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. A MATLAB code for solving this problem is given in Appendix \(D\).

Short answer

The thickness of the crust beneath the mountain range is approximately 47.1 km.

Step-by-step solution

Understand the Concept of Isostasy

Isostasy is a concept in geology that describes the equilibrium between the Earth's crust and mantle. The crust 'floats' at an elevation that depends on its thickness and density, much like an iceberg in water. In this problem, we assume that the mountain range is in isostatic equilibrium, meaning the weight of the crust is balanced by the buoyant force from the mantle beneath.

Write the Isostasy Equation

The isostasy principle can be summarized using the equation \[ \rho_m (H_m + H_c - h) = \rho_c H_c \]where \( \rho_m \) and \( \rho_c \) are the densities of the mantle and crust, respectively, \( H_m \) is the elevation of the mountain range (5 km), \( H_c \) is the thickness of the crust, and \( h \) is the thickness of the crustal root beneath the mountain.

Set Known Values

Substitute the known values into the equation:- \( H_m = 5 \text{ km} = 5000 \text{ meters} \)- \( \rho_m = 3300 \text{ kg/m}^3 \)- \( \rho_c = 2800 \text{ kg/m}^3 \)- Normal thickness of the crust \( = 35 \text{ km} = 35000 \text{ meters} \).

Solve for Unknown Thickness

Rearrange the equation to solve for the thickness of the crustal root \( h \): \[ h = H_c - \frac{\rho_c H_c - \rho_m H_m}{\rho_m} \].Given that we are trying to find the actual thickness under the mountain, modify the equation: \[ H_c' = h + 5000 \]. Calculate \( h \) using substitution and obtain \( H_c' \).

Calculate the Thickness

First, calculate the crustal root \( h \):\[ h = 35000 - \frac{2800 \times 35000 - 3300 \times 5000}{3300} \].Calculate this value to determine \( h \), then calculate the full thickness \( H_c' = h + 5000 \).

Final Calculation

Using the formula:\[ h = 35000 - \frac{2800 \times 35000 - 3300 \times 5000}{3300} \approx 42121.21 \text{ meters} \].Then, the thickness of the crust beneath the mountain range:\[ H_c' = 42121.21 + 5000 \approx 47121.21 \text{ meters} \].Convert back to kilometers: \( H_c' \approx 47.1 \text{ km} \).

Key concepts

Hydrostatic Equilibrium

Hydrostatic equilibrium is a fundamental concept in understanding how different layers of Earth interact with each other. Imagine it like a see-saw balancing act, where the Earth's crust sits on the mantle. In this context, the mantle is like a dense, heavy fluid supporting a less dense crust floating on top. In terms of physics, hydrostatic equilibrium happens when there is an equilibrium between gravity pulling the crust down and the buoyant force pushing it up. This keeps the crust at a stable height, rather like how a piece of wood floats on water. The balance is essential not only for determining the crust's position but also for calculating variations in crustal thickness under different geographic features. When a mountain range is involved, its enormous mass pushes the crust downward into the mantle more than a valley or a plain would. But due to the principle of hydrostatic equilibrium, the mantle pushes back, helping to support the mountain's mass in a stable position.

Crustal Thickness

The thickness of Earth’s crust can vary greatly depending on the geological features present. A notable factor contributing to this is the principle of isostasy, which suggests that changes in crustal thickness can balance different surface elevations. For example, under mountain ranges, the crust tends to be much thicker compared to areas of lower elevation. This is partly because mountains exert more downward force, which means they need a 'root' extending into the mantle for support, much like an iceberg has a portion submerged underwater. By analyzing the thickness of the crust, as seen in our exercise, we can determine the depth of this crustal root. This is critical for geologists to understand because it helps map internal Earth processes and can even unveil insights into past tectonic activities. Calculating the crust's thickness beneath certain features using known densities and elevations is a practical application of such gravitational balances in geology.

Density of Earth's Layers

The density of Earth's layers is a key factor affecting how the crust floats on the mantle. Density differences between the crust and mantle embody why isostatic balance is possible. Normally, the mantle, with a density around 3300 kg/m³, is denser than the crust, which has a density about 2800 kg/m³. These density values allow the crust to "float" atop the denser mantle, with dense areas, such as mountains, sinking lower and requiring a thicker crustal portion underneath to maintain balance—this is part of what we call the crustal root. Understanding these density variations allows scientists to use mathematical equations to predict how thick the crust should be under various earth features. This not only helps in constructing accurate geological models but also plays a significant role in resource exploration and understanding seismic activities.