Mathematical modelling of the dynamics and morphology of aeolian dunes and dune fields

M ath em atical m od ellin g o f

th e dynam ics and m orphology o f

aeolian dunes and dune fields

Hiroshi Momiji

University College London

Thesis submitted to the University of London

ProQuest Number: U641913

All rights reserved

INFORMATION TO ALL USERS

The quality of this reproduction is dependent upon the quality of the copy submitted.

In the unlikely event that the author did not send a complete manuscript

and there are missing pages, th ese will be noted. Also, if material had to be removed, a note will indicate the deletion.

uest.

ProQuest U641913

Published by ProQuest LLC(2015). Copyright of the Dissertation is held by the Author.

All rights reserved.

This work is protected against unauthorized copying under Title 17, United States Code. Microform Edition © ProQuest LLC.

ProQuest LLC

789 East Eisenhower Parkway P.O. Box 1346

A b stract

The aim of th is thesis is to model th e dynamics of free sand dunes.

In the first p a rt, a new theoretical scheme is presented to m odel th e shape and m igration speed of a sand dune a t equilibrium . Unlike earlier models it does not require iterative calculations of th e interaction between the wind flow and the topography. As th e first step, a self-consistent m odel which describes two- dim ensional dune m igration is introduced, which is com prised of a grain-scale model of sand deposition in th e lee of dune and th e assum ption of equilibrium. The model gives quantitative relations between sand grain diam eter, wind velocity on level ground and dune height. By further incorporating theory based on aero­ dynamics, th e w ind-directional profile of barchan dunes can be estim ated.

The thesis goes on to develop a com puter sim ulation m odel th a t describes the three-dim ensional m orphology and dynam ics of an aeolian dune field. Following recent advances, the proposed model is based on an approach using discrete la t­ tice dynamics. In th e model, dunes are treated as accum ulations of ‘sand slabs’ on a two-dim ensional lattice, whose m otion is the result of w ind-directional sand tra n sp o rt and avalanching. By incorporating new features, which reflect physi­ cally observed mechanisms, the model can sim ulate dunes whose individual shape and collective p a tte rn s are sim ilar to those observed in n ature. T he model can also quantitatively sim ulate dune growth and dune m igration. Some dune p attern s can be explained by the model in term s of seasonally changing wind direction and sand availability (initial sand depth).

A cknow ledgem ent

F irst of all, I would like to th an k my supervisors, Professors Andrew W arren and Steven R, Bishop for th eir continuous supervision and encouragem ent. Very frequent and intense discussion w ith them m ade it all possible. T he au th o r knows well th a t this is not always the case am ongst PhD students.

Discussion also w ith two of my collaborators: D r R icardo C arretero-G onzalez and Professor H iraku Nishimori has been a v ital p a rt of this stu d y thro u g h ou t these three years.

This work has greatly been improved w ith com m ents to my m anuscripts, which were subm itted to journals or are in preparation, from b o th anonym ous and unanonym ous reviewers: Professors G ary Kocurek, Jim L. Best, Julian C.R. H unt, Nicholas Lancaster, John H. van Boxel, and D r Giles F.S. Wiggs.

I owe a lot to Professor Yasuji Sawada, D r M asaaki Futam oto, Mr Kazumi K awamoto and th e International Office, UCL, who assisted w ith my original application to UCL.

D uring the prep aratio n of this thesis, th e com puter environm ent has greatly been improved w ith support by D r Julian R. Thom pson.

Life in London has been fruitful w ith my friends, in particular: Miss Tomoko Y oshitani, M r Taro Niikura, Mr and Mrs K um agai, and Mr and Mrs O kum ura.

Finally, m any th an k s to my parents and brother for th eir continuous su pp o rt and encouragem ent throughout this study.

C on ten ts

A b s t r a c t ... 2

A c k n o w le d g e m e n ts... 3

List of f i g u r e s ... 8

List of t a b l e s ... 12

List of s y m b o l s ... 13

1 A im s of th e study

18

1.2 M odelling p h ilo s o p h y ... 21

1.3 Scope ... 27

2 Introduction to desert dune geom orphology and aeolian pro­

cesses

28

2.2 Classification of simple (elementary) d u n e s ... 29

2.3 Form ative environm ent of each type of d u n e ... 32

2.4 Dune dynam ics and sand tra n sp o rt induced by w ind and by gravity 33 2.4.1 Sediment continuity e q u a t i o n ... 33

2.4.2 Sand m otion induced by w i n d ... 34

2.4.3 Sand tra n sp o rt f o r m u l a e ... 35

2.4.4 W ind fiow over a d u n e ... 37

2.4.5 Avalanche and sand deposition on th e lee f a c e ... 39

2.4.6 Dune m ig r a tio n ... 40

2.5 The stru ctu re and evolution of a dune f i e l d ... 41

2.6 A nalytical studies of emergence and shape of d u n e s ... 43

2.6.2 K inem atic a n a l y s i s ... 44

2.6.3 Difficulties in analytical s t u d i e s ... 46

2.7 C om puter m odelling of dunes and dune fields ... 46

2.7.1 Models dealing w ith an isolated dune on a fiat surface . . . 47

2.7.2 Models dealing w ith dune field d y n a m ic s ... 51

3 M odelling a transverse and a barchan dune

56

3.2 M igration speed of transverse dunes and sand trap p in g efficiency . 61 3.2.1 M igration speed of transverse dunes a t equilibrium . . . . 61

65 70 72 74 81 3.2.2 Sand trap p in g e f f ic ie n c y ... 3.3 W indw ard slope profile of a barchan dune ... 3.3.1 W indw ard slope profile of barchan dunes in the field 3.3.2 Wind-fiow modelling w ith the Jackson-H unt theory . 3.4 Conclusions in this c h a p te r ...

