functions (Sadler and Strauss, 1990) and it is

markedly different from the observed scaling in

Figures 8.7a--8.7d.

Stochastic models have featured prominently in

The results of Figure 8.7 suggest that eolian

our understanding of drainage networks. Hor-

dust accumulation on alluvial fans located close

ton (1945) was among the ¬rst to quantitatively

to playa sources (Figures 8.7e--8.7f) is fundamen-

study the geometry of drainage networks using

tally different from that on fans located far from

Strahler™s classi¬cation scheme. In this scheme,

playa sources (Figures 8.7a--8.7d). The fan study

channels with no upstream tributaries are classi-

sites close to Silver Lake Playa exhibit a linear

¬ed as order 1. When two streams of like order

accumulation trend on the lower fan site and

join together they create a stream of the next

a cyclical-climate trend on the upper fan site

highest order. Horton showed, remarkably, that

while the remaining fan sites exhibit accumu-

the ratios of the number, length, and area of

lation trends consistent with the bounded ran-

streams within a drainage basin nearly always

dom walk model. These results suggest that the

follow common values independent of Strahler

bounded random walk model is representative of

order, i.e.

fans located far (i.e., greater than a few km) from

Ni A i+1 L i+1

playas, and that the classic linear-accumulation ≈ 4, ≈ 4, ≈2 (8.12)

N i+1 Ai Li

or cyclical-climate models are representative of

accumulation on fans proximal to playa sources where i is the Strahler order. In other words the

where dust deposition is rapid enough to over- ratio of the number of order 1 streams to order

whelm the effects of episodic disturbances. 2 streams is nearly the same as the number of

8.6 STOCHASTIC TREES AND DIFFUSION-LIMITED AGGREGATION 197

(a) (b) (c)

(a)

(b)

(d)

(c)

Fig 8.10 TDCNs for Shreve magnitude (a) 1, (b) 2, (c) 3,

and (d) 4.

to transform this structure into that of Figure

8.9a without lifting one of the strings off of the

table. What is nice about the TDCN framework

Fig 8.9 Illustration of the concept of Topologically Distinct

is that the entire set of all possible TDCNs can

Channel Networks (TDCNs). (a) A hypothetical channel

network of magnitude 4. (b) Two channel networks that are be identi¬ed for a given Strahler order or Shreve

topologically identical to that in (a) because they can be magnitude. In the Shreve classi¬cation scheme,

transformed into (a) by moving stream segments around

channels without tributaries are de¬ned to be

within the plane. (c) A network that is topologically distinct

magnitude 1 and when two streams join they

from (a).

form a stream with magnitude equal to the sum

of the two tributaries. Figure 8.10, for example,

shows all of the TDCNs up to Shreve magnitude

order 2 streams to order 3 streams (and so on).

4. The number of TCDNs of Shreve magnitude M

The fact that drainage basins have common ra-

is given by

tios of number, length, and area across Strahler

orders (i.e. spatial scales) is the basis for fractal (2N ’ 2)!

N= (8.13)

models of drainage network geometry.

M !(M ’ 1)!

Following the establishment of Horton™s Laws,

where M ! is equal to M — (M ’ 1)— (M ’ 2) . . . 2 — 1.

stochastic models of drainage network geometry

were proposed. One such framework is Topologi- An analysis of tree structures randomly se-

cally Distinct Channel Networks (TDCNs). TDCNs lected from all possible TDCNs indicates that

are tree structures that cannot be topologically these structures obey Horton™s Laws. Since TD-

transformed into one another within their plane. CNs are a general mathematical model for tree

This concept is best illustrated with an example. structures and do not involve any geomorphic

In Figure 8.9a, a possible tree structure of order processes, the natural conclusion is that Horton™s

2 is shown. Now, imagine that this network is Laws are not really a consequence of drainage

made of a series of strings knotted together at network evolution, but instead are a general

junctions, and that the strings are laying on a feature of tree structures (Kirchner, 1993). Fur-

table. A structure is topologically indistinct from ther studies of drainage network geometry have

another tree structure if the strings can be rear- discovered other kinds of statistical relation-

ranged without lifting any of the strings off of ships commonly found in real drainage networks.

