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 approximately 3.17 Pa.

Step-by-step solution

Understand the Parameters

We are given the viscosity of the asthenosphere \(\mu = 4 \times 10^{19} \mathrm{~Pa \cdot s}\), thickness \(h = 200 \mathrm{~km}\), upper plate velocity \(u_0 = 50 \mathrm{~mm/yr}\), and \(\partial p / \partial x = 0\) for no counterflow.

Convert Units

Convert the thickness from kilometers to meters: \(h = 200 \mathrm{~km} = 200,000 \mathrm{~m}\). Convert the velocity from mm/yr to m/s: \(u_0 = 50 \times 10^{-3} \mathrm{~m/yr} \times \frac{1}{3.154 \times 10^7 \mathrm{~s/yr}} \approx 1.585 \times 10^{-9} \mathrm{~m/s}\).

Apply the Shear Stress Formula

The formula for shear stress \(\tau\) in a region with a linear velocity profile and no counterflow is given by \[\tau = \mu \frac{u_0}{h}\].

Calculate the Shear Stress

Substitute the given values into the formula: \[\tau = 4 \times 10^{19} \mathrm{~Pa \cdot s} \cdot \frac{1.585 \times 10^{-9} \mathrm{~m/s}}{200,000 \mathrm{~m}}\]. Calculate \(\tau = 3.17 \mathrm{~Pa}\).

Key concepts

Asthenosphere Viscosity

Asthenosphere viscosity is a crucial factor when studying Earth's mantle dynamics. Viscosity itself is defined as a fluid's resistance to flow. In the case of the asthenosphere, a layer in the Earth's upper mantle, it behaves like a highly viscous fluid allowing the tectonic plates to move.

• The given problem specifies the viscosity as \(\mu = 4 \times 10^{19} \,\text{Pa\cdot s}\). This means that the asthenosphere is extremely resistant to flow, yet still fluid enough for geological time scale movements.
• High viscosity in geodynamics implies slow movements of materials beneath the Earth’s lithosphere.

Understanding asthenosphere viscosity helps in explaining how continents drift and how mountains and ocean basins form over millions of years.

Shear Stress Calculation

Calculating shear stress at the base of the lithosphere helps to predict tectonic movements. Shear stress results from forces that cause layers within something like the earth to slide past each other. For a simple model:

• In our exercise, knowing the viscosity \(\mu\) and the relative velocity \(u_0\), we apply the formula \[ \tau = \mu \frac{u_0}{h} \].
• This relates directly to how the lithosphere's creeping motion is resisted by the underlying asthenosphere.
• The units of shear stress are Pascals (Pa), which is the standard unit for pressure.

By substituting converted values, it results in \(\tau = 3.17 \,\text{Pa}\). This calculation gives insight into the stresses involved in plate tectonic processes.

Lithosphere Dynamics

The lithosphere is Earth's rigid outer layer, consisting of both the crust and the uppermost mantle. Understanding its dynamics involves studying how it interacts with the asthenosphere below.

• The exercise assumes a no counterflow condition \(\frac{\partial p}{\partial x} = 0\), simplifying interactions by removing pressure gradients.
• The movement of the lithosphere is influenced by temperature, composition, and the viscosity of the underlying asthenosphere.
• In geodynamic models, the lithosphere acts like a solid shell floating on the softer, more fluid-like asthenosphere.

These dynamics are crucial for plate tectonics, explaining phenomena such as earthquakes, volcanic activity, and continental drift.