Question

A mountain range has an elevation of \(5 \mathrm{~km}\). Assuming that \(\rho_{m}=3300 \mathrm{~kg} \mathrm{~m}^{-3}, \rho_{c}=2800 \mathrm{~kg}\) \(\mathrm{m}^{-3}\), and that the reference or normal continental crust has a thickness of \(35 \mathrm{~km}\), determine the thickness of the continental crust beneath the mountain range. Assume that hydrostatic equilibrium is applicable.

Short answer

The thickness of the continental crust beneath the mountain range is 63 km.

Step-by-step solution

Understand Hydrostatic Equilibrium

In hydrostatic equilibrium, the pressure exerted by the weight of the overlying material is balanced by the pressure supporting it. For the mountain range, the extra elevation is compensated by an increase in crust thickness. Here, \[ ho_m \] is the density of the mantle (3300 kg/m³) and \[ \rho_c \] is the density of the continental crust (2800 kg/m³).

Write the Hydrostatic Equilibrium Equation

We use the principle of hydrostatic equilibrium to write:\[ \rho_c h_c = \rho_m (H_m - h_m) \]where \( h_c \) is the thickness of the continental crust with the mountain range, and \( h_m \) is the thickness of the 'root' beneath the mountain required to maintain equilibrium.

Express the Variables

Start by expressing the thickness of the root \( h_m \): Since the mountain is 5 km high, the root \( h_m \) is \[ h_m = \left( \frac{\rho_c}{\rho_m - \rho_c} \right) imes (5 \text{ km}) \].Thus, \[ h_m = \left( \frac{2800}{3300 - 2800} \right) \times 5 \text{ km} \].

Calculate the Root Thickness

Calculating the root thickness:\[ h_m = \left( \frac{2800}{500} \right) \times 5 = 28 \text{ km} \].

Calculate Total Crust Thickness

The total thickness of the continental crust with the mountain is the normal crust thickness plus the additional root thickness:\[ h_c = 35 \text{ km} + h_m \]Substitute \( h_m \):\[ h_c = 35 + 28 = 63 \text{ km} \].

Key concepts

Continental Crust Thickness

The thickness of the continental crust can vary but typically averages about 35 kilometers in regions that don't have mountains. When we talk about continental crust beneath mountain ranges, it’s crucial to understand that it’s not just the mountains pushing upwards, but also something known as the 'root' below them.

Mountain ranges are elevated much like icebergs floating on water, with both visible and submerged portions. A mountain range with visible elevation causes an increase in crust thickness underneath, forming this root to maintain balance.

When calculating the extra thickness beneath a mountain, we consider hydrostatic equilibrium, which means that the different pressures and weights of the crust are in balance. This balance helps keep the crust floating on the mantle, preventing it from sinking or rising too much.

Mountain Range Elevation

Mountain range elevation refers to how high a mountain rises above sea level. In our example, a mountain range rises 5 kilometers. This isn't only significant for its height, but also for what it implies beneath the earth’s surface.

For a mountain to maintain such an elevation without crumbling or sinking, it needs support below, which is the ‘root’ of the mountain. The root is a thicker portion of the crust that extends downwards, providing the necessary mass and balance to support the mountain’s height.

To calculate the root’s thickness required for a mountain of 5 km, we make use of the densities of the crust and mantle. The differing densities ensure the mountain stays buoyant and balanced at its elevation.

Density of Mantle

The mantle, which lies beneath the crust, has a substantial impact on geological features like mountains due to its density and the pressure it exerts upwards. The density of the mantle is a key factor when calculating how crust 'floats' or 'sinks'. In our case, the mantle's density is 3300 kg/m³.

This density contrasts with that of the continental crust, which has a density of 2800 kg/m³. This difference in density is critical for understanding how and why mountain roots form.

The greater density of the mantle allows the less dense continental crust to float on top, similar to wood floating on water. When mountains push upwards, they displace more of the mantle, requiring a thicker crust to maintain equilibrium. Knowing these densities allows us to calculate the thickness of the crust required to support mountainous regions, ensuring they remain stable over time.

• The mantle's density is greater than the crust's
• This disparity causes the crust to float, much like oil on water
• Balance is achieved through hydrostatic equilibrium, allowing the crust to maintain its structure