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Ocean Waves (continued) + ward ' s science has a single maximum somewhere in the body of the fluid; in the second mode, there are two such maxima (180° out of phase), with a node between; and so on. The actual motion usually consists of a superposition of modes. In the ocean, the Väisälä-Brunt frequency varies from a maximum, commonly near 0.2 cycle per minute in the steepest part of the thermo- cline to negligible values at the bottom of the deep seas and in mixed layers such as those at the sea surface. Amplitudes of waves are oscillatory with depth and appreciable only where their frequency is less than the local value of the Väisälä-Brunt frequency. For this reason, internal waves of high frequency are limited in depth to the thermocline. Long-period (low- frequency) internal waves are affected by Earth rotation and the limiting periodicity is the local value of the pendulum day (one-half sidereal day divided by the sine of the latitude). These longest-period oscillations are inertial motions, in which the orbital path no longer has any vertical component but forms a circle in the horizontal. In a continuously stratified fluid such as the ocean below the thermocline, internal waves propagate diagonally upward and downward, thus distributing energy throughout the ocean depths. Their ray paths are easily distorted because the group velocity of internal waves is comparable to the differences in velocity in the vertical of ocean currents and inertial motions. As a consequence, bundles of internal wave rays suffer severe refraction, sometimes to the extent that they form caustics known as critical layers, where the wave energy is absorbed by breaking. Where the sea bed is level, internal waves can be reflected by surface and seafloor to form normal modes within the body of the ocean. On a sloping seafloor, reflection at the bottom causes a change of wavelength. Internal waves in the atmosphere have been detected by a variety of instruments: microbarographs and wind recorders at ground level and long-term recordings of the scattering of radar or sonar beams by sharp density gradients in the high atmosphere. In the ocean, internal waves have been found by recording fluctuating currents in middepths by moored current meters, by acoustic backscatter Doppler methods, and by stud- ies of the fluctuations of the depths of isotherms as recorded by instruments repeatedly lowered from shipboard or by autonomous instruments floating deep in the water. Internal waves are thought to be generated in the sea by variations of the wind pressure and stress at the sea surface, by the interaction of surface waves with each other, and by the interaction of tidal motions with the rough seafloor. Their importance is that they can transmit energy and momentum throughout the ocean, not only laterally but also vertically. They can, therefore, transmit energy from the surface to all depths. In this way the otherwise sluggishly moving water at great depths can be agitated. Long-period waves Long-period waves are those that exist when the period (T) is longer than one-half of a pendulum day, that is, 12 h/sin θ, where θ is the latitude of the location of interest. The main types of low-frequency ocean waves are Rossby waves, topo- graphic Rossby waves, coastal Kelvin waves, and equatorial Kelvin waves. Rossby waves Rossby waves, named after meteorologist C. Rossby, are fundamental to the large-scale dynamics of both atmosphere and ocean. They can exist at periods from a few days to several years and help to describe, for example, seasonal and climatic fluctuations in the oceans. Since they exist at long periods, the Earth rotates several times during a wave period and the rota- tion of the Earth therefore plays a central role in Rossby-wave dynamics. In order to understand Rossby waves, it is necessary to consider rotation, angular velocity, and vorticity. The angular velocity of the Earth is defined as a vector of magnitude Ω and direction northward along the axis of rotation. An angular velocity can be similarly defined for all solid rotat- ing bodies: the magnitude of the angular velocity is 2π radians divided by the period of rotation and the direction is along the axis of rotation in the direction analogous to that for the rotating Earth. The vorticity of a particle of water in solid-body rotation is defined to be twice the angular velocity. In general, the velocity shear for a water particle will not correspond to that for solid- body rotation and the effective rotation and vorticity will conse- quently be modified. However, this does not affect the following explanation for the Rossby-wave mechanism. Because the gravitational force perpendicular to the Earth's surface is so dominant, water particles tend to remain on the same horizontal level; that is, long-period ocean flows tend to be parallel to the Earth's surface. For this reason, rotational ef- fects in the horizontal plane are of prime significance; conse- quently the vertical component of vorticity is of greatest impor- tance to ocean dynamics. This vertical component of vorticity has two contributions. One, which exists even when the water is at rest relative to the Earth, is the local vertical component of the Earth's rotation at latitude θ. The remaining contribution is due to rotational effects in the horizontal currents and its value is positive when rotation is counterclockwise as viewed locally above the Earth's surface. Topographic Rossby waves Another type of long-period ocean wave is the topographic Rossby wave, whose mechanism depends on variations of bot- tom topography and hence on the water depth. An example is the continental-shelf wave in which water depth varies strongly perpendicular to the coastline across the continental shelf and slope. Continental-shelf waves propagate with the coast on