VOL. 84, NO. B9
JOURNALOF GEOPHYSICALRESEARCH
AUGUST 10, 1979
Soil Transportby Winds on Mars
BRUCE R. WHITE
Universityof California,Davis,California95616
Theeoliantransport
of surface
materialon theplanetMarsisestimated
fromresults
of low-pressure
windtunneltestingandtheoretical
considerations.
A semiempirical
relationis developed
that will
estimatethetotalamountof surfacematerialmovingin eoliansaltation,suspension,
andsurface
traction.
The estimated
totalmassmovementof surfacematerialperunit widthtimeon thesurfaceof Mars isq =
2.61p(V,- V,t)(V,+ V,t)2/g(g/cms),wherepisthedensity
of theatmospheric
gas,g istheacceleration
duetogravity,
andV, andV,t arethefriction
speed
andsaltation
threshold
friction
speed,
respectively.
A
flat surface
composed
of particles
of nearlyuniformsizeisassumed.
A change
in themeanparticle
size
changes
thethreshold
friction
speed
V,t.Thepathlengths
ofsaltating
particles
andwavelengths
ofsurface
ripples
canvaryasmuchasa factorof 2 if thesurface
temperature
varies
from150to250K. Theangles
between
particle
pathsandthehorizontal
surface
arecalculated
tobeloweronMarsthanonearth,and
particles
travelmuchfaster
onMarsthanonearth.
Theratiooffinalparticle
velocity
tothreshold
friction
speed,VF/V,t,is foundto be several
timesthatof saltation
onearth.
•'t = ,40(,%,- o )gD•,
l NTRODUCTION
(1)
This paperpresents
threeaspects
of the movement
of sur- wherert is the surfaceshearstressat threshold,A is a varying
p•, is the particledensity,p is the fluid
facematerialby windon Mars andassembles
themto estimate empiricalcoefficient,
density,
g
is
the
acceleration
dueto gravity,andD•,isthemean
transportrates.First, the governing
equationsof motionare
is commonlyrewrittenin
presented
andtheir numerical
solutions.
Theseare discussedparticlediameter.This expression
with applicationsto Mars. Second,an analyticaltheoryis termsof the frictionspeedat thresholdV,t2 = r/p as
presented
fromwhichonemayestimatethesurface
flux.Last,
(2)
V,t : `4{[(jo- av)/a]gDv}•/0'
resultsfrom low-pressurewind tunnel experimentsare presentedand are usedto completethe semiempiricalrelationWhen the particledensityis much greaterthan the fluid
ship.The lasttwo sections
are in goodagreement
with basic density,this expressionreducesto
characteristics
of particlemotionasdetermined
bythenumerical solutions.The numericalresultspertain directly to Mars,
whilethe experimental
work isperformedat reducedpressure Bagnoldas wellasotherinvestigators
demonstrated
thatA is
to simulateMartian conditions.The resultingtransportequa- onlya functionof theparticlefrictionReynolds
number,B =
tion is, however,applicableto both earthand Mars.
V,tDt,/v, for thethreshold
condition.Herev is the kinematic
viscosityof the fluid.This conditionof A = A(B) is trueonly
SALTATION ON MARS
only of terrestrialsaltation.Figure I showsthe A versusB
The movementof materialby windblown(eolian) processes curvedisplaying
terrestrialwindtunneldata.Heretheparticle
canbe by surfacetraction,saltation,andsuspension.
Surface diameters
rangefroma fewmicronsto approximately
1.4mm,
tractionis the movementalongthe surface.Saltationrefersto and the particledensityvariesfrom 0.2 g/cm to 11.4g/cm.
a leapingmotionof particles
wheretheylift offthesurface
and Thisgivesa fairlywiderangeof conditions
to demonstrate
the
travelin ballistictypetrajectories
returningto the surface.In terrestrial uniquenessof the curve. Recently, low-pressure
the caseof suspension
the particlesleavethe surfacein the wind tunneltests[Greeleyet al., 1977]havedemonstrated
that
samemanner as that of saltation;however,they do not return A = A(B) is not uniquefor atmospheric
pressures
different
directlyto the surface.
The verticalturbulentvelocityfluctua- from the earth's.Other forces,previouslynot considered,
seem
tionsare largeenough(greaterthan the terminalfalling to havea strongfunctionaldependence
on the ambientpres-
speeds)
to keeptheparticles
suspended.
For everysizeparticle sure. Such forcesinclude electrostaticand interparticle reacthere is a minimum wind speednecessaryto createmotion. tions; these seem to alter the empirical coefficientA in a
Associatedwith the wind speedis the surfaceshearstressr manner not fully understood.Figure 2 showsa plot of a
that is a result of the gas motion above the surface.The dimensionless
frictionspeed,(A:B)TMversusa dimensionless
velocitydiminishes
to zeroat thesurface
for thecaseof contin- diameter(B/A):/3. Thesedata weretakenfrom experimental
uumflow (e.g.,wherethe Knudsennumber,the ratio of the work usinga low-pressure
windtunnellocatedat NASA Ames
meanfreepath lengthbetweenmolecularcollisions
of mole- ResearchCenter, Moffett Field, California. The test material
culesto a characteristic
lengthof the flow, is very small).For wasgroundwalnutshellthatwassieved
to theshownparticle
all terrestrialflows containingsaltatingmaterialthe contin- size distributionon the figure. If A is only a functionof B
uumassumption
isvalid,butit isnotalwaysvalidon Mars.As regardless
of pressure.
all data shouldbe fittedby oneuniverthegaspressure
anddensityare reducedin theMartian atmo- salcurve.However,asshownon the graph,thisis not thecase,
to depend
strongly
onparticle
diameter;
sphere,
theKnudsen
numberbeginsto increase.
