What Are the Key Characteristics of Inertial Waves in Uniform Rotation?

[Angelo Pascal]

Homework Statement

Consider an inviscid fluid in a uniformly rotating frame of reference. The effect of
gravity is neglected. In the frame rotating with an angular velocity Ω, the momentum
equation governing the fluid flow is given by:
du/dt = ∂u/∂t + (u · ∇) u = −1/ρ (∇pr) − 2Ω × u
where u is the velocity field, ρ is the density of the fluid and pr= p − ((|Ω| |x⊥|)^2)/2
is the reduced pressure, where p is pressure and x⊥ is the perpendicular distance from
the position vector x to the rotation axis.
We consider the dynamics of inertial waves, i.e. small perturbations to the state of
uniform rotation.

Relevant Equations

The questions are below.

(a) Write down an expression for the velocity field corresponding to uniform
rotation. Find the vorticity corresponding to this flow.

(b) Consider a small perturbation u' to the state of uniform rotation with angular
velocity Ω, which has the form of a plane harmonic wave
u'= A exp i(k·x−ω t) + A*exp -i(k·x−ω t)
where k is the wavenumber vector, ω is the frequency, A is the complex
amplitude and A* is its complex conjugate.
How small does the amplitude A need to be for the nonlinear (with respect to A) terms to be much smaller than the linear ones in the momentum equation?

(c) Consider the linearised momentum equation and derive the dispersion relation
ω = ω (k). (Hint: Take the curl of the momentum equation and use the identity
∇ × (Ω × u) = − (Ω · ∇) u.)

(d) What is the polarisation of the inertial waves?

(e) Are these waves dispersive or non-dispersive? Isotropic or anisotropic? Explain why. Find the group velocity for the inertial waves and comment on its
relative direction with respect to the wavenumber vector.

 

[Chestermiller]

Please show us what you've done on this so far.

 

[guv]

Did the original problem say anything about the fluid being incompressible?