(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

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

108 THE ADVECTION/WAVE EQUATION

and talus slope both have height H /2. Assume Assume that the rock and talus slope have equal

that the cliff retreat can be modeled advec- densities.

tively with retreat rate c = 1 m/kyr. Model the 4.6 Repeat the above for a case in which the channel

evolution of the cliff and talus slope through crosses a major lithologic boundary (e.g. the Kaibab

time. Impose conservation of mass at the base limestone at the rim of the Grand Canyon). Choose

of the cliff (i.e. the volume of rock removed a smaller value of c to represent the more resistant

from the slope must be deposited on the slope). unit.

119° W 118° W

along-dip profile

3.5

(b) 1°

(a)

h

(km)

along-strike

2.5

profile in (b)

0 50

x (km)

San Joaquin R.

along-dip

profile in (b)

37° N

inset in (c)

Kings R.

Kaweah R.

Chagoopa

Boreal

along-strike profile

36° N

Boreal

3.0

Kern R.

h

(km)

Chagoopa

1.0

100 150

0 50

x (km)

3.0 Boreal

(e)

(d)

(c)

longitudinal

h2.5

profile in (f)

(km)

2.0 Chagoopa

Boreal

Kern R.

1.5

x (km)

15

0 30

Chagoopa

(f)

3.0 knickpoints

h

(km)

linear 2.0

Kern R.

profile in (e)

1.0

slopemap

x (km)

50

0 100

S (m/m)

h (km)

0.0 0.5 >1.0

3.0 4.0

1.0 2.0

Plate 1.8 Major geomorphic features of the southern Sierra Nevada. (a) Shaded relief map of topography indicating major rivers

and locations of transects plotted in b. (b) Maximum extents of the Chagoopa and Boreal Plateaux based on elevation ranges of

1750“2250 m and 2250“3500 m a.s.l. Also shown are along-strike and along-dip topographic transects illustrating the three levels

of the range in the along-strike pro¬le (i.e. incised gorges, Chagoopa and Boreal Plateaux) and the westward tilt of the Boreal

Plateau in the along-strike transect. (c) and (d) Grayscale map of topography (c) and slope (d) of the North Fork Kern River,

illustrating the plateau surfaces (e) and their associated river knickpoints (f). Modi¬ed from Pelletier (2007c). Reproduced with

permission of Elsevier Limited.

116° W

117° W Plate 2.18 (a) Location map and

(b)

(a)

Yucca LANDSAT image of Eagle Mountain

Mtn.

Eagle piedmont and adjacent Franklin

Mtn.

Nevada Lake Playa, southern Amargosa

Valley, California. Predominant wind

N direction is SSE, as shown by the

e Amargosa

Ca vad Valley

wind-rose diagram (adapted from

Funeral li a

36.5° N Mtns. fo

January 2003“January 2005 data

rn

ia

Eagle from Western Regional Climate

Mtn. Center, 2005). Calm winds are

Death de¬ned to be those less than 3 m/s.

Valley Qa2

(b) Soil-geomorphic map and

Qa3

oblique aerial perspective of Eagle

Qa4

Black

Mtns. Qa5“Qa7 Mountain piedmont, looking

36.0° N N 10 km Soil-geomorphic map and predominant playa

southeast. Terrace map units are

N pits

sample-pit locations wind direction

based on the regional classi¬cation

by Whitney et al. (2004).

outline of active

wind (c)

Approximate ages: Qa2 “ middle

modern playa

direction

Pleistocene, Qa3 “ middle to late

Eagle

3“6 m/s

calm

Pleistocene, Qa4 “ late Pleistocene,

Mtn.

6“9 m/s

90%

Qa5“Qa7 “ latest Pleistocene to

Franklin Lake

active. (c) Map of eolian silt

1% Playa

thickness on Qa3 (middle to late

2%

Pleistocene) surface, showing

3% maximum thicknesses of 80 cm

close to the playa source,

Amargosa Eagle

River decreasing by approximately a

Mtn.

factor of 2 for each 1 km

secondary hotspot

downwind. Far from the playa,

background values of approximately

1 km 20 cm were observed. Modi¬ed

0 10 20 40 80 cm

Silt thickness on Qa3

from Pelletier and Cook (2005).

