;iii':':.,....:...L.•?::
:';:-- .....
'•:-' -'::;-'::::ii::•:•:.,.•i
..:::.•½•:i!ilE..::,:;•....:-•{:..--.-yS;;:.,:?•:
.........
.•.,---:- .::•:...:,•;i..,;.!:::::::iiii•i::•?:i{i
5.0
qP:0.46
arm
I
,0]
....
::.?:.-?•'•'-::."
...:.,,D•::½:::ii::•i::•?:'•::;.t:•t..:.:.-.:'
.... .:.::::::.!i•;::•:•;•i•:•i•½•.'-'.•.::i!i':;'.-':.-•?.?..•::•
..?:..'?•:':;ii•;!?:"
':'•:•------"•;:'""'"""'"'"•:•i;i•:,
::':"•"';'•'
,•':•'•;;i
.••••••!:i!•.:.......,...•!i....-.-.-.-•....
.....................
½.........•ii.....:..-.•-:......•;i•!•:,
,.:.:.•:..•.••••:....
km/s :••'""-"•:•:•:•:'"
4.0
t V:2.17
..........
':'""
•:"
':''
.•.
..:::•½:-'.-:!'
•<•.•".::...:"•,•-:•s.-..;;...-'•
½'".
•"":.'.':•?"
•.-'..'i•::.
.....
:::.•.½:.-'.:
.:.'5!::."
.:-,.•.•::!:;.;::'
"i•• v.-:..%,.':::::
I•••.:.:'
• '":i•i!•
i.i:i:$...
' :...'.-'-...-'.:-:•:'
..-::::i!!
....'-':
•si:5::.5:!:•:!:i!i.:5:i•5:i:•:5
2.0'i
an,
ß]
0.0
I0
a'.0
s.0 0.0
TIME (t/T)
..?:.!:-•.
'::½..'..-i:;-':...-"F'
"::;
'? ß....•.-.,•
-?'"':'"'"•;•i:':!•:.:..-.'.,.'•'.*•*>'-':-'•:'.................
'......................
":?":;:'
•:::.::::::;::.•
T•.•.
....
.•............
a.0
s.0
..........
,.0
TB4E (t/T)
Figure 6. Observed(solid squares)and calculated(curves)radius of vortex core as a function of dimensionless time (see figure 5). The top three curves show the turbulentflow models (TMs) and the bottom two curves
(actually three) show the laminar models (LMs). The solid lines depict modelsthat do not considerthe effects
of the lateral expansionof the ring vortex (ITM or ILM). The dashedcurves correspondto models that consider only the conservationof energy (ETM or ELM). In the laminar flow case,the curve for energyconservation coincideswith its solid curve (i.e., ILM = ELM). The dottedcurvestrace model predictionswhen both energy and angular momentumare conservedwhile the ring vortex expands(RTM and RLM). Shadedregions
showtherangein modelestimates
dueto observational
uncertainties
forL, U andtpusedin calculating
theradius of the vortex core. These regionsoverlay each other, with the RTM or RLM region overprinted. Because
the vortex core possessesan elliptical shape, the error bars indicate the size range of its semimajor and
semiminor
axes.
other. At laboratory scales, observationssuggestthis time
periodis small. We thereforeassumethat the changefrom an
The Plexiglassheetusedin quarter-space
experiments
cuts
the crater and ejecta curtain in two but interfereswith the adimpermeable
to permeable
curtainoccursinstantaneously,
and vancingmotionof the portionof the ejectacurtainonly where
wedefine
t•,asthetimewhenthischange
begins.Repeatedit contacts.
As a result, we focus on the curtain and the flow a
measurements
showthat this time can vary by 0.002 to 0.01 s.
few centimeters
behindthe sheet. This is achievedby placing
The timeof craterformationat laboratoryscalesapproximates the microspheretracers a few centimetersaway from the
0.1 s.
Plexiglas. Observationsdemonstratethat the presenceof the
Uncertaintiesalsoarisewhenmeasuringthe vortexcorera- Plexiglas does not significantly affect the flow at this disdius. To determinethe size of the vortex core, we assumethat
tance, consistentwith boundarylayer theory.
Additional measurementuncertaintymay arise when we attempt to distinguish between flows that occur a few centimeters from the Plexiglas and those that occur farther behind the
thevortexcore,sincethevelocity
in thevortexis a maximum Plexiglas. Theseuncertaintiesarisebecausethe flows separatat this point.
ing from the top edgeof the half ejectaplumepossessa three-
the distancefrom the vortexcenterto the regionwheremostof
the microspheres
are entrainedequalsthe radiusof the core of
the vortex. This shouldbe a goodassumption
for the radiusof
BARNOUIN-JHA AND SCHULTZ: EJECTA ENTRAINMENT BY RING VORTICES
I ALarninarfiow
ILM
•:.....
"•;i::'--'":.•i•:'
•'.
......
I co,
304 P=0.14
atms
'
ß..•i:?;..'?::
....
•Z-;i•i?'i
!iiii•::'::'
...:.:::s::
......,• ?
P I.
"
•
..::•::.'.-".
?g
'" i :'•:•'
....
• 2.0
1.0
iirl•
ELM
and
RIM
.....
