<< . .

. 11
( : 51)



. . >>

depositional landforms have complex internal
(a)
processes that are not captured by the diffusion
model. For example, deposition in deltas and al-
luvial fans takes place by a series of avulsion
h
events. At short time scales, deposition is local-
ized along one or more lobes that are active un- t2
t1
til the lobe becomes superelevated with respect
to the surrounding terrain. A large ¬‚ood can
cause the lobe to shift rapidly to another portion
of the fan or delta. If the diffusion equation ap-
plies at all to the large-scale evolution of these
x
systems, it applies only at a time scale that av-
1.0
erages the effects of many avulsion events. Sec- (b)
ond, variations in sediment texture can produce
0.8
spatial variations in sediment mobility. The sedi-
ment texture in the proximal portion of an allu-
vial fan, for example, can be one to two orders of 0.6
h
magnitude larger than sediment in the distal re- h0
gion (i.e. boulders compared to sand). Variations 0.4
in channel morphology (and hence ¬‚ow depth)
and sediment texture both combine to create spa-
0.2
tial variations in the effective diffusivity that are
not well constrained. 0.0
2.0
1.0 3.0
0.0
Kenyon and Turcotte (1985) applied the 1D 4.0
U0
(x ’ U0 t)
diffusion equation to model the progradation of
k
a single delta lobe. Their model does not ap-
ply to the long-term 3D morphology of deltas. Fig 2.16 (a) Schematic diagram of the 2D model for a
Here we review their approach and propose a prograding delta lobe in the Kenyon and Turcotte (1985)
model for the 3D evolution of deltas over geologic model. (b) Solution for the pro¬le shape of a prograding delta
time scales. Both models are limited in that they (Eq. (2.61)).
neglect the ¬‚exural--isostatic subsidence of the
basin in response to sediment loading. This lim-
diffusion equation gives
itation will be corrected in Chapter 5 where we
consider the geometry of foreland basins within d2 h
dh
’U 0 =κ 2 (2.59)
the context of a model that couples the diffusion dξ dξ
equation to ¬‚exural subsidence.
Equation (2.59) must be solved with boundary
The Kenyon and Turcotte model assumes that
conditions h(0) = h 0 and h(∞) = 0. The solution is
sediment is introduced to the basin at an eleva-
tion h 0 above the basin ¬‚oor. As the basin ¬lls, the U0ξ
h(ξ ) = h 0 e’ (2.60)
κ
position of the river mouth migrates towards the
basin at a rate U 0 . Generally speaking, solving dif- Substituting Eq. (2.58) into Eq. (2.60) gives
fusion problems with moving boundaries poses a
U0
considerable challenge. In this case, however, the h(x, t) = h 0 e’ (x’U 0 t)
(2.61)
κ

moving boundary can be handled by introducing
a new variable ξ given by The solution to this model is given in Figure 2.16.
The solution is an exponential function with a
ξ = x ’ U 0t (2.58)
width controlled by a balance between diffusion
The sediment source is ¬xed in this new coor- (which transport sediment away from the source)
dinate system. Substitution of Eq. (2.58) into the and the movement of the source. This solution
2.2 ANALYTIC METHODS AND APPLICATIONS 53


does a good job of matching the observed stratig- 1.0 kt =
raphy of the southwest pass segment of the Mis-
104 km2
sissippi River delta (Turcotte and Schubert, 2002). 2 — 104 km2
0.8
More generally, however, the exponential form of 4 — 104 km2
Eq. (2.61) is inconsistent with the generally sig- 8 — 104 km2
0.6
h
moidal nature of deltaic deposits.
h0
Over geologic time scales delta lobes will
0.4
avulse and a radial, 3D landform will be cre-
ated. In order to model this 3D development, we
consider the diffusion equation in polar coordi- 0.2
nates (Eq. (2.52)). We also assume a ¬xed sedi-
0.0
ment source (in contrast to the moving sedi-
0 100 200
r (km)
ment source of Kenyon and Turcotte) and a spa-
tially variable diffusivity that is greatest near the
Fig 2.17 Pro¬les of the 3D radially symmetric model of
river mouth and decreases with distance from
delta formation (Equation (2.67)).
the mouth as 1/r :
κ
κ(r ) = (2.62)
r deposits, which is more consistent than the
This spatially variable diffusivity represents the exponential form of the Kenyon and Turcotte
loss of transport capacity as rivers reach the model.
ocean. The speci¬c 1/r dependence we have
assumed is not well constrained, but it pro-
2.2.12 Dust deposition downwind
vides a useful starting point for representing the
of playas
decrease in sediment mobility as distance from
The dust cycle in arid environments is character-
the shoreline increases.
ized by a net eolian transfer of dust from playas
The 3D diffusion equation with radial symme-
to piedmonts (Pye, 1987). Understanding the pro-
try is given by Eq. (2.52). Substituting Eq. (2.62)
cesses and rates of this transfer is important for
into Eq. (2.52) gives
many basic and applied geologic problems. Dust
‚h κ ‚ 2h transport in the atmosphere also provides a nice
= (2.63)
‚t r ‚r 2 illustration of diffusion acting in concert with
advection. In this section we describe the phe-
Equation (2.63) can be reduced to an ordinary dif-
nomenology of dust deposition in a well-studied
ferential equation by introducing the similarity
¬eld site in California, and then apply an analytic
variable
solution for the advection, diffusion, and gravi-
r3
·= (2.64) tational settling of dust from complex, spatially-
9κt
distributed sources, following the work of
In terms of ·, Eq. (2.63) becomes Pelletier and Cook (2005).
d2 θ The study area is located in southern Amar-

