San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
USA

The Formation of Mountains
Through Crust Crumpling

The is an obvious pattern to a range of mountains when viewed from an airplane. The ridges are generally parallel to each other and parallel to the coast and hence to the techtonic plate of the continent. The separations between the ridges are roughly equal. It appears that the mountains arose from the crumpling of the continental crust above the techtonic plate. Here is an analysis of such crumpling.

Consider a span of crust of length L and thickness H. The width is irrelavant. Suppose the span is compressed by techtonic forces to a length (L−ΔL) and breaks into n subspans. Although n has to be an integer that requirement will be ignored in the following analysis.

The distance from trough to peak is along the mountain surface is then L/n, but horizontally it is (L/n)cos(φ), where φ is the angle at which the crust is tilted upward. Then

n(L/n)cos(φ) = (L−ΔL)
and hence
Lcos(φ) = L(1 − ΔL/L)

Therefore

cos(φ) = 1 − ΔL/L

Thus φ) is entirely determined by ΔL/L

There is also the matter of maintaining the amount of crust material. Before the crumpling that amount was LH. An amount ΔL is displaced and supports the rise of the mountains.

After the crumpling the height of the troughs is still H but there is the height of the peaks of

r (L/n)sin(φ)
and hence
the material
of the mountains is
n[½(L/n)sin(φ)(L/n)cos(φ)] = (L²/(2n)sin(φ)cos(φ)
= L²/(2n)sin(φ)cos(φ)

This must match the material displaced; i.e.,

L²/(2n)sin(φ)cos(φ) = HΔL
which may be
expressed as
(1/2n)sin(φ)cos(φ) = (H/L)(ΔL/L)
or, equivalently
n =½sin(φ)cos(φ)/[(H/L)(ΔL/L)]

The upper profile is the crust with no crumpling. The middle profile is what would occur if there was only one point of fracture of the crust. The problem is that there is not enough material to fill the space under the mountain surface. The third profile depicts the case of multiple fractures (mountain ridges) in which the is exactly enough material to fill the space under the mountain surfaces.

There were two unknowns n and φ and two conditions with two determing parameters (H/L) and ΔL/L; which determined n and φ .

Here is a plot of n versus ΔL/L for H/L=0.1.

Without any degree of compressibility of the rock any amount of compression would result in a large number of tiny ridges.

The peak height is (L/n)sin(φ). Here is a plot of peak height relative to L as a function of ΔL/L for H/L=0.1.


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