4.3 CALCULO DIMENSIONAMIENTO Y SELECCION DE ELEMENTOS
4.3.2 SISTEMA DE SUJECION DE LAS CARCAZAS
linguoid
(low)
barchanoid
(high)
\
crest
>
+xI avalanche
slip face
bottom
/
Figure 4 .16: Schematic plan view of the transverse dune system comprised of barchanoid and linguoid parts. Barchanoid parts are higher th an linguoid parts {H{B) > H{L)). In this configuration of barchanoid/linguoid sequence in the wind direction, windward slopes of barchanoid parts are shorter than those of linguoid parts ( L w ( 5 ) <
L ^ L ) ) .
decrease the m igration speed of linguoid parts, which is another mechanism th a t enables dunes to be a t equilibrium.
Extending equations (3.7) and (3.2), and by considering the symmetries at the apexes of barchanoid and linguoid parts, the dynamics of barchanoid(B )/linguoid
(L) p attern can be described as
q(0;B) = q{- ^oo]B) / ( l - T e( B ) )
ç(0;L ) = q { +oo] L) / ( l - T e{ L) )
ç(+oo; 5 ) + ç ( + c x d ; L) = q( - oo; B ) q ( - o o ; L),
enhanced by wind diversion. In a b archanoid(B )/linguoid(L ) downwind sequence as in Figure 4.16, we can expect
ç ( H - o o ; B ) ~ ç ( - o o ; L)
g (+ o o ;L ) ~ q{—c o \ B) .
A barchan dune can be considered as th e special type of sinuous transverse dune, where the linguoid height is zero.
In this section, th e w ind-directional profile of barchan dunes has been discussed. T he full m odel th a t incorporates th e Jackson-H unt theory was found to be capable of explaining why th e w indward slope of higher dunes are steeper. By extending th e model to suit sinuous transverse dunes, th e m odel m ay also be able to explain why barchan dunes have a height-independent cross section, a t right angles to th e wind direction.
4.3
Shape and m igration speed o f proto-dunes
^Despite th eir simple shapes and form ative environm ent, th e in itiatio n process of barchan dunes is not well understood. Some barchan dunes are th o u g h t to have been formed in th e dense stream of sand downwind of th e end of a preceding dune (Cooke et al , 1993, p323). A lternatively a very low and gently-sloping sand m ound (known as proto-dune, sand patch or ephem eral dune) is also thought, by some, to be th e in itial stage of a barchan dune (Lancaster, 1996; Cooke et al, 1993, p325). According to Lancaster (1996), typical proto-dunes, observed in a dune field near G obabeb, Nam ibia, are 0.05—0.10 m high. T heir average w ind-directional length and w idth are 12.71 m and 8.14 m , respectively.
The fact th a t even proto-dunes have lengths which are com parable to those of m atu re dunes m ay have significance. Dune fields show hierarchical structures
from small to large: ripples, dunes and mega dunes (W ilson’s draas), (Wilson, 1972; Livingstone and W arren, 1996, p79). Ripples occur on m ost bare sand surfaces, including dunes, and in m any areas dunes are superim posed on mega dunes. W ilson th o u g h t th a t these th ree features co-existed in quasi-equilibrium , and ripples did no t grow into dunes, nor dunes into mega dunes. This leads to th e conclusion th a t th e nuclei of m ega dunes m ust be larger th a n fully grown dunes, of which nuclei m ust be larger th a n fully grown ripples (see section 2.1, which includes W arren and A llison’s (1998) different idea).
In the following, we a tte m p t to explain th a t proto-dunes can be seen as th e lower lim it of barchan dunes in respect b o th to their profile (cross-sectional shape in th e wind direction) and to th eir m igration speed.
In chapter 3, a m odel for th e m igration speed (cj) of a m atu re barchan dune, which has a slip face in th e lee, was developed (section 3.2). W ith sup p o rt of field studies, it predicts th e following approxim ate form ula (4.1):
br
Cd — flcd + ^ 5
where H is dune height, Ucd ^tnd bc^ are positive constants. Strictly, th e logarith mic relation:
Cd = ÛL - &L log H,
where o l and bj, are positive constants, may be a b e tte r approxim ation (see Figure 4.6). W hen p lo ttin g field d a ta in the Cd — plane, and applying linear regression analysis, coefficients and bc^ correspond to th e ^/-intercept and th e slope, respectively (see section 4.1.1). Further incorporation of th e wind-fiow theory (Jackson and H unt, 1975) allowed th e m odel to be developed such th a t it predicted the w indw ard slope profile (see section 3.3).
