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The model geometry and boundary condi- 4.8b. In these two experiments, the model param-
eters are identical (K = 10’7 yr’1 , L m = 200 km,
tions are illustrated in Figure 4.8a. h a and h b refer
L = 500 km) except for the values of κ, which dif-
to the alluvial and bedrock portions of the pro-
fer by a factor of 2. A smaller value of κ results in
¬le. Tectonic uplift is assumed to occur as a rigid
block between x = 0 and x = L m prior to t = 0. a steeper piedmont, lower bedrock relief, and a
h 0 is the maximum elevation immediately follow- smaller denudation rate. After 250 Myr, the small-
κ case has a mean elevation about twice as large
ing the cessation of active uplift. This boundary
4.5 THE EROSIONAL DECAY OF ANCIENT OROGENS 103


as the large-κ case. Varying the piedmont length bedrock and alluvial components, and a transcen-
L for a ¬xed value of κ has a similar effect. dental equation is obtained for „d :
Figure 4.8c illustrates the decline in mean
basin elevation with time on a semi-log plot for Ar /(K „d )
Lh
e’t/„d
h b (x, t) = h 0
a piedmont of ¬xed length (L = 500 km) and a (4.32)
x
range of values of K and κ. Following an ini-
L ’x
Ar /(K „d ) sin √
Lh
tial rise to a maximum value, the decay in mean κ„d
e’t/„d
h a (x, t) = h 0 (4.33)
basin elevation is exponential in each case, but Lm L ’L m
sin √
κ„d

varies as a function of both the bedrock and pied-
κ„b L ’ Lm
Ar
1’ = tan
mont parameters, illustrating their dual control √ (4.34)
K „d κ„b
Lm
on the tempo of denudation.
In nature, values of K , κ, L m , and L vary from
one mountain belt to another. Persistent moun- Equation (4.34) cannot be further reduced
tain belts can be expected to have extreme values and must be solved numerically for speci¬c val-
of one or more of these parameters, resulting in ues of the model parameters. As an example,
large values of „d compared to the global average. for the model parameters of the Appalachian-
type case of Figure 4.9a, Eq. (4.34) gives „b = 186
L and L m can be determined from topographic
and geologic maps. Values of K and κ are not well Myr. In the limit of no piedmont (i.e. L m ’ L ),
constrained, but qualitative data on climate, rock Eq. (4.34) has the appropriate limiting behavior
„b ’ Ar /K .
type, and sediment texture can be used to vary
those parameters around representative values. Shepard (1985) found actual bedrock pro¬les
The effects of unusually resistant bedrock, for in nature to be best ¬t by a power law when slope
example, are illustrated in the Appalachian-type was plotted versus downstream distance, consis-
example of Figure 4.9a. Hack (1957, 1979) argued tent with Eq. (4.32). The power-law exponent in
that the resistant quartzite of the Appalachian the model is a function of the bedrock erodibil-
highlands was a key controlling factor in its evo- ity (more resistant bedrock leads to more con-
lution. In this example, the values of L m and L cave pro¬les) but the exponent is also a func-
were set to 100 km and 500 km. The values K = tion of the basin diffusivity, κ (through „d ), il-
5 — 10’8 yr’1 and κ = 6 — 10’4 km2 /yr led to pro- lustrating the explicit coupling of the bedrock
¬les most similar to those observed (Figure 4.9a). and piedmont pro¬le shapes. This result suggests
In contrast, Figure 4.8b illustrates the Ural-type that piedmont deposition must be included to
example (i.e. an unusually broad piedmont). Here, correctly model bedrock drainage basin evolu-
L m and L were set to 600 km and 2000 km. The tion. The sine function is a poor approximation
value of K most consistent with observed pro¬les to many observed piedmont pro¬les, but this dis-
in the Urals (K = 10’7 yr’1 , κ = 2 — 10’2 ; Figure crepancy is largely the result of neglecting down-
4.9b) is a factor of two greater than the value for stream ¬ning. A more precise piedmont form
could be achieved by making κ a function of
the Appalachians. The Ural example shows that
a broad piedmont can diminish bedrock relief, downstream distance.
reducing the average basin denudation rate com- Montgomery and Brandon (2002) recently
pared to a narrower piedmont of the same slope. questioned the linear relationship between mean
The eastern Appalachian piedmont is relatively erosion rate and mean elevation (or mean lo-
modest in size, so resistant bedrock is the most cal relief) that lies at the heart of the persis-
likely explanation for its persistence and strongly tent mountain belt paradox. Their study is best
concave headwater pro¬les. The values of K and known for documenting a rapid increase in ero-
κ in Figure 4.9 are not unique; a range of values sion rates in terrain with a mean local relief
is consistent with the observed pro¬les. greater than 1 km. However, these authors also
The model equations can also be solved an- expanded upon earlier data sets that established
alytically for the decay phase of mountain-belt a linear relationship between mean erosion rate
evolution using separation of variables. Power- and mean local relief. Montgomery and Brandon
law and sinusoidal solutions are obtained for the (2002) found that the expanded data set was best
104 THE ADVECTION/WAVE EQUATION



