rate as thrust-belt migration. Basins naturally where w(x) is the ¬‚exural pro¬le, given by Tur-

evolve toward this condition because when sed- cotte and Schubert (2002) as

iment progradation lags behind thrust-belt mi- x

w(x) = ’w 0 e’x/± cos (5.36)

±

gration at any point along its pro¬le, the basin

steepens locally until the two are in balance. where w 0 is the basin depth beneath the thrust

Conversely, if sediment progradation outstrips front and ± is the ¬‚exural parameter. Equa-

thrust-belt migration, slope decreases and progra- tion (5.36) corresponds to the ¬‚exural pro¬le cor-

dation slows. This approach is analogous to that responding to a line load. This is only an approx-

of the Kenyon and Turcotte model of deltaic sed- imation to the Andes, where loading is spatially

imentation considered in Chapter 2. In this case, distributed.

however, we will also consider the pattern of Combining Eqs. (5.33) and (5.35) and as-

¬‚exural subsidence that controls accommodation suming steady-state conditions gives the time-

space in the basin. independent equation:

Basin evolution in the model is governed by

d2 h dh dw

κ +F =F

the diffusion equation: (5.37)

2

dx dx dx

‚h ‚ 2h which can be written as a ¬rst-order differential

=κ 2 (5.33)

‚t ‚x equation for the slope S = dh/dx with solution

where h is the elevation, t is time, κ is the diffu-

’e’F x/κ x

dw

e’F x /κ

S(x) = Qs ’ F dx

sivity, and x is the distance from the thrust front.

κ dx

0

Boundary conditions are needed at both ends of

(5.38)

the basin to solve Eq. (5.33). At the thrust front,

Substituting Eq. (5.36) into Eq. (5.38) gives

a constant sediment ¬‚ux Q s (in m2 /yr) enters the

’e’F x/κ ±F ±F

F x x x

e’(F /κ+1/±)x

S(x) = Qs ’ ’ sin ’ 2sin ’

cos (5.39)

κ (± F /κ)2 + 2± F /κ + 2 κ ± ± ± κ

The basin topographic pro¬le is obtained by inte-

basin, providing a ¬‚ux boundary condition

grating Eq. (5.39) analytically or numerically and

‚h Qs enforcing the basin-outlet boundary condition to

= (5.34)

‚x κ constrain the integration constant.

x=0

Model solutions are plotted in Figure 5.7, rep-

In this 2D framework, the sediment ¬‚ux is given

resenting the effects of varying Q s on basin pro-

by ELd , where E is the basin-averaged erosion rate

¬les around a reference case with parameter val-

and L d is the upstream drainage basin length.

ues F = 10 mm/yr, ± = 150 km, w 0 = 5 km, κ =

At the basin outlet, a constant base-level eleva-

5000 m2 /yr, Q s = 10 m2 /yr, and assuming an in-

tion boundary condition is prescribed. Two cases

¬nite basin. Figure 5.7a illustrates three impor-

can be considered. For an in¬nite basin, h = 0 as

tant points. First, the basin pro¬le is highly sen-

x = ∞. For a basin of length L b and base-level ele-

sitive to incoming sediment supply: the topo-

vation h b (determined by sea level or a valley-¬‚oor

graphic relief of the basin doubles when Q s is

channel), the boundary condition is h(L b ) = h b . In

increased by 50% from 8 to 12 m2 /yr. This suggests

the moving reference frame, the foreland basin

that basin pro¬les may provide useful constraints

moves towards the thrust front and enters the

on upstream sediment supply through forward

¬‚exural depression with velocity F , the thrust-

modeling and comparison with observed pro¬les.

migration velocity. This motion is represented by

Second, as sediment supply decreases, the basin

an advection equation:

undergoes a transition from over¬lled to under-

‚h ‚w ‚h ¬lled (closed) basins. As such, the model provides

=F ’ (5.35)

quantitative criteria for topographic closure in

‚t ‚x ‚x

118 FLEXURAL ISOSTASY

0.8

(b)

reference case:

overfilled

e’Fx/k F = 10 mm/yr

k=

a = 150 km

Qs = 12 m2/yr

0.6 w0 = 5 km 10 000 m2/yr

k=

Qs = 10 m2/yr k = 5000 m2/yr

5000 m2/yr

Qs = 8 m2/yr Qs = 10 m2/yr

h (km)

k=

1500 —

0.4 2500 m2/yr

underfilled

vert. exag.

