DESERT DUNE DYNAMICS AND PROCESSES
By LEVENT YILMAZ, (visited Professor at Louisiana State University, Coastal Studies
Institute, Baton Rouge, LA, 70803, USA)
Technical University of Istanbul, Civil Engineering Faculty, Hydraulic Division, 80626,
Maslak, Istanbul, Turkey, e:mail : lyilmaz@itu.edu.tr
Abstract
The development of the dunes are governed by the effects of turbulence. Turbulence is a
type of fluid flow that is strongly rotational and apparently chaotic. Turbulence separates
nearby parcels of air and thus mixed fluid properties . The evolution of sand dunes is
determined by the interactions between the atmosphere, the surface and the transport and
deposition of sand. We are concerned with this physical process and its computational
simulation from three perspectives; namely, (1) flow structure; (2) sand transport and
deposition and (3) interactions between flow structure and sand transport-deposition,
which determine the dune morphology.
Keywords: Coastal management, dunes, sediment supply
Introduction
The system of moving bedforms in a flow field can be explained by the sedimentcontinuity equation and the sediment-transport equation. It yields
h( x, t )
1 q( x, t )

t
 x
q( x, t )  f ( ( x, t ))
(1)
(2)
where h is the height of the bedforms or topographic height, t is the time,  is the
sediment density, q is the sediment transport capacity in kg.m-1.s-1 ,  is the shear stress
due to saltation.
Mathematical Model
Equation (2) shows that the sediment transport varies linearly with the shear stress (
Stam, 1994) as
q(x,t)=qo+A1  (x,t)
(3)
where qo is a constant basic sediment-transport and A1 is the linearity constant. If
topographic variations are relatively small, it can be generally stated that the shear stress
is formed of a constant basic term (o), and a correction term (1) that varies with space
and time:
(x,t)= o+ 1(x,t)
(4)
If, in a first approximation, this correction is assumed to be linear with the topographic
height:
1(x,t)=A2h(x,t)
(5)
the sediment-transport equation becomes:
q(x,t)=qo+A1o+A3h(x,t)
(6)
where
A3=A/A2
Taking the derivative of Equation (6), it yields
q ( x, t )
h( x, t )
 A3
x
x
(7)
which, substituted in the continuity equation becomes
h( x, t ) A3 h( x, t )

0
t
 x
(8)
This is called the simple wave equation, and it describes the propagation of a wave at
constant velocity. The ratio
c
A3

(9)
The simple-wave equation, in this case, results from combining a linear shear stress and a
linear sediment-transport formula. The ratio in Equation (9) is called the wave velocity. It
concerns the morphodynamics of dunes and this wave velocity is equal to the migration
rate. This wave will advance at a constant rate c without changing its shape. The solution
of Equation (8) is given as
h(x,t)=f(x – ct)
(10)
which expresses that at a certain point x at a certain time t the topography will have the
same height as it had at the start (at t=0) at a point (x – ct). The height of the bedforms
can be any function of ( x – ct ).
Kinematic Wave Approximation
Another formulation of the mathematical model of the topography is given by using
a sediment-transport formula instead of a linear relationship with the shear stress.
Bagnold’s (1941) model yields
q( x, t )  CB ( x, t )3 / 2
(11)
where CB is the Bagnold’s constants with the unit [s2m1/2kg1/2]. If the linear shear stress
are assumed as in the Equations (4) and (5), it is obtained
q ( x, t )  CB ( 0  A2 h( x, t )) 3 / 2
(12)
and
q ( x, t ) 3
h( x, t )
1/ 2
 CB  o  A2 h( x, t )  A2
x
2
x
h( x, t )
3 CB
1 / 2 h x, t 

