Download GE 11a Homework 4: Isostacy and the Geographic

GE 11a Homework 4: Isostacy and the Geographic Cycle
This homework set asks that you use the equations describing isostatic compensation and
a equations for modeling erosion given below to reach conclusions regarding the
elevation of the crust given certain facts about crustal structure and deformation. In all
cases assume:
Continental crust has a density of 2.6 g/cc
Oceanic crust has a density of 2.7 g/cc and a typical thickness of 6 km
Mantle peridotites have a density of 3.2 g/cc, regardless of whether they are in the
asthenosphere or lithosphere (not exactly true, but a simplifying assumption)
.
The equation describing isostacy is: d ρ0 = h. ρ1, where d is the depth to which a rigid
block (assume equal to the crust for most of this problem set) of density ρ1 and height h
sinks into a ductile medium (assume equal to the mantle for most of this problem set) of
density ρ0.
Assume the following equations for modeling erosion:
δE/δt = c.H
H = H0 – F.E
F = (1- ρ1/ ρ0)
E = (H0/F).(1-exp(-Fct))
where:
E is the elevation, in meters above the top surface of the dense medium in which the
eroding block ‘floats’. δE/δt refers to changes in elevation due to erosion
t is time, typically in millions of years
c is an empirical constant
H is the height of the eroding surface above sea level, in meters
H0 is the initial height above sea level before the onset of erosion, in meters
F is a unitless factor describing the extent to which elevation decreases due to erosion
are compensated by elevation increases due to isostacy
ρ0 and ρ1 are defined as for the equation describing isostacy, above.
Erosion can be ignored for problems 1-4:
(1) Assuming that rigid crust (both continental or oceanic) is isostatically compensated by
ductile mantle, and given that the mean topographic contrast between the continents
and oceans is 4 km, how thick are the continents? (Note we are not considering the
isotostatic effect of peridotitic roots of the lithosphere in this problem; i.e., the crust is
the rigid block; the mantle is the fluid, compensating medium.)
(2) Oceanic plateaus are places where voluminous basaltic volcanism caused by hotspot
activity has greatly thickened the oceanic crust. The ocean is typically 1 km deep
over these plateaus, vs. 4 km deep over typical ocean crust. How thick is the crust
beneath oceanic plateaus? (Note we are not considering the isotostatic effect of
peridotitic roots of the lithosphere in this problem; i.e., the crust is the rigid block; the
mantle is the fluid, compensating medium.)
(3) Crystallized basalt has a density of 2.7 g/cc, but at a depth of 30 km it undergoes
metamorphic reactions that convert it to eclogite, which has a density of ca. 4 g/cc.
Consider a column of continental crust having a thickness of 25 km.
• What will be the change in its altitude (i.e., height of its surface vs. some global
reference frame, like sea level) if it is ‘underplated’ by 5 km of basalt?
(‘underplating’ means magmas pond, cool, and crystallize at the moho, thickening
the crust from below).
• What will be the change in its altitude if it is underplated by 15 km of basalt, which
converts to eclogite wherever it is at great enough depth?
• What thickness of underplated basalt (converted to eclogite where appropriate) will
result in no change in surface elevation?
(Note we are not considering the isotostatic effect of peridotitic roots of the
lithosphere in this problem; i.e., the crust is the rigid block; the mantle is the fluid,
compensating medium.)
(4) Consider a block of continental crust that is initially 50 km thick and 300 km wide (its
third dimension can be neglected). This block thickens homogeneously in response
to shortening driven by tectonic convergence on its edges, with no change in the
block’s total volume. Given a rate of shortening of 10 km per million years (that is,
after 10 million years it is 290 km wide, and proportionately thicker), at what rate
does the block’s top surface increase in elevation? Assume the entire block is
continuously isostatically compensated by the mantle. (Note we are not considering
the isotostatic effect of peridotitic roots of the lithosphere in this problem; i.e., the
crust is the rigid block; the mantle is the fluid, compensating medium.)
The following three problems require that you consider erosion.
(5) A block of continental crust that is initially 30 km thick and its top surface is at sea
level (isostacy will let you calculate how high sea level is above the top of the ductile
mantle). It doubles in crustal thickness due to compression (assume this thickening is
geologically instantaneous). What is its elevation above sea level immediately after
thickening? Assuming it takes 100 million years for the block’s elevation to decrease
to 500 meters above sea level, solve for the empirical constant, ‘c’ in the erosion
equation.
(6) Using the value of ‘c’ solved for in problem 5, create a plot of elevation vs. time for a
block of continental crust that is initially an average of 20 km thick, has 2 km of
relief between hills (2.5 km above sea level) and valleys (0.5 km above sea level), and
is suddenly underplated by 5 km of basalt (density 2.7 g/cc). Assume elevation gain
due to underplating is instantaneous and even across the entire block (i.e., hills and
valleys go up together), but that erosion after uplift differs between hills and valleys,
according to their different heights above sea level. Plot the elevation histories of
hills and valleys together. If you need to estimate the average elevation of the block
at any given time, assume the block is 50 % hill and 50 % valley.
(7) In real ‘geographic cycles’, sudden uplift leads to an initial increase in relief as
valleys quickly erode while hills remain untouched, followed by a decrease in relief
as the hills erode more quickly later. Is this what your calculation in problem 6
revealed? If not, what might be wrong with the equation we are using to describe the
relation between erosion and height?