Analytical Theory of Erosion
Author(s): W. E. H. Culling
Source: The Journal of Geology, Vol. 68, No. 3 (May, 1960), pp. 336-344
Published by: The University of Chicago Press
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ANALYTICAL
THEORY OF EROSION1
W. E. H. CULLING2
ABSTRACT
A mathematicaltheoryof erosionis developedalong similarlines to the classicalFouriertheoryof heat
flow in solids. The generaltheory is consideredin some detail, and a few representativeexamplesof its
applicationto streamprofilesand valley slopesare suggestedin outline.
ANALYTICAL THEORY OF EROSION
In attempting to fit a mathematical model to the physical conditions of flowing
streams and valley slopes, it will prove to be
too ambitious an undertaking at present to
construct a theory from fundamental physical principles, whether at the hydrodynamic
or hydraulics level. This article will be concerned with the development of a macroscopic theory at the geomorphic level, dealing with large aggregates of material over
extended periods of time. We shall aim at a
coherent structure of as wide a generality as
possible from which we may be able to deduce quantitative natural laws (Strahler,
1952, p. 937).
GENERAL
THEORY
Elevation z is regarded as a function of
the geometrical and time variables (x, y, t).
Material flow is assumed to take place at a
rate proportional to the gradient.
Consider a landform upon which two contour lines trace parallel linear paths. That is,
we can take two vertical parallel planes reasonably close together so that the intersection of the planes with the landform will produce two straight lines parallel to each other
at constant elevation. We assume that the
planes are of sufficient extent that points
near the center may be regarded as parts of
infinite planes. Then, over a period of years
the surface of the landform will assume a
steady distribution, and a series of vertical
planes parallel and within the range of the
original planes will intersect the landform in
1ManuscriptreceivedFebruary19, 1959.
2 Presentaddress:22, OrchardClose, New
Den-
straight lines of constant elevation, and all
contour lines will be parallel. Let the bounding planes intersect the landform at elevations z1and z2, constants, and let these elevations be maintained. At steady state the
quantity of material flowing from the higher
elevation to the lower in time t is given by
K ( z - z2)
df
It,
(1)
where 1 is the length of the contour considered and d the distance separating the two
planes. K is a constant and represents the
mobility of the surface cover.
The material flow is the rate at which material is transferred across any vertical plane
at the point P(x, y), per unit area, per unit
time. In general, it is dependent upon the
position of the point P and the time. We
assume that the flux across a plane through
P varies continuously with the position of
the point, and so at every point of the landform surface a material flow vector is defined
with components f/ and f,. The flow in the z
direction is nil, as all flow takes place within
the movable cover. The vector has magnitude
_+ (fU) 2] 1/2,
fm = [ (f)
and its direction is given by the direction
cosines f/f,. and f,/f,,. We assume that the
parent rock and the movable cover are
homogeneous and isotropic so that flow is
directed along a normal to the contours of
the surface. If the contours are at elevations
z and z + 6z and are separated by ax, then
the rate of material flow per unit time in the
direction of x increasing is -K(8z/8x); and
at the limit as 6x - 0, we have
ham, Uxbridge,Middlesex,England.
/. = -K(z/Ox).
336
ANALYTICAL THEORY OF EROSION
This is the fundamental assumption, that
material flow is proportional to the surface
gradient and is analogous to the assumption
made in the theory of heat conduction
(Fourier, 1882, p. 41).
It is extremely unlikely that material flow
on valley slopes will follow precisely a linear
relationship to the surface gradient, and it is
almost certain that such a relationship never
holds in flowing streams. But it may be possible to regard it as a valid first approximation, at least for certain classes of landforms,
from which we may derive an idealized model of erosional processes, occupying a similar
position in relation to the relevant physical
phenomena as that held by the theory of
Newtonian viscous flow. The actual flow on
hillside slopes is more akin to one (Bingham), or possibly, with varying circumstances, to several of the various types of
plastic flow. From the table with graphs
plotting shear stress against rate of shear
given by Scott-Blair (1949, p. 64), it can be
seen how close an approximation a linear
relationship will give in the various cases. In
the case of Bingham flow above the yield
point, the flow is as for a highly viscous
(Newtonian) fluid (Bingham, 1922; Strahler, 1952, p. 926).