At a valueof andthedataseem
0.1 or so,noncontinuum
effectsof theflow mustbeconsidered eachparticularparticlediameteryieldsits own curveon the
in the analysis.
plot.
Bagnold[1941]had demonstrated
that at the initiationof
EQUATIONSOF PARTICLEMOTION
saltation,
A one-dimensional flow situation is assumed to calculate the
Copyright
¸ 1979by theAmerican
Geophysical
Union.
Paper number9B0222.
0148-0227 / 79/009B-0222501.00
trajectories
in whichthe velocityin the vertical(y) directionis
4643
4644
WHITE: SOIL TRANSPORTBY WINDS ON MARS
1.2
[
i
I
i
i
0.3
1
3
10
30
1.0
0.8 -
0.6 -
0.4 -
0.2 -
0
0.1
60
Fig. 1. Bagnold'scoefficientA as a functionof the thresholdparticlefrictionReynoldsnumberB = V,t Dp/v. This is a
,uniquecurvefor atmosphericpressure.A obtainsa constantvalue of 0.118 for valuesof B greaterthan 10.
zero and the velocityu in the flow (x) directionis a functionof
height above the surfaceonly. A viscoussublayerexistsfor
flow over a surfaceof like particleson Mars when the wind
speedis below threshold[White et al., 1976].However, when
minimum thresholdspeedsare reached,the optimumsizeparticlesbeginto saltate.This consequentlybreaksup the viscous
sublayer,and a fully turbulentboundarylayer is formed.The
presenceof saltatingparticlessignificantlyaltersthe natureof
the velocity profilesas describedby White and Schulz[1977]
and is representedby
U
V*t
V, - 2.5In(y/Ds)- 2.29+ 1.79V,
(4)
where u is the velocity in the flow direction, V. is the friction
speed, and Dp is the average particle diameter. As V. approachesV.t, this equationapproachesthe velocityprofileof
the fully developednonsaltatingturbulent boundarylayer.
0.7
I
The forcesacting on a saltating particle are a downward
force due to gravity, the aerodynamicdrag force D acting
oppositethe direction of relative particle velocity Vr, and the
lift forceL actingnormal to the drag force.The equationsfor
translationalmotion of a particlecan be written as
mp.•=
L-f-- D'•- u
mpj)
= -L '•Vr
- u- D -•y - mpg
(5)
(6)
where m• is the particle's mass, (•, •) and (2, •) are the
particle'svelocity and accelerationcomponents,and g is the
accelerationdue to gravity.
In (5) and (6),
=
u) +
(7)
The effectsof lift and drag are expressedin terms of lift and
drag coe•cients C• and C•, definedby
I
L = ••D/g/
D = ••D/g/
(8)
0.6
Assumingthe particlesare sphericaland of uniformdensity,
the equationsof motion reduceto
Dp pm
0.4
0.3
m
ß
12
ß
32
ß
52
•
69
ß
150
o
355
o
718
• [C•(•- u)- C•9]
• = - •• •• D•
f = 4• D•
0.1
0.01
0.1
1
10
(B/A)2/3
Fig. 2. Dimensionless
thresholdfrictionspeed(A:B)TMas a function of dimensionless
particle diameter (B/A W• for severaldifferent
particledistributions
all havingan equalparticledensityof 1.1g/cma.
The pressure
rangedfroma minimumof 5 mbar(0.5 kPa) to I atm.If
Bagnold'scoctficicnt
A is uniquelyonly a functionof B for the range
of pressures
tested,thenonlya singlecurveshouldsufficeto fit all the
experimentaldata. This is not the case,sinceeachseparateparticle
diameterrangeexhibitsits own curve.HenceA = A(B) onlyis not the
casewhen the atmosphericpressureis varied or altered.
+
(•)
- u)]- g
wherep• is the particledensity.
The drag coe•cient of a sphereis stronglydependentupon
the Reynolds number. The empirical formulas of Morsi and
• lexander[ 1972]are usedfor its determinationin the numerical calculations.
The initial conditions are same as those used
by White et al. [1976]. Initially, the particle is at rest on the
surface,and as the wind speedis increasedabovethreshold,
particlesbeginto move.The expressions
for the lift coe•cients
are presentedby White et al. [1976]. The reader may wish to
reviewan in-depthdiscussionof this numericalsolutionprocedure, includinga completediscussion
of this technique,in the
work of White et al. [1976].
