Question

PROBLEM 5-4 Assume a large geoid anomaly with a horizontal scale of several thousand kilometers has a mantle origin and its location does not change. Because of continental drift the passive margin of a continent passes through the anomaly. Is there a significant change in sea level associated with the passage of the margin through the geoid anomaly? Explain your answer. The anomaly in the potential of the gravity field measured on the reference geoid \(\Delta U\) can be related directly to the geoid anomaly \(\Delta N\). The potential anomaly is defined by $$ \Delta U=U_{m 0}-U_{0} $$ where \(U_{m 0}\) is the measured potential at the location of the reference geoid and \(U_{0}\) is the reference value of the potential defined by Equation \((5-54)\). The potential on the measured geoid is \(U_{0}\), as shown in Figure \(5-6 .\) It can be seen from the figure that \(U_{0}, U_{m 0},\) and \(\Delta N\) are related by $$ U_{0}=U_{m 0}+\left(\frac{\partial U}{\partial r}\right)_{r=r_{0}} \Delta N $$ because \(\Delta N / a \ll 1 .\) Recall from the derivation of Equation \((5-53)\) that we obtained the potential by integrating the acceleration of gravity. Therefore, the radial derivative of the potential in Equation \((5-69)\) is the acceleration of gravity on the reference geoid. To the required accuracy we can write $$ \left(\frac{\partial U}{\partial r}\right)_{r=r_{0}}=g_{0} $$ where \(g_{0}\) is the reference acceleration of gravity on the reference geoid. Just as the measured potential on the reference geoid differs from \(U_{0}\), the measured acceleration of gravity on the reference geoid differs from \(g_{0}\). However, for our purposes we can use \(g_{0}\) in Equation \((5-69)\) for \((\partial U / \partial r)_{r=r_{0}}\) because this term is multiplied by a small quantity \(\Delta N\). Substitution of Equations \((5-69)\) and \((5-70)\) into Equation \((5-68)\) gives $$ \Delta U=-g_{0} \Delta N $$ A local mass excess produces an outward warp of gravity equipotentials and therefore a positive \(\Delta N\) and a negative \(\Delta U\). Note that the measured geoid essentially defines sea level. Deviations of sea level from the equipotential surface are due to lunar and solar tides, winds, and ocean currents. These effects are generally a few meters. The reference acceleration of gravity on the reference geoid is found by substituting the expression for \(r_{0}\) given by Equation \((5-62)\) into Equation \((5-50)\) and simplifying the result by neglecting quadratic and higher order terms in \(J_{2}\) and \(a^{3} \omega^{2} / G M\). One finds $$ g_{0}=\frac{G M}{a^{2}}\left(1+\frac{3}{2} J_{2} \cos ^{2} \phi\right)+a \omega^{2}\left(\sin ^{2} \phi-\cos ^{2} \phi\right) $$ To provide a standard reference acceleration of gravity against which gravity anomalies are measured, we must retain higher order terms in the equation for \(g_{0}\). Gravity anomalies are the differences between measured values of \(g\) on the reference geoid and \(g_{0} .\) By international agreement in 1980 the reference gravity field was defined to be $$ \begin{aligned} g_{0}=& 9.7803267715\left(1+0.0052790414 \sin ^{2} \phi\right.\\\ &+0.0000232718 \sin ^{4} \phi \\ &+0.0000001262 \sin ^{6} \phi \\ &\left.+0.0000000007 \sin ^{8} \phi\right) \end{aligned} $$ with \(g_{0}\) in \(\mathrm{m} \mathrm{s}^{-2}\). This is known as the 1980 Geodetic Reference System (GRS) (80) Formula. The standard reference gravity field given by Equation \((5-73)\) is of higher order in \(\phi\) than is the consistent quadratic approximation used to specify both \(g_{0}\) in Equation ( \(5-72\) ) and \(r_{0}\) in Equation \((5-67)\). The suitable SI unit for gravity anomalies is \(\mathrm{mm} \mathrm{s}^{-2}\).

Short answer

No significant sea level change is associated with passing through the geoid anomaly due to large spatial scales and gradual adjustment.

Step-by-step solution

Understand the Problem

The problem asks if there is a significant change in sea level when a passive margin of a continent passes through a geoid anomaly. The anomaly affects the gravitational potential at the geoid, which is factored into how sea level is defined in relation to the geoid.

Identify Key Equations and Variables

The key relationship is \[ \Delta U = -g_0 \Delta N \]where \( \Delta U \) is the potential anomaly, \( \Delta N \) is the geoid anomaly, and \( g_0 \) is the reference acceleration of gravity. The geoid essentially defines sea level, which can be affected by gravitational changes.