4 M odel results and comparison to nature

83

4.1.1 Numerical calculations and com parison to n a t u r e ... 83

4.1.2 Model applicability to subaqueous and terrestrial aeolian dunes (section 2 . 6 . 2 ) ... 92

4.1.3 Model lim itations and future work ... 94

4.2 W indw ard surface m o rp h o lo g y ... 95

4.2.1 G eneral c h a r a c te r is tic s ... 97

4.2.2 C alculations for dunes in southern P e r u ...101

4.2.3 C alculations for dunes in C a lifo rn ia ...106

4.2.4 Towards a model of a three-dim ensional d u n e ...107

4.3 Shape and m igration speed of p r o to - d u n e s ...110

4.4 Conclusions in this c h a p te r ...114

5 D eveloping dune field m odel w ith discrete dynam ics

117

5.3 Sim ulated dune shape and dynam ics w ith W erner’s model . . . . 122

5.4 No erosion in shadow zones ... 125

5.5 Introducing wind s p e e d u p ... 126

5.5.1 Linear wind speedup: K inem atic fo rm u la tio n ...127

5.5.2 Non-linear wind s p e e d u p ... 129

5.6 Sand availability and reference num ber of slabs c o rre c tio n ... 129

5.7 Introducing wind directional c h a n g e ... 130

5.8 Conclusions in this c h a p te r ... 132

6 Comparisons w ith field studies

133

6.2 The effects of wind speedup over a d u n e ... 135

6.2.1 The effects of linear wind s p e e d u p ...136

6.2.2 The effects of non-linear wind speedup and equilibrium dune f i e l d ... 138

6.3 The effects of sand availability and wind v a r i a b i l i t y ... 143

6.3.1 Dune types in a uni-directional w ind regime: p a tte rn and sand a v ailab ility ... 143

6.3.2 Spatial and tem poral scales in the model: q u an titativ e com­ parison to n a t u r e ...146

6.3.3 Dune types in a bi-directional wind r e g i m e ...149

6.4 Conclusions in this c h a p te r ... 151

7 Conclusions

153

7.2 Assessment of models and m e th o d o lo g y ...155

7.2.1 Reductionism / u n iv ersalism ... 156

7.2.2 Four criteria for a good model (Kirkby, 1996) ... 157

7.2.3 Engineering a s p e c ts ...160

7.2.4 C ontributions to geomorphology in general and to other d is c ip lin e s ... 161

7.3 Future w o r k ...162

7.3.2 Dune field m o d e l ... 163 7.3.3 C o lla b o ra tio n ...165

List o f figures

2.1 Schematic views of typical dunes: (a) dome dune, (b) barchan dune, (c) transverse dune, (d) linear dune, (e) sta r dune and (f) network dune. 31 2.2 Dune type diagram w ith regard to sand availability and w ind-direction

variability... 33 2.3 Flow chart for dune modelling in th e conventional m e th o d ... 48 3.1 Schem atic views of a barchan dune (a) from to p and (b) from side. . 57 3.2 Flow chart for dune m odelling in the conventional m e th o d ... 58 3.3 Cross-sectional geometry of a m igrating shape-invariant transverse

dune: (a) global view (b) magnified view of windw ard surface (c) sand flux (ç(a;)) over a dune... 63 3.4 Model and co-ordinate system used in calculation of sand trap p in g

efficiency ( % ) ... 66

3.5 Flow chart for dune modelling in the new approach exam ined in this thesis... 71 3.6 M easured d a ta of dune height ( H) and th e average angle of wind­

w ard slope (^wavg) taken from Finkel (1959), Long and Sharp (1964), H astenrath (1967) and Sauerm ann et al. (2000)... 73 3.7 T he norm alised surface profiles (/(^ )) of (a) a cosine hill and (b) a

G aussian hill, and examples of norm alised surface shear stress profiles over th e m ... 78 4.1 R elations between sand trap p in g efficiency (Te) and dune height (H)

calculated for various shear velocities a t the dune crest (u*(0)). . . . 85 4.2 R elations between sand trap p in g efficiency (Te) and shear velocity on

4.3 R elations between dune m igration speed (cd) and shear velocity on a level surface (u*(—oo)), for various dune heights ( H )... 87 4.4 R elations between dune m igration speed (cd) and w ind velocity a t

10 m height on a level surface (uiom(~oo)) for various dune heights {H). 88

4.5 R elations between shear velocity a t the dune crest (u*(0)) and dune height ( H )... 88

4.6 R elations between dune m igration speed (cd) and dune height (H),

for various shear velocities on a level surface (u*(—o o ))... 90 4.7 An exam ple relation between dune m igration speed (cd) and the in­

verse dune height ( l / H ) . D a ta collected by H asten rath (1967) are analysed... 92 4.8 Bedforms formed in th e wind tunnel th a t sim ulated Venusian atm o­

sphere... 93 4.9 T he calculated relation between the norm alised increase in th e shear

stress a t the dune crest and dune height (H) for various shear velocities on the level surface (u*(—oo))... 97 4.10 T he calculated relation between the average w indw ard slope angle

(^wavg) dune height (H) for various shear velocities on th e level surface (u*(—oo)). The sand grain diam eter {Dg) is taken as 0.15 mm. Field d a ta collected by H asten rath (1967) are also p lo tte d ...102 4.11 T he relation between the m axim um windward slope angle (^wmax)

and dune height {H) for various shear velocities on th e level surface (u * (-o o ))... 104 4.12 T he relation between dune m igration speed (cd) and dune height {H)

for some shear velocities on a level surface (u*(—oo)). Sand grain diam eter (Dg) is assumed to be 0.15 mm. Field d a ta collected by H astenrath (1967) are also p lo tte d ...104 4.13 Schem atic explanation of the origin of (a) barchan and (b) parabolic

4.14 The relation between windward slope angle (^wavg) dune height

(H) for various shear velocities on a level surface (u*(—oo)). Sand grain diam eter {Dg) is 0.250 mm. Field d a ta collected by Long and Sharp (1964) are also p lo tte d ...107 4.15 T he relation between dune m igration speed (cd) and dune height {H)

for d a ta by Long and Sharp (1964) and for calculations w ith sand grain diam eter {Dg) of 0.250 m m ... 108 4.16 Schematic plan view of th e transverse dune system com prised of barchanoid

and linguoid p a rts ...109 4.17 T he calculated relation between the windward slope length (Lw) and

dune height {H) for various shear velocities on th e level surface (u*(—oo)). T he sand grain diam eter {Dg) is taken as 0.25 m m ... 113 4.18 The calculated relation between dune m igration speed (cd) and dune

height {H) for some shear velocities on a level surface (u*(—oo)). Sand grain diam eter {Dg) is assumed to be 0.25 m m ... 115 5.1 L attice m odel configuration and algorithms: a) w ind-directional sand

tra n sp o rt, b) shadow zone and c) avalanching...120

5.2 Transverse dunes (a) in {i, j) space sim ulated w ith W erner’s original model, from a random initial morphology, and (b) in n atu re (W ahiba Sands, O m an )... 123 5.3 Evolution of an initially introduced, isolated transverse dune, simu­

lated w ith W erner’s original m odel... 124 5.4 C alculated relation between saltatio n length (A) and shear velocity (u*).130 5.5 M odelling scheme for a dune field where th e wind changes direction.

T he lattice is ro tated in response to the wind directional change. . . . 131 6.1 Evolution of an initially introduced, isolated transverse dune sim ulated

w ith o u t erosion in shadow zones...135 6.2 Transverse dunes sim ulated from an initially random morphology. No

erosion occurs in shadow zones... 136 6.3 Evolution of an initially introduced, isolated transverse dune sim ulated

6.4 A sym m etric transverse dunes formed from a random in itial m orphol­ ogy by introducing wind speedup... 137 6.5 Tim e evolution of a 1-dimensional dune field, from a random initial

m orphology... 139

6.6 Tim e evolution of transverse dunes sim ulated on th e 1-dim ensional lattice w ith non-linear increase of the tra n sp o rt length, from a random in itial morphology...139 6.7 R elations between m axim um dune height and sim ulation tim e (t). The

relations w ith and w ithout th e non-linear increase of th e tra n sp o rt length are averaged from 13 and 15 runs on th e 1-dimensional lattice, respectively...141

6.8 Transverse dunes sim ulated w ith non-linear increase of th e tra n sp o rt length, on th e 2-dim ensional lattice, from a random initial morphology. 142 6.9 Different types of dune sim ulated w ith various sand availabilities (havg) ;