the table. Figure 8.9b illustrates two possible tree Tokunaga (1984) and Peckham (1996) expanded

structures that are topologically identical to the the Horton analysis to quantify not just the num-

structure in Figure 8.9a. Figure 8.9c, in contrast, ber of streams of a given order, but the num-

is topologically distinct because there is no way ber of ˜˜side branches™™ of a given order i that

198 STOCHASTIC PROCESSES

feed into streams of order j. This approach recog-

nizes the fact that small streams that join with Seed cells

small streams in the headwaters of the Missis-

Accreted cells

sippi River basin in Minnesota are fundamentally

different from small streams that join with the

Newly added cell

mainstem Mississippi River in Louisiana.

Diffusion-Limited Aggregation (DLA) provides Random walk

one simpli¬ed stochastic model of drainage net-

work evolution that matches the observed Toku- Prohibited sites

naga statistics of real drainage networks. In the

Other allowed sites

DLA model for drainage networks (Masek and for accretion

Turcotte, 1993), incipient channels are de¬ned on

Fig 8.11 Illustration of the mechanism for network growth

a rectangular grid using ˜˜seed™™ pixels along one

in the DLA model. A particle is randomly introduced to an

boundary (Figure 8.11). Then, a particle is intro-

unoccupied cell. The particle undergoes a random walk until

duced into an unoccupied cell chosen randomly

a cell is encountered with one (and only one) of the four

from all unoccupied cells on the grid. The par-

nearest neighbors occupied (hatched cell). The new cell is

ticle then undergoes a random walk until it en- accreted to the drainage network. From Masek and Turcotte

counters a cell with one of the four nearest neigh- (1993). Reproduced with permission of Elsevier Limited.

bors occupied with a channel pixel. The new cell

is then accreted to the network at this position

(Figure 8.11). In Dunne™s model of drainage network evolution,

Clearly, the DLA model is a very abstract headward-growing channels cause groundwater

model of the processes involved in the evolution to be de¬‚ected towards the channel head, thereby

of real drainage networks. So, how are we to inter- increasing the headward growth rate of ˜˜master™™

pret the fact that the model gives rise to drainage streams at the expense of smaller streams nearby.

networks that look realistic and also match some The DLA model works in a broadly similar fash-

of the statistical features of real drainage net- ion. Random walkers dropped randomly on the

works (Figure 8.12)? One explanation is that surface will tend to accrete new channels close

the random walk process mimics the particle to the tips of existing channels because these

pathways taken by groundwater ¬‚ows that drive channels extend out farther into the grid. The

headward growth of channels by spring sapping. behavior of random walkers on the grid acts as a

(b) Fig 8.12 (a) Drainage network of

(a)

the Volfe and Bell Canyons, San

Gabriel Mountains, near Glendora,

California, obtained from ¬eld

mapping. (b) Illustration of one

realization of the DLA model. From

Masek and Turcotte (1993).

Reproduced with permission of

Elsevier Limited.

0.5 km

8.7 ESTIMATING TOTAL FLUX BASED ON A STATISTICAL DISTRIBUTION OF EVENTS 199

simpli¬ed model for the groundwater ¬‚ow path- dust emissions. Our model aims to quantify the

ways in Dunne™s model. range of water-table depths over which this tran-

sition from a wet to dry playa takes place. An

important assumption in this analysis is that pre-

8.7 Estimating total ¬‚ux based on a cipitation events are rare enough that the surface

is near its steady-state moisture value most of the

statistical distribution of events:

time.

dust emission from playas The vertical transport of moisture in an un-

saturated soil is governed by Richards™ equation:

