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diffusion-limited aggregation
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-

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( : 51)



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