**Yuri Lvov ^{1}**

*Department of Mathematical Sciences*

Rensselaer Polytechnic Institute

**Esteban G. Tabak ^{2}**

*Courant Institute of Mathematical Sciences*

New York University

A novel canonical Hamiltonian formalism is developed for long internal
waves in a rotating environment. This includes the effects of
background vorticity and shear on the waves. By restricting
consideration to flows in hydrostatic balance, superimposed on a
horizontally uniform background of vertical shear and vorticity, a
particularly simple Hamiltonian structure arises, which can be thought
of as describing a nonlinearly coupled infinite collection of shallow
water systems. The kinetic equation describing the time evolution of
the spectral energy of internal waves is subsequently derived. In the
high frequency limit, the Coriolis effects may be neglected, and a
family of stationary Kolmogorov solutions can be found, which includes the
Garrett-Munk spectrum of oceanic internal waves.

*submitted to Physica D*

- Introduction
- Hamiltonian formalism for long internal waves
- Linear, Non-Rotating Shallow Waters
- Nonlinear, Non-Rotating Shallow Waters
- Nonlinear, Non-Rotating, Internal Waves
- Linear Shallow Waters in a Rotating Environment
- Rotating Nonlinear Shallow Waters
- Nonlinear Internal Waves in a Rotating Environment

- Weak turbulence theory
- The High Frequency Limit of the Kinetic Equations
- Conclusions
- Bibliography
- About this document ...