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range from about 1 to 10 kyr.
pulse at 7 Ma. Also shown are transect locations and stream
Geomorphic and stratigraphic observations
location identi¬ers corresponding to the erosion-rate data in
suggest that isostatic rock uplift takes place by
Figure 4.6. For color version, see plate section. Modi¬ed from
Pelletier et al. (2007c). Reproduced with permission of westward tilting along a hinge line located west
Elsevier Limited. of the range front. Sediment loading of the San
Joaquin Valley and the presence of a major high-
angle fault along the eastern escarpment suggest
that isostatic rebound is a maximum at the range
This m/n ratio implicitly includes the effect of in-
crest and decreases uniformly towards the hinge
creasing channel width with increasing drainage
line (Chase and Wallace, 1988). Therefore, we as-
area. In this model experiment, n values of 1
sume here that isostatic uplift has a mean value
and 2 were used, scaling m accordingly to main-
tain a constant m/n = 0.5 ratio. Sediment ¬‚ux equal to Eq. (4.26) but is spatially distributed us-
ing a linear function of distance from the range
is computed in the sediment-¬‚ux-driven model
crest:
using erosion rates computed during the previ-
ous time step. Sediment ¬‚ux and drainage area ρc
L xc
Ui = 1’ E av (4.27)
are both routed downstream on hillslopes and in < xc > ρm ’ ρc
L
98 THE ADVECTION/WAVE EQUATION


uplift was imposed from time t = 0 to 1 Myr (out
where xc is the distance from the range crest
along an east--west transect, L is the distance be- of a total duration of 60 Myr) and a second 0.5-km
tween the crest and the hinge line, and < xc > pulse of late Cenozoic surface uplift was imposed
from t = 50 to 50.5 Myr. Model experiments with
is the average distance from the range crest. For
the results of this section I assumed L = 100 km n = 2 were also performed and yielded a similar
(Unruh, 1991) and ρc (ρm ’ ρc ) = 5. best-¬t uplift history, but the knickpoint shapes
Calibration of K w and K s was performed for in that model were a poor match to the observed
each uplift history by forward modeling the re- knickpoint shapes along the North Fork Kern
sponse to a late Cenozoic pulse of uplift and River. Figures 4.5b--4.5d present color maps of
the model topography at t = 20, 40, and 60 Myr.
adjusting the values of K w and K s to match
the maximum incision rate of 0.3 mm/yr mea- Figure 4.5e illustrates the modern topography
sured by Stock et al. (2004, 2005) along the South of the southern Sierra Nevada with the same
Fork Kings River. For each uplift history consid- color scale for comparison. Figure 4.6a plots the
ered, the values of K s and K w were varied by channel incision rate at three points along the
trial and error until the maximum value of late Kings and South Fork Kings Rivers as a func-
Cenozoic incision matched the measured value tion of time. Location 2 is of particular signi¬-
of 0.3 mm/yr on the South Fork Kings River to cance since that point coincides with the South
within 5%. Late Cenozoic incision rates in the Fork Kings River where Stock et al. (2004, 2005)
model increase monotonically with K s and K w , measured maximum incision rates of 0.3 mm/yr.
so trial values of K s and K w were raised (or low- Each point along the Kings River experiences two
ered) if predicted incision rates were too low (or pulses of incision as knickpoints propagate up-
high). All uplift histories considered in the model stream as topographic waves. Figure 4.6a indi-
experiments assumed two pulses of range-wide cates that incision rates decrease with increas-
surface uplift with the late Cenozoic phase con- ing distance upstream and that knickpoint prop-
strained to occur on or after 10 Ma and on or be- agation is slower following the ¬rst phase of
fore 3 Ma to be consistent with ¬rm constraints uplift compared to the second phase of up-
imposed by previous studies. The magnitude of lift. Both of these behaviors can be associated
uplift during each pulse was tentatively con- with the ˜˜tools™™ effect of the sediment-¬‚ux-driven
strained using the offset of the two major river model. First, larger drainages have more cutting
knickpoints of the North Fork Kern River, which tools to wear down their beds, thereby increas-
suggests an initial pulse of ≈ 1 km surface uplift ing incision rates and knickpoint migration rates.
corresponding to the upstream knickpoint, fol- Second, knickpoint propagation occurs slowly fol-
lowed by a smaller (≈ 0.5 km) pulse correspond- lowing the ¬rst pulse of uplift because of the
ing to the downstream knickpoint. Forward mod- lack of cutting tools supplied by the low-relief
eling indicates that knickpoint relief during the upland plateau. Figure 4.5b, in particular, illus-
model remains nearly equal to the magnitude trates that canyon cutting has affected only a very
of the range-wide surface uplift that triggered small percentage of the landscape even as late as
knickpoint creation except for late stages of the 20 Myr following the initial uplift. In contrast,
model when the uplift of isolated plateau rem- knickpoint and escarpment retreat following the
nants can greatly outpace stream incision. Best- second phase of uplift is enhanced by the up-
¬t uplift histories were determined by visually stream sediment supply associated with the ¬rst
comparing the model-predicted topography with phase of uplift. As a result, the second knickpoint
those of the modern Sierra Nevada, including the travels almost as far upstream as the ¬rst knick-
extents of the Chagoopa and Boreal Plateaux and point in less than 20% of the time.
the major river knickpoints. The results of Figures 4.5b--4.5d and Figures
Figures 4.5b--4.5d illustrate results of the best- 4.6a--4.6b also provide several consistency checks
¬t sediment-¬‚ux-driven model experiment, with that aid in con¬dence building. First, the maxi-
n = 1 and K s = 2.4 — 10’4 (m kyr)’1/2 . In this ex- mum incision rate observed by Stock et al. (2004,
periment, a 1-km pulse of late Cretaceous surface 2005) from 3 to 1.5 Ma along the South Fork Kings
4.4 FLUVIAL-GEOMORPHIC RESPONSE OF THE SOUTHERN SIERRA NEVADA TO UPLIFT 99