Figure 4.17 shows th e calculated relation between th e w indw ard slope length (L^) and dune height ( H) for three shear velocities on a level surface (u*(—oo)). The sand grain diam eter (Dg) is taken as 0.25 mm. These values are adjusted to those in the dune field near G obabeb, N am ibia (Lancaster, 1996). T he roughness
length (zo) is calculated w ith Owen’s formula (2.8). T he n a tu ra l extrapolation of th e curve for ?/*(—oo) = 0.35 m s~^ is 7.8 m, which is close to th e average half length of a proto-dune m easured by Lancaster (1996), which is 12.7/2 = 6.4 m. Now we consider a dune th a t m igrates downwind { +x direction) a t a constant speed (cd) w ithout changing its shape. The dune crest, height 77, is set to T = 0. For m igrating shape-invariant dunes, equation (3.5) holds:
q{0) - q { - o o ) = 7Cd77,
where q{x) is sand flux and 7 is th e sand bulk density in th e dune (for derivation, see section 3.2.1),
In the analysis of wind flow developed by Jackson and H unt (1975), th e dune is characterised w ith two length scales: dune height (77) and th e h alf distance (L) between two sites whose height is half th e dune height. L can be related to half th e w indward surface length (7/^/2) (section 3.3.2). According to Jackson and H unt (1975), th e increase of shear velocity a t the crest:
(^u*(0) = u*(0) - u*(oo) can be related to the average dune slope:
5«.(0) ~ (4.2)
Jjyf
A lthough sand flux (g) is proportional to cube of shear velocity (u*), if th e shear velocity increase {5u^) is small enough, the sand flux (q) can be linearised w ith regard to th e shear velocity increase (6u*), so th a t,
q(0) - q ( - o o ) rv âu„(0). (4.3) From equations (4.2) and (4.3),
g(0) - q ( - o o ) ^ (4.4) i>w
S ub stitu tin g equation (4.4) into equation (3.5), we get
Cd ~ (4.5)
0.35 m s y = -4E-15x^ - 0.0455x2 + 2.5x + T 8 R2 = 0.9985 -1 30 ■■ 0.30 m s
f
I
M 20 ■ ■ >g = 0.250 mm *, = 0.23 m s"* 8 10 0 2 4 6 D une height, H [m]Figure 4 .1 7 : The calculated relation between the windward slope length (Lw) and dune height ( H) for various shear velocities on the level surface
oo)). The sand grain diameter {Dg) is taken as 0.25 mm and the roughness length (zq) is calculated with Owen’s formula (2.8).
Dune m igration speed (cd) is usually discussed in respect to th e dune height (if). E quation (4.5), however, implies th a t for low dunes th e w ind directional length (LpD = 2 Lw) m ay be a b e tte r index.
Since a proto-dune, whose height (if) is nearly zero, has a length (Lpd) of more th a n 10 m, from equation (4.5), dune m igration speed (cd) m ust approach a finite value as dune height (H) approaches zero. The m igration speeds (cd) of a proto dune observed in th e dune field near Gobabeb, N am ibia were ab o u t 1.0—1.4 m day“ ^ = 365—511 m year“ ^ when winds were strong (L ancaster, 1996). This may be regarded as a finite value, since according to Cooke et al. (1993, figure 23.24), th e m igration speed of 5-m high dunes is 10—60 m year“ ^.
Figure 4.18 shows th e calculated relation between dune m igration speed (cd) and dune height (if ) for three shear velocities on a level surface (u*(—oo)). A sand grain diam eter (jDg) of 0.25 mm, the same value as in Figure 4.17, is assumed. For th e bulk sand density (7), a figure of 1670 kg m ” ^ is used (see section 4.2). The n a tu ra l extrapolations of th e curves in this figure are 135, 295 and 423 m year“ ^
for th e shear velocity on a level surface (u*(—oo)) of 0.25, 0.30 and 0.35 m s“ ^, respectively. T he last estim ation is com parable to field d a ta referred to above, in which m igration speeds were 365—511 m year“ ^ when winds were strong. Figures 4.17 and 4.18 im ply th a t the present model, which was developed by assum ing a slip face, m ay be a good predictor even for dunes w ith o u t slip face.
According to equation (4.1), however, the dune m igration speed (cj) tends to infinity as dune height {H) approaches zero. G eneralising equation (4.1), we therefore propose th e revised approxim ation formula:
Cd = O^Cd + rr / 5 ('^•^)
i l +Ccd
where is also a positive constant. C onstants and can be determ ined from field d a ta in th e same m anner as in equation (4.1). T he constant can be determ ined by tak in g the m igration speed of a proto-dune { H = 0) as Ucd+^Cd/ccd- E quation (4.6) produces a finite value for dune m igration speed (cd) as the dune height {H) approaches zero. Furtherm ore, the inverse of equation (4.6) resembles th e curves in Figure 4.17, which is consistent w ith equation (4.5).