(b) 1.0
81
82 80
83° W
h model
36° N
(km)
(a) 250 Myr
0.5

35
0
1.0
K = 5 — 10’8 yr’1
k = 6 — 10’4 km2/yr
h/h0
34
model
0.5 250 Myr


33
0
50 km 400
0 200
x (km)
48° E 60
54 0.7
(d)
h
60° N
(c)
model
(km)
250 Myr
0.35

56

0
1.0
K = 10’7 yr’1
k = 2 — 10’2 km2/yr
h/h0
52 model
0.5 250 Myr


200 km
0
2000
0 1000
x (km)
represented by a power law:
Fig 4.9 (a) and (b) Appalachian-type example. (b)
Longitudinal pro¬les of channels in the Santee and Savannah
E av = 1.4 — 10’6 R z
1.8
(4.35)
drainage basins (location map in (a)), plotted with a model
pro¬le (K = 5 — 10’8 yr’1 , κ = 6 — 10’4 km2 /yr) at where E av is in units of mm/yr and R z is
t = 250 Myr for comparison. Evolution of the model pro¬le the mean local relief (de¬ned as the average
in intervals of 50 Myr is also shown. (c) and (d) Urals-type difference between the maximum and minimum
example. (d) Pro¬les in the Volga drainage basin (location map
elevation over a 10 km radius) in m. This re-
in (c)), plotted with a model pro¬le (K = 10’7 yr’1 ,
sult necessitates a reassessment of the persistent
κ = 2 — 10’2 km2 /yr) for comparison. From Pelletier
mountain-belt paradox.
(2004c).
4.5 THE EROSIONAL DECAY OF ANCIENT OROGENS 105



