Dynamics of Internal Gravity Waves in the Ocean

Investigations in the field of internal gravity waves has occupied one of the central places in oceanology in recent years. The topicality of the problems is caused first of all by propagation of internal wave motions in the entire depth of the World Ocean, and, in this connection, by their important role in all oceanic dynamic phenomena.

In this chapter the principles of oceanic thermohydrodynamics are presented briefly. For the relatively short internal waves with length scales less than 100 km, considered in this hook, it is obviously possible to neglect the radius of the Earth restricting the local description.

This chapter discusses equations describing oceanic internal waves. These equations are obtained by different simplifications of the basic equations of oceanic thermodynamics derived in Chapter 1. A list of different forms of these equations for special cases plane (twodimensional) stationary and nonstationary problems, equations linearized relative to the state of rest and to background flow — is given here for convenience.

This chapter considers the simplest example of free harmonic internal waves of small amplitude propagating in an undisturbed horizontally homogeneous ocean. A detailed analysis of the basic boundary value problem for amplitude functions of these internal waves for a finite depth is given in Section 3.1. Examples of solutions of the basic boundary value problem for typical vertically undisturbed density distributions in the ocean are presented in Section 3.2. If the vertical scale of internal waves is small, the ocean may be considered as infinitely deep. In this case internal waves propagate not only horizontally, but also in the vertical direction. This case is examined in Section 3.3. A theory of the propagation of linear internal wave packets is worked out in Section 3.4 based on geometric optics’ approximations. The influence of a fine structure of the ocean’s density field on the propagation of free infinitesimal internal waves is studied in Section 3.5. The influence of the Earth’s rotation (taking into account all components of the Coriolis force) on the propagation of free internal waves is considered in Section 3.6.

Shear flows slightly varying in time and horizontal coordinates are almost always found in the ocean¹. Such shear flows are especially pronounced in the seasonal thermocline zone and may strongly influence the internal waves’ dynamics. If a characteristic period of a shear flow variation is considerably greater than the internal waves’ period and the horizontal variations are small, then the flow may be considered to be stationary and horizontally homogeneous.

The real ocean has, as a rule, different horizontal inhomogeneities which sometimes considerably affect internal waves’ propagation. Among the most characteristic horizontal inhomogeneities are: bottom unevenness; horizontal inhomogeneities of the density field; and horizontal variations of the mean flows. The influence of these phenomena on internal waves’ propagation will be studied in this chapter.

There are different sources which induce internal waves in the ocean: they may be caused by oscillations of atmospheric pressure, wind, regions of the ocean’s bottom (during underwater earthquakes), by tidal forces etc.. Nonlinear mechanisms of internal waves generation, e.g., resonant interaction of surface waves for a flow over large bottom obstacles, are quite important. The generation of internal waves by oscillations of wind and atmospheric pressure is thoroughly studied in this chapter, the main attention being paid to resonant effects. The generation of internal waves by local initial disturbances is also considered in this chapter. Such disturbances may be induced by both atmospheric sources and different deformations of density and velocity fields. A brief overview of the other sources of the generation of internal waves is given in the conclusions of the chapter. Specific questions of wave generation by underwater earthquakes, tidal flows, and bottom obstacles are presented in [49], [179] and therefore are not considered here.

Equations describing the dynamics of internal waves in incompressible stratified fluid may be written in the form of Hamiltonian equations, widely used in modern physics. The Hamiltonian formalism appears to be especially convenient for the investigation of weakly nonlinear wave fields.

In this chapter we study trivial nonlinear effects appearing while long internal waves propagate. There may be two situations and each of them needs its own analysis. The first is the propagation of internal waves in shallow water (e.g., in a shallow sea like the Baltic Sea) when the wavelength λ considerably exceeds the sea’s depth H. In this case equations of internal waves dynamics are reduced to the Korteweg-de Vries equation (KdV) (see a review of its evolution in the Appendix). Derivation and analysis of this equation for long internal waves is given in Section 8.1. In Section 8.2 stationary solutions of the KdV equation describing long nonlinear internal waves of the stable type are analyzed. Here we also give the numerical results for particular stratification profiles. The second possible situation is propagation of long waves within a thin pycnocline (the wavelength λ is much greater than the pycnocline’s thickness h). This is quite typical for natural conditions and is studied in Section 8.3. In this case internal waves are described by an equation of the KdV type but with integral dispersion. An analysis of the stationary solutions for this equation is presented as well.

The basic nonlinear effects caused by the propagation of narrow spectrum wave packets are considered in this chapter. The first of such effects interesting from the viewpoint of oceanology is that of a weakly nonlinear internal wave packet with the mode structure generating velocity, density, and pressure fields which do not vanish when averaged over the wave period. In Section 9.1 the main characteristics of such fields are obtained. In Section 9.2 the problem of the stability of an internal wave packet with respect to its longitudinal modulation is studied. The long wave packet is shown to be always stable, and wavelength reduction causes the appearance of instability zones, whose parameters depend upon the stratification profile. In Section 9.3 it is shown that if a weakly nonlinear packet is formed by internal waves of relatively short period, then induced mean fields oscillate strongly in the vertical direction. The theory of the formation of oceanic fine structure, based on this phenomenon, is proposed. A rather good agreement between the evidence and numerical results for certain of some of the stratification profiles was obtained. In Section 9.4 propagation of a weakly nonlinear wave packet in the fluid with a constant Brunt- Väisälä frequency is studied.

Nonlinear effects while internal waves propagate in the ocean are caused not only by the self-modulation of almost monochromatic internal wave packets, but also by the interaction of internal waves with different wave numbers, frequencies and mode numbers.

In this chapter the methods of measurements of oceanic internal waves both in a fixed point (by an anchored buoy) and in moving coordinates (by a towed chain of thermistors) are described. In the final section the problem of the discernment of internal waves and turbulence is considered. Existing statistical methods of the description of internal waves’ field as well as the methods of interpretation of evidence are valid only for small amplitude (linear and weakly nonlinear) waves. When natural internal waves have strong slopes and nonlinear effects take place, traditional spectral and correlation statistical methods are hardly applicable.

In this chapter characteristic properties of natural internal waves and some problems of the interpretation of evidence are considered’. A brief list of basic properties of frequency and spatial spectra of internal waves measured in the ocean is given at the beginning. Then local properties of measured short period internal waves (in particular, wave packets) are analyzed. One of the important characteristics of an internal wave field is the probability partition function of wave components. Peculiarities of empirical partition functions are considered in Section 12.3. In Section 12.4 brief data from laboratory experiments on the study of internal waves are given. Hypothetical mechanisms of internal waves’ breaking and dissipation, based on the evidence, are discussed in the final Section 12.5.