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

1
Q

Stratified Fluids

A
• 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
2
Q

Governing Equations for Perturbation in Stratified Fluid

A

∂ρ/∂t + u∂ρ/∂x
ρ [∂u_/∂t + (u_.∇)u]
= -∇p - ρgez

∇_ . u_ = 0

3
Q

Brunt-Vaisala Frequency

Description

A

-consider a particle with density ρsuch 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)

4
Q

Brunt-Vaisala Frequency

Equations

A
```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```
5
Q

Brunt-Vaisala Frequency

Assumption

A
• 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
6
Q

Boussinesq Approximation

A
```-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```
7
Q

Small Amplitude Dynamics

Governing Equations

A

-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

8
Q

Small Amplitude Dynamics

Linearisation

A

-linearisation:
u_ = 0 + u~
ρ = ρo(z) + ρ~
p = po(z) + p~

9
Q

Small Amplitude Dynamics

Background State

A
```u_ = 0
ρ = ρo(z)
p = po(z), dpo/dz = -ρg```
10
Q

Small Amplitude Dynamics

Solution

A

-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

11
Q

Wave Guide

Ocean

A
• 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
12
Q

Wave Packet

Definition

A

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
13
Q

Wave Packet

f(k)

A

-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|

14
Q

Wave Packet

Linearisation of ω(k)

A

-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

15
Q

Wave Packet

Deriving Group Velocity

A

-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

16
Q

Group Velocity

Definition

A

-the velocity of the envelope, the envelope encompasses as sinusoidal function
vg = ω’(ko) = ∂ω/∂k

17
Q

Simplest Wave Packet Composition

Wave Packet Equation

A

Ψ = e^[i(k1x-ω(k1)t] + e^[i(k2x-ω(k2)t]

-where k1=ko-ε and k2=ko+ε and ε«1

18
Q

Simplest Wave Packet Composition

From Wave Packet Envelope Equation

A
• let f(k) = δ(k-k1) + δ(k-k2)

- where δ is a dirac-delta function

19
Q

Simplest Wave Packet Composition

Wave Packet Envelope Equation

A

Ψ = e^[iko(x-cp(ko)t] { e^[i(x-cg(ko)t)(k1-k)] + e^[i(x-cg(ko)t)(k2-ko)]}
-where k1-ko=ε and k2-ko=-ε
= e^[iko(x-cp(ko)t] 2cos[ε(x-cg(ko)t]

20
Q

Simplest Wave Packet Composition

Envelope and Beats

A
• sinusoidal with wavelength λ=2π/ε
• so in the limit ε->0, the wavelength of the envelope is larger
• for sound, this envelope is heard as beats
21
Q

When does cp=cg ?

A

-for non-dispersive waves, ω∝k

22
Q

Rate of Energy Transport

A

-rate of energy transport Ce is equal to the group velocity cg