Introduction

The term ocean waves typically evokes images of surface waves shaking ships during storms in the open ocean, or breaking rhythmically near the shore. Yet much of the ocean wave action takes place underneath the surface, and consists of modulations not of the air-water interface, but of invisible surfaces of constant density. These internal waves are ubiquitous in the ocean, contain a large amount of energy, and affect significantly the processes involved in water mixing and transport.

Our knowledge of the typical scales and energy content of oceanic internal waves advanced significantly through improved and more widespread observations in the last few decades. In particular, an empirically based formula that Munk and Garrett developed in the seventies (now called the Garrett Munk spectrum of internal waves) synthesizes magnificently a seemingly universal distribution of energy among scales [1,2,3]. Description of modern observational work can be found in [4,5,6,7,8]. In particular, in [6] the deviation from the Garrett-Munk spectra are documented, and in [7] the dissipation rate of turbulent kinetic energy is measured. On the theoretical side, this distribution has been generally understood as due to the effects of nonlinear interaction among waves, amenable to a description based on kinetic equations analogous to the ones of statistical mechanics. Probably the first kinetic equation for internal waves was written in [9], though not within the frame of a Hamiltonian formalism.

A comprehensive review of a significant line of work developed in the seventies and eighties is provided in [10] and references therein. Some important references not cited there are [11,12,13]. More recent work includes, for example, [14] where a thorough perturbative Eulerian-Boussinesq approach in a nonrotating environment is developed. See [15,16,17] for a detailed discussion of the relation between the spectral tails of Lagrangian and Eulerian flow descriptions. One can also use ray theory to study internal wave scattering, as in [18], where the Doppler-spreading of short internal wave packets in the atmosphere and the ocean is studied or, as in [19], where the refraction of short oceanic internal waves by a spectrum of large amplitude inertia waves is considered.

Our work differs from the line reviewed in [10] in various ways. In [10] the wave dynamics is formulated in a fully Lagrangian framework, while our isopycnal formulation is Eulerian in the horizontal coordinates and Lagrangian in the vertical. To write down the equations of motion in the Lagrangian framework, the system's Lagrangian is expanded in powers of the assumed small displacement of the fluid parcels. This description is therefore approximate even at the level of the dynamic equations of motion. Such a description fails to adequately describe the advection of small scale waves by larger scale flows, as well as the interaction of waves with the vortical part of motion. This is acknowledged in [10], which proposes as a challenge the rederivation of the kinetic equation in an Eulerian framework. In the present article, we fulfill this program, and use therefore as a small parameter not the small displacement of fluid parcels, but only the weakness of the nonlinear interaction among waves.

A fundamental question posed in the eighties is whether the GM spectra is close to the statistical steady state solution of the kinetic equation. The discussion in [10] indicates that GM is not inconsistent with the kinetic equation proposed, and may be close to being a steady state solution. Recently, the authors have put forward two studies. One [20] shows that a power law spectrum which, in the high frequency limit, is very close to GM, constitutes an exact analytical steady state solution of a kinetic equation for hydrostatic flows described in isopycnal coordinates. The later study [21] shows that, in fact, GM itself is a member of a larger family of exact solutions to these kinetic equations, which also include much of the pattern of observed deviations from GM that have accumulated over the last few decades. The present article extends these results, by starting to build a theory that includes frequencies comparable to the rate of rotation of the Earth, and that accounts for the existence of large-scale horizontal eddies and vertical shear in the ocean, over which the internal waves are superimposed.

The main contribution of this paper is the development of a novel Hamiltonian formalism for the description of internal waves. Our approach is general enough to include the effects of the Earth's rotation, of large-scale eddies and of vertical shear on the waves, yet exclusive enough in its assumptions to yield a relatively simple, manageable model. The main assumption is that the waves are long enough to be in hydrostatic balance, yet they live in horizontal scales shorter than those characterizing the underlying eddies. This allows us to consider as unperturbed flow an arbitrary layered distribution of potential vorticity and vertical shear -that is, potential vorticity and horizontal velocity profiles which adopt independent, horizontally uniform values at each depth. Such hydrostatically balanced, horizontally uniform, vertically varying profiles are quite representative of long waves in the real ocean; they arise due to the highly anisotropic nature of the ocean's eddy diffusivity, which tends to homogenize the flow along isopycnal surfaces.

Hamiltonian structures for stratified incompressible flows have been the subject of active research over the last few decades. Though a complete review of the subject is outside the scope of this paper, we list here some of the most important results. The first paper to derive a Hamiltonian structure for stratified internal waves is probably [12], where a representation is proposed based on Clebsh-like variables. The resulting Hamiltonian is an explicit infinite power series of canonically conjugated variables. In [22], a Hamiltonian formalism for internal waves in isopycnal coordinates is developed. No hydrostatic approximation is invoked, and thus the resulting Hamiltonian is expressed as an explicit power-series in the powers of the assumed small nonlinearity. Potential problems in using Clebsh variables for stratified flows have been addressed in [23,24]. Problems of describing the Hamiltonian structures to describe the interaction between wave and vortex modes are addressed in [25]. In addition to the references above, a noncanonical Hamiltonian structure based on a Lie-Poisson framework has been developed in [26,27]. More recently, two broad reviews on Hamiltonian structures for fluids have been published [28,29].

We choose to describe the flow in isopycnal coordinates, replacing the depth $z$ by the density $\rho$ as the independent vertical coordinate. The advantage of such semi-Lagrangian description is manifold. First, it eliminates the need to handle the vertical velocity explicitly, which renders the equations much more tractable. At a deeper level, it greatly simplifies the description of the interaction between waves and vorticity, since potential vorticity is preserved along particle trajectories, and these remain on isopycnals in the absence of vertical mixing. In particular, if the potential vorticity is uniform throughout an isopycnal surface, it remains so forever. This is the situations we choose to describe: a profile of vorticity which varies across surfaces of constant density, but is homogeneous along them. Such ``pancake-like'' profiles are quite representative of the intermediate scales in the ocean, smaller than the dominant eddies, but still containing a significant wealth of internal waves. Modeling this scenario constitutes an intermediate step between considering irrotational frameworks, and studying the fully turbulent interaction between arbitrary profiles of vorticity and waves.

Finally, the equations in isopycnal coordinates are highly reminiscent of the equations for a shallow single layer of homogeneous fluid. We exploit this formal analogy to develop a Hamiltonian formalism for internal waves which extends naturally similar formalisms for shallow waters.

The plan of this paper is the following. In section (), we develop a hierarchy of Hamiltonian descriptions for long waves. Our goal here is to develop a general Hamiltonian formalism for nonlinear internal waves in a rotating environment, superimposed on a background of layered potential vorticity and horizontally uniform shear. However, we choose to do so in stages, starting in the simple context of linear, irrotational shallow water equations, and adding progressively nonlinearity, stratification, the Coriolis effect, and nontrivial potential vorticity. This not only clarifies the logic and essential simplicity of the structure of the final Hamiltonian, but also yields along the way a series of Hamiltonian structures for problems of intermediate simplicity. In section (), we derive kinetic equations for the time evolution of the energy spectrum of internal waves, based on the Hamiltonian description developed in the previous section. For this, we constrain ourselves to consider a neutral background with neither shear nor vorticity, yet keeping the Coriolis effect. Then, in section (), we discuss the high frequency limit of these kinetic equations, where the Coriolis parameter becomes negligible. This was the situation considered in [20] and [21], where exact stationary solutions were found, in full agreement with the high frequency range of the GM spectrum. Finally, in section (), we provide some concluding remarks, and discuss open problems for further research.