0.6 0.03 (a) (b)
E
E (mm/yr)
4 0.2
(mm/yr) 0.02
E, U
h
(mm/yr)
(km)
0.01
max h
0.3
0.3 0 50 100
calibrated with x (km)
1 2 0.1
mean h
Stock et al., Riebe et al.
2
3 calibrated with
mean E
Stock et al. (2004)

mean U
0.0 0 0.0
20 t (Myr) 40 60 20 t (Myr) 40 60
0 0
0.9 0.03 (c) (d)
E
E (mm/yr) 4 0.2
(mm/yr) 1 0.02
E, U
h
0.6
2
(mm/yr)
(km)
0.01
max h
0 50 x (km) 100
2
3 0.1
mean h
0.3
0.3 calibrated with
Stock et al. (2004)
mean E
mean U
0 0.0
0.0
20
10 30
20 0
10 30
0
t (Myr) t (Myr)
Fig 4.6 Plots of erosion rate versus time for the best-¬t rates before and after knickpoint passage. The in-
sediment-¬‚ux-driven model (a) and (b) and stream-power
set graph in Figure 4.5d plots the erosion rate of
model (c) and (d). (a) Plot of erosion rate versus time for
the upland plateau along a linear transect shown
three locations along the Kings and South Fork Kings River
in Figure 4.5b. The hillslope erosion rate E h was
(locations in Figure 4.5), illustrating the passage of two
prescribed to be 0.01 mm/yr based on measured
knickpoints corresponding to the two uplift pulses. The
cosmogenic erosion rates (Small et al., 1997; Stock
late-Cenozoic pulse of incision at location 2 provides a
et al., 2004, 2005). The plot in Figure 4.6a shows
forward-model calibration based upon the cosmogenically
a range of erosion rates on the upland surface
derived incision rates (Stock et al., 2004, 2005) at this
location. The maximum value of the erosion rate at this from a minimum of 0.01 mm/yr (on hillslopes) to
location is matched to the observed rate of 0.3 mm/yr by a maximum value of 0.03 (in channels). Erosion
varying the value of K s . Inset into (a) is the upland erosion
rates in upland plateau channels are controlled
rate along the transect located in Figure 4.5b, illustrating
by both E h and K s . The fact that the model pre-
minimum erosion rates of 0.01 mm/yr and maximum values of
dicts an average erosion rate of ≈ 0.02 mm/yr,
0.03 mm/yr, consistent with basin-averaged cosmogenic
consistent with basin-averaged rates measured
erosion rates of ≈ 0.02 mm/yr. (b) Maximum and mean
cosmogenically (Riebe et al., 2000, 2001), provides
elevations of the model, illustrating the importance of
additional con¬dence in this approach.
isostatically-driven uplift in driving peak uplift. (c) and (d)
Figure 4.6b plots maximum and mean eleva-
Plots of erosion rate and elevation versus time for the
stream-power model, analogous to (a) and (b). Modi¬ed from tion values as a function of time in the model.
Pelletier et al. (2007c). Reproduced with permission of Mean surface elevation increases during active
Elsevier Limited.
uplift but otherwise decreases. Maximum eleva-
tion continually increases through time to a ¬-
River was preceded and followed by much lower nal value of over 4 km as a result of isostatic up-
rates of incision (≈ 0.02--0.05 mm/yr) from 5 to lift of Boreal Plateau remnants driven by canyon
3 Ma and 1.5 Ma to present. The model predicts cutting downstream. Mean erosion and uplift
the same order-of-magnitude decrease in incision rates are also plotted in Figure 4.5e, showing
100 THE ADVECTION/WAVE EQUATION



