5 6 Sand availability and reference num ber o f slabs correction

To correctly model th e effects of sand availability, which is tho u g h t to be th e second m ost im p o rtan t control of dune type (see section 2.3), it is necessary to define th e reference num ber of slabs (href) in equations (5.9) and (5.10) carefully. This is because th e wind structure over a dune is determ ined no t by th e height

0.50 T ? 0.40 • ■ Da = 0.25 mm r< £ 0.30 ■ ■ g₀₎ 0.20 ■ ■ c o

§

Figure 5.4: Relation between saltation length (A) and shear velocity (u*). Data are plotted following Anderson (1988). Other parameters, not specified in his paper but used here, are sand grain diameter {Dg) of 0.25 mm and lift-off angle of 50° (White and Schulz, 1977).

m easured from the hard ground beneath sand, b u t by the relative height measured from th e level ground, on which dunes sit. Such a ground level falls as dunes grow w ith tim e. T h e reference num ber of slabs (href) is th en defined here as

^ ( L s — l , L s — 1 )

I h{i,j',t) h a v g | j

href(^) — havg _{2L 2}

where Lg x Lg is th e lattice size. The second term of th e right-hand side of this equation expresses how m any slabs a t a site on average incorporate dune construction b o th erosionally and depositionally.

5.7

Introducing w ind directional change

As seen earlier in section 2.3, the prim ary factor th a t controls type and m obility of dunes is w ind directional variability (how w ind direction changes seasonally). Figure 5.5 shows how the lattice may be adjusted to m odel a dune field w ith bi- or m ulti-directional wind. W hen the wind is initially blowing parallel to the i

Winter

Figure 5.5: Modelling scheme for a dune field where th e wind changes di­ rection. T h e lattice is rotated in response to the wind directional change. After th e rotation, neighbouring sites are checked to see if they break th e angle-of-repose criterion.

be rotated by an angle 9 keeping the topography unchanged.

I = ( W w C O S 0 — ( j n e w s l n 0 -j-

j — ( î n e w sin 0 + ( j n e w COS ^ —

^ ( ^ n e w j j n e w ) ~ ^ ( ^ > j ) -

Consequently in the com putations, the wind always blows parallel to the i axis. After the rotation, it is checked to see if any neighbouring sites break the angle- of-repose criterion. The drawback of this scheme is th a t the num ber of slabs is not conserved. If the average number of slabs (/iavg) is small, the situation is worse. In the case of havg = 2.0 the difference between the initial and the final numbers of slabs after t = 1500 is about 10%. However, if we choose an optim al rotation (pivot) centre for each wind-direction change, which is usually different from ( L s / 2 , L s / 2 ) , such a difference in number of slabs can be reduced to the

order of 0.1% with m odest additional com putational time. This is a practical remedy. An ideal solution is to use an infinitely large lattice {i.e. Ls Too) to increase uniform ity in spatial distribution of dunes.

5.8

C onclusions in this chapter

A sim ulation model for a transverse-dune field was developed based on W erner’s m odel (1995). A portion of a large dune field is sim ulated on a two-dim ensional square lattice, whose edges are connected w ith periodic bou n d ary conditions. In th e model, dunes are the accum ulation of three-dim ensionally discrete ‘slabs’, whose m otion represents sand m otion in nature. There are two types of slab motion: w ind directional tran sp o rt and avalanche. Furtherm ore, based on obser­ vational facts, a shadow zone in a lee of a dune and bouncing p robability difference between bare and sandy surfaces were introduced (section 2.7.2; section 5.2).

A pplication to an infinitely deep sand field revealed th a t th e original W erner m odel cannot correctly sim ulate transverse dunes w ith d istin ct shape, or w ith asym m etric w ind-direction profile. Model analysis showed th e im portance of the shadow zone, w ithout which dunes cannot grow (section 5.3). Two revisions were m ade by considering the wind stru ctu re over a dune. From consideration about the separation zone in the lee of the dune, a no-erosion criterion in shadow zones was introduced (section 5.4). Noting th a t the shear velocity increases over the windw ard surface of a dune, a wind speedup effect was introduced by adding a height-proportional increase term to th e original slab tra n sp o rt length (Lq)

(section 5.5.1). Furtherm ore, by considering a non-linear dependence of shear ve­ locity on th e saltatio n length of a sand grain, a non-linear shear-velocity-increase term was introduced into the slab tra n sp o rt length (section 5.5.2). Sand avail­ ability and w ind directional variability, th e two m ost im p o rtan t controls of dune types, can now be studied w ith the introduction of reference num ber of slabs

C H A P T E R 6