’ = (2.65)
gosa Valley, California, where Franklin Lake Playa
d·2

abuts the Eagle Mountain piedmont (Figure
where θ = h/ h 0 . The solution to this equation is
2.18a). The water table in Franklin Lake Playa is
θ = e’· less than 3 m below the surface (Czarnecki, 1997).
(2.66)
Czarnecki identi¬ed several distinct geomorphic
Transforming back to the original coordinates: surfaces on Franklin Lake Playa that can be read-
ily identi¬ed in LANDSAT imagery (Figure 2.18a).
r3
h(r, t) = h 0 e’ 9κt (2.67)
For the purposes of numerical modeling, the ac-
Figure 2.17 presents plots of Eq. (2.67) for tive portion of the playa was mapped based on
different values of κt. The model predicts the Czarnecki™s map of playa surfaces with signi¬-
classic sigmoidal (S-shaped) geometry of deltaic cant dust-emitting potential (Figure 2.18a).
54 THE DIFFUSION EQUATION



(a) (b)
116°W
117°W
Yucca
Mtn.
Eagle
Mtn.
Nevada

Ne Amargosa
va Valley
da
Ca
Funeral
36.5°N
lif
Mtns.
or
ni Eagle
a
Mtn.
Death
Qa2
Valley
Qa3
Qa4
Black
Qa5“Qa7
Mtns.
N 10 km
36.0°N predominant
Soil-geomorphic map and playa
N
sample-pit locations wind direction pits



(c)
outline of active
wind
modern playa
direction
Eagle
3“6 m/s
calm
Mtn.
6“9 m/s
90%

Franklin Lake
Playa
1%


2%


3%


Amargosa Eagle
River Mtn.
secondary hotspot



1 km
10 20 40
0 80 cm
Silt thickness on Qa3


Fig 2.18 (a) Location map and LANDSAT image of Eagle conditions (see wind-rose diagram in Figure
Mountain piedmont and adjacent Franklin Lake Playa,
2.18a). Silt-rich eolian deposits occur directly un-
southern Amargosa Valley, California. Predominant wind
derneath the desert pavements of Eagle Moun-
direction is SSE, as shown by the wind-rose diagram (adapted
tain piedmont, varying in thickness from 0 to
from January 2003“January 2005 data from Western Regional
80 cm based on soil-pit measurements. The tech-
Climate Center, 2005). Calm winds are de¬ned to be those
nical term for these deposits is cumulic eolian
less than 3 m/s. (b) Soil-geomorphic map and oblique aerial
epipedons (McFadden et al., 2005), but here we
perspective of Eagle Mountain piedmont, looking southeast.
Terrace map units are based on the regional classi¬cation by refer to them as eolian or silt layers for sim-
Whitney et al. (2004). Approximate ages: Qa2 “ middle plicity. These layers are predominantly composed
Pleistocene, Qa3 “ middle to late Pleistocene, Qa4 “ late of silt but also include some ¬ne sand and sol-
Pleistocene, Qa5“Qa7 “ latest Pleistocene to active. (c) Map
uble salts. The homogeneity of these deposits,
of eolian silt thickness on Qa3 (middle to late Pleistocene)
combined with their rapid transition to gravelly
surface, showing maximum thicknesses of 80 cm close to the
alluvial-fan deposits below, makes the layer thick-
playa source, decreasing by approximately a factor of 2 for
ness a reasonable proxy for the total dust content
each 1 km downwind. Far from the playa, background values
of the soil.
of approximately 20 cm were observed. For color version,
Alluvial-fan terraces on Eagle Mountain pied-
see plate section. Modi¬ed from Pelletier and Cook (2005).
mont have a range of ages. Eagle Mountain pied-
mont exhibits the classic sequence of Quaternary
alluvial-fan terraces widely recognized in the
The Eagle Mountain piedmont acts as the
southwestern United States (Bull, 1991). Mapping
depositional substrate for dust emitted from
and correlation of the units was made based on
Franklin Lake Playa under northerly wind
2.2 ANALYTIC METHODS AND APPLICATIONS 55


