1

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@bgumail.bgu.ac.il

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

QQ

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