L. G. Redekopp

On two-dimensional packets of capillary-gravity waves

The motion of a two-dimensional packet of capillary-gravity waves on water of finite depth is studied. The evolution of a packet is described by two partial differential equations: the nonlinear Schroedinger equation with a forcing term and a linear equation, which is of either elliptic or hyperbolic type depending on whether the group velocity of the capillary-gravity wave is less than or greater than the velocity of long gravity waves. These equations are used to examine the stability of the Stokes capillary-gravity wave train. The analysis reveals the existence of a resonant interaction between a capillary-gravity wave and a long gravity wave. The interaction requires that the liquid depth be small in comparison with the wavelength of the (long) gravity waves and the evolution equations describing the dynamics of this interaction are derived.

Sediment Resuspension and Mixing by Resonantly Generated Internal Solitary Waves

The observation of internal solitary waves (ISWs) propagating upstream along a strongly stratified bottom layer on the California shelf is reported. An increased concentration of particulates in the water column accompanies the passage of these ISW packets. The estimated local Richardson number in the bottom vicinity is around 1/4, and a vertical coefficient of eddy diffusivity of order 1022 m2 s21 is associated with the upstream propagating leading ISW. The leading ISW gave rise to reversed flow in an 8-m layer above the bottom. It is argued that the upstream propagating ISWs were generated by resonant flow over bottom topography. Internal waves generated in this way seem to be frequent in the record of a month-long experiment. Model results suggest that the ISWs can carry up to 73% of such generated long wave energy. The ocean conditions at the site are similar to those of other coastal sites, which suggests that the phenomenon described here may be common.

The nonlinear capillary instability of a liquid jet. Part 1. Theory

Nonlinear capillary instability of an axisymmetric infinite liquid column is investigated with an initial velocity disturbance consisting of a fundamental and one harmonic component. A third-order solution is developed using the method of strained co-ordinates. For the fundamental disturbance alone, the solution shows that a cut-off zone of wavenumbers ( k ) exists such that the surface waves grow exponentially below the cut-off zone, linearly in the middle of the zone (near k = 1), and an oscillatory solution exists for wavenumbers above the boundary of the zone. For an input including both the fundamental and a harmonic, all wave components grow exponentially when the fundamental is below the cut-off zone. Using a Galilean transformation, the solution is applied to a progressive jet issuing from a nozzle. The jet breaks into drops interspersed with smaller (satellite) drops for k k > 0·65. It is shown theoretically that the formation of satellites can be controlled by forcing the jet with a suitable harmonic added to the fundamental.

On the theory of solitary Rossby waves

The evolution of long, finite amplitude Rossby waves in a horizontally sheared zonal current is studied. The wave evolution is described by the Korteweg-de Vries equation or the modified Korteweg-de Vries equation depending on the atmospheric stratification. In either case, the cross-stream modal structure of these waves is given by the long-wave limit of the neutral eigensolutions of the barotropic stability equation. Both non-singular and singular eigensolutions are considered and the appropriate analysis is developed to yield a uniformly valid description of the motion in the critical-layer region where the wave speed matches the flow velocity. The analysis demonstrates that coherent, propagating, eddy structures can exist in stable shear flows and that these eddies have peculiar interaction properties quite distinct from the traditional views of turbulent motion.

The stability of a sheared density interface

This study investigates the stability of stratified shear flows when the density interface is much thinner than, and displaced with respect to, the velocity interface. Theoretical results obtained from the Taylor–Goldstein equation are compared with experiments performed in mixing layer channels. In these experiments a row of spanwise vortex tubes forms at the level of maximum velocity gradient which, because of the profile asymmetry, is displaced from the mean interface level. As the bulk Richardson number is lowered from a high positive value the effects of these vortex tubes become more pronounced. Initially the interface cusps under their influence, then thin wisps of fluid are drawn from the cusps into asymmetric Kelvin–Helmholtz billows. At lower Richardson numbers increasingly more fluid is drawn into these billows. The inherent asymmetry of flows generated in mixing layer channels is shown to preclude an effective study of the Holmboe instability. Statically unstable flows (negative Richardson num...