(a) 80 (b)

Qa2

u = 5 m/s

Qa3

K = 5 m2/s

Qa4

p = 0.05 m/s

60

z

analytic

silt

depositional

thickness

topography

(cm)

40

(x', y')

cross-wind

direction

y

20

a

u = 5 m/s

K = 10 m2/s source

p = 0.075 m/s

wind x

0

2 3 4

1

0 direction

distance downwind from playa (km)

(c)

2 km

0.1 0.2 0.4 1.0

0

wind direction

numerical model results

u = 5 m/s, K = 5 m2/s, p = 0.05 m/s

wind direction

(d) (e)

numerical model results

numerical model results

u = 5 m/s, K = 5 m2/s, p = 0.05 m/s

1 km

u = 5 m/s, K = 5 m2/s, p = 0.05 m/s

q = 10

q = ’10 q = 30

10 20 40

0 80 cm

Plate 2.19 (a) Plot of eolian silt thickness versus downwind distance, with analytic solutions for the two-dimensional model for

representative values of the model parameters. (b) Schematic diagram of model geometry. Depositional topography shown in this

example is an inclined plane located downwind of source (but model can accept any downwind topography). (c) Color maps of

three-dimensional model results, illustrating the role of variable downwind topography. In each case, model parameters are

u = 5 m/s, K = 5 m2 /s, and p = 0.05 m/s. Downwind topography is, from left to right, a ¬‚at plane, an inclined plane, and a

triangular ridge. Width and depth of model domain are both 6 km. (d) Color maps of three-dimensional model results for

deposition downwind of Franklin Lake Playa, illustrating the role of variable wind direction. From left to right, wind direction is

θ = ’10—¦ (0—¦ is due south), 10—¦ , and 30—¦ . (e) Map of three-dimensional model results obtained by integrating model results over a

range of wind conditions, weighted by the wind-rose data in Figure 2.18a. Modi¬ed from Pelletier and Cook (2005).

(a) Plate 3.1 Comparison of steepest-descent and MFD

¬‚ow-routing algorithms. (a) Steepest descent: a unit of

precipitation that falls on a grid point (shown at top) is

successively routed to the lowest of the eight nearest

neighbors (including diagonals) until the outlet is reached.

Precipitation can be dropped and routed in the landscape in

steepest descent MFD method

any order. MFD: all incoming ¬‚ow to a grid point is

117.05° W 117.00 116.95 116.90

(b) distributed between the down-slope pixels, weighted by bed

2 km slope. To implement this algorithm ef¬ciently, grid points

36.225° N

should be ranked from highest to lowest, and routing should

be done in that order to ensure that all incoming ¬‚ow from

up-slope is accumulated before downstream routing is

36.20

calculated. (b) Map of contributing area calculated with

steepest descent for Hanaupah Canyon, Death Valley,

California, using USGS 30 m DEMs. Grayscale is logarithmic

36.175

and follows the legend at ¬gure bottom. (c) Map of

steepest descent

contributing area computed with bifurcation routing, for the

(c) same area and grayscale as (b). Multiple-¬‚ow-direction

routing results in substantially different and more realistic

¬‚ow distribution, particularly for hillslopes and areas of

distributary ¬‚ow. Modi¬ed from Pelletier (2004d).

MFD

106 107 m2

100 104

102

scour/mixing depth tephra concentration

10%

1% 100%

1.0 m

0.5

0.1 0.2

(a) (b)

vent

location

Plate 3.14 Digital grids output by the modeling of tephra outlet

redistribution following a potential volcanic eruption at Yucca concentration

southwesterly = 1.97%

Mountain. (a) Map of scour-depth grid, calculated from

winds

contributing area grid. (b) Grayscale map of tephra

concentration in channels, calculated using the

scour-dilution-mixing model. Modi¬ed from Pelletier et al.

N 3 km

(2008). Reproduced with permission of Elsevier Limited.