the regional chronology of Whitney et al. (2004). material deposited over time) and that Qa4 sur-
In this example, we focus on silt thicknesses of faces have received a higher average dust ¬‚ux
the Qa3 terrace unit (middle to late Pleistocene) because they have existed primarily during the
because of its extensive preservation and lim- warm, dry Holocene. The Qa5 unit (latest Pleis-
ited evidence of hillslope erosion. Silt thicknesses tocene) exhibited uniformly thin deposits under-
were measured at locations with undisturbed, lying a weak pavement, independent of distance
planar terrace remnants to the greatest extent from the playa, suggesting that the trapping abil-
possible. A color map of silt layer thickness for ity of young surfaces is limited by weak pavement
this surface is shown in Plate 2.18b, draped over development.
the US Geological Survey (USGS) 30 m digital el- As discussed in Chapter 1, atmospheric trans-
evation model and an orthophotograph of the port of particulate matter can be modeled as a
area. Silt thickness is observed to decrease by combination of turbulent diffusion, downwind
roughly a factor of 2 for every 1 km of distance advection, and gravitational settling. The 3D con-
from the playa. Several kilometers downwind, a centration ¬eld for a point source located at
(x , y , 0), obtained by solving Eqs. (1.17) and (1.18),
background thickness of 15--20 cm was observed
on Qa3 surfaces. The strongly localized nature of is given by


exp ’ 4K (x’x )
uz
u(y ’ y ) p2 (x ’ x )
2
Q p pz

c p (x, y, z, x , y ) = exp ’ ’ +
exp
4K (x ’ x ) πuK (x ’ x ) uK K Ku
4π K (x’x )
u
¤
x’x
u ¦
—erfc z+ p (2.68)
4K (x ’ x ) Ku


where Q is the source emission rate and erfc
downwind deposition in this area suggests that
is the complementary error function. Equation
Franklin Lake Playa is the source for nearly all
(2.68) combines the two-dimensional solution of
of the eolian deposition on Eagle Mountain pied-
Smith (1962) with an additional term required to
mont. Localized deposition also implies that dust
describe crosswind transport (Huang, 1999). Equa-
deposition rates may vary regionally by an order
tion (2.68) assumes that the settling velocity q
of magnitude or more, down to spatial scales of
is small compared with the deposition velocity
1 km or less.
p. Stokes™ Law (Allen, 1997) implies a settling ve-
Figure 2.19a illustrates plots of silt thickness
locity of less than 1 cm/s for silt particles (i.e.
as a function of downwind distance, for compar-
particles less than 0.05 mm) in air. Deposition
ison with two-dimensional model results. This
velocities consistent with the spatial distribu-
plot includes all measurements collected from
tion of deposition on Eagle Mountain piedmont,
terrace units late Pleistocene in age or older
however, are approximately 5 cm/s, or at least
(Qa4--Qa2) within a 1 km wide swath of western
¬ve times greater than q. For nonpoint sources,
Eagle Mountain piedmont. Data from the Qa4
Eq. (2.68) can be integrated to give
and Qa2 units show a similarly rapid downwind
decrease in silt thickness, illustrating that the ∞
c(x, y, z) = c p (x , y )dx dy
Qa3 pattern is robust. Silt thicknesses were rela- (2.69)
’∞
tively similar on the three terrace units at com-
parable distances from the playa. This similar- Using Eqs. (2.68) and (2.69), the deposition rate
ity was not expected, given the great differences on a ¬‚at surface is given by pc(x, y, 0). Deposition
in age between the three surfaces. This may be on a complex downwind surface can be estimated
partly explained by the fact that Qa2 surfaces as pc[x, y, h(x, y)], where h(x, y) is the elevation
have undergone extensive hillslope erosion (and of the downwind topography. This approach is
hence preserve only a portion of the total eolian only an approximation of the effects of complex
56 THE DIFFUSION EQUATION



80
(a) (b)
Qa2
u = 5 m/s
Qa3
K = 5 m2/s
Qa4
p = 0.05 m/s
60
z
analytic
silt z depositional
thickness
topography
(cm)
40
(x', y')
cross-wind
direction
y
20
a
u = 5 m/s
K = 10 m2/s source
p = 0.075 m/s
wind x
0
2 3 4
1
0 direction
distance downwind from playa (km)


(c) 2 km
0.1 0.2 0.4 1.0
0



wind direction




numerical model results
u = 5 m/s, K = 5 m2/s, p = 0.05 m/s
wind direction
(d) (e)
numerical model results
numerical model results
u = 5 m/s, K = 5 m2/s, p = 0.05 m/s

<< . .

. 11
( : 51)



. . >>