A2  o  A2 h( x, t ) 
t
2 
x
(13) and (14)
In this mathematical model it can be seen that the migration rate is not a constant but a
more complicated expression that varies with the topographic height, and the wave
velocity is given as
c
3 CB
1/ 2
A2  o  A2 h( x, t ) 
2 
(15)
In this mathematical model the dune shape changes. The migration rate increases with
height, which means that the top of the dune will advance more rapidly than the base. The
lee side of the dune will tend to become steeper and the peak will eventually overtake the
slipface. In observations, according to this mathematical model, the maximum angle of
repose for sediment will be surpassed and avalanching will occur at the slipface, limiting
this asymmetrical shape. This is a well known type of equation called a breaking-wave
equation which shows the breaking-wave behaviour at the mathematical solution. This
behaviour of the solution will be similar for any non-linear sediment-transport equation,
as long as the peak has a higher velocity than the base which is given in mathematical
formulation as
ch 
0
h
(16)
The deformation of the bedform in a “dune-like shape” is partly due to the non-linear
relationship between sediment transport and shear stress.
length/width
Comparison of Dunes Dimensions
2
0.2
1.5
0.15
1
0.1
0.5
0.05
height/length
25
21
17
13
9
5
0
1
0
length/width
Length (m)
Figure: 1 The comparison of dimensionless lengths of barchan dunes (Stam, 1994)
Development of an analytical solution for bedform migration and growth
The expression of the shear stress of the velocity in terms of the topography makes it
adequate for the development of an analytical solution. The analytical solution becomes
simpler if only one wave number is considered, so that the summations in the Fourier
series are reduced to only one term.
2
1.5
1
0.5
0
0.15
0.1
height/length
0.05
length/weight
0
1
5
9
13
17
21
25
height/whole
length
Comparison of the Dimensionless
lengths of the Dunes
length
Figure:2 Comparison of the Dimensionless lengths of the Dunes (Stam, 1994)
The general line is analogous to the development of the simple and kinematic wave
equations . The continuity equation is given as
 
 
h* x* , t
1 q x* , t
 *
t

x*
Dimensionless coordinates have been used (indicated by an asterisk *), so that
(17)
x* 
x
L
h( x* , t )
(18), (19) and (20)
H
 *  HL[kg.m 1 ]
where H is the maximum topographic height [m].
A spatial derivative expression of the sediment transport has to be substituted in the
continuity equation. A linearization of Bagnold’s sediment-transport formula will be used
(Stam, 1994). For the linearization it has to be considered that the shear stress ( (x*,t))
results from Prandtl’s logarithmic profile (o) with Jackson and Hunt’s (1975)
dimensionless first-order correction (1(x*,t))
h* ( x * , t ) 
 ( x* , t )   o 1  1 x* , t 
(21)
Bagnold’s linearized sediment-transport formula then becomes
 3

ql x* , t   CB 03 / 2 1  1 x* , t 
(22)
 2

where:
o = shear stress from logarithmic profile [Pa]
1 = first order correction to the shear stress from the logarithmic profile [dimensionless]
 = perturbation factor. This is dimensionless number smaller tan 1. (Stam, 1994)
Differentiation of the linearized transport equation to the dimensionless coordinate x*
gives
 
 
*
ql x* , t
3
3 / 2  1 x , t

C

(23)
B
o
x*
2
x*
Jackson and Hunt’s (1975) expression for the correction of the shear stress can be used
for the developing of this equation. Expressed as a Fourier Transform 1(k*,t) the shear
stress correction is given as


i
*
* ˆ* *
i K1 z k e
ˆ
 1 (k , t )   k h k , t zk e
K o z k e i
 
zk  2


zr k *
l
and  = /4 if k*>0
(positive wave number)
 = -/4 if k*<0
(negative wave number)
where:
l = thickness of the inner region [m]
Ko= Modified Bessel function of the zero-order
(24) and (25)
(25a)
K1= Modified Bessel function of the first – order
k*= dimensionless wave number. It should be noted that by introducing the dimensionless
coordinate x*=x/L, the maximum wave length () has become equal to one and therefore
the wave number (k=2 / ) has become dimensionless also. For small arguments, the
2.5
2
1.5
1
0.5
0
1.5
1
length/width
0.5
height/length
43
37
31
25
19
13
7
0
1
height/length
Comparison of the Dunes Dimensions
Length(m)
Figure:3 Comparison of the Dune Dimensions (Yilmaz, 1997)
Conclusions
The evolution of desert dunes is determined by the interactions between the atmosphere,
the surface and the transport and deposition of sand while the morphology and dynamics
of Mediterranean Aeolian sand dunes are governed by sand movement induced by shore
wave shear. In conditions of unidirectional constant winds and sand supply, it is well
known that transverse and Mediterranean shore dunes migrate downwind without
changing their shapes in comparison with the desert barchan dunes. Beach ridges or
coastal dunes consist of also compound dunes made up of two or more dunes of the same
basic type, coalescing or overlapping, and complex dunes in which two or more different
basic types are combined or superimposed.
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