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 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)
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