(closed basin)

0.2 forebulge k=

1000 m2/yr

sensitivity to Qs sensitivity to k

0.0 infinite basin

infinite basin

(a) no forebulge erosion

no forebulge erosion

0.8 Ld

sensitivity to Qs

h

finite basin

(c) (d)

no forebulge erosion

0.6 Qs = 12 m2/yr

Qs = 10 m2/yr flux boundary

Q

h (km)

Qs = 8 m2/yr

dh/dx = Qs /k sediment flux s

F k

0.4 diffusivity

migration

basin-fill profile

rate

(diffusion)

decollement

x=0

0.2 slope

x

flexural profile

w0

0.0

w

800 1000

400 600

200

0

x (km)

cause sediment back¬lling at the thrust front to

Fig 5.7 (a) and (b): Model solutions for the topographic

produce steep, short basins. Surprisingly, higher

pro¬le of an in¬nite basin assuming no forebulge erosion,

diffusivities (e.g. more humid conditions) pro-

showing sensitivity to variations in Q s and κ, respectively,

around a reference solution with F = 10 mm/yr, ± = 150 km, mote basin closure if sediment supply is held

w 0 = 5 km, κ = 5000 m2 /yr, and Q s = 10 m2 /yr. (c) Solutions constant. Naively, we might expect more humid

for a ¬nite basin of length L b = 1000 km for the same model conditions to result in steeper, more over¬lled

parameters as in (a). (d) Schematic diagram of the model.

basins. However, to understand the in¬‚uence of

Modi¬ed from Pelletier (2007b).

climate on basin geometry we must consider

the effects of climate on sediment supply (i.e.

foreland basins. Third, the distal basin pro¬le is weathering) and basin transport rates indepen-

approximated by a simple exponential function dently. Greater sediment supply puts more total

with length scale κ/F . This result suggests that sediment into the basin and therefore promotes

distal basin pro¬les can be used to uniquely and steeper and more over¬lled basins. Diffusivity val-

easily constrain κ values if thrust-belt migration ues, on the other hand, control how uniformly

rates are known. that sediment is spread across the basin. Higher

Figure 5.7b illustrates the sensitivity of model diffusivity values promote closed basins because

solutions to variations in κ around the reference a larger fraction of the total sediment is trans-

value of κ = 5000 m2 /yr. Diffusivity values are ported past the foredeep, leaving behind a depres-

sion if the ratio Q s /κ is suf¬ciently small. These

controlled by upstream basin length, precipita-

tion, and sediment texture (with longer basins, results emphasize that the key to understanding

higher precipitation rates, and ¬ner textures pro- basin geometry is not simply the sediment sup-

moting greater diffusivity values) (Paola et al., ply to the basin, but the ratio of sediment supply

1992). Low diffusivity values (e.g. κ = 1000 m2 /yr) to transport capacity (e.g. as emphasized in the

5.4 MODELING OF FORELAND BASIN GEOMETRY 119

geomorphic context by Bull (1979)). Figure 5.7c on crustal shortening rates. Flexural modeling

illustrates model solutions for a ¬nite basin us- coupled with geophysical and geomorphic obser-

ing the same model parameters as in Figure 5.7a. vations constrain the ¬‚exural parameter to be

Shorter basins have lower basin relief, all else be- approximately 150 km (resulting in a foredeep ap-

ing equal. proximately 250 km wide and a forebulge at a

Equation (5.40) assumes no forebulge erosion distance 400 km from the thrust front) (Coud-

(i.e. that the forebulge is buried or, if exposed, ert et al., 1995; Watts et al., 1995; Horton and

that the erosion rate is low). In some cases, it may DeCelles, 1997; Ussami et al., 1999). These data

be more appropriate to assume that the forebulge provide relatively ¬rm constraints on three of

the ¬ve model parameters: F = 10 mm/yr, ± =

erodes completely to sea level. In this alternative

150 km, w 0 = 4 km.