In certain cases investigated, where transport takes place by a series of "jumps"
rather than continuous flow, such as in the
movable bed load, it appears that the rate of
transport is an exponential function of the
force (Denbigh, 1951, p. 66). The simple
linear law is obtained by neglecting all but
the first term in an expansion of the exponential function.
To derive the basic equations, we consider an element of area of the landform at
P with edges 28x and 28y. Then, following
the usual analysis, we find that the rate at
which material flows across the plane x - 8x
in the direction of x increasing will ultimately be given by
The rate of gain of material is given by the
difference, that is,
ax
Similarly, across the planes y y + by we have
-4
(i-.
6x)
(f
x +-¥
5xy.
-
ay
and so
tand
(2)
y
ax
1
If the movable cover is isotropic as regards material flow, fx and f, become
- K(az/ax) and -K(z/y),
and so, provided no material is added or subtracted,
at
ax2
ay2
(3)
If material is added, a further term is required on the right-hand side:
-+-2-
z-K
+ A (x, y, ),
(4)
ax2 ay2
at
and, finally, for the steady state case we
have
ax2Qa.
2
=2
0
(5)
Where the landform possesses parallel
linear contours, no flow takes place in the y
direction, and we obtain the equation for
linear flow. Because in this article we will be
exclusively concerned with this type of flow,
we will use the two variables x and y and not
z, and so the equation with which we will be
concerned is
_
1
y
(6)
6y
The three-dimensional case,
and across the plane x + 5x by
2(/+y.)x
y and
The element of area gains material and,
therefore, elevation, for we neglect such factors as compaction, etc., at the rate
8azz 4xy,
a2y
2
337
.
--+
ax
-+ay2 az2 K at
--=
0,
(7)
W. E. H. CULLING
338
is commonly known as the equation of the
conduction of heat, and in the steady state
case it becomes Laplace's equation
2
vv--
ayv
&2v
02v
v ax" + y2+ a2-0.
(8)
These equations occur in other physical contexts and have been extensively studied.
With suitable changes of notation we can
use well-known results.
As the boundary problems are similar, we
will follow closely the mathematics of the
theory of heat conduction, referring to
Carslaw and Jaeger (1959) as standard authority. For cases where K is not a constant,
we will have to refer to the theory of nonFickian diffusion as a parallel, for this
branch is not so well developed in the theory
of heat flow.
BOUNDARY CONDITIONS
The boundary conditions are derived
from physical considerations and usually
concern the value or variation at terminal
points or surfaces. With regard to stream
profiles the essential principles were given in
the discussion on base-level in Culling
(1957b, p. 466). The stream profile is regarded as a vertical section of an infinite
solid generated by the stream profile in the
time dimension. Thus boundary conditions
are presented at both terminal points of the
stream segment and also at the initial state.
In the simplest case the boundary conditions are constant elevations. In general, the
two end points may vary in position (both x
and y directions) with time. Variation in the
y direction may be reduced to the simpler
problem of constant elevation by the use of
Duhamel's theorem, and it presents relatively little difficulty compared with variation in the x direction. In dealing with this,
we may regard the stream segment as moving in a stationary medium or the medium
as moving past the stationary segment. The
former is probably the easier method and
involves the introduction of a term analogous to "differentiation following the mo-
tion" in hydrodynamics (Lamb, 1952, p. 2).
Thus, if the velocity of the segment is U in
the x direction, equation (6), becomes
32y
9x2
U Oy
K0x
1 y
K 3t"
(9)
Boundary conditions in terms of variation are normally for zero flow at end points,
as in the case of valley slopes where no flow
of material takes place across the watershed.
INITIAL CONDITIONS
If, as is invariably the case in any problems dealing with landform profiles, the initial conditions can be represented by a continuous function of
y=f(x)
t1=0,
then we require a solution where
lim y = f (x).
t-)o
It is too restrictive a condition that y must
equal f(x) at I = 0. The initial conditions
may be supplied by the consequent profile or
the profile at any subsequent stage. The
function y = f(x) may be regular or irregular, but of physical necessity it will be continuous; and so the initial profile may always be represented by a Fourier series.
From the foregoing analysis we are free to
construct a model of stream and slope erosion, but we will soon arrive at the paradox
that for steady states the profile is linear.
This may be roughly correct for valley slopes
under ideal conditions, but it is certainly at
variance with observation as regards stream
profiles. In order to rectify this discrepancy,
the basic equations must be amended so that
they predict a concave curvature for steady
state stream profiles. There is a body of evidence (Shulits, 1941) that the profile ultimately adopted by large rivers will be of an
exponential nature; and by adding an exponential term to the equation of material
flow, we can insure that the steady state
profile is of a form more in accordance with
observation.
