(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-

dθ

’ = (2.65)

gosa Valley, California, where Franklin Lake Playa

d·2

d·

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