Global dynamics of symmetric and asymmetric wakes

The two-dimensional wake-shear layer forming behind a rectangular-based forebody with independent ambient streams on either side of the forebody is examined by direct numerical simulation. Theoretical aspects of global modes and frequency selection criteria based on local and global stability arguments are tested by computing local stability properties using local, time-averaged velocity profiles obtained from the numerical simulations and making the parallel-flow approximation. The theoretical results based on the assumption of a slightly non-parallel, spatially developing flow are shown to provide a firm basis for the frequency selection of vortex shedding and for defining the conditions for its onset. Distributed suction or blowing applied at the base of the forebody is used as a means of wake flow modification. The critical suction velocity to suppress vortex shedding is calculated.

Long nonlinear waves in stratified shear flows

The propagation of finite-amplitude internal waves in a shear flow is considered for wavelengths that are long compared to the shear-layer thickness. Both singular and regular modes are investigated, and the equation governing the amplitude evolution is derived. The theory is generalized to allow for a radiation condition when the region outside the stratified shear layer is unbounded and weakly stratified. In this case, the evolution equation contains a damping term describing energy loss by radiation which can be used to estimate the persistence of solitary waves or nonlinear wave packets in realistic environments. A continuous three-layer model is studied in detail and closed-form expressions are obtained for the phase speed and the coefficients of the nonlinear and dispersive terms in the amplitude equation as a function of Richardson number.

Transitions to chaos in the Ginzburg-Landau equation

Abstract The amplitude evolution of instability waves in many dissipative systems is described close to criticality, by the Ginzburg-Landau partial differential equation. A numerical study of the long-time behavior of amplitude-modulated waves governed by this equation allows the identification of two distinct routes of the Ruelle-Takens-Newhouse type as the modulation wavenumber is decreased. The first route involves a sequence of bifurcations from a limit cycle to a two-torus to a three-torus and to a turbulent regime, the last stage being preceded by frequency locking. The turbulent regime is itself followed by a new two-torus. In the second route, this two-torus exhibits a single subharmonic bifurcation which immediately results in transition to chaos. A description of the various possible dynamical states is tentatively given in the plane of the two control parameters c d and c n .

Local and global instability properties of separation bubbles

Abstract A family of velocity profiles with reversed flow, typical of those found in separated flows, are examined for their linear instability properties. Specifically, a family of modified Falkner-Skan profiles are analyzed for the onset of absolute instability as the magnitude of the reversed flow increases. The mode of instability associated with the inflection point in the vicinity of the dividing streamline is found to become absolutely unstable as the peak reversed flow approaches about thirty percent of the free stream value. The family of profiles were used to construct generic models of separation bubbles and study the possible onset of global instability in the representative spatially-developing flows.

Supersonic interference flow along the corner of intersecting wedges

Abstract : An experimental investigation of the inviscid supersonic flow field in a corner formed by two intersecting wedges is reported. The study includes measurements of the distribution of the surface pressure and the wave structure of the flow field in the corner region both for a symmetrical corner configuration (e. e., equal wedge angles) in the Mach-number range between 2.5 and 4 and for an asymmetrical model at one Mach number as a function of the incidence angle of one wedge. A clearly identifiable shock structure is discovered that divides the interference field into four zones. The total extent of the disturbance near the surfaces of the model is approximately twice the extent indicated by available small-perturbation analyses. The highest pressure rise occurs in the inner zone near the corner; the magnitude of this rise is related successfully to the second-order linearized theory. The inviscid flow field is shown to involve strong cross-stream pressure and entropy gradients both at the corner proper and away from it, which cannot be disregarded in analyses of the viscous boundary layer. (Author)