ANALYTICAL THEORY OF EROSION
Thus, if we put A (x, y, t) = - Aoex
the linear form of equation (4), we get
2y Ao e_y-- _ 1
_y_Ao
10y
K ''
9x2
K
in
(1)O
for which the steady state solution is
Y= y
(i-x
e-ax_1
+
(1-
e-a) X1],
which is equivalent to the more usual expression
y= Yoe-a .
In seeking the physical significance of the
term A 0e""x,we recall that the exponential
expression for the profile has been derived
from the Sternberg abrasion law by assuming that the bed slope is proportional to the
pebble size. Despite experimental support
for the abrasion law, the exponential expression as applied to rivers in general is an empirical formula (Rubey, 1941). It would appear that the coefficient is composite
y = Yoe-[al,++
+
.+ax
,
and in large rivers from which most of the
observational support derives, we are dealing with a special case in which the abrasion
coefficient a1 becomes predominant. It is
possible, however, to fit exponential curves
to the profiles of smaller streams and to alluvial fans, and the over-all profile curvature
of the multicyclic streams of the Chilterns is
also of an exponential nature. In such cases
the other terms in the expanded coefficient
assume significance, and it is probable that
some of these arise from processes other
than abrasion that also tend to decrease the
size of the bed-load particles as they travel
downstream (Culling, 1957a, p. 263).
Mathematically, the exponential term in
equation (10) represents the abstraction of
material from the system defined by the
equation, and this may be interpreted as an
elimination of energy demand on the stream.
Thus a decrease in particle size will result in
a decrease in the energy demand on the
339
stream by the bed load. Assuming that part
of the energy thus made available is dissipated in corrasion of the bed, then with the
resulting increase in the bulk and also caliber
of the bed load, together with the attendant
alteration in the bed slope and channel
form, the stream will tend to regain a steady
state where the energy demand of the bed
load is equal to the energy made available
for sediment transport. Throughout the
stream segment material will pass into the
system from below; but in terms of energy
demand, the bed load is unchanged. It is as
if material passed out of the system in an
analogous manner to the loss of heat by
radiation or by endothermic reaction.
This interpretation is not restricted to the
effect of decreasing particle caliber but is
equally applicable to all processes whereby
the energy demand of the bed load is diminished; and in order to regain equilibrium,
material is corraded from the bed. We will
therefore assume provisionally that the exponential term represents to a first approximation the combined effect of several factors
that tend to systematically decrease the
energy demand of the bed load downstream.
Of these factors it is probable that the decrease in particle size is always significant
and becomes relatively more important as
the size of the stream increases and also as
the stream progresses toward steady state.
The method may be extended by adding
further terms to equation (10) if it is seen to
be necessary upon development of the theory or if sufficient knowledge of any factor
becomes available that it may be represented in this manner. An external process,
such as the introduction of significant
amounts of debris direct into the channel
from the valley slopes, may be readily accounted for by the addition of a suitable
term.
Changes in the discharge insofar as they
affect the intensity of sheet-wash erosion or
the depth, width, roughness, and stream
velocity will influence the rate of material
flow. It may eventually be possible to represent some of these factors by additional
terms, but until that is so, the effect of sys-
340
W. E. H. CULLING
tematic variations in the transporting capacity will have to be accounted for by taking
K as a function of x. The solution of the resulting equation presents a far more advanced problem, and solutions are available
only for a number of simple functions, i.e.,
K = Kox". The introduction of Ko(t) a variable in time presents even greater difficulty
for this will transform the equation into a
non-linear form which has been little studied.
SOLUTIONS
We now proceed to give a number of examples to illustrate the type of problem that
falls within the scope of the theory and the
form of the solutions. The derivation of
these solutions will have to be deferred until
we deal with the various applications in
detail.
For the semi-infinite region extending
from the plane x = 0, initially at zero elevation and with the point x = 0 maintained at
a constant elevation y = Yo, the solution
may be stated in the form
y=
Yoerfc( 2VKt
,
where erfc x is the complementary
function, or
erfc x =-
2
00
e-'du
(11)3
error
= 1 - erf x,
3 Carslaw and Jaeger, 1959, eq. 2.4 (10), p. 60.
where erf x is the error function, or
=x 2
erf x= =-
/
ro
V
Tables are available for both functions, and
a series of curves for equation (11) are given
in figure 1 for several values of time (t).