(a) 2, (b) 7 and (c) 20...144 6.10 Downwind m igration of barchan dunes in two sim ulations... 148 6.11 Dune types and wind directional change. Sand availability and angle

between two directions of wind (havg, 0) are a) (100,60°), b) (100,120°), c) (2,60°) and d) (2,120°)... 150 6.12 Dune type diagram w ith regard to sand availability and w ind direction

List o f tab les

4.1 Values and constants used for num erical calculation... 84

4.2 Some published dune m igration d a ta and m odel resu lts... 91

4.3 Exam ple results w ith a cosine hill and a G aussian hill... 98

4.4 Some published field d a ta of the flow over a low h ill...102

6.1 Sim ulation p aram eters...134

6.2 M axim um dune height when sand availabilities are below and above the th resh o ld ... 145

List o f sym bols

a coefficient in Anderson’s set of equations ab o u t saltatio n length coefficient in a Fourier transform form ula

Ucd coefficient in the proposed formulae for dune m igration speed against dune height

üL coefficient in the logarithmic relation between dune m igration speed and dune height

o w indward slope angle

a-i sand grain incident angle (10°) B barchan dunes

B barchanoid p a rt

b coefficient in Anderson’s set of equations ab o u t saltatio n length

6cj coefficient in the proposed formulae for dune m igration speed against dune height

6l coefficient in the logarithmic relation between dune m igration speed and dune height

j3 tem poral variable for calculation of xqc

Cb coefficient in B agnold’s sand tra n sp o rt form ula

c coefficient in A nderson’s set of equations ab o u t saltatio n length Cll coefficient in Lett au and L ettau ’s sand tra n sp o rt form ula

Ccj coefficient in the proposed formula for dune m igration speed against dune height

Cd dune m igration speed (celerity)

Cl linear wind speedup coefficient of slab tra n sp o rt length

D dune p a tte rn

d coefficient in A nderson’s set of equations ab o u t saltatio n length

Dg sand grain diam eter

Dj. reference sand grain diam eter (0.25 mm)

e p e rtu rb a tio n variable in th e Jackson-H unt theory (1975) / surface undulation function

surface profile

/ Fourier transform of surface profile ( / )

frj function which relates 7]{x) and (u*(—oo), 77, Dg)

fo function which relates u*(—oo) and H, Dg)

/_oo function which relates u*(0) and {u^{—o o ) , H, Dg) Gi coefficient in a gam m a function form ula

G2 coefficient in a gam m a function formula g g rav itatio n al acceleration (9.8 m s” ^)

r gam m a function

7 sand bulk density

tem p o ral variable for calculation of Xqc

H dune height

h num ber of sand slabs

h Fourier transform of num ber of sand slabs (h)

havg average num ber of slabs

Hmax th e largest dune height in a dune field hmax m axim um num ber of slabs

77min th e sm allest dune height in a dune field

hmin m inim um num ber of slabs href reference num ber of slabs

T] (dune) surface height

r}' modified dune surface

T/avg average (dune) surface height

77a m axim um height of undulation ( / ) I isolated dunes

i u n it of im aginary num bers (V ^ ^ )

Znew lattice suffix in wind direction after lattice ro tatio n

j lattice suffix in transverse direction to th e wind

jnew lattice suffix in transverse direction to th e w ind after lattice ro tatio n

k tem poral param eter for counting i

Fourier wave num ber

K, von K arm an ’s constant L linear dunes

L linguoid p a rt

horizontal characteristic length of a dune sand slab tra n sp o rt length

I inner-layer thickness

Lq sand slab tra n sp o rt length a t sites where h = href LpD proto-dune length in the wind direction

Z/g lattice size

Lw w indw ard slope length of a dune A wavelength of undulation

A saltatio n distance of a sand grain N no p a tte rn

Nxo num ber of sand grains arriving at xq

No num ber of sand grains departing a t x ' P probability function of saltatio n distance

Pns slab deposition probability a t sites w ith o u t sand slab (0.4) Pg slab deposition probability a t sites w ith a t least one slab (0.6)

(f) phase constant in the expression of fd (H-7r/ 4 or —tt/4 )

q sand flux

B? R -squared value, correlation measure of linear regression analysis Pa air density (1.2 kg m “ ^)

Pg quartz-sand grain density (2,650 kg m “ ^) T transverse dunes

Teq function which relates Te and (u*(0), H , Dg) Te-oo function which relates Te and (u*(—oo), H, Dg)

9 slip face angle

angle of lattice rotation ^wavg average windward slope angle

9c angle of repose

m axim um windward slope angle

0ph phase difference between topography and surface shear stress

9sz angle of shadow zone (15°) r surface shear stress

To surface shear stress on the flat surface Td sp atial variation in the surface shear stress fd Fourier transform of Td

u w ind speed

Uiom w ind speed a t the height of 10 m

Uz w ind speed a t the height of m easurem ent {z)

u* w ind shear velocity

u*t threshold wind shear velocity

vq sand grain incident velocity

u* incident velocity of sand ejected a t th e dune crest

vIq m ean grain lift-off velocity

wIq m ean vertical grain lift-off velocity

X horizontal distance in wind direction

x ' horizontal position of saltation entrainm ent

xq horizontal position on z = 0 plane corresponding to Xp

xqc horizontal position on z = 0 plane corresponding to Xpc Xp horizontal position on the slip face in w ind direction

Xpc horizontal position of the bo tto m end of slip face in wind direction f norm alised horizontal distance {x/ L)

y horizontal distance z vertical distance

Zp vertical position on the slip face

C H A P T E R

1

A im s o f th e stu d y

1.1

M otivation and aims

This stu d y aims to m athem atically model th e morphology and dynam ics of desert dunes and dune fields. The ultim ate goal is to provide geomorphology and other disciplines w ith novel concepts and methodologies, as well as to give new inter­ p retatio n s to long-debated issues in geomorphology.

A t th e s ta r t of the thesis, it is worth reviewing past studies of modelling in geomorphology as a whole and in aeolian geomorphology in particular, in order to asses th e current state of the art, i.e., w hat has been understood, in which directions th e study is going and w hat is still needed. In th e rest of this chapter, only a conceptual discussion, as regards th e choice of m ethod, is given. The detailed review of models in aeolian geomorphology is given in chapter 2.

vant here, some of it is directly related to the choice of th e appropriate m ethod for th e present study.

G eomorphology has followed a p a th of change in its m ethodology from extensive and em pirical studies, which rely on th e statistical consideration of d a ta from large samples, to intensive and in-depth studies of sm all num bers of cases, as seen in the investigations of river m eandering (Richards, 1996). O th er in-depth studies have considered micro-scale aeolian sand tra n sp o rt, where precise experim ents using th e state-of-the-art technologies have predom inated (M itha et al, 1986; W illetts and Rice, 1989; Butterfield, 1991; Rice et al , 1995, 1996). T he inclusion of m ath em atical modelling in these in-depth studies (Richards, 1996) has greatly deepened our knowledge about micro-scale phenom ena.