The relationship between the hydrological state ‚ψ ‚ ‚ψ

= ’1

Kψ (8.14)

of a playa and its dust-emitting potential is a nice

‚t ‚z ‚z

example of the coupling of hydrology and geo-

where ψ is the suction, t is time, z is height above

morphology. In this section we consider a model

the water table, and K ψ is the hydraulic conduc-

for the long-term average dust emission from

tivity. For steady upward ¬‚ow, Eq. (8.14) can be

playas. Speci¬cally, the goal of the model is to cal-

written as

culate the long-term average dust emission rate

‚ ‚ψ

and the role of the water table depth in control-

E= ’1

Kψ (8.15)

‚z ‚z

ling dust emission rate. This example illustrates

the use of probability distributions to calculate where E is the steady-state evaporation rate.

long-term geomorphic rates. Gardner (1958) proposed the following relation-

In nature, dust storms often originate when ship between K ψ and ψ to solve Eq. (8.15):

coarse sand from the playa margin enters into

a

Kψ =

saltation. As sand blows across the playa sur- (8.16)

ψn +b

face, the crust is disturbed, releasing both sand

where a and b are empirical constants for each

and ¬ne-grained sediments (silt and clay). Sand

soil. Gardner (1958) provided analytical solutions

from the playa surface becomes part of a self-

for suction pro¬les at several values of n. Most

sustaining saltation cloud that drives dust pro-

soils have n values between 2 and 3 (Gardner

duction from the surface. The model we consider

and Fireman, 1958). Here we assume n = 2, the

in this section does not resolve different compo-

most appropriate value for ¬ne-grained playa sed-

nents of that playa. Rather, it assumes that the

iments according to Gardner (1958). The solution

¬‚ux of blowing sand along the playa margin can

to Eq. (8.15) for n = 2 and with boundary condi-

be used as a proxy for dust emission from the

tion ψ = 0 at z = 0 is (Gardner, 1958):

playa. This assumption is supported by data show-

ing a strong correlation between horizontal salta-

E /a

1

tan’1

z= ψ

tion ¬‚ux and the vertical dust ¬‚ux (Gillette et al., (8.17)

(E /a)b + 1

b+1

E E

2003). However, the model is not applicable to a a

playas with no upwind sand available for trans-

The value of E is determined by the maximum

port.

soil moisture ¬‚ux within the pro¬le, given by

Soil moisture affects dust emission from

Gardner (1958) and Warrick (1988) as:

playas through its effects on soil cohesion. In or-

π

E 2

der to model soil moisture at the surface of a

= (8.18)

a 2d

playa, we consider the physics of steady-state cap-

illary rise from a water table. For a shallow wa- where d is the water table depth. Substituting Eq.

ter table, capillary rise leads to a moist surface (8.18) into Eq. (8.17) and solving for the suction at

z = d gives

that suppresses dust emissions under all but the

most extreme wind conditions. For a deep wa- 2

π π

2d 2 2

ter table, the dry surface is at or near its max- ψz=d = b+ b+1

tan

π 2 2d

imum dust-producing potential and changes in

subsurface moisture have little or no effect on (8.19)

200 STOCHASTIC PROCESSES

Van Genuchten (1980) predicted the surface soil velocities (Takle and Brown, 1978; Bowden et al.,

moisture from the suction to be 1983), was used for the probability distribution

θz=d ’ θr of friction velocities:

»

’ »+1

= 1 + (±ψz=d )1+» (8.20)

θs ’ θr γ

fw (u— ) = γβ ’γ uγ ’1 e’(u— /β) (8.24)

—

where θr is the residual soil moisture, θs is the

where γ and β are parameters ¬t to measured

saturated soil moisture, ± is the inverse of the

bubbling pressure, and » is the pore-size distri- wind data. The long-term average saltation ¬‚ux

is obtained by integrating Eq. (8.22):

bution parameter. Equations (8.19) and (8.20) pro-

vide an analytical solution for the surface soil ∞

ρ

< q >= du— fw (u— )u— (u2 ’ u2 ) (8.25)

moisture as a function of water-table depth and — —t

g u—t

soil-hydrologic parameters.