1.0
2;
310
•ME
I
]v
4.0
5;
6.0
II
0.0
'
0.0
I
-
III
'
1.0
I
'
IN/
I
2.0
'
3.0
(fid
I
'
4.0
I
5.0
6.0
TIME (fid
4.0] • . • , • •, 4.0
ß
,1.
,
.
DLarninarfiow
I
Flow
Ar
3.0
-
ELM
and
RIM
ß
ß [,
C Laminar
A:i'..-:?•iii
'*'""':"'"'::•:•::::'•
:•
:'-""
S•l"n
0.0
....
•::5":•i}iiii'""
"'"•"•'"'"'••••""
V=].97krn/s
1.0d.-:.%.!
,....,.,.::;.
:':' /
0.0/
e=1.0
atms
'%3.0
' l r=0.238
cm
JI
21,107
P=0.46
atms
..•:?.::.......:.:.#:.-•.•?....'...
I
'• 3.0
r = 0.3175cm
,1.
V =4.91kn•s
..-..........
.•......4::•?:.:-:>•..•?iii•i!!i•!i!i;•::i•i
.: ....
..!*:•*:':'::":"..
F!':'
.....
•.o.
............
--:::
':':;:"•
.'•.
.
.
.
. ......
ELM
o.o
1• I
0.0
"'"'
I
and RLM
'
S
• •.o.
III
1.0
?'""
• .:.:.s::i:
::i:::'"'
--:"•-:":•':
............
I
2.0
IV
3
i0
0.0
4.0
t
o.o
';':::'"
......
II
......
ELM
and
RLM
III
i
?.o
IN/
I
3'.0
;.o
TIME (fid
TIME (fid
Figure 7. Measured
(solidsquares)
andcomputed
(curves)resultsof thenormalized
toroidradius(R/Rc,where
R is the toroidradiusandRc is the craterradius)as a functionof dimensionless
time (seeFigure 5). Calculated
results for the laminar flow cases(LMs) are shown. The solid lines, the dashedlines and the dotted lines are defined in Figure 6. The advancingmotion of the vortex can be split into four stagesbasedon observedphysical
phenomena:stageI, flow separation;stageII, constantoutward velocity until t/T •-2-3; stageIII, acceleration
of motion of vortex to a new higher velocity; and stage IV, possibledecelerationof vortex by surface shear
drag and viscousdecayafter t/T-3-4. Shadedregionsshowthe rangein calculatedmodelresultsdue to observa-
tionaluncertainty
for L, U, andtpusedin calculating
thevortextoroidradius.Lightgrayshaded
areashows
uncertaintyrangefor the ILMs; dark shadedregionshowsuncertaintyrangefor the ELMs and RLMs. Scatterin
the measuredtoroid radius of the ring vortex reflects the measurementrepeatability error.
dimensional character. As a solution, the vortex furthest from
pectedif the flow in the vortex is turbulent("turbulentmodel"
the impact point traveling parallel to the plane of the
Plexiglas is measured.
or TM). The turbulent model that conservesboth momentum
and kinetic energy ("realistic turbulent model" or RTM) best
fits the observedgrowth of the core of the vortex ring.
Results
Motion
of
core
of
vortex.
The
observed
motion
of
the core of the ring vortex exhibits four stages that are
In order to determine how well theory predicts the observed matchedto differing degreesby the theory. During stageI, the
behavior of the ring vortex during its creation and initial de- curtain is impermeable. The vortex core closely follows becay, we compareobservationwith theoreticalestimatesof (a) hind the curtain acceleratingrapidly to position S of Figures 7
vortex initiation and growth, (b) vortex motion, and (c) vortex and 8. Position S(for Start) indicatesapproximatelywhere the
wind velocities.
ejecta curtain becomes permeable. In stage II, the vortex
Vortex evolution. Within experimental error, the the- toroid radius increasesat a constant velocity along a straight
oretically estimatedstart-upradius and the observedradius of line stretchingfrom S to a time 2-3 timeslongerthan the time
the coreof the ring vortexare very nearlyidentical(Figure4). of crater formation (T). All the theoretical models considered
Oncethe curtainbecomespermeable,the observedradiusof the in this study adequatelymatch the positionof the vortex core
core of the vortex grows (Figure 6). Indeed, it grows more center during this stage and each model is indistinguishable
rapidlythan'expected
theoretically
if theflowin thevortexis from the next. Stage III begins at t/T •- 3 when the outward
laminar ("laminar model" or LM) but more slowly than ex- motion of the core of the ring vortex accelerates,eventually
21,108
BARNOUIN-JHA AND SCHULTZ: EJEC'!'AENTRAINMENT BY RING VORTICES
4.0
4.0 I
'
I
' I I
'
I
'
I
'
. ATurbulent
flow
[ ' ' '[ '
'
ITM
•
co2
- P=0.14
atms
3.0-
I
ß
ß
ii•lI
-
'
' ' '
BTurbulent
flow
' ' I
ITM
^r
P=1.0
atms
r=0.238
cm I
3.0
•o
I.:.:::,
'-'•
r=0.238
cm
I•llll
• 1-------.....::•••
Il
•
V=1.97
km/s
, .. '-'"-"-'-"-•:'•
ß::iii::!i:•?:.......•i•iifli•i:2•-.:•ii•.ii•i•?•i::g?-!•
........