WHITE: SOIL TRANSPORTBY WINDS ON MARS
4645
TABLE 1. Values of Gas Density, Viscosity,Mean Free Path, and Particle Fall Speedfor Various SurfacePressuresand Temperatures
Surface Pressure,kPa
0.25
150 K
Gas densityp, x10 • g/cm8
Kinematic viscosityv, cm:/s
Mean free path 3,,* •tm
Particle fall speed,•'cm/s
0.5•tm
1.0•tm
2.0 •tm
10.0•tm
50.0 •tm
0.882
8.59
9.59
1.11X 10-x
2.20 X 10-x
4.36 X 10-x
2.46
26.1
0.75
250 K
0.529
23.8
20.6
1.44X 10-x
2.86 X 10-x
5.68 X 10-x
2.85
21.6
150 K
2.65
2.86
3.20
3.64X 10-•7.27 X 10-•1.50 X 10-x
1.27
20.4
1.5
250 K
1.59
7.94
6.87
4.75X 10-•9.42 X 10-•1.88 X 10-x
1.16
14.2
150 K
250 K
5.29
1.43
1.60
1.82X
3.76 X
8.59 X
9.87 X
19.0
3.18
3.97
3.43
10-•10-•10-•10-x
2.35X
4.69 X
9.65 X
7.88 X
12.4
10
10-•l0
10x
*The meanfreepathsare calculatedby Maxwell'stechnique,as describedby Bird et al. [1960,p. 21];here3•t = pdX,whered is the molecular
velocityrelative to the fluid velocity and •t is the absoluteviscosity.
pTheparticlefall speeds
includethe effectsof slipflowaroundthe particle.The correctionfactorof Davies[ 1945]is usedin the determination
of the drag coefficient.
For calculationsin the Martian atmosphereit is necessary
to rates associatedwith the winds and thus greatly alter the
include the effectsof slip flow causedby the rarefied atmo- saltation characteristics of eolian movement.
sphere.The correctionfactor of Davies[ 1945]is usedto calcuFigure 4 showsthe estimatedthresholdwind speedsneceslate the slip flow effects.The mean free path neededin the sary to initiate particle motion for various surfacepressures
Davies equation is calculated by a conventionalChapman- for a range of particlesfrom a diameter of 10 •m to 1 mm.
Enskog technique (see Table 1). The slip flow factor only Also shownis the speedof sound,abovewhich the wind speed
becomessignificantfor the smaller-sizedparticlesin the calcu- would not likely be reachedin a planetaryatmosphere.Such
lations. Figure 3 displays a typical particle trajectory and wind speedsare improbable,sincethe largeamountsof turbuidentifiesthe important parameters.
lenceassociatedwith them would not allow transonicor supersonic flow [Pollack et al., 1976a].
COMPUTATIONAL RESULTS
Figure 5a showsthe dependence
of the 'typical' path length
This sectionexaminesthe dependenceof the particles'tra- of saltatingparticleson the particle diameterDp for valuesof
jectory path lengths as functions of severalkey parameters friction speedV, rangingfrom 5 to 10 m/s. The numerically
presentedin Table 2, which summarizesthe nominal and simulated surface conditions correspondto the values obalternative parameter values used in the calculations.One tained by the Viking Lander l, with a meansurfacepressureof
parameter is allowed to vary at a time. A broad range of 7.5 mbar (0.75 kPa) and a minimum gastemperatureof 187.2
surfacepressuresand temperaturesis explored.The nominal K. The averageheightsthat saltatingparticlesobtainedare 4caseis an approximateaverageof the surfacedata received 5 m abovethe surfacefor a friction speedof 10 m/s. This value
from Viking Lander 1 site.The atmosphericgas composition of V, would be encounteredrarely on Mars, consideringgenusedin the nominal calculationis equal to 100%carbon diox- eral circulationtheories[Pollack et al., 1976b].It shouldact as
ide. A sensitivitystudy showsthis to be valid within a few an estimatein obtaining an upper limit value of the saltating
percentof the actual atmosphericgas composition.
grains.A more realisticcaseis that of V, equal to 2.5 and 5
AtmosphericsurfacepressuresP vary greatly with time and m/s. The first case,near thresholdconditions(not displayed
location on Mars. At the bottom of the larger basinssuchas on the figure), showsonly minute leaps.ThesesaltatingpartiHellas Planitia the pressuremay be as high as 10-15 mbar (1- cleshave relativelyhigh collisionimpact anglesa, with many
1.5 kPa) with someseasonalvariation.The surfacepressureis of the larger particlesnot moving. These may be of primary
only a few millibars atop the large shieldvolcanoes.Kliore et importancein the final ripple pattern geometrywhen either a
al. [ 1972, 1973]have estimatedan averagepressurefor Mars of major sandstorm subsidesor when windjust surpasses
threshabout 6 mbar. Data from Viking indicatethat the mean pres- old but fails to continueto grow in strength.The friction speed
sure may shift upward slightly. There will be substantial of 5 m/s showssaltationaveragemaximumheightsfrom 0.4 to
changesin gas densityp and gas temperatureT with such 1 m. Theseheightsof saltatinggrainsare comparableto those
dramaticchangesin pressure.The largechangesin gastemper- found on earth. The path lengths are from 3 to l0 m. The
ature and density may greatly alter the prevailing shearing particlessmallerthan 10# m would, if injectedinto the wind by
Trajectory
Fig. 3. A typicalpath of particle'strajectory.H is the maximumheightthe particleobtainsin the trajectory,whileL is
the maximumjump lengthor pathlengthobtained;a is the collisionanglewhichtheparticle'spath makeswith thesurface
upon impact.
