Geology Reference
In-Depth Information
where
Q
x
is the discharge of mass (formally, the
specific discharge, or mass per unit contour
length per unit time) in the
x
direction, through
whatever processes are operating,
r
b
is the bulk
density of the material being transported, and
z
is the elevation of the surface. The surface will
decline in elevation (erosion) if the gradient in
mass discharge is positive (more mass leaves
the element than arrives), and it will increase
in elevation (deposition) if the gradient in
discharge is negative (more mass arrives than
leaves). This statement is very general and
broadly applicable; it holds for hillslopes, river
beds, and sea floors alike. What changes from
one system to the next are the surface process
and how that process is governed by measura-
ble quantities in the landscape, such as local
slope, the distance from a ridge crest, the
distance downstream, and so on. What is needed
to close the system is a process rule that
describes what controls the hillslope fluxes,
Q
x
.
In the simplest case on a single hillslope, of
which a scarp is an example, the discharge of
mass is taken to be simply proportional to the
local slope. Stated mathematically,
1.5
Scarp profile
A
1.0
0.5
0
0
2
4
6
8
10
Distance (m)
Evolution of a Scarp
50
25
40
20
30
15
20
Maximum slope
B
10
10
0
2
4
6
8
10
Time (ky)
Fig. 11.6
The evolution of a scarp profile by
diffusion.
A. The top half of a scarp profile at several times after
initiation of a 2-m vertical step in the surface.
B. Evolution of the maximum slope from 1 to 10 kyr.
Note that the maximum slope falls rapidly at first, and
more gently later, a change reflecting the dependence of
diffusive processes on the square root of time. Modified
after Hanks
et al
. (1984).
of the mass flux and mass conservation equations
results in the following diffusion equation:
∂
∂
z
2
∂
z
∂
z
=−
Qk
(11.7)
=
κ
(11.9)
x
x
∂
∂
2
t
x
where
k
is a proportionality constant reflecting
the efficiency of the process. Note that the
negative sign reflects the fact that mass is being
fluxed in the downslope direction. The local
slope in two dimensions is
∂z
/
∂x
, whereas in
two horizontal dimensions
x
and
y
the slope is
∇
The rate of change of the elevation of the surface
depends solely on the local curvature (or rate of
elevation change) of the surface. Here, the
diffusion coefficient, often called the landscape
diffusivity,
k
, reflects both the bulk density of
the material being transported and the efficiency
of the transport process:
k
=
k
/
r
b
. Diffusivity
always has units of L2/T.
2
/T. Note that this
formulation is easily extended to both
x
and
y
directions, although we will focus on simple
landforms described by a single profile,
z
(
x
).
Solution of the diffusion equation, a second-
order partial differential equation, requires
specification of initial conditions. In addition,
estimation of the age of a feature requires knowl-
edge of the diffusion constant. In analogy with
the problem of thermal evolution (see Turcotte
and Schubert, 2002) within a slab with an initial
step in temperature, scarp evolution may be
represented as an error function (Fig. 11.6).
z
. This expression is a subset of a large suite
of possible models for the flux of mass on a
hillslope, in which a general dependence exists
on
x
, possibly to some power, and in which
slope may come in to some power (e.g., Carson
and Kirkby, 1972).
In general, then, the rule may be stated as:
n
∂
∂
z
⎛⎞
(11.8)
Qkx
=−
m
⎜⎟
x
⎝⎠
Under conditions in which the transport
efficiency,
k
, is uniform with
x
, and the dis-
charge of sediment is dictated linearly by the
local slope, i.e.,
m
= 1 and
n
= 0, the combination