(a) (b)

(c) (d)

Plate 3.7 Illustration of a successive ¬‚ow-routing algorithm applied to Fortymile Wash alluvial fan in Amargosa Valley,

Nevada. (a) Shaded-relief image of a high resolution (1 m/pixel) photogrammetric DEM of Fortymile Wash, (b)“(d) ¬‚ow depths

predicted by the model for different values of the prescribed upstream discharge: (b) 300 m3 /s, (c) 1000 m3 /s, and (d) 3000 m3 /s.

(a) (b)

(c)

Plate 3.11 Field photos illustrating the three-dimensional pattern of contaminant dilution near Lathrop Wells tephra sheet.

(a) and (b) Channel pits on the tephra sheets expose a ¬‚uvially mixed scour zone ranging from 12 to 29 cm in thickness. Two types

of deposits occur beneath the scour zone: the tephra sheet itself (exposed on the upper and lower sheet) and debris-¬‚ow

deposits comprised predominantly by Miocene volcanic tuff and eolian silt and sand transported from the upper tephra sheet.

(c) The effects of dilution visible as high-concentration channels draining the tephra sheet (channel at right, 76% tephra) join with

low-concentration channels (at left, 26% tephra). Tephra concentration in these channels correlates with the darkness of the

sediments, with the dark-colored channel at right joining with the (larger) light-colored channel at left to form a light-colored

channel downstream. From Pelletier et al. (2008). Reproduced with permission of Elsevier Limited.

(b)

116.52° W 116.50° 10 ’2 10’1

116.48° 101 km

100

(a)

N1 km

36.74°N

3%

9%

(d)

36.72°

tephra sheet

36.70° slope*area1/2

(c)

36.68°

73%

Lathrop Wells cone

relative tephra

concentration

36.66°

0.01 0.1 1.0

0.001

topography and source mask

relative

scour/mixing depth 0.1 0.3 1.0

900 1000 m

800

Plate 3.12 Model prediction for tephra concentration and scour/mixing depth downstream from the tephra sheet of the

Lathrop Wells volcanic center. (a) Model inputs include a 10-m resolution US Geological Survey Digital Elevation Model (DEM) of

the region and a grid of source and background concentrations. Input concentration values are: 73% (Lathrop Wells tephra sheet),

9% (distal source region), and 3% (background). (b) Map of stream power in the vicinity of the source region. Scale is logarithmic,

ranging from 10’2 to 101 km. (c) Map of scour/mixing depth (scale is quadratic, ranging from 10 cm to 1 m). (d) Map of tephra

concentration (scale is quadratic, ranging from 0.001 to 1). From Pelletier et al. (2008). Reproduced with permission of Elsevier

Limited.

slope*area1/2

tephra thickness slope

10 ’3 10 ’2 10 ’1 100 km

0.01 0.1 1m 0.01 0.03 0.1 0.3 1.0

(b)

(a) (c)

vent

location

southwesterly

winds

N 3 km

slope*area1/2

slope > 0.3 (17°) > 0.05 km

Plate 3.13 Digital grids used in the modeling of tephra redistribution following a volcanic eruption at Yucca Mountain.

(a) Shaded relief image of DEM and Fortymile Wash drainage basin (darker area). Tephra from only this drainage basin is

redistributed to the RMEI location. (b) Grayscale image of DEM slopes within the Fortymile Wash drainage basin.

(c) Black-and-white grid of areas in the drainage basin with slopes greater than 17—¦ . (d) Black-and-white grid of active channels in

the drainage basin (de¬ned as pixels with contributing areas greater than 0.05 km2 ). All tephra deposited within the black areas of

(c) and (d) are assumed to be mobilized by mass movement, intense rilling, or channel ¬‚ow. Modi¬ed from Pelletier et al. (2008).

Reproduced with permission of Elsevier Limited.

(b) (c)

(a)

Plate 3.15 Maps of tephra concentration in channel sediments following a hypothetical volcanic eruption at Yucca Mountain

with (a) southerly, (b) westerly, and (c) northerly winds. Tephra concentration at the basin outlet varies from a maximum of 0.73%

(for southerly winds) to a minimum value of 0.076% (for northerly winds) for this eruption scenario. Southerly winds result in

higher concentrations because the high relief of the topography north of the repository is capable of mobilizing more tephra than

other wind-direction scenarios. Modi¬ed from Pelletier et al. (2008). Reproduced with permission of Elsevier Limited.

