Page 217 - Physical Principles of Sedimentary Basin Analysis

7.2 Thickness of crustal roots 199

continental crust h
water w
oceanic crust c
d

lithospheric mantle

Figure 7.4. The continent has a height h above sea level and a crustal root of thickness d below the
oceanic crust. The oceanic crust has a thickness c with water of depth w above.

h c = d ( m − c ) − w( c − w ), (7.13)

which says that weight of the continent above sea level is equal to the buoyancy from the
mantle displaced by the root of the continental crust minus the buoyancy from the water
being displaced by the continental crust. The thickness of the crustal root of the continent
becomes
c ( c − w )
d = h + w. (7.14)
( m − c ) ( m − c )
As an example we can calculate the crustal root of a mountain range with an elevation 3 km
above sea level when there is an ocean with depth 1 km over the oceanic crust. The densities
of the crust and mantle are c = 2900 kg m −3 and m = 3300 kg m −3 , respectively. The
thickness of the crustal root is then d = 26.5 km. The total thickness of the continental
crust becomes h tot = h + w + c + d = 37 km, when an average thickness c = 6kmis
used for the oceanic crust. This estimate is close to the average thickness of the continental
crust, which is 35 km.

Exercise 7.4 A floating iceberg is similar to a continent floating on the asthenosphere. Use
the same reasoning as above to show that the submerged base of an iceberg is nine times
the size of its top when the density of ice is 90% of the water density.

Exercise 7.5 A mountain erodes and becomes 1 km lower. How much has then been
eroded from the mountain?
Solution: A mountain of height h has a crustal root of depth b. Airy isostasy gives that
(h + b) c = b m , where c and m are the crustal and mantle densities, respectively. The
crustal root becomes b = h c /( m − c ). The height of the mountain is reduced with h,
and the crustal root is therefore reduced with b = h c /( m − c ). The total amount of
crust that is eroded from the mountain is then

m
h + b = h. (7.15)
m − c
A crustal density c = 2800 kg m −3 and a mantle density c = 3300 kg m −3 give that
6.6 km must be eroded from the mountain to reduce its height by h = 1km.