4646
WHITE: SOIL TRANSPORTBY WINDS ON MARS
TABLE 2. Values of the ParametersUsed in the Calculationsof Particle Trajectoriesand Ripple
Wavelengths
Nominal
Parameter
Symbol
Surfacepressure
P
7.5 mbar (0.75 kPa)
Gastemperatureneartheground*
Accelerationofgravity
T
g
187,242 K
372 cm/s:
Von Kfirmfin
k
0.4
Roughnessheight
Dynamic viscosity•'
y0
#
Do/30
Molecular weight
Atmosphericcomposition
...
...
44.01
100% CO:
Particledensity
Particlediameter
Friction speed:l:
Po
Do
V,
2.58g/cm8
10x-104
2.5, 5.0, 7.5, 10.1m/s
constant
Alternative
Value
Value
2.5, 5, 10, 15 mbar (0.25, 0.5, 1.0,
1.5kPa)
150, 250 K
..o
o.o
...
immovableroughnesselements
adjusted to the value appropriate
for T
43.65
96.1% CO:, 2.2% N, 1.5% Ar, 0.1%
0:, 0.1% CO
1.55, 3.54 g/cm 8
can account for varying
atmosphericstabilities
*Thesegastemperaturevaluesrepresentthe minimumand maximumtemperaturesexperiencedby the
Viking Lander I spacecraftduring the first 20 sols as reported by Hess et al. [1976]. These are not
intendedto representthe absolutevariation at the Viking Lander 1 site but rather provide a reasonable
estimate of a minimum and maximum value of temperature.
•'The values of # were obtained from the proceduredescribedby Bird et al. [1960, pp. 19-26]. The
kinematic viscosityv equals#/p. Note that in contrast to v, # dependsonly on gas temperatureand
composition.
:]:Thefriction speedcan account for changesin the atmosphericstability. Once the saltation is
initiated, the prevailingprofileswithin the saltationlayer appearto be independentof the atmospheric
stability (see (4)). Hence all that is necessaryto accountfor the atmosphericstability is to know the
proper ratio of V,/Vo, where V0 representssome characteristicvelocity at a particular height. Such
information can be obtained from a recentpaper by Pollacket al. [1976a],which presentsfriction velocity ratios for a variety of stability conditions.
impactingcollisions,likely go into suspension,
sincetheir terminal falling speedsare small(seeTable 1). This is shownby
the long path lengthsof smaller-particletrajectories.
Impact anglesof particlescolliding with the surfacehave a
wide variation in values. The collision angle a decreasesin
value with increasingsurfaceshear.A 0.15-mm particle has an
impact angle of approximately7ø-10ø near threshold(Figure
5b); and if the friction speedis increasedto 5 m/s, the collision
angle decreasesto 3.6ø. In comparisonto earth these angles
are unusuallysmall for similar ratios of surfaceto threshold
stress.The typicalrangeof anglesfor particlesgreaterthan 0.4
mm is from 5ø to 15ø, which agreeswith saltation on earth.
The larger particlesseldomsaltatemore than 6 to 10 m in path
lengths.
Figure 6 showsresultsfor a caselike that of Figure 5a except
the temperature is 241 K, the maximum measuredby the
Viking Lander 1 duringthe first sols(a solis a Martian day). A
drop in temperaturefrom 241.8 to 187.2 K reducesthe dynamicviscosityof the atmosphericgason Mars by 22.7%.At a
constantpressureof 0.75 kPa the gasdensitydecreases
22.6%,
drasticallyaffectingthe shearstresson the surface.The Reynolds number, ratio of inertia to viscous forces, is altered
substantially.The changein flow around individual particles
and the surface alters the pressuredistribution around the
particles.Hence this changesthe typical path lengthsof the
saltating particles.
As the temperatureis increased,the maximum heightsand
lengthsof trajectoriesare generallycurtailed.For the majority
of the casesthe characteristicpath length is substantially
shorter at the higher temperature.The maximum heightsoccurring in saltation on Mars have only minor deviationwith
changein temperature.Correspondingly,there are only small
P ,mb
differencesoccurringin impact angles.