Boreal
(b)
3.0
(a)




h (km)
2.0
(c) Chagoopa

1.0
Kern R.
x (km) 20
10
0
Late Cretaceous knickpoint (c)
3.0




h (km)
Late Cenozoic knickpoint
2.0

1.0 n = 1
x (km)
0 40 80
Late Cretaceous knickpoint
(d)
3.0
h (km)



Late Cenozoic knickpoint
2.0
(b)

h (km) 1.0 n = 2
2.0 3.0 4.0
0.5 1.0
x (km)
0 40 80
Fig 4.7 Best-¬t sediment-¬‚ux-driven model results for the
this model occurs as two pulses: a 1-km pulse
North Fork Kern River basin (i.e. 1 km of late Cretaceous
in the late Eocene (t = 35 Ma) and a 0.5-km pulse
uplift and 0.5 km of late-Cenozoic uplift), illustrating broadly
in the late Miocene (t = 7 Ma). Figure 4.6c illus-
similar features to those of the actual North Fork Kern River
trates the knickpoints as they pass points along
basin illustrated in Chapter 1. Modi¬ed from Pelletier et al.
the Kings and South Fork Kings River. In this
(2007c). Reproduced with permission of Elsevier Limited.
model, knickpoint migration occurs at a simi-
lar rate in the ¬rst and second pulses of up-
lift, as illustrated by the equal durations between
a gradual increase in mean erosion rate over a
knickpoint passage following the ¬rst and second
40 Myr period to a maximum value of approxi-
pulses. The rates of vertical incision are lower in
mately 0.05 mm/yr. Isostasy replaces 80% of that
the second phase because the knickpoint has a
erosion as rock uplift, as prescribed by Eq. (4.27).
gentler grade. Model results are broadly compa-
Figure 4.7 illustrates the ability of the model to
rable to those of the sediment-¬‚ux-driven model,
predict details of the Sierra Nevada topography
except that the propagation of the initial knick-
using the North Fork Kern River as an example.
point requires only about half as long as the
The model predicts the elevations and extents of
sediment-¬‚ux-driven model to reach its present
the Chagoopa and Boreal Plateaux as well as the
location.
approximate locations and shapes of the two ma-
The results of this model application sup-
jor knickpoints along the North Fork Kern River.
port two possible uplift histories corresponding
The best-¬t results of the stream-power model
to the best-¬t results of the sediment-¬‚ux-driven
are shown in Figures 4.5f--4.5h and 4.6c--4.6d,
with n = 1 and K w = 8 — 10’5 kyr’1 . Uplift in and stream-power models. Only one of these
4.5 THE EROSIONAL DECAY OF ANCIENT OROGENS 101