The study of desert dunes and related aeolian features has a history of more th an a hundred years, in which changes in m ethodology from extensive to intensive can be seen a t different tim ings for different scales of interests. Field studies of desert dunes began in late 19th century, and th e m ajo r desert dune areas were all explored by 1980, as sum m arised in Goudie (1999, tab le 1.1). Em pirical field studies have contributed basic knowledge, for example ab o u t dune shape and m obility (see for example Cooke et al , 1993; L ancaster, 1995; Goudie et al,

1999).

dune form and alignment to wind-flow p attern .

Extensive and empirical study has continued nonetheless, not only a t a dune-held scale b u t a t th e global scale. A comprehensive description of worldwide dune helds enabled W ilson (1972) to develop th e idea of bedform hierarchy and the role of sand grain size. W ith the aid of satellite images, m ore detailed descriptions of dune helds have been made available (McKee, 1979a). Based on th e accumula­ tion of such knowledge, together w ith m eteorological considerations, attem p ts to classify and understand dune forms in relation to its controls have been made

{e.g. Wasson and Hyde, 1983). A t th e dune-scale, intensive m orphom etric inves­ tigations have revealed dune-form sensitivities to dune size and wind, which has become one of m ajor interests in aeolian geomorphology {e.g. Hesp and Hastings, 1998).

on laboratory experim ents was not advisable.

It is because of this th a t th e field study of dunes has been th e prim ary means of study. For example, extensive field observations m ade it possible to compile a diagram of the relation between dune form and its controls: wind variability and sand availability (Wasson and Hyde, 1983). In addition, a field experim ent th a t m onitored the decay of an artificially constructed barchan dune successfully highlighted the im portance of incoming sand flux for a dune to m aintain its shape (Cooke et aL, 1993, p322). However, “Field studies of dune dynam ics are tim e consuming, expensive to carry out, and sample short tim e periods on the scale of minutes. Models of the behaviour of dunes under varied air flow and tra n sp o rt conditions are necessary therefore to be able to predict th e long term evolution of dunes.” (M cKenna Neum an et ai, 1997, p i 112). D espite th e firm recognition of its necessity as shown in th is quotation, theoretical analysis still lags far behind em pirical studies (Nickling and M cKenna Neum an, 1999), if only in the volume of published m aterial.

1.2

M odelling philosophy

q u antitatively b u t some qualitative comparisons are made.

There is no definitive m ethod for quantitative modelling, though K irkby (1996) listed the following criteria for a good model. 1) physical basis: if th e model has a strong physical basis, 2) simplicity: if the model avoids an excess of complexity, 3) generality and richness: if the model can be applied to other fields of study (generality) an d how much th e model can explain in th e field of interest (richness), and 4) p o ten tial for scaling up and down: if the m odel can apply to problems whose scale is larger or sm aller th a n th a t of interest. A nalytical (pen-and-paper) study, if possible, is thought to be b etter th an com putational modelling, since analytical models identify th e physics involved {e.g. th e morphology and dynamics of dunes and dune fields) and make long-term predictions possible.

There are two distinct approaches in n atu ral sciences; reductionism and univer­ salism. R eductionism is a classic and probably the m ost convincing approach, in which observed phenom ena are repeatedly reduced to a deeper level, hopefully to th e fundam ental processes (first principles). To how deep a level it is satisfactory to reduce is a question. In addition, the reductionist approach is alm ost always ham pered by m athem atical difficulties both analytically and numerically, due, for example, to the non-linearity of the system and lim itatio n s of com putational power, respectively. Scaling-up processes are usually associated w ith some as­ sum ptions (simplifications), such as continuity, approxim ations and m any kinds of sym m etry (Richards, 1996). However, any step up from such simplifications makes th e problem very difficult in a m athem atical respect.

uni-versai aspect, which is independent of microscopic details, we only need to study th e sim plest model (prototype) th a t shows such a universal property (Bak, 1996). Some key words are chaos, self-organised p a tte rn , fractals and self-organised crit- icality (Phillips, 1996).

Self-organised pattern: some system s which are com prised of m any interacting degrees of freedom can spontaneously form a distinct p a tte rn (self-organised p a t­ tern) such as periodic rolls and grids. M athem atically, lattice dynam ics, such as Coupled M ap L attice (CML) (Kaneko, 1993) and C ellular A u to m a ta (CA) (Wol­ fram , 1986), have been studied as prototypes th a t generate such p attern s. CML can correspond to th e spatially and tem porally discretised p a rtia l differential equations, where th e tim e evolution of quantities (state) defined on each lattice site is sim ulated following rules. If th e state of the system is also discretised, th a t is the CA. However it is not an easy task to derive CML or CA from th e original fundam ental equation of th e system. Consequently, we set th e rules based on physical considerations for each case {e.g. Bak, 1996). T here is much circum stan­ tial evidence th a t geomorphic system s do show chaos a n d /o r self-organisation, such as in th e form ation of periglacial and nonperiglacial p attern ed ground, the evolution of beach cusps, and drainage network evolution (H allet, 1990; Phillips, 1996, tab le 13.1, 13.2). Some of these p a tte rn s have been sim ulated by lattice dynam ics {e.g. W erner and Fink, 1993; M urray and Paola, 1994). Universalism may provide us w ith insights into these kinds of geomorphology.

Fractals: fractals are another type of self-organised p a tte rn . T he word ‘fractal’ was coined by M andelbrot (1983) to refer to scale-invariant geometries. For ex­ ample, rivers in the Amazon (Takayasu, 1990) and th e Norwegian coast (fjords) (Bak, 1996) have been shown to have fractal structures. Fractal stru ctu res can be seen in so m any fields other th a n landscapes, for example, in th e tim e series of price of commodity, th a t fractal-ity is thought to be one of th e universal aspects of n ature. However, th e concept of fractals cannot address th e dynam ic origins of a phenom enon.

insensitive to order param eters [e.g. tem perature, m agnetic field in phase tra n ­ sition in solids). For this reason, SOC can be expected to be observed widely in m any different systems. In geomorphology, SOC has been discussed in reference to tu rb id ites (Bak, 1996).

Since there is some circum stantial evidence, it can be expected th a t our under­ standing of geomorphology may deepen w ith the concepts of chaos, self-organised p a tte rn , fractals and self-organised criticality. The problem is th a t quantitative discussion and prediction is difficult if it is confined to these concepts alone. For system s whose basic equations are not known or slowly change w ith tim e, th e im­ pact of chaos rem ains a t th e level of scientific philosophy (Ruelle, 1991, chapter

12).

In th e study of aeolian geomorphology, however, the reductionist approach does not seem to work well even for simplified problems (W erner, 1995, 1999). M ath­ em atical difficulties, such as non-linearity in the Navier-Stokes equation, three- dim ensional features of the system, and the discrete n atu re of sand grain motion, lim it our choice of m ethods as follows.

dim ensional space. Tracking dune evolution through conventional analysis th a t calculates wind flow over a dune fleld each tim e is therefore practically impossible.