•ll-• t
' V=1.85
km/s
2.0
2.0
ß
• • S ""
1.0
E•
1.C
•l I
II
III
IV
0.C
.......
0.0
.....
'
I
0.0
'
1.0
I
'
2.0
I
'
3.0
'
4.0
I
'
,.•
6.0
5.0
0.0
1.0
2.0
3.0
TIME (t/T)
.
I
.
T•E
I
I
C Turbulentflow
P = 0.46 atms
r = 0.3175
ß
6.0
4.0
]
D
'
Turbulent
I'
'
flow
[
•
I•• _.
3.0 '
•l•..:•i::::::::::::•i•t::
•.:.,::
''....
:-::::-:-Air
Ar
3.0
5.0
.
4.0
.
4.0
(•)
m•
cm
•
V = 4.91 km/s
2.0 -
ITM
I
.....
•:•f•:•a•w:::,-
I..............
•:aa•::•:•
•:.•:
......
m=•::•i•
:::::::::::::::::::::.'
.:• ß .-.-::.
r = 0.238 cm
I
•.. ::•::32•::•?:•r":')
- ::"-:..}:5:
:"•
.....:'}....
V=2.17
k•s
-.•:::::"7:...;.:.•:•:•g•
.•-:*•:-•:•:•::':
...............
.........
_.'"""•
'"'"'
[
ß
P= 0 67atms
•
-:-:-'-'.
•..•.
ß ......
•f:•-
I
•
•<•[•:•:•::::':
....
.-'•::;' •.:• •:'
.....
S ....
-....
":'•"':'•
.... "C'c:'
'....
1.o
ETM
ß
I
oo
0.0
II
[
1.0
'
III
I
2.0
TIME
(fiT)
IV
I
3.0
4.0
o.0/
0.0
.
,
1.0
.....
210
310
4.0
TIME
(fiT)
510
/
6.0
Figure 8. Measured(solidsquares)
andcomputed
(lines)of thenormalized
toroidradiusas a functionof dimensionless
time (seeFigure5 and7). Calculated
resultsfor theturbulentflow cases(TMs) are shown.The
solid,dashedlinesand dottedcurvesare definedin Figure6. Shadedregionsshowthe rangein modelestimates
dueto observed
rangeof L, U, andtp. Lightgrayshaded
region
shows
uncertainty
rangefortheITMs;gray
shaded
regionshowsuncertainty
rangefor theETMs;anddarkgrayshaded
regionshowsuncertainty
rangefor
the RTMs. The ETM rangeoverprints
theRTM range,andbothoverprinttheITM range. Scatterin the measuredtoroidradiusof the ring vortexshowsthe measurement
repeatabilityerror.
reaching a greater but constant velocity. All the theoretical
models predict qualitatively this observed increase in the
expansionvelocity of the toroid radius of the ring vortex as
well, but differ in the exact position of the core of the ring
vortex during this stage. The estimatesfrom turbulentmodels
(TMs) place the vortex core closer to the impact center than
observed,while the laminar models(LMs) place it fartherthan
observed.
Some experimentsappear to exhibit a fourth stage in the
motionof the vortexthat occursafter t/T ~ 4 (seeFigures8 b, c
and maybe d). In this stage,the observedexpansionvelocity
of the toroid radiusof the ring vortex decreases.Only the turbulent models qualitatively predict such a motion.
Wind Velocities. The Styrofoam balls placed on the
targetsurfaceact as tracersof the flow velocityin the ring vortex at the target surface. Their velocity decreasesas a function
of time consistentwith the absoluteflow velocitiespredicted
by the turbulentflow models(Figure 9). Small scalevariations
in the velocity of the tracersreflect other secondaryprocesses
that affect the advanceof the particlesas they move along the
target surface. The LMs predict slightly greater absolutevelocities
and an overall
increase
in these velocities
in time.
The ideal turbulent model where neither energy nor angular
momentum is conserved (ITM) and the RTM best fit the observations.
Discussion
The resultsindicatethat the theorygenerallyaccommodates
the observations.Consequently,
the theoryis believedto take
into accountthe primaryphysicalmechanism
controllingthe
behaviorof the ring vortexat the time of flow separation
and
immediately thereafter. Furthermore, results show that the
flow in the ring vortexis mostlikely turbulent.The difference
betweenlaminaror turbulentflow in the ring vortexcannotbe
readilydeterminedby simplyevaluatingthe Reynoldsnumber
of its flow becauseturbulencehasseveraloriginsin a vortex
ring [e.g. Sallet and Widrnayer,1974]. Small discrepancies
BARNOUIN-JHAAND SCHULTZ:EJECTAENTRAINMENTBY RING VORTICES
B
A
1.6'
21,109
Turbulent
flow
Turbulent
flow
...
1.4
co2
P=0.11atms
----rm
---- -EEIM
1.2.
r=0.3175
a•n
ß data
V = 1.81 km/s
Air
P = 0.30 atms
r =0.3175
oxt
V = 1.82 knfs
1.0'
0.8,
0.6,
0.4
0.2
0.0,
1.8,
1.6•
/,:•..'..:•::.'::::.-::•:.:..-'..-:...'.'-:.'::i*-Y.,
E:.•..::::*
-;'•......
•.
-•
'•'... •,-::-.','.;.-.'::..
',-
/
....
•:...••...•-..
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•--?•••,.....
-•-o•-•-..
