PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Fluids & Continuum Physics · Waves in two and three dimensions, dispersion and wave packets
Signal Speeds in Fluids, Ch Waves in a rotating fluid: inertial, Poincare and Kelvin waves
Problem
A large rotating tank, 5.0m across, of homogeneous, inviscid fluid spins about a vertical axis at Ω=0.8rads−1. Because the axis is vertical, the local Coriolis parameter for horizontal motion in the tank is f=2Ω.
((a)) A small oscillating paddle launches a plane inertial wave whose wave vector makes an angle φ=25∘ with the horizontal. Starting from the linearised, rotating Euler equations for a homogeneous fluid, derive the inertial-wave dispersion relation and use it to find ω for this wave.
((b)) The tank is now filled instead with a fluid of uniform buoyancy frequency N=3.0rads−1. Extend the derivation to include the buoyancy restoring force, obtain the general dispersion relation for a wave vector at angle φ to the horizontal, and evaluate ω again for φ=25∘, checking that it lies in the band the relation predicts.
((c)) The stratification is removed and the tank instead holds a single homogeneous layer of depth h=0.15m. A partition is suddenly removed, leaving a step of height Δ=0.010m across a diameter of the tank. Using conservation of the linear potential vorticity q=ζ−fη/h, find the half-width of the adjusted geostrophic front and the peak current within it, and state what fraction of the released potential energy remains there once the tank has settled.