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sion equation in the troposphere. Also, Hofmann
ical time series data have power spectra of the
form S( f ) ∝ f ’1/2 from time scales of decades and Rosen (1987) argued that turbulent diffusion
of aerosols from the El Chichon volcanic erup-
to centuries, we must look for some general un-
tion was consistent with the exponential decay
derlying climatological mechanism. In this sec-
of aerosol concentrations with time (i.e. turbu-
tion we present one possible model of ¬‚uctua-
lent diffusion and gravitational settling combine
tions in local temperature and water vapor based
192 STOCHASTIC PROCESSES



(a) (b)




b=0 b=1

(c) (d)




b=2 b=3

Fig 8.5 Examples of 2D fractional Gaussian noises with β =
(a) 0, (b) 1, (c) 2, and (d) 3.
is given by

c in+1 ’ c in ∝ c i+1 ’ 2c in + c i’1
n n
(8.2)
to predict exponential concentrations in time).
Vertical atmospheric turbulent diffusion is super- where i is an index for space and n is an in-
imposed, as it is in oceanic diffusion, on large- dex for time. In our model we establish a one-
scale Hadley and Walker circulations (Peixoto and dimensional lattice of 32 sites with periodic
Oort, 1992). boundary conditions at the ends of the lattice.
To see how time series with power-law power At the beginning of the simulation, we place
spectra arise from a stochastic diffusion pro- ten particles on each site of the lattice. At each
cess, we explore the behavior of a discrete, one- timestep, a particle is chosen at random and
dimensional stochastic diffusion process. A dis- moved to the left with probability 1/2 and to the
crete version of the diffusion equation for the right if it does not move to the left. In this way,
density of particles, c, on a one-dimensional grid the average rate at which particles leave a grid
8.4 RANDOM WALKS 193


point is proportional to the number of particles
20
at the grid point. The average rate at which par-
ticles enter a grid point is proportional to the
10
number of particles on each side multiplied by
one half (since the particles to the left and right
of site i move into site i only half of the time). 0
This is a stochastic model satisfying Eq. (8.2). The
probabilistic nature of this model causes ¬‚uctu- 10
ations to occur in the local density of particles
that do not occur in a deterministic model of dif- 20
fusion. This simple model produces ¬‚uctuations
40 60
20
0 100
80
in the number of particles in the central site of
the 32-site lattice that have a power-law power
Fig 8.6 Five examples of a simple random walk process.
spectra with β = 1/2 (Pelletier and Turcotte, Plotted in Gnuplot by mdoege@compuserve.com.
1997). Reproduced from http://en.Wikipedia.org/wiki/Image:
A stochastic diffusion process can be studied Random Walk example.png.
analytically by adding a noise term to the ¬‚ux of
The power spectrum of variations in E (t),
a deterministic diffusion equation (van Kampen,
S E (ω) =< |E (ω)|2 > is
1981):

sin2 (kL ) 1
dk ∝ ω’ 2
S E (ω) ∝ (8.7)
‚T ‚J 2 k 4 + ω2
D
ρc =’ ’∞
(8.3)
‚t ‚x
for low frequencies. Since T ∝ E , S T (ω) ∝
ω’1/2 also. This is the same result as the discrete
where the ¬‚ux, J , is given by
model.
‚T
J = ’σ + ·(x, t) (8.4)
‚x
8.4 Random walks
The variable T represents the ¬‚uctuations in
temperature from equilibrium and ·(x, t) is a The simplest type of random walk is a stochastic
process consisting of a sequence of discrete steps
Gaussian, white noise. Equations (8.3)--(8.4) are
of ¬xed length. Figure 8.6 illustrates ¬ve exam-
one example of a Langevin equation (i.e. a partial
ples of a random walk which begins at zero and
differential equation with stochastic forcing).
involves unit jumps along the y axis up or down
We will use Eqs. (8.3)--(8.4) to calculate the
with equal probability for each step along the x
power spectrum of temperature ¬‚uctuations in
axis. Each realization of the random walk is dif-
a layer of width 2L exchanging heat with an in¬-
ferent, but the fact that the distance from the
nite, one-dimensional, homogeneous space (Voss
origin tends to increase with each step in all of
and Clarke, 1976). The Fourier transform of the
the examples gives the random walk a certain ele-
heat ¬‚ux of the stochastic diffusion equation
ment of predictability. What is remarkable about
is
random walks and other stochastic processes is
iω·(k, ω) that the average behavior of a large collection of
J (k, ω) = (8.5)
D k 2 + iω random walks is entirely predictable. In a simple
random walk, the expected distance from the ori-
The rate of change of heat energy in the layer
gin, | y| is proportional to the square root of the
will be given by the difference in heat ¬‚ux
number of steps along the x axis, x:
out of the boundaries, located at ±L: dE (t)/dt = √
J (L , t) ’ J (’L , t). The Fourier transform of E (t) is < | y| >∝ x (8.8)
then
By expected, we mean that this is the average be-

havior obtained by many realizations of the ran-
1
E (ω) = sin(kL )J (k, ω) dk (8.6)
1
dom walk process.
(2π) 2 ω ’∞
194 STOCHASTIC PROCESSES