Analyze Gravitational Effects on Sea Level

Since \( \Delta N / a \ll 1 \), the potential anomaly \( \Delta U \) is proportional to the product of the gravitational acceleration and the geoid anomaly. A positive \( \Delta N \) results in a negative \( \Delta U \), indicating a mass excess protrudes the geoid upwards, causing potential energy to decrease.

Consider Temporal and Spatial Scales

The geoid anomaly has a large scale and does not move as the continent drifts. The temporal scale of a continent crossing this anomaly is slow enough such that gravitational equipotential surfaces are adjusted gradually, minimizing abrupt sea level change.

Conclusion on Sea Level Change

Sea level, as defined by the geoid, adjusts locally to these anomalies over such large horizontal scales. However, significant sea level deviations primarily arise from external factors like tides and currents rather than static geoid anomalies. Thus, any sea level changes due to the passage over the geoid anomaly are negligible.

Key concepts

Gravity Field

A gravity field encompasses the gravitational forces exerted by a massive object, like the Earth, which influences objects within its vicinity. This gravitational force is what keeps us grounded and affects all celestial bodies. Essentially, the gravity field shapes how potential energy is distributed across any terrain on the planet. The Earth's gravity field is not uniform, due to variations in density and composition within the Earth's structure.

The Earth's gravity field can be visualized as a series of equipotential surfaces, which are imaginary levels of constant gravitational potential energy. The geoid represents one such equipotential surface, roughly aligning with the mean sea level, though irregular due to the Earth's internal mass anomalies.

Understanding the Earth's gravity field is crucial in geophysics, helping to map out gravitational anomalies which can give insights into underground structures. These anomalies often indicate mass excesses or deficits, leading to variations in gravity measurement and influencing the geoid's shape.

Potential Anomaly

Potential anomaly refers to the difference in gravitational potential energy at a specific location compared to a reference level. In geophysics, it is quantified as \[ \Delta U = U_{m0} - U_{0} \]where \( U_{m0} \) is the measured potential and \( U_{0} \) is the reference potential on the geoid. This difference, \( \Delta U \), is often a result of local variations in mass distribution beneath the Earth's surface, such as mountains, valleys, or mineral deposits.

Potential anomalies are pivotal in understanding geoid anomalies, as they reflect the disturbances in the regular gravitational field. When a mass protrudes above the normal geoid level, we observe an upward bulge in equipotential surfaces, resulting in a negative potential anomaly (\( \Delta U < 0 \)). This occurs because the extra mass causes the gravitational potential energy to decrease in that area, fascinatingly highlighting the dynamic nature of the Earth's geophysical properties.

Grasping potential anomalies aids scientists in interpreting geological formations and their intrinsic mass properties, contributing significantly to fields such as resource exploration and natural hazard assessment.

Continental Drift

Continental drift is the gradual movement of continents across the Earth's surface due to plate tectonics. This theory, first proposed by Alfred Wegener in 1912, suggests that the continents were once part of a giant supercontinent called Pangaea, which slowly drifted apart over millions of years. Geologists now explain this phenomenon through plate tectonics, where the Earth's lithosphere is divided into tectonic plates that float on the semi-fluid asthenosphere beneath.

This drift is responsible for shaping our current continents and ocean basins. It exerts profound effects on climate, sea level, and biodiversity over geological timescales. For instance, as a continent drifts over a geoid anomaly—a region of mass excess or deficit in the Earth's mantle—the gravitational field shifts, though this happens so gradually that its impact on immediate sea level is minimal.

Continental drift also induces geological activity, such as earthquakes and volcanic eruptions, along plate boundaries. Its recognition laid the groundwork for our understanding of dynamic Earth processes and emphasizes the ever-evolving nature of our planet.

Sea Level Change

Sea level change refers to the changes in the average height of the world's oceans relative to the Earth's surface. These changes can occur due to various factors, both global and local.

• Global Factors: Thermal expansion of seawater due to global warming, melting glaciers, and ice sheets are major contributors.
• Local Factors: Local sea level variations can be influenced by tides, ocean currents, and atmospheric pressure, as well as land movements such as subsidence or uplift.

Understanding sea level change is crucial for coastal planning and climate change adaptation strategies.

Geoid anomalies, while affecting gravitational potential, have a relatively small direct impact on sea level. This is because the scale and persistence of a geoid anomaly allow sea levels to adapt over time locally, reducing immediate impacts. Sea levels tend to maintain equilibrium with the geoid, from which they are measured. Thus, while geoid anomalies can affect the sea level, their effects are typically overshadowed by dynamic and temporal factors like ocean currents and climate-induced changes.