Question

For an asthenosphere with a viscosity \(\mu=4 \times\) \(10^{19} \mathrm{~Pa} \mathrm{~s}\) and a thickness \(h=200 \mathrm{~km},\) what is the shear stress on the base of the lithosphere if there is no counterflow \((\partial p / \partial x=0)\) ? Assume \(u_{0}=\) \(50 \mathrm{~mm} \mathrm{yr}^{-1}\) and that the base of the asthenosphere has zero velocity.

Short answer

The shear stress is \(1.0 \times 10^{10}\) Pa.

Step-by-step solution

Understand the problem and given values

We are given the viscosity \( \mu = 4 \times 10^{19} \text{ Pa} \cdot \text{s} \) and thickness \( h = 200 \text{ km} = 200,000 \text{ m} \). The velocity at the top of the asthenosphere (the base of the lithosphere) is \( u_0 = 50 \text{ mm/yr} = 50 \times 10^{-3}/(365.25 \times 24 \times 3600) \text{ m/s} \), and the base of the asthenosphere is at zero velocity. We need to find the shear stress at the base of the lithosphere, assuming no counterflow (\( \partial p / \partial x = 0 \)).

Calculate the velocity gradient

The velocity gradient \( \frac{du}{dz} \) can be defined as the change in velocity with respect to the change in height. Since the base of the asthenosphere (bottom) has zero velocity and the top has a velocity \( u_0 \), the velocity gradient is given by:\[\frac{du}{dz} = \frac{u_0 - 0}{h} = \frac{50 \times 10^{-3} \text{ m/s}}{200,000 \text{ m}}\].

Calculate the shear stress using viscosity

Shear stress \( \tau \) is calculated using the formula:\[\tau = \mu \cdot \frac{du}{dz}\]Substitute the given values into the equation:\[ au = 4 \times 10^{19} \text{ Pa} \cdot \text{s} \times \frac{50 \times 10^{-3}}{200,000}\]

Simplify and solve for shear stress

Simplifying the above expression:\[ au = 4 \times 10^{19} \times 2.5 \times 10^{-10}\]Perform the multiplication:\[ au = 1.0 \times 10^{10} \text{ Pa}\]Therefore, the shear stress at the base of the lithosphere is \( 1.0 \times 10^{10} \text{ Pa} \).

Key concepts

Asthenosphere

The asthenosphere is a layer of the Earth's mantle that lies beneath the lithosphere. This zone is key in geodynamics due to its semi-fluid nature, allowing it to flow slowly over geological time. Unlike the rigid lithosphere, which consists of all the Earth’s surface plates, the asthenosphere behaves in a more ductile manner.

One of the primary characteristics of the asthenosphere is its ability to deform under pressure, accommodating movements of tectonic plates. Its partial melt helps facilitate this flow, making it crucial for understanding plate tectonics, continental drift, and volcanic activity. The ability of the asthenosphere to undergo slow movement is important for the redistribution of heat from the Earth's interior towards the surface.

Shear Stress

Shear stress is a force that causes layers within a material, like rocks or soil, to slide past one another, which is in this case observed between the lithosphere and asthenosphere. It's defined as the force per unit area acting parallel to the plane where the force is applied. Mathematically, shear stress (\( \tau \)) is expressed in Pascals (Pa).

Shear stress plays a significant role in geological processes, influencing how different earth layers behave under various conditions. In geodynamics, it helps in understanding how tectonic plates can slide or deform due to applied forces. This stress impacts structures and is a critical consideration in engineering geology to anticipate how geological formations will behave under load.

Viscosity

Viscosity is a measure of a fluid's resistance to flow. In geological terms, it refers to how the asthenosphere behaves as it flows under pressure. High viscosity means the substance is thick and resists movement, whereas low viscosity indicates a more fluid, easily flowing substance.

The viscosity of the asthenosphere significantly affects mantle convection and the motion of tectonic plates. As seen in the exercise, the viscosity \( \mu = 4 \times 10^{19} \text{ Pa} \cdot \text{s} \) indicates a highly viscous medium that resists flow, yet permits geological movement over time. This property is crucial in modelling mantle convection and understanding the energy required for tectonic processes.

Lithosphere

The lithosphere is the outer shell of the Earth, comprising the crust and the upper mantle. It's characterized by its rigidity and lack of flow, unlike the asthenosphere below. Making up tectonic plates, the lithosphere is divided into segments that move over the more pliable asthenosphere.

This movement of lithospheric plates is driven by forces such as mantle convection, slab pull, and ridge push, all of which are influenced by the properties of both the lithosphere and asthenosphere. Understanding the characteristics of the lithosphere is essential for geodynamics as it helps explain phenomena like earthquakes, plate tectonics, and mountain-building processes.

Velocity Gradient

The velocity gradient is a measure of how speed changes with position within a fluid, and in geodynamics, it's used to describe the movement between layers of the Earth. It's particularly relevant when assessing how the asthenosphere flows under the lithosphere.

Mathematically, the velocity gradient (\( \frac{du}{dz} \)) is the change in velocity with respect to the change in height between the lithosphere and asthenosphere. For the exercise, it was calculated as the velocity difference between these layers divided by their distance. This gradient is vital for determining shear stress and understanding the dynamics of the Earth's interior movements.