10 (c)
(a) (b)
RT
(g/cm2/yr)



10
R T
RT
R




10


10

Fortymile Wash fan Whipple Mtns. fan (clay)
Coyote Mtns. fan
10
(f)
10 (d) (e)
RT
RT
R T 0.64
(g/cm2/yr)




10
R




10


10

Silver Lake fan (lower)
Yucca Wash fan Silver Lake fan (upper)
10
106 102 103
106 102 103 106
104
104 105
105
102 103 104 105
T (yr)
T (yr)
T (yr)

dust accumulation on alluvial fan terraces. Low
Fig 8.7 Plots of silt accumulation rates versus time interval
rates of long-term (105 --106 yr) dust accumulation
on alluvial fan terrace surfaces from data in Reheis et al.
(1995): (a) Fortymile Wash fan, (b) Coyote Mtns. fan, (c) in the southwestern US, for example, have been
Whipple Mtns. fan (clay fraction shown), (d) Yucca Wash fan, interpreted as a direct result of low dust de-
(e) lower fan near Silver Lake Playa, and (f) upper fan near position rates during the predominantly cool,
Silver Lake Playa. Best-¬t lines are also shown indicating
wet Pleistocene, whereas high rates of shorter-
power-law scaling with exponents close to ’0.5 for (a)“(d).
term (103 --104 yr) dust accumulation have been
Trends in (e) and (f) are more consistent with linear
interpreted as an early Holocene pulse of dust
accumulation or cyclical-climate models. Modi¬ed from
deposition related to pluvial-lake dessication
Pelletier (2007a).
(McFadden et al., 1986; Chadwick and Davis, 1990;
Reheis et al., 1995). In this section we compare the
observed eolian dust accumulation rates at six al-
8.5 Unsteady erosion and luvial fan study sites with the predictions of a
climatically controlled deterministic model and
deposition in eolian
a simple random-walk model of eolian dust ac-
environments cumulation. The results suggest that eolian ero-
sion and deposition on alluvial fans is a highly
episodic process that closely resembles a random
The accumulation of dust on alluvial fan terraces
walk.
in desert environments provides a good example
Reheis et al. (1995) synthesized data on eo-
of the application of a simple random walk in
lian dust accumulation from alluvial fan terrace
geomorphology. In this example, the y axis in
study sites in the southwestern US with opti-
Figure 8.7 represents the accumulation of wind-
mal age control. These sites are located on gen-
blown dust (as a thickness or mass per unit area)
tly sloping, planar alluvial fan terraces that have
underneath the desert pavement, and the x axis
not been subject to ¬‚ooding since abandonment
represents time.
by fan-head entrenchment. The time since fan-
Dust accumulation on desert alluvial fan ter-
head entrenchment is the surface ˜˜age™™ and it
races can be affected by changes in either depo-
corresponds to the time interval of eolian dust
sition or erosion. Previous studies on dust accu-
accumulation. Figure 8.7 plots silt accumulation
mulation of alluvial fans terraces have generally
rates versus time interval on logarithmic scales
argued that climatically controlled dust deposi-
for these locations. Error bars represent time
tion plays the predominant role in controlling
8.5 UNSTEADY EROSION AND DEPOSITION IN EOLIANENVIRONMENTS 195


2 1 10
(a) (b) (c)
(g/cm2)
h




RT
10




(g/cm2/yr)
R
1 0.5

10
bounded random walk
model
0 0 10
101 105
10
0 5 15 103 104
100
4 102
2 T (k.y.)
0
T (k.y.)
T (yr)
2
(d) (e) (f)
10
(g/cm2)