1
00o
The trendsexhibited in the path length, maximum height,
500
and impactanglefor a pressureof 7.5 mbar(0.75 kPa) are also
presentfor other surfacepressures.As an example,an ex400
2.5
of Sound
tremely low pressurethat may exist on the surfacewould be
300
2.5 mbar (0.25 kPa); Figures 7a and 7b demonstratethat the
relationshipbetweenpath length and height is a function of
200
temperature.In thesefiguresa low temperatureof 150K and a
100
maximumof 250 K are used.The path lengthsare curtailedas
muchas50%with an increasein surfacetemperatureof 100K.
o
I
I
J
O.Ol
o.1
1
lO
An examinationof the final velocitiesthe particlesobtain
shows increasedvalues at the low temperatures.The final
Dp,mm
particle speedsare typically greater than V,. The value deFig. 4. Thresholdwind speedfor Mars as a functionof particle
diameterDo for severalvaluesof the surfacepressure
p. The horizon- pendson the boundarylayer flow and V,/V•. (V• is the wind
tal line representsthe speedof soundat a temperatureof 239 K [after speednear the top of the atmosphericboundarylayeron Mars
Pollack et al., 1976a].
as describedby Pollack et al. [ 1976a].)The valueof Vs would
WHITE:
SOIL
TRANSPORT
BYWINDS
ONMARS
I
I
I
4647
I
\%V,:];0
m/s
\
V,: ];0m/s
V,=7.5
m/s
_
\
\
V, = 7.5 m/s
zo
- •V,=5
m/s
\
•
0
V, = 5 m/s
i
i
i
I
I
0.2
0.4
0.6
0.8
1.0
PARTICLE
DIAMETER
Dp,mm
Fig.5a. Typical
path
lengths
L (dashed
lines)
andmaximum
vertical
heights
H (solid
lines)
forsaltating
particles
as
functions
ofparticle
diameter
Dpforseveral
values
ofsurface
friction
speed
V,.Thesurface
conditions
correspond
toa
pressure
of7.5mbar
(0.75
kPa),
aparticle
density
of2.58
g/cm
s,and
agas
temperature
of187.2
K.Note
thedifferent
scales
for L and H.
be the maximumlimit that the final speedof particlescould
obtain.
carrythese
particles
alofttohighaltitudes
andarecapable
of
transporting
these
particles
longdistances
before
theyslowly
returnto the surface.
Thisis in contrast
to
As theparticlediameter
decreases,
it becomes
moresuscep- andgradually
wheretheheightandlengthof particles
areonlya
tibleto turbulentvelocityfluctuations
in theboundarylayer saltation,
particles
in suspension
maybe
flow. When the terminalfallingspeedof the particlesis less fewmetersin value;however,
carried
up
to
altitudes
of
several
kilometers
and
carried
downthan the verticalcomponent
of turbulentfluctuations,
the
stream
distances
of
several
thousand
kilometers.
A
terrestrial
particles
arein suspension.
Theturbulent
eddies
of theflow
15
x
I
I
I
I
I
_•\ \\V,=
10
m/s
\
3
E 30
_[
•
\
8
•
20
-
\ V.=7.5m/s
\
\
• •
V,=7.5m/s
]•0 •
V, = 5 m/s
-• • • ....
o
o
•
I
--I ....
T-
0.4
0.6
0.8
V, = 5 m/s
- ---T---1.0
PARTICLE
DIAMETER
Dp,mm
0
0.2
0.4
0.6
0.8
1.0
Fig.6. Typical
pathlengths
L (dashed
lines)
andmaximum
verticalheights
H (solid
lines)
forsaltating
particles
asfunctions
ofparticle
Dpforseveral
values
ofsurface
friction
speed
V,.Thesurface
Fig.5b. Impact
angles
a asa function
ofparticle
diameter
Dpfor diameter
correspond
to a pressure
of 7.5mbar(0.75kPa),a particle
severalvaluesof surfacefrictionspeedV,. The surfaceconditions conditions
of 2.58g/cm
a,anda gastemperature
of 241.8K. Notethe
correspond
toa pressure
of7.5mbar(0.75kPa),a particle
density
of density
different scales for L and H.
PARTICLE
DIAMETER
Dp,mm
2.58g/cma,anda gastemperature
of 187.2K.
4648
WHITE:SOILTRANSPORT
BYWINDSON MARS
I
'=
1.0
•-
0.5
I I/l
I
--
V,=
7.5
m/L.,,.,.
--•
x•. -•,
V :7 5m/s
,,,,•-Z,
v* '•
-
V=5m/s
0.5
V - 10m•s
0
than unity on Mars for equal ratios of V,/V,t, where subscriptsM and E indicateMars and earth, respectively.From
V, •' = r/O and wind tunnel testsat low pressures[Greeleyet
al., 1977] the threshold stressis about equal to that of the
atmosphericcase.Thus oEV,•'/pMV,M•'is approximatelyequal
to unity. Hence V, is roughly 7 timesgreater;and comparing
the ratiosof Vg/V, indicatesthat Vr on Mars is substantially
larger than VF on earth. Theseare alsoindicatedin the numerical results;the ratio Vr/V, is generallygreaterthan unity for
typical saltationand can go as high as 10, the latter definitely
indicatingparticlesgoing into suspensionin numericalcalculations.
