48. Seguimiento de Incidentes de Commons Weaver
2.29. Ejemplo de Prueba Unitaria
In section 3.2.2, th e shear velocity a t th e dune crest (u*(0)) was linked to dune height, sand grain diam eter and shear velocity upw ind of th e dune (if, Dg, u*(—00)). The calculation in section 3.2.2 is perform ed only by estim ating th e sand trapping efficiency of th e dune. T he windward slope profile is unknown throughout the calculation, and to estim ate th e profile a discussion based on fluid dynamics is strictly necessary. For this inverse problem of predicting th e topography from the wind flow, th e boundary conditions on surface height and shear velocity profiles
{t){x) , u^{x)) are as follows: r}{—oo) = 0 ,7^(0) = f f ;u * ( —oo),u*(0) (Figure 3.5). A lthough a tte m p ts based on com putational fluid dynam ics have been m ade to calculate th e wind flow over a dune (most recently by Van Boxel et al, 1999; see also section 2.7.1), the flow is known to be described approxim ately by the (modified) Jackson-H unt theory (1975). The Jackson-H unt theory gives an ana lytical solution for th e wind flow over a dune, so th a t it can highlight th e physics more clearly. The applicability of th e Jackson-H unt theory to a barchan dune was investigated by W almsley and Howard (1985), using th e topographic d a ta of a real dune. They concluded th a t the theory is reliable for th e wind flow over the w indward slope of the dune (for details, see section 2.7.1). Since its presentation,
(m* (0), 11(0) = H,Dg)-^ q(+°°) with the grain scale model (Anderson, 1988, modified)
out-going flux estimation
.à Equilibrium condition:
%
r q ( - o o ) = ^(+oo)
U* = /o (m* (0), H,Dg) with a sand flux formula
{e.g. Lettau & Lettau, 1978)
Shear velocity calculation
[■ M*(0) = / ^ ( m . ( - o o ) , H,Dg) Boundary conditions:
(n,
mo1x=— = (0, u* (-oo)) lx=o =(H,u*(P)) 4 r Equilibrium profile: 'n W = fr \ (m* (-~ ), H,Dg) Inter-variable conversion Section 3.2Wind flow model:
(e.g. Jackson & Hunt, 1975)
Figure 3.5: Flow chart for dune modelling in the new approach examined in this thesis.
th e Jackson-H unt theory has been continuously revised (see for exam ple H unt et al, 1988; Weng et al, 1991; for reviews see Taylor et al, 1987; Belcher and H unt, 1998). T he m ost recent revision is th a t developed by W eng et al (1991), who also discuss th e applicability of th e theory to a barchan dune. However, the use of th e expression for th e surface shear stress derived by W eng et al is questionable, since it does not satisfy the sym m etry condition of th e Fourier transform (Weng
et al, 1991, equation (2.14a)).
In th e following, th e original Jackson-H unt theory (1975) is incorporated into the dune-m igration model developed in the previous section (3.2) for th e estim ation of the w indward slope profile from three quantities: dune height, sand grain
diam eter and shear velocity upwind of the dune { H, Dg , u ^ { —oo)). The Jackson- H unt theory considers a hill th a t has gentle windw ard and leeward slopes unlike a dune which has a slipface in th e lee. Nonetheless, th e theory is thought to be applicable to a dune w ith a few allowance for th e air flow p a tte rn over the dune, which is described in section 3.3,2 (see also section 2.7.1).
3.3.1
W indw ard slope profile o f barchan dunes in th e field
T he w indward slope angle of a barchan dune is typically said to range between 4-11° (Hesp and Hastings, 1998) or 2-10° (Lancaster, 1995, p52). The slope is concave a t the base, steepest in mid-slope and convex a t th e crest (Figure 3.1). T he relationship between the average windw ard slope angle (^wavg) dune height {H) for some dune flelds are p lotted in Figure 3.6, using fleld d a ta from Finkel (1959), Long and Sharp (1964), H asten rath (1967) and Sauerm ann et al
(2000). The average windward slope angles (^wavg) were calculated simply by dividing dune height {H) by th e w indward slope length (Lw) (see Figure 3.1). All sets of d a ta show in general th a t th e w indw ard slope angle (^wavg) increases as dune height ( H) increases.
