Isostacy : a balance problem
Reminder :
Principle :
The lithosphere (upper mantle topped with earth’s crust, stiff) rests on the asthenosphere
(lower mantle, a bit less stiff (cf. Article “Earth is made of hard rock”)). Because the
asthenosphere is a bit less stiff than the lithosphere, it can lose its shape if the lithosphere’s
weight is important (under a mountain for example, given that there is a big crustal root),
which makes the Moho (boundary between the earth’s crust and the mantle, which is called
the lithosphere here) sinking deeper under the mountains ; consequently the lithosphere is
sinking in the asthenosphere too. If the asthenosphere was as stiff as the lithosphere, there
would be no deformation.
This deformation takes place on a geological scale, the unity of which is the million of years
(!). Thus a permanent balance exists.
Isostacy definition : state of balance of the stiff lithosphere on a deeper and more plastic
layer, the asthenosphere.
So there is a permanent balance between the thickness of the earth’s crust and the sinking
which takes place in the asthenosphere below.
Given that the oceanic crust is much less thick than the continental crust (5 to 7 km on
average vs. approximatively 30), the asthenosphere (and so the mantle) will generally be a bit
more sunk below the continents.
It’s the same thing when a very vast glacier (as in the latest ice period) melts : the mantle raise
up. On the other hand, if a very extensive mountain or a glacier forms, the asthenosphere will
end up sinking below, thanks to the weight’s increase.
Local isostacy : case of a mountain
Reminder on a mountain formation
The formation of a mountain (here, I’m not talking about volcanic chains like the Andes or
the Rocky Mountains, rather the Alps or Himalaya) is due to a collision between to
continental masses because of compression zones existing locally. It occurs when two plates
collide. The oceanic crust, which formed the bottom of the ocean and which, at first, separated
these two parts of continents, goes into subduction and ends up disappearing completely
thanks to it (obduction). If the compression movement continues, the two continental masses
collide, which produces deformations, piles of layers that lead to mountain formation. If this
movement continues, the mountain will keep growing.
Of course, this takes place on a geological scale, namely several millions years (or even tens
of millions of years).
While a mountain is forming, thus there is a progressive sinking of the asthenosphere (and so
the lithosphere that tops it, and the mantle in general) under the weight of the forming
mountain. As that mountain keeps growing, the sinking continues, proportionally.
Calculation of crustal root thickness thanks to altitude and isostacy
When a mountain is “ready”, its crustal root can reach up to 70 km depth, which is huge. But,
the more important the crustal root is, the more important the altitude is too. So we can
calculate the deepness of the Moho below this mountain if we know its altitude.
The continental crust’s density is approximatively 2.7 because it is principally composed of
granitic rocks. The lithospheric mantle’s density is 3.3.
Calculation is easy : given the principle of isostacy, the column “without mountain” is as
heavy as the column “with mountain”.
We can write the following equality :
Continental crust thickness x density + crustal root thickness (in green on this side because
it’s made of lithospheric mantle*) x lithospheric mantle’s density = (altitude + continental
crust thickness + crustal root thickness (in yellow here because it’s about a crustal root that
belongs to continental crust) ) x continental crust’s density
* it is just an height equivalence to reach isostatic balance at the compensation surface area.
So : 30 x 2.7 + r x 3.3 = (a + 30 + r) x 2.7
81 + 3.3r = 2.7a + 81 + 2.7r
3.3r = 2.7a + 2.7r
0.6r = 2.7a
R = 2.7/0.6 h = 4.5 a
So the crustal root equals 4.5 times the altitude
Thus, for a 3 kilometers altitude :
r = 4.5 x 3 = 13.5 km
The Moho will be at a depth of (13.5 + 30 + 3) = 46.5 km
Moutain erosion and Moho’s rise thanks to isostacy
If a mountain is under erosion, the asthenosphere will gradually rise, as the erosion goes along
erosion, because the weight decrease it’s undergoing, proportionally. Therefore there will be a
progressive rise of the continental crust to restore the isostatic balance : we are talking about
isostatic rebound.
Then, plutonic rocks like granite will surface not only because of the erosion (removal of
materials above them) but because of the continental crust’s rise due to isostatic rebound too.
We can estimate that for a hundred meter of erosion, there is a chain rise up about 80 m.
That’s why mountain chains keep “young” for very long time : the altitude’s decrease created
by erosion is largely compensated by the isostatic rise.
Regional isostacy : case of Scandinavia
Observations in Scandinavia : the coast is lower today than it was before (+15m in 500 BC
and +30m in 1700 BC). Scandinavia’s average vertical displacement is about 4 mm/year.
The lithosphere can be affected by vertical movements resulting from the implementation or
the disappearance of an excess load. So, in Scandinavia, we can at present observe an upthrust
of the lithosphere, following the icecap’s melting between 15 000 and 17 000 BC.
These movements give evidence of the temporary break of the balance between the
lithosphere and the asthenosphere : the return to balance occurs gradually, on a geological
scale.
At the end of a glacial era, ocean’s level tends to rise because of glaciers’ melting that brings
liquid water into oceans (level that had fallen before, because of that lack of liquid water,
which was frozen). And we now know that this water’s rising is partially compensated, on the
long term, in regions like Scandinavia, by the fall of ocean’s level due to the isostatic rebound.
Summary in images:
Calculation about Moho’s depth :
Calculation of x1 :
Thickness of the continental crust x continental crust’s density + crustal root’s thickness x1 x
mantle density = (height + continental crust’s thickness + crustal root’s thickness x1) x
continental crust’s density
So with figures :
30 x 2.7 + x1 x 3.2 = (3 + 30 + x1) x 2.7
81 + 3.2 x1 = 89.1 + 2.7 x1
(3.2 – 2.7) x1 = 89.1 – 81
0.5 x1 = 8.1
x1 = 16.2
The crustal root’s thickness is 16.2 km
A bit more complicated : calculation of x2 :
Thanks to the picture, we can see that:
Water height x water density + height x2 x continental crust’s density + height y x mantle
density + height x1 x mantle density = continental crust’s height x continental crust’s density
+ height x1 x mantle density
4 x 1 + x2 x 2.7 + y x 3.2 + x1 x 3.2 = 30 x 2.7 + x1 x 3.2
If we watch carefully, we can notice that :
4 + x2 + y = 30
so y = 30 – x2 – 4 = 26 - x2
If we simplify (we remove x1 which is in the two sides) and replace y by its equivalent,
calculated just above :
4 + 2.7 x2 + (26 – x2) x 3.2 = 81
4 + 2.7 x2 + 83.2 – 3.2 x2 = 81
87.2 – 81 = 3.2 x2 – 2.7 x2
6.2 = 0.6 x2
x2 = 10.33
So x2 is 10.33km high