Plate 4.5 Maps of best-¬t

model results for the

sediment-¬‚ux-driven model

(b)“(d) and stream-power

model (f)“(h) starting from

the low-relief, low-elevation

surface illustrated in (a). The

actual modern topography

is shown in (e). The best-¬t

uplift history for the

sediment-¬‚ux-driven model

occurs for a 1-km pulse of

uplift starting at 60 Ma

(t = 0) and a 0.5 pulse

starting at 10 Ma (t = 50 yr

out of a total 60 Myr). The

best-¬t uplift history for the

stream-power model

occurs for a 1-km pulse of

uplift at 35 Ma and a 0.5 km

pulse at 7 Ma. Also shown

are transect locations and

stream location identi¬ers

corresponding to the

erosion-rate data in

Figure 4.6. Modi¬ed from

Pelletier et al. (2007c).

Reproduced with

permission of Elsevier

Limited.

103° W

103° W 117° W

117° W 49° N

49° N

(a) (b)

300 km

42° N

uniform erosion above glacial-limit erosion concentrated near glacial-limit

109° W

109° W

0 compensation 1

Plate 5.11 Map of compensation of glacial erosion assuming (a) uniform erosion within glaciated areas and (b) erosion

concentrated within 500 m altitude of glacial limit. Modi¬ed from Pelletier (2004a).

DEM

(a) Wild Burro

(c) Vamori

downstream

300 m

B

downstream

photo

I

N B I

B I B I

>100 cm

1988 flood depths

50“100

30“50

10“30

0“10

w/h long./width

h

120/ w profile B

0.3

60/

0

I

0/

2500 0

1000 500

0.3 2000 1500 x (m)

headcuts

(b) Dead Mesquite headcuts

incised

B

channel fans

300 m

downstream

I

N

5 km B

N

B I B I B I

Plate 7.9 Examples of oscillating channels in southern Arizona. Alternating reaches: B“braided and I“incised. (a) Wild Burro

Wash data, including (top to bottom) a high-resolution DEM (digital elevation model) in shaded relief (PAG, 2000), a color Digital

Orthophotoquadrangle (DOQQ), a 1:200-scale ¬‚ow map corresponding to an extreme ¬‚ood on July 27, 1988 (House et al., 1991),

and a plot of channel width w (thick line) and bed elevation h (thin line, extracted from DEM and with average slope removed).

(b) Dead Mesquite Wash shown in a color DOQQ and a detailed map of channel planform geometry (after Packard, 1974). (c) Vamori

Wash shown in an oblique perspective of a false-color Landsat image (vegetation [band 4] in red) draped over a DEM. Channel

locations in Figure 7.11a. Modi¬ed from Pelletier and Delong (2005).

early

(a)

40 — vert.

l = 0.2

exag.

middle expansion

contraction

(b)

late porewater

(c)

sediment

Plate 7.14 2D model evolution, with 40— vertical exaggeration. Color plots of compaction rate C shown at (a) early,

(b) middle, and (c) late times in the model. In the early phase, alternating zones of matrix expansion and contraction develop near

the upper boundary as zones of initially higher porosity expand, capture upwardly migrating porewater, and further expand in a

positive feedback. In the second stage, zones of expansion ascend and drive converging ¬‚ow of the matrix into the high-porosity

drumlin core. In the third stage, porewater is squeezed from the matrix until the sediment is fully compacted.

(a) (c) Plate 7.17 Grayscale

maps for average drumlin

43.4° N

width and bedrock depth in

43.6° N

(a) and (b) north-central

New York and (c) and (d)

Wisconsin, east of Madison.

43.2

43.4 (a) and (c) Maps of drumlin

width constructed by

averaging widths within a

2-km square moving

43.0

window. (b) and (d) Maps of

43.2

bedrock depth also

constructed with a 2-km

square moving window and

42.8

drumlin width using USGS groundwater

well data. Curves are drawn

76.0

76.4

76.8 89.2° W

77.2° W 88.8

to highlight areas in which

0 1000 m 0 500 m thick sediment and wide

drumlins (including Rogen

(b) (d)

moraine) coincide.