10’1
(a) (b) Eav ∝ H

Eav ∝ H 2
10’2 1.3
Eav∝ Rz
Eav
(mm/yr) SP


10’3 SFD
1.8
Eav∝ Rz SFD
SP

10’4
10’1 10’3 10’2 10’1 100
100
Rz (km) H (km)
uplift of 0.05 mm/yr for the ¬rst 100 Myr of the
Fig 4.10 (a) Plots of mean erosion rate vs. mean local relief
simulation (suf¬cient to develop steady state), fol-
following uplift for the stream-power and
sediment-¬‚ux-driven models. The stream-power model lowed by a long period of erosional decay and
isostatic response to erosion. The values of K =
closely approximates a power-law relationship with an
2 — 10’3 kyr’1 and K s = 4 — 10’2 (m kyr)’1/2 used
exponent of 1.3 (plots closely overlap), while the
sediment-¬‚ux-driven model also follows a power law but with
in these runs predict identical steady state condi-
an exponent of 1.8. (b) Plots of mean erosion rate versus
tions at the cessation of tectonic uplift at 100 Myr.
mean elevation following uplift for the stream-power and
Figure 4.10a plots the relationship between mean
sediment-¬‚ux-driven models. Following a transient phase,
erosion rate E av and mean local relief for the two
erosion rates in the stream-power model are proportional to
models following tectonic uplift. The sediment-
mean elevation H while in the sediment-¬‚ux-driven model
¬‚ux-driven model closely follows the power-law
they are proportional to the square of H .
trend documented by Montgomery and Brandon
(2002) with an exponent of 1.8 (model results and
power-law trend strongly overlap). The stream-
In order to determine the relationships be-
power model also follows a power-law trend but
tween erosion rate, mean local relief, mean el-
with a smaller exponent of 1.3. Figure 4.10b plots
evation, and speci¬c models of bedrock channel
the relationships between mean erosion rate and
erosion, we must work with a model designed
mean elevation for the stream-power (SP) and
to use the stream-power model and a sediment-
sediment-¬‚ux-driven (SFD) models. Following an
¬‚ux-driven model interchangeably. The model we
initial transient phase, mean erosion rates in the
will consider is similar to that of Figure 4.4 but
stream-power model are proportional to mean el-
it assumes that each pixel in the model is above
evation while in the sediment-¬‚ux-driven model
the threshold for channelization. As such, it does
they are proportional to the square of mean ele-
not include hillslope processes. The stream-power
vation. The fact that the trends between mean
version of the model uses Eq. (4.4) while the
erosion rate and mean local relief differ from
sediment-¬‚ux-driven version uses Eq. (4.25). The
those between mean erosion rate and mean eleva-
values of m and n used in the model are 0.5 and
tion indicates that declines in mountain-belt re-
1.0, respectively (Kirby and Whipple, 2001). Iso-
lief and mean elevation are not identical. For the
static rebound in the model is estimated using
purposes of modeling erosional decay it is mean
Eq. (4.26). Sediment ¬‚ux Q s in Eq. (4.25) is com-
elevation, not relief, that is most signi¬cant
puted in the model by downslope routing of the
because mean erosion rate is the derivative of
local erosion rate computed during the previous
mean elevation with respect to time, while ero-
time step, multiplied by the pixel area.
sion rates and relief are not as simply related.
Model runs reported here use a square do-
Nevertheless, observed relationships between
main of width 128 km subject to uniform vertical
106 THE ADVECTION/WAVE EQUATION



(a)
100 max, sediment-flux-
h driven (SFD)
(km) 50 km

10
mean (H), SFD

10
max, stream-power
mean (H), SP
(SP)
10
800
0 600
400
200
t (Myr)
(b) (c)
0
10 mean relief (Rz )
100
mean relief (Rz )
h
h max
(km) max
mean (H)
(km)
10
mean (H)
10
H ∝ 1/t
10
H ∝ ed’t/t
td = 30 Myr
10
10 stream-power model (SP) sediment-flux-driven model (SFD)
400
200 300
0 800
100 100 t (Myr)
t (Myr)
Fig 4.11 (a) Semi-log plots of maximum and mean elevation where c is a constant. Equation (4.36) has the so-
versus time following the cessation of tectonic uplift for the lution
stream-power and sediment-¬‚ux-driven model of a uniformly
’1
1
uplifted square domain of width 128 km. The bedrock
H = ct + (4.37)
erodibilities of each model have been scaled to predict the H0
identical steady-state landscapes. The sediment-¬‚ux-driven
where H0 is the initial mean elevation following
model exhibits persistent mountain belts relative to the
tectonic uplift.
exponential decay of the stream-power model. (b) Semi-log
The inverse power-law dependence in Eq.
plots of maximum and mean elevations and mean local relief
versus time following uplift for the stream-power model (4.37) predicts a fundamentally different his-
(same model output as in (a)). Also plotted is an exponential tory of mountain-belt decay than the exponen-
decay model with a time constant „d = 30 Myr. (c) Log“log tial model that follows from Eq. (4.29). The in-
plot of maximum and mean elevations and mean local relief
verse power law has a broad tail in which to-
versus time for the sediment-¬‚ux-driven model, illustrating
pography decays much more slowly than pre-
the asymptotic approach to 1/t behavior at large time scales.
dicted by the exponential model at long time
scales. Figure 4.11 compares the topographic de-
mean erosion rates and mean relief are still cay of the stream-power and sediment-¬‚ux-driven
very useful for distinguishing between different models. Figure 4.11a plots the mean and max-
models of topographic evolution. imum elevations and mean local relief of each
Data plotted in Figure 4.10b suggest that topo- model on a semi-log scale as a function of time
graphic decay in the sediment-¬‚ux-driven model following tectonic uplift. For large times, the
is described at large times by mean elevation in the stream-power model de-
cays as e’t/„d with „d ≈ 30 Myr. The sediment-¬‚ux-
‚H
= ’c H 2 (4.36) driven model, however, decays much more slowly
‚t
EXERCISES 107