In comparisonwith the nominal test case of 0.75 kPa the
reduction
in surfacepressureto 0.25 kPa resultsin thesedifferPARTICLE
DIAMETER
Dp,mm
Fig. 7a. Typical path lengthsL (dashedlines) and maximumverti- ences:(1) the saltating particles are roughly decreasedin
cal heightsH (solid lines) as functionsof particle diameterDp for height and path length by a factor of 2-4, (2) similarly, the
several values of surface friction speed V,. The surface conditions ratio of maximum impact speed to V, is reduced with decorrepsondto a pressureof 2.5 mbar (0.25 kPa), a particle densityof creasesin pressure,and (3) the overall trajectoryand scaleof
0.2
0.4
0.6
0.8
1.0
2.58 g/cm8, and a gas temperatureof 150K.
flux movement
are reduced.
ANALYTICAL DETERMINATION
example of this is reported by YaMon and Ganor [1977], in
which Sahara dust (a few microns or less in diameter) was
carried to the far easternpart of Asia, some 5000 km.
An analysisof the infrared interferometerspectrometerexperimenton board Mariner 9 sizedthe majority of suspended
particlesfrom 0.1 to 10 •m in diameter in the Martian atmosphere.The boundary betweensuspendable
and saltatingparticles is determinedby a constantratio of VF/V,, where VF is
the final particle speedin its trajectory[Pollacket al., 1976a].
Toonet al. [ 1976]show that the Martian planetaryboundary
layer can supportparticleswith fall speedsas large as 1 cm/s
(10-urn-diameterparticles at 0.75 kPa). The calculatedratio
Vr/V, is severaltimes unity for theseconditions.This is at
variancewith SaganandBagnold[ 1975],who suggesta ratio of
OF SURFACE MATERIAL
MOVEMENT
The rate of movement of surfacematerial under specified
storm conditionsis of primary interestfor Mars. An estimate
of the total amount of surfacematerial transportedby the
wind may be made by examiningthe physicsof the process.
This will accountfor the changesin the Reynoldsnumber and
interparticleforcesthat occur with reductionof pressure.In
order to estimatethe surfaceflux the typicalpath lengthsof
saltationmustbe known, asdescribedin the previoussection.
The key parameterin estimatingsurfacematerialmovement
rates is the characteristicpath length of saltatingparticles.
Severalinvestigatorshave developedboth empiricaland theoretical methods to calculate path lengths[Tsuchiya,1969;
unity.It seems
reasonable
thattheratioVr•,/V•Ewillbelarger Kawarnura,1951;Bagnold, 1941]. Tsuchiyahas a theoretical
developmenton the successive
saltatingleaps.However, it is
necessary
to know the initial or maximumverticalcomponent
of velocity at lift-off for the particle in order to calculate its
path length.This componentis generallya functionof the flow
7
0.7
conditions.Formulationof this type,with the verticalvelocity
componenta functionof the path length,doesnot lend itself
6 --
5 --
-- 0.6
well to calculations
-
favorableexpressionis one wherethe path lengthdependson
the ratio of surfaceto thresholdstress.One suchanalysisis a
force balance on a control volume [Kawarnura,1951]. The
0.5
of surface material
movement.
A more
surface shear stressro consistsof two elements. The first is the
---4--•
-0.4
:•
Z:
''"
'.r-
\
3
0.3 =
'"
•
\v(:
0
-
0.1
o
0.4
0.6
(1 1)
or
,
0.2
TO -- T8snd+ Tw,nd
--0.2
1 •\ X'*••
V*=
7'5
m/s
% •v,=s,,,/,
o
ordinary wind shearrw,,d developedfrom the motion of the
gas flow over the surface.The secondstress,r,a,d, stemsfrom
the impact of particlescolliding with the surface:
0.8
!.0
PARTICLE
DIAMETER
Dp,mm
r,...d = p(V. + V.t)(V.-
V.t)
(12)
Tsan
d isalsoequalto the momentumlossin theflow direction,
r,,,a = M(Vr - V,)
(13)
whereM is the massof materialfallingper unit areatime, Vt is
Fig. 7b. TypicalpathlengthsL (dashedlines)andmaximumverti- the meanfinal horizontalvelocitycomponentof the particles
cal heights
H (solidlines)asfunctions
of particlediameter
Dp for at collision with the surface, and V• is the horizontal comseveralvaluesof surfacefrictionspeedV,. The surfaceconditions ponent of velocityduring the initial lift-off of the particles
correspond
to a pressure
of 2.5mbar(0.25kPa),a particledensityof from the surface.Assumingan inelasticcollision
2.58g/cm8, anda gastemperature
of 250K. Note thedifferentscales
for L and H.
wM • p(V, •' - V,t:)
(14)
WHITE: SOIL TRANSPORTBY WINDS ON MARS
from experience,
M ... p(V, - V,t)
(15)
Hence combiningyields
w•
V, + V,t
(16)
4649
flow to enter freely. The downstreamend is sealedwith #100
mesh wire to catch the particlesyet still to allow a sizable
amount of air to passthrough so as not to alter greatly the
flow field streamlines
aroundthe trap. The lengthof the trap is
approximatelyI m long to preverntreboundingparticlesoff
the back screenfrom bouncingout of the trap. The flow field is
carefully
observed,
andnoflowanomalies
arefoundowingto
or
the presenceof the trap. The trap catchessuspendable,
saltating, and surfacecreepingparticles.Knowing the width of the
whereg is the accelerationdue to the gravity of the planet. trap and the time of saltating, a material flux rate can be
calculated.