For fu rth er analysis, more field studies on shape and m igration speed of bo th proto-dunes and m atu re barchan dunes are necessary.
4.4
C onclusions in this chapter
In th is chapter, th e single-dune model which was developed in th e previous chap ter was evaluated (as in th e flow chart in Figure 3.5).
In section 4.1, th e basic model was discussed w ith num erical calculations, before incorporating wind-fiow theory. For low shear velocities a t th e dune crest, such as u*(0) = 0.4 m s " \ sand trap p ing efficiency (Te) rapidly increases as dune height increases. However as u*(0) increases further, th e ra te of increase of Te
„ 500 T Ê 400 ■ • U'{— ««) = 0.35 rn s S « 300 S' § % 200 O) Ê g 100 3 Q 0.30 m s 0.25 m s y = -0.4355x^ + 9.938x^ - 83.23x + 422.92 = 0.9973 y = -0.2347x^ + 5.5179x^ - 49.206x + 295 = 0.9985 y = -0.0012x^ + 0.3403x^ - 8.6777x + 134.63 = 0.9996 Dg = 0.250 mm u*t = 0.23 m s ’’ porosity: 37% 4 6 Dune height, H [m] 10
Figure 4 .1 8 : The calculated relation between dune migration speed (cd) and dune height ( H) for some shear velocities on a level surface
oo)). Sand grain diameter {Dg) is assumed to be 0.25 mm.
even for dunes higher th a n 150 m. Prom th e assum ption of equilibrium , where Te = 0.0, there is no w ind speedup, hence no dune m igration. A lthough sand trapp in g efficiency (Te) depends on dune height. Te is m ainly determ ined by shear velocity on a level surface (u*(—oo)), and rapidly decreases as u*(—oo) increases. For each dune height, dune m igration speed first increases, and then decreases m onotonically after reaching the m axim um , as shear velocity on a level surface (u*(—oo)) increases. The relation between shear velocity a t th e dune crest (u*(0)) and dune height has been estim ated for a given shear velocity on a level surface (u#(—oo)). D une m igration speed (cd) is no t inversely proportional to dune height as was previously thought. For low dunes, sm all sand trap p in g efficiency (T e) suppresses Cd, whereas for high dunes, wind speedup and large Te resist th e decrease of Cd. Some field d a ta show th e same tendency (section 4.1.1). It has been suggested th a t th e dune-to-plane-bed tra n sitio n observed in subaqueous and Venusian bedform s could be associated w ith th e decrease of sand trap p in g efficiency (section 4.1.2).
In section 4.2, th e average windward slope angles of barchan dunes have been estim ated w ith th e Jackson-H unt theory (1975). W ind flow over a dune was re
placed by th a t over either a cosine hill or G aussian hill. T he m odel has succeeded in explaining th e above relation between dune height and th e w indw ard slope, as a consequence of the balance between shear velocity increase over th e windward surface and the sand trap p in g efficiency, since the dune is a t equilibrium . Fur therm ore th e calculations show th e following. There is no significant difference between results estim ated w ith the roughness length from th e Owen form ula and th a t of 1 m m (section 4.2.1). There is no significant difference between results estim ated w ith a cosine hill and a G aussian hill (section 4.2.1). As the surface shear velocity on a fiat surface increases, th e average w indw ard slope of dunes decreases (section 4.2.1). By assuming a shear velocity on th e level surface of 0.22 m s“ ^, d a ta for the w indward slope angle and m igration speed of dunes in south ern Peru, where sand grain diam eter is ab o u t 0.150 m m , fit th e m odel (section 4.2.2). The m odel implies th a t the upper-lim it in dune height m ay be higher in windier environm ents (section 4.2.2). W ith sm all shear velocities th a t are close to the threshold value, and small dune heights, dune m igration speed increases w ith dune height. These positive relations m ay associate w ith zibar parabolic dunes (section 4.2.2). W ith sand grain diam eter of 0.250 mm, which was chosen by con sidering table 2 in Long and Sharp (1964), windward slope angle and m igration speed of dunes in Galifornia cannot be explained a t th e sam e tim e. Field d a ta of w indward slope angle can be approxim ated w ith good accuracy w ith a shear velocity on th e level surface of 0.55 m s~^, b u t this selection of shear velocity results in m igration speed more th a n ten tim es larger th a n shown in field d a ta (section 4.2.3). This has yet to be resolved.
In section 4.3, a short discussion was m ade on proto-dunes. B oth th e finite wind- directional length and m igration speed of proto-dunes could be estim ated w ith th e model and are shown to be a t th e lower lim it of barchan dunes. This implies th a t th e present model, which was developed by assum ing a slip face, may be a good predictor even for dunes w ithout a slip face.