Figure 3.6 shows th a t th e change in th e average w indw ard slope angle as dune height increases in th e d a ta of Long and Sharp (1964) is ab o u t h alf th a t shown by the d a ta of H asten rath (1967). In a uni-directional w ind regime, th e sand grain size and w ind speed are thought to be the two m ost im p o rtan t factors th a t govern dune dynam ics (section 3.2.2). H asten rath (1967) reported th a t in the dune field in southern Peru which he studied, th e sand was well sorted and its grain diam eter (Dg) was between 0.125 and 0.177 mm. In contrast, in the dune fleld near Salton Sea, California, the sand was poorly sorted w ith Dg = 0.250 mm (Long and Sharp, 1964, tab le 2). N either pap er provided details of the wind speed.
I
» g 15 10 - ■ 5 -■ A A A ■ ■ Finkel (Peru, 1959) aSauerm ann e t al. (Morocco, 2000) x
:
:
. — O oO X X o o ■_ ■ Hastenrath (Peru, 1967) Long & Sharp (California, 1964) ■ o
5 10
Dune height, H [m]
15
Figure 3.6: Measured data of dune height {H) and the average angle of windward slope (^wavg) taken from Finkel (1959), Long and Sharp (1964), Hastenrath (1967) and Sauermann e t al. (2000).
the grain diam eter (Dg) was between 0.208 and 0.420 m m, which gives a mean value of 0.314 mm. The threshold shear velocity (u*t) corresponding to 0.314 mm is calculated as 0.26 m s“ ^ from the Bagnold (1941) form ula for the threshold shear velocity (2.6):
Pa Pa
where A is a constant (% 0.1), ps is sand density and g is gravitational acceler ation. Finkel introduced a ‘shape facto r’ of 0.75, w ith o u t definition, which gave th e ‘effective particle diam eter’ of 0.24 mm, with u*t = 0.23 m s“ ^. Finkel also analysed th e wind d a ta recorded a t th e nearest m eteorological statio n, which was 33 km north east, and found th a t th e ‘equivalent effective w ind’ velocity could be calculated as 5.32 m s“ ^. Surface shear velocity over a fiat surface can be estim ated w ith th e K a rm a n /P ra n d tl relations (2.3):
Ps
5.75 lo g (^ ) ’
where is th e velocity at the height of m easurem ent (z) and zq is the roughness
length. Finkel did not m ention the m easurem ent height (z), although to analyse his field d a ta Finkel used B agnold’s (1941) theory in which the stan d ard height
is 1 m. If assum ing th e stan d ard height of 10 m, which was specified by the W orld M eteorological O rganization (Fryberger, 1979, p l4 3 ), together w ith the roughness length (z q) of 1 mm, the wind velocity of 5.32 m s“ ^ results in a
surface shear velocity (u*) of 0.23 m s~^, which is less th a n or equal to the threshold shear velocities estim ated above. As th e sand grain diam eter reported by H asten rath showed th e ‘good agreem ent’ w ith an o th er survey conducted by A m stutz and Chico in the same region (H astenrath, 1967, p311), this thesis utilises H a ste n ra th ’s data.
Since Sauerm ann et al. (2000) did not report any d a ta on sand grain size, grading or w ind speed, dunes in southern Morocco will not be discussed more in this thesis (D ata on wind direction and speed during 1956-1959 and sand grain diam eter in the same area are in Oulehri (1992).).