This solution is applicable to the building
of an alluvial fan out from the mouth of an
up-faulted valley onto a plain large enough
for the region to be regarded as infinite. The
profiles for small t are probably wide of the
mark, for at such an early stage the stream
will be a waterfall or cascade, and our hypotheses cannot be expected to cover the
physical conditions. The solution is artificial
in other ways. For a large alluvial fan the
progressive diminution of particle size as the
bed load moves downstream will be significant. A more realistic problem would be to
take the boundary condition at x = 0 as
time variant, increasing from zero at t = 0.
We could then apply the theory to alluvial
fans resulting from aggradation in the lower
courses of streams. Finally, the problem of
alluvial fans is really a three-dimensional
one, for the shape of the boundaries and the
presence of a gradient in the y direction will
influence the form of the profile down the
axis of the fan.
If the plain onto which the fan is built is
of limited extent or if the fan extends down
to a river which supplies a stable base level,
x
FIG. 1.-A
du.
e-^'du.
series of curves for equation (11), for selected values of time
ANALYTICAL THEORY OF EROSION
then the solution for the semi-infinite region
is no longer appropriate. The solution required must fit a finite region 0 < x < 1,
with boundary conditions specified at both
end points. This region will also be appropriate to stream segments and valley-slope profiles where the flow is confined to the x direction. In order to find solutions to fit a finite
region, we will require the mathematics developed by Fourier (1822).
Unless the boundary conditions are both
zero, the complete solution will be of the
form y = u + w, where u is the steady-state
term and w the transient term. If the boundary conditions are zero, then the steadystate term is also zero and so also is the complete solution once a sufficient time has
elapsed. The steady-state term is a solution
of
d2u
0<x<l,
dx2
S=0
341
where a, is the Fourier coefficient for the
sine series fitted to the initial profile y =
f(x'). Thus, if the initial profile can be represented by
n rx
2
a sin n
(19)
then an determinedin the usual way is given
by
a, =
n7rx'
dx'. (20)
7
2 o
ff(x')sin
The complete solution is obtained by adding
the two terms
+(Y2-Y-y
y=y
X
Xsin
(12)
X
/(x)
(13)
giving
nrx'
1yl , 1
)
.
sin
-
n x2K,,t/1,
e--Knt/
yi+ yi)n
(
-
d x,
that is
u= Ax+B
,
where A and B are arbitrary constants determined from the boundary conditions. If
we take the case of a stream segment with
fixed control points at either end
y=yi
x= O ,
y=y2
x=
y= yi+ (y2- yi)-
(21)4
x
X
x=0
-u]
w= [f(x')
t=0
0<x<l
(15)
and x=l
(16)
x>0,
(17)
1i
y=O
t=0
0<x<l
y=yl
x=0
1>0
y=0
x=l
t>0,
and
and we have immediately
y= y
1
- Y(22)
X 1 sin n
0o
Ca
sin nix e-Kn'lt/,
d x'.
For the aggradation of an alluvial fan,
we have the initial elevation zero
where f(x') is the initial condition.
The nature of the boundary conditions
suggest the use of a sine series and a solution
of equation (5) is
w=
nr
/ (x')sin
(14)
The transient term is a solution of
w= 0
t/l
1
yi + (ya - yi)y7
1 w
-0
K at
er
xe
sin
;
then we get
32w
ax
I
n
+2
=
+
(18)
n
1
e-Kn
'2t/12
* Carslaw and Jaeger, 1959, eq. 3.4 (1), p. 100.
FIG. 4.-A
FIG. 2.-A
series of curves for equation (22) for selected values of time
FIG. 3.-A
series of curves for equation (23) for selected values of time
series of curves for equation (25) for selected values of a, with A o/a2 kept constant
343
ANALYTICAL THEORY OF EROSION
illustrated in figure 2, for selected values at
time (t). For small fans, e.g., Hambleden
Brook (Culling, 1956, p. 322), this is probably substantially correct for the later
stages, but, as before, if the fan is large, the
influence of decreasing caliber with distance
must be taken into account.
From equation (18) we can also derive the
expression for the degradation of a stream
segment from, say, an initial parabolic profile yo(l2 - x2)/2 with boundary conditions
as before. The term f(x') - u gives
yo x (l-
x),
1
A0
(-L
we get
y = Ax+B+
A
A2-e-".
