The relation between the duration and shape of internal wave groups
Abstract:This paper discusses the effect of the shape of internal wave groups on their “duration” or “lifetime”—how long they retain their form before their component waves disperse. The methodology devised by Smith and Brulefert (2010) to study the dispersion of surface wave groups is extended to examine the dispersion of internal wave groups. To provide tangible examples, it is supposed that wave groups of ellipsoidal shape, symmetrical about a vertical plane, are generated in a uniformly stratified thermocline by moving periodic disturbances perturbing the base of an overlying mixed layer. The dispersion relation for internal waves is used to determine the group duration, taken as the time required for the volume of the group to approximately double through the fastest separation of its component waves. As well as allowing the orientation (inclination of their larger, major axis to the horizontal) and aspect ratio (that of the minor to major axis in a vertical plane) of wave groups to vary, their lifetimes are compared in two particular cases: Case A in which the length of the initial minor axes in the vertical plane of the groups is the same, and Case B in which groups are initially composed of the same number of waves. Two-dimensional groups and “three-dimensional groups” (the latter predominantly two-dimensional but of limited extent in one horizontal direction) are considered. As has been found for surface waves, the duration of internal wave groups does indeed depend upon their shape. In both Cases, groups with relatively small aspect ratio and, in Case B, groups with many waves usually have greater lifetimes than relatively large aspect ratio groups with few waves. Two-dimensional groups have greater lifetimes than three-dimensional groups. In many cases, the groups with the longest lifetime have their longer (major) axis inclined at an angle to the horizontal that is close to the inclination of the group velocity vector; in these cases the lines of constant phase of waves composing the groups are not (as is found for some surface wave groups) slanted with respect to the major axis of the groups, but parallel. Some long-lifetime groups are found to have their major axes inclined to the horizontal at an angle that is very close to that of the front of a packet of waves generated by the moving periodic disturbance at the foot of the mixed layer. In Appendix B it is shown that the ratio, np/n or nm/n, of the number of waves that would be recorded as the group passes a fixed point or a vertical mooring, to the number of waves contained, instantaneously, within a wave group, depends on the shape of the group and on the ratio of the dominant wave phase speed to the group velocity of the group. A simple model described in Appendix C suggests how such slanted wave groups can be generated in the thermocline by moving, but transient, disturbances. The orientation of “scars,” regions left by waves breaking in the wave groups, is examined in Appendix D. Except for near inertial waves with small aspect ratio, scars are generally close to being horizontal.
Document Type: Research Article
Publication date: January 1, 2010
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