# 2. Internal Gravity Waves and Group Velocity Flashcards

## stratified fluids and the Brunt-Vaisala frequency, the Boussinesq approximation, internal gravity waves, the concept of group velocity, the phase and group velocity of internal gravity waves

Stratified Fluids

- consider a fluid with a variable density field
- assume that it is incompressible
- in the neutral state get a density gradient with denser particles at the bottom
- lines of constant density are called isopycnels
- even tiny stratification can give rise to powerful effects
- want to model perturbation of a particle from its neutral state in the stratification

Governing Equations for Perturbation in Stratified Fluid

∂ρ/∂t + u∂ρ/∂x

ρ [∂u_/∂t + (u_.∇__)u] = -∇p - ρgez__

∇_ . u_ = 0

Brunt-Vaisala Frequency

Description

-consider a particle with density ρ*such that po(z*)=ρ* where z* is the particles position in the neutral density field

-what happens if the particle is perturbed slightly?

z = z* + δ(t)

-we want to model the evolution of δ(t)

Brunt-Vaisala Frequency

Equations

F=ma -sub in for particle mass, ρ* is the particle density and V is the particle volume: ρ*Vδ'' = B - W -where B is buoyancy and W is the weight, using Archimedes principle: ρ*Vδ'' = Vρo(z*+δ)g - ρ*Vg -taylor expansion of po: ρ*Vδ'' = Vρ*g + Vδρo'(z*)g - ρ*Vg => ρ*δ'' = δρo'(z*)g δ'' = - N²δ -where: N² = - g/ρo(z*) dρo(z*)/dz -this is simple harmonic motion with frequency N where N = Brunt-Vaisala frequency

Brunt-Vaisala Frequency

Assumption

- note that in using Archimedes principle the assumption that the fluid is hydrostatic (motion is slow) has been made
- this is not true in general for fluid perturbations
- in general, fluid perturbations produce waves called internal gravity waves
- their frequency is not N BUT ω∝N

Boussinesq Approximation

-in some instances you can replace ρ in the governing equations with ρ = ρ^ = const. => -Boussinesq equations: ∂ρ/∂t + u∂ρ/∂x = 0 ρ^ Du_/Dt = -∇p - ρgez_ ∇_ . u_ = 0

Small Amplitude Dynamics

Governing Equations

-focus on 2D (x,z):

ρ^ (∂u/∂t + u∂u/∂x + w∂u/∂z) = - ∂p/∂x

ρ^(∂u/∂t + u∂w/∂x + w∂w/∂z) = - ∂p/∂z - ρg

∂ρ/∂t + u∂ρ/∂x + w∂ρ/∂z = 0

∂u/∂x + ∂w/∂z = 0

Small Amplitude Dynamics

Linearisation

-linearisation:

u_ = 0 + u~

ρ = ρo(z) + ρ~

p = po(z) + p~

Small Amplitude Dynamics

Background State

u_ = 0 ρ = ρo(z) p = po(z), dpo/dz = -ρg

Small Amplitude Dynamics

Solution

-four equations, four unknowns

-aim for equation in w~ only (by convention, this works for any variable)

=>

(∂\∂x² + ∂/∂w²) ∂²w~/∂t² + N(z)² ∂²w~\∂x² = 0

-use Boussinesq approximation, treat N as a constant

-sun in wave ansatz:

w~ = Re[w^ e^(i(kx+mz-ωt))]

-where k_=(k,0,m) is the wave vector

Wave Guide

Ocean

- at the base of the ocean, z=0, have a no penetration boundary condition: w=0
- at the water surface, z=H, since the density of the water is so much greater than the density of the air we can assume the surface is levelled by gravity so w(H=0)=0

Wave Packet

Definition

Ψ = ∫ f(k) e^[i(kx-ω(k)t] dk

- where the integral is from -∞ to +∞
- the values and interpretations of Ψ and ω are context dependent
- there is an outer envelope wave which defines the amplitude of the smaller oscillations inside

Wave Packet

f(k)

-the wave packet separates into an envelope and crests

-this separation becomes clear cut for a wave packet composed of k only very near to a given ko, i.e.:

f(k) = ~0 for |k-ko|>ε, 0 for |k-ko|

Wave Packet

Linearisation of ω(k)

-in the small region surrounding ko

-introduce perturbation variable:

k = ko + k~

-can let:

ω(k) = ω(ko) + k~ω’(ko) + ….

-neglect higher order k~ terms

Wave Packet

Deriving Group Velocity

-sub the linearization approximation for a small perturbation k=ko+k~ for ω(k):

ω(k) = ω(ko) + k~ω’(ko)

-into the wave packet definition:

Ψ = ∫ f(k) e^[i(kx-ω(k)t] dk

=>

Ψ = e^[i(kox-ω(ko)t] ∫ f(k) e^[i(x-ω’(ko)t)k~] dk~

-the prefactor exponential can be written:

e^[iko(x - ω(ko)/ko t]

-where ω(ko)/ko is the phase velocity cp, the velocity of the crests

-the integral in general can be written as a function F(x-ω’(ko)t) and represents the envelope

-the velocity of the envelope is the group velocity:

vg = ω’(ko) = ∂ω/∂k