To sim ulate dune flelds, more phenomenological models are necessary. The p a t­ tern s of dunes and ripples th a t grow on an initially flat surface are no doubt self-organised in a non-linear open system, because before dunes or ripples grow, th e wind does not have any stru ctu re which resembles th e resulting bedform p a tte rn . In this respect, universalism may be a more effective approach to m od­ elling. This m ight, in tu rn , contribute to non-linear physics. However, a wholly universal approach, which discusses only p a tte rn s qualitatively, is not adopted here. Models will be developed th a t are based on lattice dynamics. Yet they will seriously a tte m p t to account for physical processes and scales, and aim a t quan­ tita tiv e prediction. Phenom enological models, which are not based on detailed physical processes, b u t on simplified rules which, nevertheless correctly reflect observations, are necessary.

1.3

Scope

C H A P T E R

2

In trod u ction to desert dune

geom orphology and aeolian

processes

In th is chapter, basic aeolian geomorphology and processes are briefly sum­ m arised, and previous studies of dune and dune-fleld modelling are reviewed. Subaqueous bedforms, which are thought to be related to aeolian dunes, are also discussed for th e following two reasons: ripples and dunes in rivers resemble ae­ olian dunes in view b o th of th eir dynamics and morphology; and th e study of subaqueous bedforms has progressed further th a n th a t of aeolian dunes.

2.1

Hierarchies in aeolian geom orphology

next largest member. In m any areas these dunes are superim posed on larger draas. W ilson thought th a t these three features co-existed in quasi-equilibrium , and ripples did not grow into dunes, nor dunes into draas. This leads to the conclusion th a t the nuclei of draas m ust be larger th a n fully grown dunes, of which nuclei m ust be larger th a n fully grown ripples. A nother explanation of the hierarchy m ay be the superim position of dunes from different geological periods w ith different wind regimes (W arren and Allison, 1998). D une size may however also be related to wind velocity (Cooke et al, 1993, p346), and superim position m ay have aerodynam ic explanations (Lancaster, 1995, p l9 1 ). In th is thesis, only dunes and draas (mega dunes) are discussed further, because different dynamics work on ripples (see section 2.4.2).

2.2

C lassification o f sim ple (elem entary) dunes

Dunes can be classified into two groups, free dunes and anchored dunes. Free dunes can grow and m igrate freely in response to changes in wind speed and direction. Anchored dunes are pegged by vegetation or topography, and cannot m igrate freely, though th eir forms are shaped by changes in w ind speed and direction. Here, only free dunes are reviewed, since th ey are more fundam ental aeolian features. Some findings from the study of free dunes m ay however also be applied to anchored dunes.

Transverse dunes

D o m e d u n e : an isolated, usually transverse dune w ith no slip face (Figure

2.1(a)).

B a r c h a n d u n e : a free transverse dune w ith a crescentic plan-shape in which th e crescent opens downwind (Figure 2.1(b)). More detailed mor- phom etric descriptions are given in the next chapter (section 3.3.1).

T r a n s v e r s e d u n e : dune ridges w ith crests transverse to th e dom inant wind, which m igrate, for th e m ost p art, in th e direction of th e dom inant wind (Figure 2.1(c)).

R e v e r s in g d u n e s : transverse dunes th a t reverse in th e direction as the wind changes through 180°.

Linear dunes

S e lf d u n e : a sinuous, sharp-crested, linear dune.

S a n d rid g e : p artly vegetated linear dune, usually larger th a n a seif.

Star dunes

S t a r d u n e : a large pyram idal or dome-like dune w ith arm s (Figure 2.1(e)). N e tw o r k d u n e : result of the overlap of a num ber of transverse dune sys­ tem s, each aligned to a different wind in a complex annual regime (Figure

(a) Dome dune b) Barchan dune

(c) Transverse dune (d) Linear dune

(e) Star dune (f) Network dune

Figure 2.1: Schematic views of typical dunes: (a) dome dune, (b) barchan dune, (c) transverse dune, (d) linear dune, (e) star dune and (f) network dune ( ( a ) - ( e ) after McKee, 1979b; (f) after Cooke e t al.,

2.3

Form ative environm ent o f each ty p e o f dune

A lthough m any factors, such as grain size and hum idity, affect dune shape and alignm ent, th e type of dune a t a site is prim arily determ ined by the wind regime. A useful term inology was given by Fryberger (1979). He categorised annual wind p a tte rn s into three m ain regimes; ‘unim odal’, ‘bim odal’ and ‘com plex’, and subdi­ vided ‘unim odal’ and ‘bim odal’ into two subregimes each. In ‘unim odal’ regimes winds blow from nearly one direction throughout th e year. If th e distribution is restricted to 45° it is called ‘narrow unim odal’, while if a d istrib u tion is larger th a n 45° it is called ‘wide unim odal’. In ‘bim odal’ regimes th e wind d istribution comprises two modes. If th e angle between th e two modes is less th a n 90° it is called ‘acute bim odal’, while if the angle is greater th a n 90° it is called ‘obtuse bim odal’. In ‘com plex’ regimes the wind distribution comprises either more th an two modes or no distinct mode.

T he second im p o rtan t factor is thought to be sand availability; i.e. th e am ount of sand available for dune building. Wasson and Hyde (1983) developed a diagram of dune types w ith respect b o th to wind variability and to sand availability. L ater this diagram was refined by Livingstone and W arren (1996, reproduced in Figure 2.2). These diagram s are accepted by m any researchers.

CO

1

■o_c s O) c

1

Complex Uni-directional

Wind directional variability

Figure 2.2: Dune type diagram with regard t o sand availability and wind- direction variability (after Livingstone and Warren, 1996, figure 5.22).

2.4

D une dynamics and sand transport induced

by wind and by gravity

2.4.1 Sedim ent continuity equation

Dune dynamics, which describes changes of dune shape and dune m igration is governed by the sediment (sand) continuity (conservation) equation:

= 7 ^ + V ç , (2.1)

where t is time, x and y are horizontal distances, 7 is the bulk density of sand in kg m “ ^, 77 is surface elevation in m and q is sand flux in kg m “ ^ s“ h According to equation (2.1), erosion occurs when the sand flux {q) diverges (Vç > 0), and deposition occurs when it converges [Vq < 0).

In this approxim ation, combining th e geometric considerations and th e sedim ent conservation equation leads to the well-known condition th a t a shape-invariant dune m ust follow;

= (2.2)

where Cj is dune m igration speed (celerity) in m s“ ^ (Bagnold, 1941; Zeman and Jensen, 1988; see section 3.2.1 w ith Figure 3.3 for derivation). E quation (2.2) is applicable also to ripples and mega dunes, if they m igrate w ithout change in th eir shape. In th e cases of dunes and mega dunes, equation (2.2) results in th e well-known wind speedup over th e w indward slope of a dune (see section 2.4.4), through, for example, B agnold’s sand tra n sp o rt equation described below (section 2.4.3). However, it should be noted here th a t equations (2.1) and (2.2) are universal, so th a t they can be applied also to bedform s in other environments, such as ripples and dunes in a river.