1.2
'"'"?"'"'•'"'•'"'•'••
':•"'•'
••••••½i
•-*•'
•--.•-'.•..'...•,.....,:'•:•.:<
..........
....•..:•:•:•.:.•:.:•..•:•.•:•.•.•.:...:•::..::•.:•.•:•.•.:•!!•5i•??
:.
/
...
,x<..•?.j..:¾,•...:.
,.,.::.:2:-:.:------.ß
:...:-.::
:::::::::::::::::::::::::::
:-:
..:
.:::..:-:
:.:.::
-:.:
:::::::::::::::::::
ß ..•
0.8'
ßx-•.::l•
•..--.E:i:!.-.
•:i:i-i:•:i:!:•:i:2:::•>.':•.
::::::::::::::::::::::::::
.•:!:ii:i.5::
.:::•:!:!i:!.•
.>:::•,>'---,?".'.::
........ i.:.-i!.
.-.:.
.':i
•?:•g•!•'•::•!•'.'•!•!•½•::•i:..•½.i•'.•i:::i::
:•:::•':"•
'":'">':f
?"•
........=
.....
. .i•':':! :::..... :" .
•'•.•'•?•-'•:•-.-.'-".•¾
' :'•-•'......
'...•.-•.•:"
....'::..':•....fi.:
•'•..;•:
'"i•'?:•:'.
:.11'•¾
•..---?"•.
0.6'
0.4
0.2
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
2.6
2.8
3.0
3.2
3.4
Figure9. Observed
andtheoretical
windvelocities
created
by theringvortexasa functionof dimensionless
time(asin Figure5). Toptwographs
illustrate
TM results;
thebottomtwographs
illustrate
LM results(see
Figure6). Thesolid,dashed
anddotted
curves
aredefined
in Figure6. Shaded
regions
showtherangein model
estimates
duetoobserved
range
ofL, U,andtp;theirshading
scheme
isdefined
in Figure
8. TheRTMorRLM
rangeoverprints
theETM or ELM rangeandbothoverprint
theITM or ILM range.Observed
windvelocities
are basedon the motionof entrainedlow-densityStyrofoamballsplacedon the surface. Small-scalevaria-
tionsin the velocityreflectsecond-order
perturbations
affectingthe balls'advance
alongthe targetsurface.
Error barsreflectrangeof velocitiesobtainedfor Styrofoamballsfrom measurements.
neverthelessarise betweentheory and observations.Such discrepanciescan be largely attributed to additional physical
phenomenaof secondaryimportanceaffecting the ring vortex.
Comparisons between observation and theory.
exists between
estimates
of the vortex
core radius obtained
from the RTM and observation. Second, the observed wind ve-
locities (from the Styrofoam tracers) and the flow velocities
estimated by the turbulent flow model behave similarly, deThe observed and estimated radii of the start-up vortex present creasingin time as expectedby viscousdecay. Third, the conthe best match between observationand theory. Both radii are stant outward velocity of the ring vortex during stageII results
essentially identical within experimental uncertainty and from expectedviscousflow behavior. The large height of the
demonstratethat flow separationis most likely the primary ring vortex with respect to its core radius and the short time
physicalmechanismcontrollingthe creationof the ring vor- since the curtain became permeable minimizes viscous decay
tex when the curtain remains impermeable. Moreover, shear effects upon the advancing velocity of the ring vortex.
stressesacting at the inner surfaceof the ejecta curtain do not Fourth, the reduction in outward velocity of the vortex core at
seem to play an important role at the early stagesof vortex late time (t/T > 3-4) reflects how viscousdiffusion reducesthe
generation(beforeS in Figures5 and7). Theorythereforepre- strengthof the flow in a turbulent decayingvortex well after
dicts a reasonablemagnitudefor the circulationF generatedby its creation. In the turbulent model estimates, the velocity reductionresultsuniquely from viscousdiffusionin the ring vorthe advancing curtain.
The comparisonbetweentheory and observationconfirms tex. In reality, both viscous diffusion and surface shear
that viscousdecay plays an important role in controlling the stressescontribute to velocity decay.
The expansionof the radius of the vortex toroid seenboth
behavior of the flow in the ring vortex. First, a good match
21,110
BARNOUIN-JHA AND SCHULTZ: EJECTA ENTRAINMENT BY RING VORTICES
in theory and observationsdemonstratethat the motion of the
ring vortex is self-induced. For a self-induced vortex, the
magnitude of the expansion velocity of the ring vortex depends (besidesviscousdecay) on the height of the vortex core
center. During stageII, becausethe vortex height is large and
varies only slightly (first increasing and then decreasingimmediately after S in Figure 5), the velocity of the vortex remains constant. The accelerationof the core of the ring vortex
at the beginning of stage III reflects its decreasein height
(Figure 5).
Turbulent flow in the ring vortex.
Observations
provide evidence that the flow in the ring vortex is turbulent.
