On waves generated in dispersive systems by travelling forcing effects, with applications to the dynamics of rotating fluids

Abstract

A theory of the generation of dispersive waves by travelling forcing effects, that may be steady, oscillatory or transient in character, is given for a general homogeneous system. Small disturbances to the system are supposed stable, and governed by a linear partial differential equation with constant coefficients which admits solutions in the form of plane waves satisfying an, in general, anisotropic dispersion relation P(σ, k) = 0 between frequency σ and wave-number vector k.

If the forcing region, supposed of limited extent, travels with constant velocity U, then oscillatory forcing terms of frequency σ₀ (which would be replaced by 0 in the limiting case of a steady forcing effect, while taking, for a typical transient one of duration T, values from 0 to about 10/T) produce waves of frequency σ₀ + U. k (the Doppler effect). For any such waves, the wave-number k satisfies the equation P(σ₀ + U.k,k) = 0, representing a surface in wave-number space here called S(σ₀), and their position relative to that of the forcing region is determined by having been generated when that region was in an earlier position, and having subsequently progressed with the group velocity. This implies the rule, also derived analytically in §§2 and 3, that waves with a particular value of k on S(σ₀) are found in a direction, stretching out from the forcing region, which is one of the directions normal to S(σ₀) at k, namely the one pointing towards S(σ₀ + δ). This rule is supplemented by results on wave amplitudes and shapes of crests.

The theory is applied (§§4 and 5) to Rossby waves excited in a beta-plane ocean by travelling patterns of wind stress. If a steady wind-stress pattern moves westward, semicircular waves of length 2π √(U/β) trail behind it, but signals are found also directly ahead, consisting of the disturbance integrated in the west-east direction and subjected to a ‘low-pass filter’ with respect to its north-south components of wave-number. An eastward-travelling pattern, by contrast, produces only a wake-like disturbance, calculated in detail in §4. The waves generated for intermediate directions of travel are identified, and the strong tendencies in all cases for westward intensification of transient currents are noted.

For example, a wind-stress pattern travelling 30° N. of E. leaves a trailing wedge of currents from W. to 30° S. of W. in the steady case. The influence on this conclusion of a finite duration T of such a pattern is investigated in §5 by Fourier analysis in time. The fate of Fourier components of frequency σ₀ depends on the ratio L =σ₀/√(Uβ). If this is less than 1 for all (T, up to about 10/T, then the disturbance retains its trailing character; on the other hand, any components with L > 1 have a much greater directional spread. Tidal terms make fairly small changes to the results, except that (in an ocean of depth H) they make the directional spread disappear for L greater than about √(βgH/4f²U).

Excitation of gravity waves in non-rotating fluid is briefly considered, includ-ing generation on deep water by a travelling oscillating disturbance (§6), and generation in a uniformly stratified fluid by a vertically moving obstacle (§7). The predicted wave shapes in the latter case, with cusps at a finite distance behind the obstacle, agree excellently (figures 7 and 8) with experiments by Mowbray (1966).

An exceptional case, in that part of S(σ₀) is doubly covered, is generation by steady (σ₀ = 0) motion of an obstacle along the axis of uniformly rotating homo-geneous fluid, the surface S(0) being a sphere and two coincident planes. Whereas waves corresponding to points on the sphere trail behind the obstacle, the appro-priate normals on the two planes point in opposite directions inside the sphere (§8), permitting the well-known formation of the ‘Taylor column’ ahead of the obstacle at low Rossby numbers.

Still more complicated, because fully three-dimensional, is the case when the obstacle moves at right angles to the axis of rotation (§9). At finite though small Rossby number it is impossible for the Taylor column formed near the body to extend to large distances from it, where on the contrary the disturbance is shown to take the form of slightly trailing cones, shown in cross-section in figure 12, containing waves whose crests have cusps on the boundaries of the cones. An estimate of the length of the Taylor column, as body dimension divided by Rossby number (for small enough kinematic viscosity), is made by considering the fit between the Taylor-column and wave-cone regions.