3.3.2
W ind-flow m odelling w ith th e Jackson-H unt theory
In the analysis developed by Jackson and H unt (1975), th e topography of a dune is characterised by two length scales: dune height {H) and th e half distance (L) between two sites whose height is half the dune height (see Figure 3.1). The surface profile of the topography (r){x)) is then expressed in the following non- dim ensional form:
r)(x) = H f { x / L )
= H m ,
where th e dune crest is set to rr = 0. In the Jackson-H unt theory, each quantity, for exam ple th e surface profile (/(O )? is given in th e form of Fourier transform (/(A;)), which is defined as
1 /*+o°
/ ( f ) = ^ y (3.35)
The Jackson-H unt theory is a linear p ertu rb atio n m ethod of analysis, where each q uantity is expanded about th e p ertu rb atio n variable (s ), which is much less th an
1.0. T he surface shear stress ( r (f)) is w ritten as
T(() = To(l + 67d(0), (3.36)
where tq is th e surface shear stress on th e flat surface far enough upwind of
th e dune an d Td(f) represents th e sp atial variation in th e surface shear stress. According to Jackson and H unt, the p ertu rb a tio n variable (e) can be calculated from th e following relation:
I " ' "
where k is von K arm an 's constant (~ 0.4), zq is th e roughness length, which is a function of shear velocity (see section 2.4.3), and I is defined as th e inner-layer thickness, which, according to Jackson and H unt, can be num erically calculated from th e following relation:
Y ln (— ) = 2k^ (3.38)
L Zq
Prom equation (3.37), equation (3.36) can be re-w ritten a t ( = 0 as
— ----— = Td(0) 6 Td(0) y .
To L
Accordingly, if the windward slope is height-independent { H / L = const.) ^ the norm alised increase in th e shear stress at th e dune crest ( ( t (0) —ro)/ro) is expected to be alm ost constant.
St am (1996, 1997) developed a two-dim ensional dune m odel combining a linear approxim ation of B agnold’s sand tra n sp o rt form ula (2.5) and th e Jackson-H unt theory (see section 2.6.1). According to St am (1996), th e Fourier transform of the surface shear stress (Td(0 ):
1 r+oo
Td( 0 = ^
J
Td{^)e"^^dk (3.39) can be approxim ated aswhere i =
0 = + f ( ^ > 0)
= ~ j < 0). (3.41) and
z{k) = 2W y|Â :|. (3.42)
In the lee of th e slipface of the dune, a reverse flow occurs, which the Jackson- H unt theory cannot model. The leeward wind flow is very difficult to calculate correctly even using a com putational fluid model (Van Boxel et al , 1999). A practical rem edy is to define the separation zone downwind of th e slipface, and calculate th e wind flow over th e topography comprised of th e windw ard surface and this separation zone (Zeman and Jensen, 1988; Van Dijk et al , 1999). This thesis adopts a similar treatm ent. The wind flow over a dune w ith a slipface is replaced w ith th a t over a sym m etrical gentle topography, whose w indward surface represents th a t of the dune in consideration. The following subsections consider a cosine and G aussian function to represent the w indw ard slope of the dune, which changes from concave to convex between th e base and th e crest (Figures 3.1 and 3.7). One m erit of using these functions is th a t th e Fourier transform of th e surface profiles {f{k)) can be analytically solved.
T he w indw ard slope length (Lw), which will be estim ated in the present paper, is defined as double the horizontal length scale (2L ). T his definition is precise in th e case of a cosine hill, and acceptable in th e case of a G aussian hill because
T]{2L) = 0 m H .
T he shear velocity ratio (u*(0)/u*(—oo)) can be estim ated for a given set of dune height, sand grain diam eter and shear velocity upwind of th e dune {H, Dg, u*(—oo)) (section 3.2.2; Figure 3.5). Since shear velocity (u*) and shear stress (r) are linked by equation (2.4):
where is air density, from equation (3.36), th e shear velocity ratio can be w ritten as
u.{0)
u ^ { - o o ) — \ / l + e Td(0). (3.43)