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kt = 25 m2
= 0.8
= 2.5


0.003
0.0005
(c) (f)

2
2

2
2


0.0
0.0




5 0 5 10 15
0 5 10 15
x (m)
x (m)
In many applications, diffusion is one com-
Fig 2.1 Evolution of a topographic scarp, illustrating (a)
ponent of a more complex model. The transport
elevation, (b) slope, and (c) curvature. In (a), arrows of
of contaminants in groundwater ¬‚ow, for ex-
varying length represent the sediment ¬‚ux at each point. In
the diffusion model, the ¬‚ux is proportional to the local ample, can be quanti¬ed using a combination
slope, and the resulting raising or lowering rate of the surface of advection (i.e. the translocation of contami-
is proportional to the change in ¬‚ux per unit length, which, in nants as they are carried along with the ¬‚uid)
turn, is proportional to the curvature. (d) and (e) Graphs of
and diffusion (i.e. the spreading of contaminant
elevation, slope, and curvature for ¬ve times following scarp
concentration from hydrodynamic dispersion as
offset (κt = 0.01, 0.03, 0.1, 0.3, 1.0.)
the contaminant travels along variable ¬‚ow paths
through a porous aquifer). Similarly, the concen-
tration of dust downwind from a playa is the re-
weathering-limited slopes and channels evolve
sult of both advection and turbulent diffusion.
advectively. Analytic and numerical techniques
Advection describes the motion of the dust as it
for solving the advection equation will be pre-
is carried downwind, while diffusion describes
sented in Chapter 4.
2.1 INTRODUCTION 33


the spreading of the plume as it is mixed by at-
mospheric turbulence. In many of the examples
in this chapter, we will assume a ¬xed chan-
nel head at the base of a diffusing hillslope. Of
course, channels have their own complex evolu-
nonlinear
q
tion and are rarely ¬xed at the same elevation transport
over time. As such, a complete understanding of
hillslopes or channels requires a fully coupled
model of the ¬‚uvial system of which hillslope
diffusion is one component. Although the tech- linear (slope-proportional)
transport (i.e. diffusion eqn.)
niques used in this chapter are framed around
solving a single equation that has limited ap-
Sc
plication in geomorphology, the techniques we
will discuss form the basis for solving many real-
world problems in which diffusion is one part of Fig 2.2 Schematic graph of ¬‚ux versus slope. The diffusion
a more realistic model. equation assumes a linear relationship between ¬‚ux and
The applicability of the diffusion model to slope. More generally, sediment ¬‚ux increases nonlinearly
hillslope evolution depends on the processes act- with slope, diverging at a critical slope value Sc . However,
even the most general ¬‚ux relationship can be approximated
ing to move sediment on the hillslope. Creep
as linear for small slopes. As such, the predictions of linear
and rain splash are examples of hillslope pro-
and nonlinear transport models are the same for
cesses that are accurately represented by the dif-
gently-sloping topography.
fusion equation for gentle slope angles (Carson
and Kirkby, 1972). In these processes, particle
movement takes place by a series of small jumps
the divide (Carson and Kirkby, 1972):
triggered by freeze/thaw cycles, wetting/drying,
or momentum imparted from rain drops. In these ‚h
q = ’κ x (2.6)
jumps, particles may move both upslope and ‚x
downslope but the distance traveled by a parti-
cle moving downslope is slightly greater than the The evolution of hillslopes governed by this
distance of those traveling upslope. Moreover, the ¬‚ux relationship is considered in Section 2.3.2.
increase in distance traveled by a particle mov- Mass movements are not diffusive because the
ing downslope increases linearly with the hill- sediment ¬‚ux increases rapidly and nonlinearly
slope angle or gradient for relatively small slopes. with small increases in hillslope gradient near
the point of slope failure Sc , as illustrated in
Bioturbation is another important hillslope pro-
Figure 2.2.
cess that can often be approximated as diffusive,
The value of κ controls how fast diffusion
particularly for gravelly hillslopes of arid envi-
takes place. In the Basin and Range Province of
ronments where processes such as slope wash
the western US, an approximate value of κ =
are relatively ineffective due to the coarse tex-
1 m2 /kyr has often been used based on studies of
ture of the slope material. Slope wash, rilling,
pluvial shorelines and other landforms of known
and mass movements are examples of hillslope
age (Hanks, 2000). It is widely recognized, how-
processes that are not diffusive. Slope wash is
ever, that the 1 m2 /kyr value varies spatially ac-
not diffusive because the shear stress exerted by
cording to climate, vegetation, soil texture, and
overland ¬‚ow increases with distance from the di-
other variables, even if that variation is not yet
vide as ¬‚ow accumulates along the slope. While
well characterized. One drawback of the diffusion
hillslopes dominated by slope wash are not gov-
approach in geomorphology is that we often do
erned by the diffusion equation, they can be de-
not know the value of κ for a given location with
scribed by a more general, spatially variable dif-
any precision. This is a valid criticism, but it is
fusion equation in which the ¬‚ux is proportional
still important to study hillslope evolution even if
to the hillslope gradient and the distance from
34 THE DIFFUSION EQUATION