end-member scenario, Eq. (5.36) is amended to

w(x) = 0 for x > 3π±/4 (i.e. the point at which Figure 5.9a illustrates a map of the topog-

the forebulge rises above sea level). In this case, raphy of the central Andes and the adjacent

foreland basin from 17—¦ S to 31—¦ S. This region

the slope is given by Eq. (5.36) for x < 3π±/4,

and by

’e’F x/κ ±F ±F

F 1 3π ±

F

1.414e’( κ + ± ) 4

S(x) = Qs ’ 1+ + (5.40)

κ /κ)2 + 2± F /κ + 2 κ κ

(± F

for x > 3π ±/4. Model solutions assuming fore- corresponds to a humid-to-arid transition in the

Andes based upon mean winter precipitation val-

bulge erosion are given in Figure 5.8.

ues above 1500 m elevation. Associated with this

This model framework also provides a simple

climatic transition is a geomorphic transition

means to estimate the basin sediment delivery

from strongly over¬lled (700 m in relief) to weakly

ratio (the ratio of the sediment yield to erosion

over¬lled (100 m in relief) basins. Closed basins

rate). The sediment ¬‚ux at the entrance to the

(e.g. L. Ambargasta and L. Mar Chiquita) also oc-

foreland basin is Q s . For a basin of length L b ,

cur near the southernmost transect, but they are

the ¬‚ux leaving the basin is computed by evaluat-

ing Eq. (5.40) at x = L b and multiplying by κ. For located in wedgetop depozones of a broken fore-

land and hence are not well described by the

basins that have a base-level control located be-

model.

yond the forebulge distance, Eq. (5.40) reduces to

Three topographic pro¬les were extracted

a simple exponential function with length scale

κ/F . In the reference case, for example, a 500-km from the Shuttle Radar Topographic Mission

(SRTM) data and are plotted in Figure 5.9b. To

wide basin has a sediment delivery ratio of 1/e

= 0.37, and a 1000-km wide basin has a value model these pro¬les, it is most accurate to use

of 1/e2 = 0.14. This latter value is broadly consis- ¬nite-basin solutions constrained by values of h b

and L b for each pro¬le. The base level eleva-

tent with measured sediment delivery ratios for

tion h b of the Andean foreland is controlled by

large drainage basins in areas of active tectonism

Rio Paraguay, which ranges in elevation from 80

(Schumm, 1977).

to 10 m (decreasing from north to south) at dis-

As a test of the model, we consider the mod-

tances L b varying from 350 to 720 km from the

ern foreland basin of the central Andes. The

thrust front. Model results in Figure 5.7a sug-

latest phase of Andean deformation responsible

gest that the distal basin pro¬le is an exponential

for Eastern Cordilleran and Subandean uplift is

function with length scale κ/F . The inset plot in

generally considered to be late Miocene in age

Figure 5.9b showing hh b (note vertical logarith-

(Gubbels et al., 1993). The foreland basin is ap-

mic scale) as a function of x con¬rms this pre-

proximately 3.5--4.5 km thick beneath the thrust

diction for the central Andean foreland. Best-¬t

front based on seismic re¬‚ection data (Horton

exponential pro¬les (straight lines on this loga-

and DeCelles, 1997). DeCelles and DeCelles (2001)

rithmic scale) provide estimates of κ = 1500 m2 /yr

estimated the thrust-migration rate to be approx-

for the northern pro¬le and κ = 4000 m2 /yr for

imately 10 mm/yr for the central Andes based

120 FLEXURAL ISOSTASY

result of assuming uniform κ values. Down-

(a) 0.8 reference case:

stream ¬ning occurs in all basins, and therefore

F = 10 mm/yr

overfilled

± = 150 km

κ values are likely to increase somewhat down-

0.6 w0 = 5 km

h stream, resulting in gentler distal-basin slopes

Qs = 12 m2/yr κ = 5000 m2/yr

Qs = 10 m2/yr

Qs = 10 m2/yr

(km) compared to the model solutions of Figure 5.9b.

