A Simple Model of Aeolian Megaripples

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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 yiyeh_at_bgumail.bgu.ac.il
2
Megaripples gallery
Death Valley
Nahal Kasui, Negev, Israel
Peru
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Talks Outline

• Ripples and megaripples characteristics
• Sand transport mechanisms
• Mathematical model of normal sand ripples
• Mathematical model of megaripples
• Conclusions and suggestions for future reserch

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h
stoss
lee
cross section
Journal of Geology, 1981, 89, 129
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Megaripples Characteristics
Ripple index Wavelength/height 15
From Zimbelman et al. 2003
Williams et al. 2002
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Megaripples on Mars
wind
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
l12m
l13m
3 km
Aeolian activity on Mars was first mentioned in
1909 by E.M Antoniadi.
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Sand Transport by the wind
Saltation high-energy population of grains in
motion.Reptation low-energy population of
grains in motion.
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The impact and ejection process during sand
transport by wind ( after Anderson 1987).

• Reptation length empirical formula
• (after Anderson 1987)
• Sedimentology (1987) 34, 943-956

21m/s
170
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.
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Andersons model Eolian sand ripples as a
self-organization phenomenon. Sedimentology, 34
(1987) 943 Earth-Science Reviews, 29 (1990) 77-96

• Simplifications
• 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.

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A model for normal sand ripples
Approximation is spatially constant
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The instability is due to geometrical effects an
inclined surface is subject to more abundant
collisions than a flat one.
wind direction
reptating grain
saltating grain
f
g
q

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Reptation flux on flat surface
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
Ballistic effect Rolling effect
Bed slope
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The integro-differential equation
Local shadowing effect
Yizhaq et al. submitted to Physica D.
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Linear stability analysis
Anderson model
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Numerical Results
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Coarsening process
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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
II. Near the instability onset
III. Taylor series expansion of
IV. Assuming me and Te2t define
and we add sand transport in the lateral
direction
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Two Dimensional Ripples long-wave expansion
equation
z
wind
y
x
We assume pure rolling in the transverse direction
The model
Animations were done with the help of Jost von
Hardenberg. (CNR-ISAC)
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2D simulation of normal sand ripples (long-wave
approximation)
em0.2 f100
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A Mathematical Model for Megaripples

• Bagnold (1941) necessary conditions for
  megaripples formation
• Availability of sufficient coarse grains.
• A constant supply of fine sand in saltation to
  sustain forward movement of coarse grains.
• Wind velocity below the threshold to remove
  coarse grains from the megaripples crest.

Fine-fraction impact ripples (Elwood et al.
1975) Fine particles
saltation Coarse particles
reptation
Mean saltation length can be very large for fine
particles which rebound from coarse grained
surface and for strong winds. (up to 20 m)
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Extension of Bagnolds 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.
1.8 m/s
1 m/s
50 cm
10 m
5 cm
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Integro-differential equation for 1D megaripples
Sand flux saltation flux of fine grains
reptation flux of coarse grains
crest close-up
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(No Transcript)
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Exner equation
e ratio between coarse grains to fine grains at
the surface e0 unimodal fine sand e1 equally
distribution of fine and coarse grains
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Linear Stability Analysis (megaripples)
Infinitesimal perturbation
the bed is unstable for
normal ripples mode 4 cm
megaripples mode 419 cm
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megaripples
wind
normal ripples
Megaripples formed in a patch of coarse sand.
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Growth rates curves for different values of e
No megaripples appear for
Sharp (1963) A concentration of coarse grains of
at least 50 percent in the crestal area is needed
for granule ripples formation.
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Megaripples on Mars
Paths lengths are from 3 to 10 m for 0.1 to 1 mm
particles (White, 1979)
64 m
12.5 m
This result can explain the observation that at
some locations on Mars several wavelengths scales
occur
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Conclusions and suggestions for future studies

• 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.
• Linear stability analysis indicates that the
  megaripples wavelength is about 4 times the mean
  saltation length of fine grains .
• Numerical simulations of the integro-differential
  equation are needed in order to find megaripple
  evolution and profiles.
• Careful experimental work is needed in order to
  estimate the values of the models parameters.

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