Question

The relevant nondimensional number in deciding deepening of the thermocline driven by instability of ocean currents is

• A. Rossby number.
• B. Reynolds number.
• C. Richardson number.
• D. Ekman number.

Hint:

In oceanography, the Richardson number helps determine the stability of fluid layers like the thermocline by comparing buoyancy forces and shear forces.

Answer 1

The Correct Option is C

In oceanography and fluid dynamics, the Richardson number (Ri) is a dimensionless number that plays a crucial role in determining the stability of stratified flows. It is particularly important in oceanography when studying the thermocline – the thin layer in the ocean where the temperature gradient is steepest. The Richardson number compares the buoyancy force (driven by density differences) to the shear force (due to current velocity differences). The Richardson number is defined as: \[ \text{Ri} = \frac{g \Delta \rho L}{\rho (du/dz)^2}, \] where:
- \( g \) is the acceleration due to gravity,
- \( \Delta \rho \) is the density difference between two layers,
- \( L \) is the characteristic length scale (typically the depth of the ocean layer),
- \( \rho \) is the density of the fluid (in this case, water),
- \( \frac{du}{dz} \) is the vertical shear of the current.
High Richardson numbers (Ri > 1) indicate stable stratification, where buoyancy forces dominate over shear forces, preventing mixing. Low Richardson numbers (Ri < 1) indicate instability, where shear forces dominate and can lead to mixing, causing the thermocline to deepen. Let's analyze the options: Rossby number (Option A): The Rossby number is a dimensionless number used in fluid dynamics to determine the relative importance of inertial forces to Coriolis forces. While important in large-scale ocean dynamics, it is not used to determine the deepening of the thermocline. Reynolds number (Option B): The Reynolds number characterizes the turbulence in a fluid flow, comparing inertial forces to viscous forces. It is essential in determining whether flow is laminar or turbulent but does not directly address the stability of the thermocline. Richardson number (Option C): As explained above, the Richardson number is used to analyze stratified flows like the thermocline in oceans. It is the correct number to determine if the thermocline will deepen due to instability caused by ocean currents. Ekman number (Option D): The Ekman number is related to rotational effects in fluid dynamics, specifically the balance between Coriolis forces and viscous forces. While important in analyzing Ekman layers in the ocean, it does not directly apply to the thermocline's instability. Thus, the correct dimensionless number that determines the deepening of the thermocline is the Richardson number. Final Answer: \[ \boxed{\text{(C) Richardson number.}}. \]