0 = 32windward surface

(Xp^, - H )

Figure 3.4: Model and co-ordinate system used in calculation of sand trapping efficiency (Te).

having originated in the upwind window of the dune crest [x', x' + dx'] is

[Nxo I x'] dxo = No(x') P[X = (xo - x')] dx'dxo

[A88, (6)], where P[X = {x q- x')] = vIq) QXYi{—{xQ — x' Y^^/ vIq} , (3.12) c = 0.36 (Dg/0.25)-'^'^ (u ,/0 .5 )" '^ d = 2.0 (u ,/0 .5 )° \ (3.13) (3.14) and Dg is sand grain diam eter in mm, it* is shear velocity in m s“ , and vIq

is m ean grain lift-off velocity in m s~^. These three equations correspond to [A88, (7); corrected], [A88, (3); a p a rt of] and [A88, (3); a p a rt of]^. vIq can be derived through th e calculation of m ean vertical lift-off velocity, wIq (Anderson

^Equation (3.12) can be derived from the following two relations:

P[wlo] ~ exp[-wZo/ < Wlo >mean] and

X = c vIq ,

which are [A88, (5); modified] and [A88, (3); a part of], respectively. In addition, P[X] ~

P[vlo] ~ P[wlo] and vIq ~ wIq are assumed, neither being described explicitly in Anderson’s (1988) paper.

and H allet, 1986):

wIq = 0.64 w* (0.25/Dg)^'^, (3.15) this being [ASS, (5b)] in A nderson’s paper. According to W hite and Schulz (1977)

vIq = wIq/ sin (50°).

In th e following argum ent shear velocity (u*) in (3.13), (3.14) and (3.15) is ap­ proxim ated w ith th a t a t th e crest (u*(0)). Equation (3.12) is different from th a t introduced by Anderson (19SS) by the factor of 1/d. It is because the condition:

f+OO

/ P(A) dX = 1

Jo

should be satisfied, th a t th e 1 / d factor is necessary.

The to ta l num ber of grains crossing all downwind windows will th en be the sum over all possible upwind origin sites, yielding the integral:

N ^ o = f N^{x' ) P[X = { x o - x ' ) ] d x ' (3.16)

J —oo

[ASS, (S)]. A ssum ing th a t th e surface ju st upwind of th e crest is gentle enough so th a t Nq{x') = Nq = const.^ and su bstitu tin g (3.12) into (3.16), we get

Nxo = No exp (-T o (3.17)

[ASS, (9); corrected]. Equation (3.17) is different from th a t derived by Anderson (19SS), because of th e factor introduced above.

O f all th e sand grains crossing the z = 0 plane, only those crossing through the window [0, xqc] are assumed to be captured on the slip face, hence contributing to sand trap p in g efficiency (Te) . Integrating ( 3 . 1 7 ) , we can calculate all those grains crossing through th e window [0, Xqc]

rxoc rxoc

/ N x o d x o = / A^o ex p (-T o dTQ.

Jo Jo

The ratio of those sand grains to all sand grains crossing th e dune crest, i.e. the sand tra p p in g efficiency (Te) is:

pXQc P + OO

Te = / Na:odxol / Nxodxo

Jo Jo

rxQc n+oo

=

/

exp(-To

uZo)dTo / /

exp(-To

uZo)dTo.

Jo Jo

T he denom inator of (3.18) can now be more analytically expressed. Replacing = G I and 1/ d = G2, and then using th e gam m a function formula:

/'+00

/ e x p ( - G i = r ( l / G 2) /{ G2 G i

Jo

we get

th e denom inator of (3.18) = T { d ) / { { l / d ) (3.19) S u b stitu tin g (3.19) into (3.18), we get th e analytical expression of sand trapping efficiency (T e) w ith regard to xoc-

rxoc r+00 T e = / N ^odxo/ / Na;odxo Jo Jo rxoc = / exp{—Xo^^ ^ / c ^^ ^ v l o) dx o/ r { d) dcv l o^ . (3.20) Jo

T he relation between dune height (H) and the corresponding integral lim it (a:oc) in equation (3.20) can now be calculated. Given th a t sand grains enter the lee region a t a point xq in the z = 0 plane w ith a speed vq and a t an angle w ith the

horizontal of ai, they land a t a site (a;p, Zp) on th e slip face, which is described by

Xp{t) = To 4- [fo cos ai\t, (3.21) Zp(t) = - [ f o s in « i ] t - ( 1 / 2 ) ( 3 . 2 2 ) where t is tim e in s. Since (xp, Zp) is on the slip face,

Zp{t) = —Tp(t) ta n 9. (3.23) Solving (3.21), (3.22), and (3.23), we get

Xp{t) = Xq + {(1 — 7 ) / ( 2 / 3 ) } + [{(1 - 7) /(2^)}^ + X q / (3.24)

where

7 = ta n Ofi / ta n 6, (3.25)

0 = g / { 2 cos^Q!i ta n 9). (3.26) E quation (3.24), (3.25), and (3.26) are different from th e corresponding expres­ sions derived by A nderson (1988), because of th e different co-ordinate system. Since Xpc (xp corresponding to xqc) is th e bo tto m of th e slip face,

S u b stitu tin g (3.27) into (3.24), we get th e relationship between xqc and I f : H / ta n 9 = xqc + {(1 ~ ? ) /( 2 /5)} + [{(1 — 7) /( 2 /3)}^ + x ^ d (3.28) Combining (3.20) and (3.28), we could draw the curves for th e relation between sand trap p in g efficiency (Te) and dune height [H). The only problem rem aining is th e derivation of sand grain speed a t {x q cZo) (uq).

The distrib u tio n of vq has its m axim um probability w ith th e sm allest value of

uo = u*, and decays alm ost exponentially (Anderson, 1988). This sm allest and m ost probable speed (u*) corresponds to th e shortest saltatio n tra je cto ry between th e crest and xqc- Since sand grains are integrated w ith respect to th eir landing points, this distribution has an influence only close to th e b o tto m of the slip face (xp). Considering (3.17), this effect is not large. All grains coming to xqc are assum ed to have been ejected a t th e crest, giving them th e same speed (u*), which is th e sm allest and m ost probable one because it corresponds to the shortest saltatio n trajectory. According to Anderson (1988),

u* = a x o d [= Vo in (3.26)], (3.29) where

o = 5.8 (u*(0)/0.5)^-^ (jDg/0.25)-°-2, (3.30)

6 = 0.4 8 (u * (0 )/0 .5 )°-l (3.31) These three equations correspond to [A88, (2)].

In summary, sand trap pin g efficiency (Te) can be described as a function of shear velocity a t the dune crest (u*(0)), dune height {H) and sand grain diam eter (Dg):

T E = T E o ( u * ( 0 ) , i 7 , D g ) . ( 3 . 3 2 )

If an equilibrium shape is assumed (as discussed in section 3.2.1) shear velocity a t th e dune crest (u*(0)) is related to th a t on a level surface (u*(—00)) by equation (3.6) and, for example, L ettau and L e tta u ’s sand flux form ula (2.7):

ç ( u ,( - o o ) ) ç(u*(0)) —

In such conditions, sand trap p in g efficiency (Te) can be rephrased as a function of shear velocity on a level surface (u*(—oo)), dune height {H) and sand grain diam eter {Dg):

Te = Te_^ (u* ( -00), i f , T>g). (3.33) Consequently, dune m igration speed (cd) is described as

c { u , { - o o ) , H, D, ) =

Note th a t th e discussion developed above does not assum e any p articu lar wind­ ward surface shape, except th a t the surface ju s t upw ind of th e crest is horizontal for a long enough distance.