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

for narrower loads. The transient response of the We can use the symmetry of the problem to spec-

lithosphere to loading and unloading can gener- ify the boundary conditions. The value of w must

ally be neglected over time scales of interest in go to zero as x goes to ∞ and dw/dx must go to

mountain building. zero at x = 0. This implies that c 1 and c 2 are zero

The equation for the 1D ¬‚exural-isostatic dis- and c 3 = c 4 to give

placement of a uniformly rigid lithosphere to ver- x x

w(x) = c 3 e’x/± cos + sin (5.10)

tical loading and unloading is: ± ±

d4w The value of c 3 is proportional to the applied load

+ (ρm ’ ρc )gw = q(x)

D (5.6)

dx 4 V . From Eq. (5.3) and Eq. (5.10) we have

This equation combines ¬‚exural bending stresses d3 w

1 4D c 3

V=D = (5.11)

(¬rst term on left side), the upward buoyancy ±3

dx 3

2 x=0

force exerted on the bottom of the lithosphere

We used 1 V for the load in Eq. (5.11) because we