of 1 cm and a diffusivity of κ = 0.1 cm2 /yr, model
with time. One can think of the inverse power-
the relative concentration of an abrupt pulse
law model as an exponential model in which the
effective time scale (i.e. „d in Eq. (4.29)) is con- of radionuclides into a semi-in¬nite soil pro¬le
with time. Plot the concentration pro¬les for t =
tinually increasing as the topography decays. As
10, 100, and 1000 yr following the fallout event.
such, the inverse-power-law model suggests that
4.2 Assume that the precipitation in a mountain range
there is no single time scale for mountain belt
increases linearly with elevation. In such a case,
decay, but rather a spectrum of time scales with
the stream-power model for a semi-circular basin
increasing values for older mountain belts. The in a humid environment (Eq. (4.12)) becomes
maximum and mean elevations in the stream-
‚h ‚h
power and sediment-¬‚ux-driven models are plot- h
= cx 1 + (E4.1)
‚t ‚x
ted in Figures 4.11b and 4.11c on semi-log and h0
log--log plots, respectively, with the exponential
where h 0 is a characteristic length scale. Using the
and power-law asymptotic trends of each model
method of characteristics, plot the evolution of
also shown.
knickpoints in this system. How do the knickpoints
Erosion in natural bedrock channels likely
change in shape and speed as they propagate up
falls between the two end-member models of
into the headwaters?
stream-power and sediment-¬‚ux-driven erosion,
4.3 Extract a ¬‚uvial channel pro¬le from a topographic
but, as previously noted, ¬eld studies suggest that
map or a DEM starting at the channel head. Using
the sediment-¬‚ux-driven model is more appro- Excel, evolve the channel forward in time accord-
priate for massive bedrock lithologies common ing to Eq. (4.12). Neglect uplift. Choose a value of
in ancient mountain belts. The results described c that produces realistic erosion rates (e.g. 1 m/kyr
in this section illustrate that the stream-power for very steep terrain).
model in its basic form and a sediment-¬‚ux- 4.4 Construct a simple model of bank retreat assum-
ing that the rate of bank retreat is inversely pro-
driven model predict power-law relationships be-
portional to the channel width w:
tween mean erosion rate and mean local re-
lief with exponents of 1.3 and 1.8, respectively.
‚h c ‚h
=
The results of the sediment-¬‚ux-driven model are (E4.2)
‚t w ‚x
more consistent with the analysis of Montgomery
and Brandon (2002). The sediment ¬‚ux-driven with c = 0.001 yr’1 . Assume a channel with an ini-
model predicts that mountain-belt topography tial width of 100 m. As the banks retreat and the
decays proportionally to 1/t, resulting in persis- channel widens, update the value of w in Eq. (E4.2).
tent mountain belts relative to the exponential- How long is required before the channel doubles
in width?
decay model. Conceptually, mountain belts per-
4.5 Cliff retreat can be modeled with the advec-
sist in the sediment-¬‚ux-driven model because
tion equation. Consider a cliff and talus slope
the cutting tools responsible for abrading chan-
as illustrated in Figure 4.12. Initially, the cliff
nel beds decrease over time, thereby decreasing
channel incision rates in a positive feedback.
This mechanism provides a new hypothesis for
mountain belt persistence in massive bedrock
lithologies dominated by the saltation-abrasion
c
process.
H


Exercises
4.1 The migration of radionuclides into a soil by diffu-
sive processes was considered in Chapter 2. Modify
30°
the model to include advective transport. This ra-
tio of the diffusivity to the advection velocity is
Fig 4.12 Schematic diagram of Exercise 4.5.
often called the dispersivity. Given a dispersivity

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