KiCa2C
(24)
Inserting the boundary conditions
y=0
x=l
y=yi
x = O,
we have
y = yl
1-
)
Ka
(25)
and the transient term is
8yo V
If we introduce into equation (6) the term
X
e-"-
1 + -(1-
e-z)],
e-K(2n+1)2.t/12
Sssin
i(2n+l)nw
X
(23)
illustrated in figure 3. The choice of a parabolic initial profile is artificial and unnecessarily restrictive, for we may express all
possible profiles as Fourier series.
As already noted, the steady-state solution is of the form
y = Ax+B
.
and in figure 4 we illustrate this curve for
various values of a and A o/Ka2 kept constant. The non-stationary case is more difficult and leads to a lengthy expression for y;
and its discussion will be deferred.
In dealing with extra-conditions, operational methods (Carslaw and Jaeger, 1948)
are to be preferred and for the more difficult
problems are essential. Often the solution
provided by transform methods is more easily computed. Thus the case giving solution
FIo. 5.-A series of curves for equation (27) for selected values of time
344
W. E. H. CULLING
(22), with the boundary condition reversed,
i.e.,
y=0
x=O
y= Yo
x=l
Y
[erfc
(2n+
(d
ax==-o
can be rendered very concisely as
y
and
(Kl)l-x
(26)5
for no flow will take place across the plane
x = 0, as this marks the watershed. If
denudation proceeds according to the hypotheses
1 &y_
02y
K t
X
0
3x2
-erfc (2n+ l)+x]
2 &/(Kt)
t>0,
=0,
K Ht
'
the solution is
by the application of Laplace transform
methods.
Finally, a simple problem in the erosional
development of valley slopes is to consider
the degradation subsequent to the up-faulting of part of a plain. Prior to faulting the
elevation is taken as zero. Faulting occurs at
x = a and raises the block 0 < x < a to an
elevation y = Y1. The appropriate boundary conditions are
y = Yi,
O<x<a,
t= 0 ,
y=0,
x>a,
t=
0 ,
=
Y y erf
S
2
x ( Kt
Ki)
2
a+(x
+erf2V/K)
(27) 6
0<x<.
The upper parts of the profiles (fig. 5) show
too great a rate of denudation, for here the
assumptions as to an equilibrium between
rate of transport and rate of weathering do
not invariably hold. A more general problem would be to consider a slow rate of upfaulting with simultaneous denudation.
REFERENCES CITED
E. C., 1922, Fluidity and plasticity: New
BINGHAM,
York, McGraw-Hill Book Co.
H. S., and JAEGER,J. C., 1948, OperaCARSLAW,
tional methods in applied mathematics: 2d ed.,
Oxford, Oxford University Press.
--1959, Conduction of heat in solids: 2d ed.,
Oxford, Oxford University Press.
CULLING,W, E. H., 1956, Longitudinal profiles of
Chiltern streams: Proc. Geol. Assoc. (London),
v. 67, p. 314.
--1957a, Multicyclic stream profiles and the
equilibrium theory of grade: Jour. Geology, v. 65,
p. 259.
1957b, Equilibrium states in multicyclic
---streams and the analysis of river-terrace profiles:
ibid., v. 65, p. 451.
* Carslaw and Jaeger, 1959, eq. 12.5 (6), p. 3 0.
6 Carslaw and Jaeger, 1959, eq. 2.2 (3), p. 54.
DENBIGH,K. G., 1951, The thermodynamics of the
steady state: London, Methuen & Co. Ltd.
J., 1822, Analytical theory of heat: Trans.
FOURIER,
A. FREEMAN,
1878; Dover ed., New York, Dover
Publications, 1955.
LAMB, H., 1952, Hydrodynamics: 6th ed., Cambridge, Cambridge University Press.
RUBEY,W. W., 1941, In discussion on SHULITS,S.,
Rational equation of river bed profiles: Am. Geophys. Union Trans., v. 41, p. 630.
ScoTT-BLAIR,G. W., 1949, A survey of general and
applied rheology: 2d ed., London, Pitman.
SHULITS,S., 1941, Rational equation of river bed
profiles: Am. Geophys. Union. Trans., v. 41, p.
627.
STRAHLER,A. N., 1952, Dynamic basis of geomorphology: Geol. Soc. America. Bull., v. 63,
p. 923.