First, the RTM best fits the observedgrowth of the ring vortex
while being physically the most reasonable. A laminar flow
model, which also conservesboth kinetic energy and momentum (RLM), predictsa vortex radiusthat reducesin size as function of time (see Figure 6), a result never observedin any of
the impact experiments. Although the other laminar flow
models also predict ring vortex growth, it is not significant
comparedto observation. Second,observedwind velocitiesat
the surface generally match estimates for the turbulent flow
models. And third, the turbulent flow modelspredict a stageIV
in the outward motion of the ring vortex consistentwith ob-
The small difference after t/T ~3 between the laminar theory
and observationsfor the outward motion of the ring vortex is
not realistic because its motion is estimated from the observed
heightof the ring vortex. This estimatemaintainsthe laminar
ring-vortexcore at a height greaterthan the radiusof the vortex core calculatedby the laminar flow models,a heightthat is
contrary to both experimental and theoretical expectations.
Indeed, the self-inducedmotion of the ring vortex shouldmove
it downward
to within
at least one vortex core radii from the
surface[Magarveyand MacLatchy,1979;Shariffand Leonard,
1992]. Because the outward velocity of the ring vortex core
variesinverselywith distancefrom the target surface,the outward motion estimatedby the laminar flow modelsin Figure 7
is slower than expected for a truly laminar ring vortex.
Therefore, artificially maintaining the laminar ring-vortex
core at the height observedin experimentsreducesfrom the
point of impact the run-outdistanceof the ring-vortexcore as
a function
of time.
Additional physical processes. The small but apparent differencesbetweenthe theory and observationsoccur becausetwo physical phenomenaof secondaryimportancealso
affect the flow in the ring vortex: first, the containmentof the
ring vortex by the particle flow in the adjacentejecta curtain
and, second,the compressionof the ring vortex as it pushes
servations for some of the data. The laminar flow models,
however, generally predict an increasein the outward motion itself againstthe target surface.
Immediately after the curtain becomespermeable,the core
of the ring vortex during stageIV; this is not observed.
The observationthat the velocity in the ring vortex is tur- of the vortex ring continuesto move upwardswhile it is near
bulent agrees with observedturbulent ring vorticesgenerated the curtain (immediatelyafter S in Figures5 and 10). This upby an acceleratingpiston[Sallet and Widrnayer,1974]. Sucha ward motion of the ring vortex resultsfrom the drag createdby
piston is roughly analogousto a deceleratingejecta curtain the upwardtrajectoryof the ejectaparticlesat the inner surface
of the ejecta curtain. Unlike the simplified theory that estijust before completionof crater growth.
It is importantto determinewhetherthe flow presentin the matesthe free growth of the ring vortex, the close proximity
ring vortex is laminar or turbulent,sincethe choiceaffectsthe of the curtain causesthe vortex growth to stagnate(Figure 10).
lifetime of the winds created by an advancingejecta curtain. The upwardmotionof the ejectaparticlesin the adjacentcurThis is especiallysignificantin future studieswherewe wish to tain could be offsettingviscousdiffusion and growth of the
examine how curtain generatedring vorticescontrol the final vortex. This curtain effect on the vortex becomes less
pronounced
asthedistance
betweenthecurtainandringvortex
ejecta emplacementprocess.
Theoretical
and
observational
differences
and
their implications. Several differencesalso exist between
the theoretical predictionsand observations(Figures 6, 7, 8
and 9). The observedradius of the core of the ring vortex and
the outward motion of the Styrofoamtracersare slightly less
than that predictedby the turbulentflow theory but significantly greater than that predicted by the laminar models.
Furthermore, the observedoutward velocity of the ring vortex
core exceedsthat estimatedby the turbulentflow models,after
t/T ~ 3, but is less than estimated by the laminar ones. All
these results indicate that the turbulent models predict excessive decayof the flow in the ring vortex;however,the laminar
flow models estimate too little decay.
The assumptions
usedin calculatingthe outwardmotionof
the ring vortexfurtherenhancethe differencebetweenthe tur-
increases.Eventually,the vortexmovesdownwardby its selfinducedmotion, and the vortex radiusgrows independentlyas
assumedin the theory. This usuallyoccursfor t/T ranging
between 2 and 3.
Once at the surface,the expansionof the vortex core sub-
sidesslightlymorerapidlythanexpected
by theRTM (Figures
6 and 10). The force of the self-inducedflow compressing
the
vortexat the targetsurfacedeformsthe shapeof the ring vortex into an ellipsoidwith its long axis paralleland its short
axisperpendicular
to the surface.In contrastwith theory,such
deformationreducesthe average growth of the ring vortex,
therebypreventing
furtherfree viscousdecay(Figure6).
It is importantto emphasize
thatthe differences
betweenthe
theoryandobservations
are small. They are the resultof the
second-order
effects,which slow viscousdecaywhencompared
to the turbulent flow models. The RTM accuratelymodelsthe
sume that once the calculatedradius of the ring vortex exceeds first-orderbehaviorof the ring vortex as exemplifiedby sucbetweentheoryand observations
for the
the vortex height from the target, the vortex remains at a cessfulcomparison
bulent flow models and the observations after t/T ~ 3. We as-
heightequalto its coreradius. This assumption
avoidstheunphysicalsituationwherethe edgeof the vortexcoreexistsunderground. Figure 5 showshow this assumptioncausesthe
generation,growth,and flow velocitiesof the ring vortex.