(The expressions
for w and L are in goodagreementwith the
First the terrestrialcaseis testedand found to be in agreenumericallycalculatedvaluesfrom the previoussection.)Thus
ment
with theory. The test is repeatedfor a Martian surface
the material flux q is
condition of 0.75 kPa and temperatureof 200 K. The results
from both experimentsare presentedin Figure 8, which shows
(18)
q "' P(V, - V,t)(V,+ V,t)2
g
the massflux q as a function of shear stress.At the lower
pressure,
substantiallylessstressis neededto move material.
OF
Furthermore, as the surfacestressis increased,the lower presL • (V. + V.t)a/g
(17)
sure data show dramatic increases in mass flux, while the
c
q= '•o(V.- V.t)(V.
+ V.t)
2
(19) terrestrial data show only small increasesin q. This notion
whereC is the constantof proportionality.
On Mars thereappear to bc differentinterparticleforcesas
well as effects from changes in Reynolds number. These
changeswill alter the saltationcharacteristics
by affectingthe
thresholdfriction speed.The flow field aroundindividualparticlcs(Knudsenand Reynoldsnumberseffects),shearingrate
within the boundarylayer, and the existenceof comparatively
large (to earth) viscoussublayer for nonsaltatingflow will
causechangein thresholdvalues.
supportsthe numerial results that show similar increasesin
particlepath lengths.The path lengthis almostdirectlyrelated
to the mass flux. It would
be useful if both cases could be
adequatelydescribedby a singleuniversalequation.According to the analyticalderivationthe only differencebetweenthe
relationsdescribingthe two casesis the empiricalconstantsC•
and C•. If the constantsof proportionalityC• and C• are the
same for both cases,this would indicate that the physicsbetween the two testsremained unchanged.This implies a universalityto the massflux process.As one would expect,the
processof movement of surfacematerial on Mars shouldbe
LoW-PRESSURE WIND TUNNEL EXPERIMENTS
nearlythe sameasthe processon earth. Hencethe constantsof
A seriesof wind tunnel tests was designedto explore the proportionalitybetweenthe two testcasesshouldbe nearlythe
relation between eolian movement of surface material between
same. From the wind tunnel data of both pressurecasesan
terrestrialatmosphericpressures
and low pressures
equivalent empiricalconstantwith a value of 2.61 was determined.
of Mars.(A discussion
of thewindtunneltestfacilityat NASA
Figure 9 displaysthe massflux versusa functionof friction
Ames ResearchCenter is given by Greeleyet al. [ 1977].)Using speeds.Here both high- and low-pressuredata collapseto a
(18), a comparisonbetweenearth and Mars can be made:
single line in support of the idea of a universal function describingthe movementprocess.Hence the followingequation
qu
= OK(v.
(V, -
+
V,t)E gE (V, + V,t)u 2 Cu
(20) canbe usedin estimatingmassflux on eitherearthor Mars
oncethefrictionspeedand thresholdfrictionspeedare known:
where the C are constantsof proportionality. Assumingthat
the ratio of friction speedto thresholdfriction speed.V,/V,t.
is the samefor both earth and Mars yields
q = 2.61V,so(1 - V,t/V,)(1 + V, t2/V,2)/g
This must be consideredthe main contribution of the present
work.
q• = o•g•,(V.•)•
•C•
q•
p• g•(V,t)• a C•
(2•)
ß
Duplicating aerodynamic forces for typical conditions on
Mars (temperatureof 200 K and surfacepressureof 0.75 kPa),
the ambient pressurein the wind tunnel should be approximately 2.3 kPa. At this air pressurethe gas densityof the
carbondioxideon Mars is equal to the gasdensityof earth air
in the wind tunnel. This createsequivalentdynamicpressures
and, consequently,aerodynamicforceson the particles.Interparticleforceequivalenceshouldexist,sincethe ambientpressure is substantiallyreduced.The effectsof gravity, not ac-
(22)
The value of 2.61 is determined
from the wind tunnel
tests.A sensitivitystudyshowsthe constantto be stableover a
range of altered conditions.
I
I
I
ß Martian Pressure
A Atmospheric
Pressure
Counted
in windtunneltests,canbe adjusted
to Mars.
The wind tunneltestsconsistof spreadinga uniformlayer 1
0
1
2
3
4
cm thick of sphericalglassbead particles(mean diameter of
0.208 mm) on the wind tunnel floor. At the end of the test
SURFACE
STRESS,
dyne/cm
2
sectiona material trap is placed. It consistsof two uniform
Fig. 8. Mass flux q as a function of the surfaceshearstressfor
thin platesspaced10 cm apart alignedparallel with the flow windtunneltests.Thecirclesareat 10• Pa, an•lthetriangles
areat 2.3
direction. The plates are from the floor to the ceiling of the kPa. Tests were performedin the Marswitt wind tunnel located at
wind tunnel.The front end(upstreamside)is opento allowair NASA Ames Research Center.
4650
WHITE:
SoilTRANSPORT
BYWINDS
ONMARS
SUMMARYANDCONCLUDINGREMARKS
,•
0,8
Numerical
solutions
of theequations
of motionforparticle
trajectories
on thesurface
of Marsarepresented.
