On the dispersion of pairs of internal inertial gravity waves
The inclination of the group velocity vector to the horizontal, , of internal inertial gravity waves propagating in a deep uniformly stratified fluid varies with wave frequency. Group velocity also depends on wavenumber. As a consequence, packets of waves with finite frequency and wavenumber bandwidths generated by intermittent processes at the sea surface or ocean floor will usually disperse.
Waves may, however, interact with one-another to suppress or entirely remove such dispersion, resulting in packets or groups of waves which retain structure during propagation. Here conditions are sought in which dispersion is entirely suppressed by the interactions between waves composing a packet. For simplicity only, a pair of waves is considered here, a dominant 'primary' wave of steepness, s1, and a smaller secondary. The effects of interaction are examined up to third order as the waves travel in an ocean of uniform buoyancy frequency, N, and Coriolis frequency, f, seeking conditions which lead to wave pairs which travel at equal group velocities and in which the waves are steady, without wave growth or decay. Excluded are conditions in which there are resonant interactions between the two waves or their interaction products, for these modify the amplitudes of the waves and the pair will not propagate steadily without change. Third order interactions lead to changes in group velocity and, in general, to a changing amplitude of the secondary unless either the azimuthal angle, α, between the two waves or F, = f/N, is zero. Numerical estimates are made for primary wave directions, 1, = 5°, 10° and 20°, α is less than 12°, and when there are only moderate differences between their wavenumbers and frequencies. In the absence of rotation when F = 0, solutions for s1 are found at which waves have the same group velocity, but only when the azimuthal angle, α, between the first order primary and secondary is zero, i.e. when the two waves propagate in the same vertical plane. With rotation, when F ≠ 0, it is generally necessary for α to be zero for there to be a steady secondary wave. It is concluded that such stable co-travelling wave pairs exist but
(i) the only wave pairs with no dispersion or wave growth are two-dimensional, travelling in the same vertical plane;
(ii) stable co-travelling wave pairs are most likely when the effects of rotation are significant. This includes near-inertial waves and M2 internal tides between latitudes of 28.9° and their turning latitude of 74.9°; and
(iii) the stable secondary waves which will co-travel with the primary or, as it passes, be 'captured' by it, have shorter wavelength and lower frequency, or longer wavelength and higher frequency, than the primary wave.
Further study is required to discover whether or not there are groups or packets of internal waves of permanent form.
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