The turbulenttheoryalsoreproduces
the observedoutwardmotion of the ring vortexuntil t/T~3, after which the assump-
vortex to travel at a height exceedingthat observedin the ex- tionsusedin calculatingthe outwardpositionof the ring vorby theturbulent
perimentsafter t/T ~ 3. The theoretically
estimated
velocityof tex amplifytheexcessviscousdecaypredicted
the ring vortexbecomessignificantlylessthanobservedafter flow models. These results indicate that the physical prothis time (Figure 8). This assumptionthereforeamplifiesthe cessesof flow separation,self-inducedmotion,and turbulent
effect of excessviscousdecaypredictedby the turbulentflow viscousdecaycontrolthe behaviorof andthe flow in the ring
models.
vortex.
BARNOUIN-JHAAND SCHULTZ:EJECTAENTRAINMENT BY RING VORTICES
ß
i
,
i
,
i
21,111
,
B
3.0.
2.0,
ß
adam
mßm ß l
I ß lll
1.0.
P=0.21
ill1
Air
am
P=0.21
alms
r = 0.238
cm
r = 0.238 cm
V= 2.17km/s
V = 2.17 km/s
0.0
3.0-
2.0.
ß
1.0
Air
Air
P = 0.21 alms
r = 0.238 cm
r = 0.238 cm
V = 2.17 km/s
V= 2.17km/s
'
0.0
1.0-
ß
•
ß
ßll••l•
0.8-
I
•
-- _•a'
0.6.
I
0.4!•
ß
Air
Air
l•.l
- -•,,,.
P =0.21
ß
P=0.21
alms
r:0.238•m
ß •1•
atms
r = 0.238cm
ß
V= 2.17
km/s
V =2.17
km/s
- - l•
1
-I •1
0.2-
0.0
'
I
1.0
'
I
2.0
'
I
3.0
TnVm
err)
•
4.0
0.0
1.0
2.0
3.0
4.0
TnVm
err)
Figure 10. (a) Measured
radiusof thevortexcore(rmax),thetoroidradius(R) andthe verticalheight(y) of
the vortexcorescaledto the craterradius(Rc) asa functionof dimensionless
time (t/T) (seeFigure5). Distinct
stagesin vortexevolutionexistnotonlyin its expansion
[Figure7 and8] but alsoin its growth.Thesevariations coincidewith observedchangesin heightof the ring vortex. They indicatethat particlestravelingupward at the inner surfaceof the curtainpull the vortexupwardand possiblyhalt viscousdiffusionwithin it.
Whenthecurtainandvortexseparate,
the vortexmovesdownwards
anddecaysfreely. Oncethe vortexreaches
the targetsurface,freeviscousdecayagainis diminished
asthevortexpushes
itselfagainstthetargetsurface.
(b) Theoretical
radiusof theringvortexandits motionusingtheRTM. Observed
stageswheretheradiusof the
ring vortexstagnates
arenot observed
because
thetheorypredictsonlyfreeviscousdecay. Nevertheless,
the
stagesdescribed
in Figure7 for the expansion
of the ring vortexdo occuras a resultof changes
in boththe
heightand viscousdecayof the ring vortex.
Implications. The confidencegainedby the comparison sities) decreasethe affectedejecta size, theseconditionsdo not
betweentheory and experimentallows one to establishthe affect either the processof flow separationor the formationof
importanceof curtaindriven winds duringplanetaryimpact a ring vortexbehindthe advancingejectacurtain.
Three forcesact uponan ejectaparticleat the top of the imevents. Suchwindsentrainejectabelow a criticalsize,thereby
permeable
portionof the ejectacurtainjust when the curtain
affectingstylesof ejectaemplacement.For this purpose,we
considersmall craterswhoseimpermeableejecta curtain does becomespermeable(1) atmosphericdrag due to the ballistic
not exceedthe scaleheightof the atmosphere.Threesimplify- motion of the ejecta, (2) atmosphericdrag due to the winds
ing butjustifiableassumptions
are thenmadeas well. First, separatingof the top edgeof the curtain,and (3) gravitational
we use the incompressible
start-upmodel,sincethe flow ve- pull (Figure 11). Balancingtheseforcespermitsone to estilocityat the pointof flow separation
remainsbelowa Mach mate the maximum ejecta particle diameter that will be ennumber of 0.4 to 0.6 for craters up to 30 km in diameter. trainedout of the curtainby curtaingeneratedwinds. EjectaenSecond,we assumethat the curtain becomespermeableat the trainmentby winds derivedfrom the top of the ejecta curtain
time of crater formationand that its length is equal to the crater will not significantlyreducethe entrainmentcapacityof these
radius. And third, we assumeambientatmospheric
conditions winds when the curtain first becomesimpermeable. On Venus,
basedon previousarguments
that late-stageejectaemplace- these winds carry ejecta up to 2 m for a 10 km apparent
craterdiameterandup to 8 m for a 30 km apparment will not be significantlyaffectedby early-timedistur- (precollapse)
bances[Schultzand Gault, 1979, 1982; Schultz, 1992a, b]. ent crater diameter (45 km rim diameter after collapse).
Althoughelevatedtemperatures
(andreduced
atmospheric
den- Smaller craterson Venus, however, are more likely affectedby
21,112
BARNOUIN-JHA AND SCHULTZ: EJECTA ENTRAINMENT BY RING VORTICES
wind flow
CurtainTop
•
ing the winds createdby the displacementof ambientatmosphereby an advancingejectacurtain.