Genessee

depth-to-bedrock

River Valley

60 m

0

0 60 m

areas of thick till and

= wide drumlins

(b) (c)

(a)

(d) (e)

Plate 7.20 Numerical model results and north-polar topography. (a) Shaded-relief image of Martian north-polar ice cap DEM

constructed using MOLA topography. The large-scale closeup indicates examples of gullwing-shaped troughs, bifurcations, and

terminations. Highest elevations are red and lowest elevations are green. (b)“(e) Shaded-relief images of the model topography,

’h, for (b) t = 10, (c) t = 100, (d) t = 1000, and (e) t = 2000 starting from random initial conditions. The model parameters are

L = 250 (number of pixels in each direction), x = 0.4, T0 = 0.3, „i = 0.05, „ f = 1. In the Barkley approximations to Eq. (7.52),

T0 is combined into two parameters, a = 0.75 and b = 0.01. The model evolution is characterized by spiral merging and

alignment in the equator-facing direction. A steady state is eventually reached with uniformly rotating spirals oriented clockwise or

counter-clockwise depending on the initial conditions. Modi¬ed from Pelletier (2004b).

Chapter 5

Flexural isostasy

Isostasy refers to the buoyant force created

5.1 Introduction when crustal rock displaces mantle rock. Isostatic

balance requires that, over geologic time scales,

the weight of the overlying rock must be uni-

Flexural isostasy is the de¬‚ection of Earth™s litho-

form at any given depth in the mantle. Because

sphere in response to topographic loading and

the mantle acts as a ¬‚uid over long time scales,

unloading. When a topographic load is generated

any lateral pressure gradient will initiate man-

by motion along a thrust fault, for example, the

tle ¬‚ows to restore equilibrium. Isostasy simply

lithosphere subsides beneath the load. The width

says that the hydrostatic pressure gradient in

of this zone of subsidence varies from place to

the mantle must be zero, otherwise the mantle

place depending on the thickness of the litho-

would correct the imbalance by ¬‚owing. Isostatic

sphere, but it is generally within the range of 100

balance requires that the hydrostatic force pro-

to 300 km. Conversely, a reduction in topographic

duced by the topographic load, ρc gh (where ρc is

load causes the lithosphere to rebound, driving

the density of the crust, g is the acceleration due

rock uplift. Flexural-isostatic uplift in response to

to gravity, and h is the elevation of the mountain

erosion replaces approximately 80% of the eroded

belt) be equal to the buoyancy force (ρm ’ ρc )gw

rock mass, thereby lengthening the time scale

(where ρm is the density of the mantle and w is

of mountain-belt denudation by a factor of ap-

the depth of the crustal root) produced by the

proximately ¬ve because erosion must remove

displacement of low-density crustal rocks with

all of the rock that makes up the topographic

higher-density mantle rocks (Figure 5.1):

load and the crustal root beneath it in order

to erode the mountain down to sea level. Given

ρc gh = (ρm ’ ρc )gw (5.1)

the ubiquity of erosion in mountain belts, it is

reasonable to assume that ¬‚exural-isostasy plays Solving for w, the thickness of the crustal root,

a key role in nearly all examples of large-scale gives:

landform evolution. Flexural isostasy also plays

ρc

w=h

an important role in the evolution of ice sheets (5.2)

ρm ’ ρ c

because the topographic load of the ice sheet

For typical crust and mantle densities, e.g. ρc =

causes lithospheric subsidence, thereby in¬‚uenc-

2.7 g/cm3 and ρm = 3.3 g/cm3 , the depth of the

ing rates of accumulation and ablation on the

crustal root is approximately ¬ve times the

ice sheet. In this chapter, we will discuss three

height of the topographic load.

broadly-applicable methods (series and integral

Flexure refers to the forces and displacements

solutions, Fourier ¬ltering, and the Alternating-

involved in bending the elastic lithosphere. Flex-

Direction Implicit (ADI) method) for solving the

ure affects isostasy because small-scale variations

¬‚exural-isostatic equation in geomorphic applica-

in the topographic load (e.g. peaks and valleys)

tions.