Substituting Eq. (8.24) into Eq. (8.25) yields a

The second model component relates the sur-

closed-form solution for the long-term average

face soil moisture to the threshold friction veloc-

saltation ¬‚ux:

ity. Chepil (1956) developed the ¬rst empirical re-

γ

ρβ

lationship between these variables. He obtained: 3 u—t

<q >= 3β 2 1 + ,

g γ2 γ β

1

2 2

θz=d

0.6 γ

u—t = u2 + 1 u—t

(8.21)

—td

’ u2 ,

ρ θw (8.26)

—t

γ β

where u—td is the dry threshold friction velocity, where (x, y) is the incomplete gamma function,

and

⎛ ⎛ ⎛ ⎞ ⎞2 ⎞ 1

⎛ ⎛ ⎞1+» ⎞’ »+1

» 2

⎜ 0.6 1 ⎜ ⎜ ⎟ ⎟⎟

2

π π

2d 2 2

u—t = ⎜u2 ⎜(θs ’ θr ) ⎜⎝1 + ⎝’± b + ⎟ + θr ⎟ ⎟

⎠ ⎠

+ b+1

tan

⎝ —td ρ θw ⎝ ⎝ ⎠ ⎠⎠

π

2 2 2d

(8.27)

In many cases of interest, including the Soda Lake

ρ is the density of air (1.1 kg/m3 ), and θw is the

example described below, the values of the in-

wilting-point moisture content (typically 0.2--0.3

complete gamma function in Eq. (8.26) are nearly

for ¬ne-grained soils (Rawls et al., 1992)).

equal to 1. In such cases, Eq. (8.26) can be approx-

The ¬nal model component is the saltation

imated as

equation for transport-limited conditions, given

ρβ

by Shao and Raupach (1993) as:

< q >≈ (6β 2 ’ u2 ) (8.28)

—t

g γ2

ρ

q = u— (u2 ’ u2 ) (8.22)

— —t

g Relative changes in saltation ¬‚ux, given by

where q is the ¬‚ux in kg/m/s, g is the gravita- Eq. (8.26), are also expected to apply to dust

tional acceleration (9.8 m/s2 ), and u— is the fric- emissions because of the proportionality between

tion velocity. The friction velocity can be obtained saltation and dust emissions observed in many ar-

from measured velocities at a height zm above the eas (Gillette et al., 2003). Dust emissions can also

ground using a modi¬ed law of the wall: be explicitly calculated using Eq. (8.26) if an esti-

κ mate is available for the K factor, or the ratio of

u— = u (8.23)

the vertical dust ¬‚ux to the horizontal saltation

zm

ln z0

¬‚ux. The value of K depends primarily on the

where u is the measured velocity, κ is von Kar- surface texture and must be determined empiri-

man™s constant (0.4), and z0 is the aerodynamic cally.

surface roughness. Three sites at the margins of Soda (dry) Lake,

A two-parameter Weibull distribution, com- California (Figure 8.13), were selected as an illus-

monly used to quantify the distribution of wind tration of the model calibration procedure. Soda

8.7 ESTIMATING TOTAL FLUX BASED ON A STATISTICAL DISTRIBUTION OF EVENTS 201

116.1 116.0 115.9

116.2 W

36.3 N

5 km

8m

North Soda Lake

36.2

20 m

Soda Mtns.

Soda Lake

Old Dad Mtn.

1m

36.1

7m

6m

MojaveR. Crucero Devil™s Playground

6m Balch

water table depths

CLIM-MET stations springs

sites has a residual value of θr = 0.05, and that

Fig 8.13 LANDSAT image of Soda (dry) Lake study site,

with locations of CLIM-MET stations and water table depths the soil moisture is within a few percent of

indicated. Modi¬ed from Pelletier (2006). Reproduced with

that value most of the time. For this reason, the

permission of Elsevier Limited.

steady-state approximation is an appropriate esti-