Atmospheric
The aboveestimates
for entrained
ejectadiameters
areprobdragdueto
curtain
ted
Atmospheric
ably conservativebecausethey representejectaentraineddirectly from the top of the impermeablecurtain (Figure 11).
The windsin the coreof the ring vortexare strongerthanthose
at the top edgeof the curtain,and potentiallycan transport
largerejectafrom the backsideof the curtain. Similarly,larger
particlesfrom the target surfacecan be transportedbecause
they are subjectto (1) winds whosevelocity sumsthe flow velocity in ring vortex and the lateralvelocityof the expanding
ring vortex; and (2) saltationby ejecta alreadyentrainedin the
winds [Schultz, 1992a]. Ejecta entrained in curtain-derived
winds may offset increasesin their entrainmentand carrying
drag
due
to •
particlemotion •
101 r p=po, T=To, p = po
-
I•eable
100 =
eje
curtain
.
wJeight,
mg
VENUS
.
' •--EARTH
10'1 r
Figure 11. Sketchof forcesaffectingparticlesat the top of
the ejectacurtainwhenthe curtainbecomes
permeable.By
balancingtheseforces,we candeterminethe maximumejecta
particlediameter
thatwill be entrained
by thecurtain-generated
winds (see text).
ß
--I--
--&--
MARS
10.2
10.3
101
.' p=po, T= 100To,p=po/100
i,
-
10-4
r
.:
.
oor
.
the disturbedatmospherecreatedby the initial shock. On
Earth,the ejectaentrainedrangesbetween2 mm for a 0.2 km
apparentcraterdiameterto 20 cm for a 30 km apparentcrater
diameter(Figure12). On Mars,suchwindscarrysandto pebble sizedejectaup to 200 gm for a 0.2 km apparent
craterdiameterand 5 mm for a 30 km apparentcraterdiameter.The es-
1
-
10-1 r
ß
EARTH
']:.
--/x-- MARS
ß
F]lllllllll
IIIIIlllllllllllllllll
ß
10'2
curtain velocity [Schultzand Gault, 1979;Schultz, 1992a].
10'4
the curtain.
VENUS
ß
10'3
permeable
curtainwhichaccelerates
the flow impingingupon
--n-.•o--
ß
timates for Mars are a factor of 10 greaterthan previousfirst
order estimates for entrainment based solely on estimatesof
Our model now includesflow separationat the top of the im-
ß
.
,
0
I
5
.
I
10
.
I
15
.
I
.
20
I
25
.
30
CRATER DIAMETER (km)
Figure 12. Diameter of ejecta particlescarriedby curtaingeneratedwinds as a function of apparentcrater diameteron
atmosphereat ambient atmosphericpressures[Schultz, Mars, Earth, and Venus for ambient (top) and reducedatmosphericdensities(bottom). Actualcraterrim-diameters
require
1992b]. However, the temperatureand hencedensityof the
a factorof 1.5 increasein size (trueonly for craterslargerthan
atmosphere
maynot recoverfromthe initialimpactshockby 5-10 km on Mars). The lengthof the impermeablecurtaindoes
this time. Figure12 showsthe effectof increasing
the ambi- not exceed the atmosphericscale height of each respective
On Mars and Venus, calculationsshow that by the time of
craterformationmostof an ejectacurtainwill be surrounded
by
ent temperature
by a factorof 100 on the size of ejectaen- planet. The limited range of craterdiametersminimizesthe eftrained. On Venus, the entrainedejecta diameterstill repre-
fects of compressible
flow effects(see text). Flow separation
sentspebble-sized
debris (from2 to 3 cm for cratersranging and a vortex sheets also should occur, however, when comfrom 10 to 30 km in apparentcraterdiameter). On Mars, the
maximumdiameterof ejectaentrainedrepresent
sand-sized
debris (from 100 [tm to 800 [tm for cratersrangingfrom 0.2 to
30 km in apparent
craterdiameter).On Earth,presented
results
assume
thatthe atmospheric
pressure
alsohasrecovered
by the
pressibility becomes important for more rapidly moving
ejecta curtainsof larger craters. For thesecalculations,the curtain becomespermeableat the time of crater formation and the
curtain length is equal to the crater radius. The two atmospheric densities used reflect (1) ambient surface conditions
and (2) higher postimpact temperatures possible in the
time of crater formation. For these conditions, entrained parpostimpactenvironment(assuminga factor of 100 increasein
ticlesrangefrom200gm to 5 mm for cratersrangingfrom0.2 temperature).Even underreducedatmosphericdensitycurtainto 30 km in apparentcraterdiameter. In reality,the atmo- driven winds are significant during an impact process. The
sphericpressure
recovers
by windsthat accelerate
inwardto- jump in particlediametersfor the Earth reflectschangesin the
wardthe crater[e.g.,Schultz,1992b],therebyactuallyenhanc- drag coefficientof a sphereas a functionof Reynoldsnumber.
BARNOUIN-JHAAND SCHULTZ:EJECTAENTRAINMENT BY RING VORTICES
capacity becauseejecta within curtain-derivedvortices should
reduce their flow velocity more rapidly than by atmospheric
viscousdiffusion and target sheardrag alone.
The large ejecta that curtain-drivenwinds can entrain even
when the atmosphereis heated suggestthat such winds most
likely play an important role in the ejecta entrainmentand deposition processon planets with atmospheres. Results suggest that on present-day Mars where the atmosphereis tenuous, ejecta entrainmentand depositionas describedby Schultz
and Gault [1979, 1982] and Schultz [1992a, b] is likely to occur.