the absolute rates of change cannot be precisely uplifting mountain range. In such cases, the
determined. diffusion equation simpli¬es to the time-
Three classic publications form the basis of independent equation:
many of the results in this chapter. Carslaw and ‚ 2h
κ +U = 0 (2.7)
Jaeger (1959) is the de¬nitive text on analytic so- ‚ x2
lutions to the diffusion equation. Although the
where U is the uplift rate. Solving for the hill-
book was written primarily for engineering ap-
slope curvature, Eq. 2.7 gives:
plications to heat conduction in solids, the solu-
‚ 2h U
tions presented in that book can be applied to =’ (2.8)
‚ x2 κ
diffusion problems that arise in any ¬eld of sci-
ence. Culling (1963) was among the ¬rst authors Integrating Eq. (2.8) gives an equation for the hill-
to apply the diffusion equation to hillslope evo- slope gradient:
lution. His paper is still the most complete ref- ‚h U
= ’ x + c1 (2.9)
erence on the subject. Culling™s paper includes ‚x κ
solutions appropriate for many different initial
where c 1 is an integration constant. In order to
conditions as well as a stochastic treatment of
constrain the value of c 1 , it is necessary to spec-
the relationship between the random motion of
ify the value of ‚h/‚ x at one end of the hillslope
sediment particles on the hillslope and the diffu-
pro¬le. The upslope end of the hillslope is cho-
sive evolution of the hillslope. Carson and Kirkby
sen to coincide with a divide in this case. Divides
(1972) is the most complete reference on the rela-
are defined as locations across which no water or
tionship between speci¬c hillslope processes and
sediment passes. As such, a no-flux boundary con-
their signature forms.
dition is appropriate for the upslope end of this
In the following sections we present solutions
pro¬le. The no-¬‚ux boundary condition is given
to the diffusion equation primarily within the
by
context of hillslope evolution. As such, the dif-
‚h
fusing variable h will represent elevation along =0 (2.10)
‚x
a topographic pro¬le. It should be emphasized, x=0

which implies c 1 = 0. Integrating Eq. (2.9) gives
however, that the solutions we describe are gen-
eral. Therefore, any solution that we obtain for h
U2
h(x) = ’ x + c2 (2.11)
(for a particular initial condition and boundary

conditions) can be applied to any other diffusion
where c 2 is another integration constant. At the
problem, whether the physical process is heat
downslope end of the pro¬le, the elevation is as-
conduction, contaminant transport, etc. That is
sumed to be constant. This boundary condition is
part of the power of applied mathematics: once
appropriate for a hillslope-channel boundary in
a set of solutions or a solution method has been
which all of the sediment delivered to the chan-
learned it can often be rapidly applied to many
nel is transported away from the slope base with
different physical systems.
no erosion or deposition. If the hillslope has a
length L and an elevation of zero at the slope
base:
2.2 Analytic methods and
h(L ) = 0 (2.12)
applications
then c 2 = U L /2κ and the hillslope pro¬le is given
by
2.2.1 Steady-state hillslopes
U2
Steady-state landscapes have received a great deal h(x) = (L ’ x 2 ) (2.13)

of attention in geomorphology in recent years.
The topography of a steady-state landform is time- Equation (2.13) shows that diffusive hillslope pro-
independent, such as when uplift and erosion ¬les are parabolas in cases of steady-state up-
are in perfect balance at every point in a rapidly lift. The hillslope relief in Eq. (2.13) is given by
2.2 ANALYTIC METHODS AND APPLICATIONS 35