Qs = 8 m 2

/yr 1500x

0.4 vert. exag.

underfilled

(closed basin)

5.5 Flexural-isostatic response to

0.2 forebulge

glacial erosion in the western

sensitivity to Qs

0.0 US

infinite basin

forebulge erosion

(b) 0.8

The pace of alpine glacial erosion increased sig-

ni¬cantly starting 2--4 Myr ago when global cool-

0.6

h ing initiated the Plio-Quaternary era of large con-

κ=

10,000 m2/yr

(km) tinental ice sheet advances and retreats (Zhang

0.4 et al., 2001). This erosion must have triggered sig-

κ=

5000 m2/yr

ni¬cant ¬‚exural-isostatic unloading of glacially

κ=

2500 m2/yr carved mountain belts. Since glacial erosion is

0.2 κ=

strongly focused in the landscape, scouring out

1000 m2/yr

sensitivity to κ

valley ¬‚oors where ice ¬‚ow is focused but erod-

0.0 infinite basin

forebulge erosion

ing the highest points in the landscapes rela-

800 1000

400 600

200

0 tively slowly, it is likely that glaciated mountain

x (km) belts have increased in relief during the Plio-

Quaternary era due to isostatic rock uplift, even

Fig 5.8 (a) and (b) Model solutions for the topographic

in areas that are tectonically inactive. Determin-

pro¬le of an in¬nite basin assuming complete forebulge

ing precisely how much glacial erosion has taken

erosion, showing sensitivity to variations in Q s and κ,

place is dif¬cult to do in most cases. Nevertheless,

respectively, for the same model parameters as Figures 5.7a

it is still useful to ask how much rock uplift oc-

and 5.7b. These results show that model results are broadly

similar to those obtained by assuming no forebulge erosion; curred for a given amount of glacial erosion. By

the only signi¬cant difference is a minor reduction in basin using the geographic extent of glacial cover as a

relief when forebulge erosion is assumed.

proxy for the spatial distribution of erosion, we

can use the 2D ¬‚exural-isostatic model to com-

pute the ratio of late Cenozoic erosionally driven

the two southern pro¬les using F = 10 mm/yr. rock uplift to glacial erosion. In this section we

Given observed or inferred values for h b , L b , and compute that ratio for the western United States,

κ, as well as estimated values for F , ±, and w 0 following the approach of Pelletier (2004a). Ar-

from the published literature, a family of solu- eas where this ratio is large have likely under-

tions corresponding to a range of Q s values can gone the greatest relief production and should

be generated for each pro¬le and compared to be the focus of future efforts to identify signa-

the observed data. Figure 5.9c illustrates the ob- tures of erosionally driven rock uplift and re-

served pro¬les plotted with their corresponding lief production. In this application we will use

best-¬t pro¬les (κ values for the two southern- the Alternating-Direction-Implicit technique, tak-

most pro¬les were also varied to ¬nd the optimal ing into account spatial variations in ¬‚exural

¬t for both κ and Q s ). Model solutions match rigidity.

the observed data very closely for the strongly The ratio of rock uplift to erosion depends on

over¬lled pro¬le, and they match the ¬rst-order the area and shape of the eroded region and on

trends of the weakly over¬lled pro¬le. Discrep- the local ¬‚exural wavelength of the lithosphere.

ancies in the southernmost pro¬les may be the If the ¬‚exural wavelength is large compared to

5.5 FLEXURAL-ISOSTATIC RESPONSE TO GLACIAL EROSION IN THE WESTERNUS 121

68° W 57° W 0.8 1

(b)

k = 1500 m2/yr

17° S h ’ hb

(km) k = 4000 m2/yr

0.6

0.1

h

sediment-rich (km)

0.4

0.01

400

0 200 600

x (km)

0.2

0.0

600

0 200 400

x (km)

Rio Paraguay

(c)

k = 1500 m2/yr

Qs = 9 m2/yr

0.6

h ’ hb

(km)

sediment-starved 0.4 k = 2000 m2/yr

Qs = 6.5 m2/yr

k = 2000 m2/yr

Qs = 5 m2/yr

0.2

100 km (a)

31° S L. Mar Chiquita

L. Ambargasta

0.0

elevation