Question

Consider an RRR triple junction of plates \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\). The ridge between plates \(\mathrm{A}\) and \(\mathrm{B}\) lies in a north-south direction (an azimuth of \(0^{\circ}\) with respect to the triple junction) and has a relative velocity of \(60 \mathrm{~mm} \mathrm{yr}^{-1}\). The ridge between plates \(\mathrm{B}\) and C has an azimuth of \(120^{\circ}\) with respect to the triple junction, and the ridge between plates \(\mathrm{A}\) and \(\mathrm{C}\) has an azimuth of \(270^{\circ}\) with respect to the triple junction. Determine the azimuths and magnitudes of the relative velocities between plates \(\mathrm{B}\) and \(\mathrm{C}\) and \(\mathrm{C}\) and \(\mathrm{A}\).

Short answer

Velocities: \(BC = 120\ \mathrm{mm/yr}\), \(CA = 104\ \mathrm{mm/yr}\).

Step-by-step solution

Understand the Ridge Orientation and Velocity

The ridge between plates A and B is oriented north-south with a velocity of \(60\ \mathrm{mm/yr}\). The azimuth for this ridge is \(0^{\circ}\). This means that the relative velocity vector between plates A and B is purely in the north-south direction.

Represent Velocities as Vectors

Represent the given velocities as vectors in a coordinate system where North is \(0^{\circ}\). The velocity \(\overrightarrow{V_{AB}}\) between plates A and B is: \(\overrightarrow{V_{AB}} = (0, 60)\), since it points due north. Similarly, we need to determine \(\overrightarrow{V_{BC}}\) and \(\overrightarrow{V_{CA}}\).

Use the Law of Cosines for Magnitudes

Since we don't have direct velocity magnitudes for BC and CA but know their angles with respect to AB, we use the property that these vectors should sum to zero around the triangular loop. The azimuth for \(\overrightarrow{V_{BC}}\) is \(120^{\circ}\), and for \(\overrightarrow{V_{CA}}\), it is \(270^{\circ}\).

Breakdown into Components

For \(\overrightarrow{V_{BC}}\) with azimuth \(120^{\circ}\), convert it into vector components: \(\overrightarrow{V_{BC}} = (-V_{BC}\cdot\cos(30^{\circ}), V_{BC}\cdot\sin(30^{\circ}))\). For \(\overrightarrow{V_{CA}}\) with azimuth \(270^{\circ}\), its vector components are: \(\overrightarrow{V_{CA}} = (-V_{CA}, 0)\).

Solve the Vector Sum

Applying vector addition, \(\overrightarrow{V_{AB}} + \overrightarrow{V_{BC}} + \overrightarrow{V_{CA}} = \overrightarrow{0}\):- For the x-components: \(0 - V_{BC}\cdot\cos(30^{\circ}) - V_{CA} = 0\)- For the y-components: \(60 + V_{BC}\cdot\sin(30^{\circ}) = 0\)Solve these equations to find the magnitudes \(V_{BC}\) and \(V_{CA}\).

Calculate Results from Equations

From the y-component equation, \(V_{BC} = -120\ \mathrm{mm/yr}\).Plug \(V_{BC}\) into the x-component equation to find \(V_{CA}\): \(V_{CA} = 120\cos(30^{\circ}) = 104\ \mathrm{mm/yr}\).Therefore, magnitudes are \(V_{BC} = 120\ \mathrm{mm/yr}\) and \(V_{CA} = 104\ \mathrm{mm/yr}\).

Key concepts

Triple Junction

A triple junction is a point on the Earth's surface where the boundaries of three tectonic plates meet. These junctions can have various configurations, such as RRR (ridge-ridge-ridge), where all the plate boundaries are divergent and characterized by mid-ocean ridges. Such configurations can be complex as they involve interactions between multiple moving plates. The RRR junction is a dynamic setting, as each ridge pushes the plates apart, creating new ocean floor.
Ridges are important features of the triple junction. They typically form at divergent boundaries where two tectonic plates move apart from each other. Understanding how each of these ridges interact is crucial in predicting the overall motion of the plates involved. Each ridge at a triple junction contributes to the movement of the plates and the shape of ocean basins.
At a broader scale, the study of triple junctions helps geoscientists understand plate tectonics and the mantle dynamics driving plate motions.

Plate Motion

Plate motion refers to the movement of the Earth's tectonic plates over the semi-fluid asthenosphere beneath them. This motion is driven by several forces, including mantle convection and slab pull. At a triple junction, each of the three plates can exhibit distinct motion patterns that depend on the ridge orientations and the velocities of the neighboring plates.
In the context of the exercise, we look at the relative motion between plates such as A and B, which move apart at a rate of 60 mm/yr due north. The directional velocity of such movements can be represented as vectors, which help us predict the movement's magnitude and direction. Understanding these movements is essential in predicting the geological consequences, such as the creation of new oceanic crust at ridges.
Accurately determining the vector magnitudes and directions at a triple junction allows scientists to map and anticipate tectonic activity, aiding in natural disaster prediction and understanding past geological events.

Vector Analysis

Vector analysis is a mathematical tool used to determine the magnitude and direction of forces, such as tectonic plate motions. In the context of tectonics, vector analysis helps in understanding how different plates interact at a triple junction by breaking down their velocities into components that can be more easily analyzed.
To perform vector analysis, we represent the velocity of each plate in a coordinate system. Earth's north can be considered the zero-degree direction, simplifying the representation of each plate's movement. For instance, the exercise represents the north-directed velocity of 60 mm/yr between plates A and B as a vector \(\overrightarrow{V_{AB}} = (0, 60)\).
In more complex situations involving angles, the vector components (x and y) are derived using trigonometric functions based on the azimuth. Solving these components provides a comprehensive understanding of the overall movement, allowing for the determination of unknown velocities, such as those of plates B and C, and C and A.

Ridge Orientation

Ridge orientation describes the alignment of mid-ocean ridges, which are divergent boundaries between tectonic plates. These orientations have significant implications for plate movements and interactions, especially at triple junctions where three such ridges converge.
In the presented exercise, the ridges have specific azimuths with respect to the triple junction: the ridge between plates A and B has an azimuth of \(0^\circ\), the ridge between B and C stands at \(120^\circ\), and the ridge between A and C at \(270^\circ\). These azimuths define the direction of the ridge alignments and, consequently, the direction of plate movement.
Ridge orientations influence the dynamics at a triple junction by guiding the direction of new crust formation and subsequent plate separation. Understanding these directions is essential for comprehending how plates move over time, forming geological features like ocean basins and potentially triggering seismic activities.