стр. 37 |

ical time series data have power spectra of the

form S( f ) в€ќ f в€’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 п¬‚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 ОІ =

(a) 0, (b) 1, (c) 2, and (d) 3.

is given by

c in+1 в€’ c in в€ќ c i+1 в€’ 2c in + c iв€’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 п¬‚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 п¬‚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 ОІ = 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 п¬‚ux of

The power spectrum of variations in E (t),

a deterministic diffusion equation (van Kampen,

S E (П‰) =< |E (П‰)|2 > is

1981):

в€ћ

sin2 (kL ) 1

dk в€ќ П‰в€’ 2

S E (П‰) в€ќ (8.7)

в€‚T в€‚J 2 k 4 + П‰2

D

ПЃc =в€’ в€’в€ћ

(8.3)

в€‚t в€‚x

for low frequencies. Since T в€ќ E , S T (П‰) в€ќ

П‰в€’1/2 also. This is the same result as the discrete

where the п¬‚ux, J , is given by

model.

в€‚T

J = в€’Пѓ + О·(x, t) (8.4)

в€‚x

8.4 Random walks

The variable T represents the п¬‚uctuations in

temperature from equilibrium and О·(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 п¬Ѓxed length. Figure 8.6 illustrates п¬Ѓve 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 п¬‚uctuations in

axis. Each realization of the random walk is dif-

a layer of width 2L exchanging heat with an inп¬Ѓ-

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 п¬‚ux of the stochastic diffusion equation

ment of predictability. What is remarkable about

is

random walks and other stochastic processes is

iП‰О·(k, П‰) that the average behavior of a large collection of

J (k, П‰) = (8.5)

D k 2 + iП‰ 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 п¬‚ux

number of steps along the x axis, x:

out of the boundaries, located at В±L: dE (t)/dt = в€љ

J (L , t) в€’ J (в€’L , t). The Fourier transform of E (t) is < | y| >в€ќ x (8.8)

then

By expected, we mean that this is the average be-

в€ћ

havior obtained by many realizations of the ran-

1

E (П‰) = sin(kL )J (k, П‰) dk (8.6)

1

dom walk process.

(2ПЂ) 2 П‰ в€’в€ћ

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-п¬Ѓt lines are also shown indicating

wet Pleistocene, whereas high rates of shorter-

power-law scaling with exponents close to в€’0.5 for (a)вЂ“(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. Modiп¬Ѓed 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 п¬‚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 вЂ˜вЂ˜ageвЂ™вЂ™ 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 п¬Ѓrst 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. Modiп¬Ѓed

Figure 8.8a illustrates two runs of this вЂ˜вЂ˜bounded

from Pelletier (2007a).

random walkвЂ™вЂ™ model. The walk is bounded be-

cause the net accumulation is always positive

interval (i.e., surface age) uncertainties. The п¬Ѓrst

(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 в€ќ T в€’О± (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 О± is an exponent but they are not linearly proportional to the

close to 0.5. Values of О± 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 п¬Ѓt 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 вЂ˜вЂ˜incompleteвЂ™вЂ™

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 в€’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)) speciп¬Ѓ-

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

стр. 37 |