Isostasy Exercises - Answers
ice
h
b
ρ≈0.9 g/cm3
water
ρ≈1 g/cm3
- For a small ice block 10 centimeters on a side, how much of the ice cube will lie beneath the
water? Hint: just substitute the given values for the density of water (ρw), the density of ice (ρi),
and the height of the ice cube (h) into the equation and solve for b.
ρi h = ρw b
(0.9 g/cm ) (10 cm) = (1 g/cm ) i h
3
3
(0.9 g/cm ) 10 cm = h
(
)
(1 g/cm )
3
3
h = 9 cm below the surface, 1 cm above
(you’ve probably heard the saying that nine-tenths of an iceberg lies under the water)
V.J. DiVenere - 2/13/12
“continental
crust”
thickness
B)
Now let's do the same thing but with rock. The Earth's solid mantle underlies the crust.
Because of the extremely high temperatures and pressures in the mantle, the solid ultramafic
peridotite rock that it is made of actually deforms very slowly in the ductile state (bends, folds,
stretches, flows in the solid state). The mantle will be our “fluid” and take the place of the water
in the example above. The crust is rigid and less dense than the mantle. In a real sense, the crust
"floats" on the mantle. The crust will take the place of our ice cube in the previous example.
The density of the continental crust (ρc) is about 2700 kg/m3 (kilograms per cubic meter), the
density of ocean crust (ρo) is 2900 kg/m3, and the density of the mantle (ρm) is 3300 kg/m3. The
density of water is 1000 kg/m3 (1 g/cm3).
“mantle”
- How much crust would sink down into the mantle and how much would remain above the
surface of the mantle if a 35 km thick block of continental crust were resting on the mantle?
ρc h = ρm b
(2700 kg/m ) (35 km) = (3300 kg/m ) i b
3
3
(2700 kg/m ) (35 km) = b
(3300 kg/m )
3
3
(
)
b = 0.82 i 35 km = 28.6 km below
35 km - 28.6 km = 6.4 km above
(that’s ~8 tenths below and 2 tenths above)
V.J. DiVenere - 2/13/12
C) - Find the thickness of the mountain root (r in the figure below) and the elevation above the
surrounding crust of the mountain block (e).
The thick crust has the same density as normal continental crust and a thickness of 65 km (hm).
The normal continental crust has a thickness of 35 km (hc). Densities were give in Part B,
above. To compare the weights we only need to consider the densities & thicknesses as above
because the length, width, and gravitational acceleration cancel out.
(weight of thick crust) = (weight of normal crust) + (weight of mantle of thickness r)
ρc hm = ρc hc + ρm r
e
thickened
continental
crust
hc
hm
r
mantle
(2700 kg/m ) (65 km) = (2700 kg/m ) (35 km) + (3300 kg/m ) i r
3
3
3
(2700 kg/m ) (65 km) − (2700 kg/m ) (35 km) = r
(3300 kg/m )
3
3
3
r = 24.5 km below bottom of normal crust
e = 65 km - 35 km - 24.5 km = 5.5 km above top of normal crust
V.J. DiVenere - 2/13/12
D) Now imagine that millions of years of weathering and erosion have removed 20 km from
the top of the thickened crust.
- How much will the crust rise up? (Consider how deep into the mantle the root of the mountain
will be, r in the figure, for 65 km - 20 km = 45 km thick crust, compared to in part C)
(2700 kg/m ) (45 km) = (2700 kg/m ) (35 km) + (3300 kg/m ) i r
3
3
3
(2700 kg/m ) (45 km) − (2700 kg/m ) (35 km) = r
(3300 kg/m )
3
3
3
r = 8.2 km below bottom of normal crust
so, mountain crust (bottom) isostatically rose up from 24.5 below to 8.2 km below
24.5 km - 8.2 km = 16.3 km
20 km erosion yielded 16.3 km uplift (or 0.82 * 20 km)
- How high above the top of normal crust, e, in the figure, will the mountain now be?
e = 45 km - 35 km - 8.2 km = 1.8 km above top of normal crust
- How much lower are the mountains after the 20 km of erosion?
old elevation - new elevation = 5.5 km - 1.8 km = 3.7 km
so, 20 km of erosion only resulted in 3.7 km lowering of the mountains!
(3.7 km lower = 0.18 * 20 km, complement of the 0.82 * 20 km uplift)
- Surprised?
Wow!
V.J. DiVenere - 2/13/12
E) Optional - Extra Credit
Now let's look at continental crust vs oceanic crust. Similar to B above, equalte the weight of a
column through continental crust to the weight of a column through the oceanic crust and mantle
to the depth of compensation (the base of the continental crust). The continental crust is 35 km
thick, oceanic crust is 7 km thick, and the ocean depth is 4.5 km.
- Calculate the excess depth of the continental crust below the bottom of the ocean crust and the
height of the continent above sea level.
ocean water
ocean crust
continental
crust
mantle
ρcc hcc = ρw hw + ρoc hoc + ρm r
(2700 kg/m ) (35 km) = (1000 kg/m ) (4.5 km) + (2900 kg/m ) (7 km) + (3300 kg/m ) i r
3
3
3
3
(2700 kg/m ) (35 km) − (1000 kg/m ) (4.5 km) − (2900 kg/m ) (7 km) = r
(3300 kg/m )
3
3
3
3
r = 21.1 km below bottom of ocean crust
e = 35 km - 4.5 km - 7 km - 21.1 km = 2.4 km above sea level
- Compare the difference in elevation of the continents and sea floor calculated above with the
hypsographic curve and bar graph of Fig. 1.2 in the textbook (Frisch et al., 2011) showing the
distribution of elevations on the continents and depths in the ocean.
the calculations are a little high for an average elevation, but in the right general
ballpark, using as we did, rough figures for density, etc.