histories is consistent with the conclusion of high of bedrock channel and coupled bedrock-alluvial
(> 2.2 km) Sierra Nevada in early Eocene time channel modeling using the stream power and
(Poage and Chamberlain, 2002; Mulch et al., 2006), sediment-¬‚ux-driven erosion models.
however. That constraint, taken together with the Analyses of sediment-load data have shown
fact that sediment abrasion is the dominant ero- that erosion rates are approximately proportional
sional process in massive granitic rocks (Whipple to the mean local relief or mean elevation of a
et al., 2000) support the greater applicability of drainage basin. In tectonically inactive areas, for
the sediment-¬‚ux-driven model in this case. example, Pinet and Souriau (1988) obtained
The results of this section are consistent with
E av = 0.61 — 10’7 H (4.28)
the basic geomorphic interpretation of Clark et al.
(2005) that the upland plateaux and associated
where E av is the mean erosion rate in m/yr
river knickpoints of the southern Sierra Nevada
and H is the mean drainage basin elevation
likely record two episodes of range-wide surface
in m. Similar studies have documented approx-
uplift totaling approximately 1.5 km. The results
imately linear correlations between mean ero-
presented here, however, suggest that the ini-
sion rates and mean local relief, relief ratio, and
tial 1-km surface uplift phase occurred in late
basin slope both within and between mountain
Cretaceous time, not late Cenozoic time. In the
belts (e.g. Ruxton and MacDougall, 1967; Ahnert,
sediment-¬‚ux-driven model, the slow geomorphic
1970; Summer¬eld and Hulton, 1994; Ludwig and
response to the initial uplift phase is caused by
Probst, 1998). Equation (4.28) can be expressed as
the lack of cutting tools supplied by the slowly-
a differential equation for mountain-belt topog-
eroding upland Boreal Plateau. If the behavior
raphy following the cessation of tectonic uplift:
of the sediment-¬‚ux-driven model is correct, the
model results suggest that 32 Myr (i.e. the time ‚H 1
=’ H (4.29)
since onset of Sierra Nevadan uplift, according to ‚t „d
Clark et al. (2005)) does not afford enough time
with „d = 16 Myr using the correlation coef¬cient
to propagate the upland knickpoint to its present
location. The self-consistency of the model re- in Eq. (4.28). Equation (4.29) has the solution
H = H0 e ’t/„d , where H0 is the mean basin ele-
sults provide con¬dence in this interpretation.
The model correctly reproduces details of the vation immediately following uplift. Isostatic re-
bound will increase „d by a factor of ρc /(ρm ’ ρc ),
modern topography of the range, including the
where ρm and ρc are the densities of the man-
elevations and extents of the Chagoopa and Bo-
tle and crust, thereby giving „d ≈ 50--70 Myr. This
real Plateaux and the elevations and shapes of
value is about ¬ve times smaller than the age
the major river knickpoints.
of Paleozoic orogens, several of which (e.g. Ap-
palachians, Urals) still stand to well over 1 km in
peak elevation. This is the paradox of persistent
4.5 The erosional decay of
mountain belts.
ancient orogens One possible reason for persistent mountain
belts is the role that piedmonts play in raising
One of the great paradoxes in geomorphology the effective base level for erosion. As a moun-
concerns the ˜˜persistence™™ of ancient orogens. tain belt ages, sediment can be deposited at the
Analyses of modern sediment-load data imply footslope of the mountain, causing the base level
that mountain belts should erode to nearly sea of the bedrock portion of the mountain belt to
level over time scales of tens of millions of years. increase (Baldwin et al., 2003; Pelletier, 2004c).
Several mountain belts that last experienced ac- To explore this effect we need to develop mod-
tive tectonic uplift in the Paleozoic (e.g. Ap- els for the coupled evolution of bedrock chan-
palachians, Urals) still have mean elevations at nels and alluvial piedmonts. Here we consider
or near 1 km in elevation. In this section we ex- a model that couples the stream power model
plore this question using 1D and 2D modeling (Eq. (4.4) to a simple model for alluvial-channel
102 THE ADVECTION/WAVE EQUATION



(b)
(a)
large-k case: k = 4 — 10’3 km2/yr
K = 10’7/yr


t=0
h/h0 h/h0
t = 50 Myr
t = 100 Myr
t = 100
t = 150
h(L) = 0
hb ha

Lm
small-k case: k = 2 — 10’3 km2/yr
K = 10’7/yr
(c) K = 10’7/yr
K = 3 — 10’8/yr
k = 10’3 km2/yr
h/h0 t = 50 Myr
k = 3 — 10’3 km2/yr h/h0
k = 10’2 km2/yr t = 100
10’1
t = 150




10’2
0

condition is applied a small distance L h from the
Fig 4.8 (a) Model geometry and key variables. (b) Plots of
divide (i.e. the hillslope length) to give h b (L h , 0) =
elevation vs. downstream distance for two values of κ. Each
plot shows a temporal sequence from t = 50 Myr“250 Myr in h 0 . Sea level is assumed to be constant, giving
50 Myr intervals. (c) Plots of mean basin elevation vs. time for h a (L , t) = 0. The two remaining boundary condi-
L m = 100 km, L = 500 km, and a range of values of K and κ.
tions are continuity of elevation and sediment
From Pelletier (2004c).
¬‚ux at the mountain front, x = L m :

‚h b ‚h a
Lm
evolution based on the diffusion equation (Paola h b (L m ) = h a (L m ), dx = κ
Ar
‚t ‚x
et al., 1992): Lh x=L m
(4.31)
‚h ‚ 2h
=κ 2 (4.30)
‚t ‚x Equations (4.12) and (4.30) were solved using
where κ is the piedmont diffusivity. The diffu- upwind differencing and FTCS techniques for the
sion equation is a highly simpli¬ed model of pied- bedrock and alluvial portions of the basin, re-
mont evolution, but enables us to make prelimi- spectively. Codes for modeling coupled bedrock-
nary conclusions regarding the role of piedmont alluvial channel evolution are given in Appendix
aggradation on the time scale of mountain-belt 3. The effect of the piedmont diffusivity value on
denudation. mountain-belt denudation is illustrated in Figure

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