Ocean waves

Books:

• Pedlosky: Geophysical Fluid Dynamics;
• Gill: Atmosphere-ocean dynamics.
• Batchelor: An intr. to fluid dynamics;
• Landau-Lifshitz: Fluid Mechanics.

Outline:

1. Motivation and some observational background
2. Basics:

Kinematics.
Derivation of the equations of motion (Navier-Stokes equations) for an incompressible fluid in a rotating coordinate frame.
Boussinesq approximation.
3. Gravity waves in uniform density, without rotation:

Surface gravity waves
Shallow water equations and waves
Adjustment under gravity
More basics: Kelvin's circulation theorem, Vorticity equation
Persistence of irrotationality, Potential flow
Ertel's potential vorticity.
Nonlinear Stokes waves, [Breaking waves]
Nonlinear wave-wave interactions, resonant wave triads [Pedlosky 3.26]
4. Waves in the presence of rotation:

Shallow water rotating equations, `f-plane' approximation
Waves in a uniformly rotating frame: Poincare and Kelvin waves
Rossby adjustment problem: adjustment under gravity in the
presence of rotation.
Rossby number; Large scale balance of oceanic and
atmospheric motions: geostrophy
Effects of the Earth sphericity: the beta-plane approximation
Waves on a beta-plane: Barotropic Rossby waves. Energy diagram,
reflection, [Rossby waves and the western intensification]
Rossby waves in a stratified ocean; Vertical modes on a beta plane,
vertical modes in a stratified ocean in the presence of a
bottm slope [Pedlosky, 6.11,6.12,6.15]
5. Instability:

Kelvin-Helmholtz instability (of a stratified shear flow)
Rayleigh-Bernard convection
Baroclinic instability: Phillips model, mechanism,
Necessary conditions for instability [from Holton, section 8.4.2]
Eddy's model [from Holton, section 8.4.3]
6. El Nino's chaos:

Equatorial wave modes [Gill, section 11.4, 11.5, 11.6]
An equatorial coupled ocean-atmosphere instability
From steady to nonlinear periodic solutions through a coupled
instability bifurcation, the delay oscillator mechanism
El Nino chaos: From periodic to chaotic solutions through resonance
overlapping. The quasi periodicity route to chaos.
7. Geostrophic Turbulence:

Fully developed turbulence
Enstrophy and energy cascades
Inertial ranges and spectra in 2D turbulence, [Coherent structures]