2.4.2

Sand m otion induced by w ind

From his wind tunnel experim ents, Bagnold (1941) proposed th a t there were three distin ct types of sand m otion induced by wind; creep, saltatio n and suspension. Creep was th e rolling or sliding m otion of sand grains in contact w ith th e sand surface. Saltation was th e ballistic, short-distance ju m p of sand grains, induced by wind shear or the im pact of another saltatin g grain. In saltation, the sand grains typically rose a few centim etres and flew forward m any centim etres, and ultim ately descended again to the surface. Suspension was th e movement of sand grains kept aloft and tran sp o rted long distances following tu rb u len t eddies in th e wind. These three forms of m otion occur if th e wind shear stress exceeds a specific value, the threshold shear stress, and co-exist. Generally, creep, saltation and suspension apply to heavy, m edium and light sand grains, respectively.

do not receive enough kinetic energy from th e saltatin g particle to enter into salta­ tion themselves (although some ejected grains do). Typically th eir m ean ejection velocities are less th an 1/10 of the im pact velocity of th e saltatin g grains. A nder­ son (1987a) found th a t ripple wavelength is not related to th e saltatio n length, as was widely believed since B agnold’s work (1941), an d th a t it is ab o u t six tim es long as the m ean reptation length.

For dune dynam ics, saltatio n is the dom inant sand tra n sp o rt mechanism. In w hat follows, only saltation-dom inated dynam ics are discussed. However, for sand grains departing a t th e dune crest, modified saltatio n approaching a state of suspension could be expected due to the turbulence effect (H unt and Nalpanis, 1985; Anderson, 1987b). T he consequent long airborne trajectories of such grains m ay have a significant im portance to dune dynam ics (see section 2.4.6).

2.4.3

Sand transport formulae

W ind velocity over a fiat surface can be expressed by th e K a rm a n /P ra n d tl rela­ tions:

“ • = 5.75 M ÿ ’ (2.3) where u* is wind shear velocity, Uz is the velocity a t th e height of m easurem ent (z), 5.75 is a constant incorporating von K arm an ’s constant (ac 0.4) and zq is

th e roughness length (referred to as roughness height in some literatu re). Shear velocity (u*) is another expression of shear stress (r) given by

u* —4 (2.4)

V

where pa is air density.

Bagnold first studied sand tra n sp o rt in connection w ith this boundary fiow p a t­ tern (1941). He established th e following sem i-em pirical form ula relating sand fiux (g(a;)) w ith wind shear velocity (u*(a:)), where x is horizontal distance.

q{x) = Cb {— ) , (2.5)

where Cb is constant, Dg is sand grain diam eter, is a reference sand grain diam eter of 0.25 m m, is air density, and g is grav itatio n al acceleration. Kawa- m ura (1951) postulated th a t the shear stress acting on sand grains is due in p a rt to th e direct wind shear and also to im pacting sand grains. T his idea resulted in a different sand tra n sp o rt form ula from B agnold’s (2.5), and it was th e first to incorporate th e threshold shear stress (u*t)- According to Bagnold (1941) the threshold shear velocity is expressed as

= « 0.1 (2.6)

Pa Pa

where A is a constant (% 0.1), pg is sand density. M any form ulae have since been proposed for sand tra n sp o rt (Greeley and Iversen 1985). Among others, L ettau and L e tta u ’s formula:

q{x) = Cll ( f ) u,(x)^ (u.(x) - u,t) u , ( x ) > u , t

[Z.7)

— 0 U-tfXX) ^ Ujut,

where C l l is constant, is currently, the m ost widely used (L ettau and L ettau, 1978; Greeley and Iversen, 1985 pp.99-100). However, which form ula best de­ scribes the actual sand tra n sp o rt is still in debate (Sherm an et al, 1998).

T he wind and th e sand grains interact w ith each o ther in tra n sp o rt. Once salta­ tion occurs, saltatin g sand grains modify th e log-linear profile of the wind ex­ pressed in equation (2.3). For example, roughness length (zq) increases from the static value of Dg/30 (Bagnold, 1941) to one th a t depends on th e surface shear velocity (u*):

Z q = 0.02 —^ (2.8)

^ 9

(Owen, 1964). Theoretical and experim ental studies of th e “Owen effect” were re­ viewed by G illette (1999), whose review included th e R aupach form ula (Raupach, 1991), which is an extension of Owen’s (2.8).

In th e same way th a t roughness length {z q) is modified once saltatio n occurs, the

view of dune dynamics. T hough bo th field and lab o rato ry experim ents show th a t threshold shear stresses on upslope surfaces differ from those on downslope sur­ faces (H ardisty and W hitehouse, 1988a; 1988b, Iversen and Rasmussen, 1994), th e significance of this effect is in debate as is m entioned again later (section 2.7.1).

2.4.4

W ind flow over a dune

The m ost characteristic wind flow features over dunes are w ind speedup on the windw ard slope and th e reverse flow in th e lee. W ind speedup has been noted, b o th in the field and in wind tunnel experim ents (see for example Mulligan, 1988; B urkinshaw et ai, 1993; Frank and Kocurek, 1996b; L ancaster et al, 1996; Wiggs et al, 1996). Here it is w orth recalling sand tra n sp o rt formulae (equations (2.5) and (2.7)). It is th e surface shear velocity (u*), or more correctly the surface shear stress (r), th a t causes sand tra n sp o rt, and is hence associated w ith th e erosion/ deposition p a tte rn over th e dune surface. Since th e log-linear wind profile (2.3) is valid only on a fiat surface, it is difficult to estim ate th e surface shear velocity on dune surfaces from th e conventional w ind d ata. T his difficulty was recognised firmly by around 1996 (Frank and K ocurek, 1996b; Lancaster et

al, 1996; Wiggs et ai, 1996). Recently direct sand-flux m easurem ent using sand tra p s has become popular, replacing th e conventional w ind-speed m easurem ent using cup-anem om eters (Lancaster et al, 1996; W alker, 1999). T he use of sand tra p s m ay be a more integrated m easurem ent which accounts for th e fluctuation in th e surface shear velocity. Since sand fiux is a cubic function of th e surface shear velocity (q ~ ul), if only th e average surface shear velocity is discussed, sand fiux m ay be under-estim ated. In this respect too, san d -trap m easurem ents are tho u gh t to be superior to th e use of anem om eters.

al, 1996, figure 10). However, this condition is not always satisfied. A non-linear increase of shear velocity was suggested by Frank and K ocurek (1996b, figure 3), though th is was deduced from th e conventional w ind-speed m easurem ent. A ccurate observation and prediction of wind speedup and th e associated increase in th e surface shear velocity, which is more im p o rtan t, is a vital p a rt of dune study. This is because, as seen in equation (2.2), th e sand-flux increase over the windw ard slope of a dune is necessary to sustain th e shape of th e w indward slope (Bagnold, 1941; Zeman and Jensen, 1988). The w ind speed profile over th e dune has been studied also by using a scaled dune m odel which was placed in a wind tu n n el {e.g. Howard et a/., 1978; Wiggs et ai, 1996).

T he surface slope changes ab ru p tly a t the dune crest. T he angle of a slipface, as defined by th e angle of repose of 30 — 33° for dry sand (Livingstone and W arren, 1996, p72; also see section 2.4.5), is too sharp for th e oncom ing wind to follow the topography. Consequently, th e wind and the topography separate a t th e dune crest and th e “separation zone” is formed, in which reverse flow occurs. The separation zone extends to the re-attachm ent point which is ab o u t four tim es the dune height in the lee of the dune crest (Frank and K ocurek, 1996a). This topic will be revisited a t th e end of this subsection.

