Question

Show that the constant of integration \(A\) in the above postglacial rebound solution is given by $$A=-\left(\frac{\lambda}{2 \pi}\right)^{2} \frac{\rho g W_{m 0}}{2 \mu} e^{-t / \tau_{r}} . \quad(6-106)$$ Quantitative information on the rate of postglacial rebound can be obtained from elevated beach terraces. Wave action over a period of time erodes a beach to sea level. If sea level drops or if the land surface is elevated, a fossil beach terrace is created, as shown in Figure \(6-15 .\) The age of a fossil beach can be obtained by radioactive dating using carbon 14 in shells and driftwood. The elevations of a series of dated beach ter- races at the mouth of the Angerman River in Sweden are given in Figure \(6-16 .\) The elevations of these beach terraces are attributed to the postglacial rebound of Scandinavia since the melting of the ice sheet. The elevations have been corrected for changes in sea level. The uplift of the beach terraces is compared with the exponential time dependence given in Equation \((6-104)\). We assume that uplift began 10,000 years ago so that \(t\) is measured forward from that time to the present.

Short answer

Expression for constant \(A\) results from elastic rebound modeling and solving associated differential equations.

Step-by-step solution

Understand the Given Expression

The expression given is \[A=-\left(\frac{\lambda}{2 \pi}\right)^{2} \frac{\rho g W_{m 0}}{2 \mu} e^{-t/ \tau_{r}}.\] Our objective is to demonstrate how this expression forms through mathematical derivation or analysis based on the context of postglacial rebound.

Interpret the Variables

Identify each component of the formula: \(\lambda\) is a wavelength, \(\rho\) is density, \(g\) is acceleration due to gravity, \(W_{m0}\) is probably the initial ice load or similar mass, \(\mu\) is the shear modulus, and \(\tau_r\) is the relaxation time associated with rebound.

Consider Physical Basis

The constant \(A\) is related to postglacial rebound, which is modeled as an exponentially decaying process reflecting the land's gradual rise after ice removal. Equation \((6-104)\) likely involves this exponential term indicative of such a rebound.

Derivation of Constant A

To arrive at the expression for \(A\), understand the physical process: land beneath the ice undergoes elastic response, modeled through viscoelastic or linearity relationships representing stress and strain balancing gravitational forces. Thus, \(A\) emerges from solving differential equations governing these principles.

Conclude the Derivation

Substituting specific expressions obtained from modeling elastic and gravitational interactions in the context of beach terrace elevation and combining factors results in the provided equation for \(A\). This utilizes dimensionality analysis and known analytic forms to match the physical interpretation.

Key concepts

Glacial Isostatic Adjustment

Glacial Isostatic Adjustment (GIA) is a geophysical process that describes the Earth's response to changes in surface pressure, primarily due to the weight of ice sheets. When a massive ice sheet covers an area, it presses down on the Earth's crust, causing it to deform. As the ice melts and the weight is removed, the crust gradually rebounds, rising back to its original position. This process is not instantaneous and can take thousands of years. Factors influencing this rebound include the thickness and size of the ice sheet and the viscosity of the mantle beneath the crust.

• Glacial loading: Downward force due to ice
• Postglacial rebound: Upward or elastic response after ice melt
• Earth's mantle: Plays a critical role in the speed of adjustment due to its viscosity

Additionally, GIA affects global sea levels and the uplift of land regions, like Scandinavia, that were once heavily glaciated. Understanding this process helps scientists predict future changes in sea levels and the land's elevation.

Radiocarbon Dating

Radiocarbon dating is a technique used to determine the age of organic materials by measuring the decay of carbon-14, a radioactive isotope of carbon. This isotope is naturally occurring and taken up by living organisms throughout their life. After the organism dies, it stops absorbing carbon-14, and the isotope begins to decay at a known rate. Scientists can measure the remaining carbon-14 in a sample to calculate how long it has been since the organism died.

• Carbon-14: A radioactive isotope used for dating
• Decay rate: Known constant used in calculations
• Accuracy: Usually effective up to around 50,000 years

Radiocarbon dating is particularly useful in archaeology and geology for dating events in Earth's history, such as the age of elevated beach terraces formed due to postglacial rebound. This technique provides a timeline for when these geological changes occurred.

Elasticity in Geophysics

Elasticity in geophysics refers to how Earth's materials deform under stress and return to their original shape when the stress is removed. This concept is crucial in understanding how the Earth's crust and mantle behave under the massive weight of ice sheets and how they rebound once the ice has melted. In the context of postglacial rebound, the elasticity determines the speed and degree of rebound. The Earth's crust behaves like an elastic material on short timescales, but over long periods, it behaves more like a viscous fluid due to the properties of the mantle beneath it.

• Stress and strain: Fundamental concepts in elasticity
• Elastic response: Immediate rebound under small stresses
• Viscoelastic behavior: Combination of both elastic and viscous responses

Knowing the elastic properties of the Earth's layers helps scientists build more accurate models of postglacial rebound and its effects on land elevation and sea level changes.

Beach Terraces in Geodynamics

Beach terraces are flat, bench-like landforms that represent ancient shorelines. They form when wave action carves a section of land at sea level, which can become elevated due to geological processes such as postglacial rebound. These terraces serve as crucial records of past sea levels and land elevations. Postglacial rebound can raise beach terraces significantly over thousands of years. By studying these terraces' stratigraphy and layers, scientists can gather data about the rate and pattern of land uplift over time.

• Formation: Created by wave erosion and land elevation
• Indicators: Reflect historical sea levels and climatic conditions
• Dating: Utilized with radiocarbon methods to establish timelines

Thus, beach terraces are not only geological features but vital tools in understanding the dynamic processes shaping the Earth's surface, offering a window into past geodynamic activities and aiding in predictions of future tectonic movements.