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A Simple Model of Aeolian Megaripples
Hezi Yizhaq1, Antonello Provenzale2 and Neil J. Balmforth3
1BIDR, Ben-Gurion University, Israel
2CNR-ISAC, Torino, Italy, CIMA; University of Genoa, Italy
3UCSC, Santa Cruz, CA, USA
Email: [email protected]
Death Valley
Peru
Nahal Kasui, Negev, Israel
Megaripples gallery
Talk’s Outline
1. Ripples and megaripples characteristics
2. Sand transport mechanisms
3. Mathematical model of normal sand ripples
4. Mathematical model of megaripples
5. Conclusions and suggestions for future reserch
Normal Ripples
Megaripples
Wavelength10-25 cm>25 cm
up to 20 m
Grain size0.06 to 0.5 mm
1 to 4 mm
Time scaleminutesdays, weeks
SortingUnimodal distribution
Bimodal distribution
stoss lee h
Journal of Geology, 1981, 89, 129
cross section
Megaripples Characteristics
From Zimbelman et al. 2003 Williams et al. 2002
Ripple index=Wavelength/height =15
Megaripples on Mars
Sand dunes and ripple patterns in Kaiser Crater. The picture shows an area about 1.9 miles (3 km) wide and is sunlit from the upper left.
Image Credit: NASA/JPL/Malin Space Science
Systems
millibars 6surface at the pressure
377.0
EM gg
Aeolian activity on Mars was first mentioned in 1909 by E.M Antoniadi.
3 km
wind
~12m
~13m
Sand Transport by the wind
Saltation: high-energy population of grains in motion.Reptation: low-energy population of grains in motion.
The impact and ejection process during sand transport by wind ( after Anderson 1987).
High-energy impact of a single 4 mm diameter steel pellet into a bed of identical pellets. The high-energy ejection leaves to the upper left. Nine low-energy ejections are shown at successive instants by a lower frequency strobe-lit.
21m/s
170
Reptation length empirical formula -)after Anderson 1987 (
-Sedimentology (1987) 34, 943-956
Anderson’s model: Eolian sand ripples as a self-organization phenomenon.
Sedimentology, 34 (1987) 943; Earth-Science Reviews, 29 (1990) 77-96
Simplifications:
1. The saltation population is uniform in space.
(i.e. it will not include in the model)
2. All saltating grains impact an horizontal surface with an identical angle (between 100 and 150).
3. The granular bed is composed of identical grains.
A model for normal sand ripples
densitygrain sand
0.35) (typically bed ofporosity
surface sand ofheight local ),(
)1(
p
p
pp
tx
Qt
sr QQQ
Approximation: is spatially constantsQ
Deposition 0
Erosion 0
0
0
x
Q
x
Q
saltating grain reptating grain
wind direction
2
00
1
cot1cos
tan
tan1)(
x
ximimim NNxN
The instability is due to geometrical effects: an inclined surface is subject to more abundant collisions than a flat one.
Our new assumption: The reptation flux depends on the bed slope, such it is decreased on the stoss slope and incresed on the lee slope, mathematically:
0)1()( rxr QxQ
Bed slopeBallistic effect
Rolling effect
)function"splash ("
particlesreptation ofon distributiy probabilit )p(
particles ejected ofnumber average n
graineach of mass
p
pm
x
x
imppr dxxNpdnmQ
')'()(0
Reptation flux on flat surface :
)cot1(0 xaa
0
0 ')'()()1(x
ax
xx
t dxxFdaapQ
)1(
cotQ ;
1
tan)(
0
02pp
impp
x
xNnm
xF
Local shadowing effect:
0 tanif 0)( x xF
Yizhaq et al. submitted to Physica D.
The integro-differential equation:
Linear stability analysis:
0 2 4 6 8 10-0.6
-0.4
-0.2
0
0.2
0.4
0.6
k
C i
Q 0
a=1
Anderson model
yinstabilitlinear 0c
on translatiforward 0
where),(
i
)(0
r
irctxik
c
icccetx
min 5time
0.9; ;10 ;2
mm; 25.0
;m 10
0
1-270
p
im
n
D
sN
Numerical Results:
tt ~)(
Coarsening process
Time (min)
Me
an
Rip
ple
He
ight
(m
m)
Long-Wave Approximation
Goal:
Getting a PDE nonlinear equation for the dynamics of sand ripples
near the instability onset from the integro-differential equation. A compact description of the dynamics.