A nother characteristic of flow over dunes is th e w ind speed decline ju s t upwind of th e dune. T his phenom enon is astonishing, since if we tre a t it sim ply this could make th e dune m igrate backwards, which is, of course, against observations in th e field. To rem edy this anomaly, enhanced turbulence ju s t upwind of the dune may probably be invoked (Wiggs et al, 1996; Van Boxel et al, 1999).

W ind fiow p a tte rn s over dunes have three-dim ensional structures. W ind diversion around a barchan dune was well docum ented by Howard et al (1978). W ind deflection a t th e crest was studied by Tsoar (1983) for linear dunes, and by L ancaster (1989) for sta r dunes.

th e field, however, the wind profile and speed are no t constant. According to Prank and Kocurek (1996a), for example, the wind fiow re-attachm ent point varies from 1.6 to 5.4 dune heights downwind of th e crest. Furtherm ore K n o tt reported th a t strong gusts blew in th e mornings a t Salah in Algeria, which may be im p o rtan t for dune initiation (K nott, 1979, referred to in Cooke et al, 1993, p323; Lancaster, 1996).

2.4.5

A valanche and sand d ep osition on th e lee face

A 10 m dune m igrates typically 10 — 30 m year“ ^ (Cooke et al, 1993, figure 23.24). The deposits created by m igration form s tra ta in which ancient dune m igrations have been recorded. According to H unter, there are three types of depositional modes: 1) ripple laminae; 2) grainfall lam inae; and 3) grain fiow lam inae (hereafter term ed avalanche) (H unter, 1977a). Types 2 and 3 occur on slipfaces. An avalanche occurs when the sandy surface slope exceeds th e angle of repose which is commonly 30 — 33° (Livingstone and W arren, 1996, p72). Theory and laboratory experim ents suggest th a t this slipface is indeed not a sm ooth b u t a com plicated surface, which is a consequence of avalanches of all sizes (Bak et al, 1987; Bak 1996; see SOC in section 1.2). T he to p end of th e slipface is called the brink. Brink and crest (the highest point of a dune) som etim es coincide, and are sometim es separate. T he reason of this brink-crest coincidence/ separation has been much debated, b u t is still obscure (see Cooke et al, 1993, pp331-333). For barchan dunes and transverse dunes, which are strongly two-dimensional, th e w indw ard face is erosional and the leeward face is depositional.

evidence of these conditions may be preserved in s tra ta , from which ancient dune m igrations m ay be deduced, no qu an titativ e com parison of this model to n atu re has yet been made.

G rainfall m ainly comprises th e saltatin g sand grains crossing th e brink and then being deposited on th e leeface. An expression for saltatio n length was developed by Anderson and Hallet (Anderson and Hallet, 1986). Following th eir study, An­ derson (1988) studied this depositional process further, and re-exam ined H u n ter’s assum ption in th e previous paragraph. He found th a t a sm all bum p forms on the slipface ju st leeside of th e brink, unlike H u n ter’s assum ption. This bum p was observed in th e field. D etailed field study of th e sand deposition p a tte rn on a slipface was reported w ith com puter sim ulations by M cDonald and Anderson (1995).

2.4.6

D u n e m igration

It is widely believed th a t aeolian sand dunes m igrate downwind w ithout changing th eir shapes (Bagnold, 1941). Dune m igration is a result of th e erosion on the windw ard surface and the deposition on th e slipface in th e lee. Dune m igration speed (celerity) is usually described by B agnold’s expression (Bagnold, 1941):

Cd = (2.9)

where g(0) is sand fiux in kg m “ ^ s~^ a t the dune crest, 7 is sand bulk density in the dune in kg m “ ^, and H is dune height in m. T his inverse relation between m igration speed (cd) and dune height {H) has been am ply confirmed by many researchers and is sum m arised by Cooke et al (1993, figure 23.24). W ilson (1972) proposed a more refined expression of m igration speed (cd):

Cd = (2.10)

7II

E stim atin g m igration speeds of dunes is one of th e m ajo r challenges in aeolian geomorphology, since m igration is one of th e m ost hazardous aspects of dunes. P ro tectin g a road from inexorably advancing dunes needs a continuous removal operation (Livingstone and W arren, 1996, pl6 2 ). A recent field stud y in n o rth ­ eastern Brazil found dunes m igrating a t speeds th a t were com pletely out of the range of those sum m arised by Cooke et al. (1993, figure 23.24). A dune of more th a n 50 m height was noted to m igrate a t a speed of more th a n 15 m year“ ^ (Jimenez et ai, 1999). This suggests th a t the deduction based only on empirical knowledge, as in th e sum m ary by Cooke et al, is som etim es misleading. However, no m odel has yet been proposed th a t can explain these field d a ta systematically.

2.5

The stru cture and evolution o f a dune field

Dune fields have sp atial stru ctu re b o th in plan and in depth. L ancaster (1988) found th a t dune-to-dune spacing (wavelength) scales w ith dune height. Dune-to- dune spacing is tho ug h t to be regulated by the secondary wind flow th a t is in tu rn generated by dunes themselves. The longitudinal spacing between transverse dunes (m easured in th e direction perpendicular to th e dune trend) is thought to be related to th e size of the wind separation zone in th e lee of th e dune (see section 2.4.4), while th e lateral spacing between linear dunes (also m easured in the direction perpendicular to the dune trend) m ay be associated w ith helical roll vortices (Lancaster, 1995, ppl85-190). D une-to-dune spacing m ay also be controlled by sand grain size as noted by W ilson (1972).

The in terp retatio n of the internal structure of dune fields is not an easy task, for it requires consideration based on the sedim ent state com prised of sedim ent supply, sedim ent availability and the tra n sp o rt capacity of th e w ind (Kocurek and Lancaster, 1999). Dune fields show m any types of s tra ta (H unter, 1977b). In m any cases of m igrating transverse bedforms, only th e sedim ent deposited on each lee slope and not eroded during th e passage of a following dune may be preserved as cross-stratified strata. Therefore cross-strata analysis needs an understanding of the collective motions of dunes.

The surface between older and newer bedforms is called a bounding surface. Brookfield (1977) related th e order of bounding surfaces and dune field dynam ­ ics. “F irst order surfaces m ark th e movement of draas. Second and th ird order surfaces are related to the m igration of dunes and local airflow instabilities respec­ tively” . The bounding surface is seldom horizontal, in which case each subsequent dune climbs over the one before. This is known as bedform climbing, which may be an im p o rtan t correction when com paring sim ulation results of dune field evo­ lution to ancient records of dune fields. R ubin and H unter (1982) considered climbing dune fields as net-depositional areas. T hey discussed th e dynam ics of two-dim ensional dune trains m igrating downwind w ith decreasing size or num ­ ber, which contribute to sand accum ulation resulting in a clim bing dune field. T hey showed th a t under certain conditions, th e thickness of a climbing tran sla­ te n t stra tu m is a function of dune height and the ra te of downwind decrease in the tra n sp o rt rate. Using this relation, they estim ated heights of ancient dunes recorded in th e Navajo, De Chelly and E n trad a Sandstones, and concluded th a t they m ight be more th a n a few hundred meters.