110 FLEXURAL ISOSTASY

V

(a)

0.0

3wD

V

0.5

1.0

0.6 1.0

0.2 0.4 0.8

0.0

x/L

V

rc

(b)

rm

0.0

forebulge

8wD

Va3

0.5

Fig 5.1 Isostatic balance requires that a topographic load,

given by ρc g h, be balanced by a much larger crustal root,

which extends to a depth given by Eq. (5.2).

1.0

can be supported by the rigidity of the litho- 2 4 6

0

x/a

sphere. It is only at length scales larger than the

¬‚exural wavelength (i.e. ≈ 100--300 km) that iso- Fig 5.2 (a) Nondimensional de¬‚ection of an elastic beam of

static balance is achieved. The equations used to rigidity D and length L subject to an applied force V at the

describe ¬‚exure were originally developed in the end of the beam. (b) Nondimensional de¬‚ection of an elastic

mechanical engineering literature to describe the beam overlying an inviscid ¬‚uid of different density, subject to

a line load at x = 0. The de¬‚ection of the elastic beam in this

response of elastic beams and plates to applied

case has a characteristic length scale given by ±.

forces. The de¬‚ection of a diving board under

the weight of a diver is a simple example of 2D

¬‚exure. If a diver stands still at the end of the

of a diver at the end of a diving board (x = 0)

board, all of the forces and torques on the diving

of length L can be obtained by integrating four

board must be in balance, otherwise the board

times to obtain the cubic function (Figure 5.2a):

would accelerate. A force-balance analysis of the

diving board indicates that the fourth derivative

V (L ’ x)3

of the displacement w is proportional to the ap- w(x) = (5.4)

3D

plied load q(x) (Turcotte and Schubert, 2002):

The ¬‚exural rigidity of the lithosphere is, in turn,

d4w

= q(x)

D (5.3) controlled by the elastic thickness T e as well as

dx 4

the elastic properties of rock by the relationship

where D is a coef¬cient of proportionality that

de¬nes the ¬‚exural rigidity of the board. The so- E T e3

D= (5.5)

lution to Eq. (5.3) corresponding to the weight V 12(1 ’ ν 2 )

5.2 METHODS FOR 1D PROBLEMS 111

where E is the elastic modulus and ν is the Pois- approximated as acting at a single point along

son ratio of the rocks in the crust. Equation (5.5) a 1D pro¬le. Within a 1D model, such a load is

indicates that the ¬‚exural rigidity is very sensi- referred to as a ˜˜line™™ load. A narrow mountain

tive to the elastic thickness T e . One common tech- range that extends for thousands of kilometers

nique for mapping T e in a region involves compar- along-strike is one example of a model well ap-

ing the topography and gravity ¬elds at different proximated by line loading. In such cases, the

length scales. At small length scales, topography term q(x) on the right side of Eq. (5.6) is equal

to zero except at x = 0 where it is equal to some

and gravity are poorly correlated, while at large

scales they have a high degree of coherence. Lo- prescribed value V . In order to solve Eq. (5.6) un-

cal values of T e can be inferred by mapping the der a line load it is easiest to set q(x) equal to

smallest length scales at which topography and zero for all x and then introduce the load V as

a boundary condition for x = 0. With q(x) = 0,

gravity are strongly correlated.

In this chapter our primary focus is on ver- Eq. (5.6) becomes

tical (i.e. topographic) loads. In some cases, such

d4 w

+ (ρm ’ ρc )gw = 0

D (5.7)

as when a subducting slab exerts a horizontal

dx 4

frictional force on the overriding plate, horizon-

The general solution to Eq. (5.7) is obtained by

tal forces play an important role in lithospheric

integration:

¬‚exure. Such cases are rare, however, in geo-

x x

w(x) = ex/± c 1 cos + c 2 sin

morphic applications. We will also ignore the

± ±

time-dependent response of the mantle to load- x x

+ e’x/± c 3 cos + c 4 sin (5.8)

ing, which can be important for some problems ± ±

(e.g. ¬‚exural isostatic response to Quaternary ice

where c 1 through c 4 are integration constants,

sheets). Once a load is applied, there is necessar-

and

ily some time delay before the crust can respond

1/4

4D

by uplift or subsidence. This time scale is gener- ±= (5.9)

(ρm ’ ρc )g

ally on the order of 104 yr, and generally longer