Ejemplo de Aplicaci ´on Parcial de Funciones

3.1

Introduction: shape and m igration speed

A lthough m any dunes are known to change th eir shape continually (Hesp and Hastings, 1998), some aspects of th eir m igration can be discussed under an as­ sum ption of equilibrium . In conditions of unidirectional constant winds and sand supply, it is well known th a t transverse and barchan dunes m igrate downwind w ithout changing th eir shapes (Bagnold, 1941; Coursin, 1964; Long and Sharp, 1964; H astenrath, 1967; Howard et al, 1978; Fryberger et ai, 1984; Al-Hinai and Moore, 1987; see also section 2.4.6). This dynam ic equilibrium is a consequence of th e interaction between th e form and th e processes, known as morphodynam- ics (Livingstone and W arren, 1996, pp77-79). Considering th e shape of barchan dunes, there are two distinct characteristics which m ay indicate th a t they are at equilibrium : a) th e cross-sectional shape perpendicular to th e wind (depicted as B -B ’ in Figure 3.1) is height independent (Hesp and H astings, 1998; Sauerm ann

et al, 2000); whereas b) th e cross-sectional shape parallel to th e wind (A-A’), i.e.

th e profile, is height dependent: the higher th e dune, th e steeper th e windward slope (Finkel, 1959; Long and Sharp, 1964; H asten rath , 1967; Sauerm ann et al,

(a)

Figure 3.1: Schematic views of a barchan dune (a) from top and (b) from side.

T he shape of a dune a t equilibrium is still not well understood. Howard et al.

(1978) deduced some requirem ents th a t m ust be satisfied by th e equilibrium shape of a barchan dune and the wind fiow over it, and checked th e validity of these requirem ents using lim ited field d a ta (see section 2.7.1). W ith o u t more empirical d a ta , however, their model cannot predict th e surface profile of th e dune, hence th e above two characteristics a) and b) rem ain unsolved.

T h e evolution of th e dune surface profile has previously been calculated by it­ eratively solving the interaction between th e topography and the wind fiow as schem atically viewed in the block flow chart of Figure 3.2 (Howard and Walms- ley, 1985; W ipperm ann and Gross, 1985; Zeman and Jensen, 1988; Stam , 1997; Van D ijk et a/., 1999; see section 2.7.1 for details). In th is type of modelling, the

(Initial) dune surface: il(x, t)

i f

Shear velocity over a dune: u* (x)

i

Sand flux over a dune: q(x) (e.g. Lettau & Lettau, 1978)

i f

Modified dune surface: r['(x, t)

YES

i

Equilibrium profile: ti(a:)

Dune surface: t+6t)

(x, t+ôt) = Tl(x, t) ?

Wind flow model:

(e.g. Jackson & Hunt, 1975; Zeman & Jensen, 1987)

Erosion/deposition: 5 ti(x , t) ~ -V q(x)

Avalanche:

i f V t) ' (x, t) > 0c

NO

Figure 3.2: Flow chart for dune modelling in the conventional method.

equilibrium surface profiles are defined as those th a t do no t show any change after successive iterations. However, complete equilibrium has n o t yet been achieved using th is approach. The three-dim ensional barchan dune sim ulated by W ipper­ m ann and Gross (1985) showed a continual decrease in height and elongation in th e w indward slope w ith tim e. A lthough Van Dijk et al. (1999) showed a height satu ratio n for th e two-dimensional dune, th e w indw ard slope continued to elongate.

W hile still restricting the problem to dunes a t equilibrium , a different approach is possible, where th e dune surface profile is solved as a boundary-condition prob­ lem. O n th e assum ption of morphological conservation, Zeman and Jensen (1988) developed such a model, b u t w ith lim ited success due in p a rt to th e difficulties in wind-flow calculation.

Dune m igration speed (cd) is usually described w ith th e Bagnold form ula for dune m igration speed (2.9):

where ç (0) is sand flux a t th e dune crest, 7 is sand bulk density in th e dune, and H is dune height (Bagnold, 1941). A more accurate expression is th e W ilson form ula (2.10):

g(0) - ç(+ oo)

^ ’

where ç(+ oo) is th e outgoing sand flux not cap tu red by th e slip face (Wilson, 1972; section 2.4.6). We can now introduce th e concept of sand trap p in g efficiency (Te, dimensionless), which is the proportion of moving sand tra p p e d in th e dune slip face to the to ta l am ount of sand crossing th e dune crest (W ilson, 1972; Cooke

et al, 1993, pp346-347). The W ilson form ula (2.10) is th en rew ritten as

C .