The calculations
show that even terrestrial
21,113
Appendix
To estimatethe effect of the outwardexpansionof the core
of the ring vortexuponthe flow velocityin the ring vortex,
we first consider the laminar Oseen line vortex:
_r 2
F
Vø=--2•:r1-
e
4¾(t-t•)
(A1)
craters as small
as Meteor Crater in Arizona shouldpossesssome evidence for
ejecta entrained and emplacedby the interactionbetween the
ejecta curtain and the atmosphere[Schultz and Gault, 1979;
Schultz and Grant, 1989, Schultz and Anderson, 1996].
where V0 is the axial flow velocity in the vortex; r is the distancefrom the centerof the core of the ring vortex;v is the
kinematicviscosity;t is time sinceimpact;ti is the time taken
by the Oseenline to reachthe radiusof the ring vortexestablishedwhen the curtainbecomespermeable.The circulationF
is replacedby F' in orderto facilitatecomputations:
Concluding remarks
r2
Comparisonbetween experimentsand theory indicatesthat
the start-up vortex model provides a physically reasonable
magnitudefor the circulationcreatedby an advancingimpermeable ejecta curtain. After the curtain becomespermeable,
the general behavior of the ring vortex is attributed to turbulent flow in its core. Turbulent flow models(in particular,the
RTM) reproducethe growth of the ring vortex, the outwardmotion of the ring vortex before t/T = 3, and the winds (flow velocity) of the ring vortex at the target surface. Although the
turbulent models underestimatethe general run-out of the ring
vortex after t/T- 3, such models reproducethe general behavior of the motion of the observed ring vortex. The laminar
models are far less satisfactoryin reproducingcritical aspects
of the ring vortex.
Small differences between the theory and the experiments
arise becausethe theory does not take into accountsheardrag
acting on the established ring vortex by the ejecta moving
upward along the inner surface of the curtain. Nor does it include the average reduction in growth of the ring vortex as it
pushesitself against the target surface. Both effects restrict
the growth and thus the decay rate of the ring vortex when
comparedwith turbulent theory.
This study demonstratesthat curtain-driven winds possess
significant ejecta entrainmentcapacity and should play a significant role in the emplacementof ejecta on planets with atmospheres. The model resultsindicate that many of the atmospheric processesresponsible for ejecta entrainment, transport, and deposition at laboratory scalesvery likely help create the ejecta morphologyof planetary craters. Future studies
will assess more completely the entrainment of ejecta by
winds created during largescale impacts in different atmospheric environment. Such studies will analyze the atmospheric and physical parameters which control the time at
which the ejecta curtain becomespermeableand will consider
craters larger than 30 km in diameter by including the effects
of supersonic flow separation and a stratified atmosphere.
These studies will also consider mechanismsby which a ring
vortex dissipatesdue to either ejecta entrainment, viscous diffusion or surfacesheardrag to assessto what extent impact-derived atmospheric vortices control ejecta emplacement processesand to what extent turbidity type flows derived from
these vortices control ejecta emplacement.
F'=F
1- e
4v(t-ti)
(A2)
The energyE in the ring vortexis givenby
wherem = 2 p n r2R is themass
in thetorusconstituting
the
ring vortex,p is the densityof the fluid in the ring vortex,R is
the radius of the ring vortex from the centerof the core of the
vortex to the centerof the impact. Combining(A2) and (A3),
the energyof the flow in the ring vortex at the time the curtain
becomespermeableis given by
Eo =
prb•o
4•
(A4)
The subscript0 pertainsto the time when the curtain becomes
permeable
andtheringvortexis firstallowed
to decayfreely.
After the ring vortex expands,equation(A4) takesthe form
En -'
pr;•n
4•
(AS)
where the subscriptn implies conditionsafter the curtain and
ring vortexdecouple.Otherthanthe lossesof energydue to
viscousdiffusionthat are alreadyincludedin the formulationof
the Oseenline vortex,En andEomustequaleachotherin order
to conserveenergy. The new circulationF'n in the ring vortex
therefore must take the form
(A6)
Conservingenergy in the ring vortex, therefore,tends to en-
hancethe decayof the circulation,
i.e. to reducethe velocityof
the flow in the ring vortex.
If the angularmomentumH in the ring vortexis conserved
as the ring vortex expands,the momentumbecomes
21,114
BARNOUIN-JHA AND SCHULTZ: EJECTAENTRAINMENT BY RING VORTICES
H0 = rn(rx V0)
2
,
(A7)
= pr c ROFO
when the curtainbecomespermeable. The variablerc is equal
to the radius of the core of the ring vortex. Again, the subscript 0 pertainsto the time when the curtainbecomespermeable. After the ring vortex moves outward, the ring vortex
possessesa new angular momentum
would also like to thank Kopal Barnouin-Jha, Seiji Sugita, Marc
Parmentierand Martin Maxey for their helpful commentsand suggestions. Sincerethanksto StephenBaloga and an anonymousreviewer
whose helpful commentsimproved this manuscript. This study was
madepossibleby a NASA SpaceGrant Fellowship,the NASA Graduate
StudentResearchers
Program,and NASA grantNAGW705.
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