<< . .

. 22
( : 51)

. . >>

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
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).
(a) Wild Burro
(c) Vamori
300 m
>100 cm
1988 flood depths
w/h long./width
120/ w profile B
2500 0
1000 500
0.3 2000 1500 x (m)

(b) Dead Mesquite headcuts
channel fans
300 m

5 km B

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).
40 — vert.
l = 0.2

middle expansion

late porewater


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.
43.4 (a) and (c) Maps of drumlin
width constructed by
averaging widths within a
2-km square moving
window. (b) and (d) Maps of
bedrock depth also
constructed with a 2-km
square moving window and
drumlin width using USGS groundwater
well data. Curves are drawn
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.

River Valley
60 m
0 60 m
areas of thick till and
= wide drumlins
(b) (c)

(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
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)




0.6 1.0
0.2 0.4 0.8


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).

can be supported by the rigidity of the litho- 2 4 6
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)
plied load q(x) (Turcotte and Schubert, 2002):
The ¬‚exural rigidity of the lithosphere is, in turn,
= 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 )

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
¬‚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-
ily some time delay before the crust can respond
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

<< . .

. 22
( : 51)

. . >>