U L 2/2κ, so the relief of diffusive hillslopes is pro- Fourier showed that nearly any function could be
represented as a series of sinusoidal functions:
portional to uplift rate and slope length, and
inversely proportional to diffusivity. Equations ∞
nπ x
f (x) =
(2.9)--(2.13) illustrate how boundary conditions an sin (2.16)
L
are used to solve differential equations. n=1

Equation (2.8) suggests that in regions of sim- where f (x) is any function that goes to zero
ilar climate and hillslope processes, hillslope cur- at x = 0 and x = L . Given a function f (x) that
vature can be a relative measure of tectonic up- obeys these boundary conditions, the values of
lift rates. Equation (2.8) may be applicable to the the Fourier coef¬cients an can be calculated using
early stage of uplift when hillslope gradients are
L
2 nπ x
still relatively low and weathering rates are suf- an = f (x ) sin dx (2.17)
L L
¬ciently rapid for hillslopes to be regolith cov- 0

ered. In general, however, Eq. (2.8) is of limited The solution to the diffusion equation with ini-
use in many areas of rapid uplift because the re- tial condition given by Eq. (2.16) is the summa-
quirements for the diffusion equation generally tion of Eq. (2.15) from n = 1 to n = ∞:
do not apply. Later in this chapter we will solve

nπ x ’κn2 π 2 t/L 2
the solution to the nonlinear diffusion equation.
h(x, t) = an sin e (2.18)
L
That equation usually provides a more accurate n=1
representation of hillslope evolution in rapidly
Substituting Eq. (2.17) into Eq. (2.18) gives the gen-
uplifting terrain.
eral solution

L
2 nπ x
2.2.2 Fourier series method h(x, t) = f (x ) sin
L L
Two general classes of solutions exist for the dif- 0 n=1
nπ x ’ κn2 π 2 t
fusion equation: series solutions and similarity
— sin e L2 dx (2.19)
solutions. First consider the one-dimensional (1D) L
diffusion equation in a region bounded by x = 0
This approach suggests a powerful means of
and x = L . By 1D, we mean a pro¬le that varies
solving the diffusion equation in bounded re-
with only one independent spatial variable, x.
gions: using the initial condition f (x), solve for
Let™s assume that the initial condition at t = 0
the coef¬cients an of the Fourier series repre-
is given by a function f (x) and that the value of
sentation of f (x); then, plug the an values into
h is set to zero at both ends of the region for
Eq. (2.18) to obtain the solution. Other boundary
all time t (i.e. h(0, t) = h(L , t) = 0). If the initial
conditions (e.g. values of h(0, t) and h(L , t) that
condition f (x) is given by
are nonzero, or boundary conditions speci¬ed by
nπ x hillslope gradients) can also be handled by using
f (x) = an sin (2.14)
L cosine, linear, and/or constant terms in the sum-
mation. For example, if we wish to place a divide
then the solution to the diffusion equation is
at x = 0 then the series solution is
nπ x ’κn2 π 2 t/L 2
h(x, t) = an sin ∞
e (2.15) L
1 nπ x
h(x, t) =
L f (x ) cos
L 2L
’L n=1
The correctness of this solution can be checked
nπ x ’ κn2 π22 t
— cos
by differentiation and substitution of the re- e 4L dx (2.20)
2L
sults into the diffusion equation. This solution
is only valid for the sinusoidal function given by Equation (2.20) is the same as Eq. (2.19) ex-
Eq. (2.14), so it may not appear to be of much use. cept that cosine terms are used and the hill-
slope domain is assumed to be from x = ’L
Equation (2.15), however, is actually a powerful
to x = L . If f (x) is chosen to be symmetric (i.e.
building block that can be used to solve the dif-
f (x) = f (’x)) then Eq. (2.20) is consistent with
fusion equation for nearly any initial condition
by using a Fourier series representation for f (x). the divide boundary condition given by Eq. (2.13).
36 THE DIFFUSION EQUATION