In addition to these stratigraphie analyses, absolute d atin g techniques can now provide us w ith significant inform ation ab o u t the form ation dynam ics of dunes and dune fields over long periods of tim e. W hile radiocarbon can d ate only to 40,000 years (Livingstone and W arren, 1996, p l3 2 ), optical (luminescence) datin g techniques “can provide reliable ages for aeolian sedim ents ranging in age from a few hundred to a few hundred thousand years” (Singhvi and W intle, 1999). Studying aeolian deposits in the Last Glacial M axim um is particularly im p o rtan t since in th a t period there were higher winds th a n a t present, so th a t m any of our present aeolian bedforms may have originated then (W arren and Allison, 1998).

2.6

A nalytical studies o f em ergence and shape

o f dunes

Im p o rtan t progress has been m ade in the theoretical stu d y of subaqueous dunes. T here are some distinct differences between the environm ents of aeolian and sub­ aqueous dunes and ripples. For example, there is th e free surface between air and w ater, which appears to play an im portant role in dune form ation under th e wa­ ter. Waves on this free surface are thought to cause anti-dunes, th e subaqueous, dune-like bedform s which, however, m igrate up current (Yalin, 1977, pp220-222). A nother m ajor difference is th a t w ater is much denser th a n air, so th a t th e fall velocity of particles is much lower, and consequently suspended sand grains play a significant role in subaqueous dune form ation. Still th ere are m any sim ilarities between th e two types of bedform in morphology and in dynamics.

2.6.1 Linear instability analysis

recently been reviewed by McLean (1990). A nderson (1987a) applied th is analy­ sis to the form ation of aeolian ripples, and succeeded in calculating th e relation between ripple wavelength and th e m ean rep tatio n length. N on-linear analyses for aeolian ripples have also been m ade (Prigozhin, 1999; Valance and Rioual, 1999).

S tam (1996) developed a two dim ensional dune m odel combining linear approxi­ m ation of B agnold’s sand tra n sp o rt form ula (2.5) and th e w ind flow theory devel­ oped by Jackson and H unt (1975), which is introduced in more detail w ith m athe­ m atical expressions in the next chapter (section 3.3.2). He showed th a t combining the Jackson-H unt theory and the linearised Bagnold form ula leads to th e phase difference between topography and shear stress, which is tho u g h t to be neces­ sary when dunes grow from a plane bed w ith a small undulatio n (Kennedy, 1963; McLean, 1990). There are three drawbacks in his model. F irst, only sinusoidal m igrating p attern s can be analysed. Second, due to linearisation, dunes show ex­ ponential growth w ithout reducing th eir m igration speeds. These two drawbacks are not avoidable in linear instability analysis. A nd last, th e m odel shows th a t all wavelength com ponents are unstable, and th a t the shorter th e wavelength, the m ore quickly its com ponent grows. Consequently shorter-w avelength com ponents dom inate, and a dune profile cannot be m aintained.

L ater Stam developed a num erical model incorporating avalanching, which he ex­ pected to suppress short-wavelength com ponents, into his analytical model (Stam , 1997). However th e resulting dunes still did not resemble n a tu ra l dunes.

2.6.2

K inem atic analysis

T his is an approach to m odelling th a t focusses on th e geom etry of moving objects, here ripples and dunes. K inem atic discussion in dune geomorphology dates back to Exner^ (refered to in Cooke et al, 1993, p328), who a tte m p ted to explain the

^in Exner, P.M. 1920. ‘Zür Physik der Diinen’, Akademie der Wissenschaften, Wien,

asym m etric profile of a dune in the wind direction. Predspe (1982) estim ated the equilibrium shape and dimensions of a sequence of subaqueous dunes including th eir height and wavelength, using th e current flow p a tte rn dow ncurrent of a step whose height is the same as the dune considered. A pplying linear stability analysis he found th a t th e estim ated shape was stable. T he resulting shape was at least qualitatively acceptable. In his paper, Predspe (1982) presented an equation th a t he conjectured the equilibrium dune m ust follow:

H q{0)’ ^ ^

where h{x) is the surface height a t x, H is th e dune height, q{x) is the sand fiux a t X and q{0) is th e sand fiux a t the crest. Using this equation, Lancaster (1985) predicted the dune surface profile for various wind speeds upw ind of th e dune, together w ith his field-measured sand tra n sp o rt data.

C om pared to th e study of aeolian dunes, well defined experim ents on subaqeous dunes are relatively easy, using flumes. W hen th e m ean current velocity is in­ creased, an originally plane bed changes to dunes m igrating dow ncurrent. Further increases of m ean current velocity lead to another plane bed called th e upper- regime plane bed. U ltim ately th e increase of m ean current velocity results in antidunes m igrating up current, which are strongly related to th e existence of the free surface between the air and the w ater, and are tho u g ht to be unlikely to occur subaerially. These changes among dunes, plane bed and antidunes also depend on th e sand grain size. Increasing grain size makes these changes occur a t higher current velocities. The morphological diagram against stream power and grain size was given by Allen (1970, figure 2.6). Engelund and Fredspe (1974) discussed th e kinem atics of subaqueous dunes and succeeded in describing the tran sitio n from dunes to plane bed. They thought th a t th e tra n sitio n occured when the dom inant sand tra n sp o rt mode changed from bedload tra n sp o rt, which corre­ sponds to saltatio n and creep in aeolian processes, to suspension. This scheme was supported by recently conducted flume experim ents (B ennett et ai, 1998). T his dune-to-plane-bed tran sitio n may be more universal. Looking wider afield,

since th e air on Venus is extrem ely dense (Greeley and Iversen, 1985, p28), ae­ olian processes and geomorphology on Venus are sim ilar to subaqueous ones on E arth . Using a wind tunnel which sim ulated Venusian atm osphere, Bougan and Greeley (1985) dem onstrated th e possibility of a tran sitio n from dunes to upper plane bed on Venus.

2.6.3 D ifficulties in analytical stu d ies

T he previously reviewed analyses have been carried out for th e two-dimensional geom etry of dunes, representing transverse dunes a t right angles to the flow. In his book, however, Allen (1968) emphasised the im portance of th ree dim ensional flow p attern s on ripple and dune pattern s. These three-dim ensional p attern s affect not only th e shape of single dunes b u t also th e alignm ent of dune systems. For example, th e p a tte rn of out-of-phase transverse bedform s (the common p attern ), where neighbouring bedforms downwind are arranged such th a t they are half a wavelength shifted sideways (see Figure 4.16), is said to be strongly associated to three-dim ensionally diverted local wind flow (W ilson, 1970 referred to in Cooke

et al., 1993, figure 26.1). These three-dim ensional current flows over bedforms have been studied w ith flume experim ents, and related observations were made in th e desert (see section 2.4.4). However, m ath em atical form ulations of these p a tte rn s have proved difficult.

2.7

C om puter m odelling o f dunes and dune fields