(3.,)

Te is related to th e p a tte rn of deposition of sand on th e slip face which was modelled by A nderson (1988) and M cDonald and A nderson (1995).

Sand flux (q{x)) can be estim ated from shear velocity (u*(x)) using th e Bagnold form ula for sand flux (2.5):

q{x) = Cb ( ^ ) u*{x)^,

where Cb is a constant, Dg is sand grain diam eter, is reference grain diam eter.

Pa is air density, g is gravitational acceleration (Bagnold, 1941), or using the L ettau and L ettau form ula (2.7):

q{x) = C l l (§f)^/^ ( ^ ) u ^ { x f {u^{x) - w*t) u^{x) > u^t

— 0 Uif{x) ^ ^*t;

where C l l is a constant and u*t is th e threshold shear velocity, below which no saltatio n occurs (L ettau and L ettau, 1978). T he two formulae (2.5) and (2.7) give alm ost identical results for th e cases where u^(x) %$> u*t (see for example Fryberger, 1979, figure 92).

B oth th e Bagnold (2.9) and th e W ilson (2.10; 3.1) form ulae for dune m igration speed were derived as a consequence of th e assum ption of shape invariance. In such conditions, as Zeman and Jensen (1988) noted, th e outgoing and th e incom­ ing sand fluxes m ust be th e same:

g(+oo) = q{ - oo) . (3.2) If otherwise, dunes shrink (ç(-Hoo) > q(—oo)) or grow (ç(-l-oo) < q{—oo)). Con­ sequently, the Bagnold form ula (2.9), which does not take into account the ou t­ going sand flux (ç(-t-oo)), suffers not only inaccuracy, b u t also inconsistency in its derivation.

The W ilson form ula (2.10; 3.1), which is consistent in its derivation, is still not satisfactory. It suffers lack of predictive power. There are m any dunes in a dune field, each being different in height, as will be seen in section 3.3.1. It is practically impossible to know the shear velocity a t the crest (u*(0)) for each dune. On an ideal flat surface, it is true, wind speed {u) and shear velocity (u*) are connected by th e K arm an -P ran d tl form ula (2.3):

“ • = 5.75 l o g ( ^ ) ’

where Uj, is the velocity a t the height of m easurem ent (z), 5.75 is a constant incorporating von K arm an ’s constant {k, ^ 0.4) and zq is th e roughness length. B ut in order to use wind d a ta collected a t nearby meteorological stations to predict qualitatively (sem i-quantitatively) dune m igration speed, a model th a t links dune m igration speed and shear velocity on th e level surface is necessary. For q u an titativ e estim ations, wind d a ta collected closer to th e dunes are necessary. The first half of th is chapter (section 3.2) shows th a t only by assum ing an equi­ librium condition and by estim ating th e sand tra p p in g efficiency (T e ), can a self- consistent model th a t describes transverse dune m igration be developed. The following section (3.2) shows how the sand trap p in g efficiency (Te) depends on wind shear velocity (u*) and dune height (H), and consequently how Te influ­ ences dune m igration speed (cj). The discussion is developed by combining and extending Zeman and Jensen’s (1988) approach and A nderson’s (1988) microscale

analysis of sand grain deposition on the slip face. T hroughout th e first half of the next section (3.2.1) the im portance of the equilibrium assum ption, w ind speedup over th e w indw ard surface of a dune and sand trap p in g efficiency (Te) are rep eat­ edly emphasised. They are shown to be strongly connected.

In the second half (section 3.3), after exam ining field d a ta of th e w indward surface profile of barchan dunes, th e model is further extended by introducing th e wind- flow theory developed by Jackson and H unt (1975). T his introduction enables the estim ation of th e surface profile of a dune a t equilibrium , as a boundary-condition problem.

3.2

M igration speed o f transverse dunes and sand