The Fourier series approach is only applicable integration. The approach is to introduce a new
to bounded regions. Of course, all geomorphic variable that is a combination of x and t:
applications are bounded, because the Earth is x
·= √ (2.22)
2 κt
not in¬nite in extent! In some cases, however, it
is useful to de¬ne mathematical models in in¬- and rewrite the diffusion equation for θ and its
nite or semi-in¬nite domains. For example, if one boundary conditions in terms of ·. This can be
is interested in modeling the ¬rst few years of done (following the approach in Turcotte and
radionuclide migration into an alluvial deposit Schubert (1992)) using the chain rule for differ-
that is hundreds of meters thick, a semi-in¬nite entiation:
model is more convenient to work with and prac-
‚θ ‚θ ‚· ‚θ 1·
1x1 dθ
= = ’√ = ’
tically speaking more accurate than a Fourier
‚t ‚· ‚t ‚· 4 κt t d· 2t
series approach.
(2.23)
The Fourier series approach works because
‚θ dθ ‚· dθ 1
= = √
the diffusion equation is linear (i.e. it does not (2.24)
‚x d· ‚ x d· 2 κt
involve squared or higher-order terms of h or
‚ 2θ 1 d2 θ ‚· 1 1 d2 θ
its derivatives). Only in linear equations can dif- =√ = (2.25)
‚ x2 2 κt d·2 ‚ x 4 κt d·2
ferent solutions be superposed to obtain other
The diffusion equation for θ becomes
solutions.
1 d2 θ

’· = (2.26)
2.2.3 Similarity method 2 d·2

Series solutions with ¬xed-elevation boundary
and the boundary conditions become θ(∞) = 0
conditions are appropriate for hillslopes with a
and θ (0) = 1. Equation (2.26) is a second-order or-
well-de¬ned base-level control (such as a nearby
dinary differential equation (ODE) that can be
channel that carries sediment away from the
reduced to a ¬rst-order ODE by introducing the
slope base to maintain a constant elevation). In
variable
some cases, however (such as when sediment is

allowed to deposit and prograde at the slope base) φ= (2.27)

it is more appropriate to assume that diffusion
occurs over an in¬nite or semi-in¬nite region. In Equation (2.26) then becomes
such cases, similarity solutions are a powerful 1 dφ
’·d· = (2.28)
approach. Let™s consider the semi-in¬nite region 2φ
given by x = 0 to x = ∞ with an initial condition
Equation (2.28) can be integrated to obtain
h(x, 0) = h 0 . Let™s also assume that at t = 0 the el-
’·2 = ln φ ’ ln c 1
evation h is instantaneously lowered to zero at (2.29)
x = 0. In the context of hillslope evolution, this
Taking the exponential of both sides of Eq. (2.29)
problem corresponds to instantaneous base level
gives
drop of an initially planar hillslope or terrace.

In order to solve this problem, it is useful to 2
φ = c 1 e’· = (2.30)
introduce a dimensionless elevation given by d·
Integrating Eq. (2.30) gives
h
θ= ’1 (2.21) ·
h0 2
e’· d· + 1
θ = c1 (2.31)
The diffusion equation for θ is identical to the 0

where · is an integration variable and the con-
diffusion equation for h. The boundary condi-
tions on θ are now given by θ(x, 0) = 0, θ(0, t) = 1, dition θ (0) = 1 was used to constrain the integra-
and θ(∞, t) = 0. Similarity solutions make use of tion constant. Application of θ(∞) = 0 requires
a mathematical trick that reduces the diffusion that

equation (a partial differential equation) to an 2
e’· d· + 1
0 = c1 (2.32)
ordinary differential equation that is solved by 0
2.2 ANALYTIC METHODS AND APPLICATIONS 37



The integral in Eq. (2.36) is given by π/2, so the on h. The initial condition for w is given by

constant c 1 has the value ’2/ π. Equation (2.31)
U2
w(x, 0) = ’ (L ’ x 2 )
becomes (2.39)


2 2

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