1. INTRODUCCIÓN
2.5 DIRECCIONAMIENTO ESTRATÉGICO
2.5.5 FORMULACIÓN DE ESTRATEGIAS
In chapter 5, after having exam ined th e original W erner (1995) model, some revi sions were m ade. In this chapter, each revision will be tested through com puter sim ulations, and m any aspects of morphology and dynam ics of dune fields will be discussed. Table 6.1 shows the param eters used in th e sim ulations th a t fol low. For a historical reason, th e concept of the reference num ber of slabs (href) in equations (5.9) and (5.10), which was defined in section 5.6, will be introduced from section 6.3. In sections 6.1 and 6.2, hence in Figures 6.1-6.8, href is replaced w ith th e average num ber of slabs (havg)-
6.1
N o erosion in shadow zones
In section 5.4, by considering observational facts, a no-erosion criterion has been introduced in th e shadow zones in the lee of dunes. Figure 6.1 shows cross- sectional views of transverse dunes sim ulated w ithout erosion in shadow zones, on a 2-dim ensional lattice from an initially isolated transverse dune w ith the num ber of slabs (h) initially set a t 30, built on a level surface {cf. Figure 5.3). The sim ulation tim es {t) are 0 to 100. W ith no erosion in th e shadow zones: a) the slip face in th e lee is sustained; b) th e dunes grow m ore quickly due to strong accum ulation of sand slabs; and c) th e dunes m igrate downwind w ith constant
p aram eter value
lattice size; Ls X Ls = 1024 X 1024
slab aspect ratio 1 /3 angle of repose
to th e nearest neighbour t a n - i (2/3) = 33.7° to th e second nearest neighbour t a n - i ( 2 / 3 / v ^ ) = 25.2° shadow zone angle 15°
tra n sp o rt length a t th e reference height Lo = 5 (unless specified in the text) shear-velocity-increase:
linear coefficient C l = 0.4 non-linear coefficient C2 = 0.002
erosion probability:
outside shadow zone same a t every site in shadow zones 0.0
deposition probability: outside shadow zone:
w ith a t least a slab Ps = 0.6
w ithout a slab Pns = 0.4 in shadow zones Ps = Pns = 1.0
CO X F 100 80
t
425 h= 1 0 0 + + initial^ dürié"^ Q ^ 475 525 575 D istance, / 625 675Figure 6.1: Evolution of an initially introduced, isolated transverse dune simulated without erosion in shadow zones. The dune’s initial number of slabs above the surface is 30. The simulation time (i) is 100.
speed during th e period of a constant height. These features are sim ilar to those of dunes in n atu re (section 2.4.6). Figure 6.2 shows transverse dunes sim ulated on a 2-dim ensional lattice from an initially random morphology. W ith o u t erosion in th e shadow zones, the dunes are higher th a n those shown in Figure 5.2(a), due to strong slab accum ulation. In the rem ainder of th is chapter we consider no erosion in shadow zones in all simulations.
6.2
The effects o f w ind speedup over a dune
In this section, the effects of wind speedup, which was introduced in section 5.5, are exam ined numerically. W ind speedup has been form ulated by adding linear and non-linear height-dependent term s to th e original constant slab tra n sp o rt length (Lo) (see equation 5.10). To highlight th e role of each term , in th e first half of th is chapter (section 6.2.1), the effects of linear w ind speedup form ulated in equation (5.9) in subsection 5.5.1 are examined. Subsequently in th e next half (section 6.2.2), the effects of non-linear wind speedup form ulated in equation (5.10) in section 5.5.2 are examined.
wind
Figure 6.2: Transverse dunes simulated from an initially random mor phology. The simulation time (t) is 500. No erosion occurs in shadow zones. Contour lines are for 100, 110, 120, 130 and 140 slabs.
6.2.1
T he effects o f linear wind sp eed u p
Sim ulations were carried out w ith and w ithout a w ind speedup term introduced in the form of equation (5.9). Figure 6.3 shows th e sim ulation results started from th e sam e initial transverse dune used in Figures 5.3 and 6.1, w ith vari ous shear-velocity-increase coefficients (C i). Introducing w ind speedup makes: a) dunes lower; b) dunes m igrate more quickly; and c) dune w indward slopes gentler, producing asym m etric dunes. For C\ = 0.4, w indw ard slopes w ith the angle of 10° are sustained. The dune did not vanish even w ith a higher shear- velocity-increase coefficient, say C\ = 0.6, after t = 1,000,000. Figure 6.4 shows asym m etric transverse dunes formed on a 2-dim ensional lattice from a random initial m orphology by introducing wind speedup. T he sim ulation tim e (i) is 1,500, and th e shear-velocity-increase coefficient (C i) is 0.4. T he resulting transverse dunes are th e m ost qualitatively sim ilar to n a tu ra l ones achieved yet.
W erner realised the problem of the sym m etric cross-sectional views of dunes simu lated w ith his original model (see Figure 5.2(a)), and th o u g h t “A possible remedy
h = 100 f = 100 Cl = 0.5 0.4 0.3 co X 0.2 0.1
S
i 3 z 100 525 675 425 475 575 D istance, /Figure 6.3: Evolution of an initially introduced, isolated transverse dune simulated with various shear-velocity-increase coefficients (C i). The dune's initial number of slabs above the surface is 30. The simulation time (t) is 1 0 0.
wind
Figure 6.4: Asymmetric transverse dunes formed from a random initial morphology by introducing wind speedup. The simulation time (t) is 1,500, and the shear-velocity-increase coefficient [ C\ ) is 0.4. Contour lines are for 100, 105, 110 and 115 slabs.
is m odifying th e tra n sp o rt rules to account for th e relationships am ong local m or phology, w ind profile, and tra n sp o rt flux in greater d etail” (W erner, 1995). His prediction is now found to be p artly correct. The asym m etry in th e cross-sectional dune shape is indeed an outcome of the interaction between local morphology, wind profile, an d tra n sp o rt flux, b u t can be form ulated w ith a very simple rule as in equation (5.9).
T he tim e evolution of a 1-dimensional dune field w ith th e shear-velocity-increase coefficient (C i) of 0.4 is shown in Figure 6.5. The wavelength of th e transverse dunes increases as tim e passes. For run tim es longer th a n th a t displayed in th e fig ure, th e equilibrium dune field seems to converge to only one isolated, asym m etric dune, which is surrounded by completely plane, bare ground. Consequently, th e dune-to-dune spacing is equal to th e lattice size, Lg = 1,024, m eaning th a t th e m odel is n o t adequate when the dune-to-dune spacing approaches th e lattice size. T he reason for th is equilibrium dune field configuration is th ou g h t to be related to sand trap p in g efficiency ( Te) , which is th e proportion of moving sand trap p ed
in th e dune slip face in the lee (Wilson, 1972; Cooke et al, 1993, pp346-347). A t equilibrium , th e sand flux coming to the dune m ust be th e sam e as th e outgoing flux (Zeman and Jensen, 1988; see also section 3.2.1). However, because the shadow zone extends more quickly th an th e slab tra n sp o rt length a t the crest of dune (L) increases as the dune grows higher, all th e slabs crossing th e dune crest will be cap tured in th e dune, m eaning the sand trap p in g efficiency ( Te) is equal
to 1.0, so th a t nothing regulates dune height. The wind speedup effect introduced w ith th e form of equation (5.9), seems to regulate only th e w indw ard slope angle.