(g/cm2/yr)
h




log R
glacials
R
interglacials
1
bounded
random walk

cyclical-climate
cyclical-climate
model
0 10
15
10 log T
5
0 101 105
103 104
100 102
T (k.y.) T (yr)

intervals of 103 --104 yr, consistent with an early
Fig 8.8 (a) Plot of dust accumulation versus time for two
Holocene dust pulse.
runs of the bounded random walk model. (b) Inset of (a)
illustrating the scale-invariance of the bounded random walk In order to interpret the temporal scaling of
model by rescaling time by a factor of 4 and accumulation by dust accumulation rates with time interval, con-
a factor of 2. (c) Plot of accumulation rate versus time sider two end-member models. The ¬rst model
interval illustrating power-law scaling behavior. (d) Plot of
assumes that the magnitude of dust deposi-
dust accumulation versus time in the cyclical-climate model.
tion and erosion is constant through time (with
(e) Plot of accumulation rate versus time interval in the
a rate equal to 0.01 g/cm2 /yr in this example)
cyclical-climate model, illustrating an inverted S-shaped curve.
and that erosion or deposition takes place with
(f) Schematic diagram summarizing the results of the two
equal probability during each time step (1 yr).
models for accumulation rate versus time interval. Modi¬ed
Figure 8.8a illustrates two runs of this ˜˜bounded
from Pelletier (2007a).
random walk™™ model. The walk is bounded be-
cause the net accumulation is always positive
interval (i.e., surface age) uncertainties. The ¬rst
(in an unbounded random walk, negative values
four plots in Figure 8.7 exhibit a power-law
are also allowed). At the longest time interval
trend (i.e., a straight line on log--log scales) given
(104 yr), both model runs have accumulated ap-
by
proximately 1.5 g/cm2 of dust (equal to a thick-
ness of 1.5 cm if a density of 1 g/cm3 is assumed).
R ∝ T ’± (8.9)
On shorter time intervals (103 and 102 yr), net
accumulations are smaller than the 1.5 value,
where R is the accumulation rate, T is the time
interval of accumulation, and ± is an exponent but they are not linearly proportional to the
close to 0.5. Values of ± for each data set were time interval. For example, net accumulation is
0.3--0.5 g/cm2 over 103 yr for the two model runs
obtained by a linear ¬t of the logarithm of accu-
shown, a much larger change per unit time than
mulation rate to the logarithm of time interval,
the 1.5 g/cm2 observed over 104 yr. Figure 8.8b il-
weighted by the age uncertainty for each point.
lustrates the temporal scaling of accumulation
Accumulation rates plotted in Figures 8.7e and
rates versus time interval in this model. The re-
8.7f (lower and upper fans near Silver Lake Playa)
sults in Figure 8.8b illustrate that accumulation
show only a weak dependence on time interval,
rates decrease, on average, at a rate proportional
however. Figure 8.7e, for example, has a nearly
to one over the square root of the time inter-
constant rate across time intervals, while Figure
val, similar to that in Figures 8.7a--8.7d. This
8.7f shows a peak in accumulation rates at time
196 STOCHASTIC PROCESSES


power-law scaling behavior can also be derived The scaling behavior of accumulation rates
theoretically. The average distance from zero in a shown in Figures 8.7a--8.7d has important impli-
bounded random walk, < h >, increases with the cations for the completeness of eolian deposits.
square root of the time interval, T (van Kampen, Sedimentary deposits are said to be ˜˜incomplete™™
2001): if unconformities or hiatuses with a broad dis-
tribution of time intervals are present (Sadler,
< h >= aT 1/2 (8.10)
1981). The accumulation rate curve can be used
where the brackets denote the value that would to quantify the completeness of a given deposit
be obtained by averaging the result of many dif- using the ratio of the overall accumulation rate
ferent model runs and a is the magnitude of ero- to the average rate at a particular time inter-
sion or deposition during each time step. As ap- val T (Sadler and Strauss, 1990). In the bounded
plied to eolian dust accumulation, the average random walk model, for example, a complete
distance < h > represents the total accumulation stratigraphic section (i.e., one with no hiatuses)
of eolian dust (in g/cm2 or cm) on the surface, of 10 000 yr duration would have a total accu-
mulation of 100 g/cm2 based on an annual accu-
including the effects of episodic erosion. The av-
mulation rate of 0.01 g/cm2 . The bounded ran-
erage accumulation rate is obtained by dividing
Eq. (8.10) by T to give dom walk model, in contrast, has an average
accumulation of 1 g/cm2 over 10 000 yr based
< R >= aT ’1/2 (8.11)
on Eq. (8.11). Therefore, a 10 000 yr sequence is
In the second end-member model, dust depo- only 1% complete (i.e., 99% of the mass that was
sition is assumed to have a high value during once deposited has been eroded). A power-law
interglacial periods (0.01 g/cm2 /yr), a low value accumulation-rate curve (e.g., Eq. (8.10)) speci¬-
during glacial periods (0.002 g/cm2 /yr) and no ero- cally implies a power-law or fractal distribution
sion. Figure 8.8d illustrates the accumulation of hiatuses, with few hiatuses of very long dura-
rate versus time interval for this model. The tion and many of short duration.
plot follows an inversed S-shaped curve shown
in Figure 8.8e. This relationship is characteris-
8.6 Stochastic trees and
tic of accumulation rate curves for all periodic

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