Thepath
0.4
wheretheminimum
andmaximum
temperature
recorded
on
theMartiansurface
byViking1 and2 areconsidered.
Typi-
lengths
of saltating
particles
canvaryasmuchasa factorof 2
cally, impactingparticlescollide at the surfacewith lower
5
o
ß
angleson Marsthan on earthandhavesubstantially
more
kineticenergy.
Theirkineticenergies
canbe up to 50 times
greateronMars thanon earthfor identical
dynamic
particle
conditions,A discussion
is presentedon variationsof these
p (V,-V,t)(V,,V,t)2/•
, •m/(cmøs)
results
withchanging
parameters.
Theratioof finalparticle
speedto the particlethresholdfrictionspeedis foundto be
Fig.9. Mass
fiuxqasa function
ofthefriction
spe
edparameter.
Hereatmospheric
andlow-pressure
datacollapse
to a single
curve severaltimeslargerthan that of earth.For earth this ratio is
which
isthestraight
line(equation
(21))ontheplot.Notetheorigin knownto be approximatelyone.
dataforbothcases
aredatapoints,
since
they
areused
indetermining Windtunneltestsperformed
at low-pressure
andtheoretical
thethreshold
frictionspeeds
analysis
areusedto establish
a transport
equation
of surface
material
onMars.Windtunneldathsupport
thenotionof a
Thefluxof material
onMarsisconsistently
higherthanon universal
functionto describe
transport
of surface
material;
earth.Thisincrease
isdueto theexistence
of typically
longer e.g., both low- and high-pressure
transportdata are adepathlengths
in theparticle's
trajectory
andalsoto thereduc- quatelydescribed
by a singletransport
equation.
The trans-
tion of gravity,whichcannotbe modeledin the wind tunnel. port equation can be usedto estimatesurfacematerial flux on
Moreover,
usingtheestablished
equation,
theincrease
in q Mars with knowledgeof both the thresholdV,t and V,, Nubetween
theplanets,assuming
equalratiosof V,/V,t, is
merically
integrated
particleequations
of motionprovidein-
formation
of meanpa.thlengths
andmaximum
heights
of
(V,::)• q:
qM= (V,tS)e
13.5
(23)
saltating
particletrajectories
on Marsfor a rangeof condi-
tions. Path lengthsare from 3 to 10 m for 0.1- to 1-mm
For instance,
therelationship
between
thematerialmovement particles
at windspeedslightlyabovethreshold
(0.75kPaand
saltatingparticles
travelmuchfaster
on earthand on Mars, for equalratiosof lz,/V,t of 1,47, 200 K). Consequently,
wouldbeqM/q• = 6, or 6 timesasgreaton Marsfor 0,2.mm and manytimesfurther•on Mars in comparison
to earth.
Hence
thetransport
equation
shows
manytimes
thefluxof
materialonMarsin .comparison
todynamically
simiComparing
themovement
rates
forsimilar
dynamic
Condi-surface
particles.
lar conditions
onearth.Thisresultisverified
bywindtunnel
tests.Typically,undersimilardynamic
conditions,
5-10times
moresurface
materialwillbemovedonMars.Thismayseem
q•t•, = 0.94PatggI/,:•s
ß
ql•l,
rth
JOe
gL,,
I/,::s
- (24) misleading
unlessone considers
that windsaboveparticle
A direct'
estimate
offluxrates
onMarscanb¸ made
by thresholdspeedsdo'not occurnearlyas oftenon Mars as
by datareceived
from the
employing
theobta'med
result.
Theresults
areshowni
nFigure comparedto earth,as discovered
tions,equalratiosof B,/V,: onthetwoplanets'
yields
10.To obtain
a numerical
Value,
thethreshold
friction
speedViking spacecraft.However, when the Martian winds are
theyare moreefficient
in movingsurface
must
beknown.
Thevalue
willvarywithsize
ofparticles,:
but abovethreshold,
material.
This
effect
stems
from
two
basic
differences
in the
onceknown,
a direct
estimate
of themass
fluxcanbemade.
First,'sincethegravityonMarsisonly38%thatof
Forexample,
assuming
a .7.5-mbar
(0.75kPa)pressure
at'200 conditions.
donotreturnto theplanets
surface
as
K on theMartiansurfacefor I/,/V,: equalto 1.25,the flux theearth,theparticles
rapidly.
Second,
the
terminal
particle
speeds
are
substantially
rateq wouldbeapproximately
0.98g/scm.Thisisassuming
fasteron Marsowingto thereduced
pressure
compared
to
V,: equal to 1.75 m/s.
earth.Thiscreates
longer
pathlength
onMars,whichdirectly
increases the flux rate.
Acknowledgments.
Theauthoracknowledges
ChiTranforsome
of thecalculations.
Theworkissupported
bytheOffice
of Planetary
Geology,
NationalAeronautics
andSpaceAdministration,
through
Interchange
Agreement
NCA 2-OR180-605
to theUniversity
of Cali-
4.5
fornia at Davis.
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