Question

Consider the formation of a sedimentary basin on the seafloor. Suppose isostatic compensation is achieved by the displacement of mantle material of density \(\rho_{m}\). Show that sediment thickness \(s\) is related to water depth \(d\) by $$s=\frac{\left(\rho_{m}-\rho_{w}\right)}{\left(\rho_{m}-\rho_{s}\right)}(D-d)$$ where \(D\) is the initial depth of the sediment-free ocean. What is the maximum possible thickness of the sediment if \(\rho_{s}=2500 \mathrm{~kg} \mathrm{~m}^{-3}, \rho_{m}=\) \(3300 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(D=5 \mathrm{~km} ?\)

Short answer

The maximum possible thickness of the sediment is 14,375 meters.

Step-by-step solution

Understanding Isostatic Compensation

Isostatic compensation involves the balance of the Earth's crust floating on the denser mantle material. When sediments accumulate on the seafloor, the added weight causes the crust to sink deeper into the mantle until a new equilibrium is reached.

Introduce Relevant Densities

We have three relevant densities in this problem: \( \rho_m \) for the mantle, \( \rho_w \) for water, and \( \rho_s \) for sediments. These densities are critical in determining how much the crust will sink when sediments accumulate.

Setting Up the Equation Balance

Initially, the total pressure caused by the water column of depth \(D\) is balanced by the displacement of mantle material. When sediments of thickness \(s\) are added, the new depth of water becomes \(d\), and the crust sinks deeper, displacing a volume of mantle equal to the volume of the sediment and lost water column.

Relationship Between Sediment and Water Depth

The thickness of the sediments \(s\) is determined by the balance of the new displaced volume of mantle against the weight of the sediments and adjusted water column. This gives us the equation:\[s = \frac{(\rho_{m} - \rho_{w})}{(\rho_{m} - \rho_{s})} (D - d)\]

Calculate Maximum Sediment Thickness

Set \(d=0\) for maximum sediment thickness, meaning all water is displaced. Substitute given values \(\rho_{s}=2500\), \(\rho_{m}=3300\), and \(D=5000\) meters to find:\[s_{max} = \frac{(3300 - 1000)}{(3300 - 2500)} \times 5000\]Calculate this to find the maximum possible thickness of sediment.

Compute the Result

Plug values into the equation:\[s_{max} = \frac{2300}{800} \times 5000\]This simplifies to:\[s_{max} = 2.875 \times 5000 = 14375 \text{ meters}\]

Key concepts

Sedimentary Basin Formation

The formation of a sedimentary basin is a fascinating geological process that occurs on the seafloor or on land, where a significant amount of sediment accumulates over time. This accumulation results from the deposition of materials like sand, silt, and clay, which are transported by wind, water, or ice. As these sediments continue to gather in a specific area, they can create a depression or basin due to their weight and the natural subsidence of the Earth's crust.

The creation of a sedimentary basin is driven by tectonic movements, such as crustal stretching or compressional forces. These movements can lead to the formation of spaces that allow sediments to accumulate. Over time, with the continuous input of sediments, these basins can become significant geological structures. They often serve as critical sites for the exploration of natural resources, like oil and gas, due to their ability to trap organic materials and generate hydrocarbons.

One key component in understanding sedimentary basin formation is the balance or isostatic adjustment. As sediments pile up and exert more pressure on the Earth's crust, the crust tends to subside or sink. This isostatic adjustment continues until a new equilibrium is achieved, where the weight of the sediments is balanced by the support provided by the underlying mantle.

Density Differences in Geophysics

In geophysics, understanding density differences between materials is crucial as it influences the behavior of the Earth's crust and mantle. Density is defined as mass per unit volume, and in the context of geology, it plays a pivotal role in how surfaces adjust and settle.

Different geological materials have distinct densities. For example:

• Water typically has a density around \( ho_w = 1000 \) kg/m³.
• Sediments, often denser than water, can have a density such as \( ho_s = 2500 \) kg/m³.
• The mantle, made up of denser material beneath the Earth's crust, might have a density of \( ho_m = 3300 \) kg/m³.

These density variations are not just numerical values; they play a critical role when examining the subsidence and uplift of the crust.

The density of a material impacts how it interacts with other layers. For instance, when heavier sediments are deposited on the seafloor, they cause the crust to move and sink because these dense sediments replace lighter water layers. As sediment density increases relative to other layers, they cause more significant crustal adjustments, often leading to deeper basins.

Sediment Thickness Calculation

Calculating the thickness of sediment that accumulates in a basin involves understanding the relationship between sediment, water depth, and the underlying mantle. The exercise we're looking at has introduced a relationship described by a specific equation:

\[s = \frac{(\rho_{m} - \rho_{w})}{(\rho_{m} - \rho_{s})} (D - d)\]

This equation connects sediment thickness \( s \) with initial water depth \( D \) and modified depth \( d \). The key to solving the equation involves substituting known densities for water, sediments, and the mantle:

• \( ho_w = 1000 \) kg/m³ (water)
• \( ho_s = 2500 \) kg/m³ (sediments)
• \( ho_m = 3300 \) kg/m³ (mantle)

The equation shows how sediment thickness is influenced by the density differences between these layers. When sediments replace an equivalent volume of water, the mantle must adjust through isostatic compensation. This process ensures that the basin maintains equilibrium.

Using these relations helps in calculating the maximum possible thickness of sediment assuming all water is displaced, represented when \( d = 0 \):

• Substitute \( D = 5000 \) meters
• Emphasize that \( s_{max} = \frac{(3300 - 1000)}{(3300 - 2500)} \times 5000 \)
• Leading to a maximum sediment thickness \( s_{max} = 14375 \) meters

Isostatic Equilibrium

Isostatic equilibrium is a concept that refers to the gravitational balance between different layers of the Earth, such as the crust "floating" on the denser, underlying mantle. This balance is crucial for maintaining the stability of various geological formations, including sedimentary basins.

When the Earth's crust accumulates added weight from sediment deposits, it experiences changes until isostatic equilibrium is reached. This process involves the downward adjustment of the crust into the mantle, or sometimes upwards if material is eroded away.

The principle of isostasy can be compared to icebergs floating on water. Just as an iceberg floats higher or lower in the water depending on its mass and the density of both the water and ice, the Earth's crust adjusts its position relative to the material it supports and the underlying mantle densities.

Isostatic adjustment is a dynamic process. It accounts for:

• The weight and density of the accumulated sediment
• The density differences between water, sediment, and mantle
• The necessity for the Earth's crust to maintain balance over geological timescales

This concept allows scientists to predict how the crust will behave under various conditions, such as the formation, growth, or erosion of sedimentary basins.