I. Nondimensional Variables :
)()~(~ ;~ ;~
;~ ;~ 0 papa
tQt
aaa
xx
II. Near the instability onset : xXtXtx , ),(),(
III. Taylor series expansion of )( and )( XFXF
IV. Assuming and T=t; define:
dpa pp )(
and we add sand transport in the lateral direction…
Two Dimensional Ripples:long-wave expansion equation
The model :
yY
QQQQQ
yryxyxrx
yyxxt
00 )1(
We assume pure rolling in the transverse direction
XXXXXXXXX
XXXXX
YYXXT
aa
a
TYX
)(2
tan)(
32
))(2(tan2
tan)tan1(),,(
22
332
22
*Animations were done with the help of Jost von Hardenberg. (CNR-ISAC)
x
y wind
2D simulation of normal sand ripples (long-wave approximation)
A Mathematical Model for Megaripples
Fine-fraction impact ripples (Elwood et al. 1975)
Fine particles saltation
Coarse particles reptationMean saltation length can be very large for fine particles which rebound from coarse grained surface and for strong winds. (up to 20 m)
Bagnold (1941) necessary conditions for megaripples formation:
1. Availability of sufficient coarse grains.
2. A constant supply of fine sand in saltation to sustain forward movement of coarse grains.
3. Wind velocity below the threshold to remove coarse grains from the megaripples crest.
Extension of Bagnold’s idea by Ellwood et al. (1975)
The mean saltation length can explain also the formation megaripples which developed in bimodal sands.
They calculated the mean saltation path for different values of wind shear velocity and different grain diameters .
50 cm 10 m5 cm
1.8 m/s
1 m/s
Integro-differential equation for 1D megaripples
Sand flux =saltation flux of fine grains +reptation flux of coarse grains
')'()()1()(0
dxxNdpnmxQx
x
imrcxrpcr
')'()()1()( dxxNdpmxQx
x
imsfxsfsf
crest close-up
)2/()(
/
2
)(
1)(
bsf
arc
Aep
ea
p
Exner equation :
)()1(
1rcsfx
ppt QQ
ratio between coarse grains to fine grains at the surface
unimodal fine sand
equally distribution of fine and coarse grains
x
x
rcxrp
x
x
sfxs
xt
dxxFdpn
dxxFdp
Q
')'()()1(
')'()()1(
0
0
0
34327
Bagnold toaccording
3
f
c
f
c
D
D
m
m
Linear Stability Analysis (megaripples)
))(exp(),( 0 ctxiktx Infinitesimal perturbation:
bkk
banak
ank
Q
csrp
rpi
sin2
1exp
)(tan
22
22
1
0
the bed is unstable for 0ic
0 1 2 3 4-4
-2
0
2
4
0 0.04 0.08-0.5
0
0.5
1
k
C Q 0
i
k r
k s
27,1,4.0
2.0,3.0,30
,100,10,1 0
prc
s
n
cmacm
cmb
megaripples mode 419 cm
normal ripples mode 4 cm
Megaripples formed in a patch of coarse sand.
megarippleswind
normal ripples
0 0.02 0.04 0.06 0.08 0.1-0.4
0
0.4
0.8
c i
Q 0f
k
Growth rates curves for different values of
No megaripples appear for 6.0
Sharp (1963): A concentration of coarse grains of at least 50 percent in the crestal area is needed for granule ripples formation .
Megaripples on Mars
0 0.004 0.008 0.012 0.016 0.02
-0.8
-0.4
0
0.4
0.8
k
c i
Q 0f 12.5 m
64 m
07,1
,27
,1,14
cmamb
Paths lengths are from 3 to 10 m for 0.1 to 1 mm particles (White, 1979)
This result can explain the observation that at some locations on Mars several wavelengths scales occur
Conclusions and suggestions for future studies
1. The proposed mathematical model takes into account both saltation flux of fine particles and reptation flux of coarse particles and can explain various field observations.
2. Linear stability analysis indicates that the megaripples wavelength is about 4 times the mean saltation length of fine grains .
3. Numerical simulations of the integro-differential equation are needed in order to find megaripple evolution and profiles.
4. Careful experimental work is needed in order to estimate the values of the model’s parameters.
TheThe