Bruno Voisin

Limit states of internal wave beams

Internal gravity waves propagate away from a localized monochromatic disturbance inside beams, which develop around a St Andrews Cross in two dimensions and around a double cone in three dimensions. The structure of the beams depends on three mechanisms, which couple together the different directions of propagation of the waves within the fluid. These mechanisms are associated with the size of the disturbance, the start-up of the motion and the viscosity of the fluid, respectively. The present paper considers each mechanism in isolation, for three-dimensional generation. The analysis is asymptotic and relies on far-field and large-time approximations. For each mechanism, three expressions of the waves are found: one, exact for an extended disturbance, that involves all the wavenumber vectors satisfying the dispersion relation; and two others, respectively uniform and non-uniform asymptotic expansions, that involve only the wavenumber vectors associated with group velocity vectors pointing toward the observer. For each mechanism, profiles of pressure and velocity are presented. A new time-independent characterization of the waves is introduced, in terms of the intensity or average energy flux; it is applied to the definition of the beam width. For an extended disturbance this width is a constant, the diameter of the disturbance. For an impulsive start-up the width increases linearly with the distance from the disturbance, and decreases in inverse proportion to the time elapsed since the start-up. For a viscous fluid the width increases as the one-third power of the distance. In all three cases, for a disturbance of multipolar order

Internal wave generation in uniformly stratified fluids. Part 2. Moving point sources

2^n

Internal wave generation by oscillation of a sphere, with application to internal tides

, at constant strength, the power output varies in inverse proportion to the

Internal waves generated by a translating and oscillating sphere

(2n+1)

Lee waves from a sphere in a stratified flow

th power of the beam width.

Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere

The Greens function method is applied to the generation of internal gravity waves by a moving point mass source. Arbitrary motion of a source of arbitrary time dependence is treated using the impulsive Greens function, while ‘classical’ approaches of uniform motion of a steady or oscillatory source are recovered using the monochromatic Greens function. Waves have locally the structure of impulsive waves, emitted at the retarded time t r , and having propagated with the group velocity; at each position and time an implicit equation defines t r , in terms of which the waves are written. A source both oscillating and moving generates two systems of waves, with respectively positive and negative frequencies, and when oscillations vanish these systems merge into one. Three particular cases are considered: the uniform horizontal and vertical motions of a steady source, and the uniform horizontal motion of an oscillatory source. Waves spread downstream of the steady source. For the oscillatory source they can extend both upstream and downstream, depending on the ratio of the source frequency to the buoyancy frequency, and are contained inside conical wavefronts, parts of which are caustics. For horizontal motion, moreover, the steady analysis (based on the monochromatic Greens function) reveals the presence of two insignificant contributions overlooked by the unsteady analysis (based on the impulsive Greens function), but which for an extended source may become of the same order as the main contribution. Among those is an upstream columnar disturbance associated with blocking.

Influence of the multipole order of the source on the decay of an inertial wave beam in a rotating fluid

A joint theoretical and experimental study is performed on the generation of internal gravity waves by an oscillating sphere, as a paradigm of the generation of internal tides by barotropic tidal flow over three-dimensional supercritical topography. The theory is linear and three-dimensional, applies both near and far from the sphere, and takes into account viscosity and the unsteadiness arising from the interference with transients generated at the start-up. The waves propagate in conical beams, evolving with distance from a bimodal to unimodal wave profile. In the near field, the profile is asymmetric with its major peak towards the axis of the cones. The experiments involve horizontal oscillations and develop a cross-correlation technique for the measurement of the deformation of fluorescent dye planes to sub-pixel accuracy. At an oscillation amplitude of one fifth of the radius of the sphere, the waves are linear and the agreement between experiment and theory is excellent. As the amplitude increases to half the radius, nonlinear effects cause the wave amplitude to saturate at a value that is 20% lower than its linear estimate. Application of the theory to the conversion rate of barotropic tidal energy into internal tides confirms the expected scaling for flat topography, and shows its transformation for hemispherical topography. In the ocean, viscous and unsteady effects have an essentially local role, in keeping the wave amplitude finite at the edges of the beams, and otherwise dissipate energy on such large distances that they hardly induce any decay.

Added mass effects on internal wave generation

Abstract At high Reynolds and Froude numbers, lee waves owing to the horizontal motion of a body in a stratified fluid are superseded by random waves generated by its wake. The origin of these waves lies in the buoyant collapse of the large-scale coherent structures of the wake, and can be modelled as a source moving at the velocity of the body and of strength oscillating at the frequency of vortex shedding. In the present paper two parallel studies of the associated wavefield are described. The first of these is theoretical and considers localized and extended models of the source, while the second is experimental and involves a vertically oscillating and horizontally translating sphere. Oscillation frequencies both smaller and larger than the Brunt-Vaisala frequency are considered, and reasonably good agreement between theory and experiment is obtained concerning, e.g. the shape of the surfaces of constant phase, the streamwise evolution of the wavelength, and the domain of existence of the waves. Calculations are then presented for a realistic turbulent wake, and comparison with available experimental results is performed.

Internal waves from oscillating objects

Two asymptotic analyses of the generation of lee waves by horizontal flow at velocity U of a stratified fluid of buoyancy frequency N past a sphere of radius a are presented, for either weak or strong stratification, corresponding to either large or small internal Froude number F = U /( Na ), respectively. For F ⋙1, the fluid separates into two regions radially: an inner region of scale a with three-dimensional irrotational flow unaffected by the stratification, and an outer region of scale U / N with small-amplitude lee waves generated by the O (1) vertical motion in the inner region. For F ⋘1, the fluid separates into five layers vertically: from the lower dividing streamsurface situated at a distance U / N above the bottom of the sphere to the upper dividing streamsurface situated at a distance U / N below the top, there is a middle layer with two-dimensional horizontal irrotational flow; from the upper dividing streamsurface to the top of the sphere, and from the lower dividing streamsurface to the bottom, there are top and bottom transition layers, respectively, with three-dimensional flow; above the top and below the bottom, there are upper and lower layers, respectively, with small-amplitude lee waves generated by the O ( F ) vertical motion in the transition layers. The waves are calculated where they have small amplitudes. The forcing is represented by a source of mass: for F ⋙1, the surface distribution of singularities equivalent to the sphere in three-dimensional irrotational flow; for F ⋘1, the horizontal distribution of singularities equivalent, in the upper (resp. lower) layer, to the flat cut-off obstacle made up of the top (resp. bottom) portion of the sphere protruding above (resp. below) the upper (resp. lower) dividing streamsurface. The analysis is validated by comparison of the theoretical wave drag with existing experimental determinations. For F ⋙1, the drag coefficient decreases as (ln F +7/4-γ)/(4 F 4 ), with γ the Euler constant; for F ⋘1, it increases as . The waves have the crescent shape of the three-dimensional lee waves from a dipole, modulated by interferences associated with the finite size of the forcing. For strong stratification, the hydrostatic approximation is seen to produce correct leading-order drag, but incorrect waves.

Wake of inertial waves of a horizontal cylinder in horizontal translation

An experimental study is presented on the spatial structure of the internal wave field emitted by a horizontally oscillating sphere in a uniformly stratified fluid. The limits of linear theory and the nonlinear features of the waves are considered as functions of oscillation amplitude. Fourier decomposition is applied to separate first harmonic waves at the fundamental frequency and higher harmonic waves at multiples of this frequency. For low oscillation amplitude, of 10 % of the sphere radius, only the first harmonic is significant and the agreement between linear theory and experiment is excellent. As the oscillation amplitude increases up to 30 % of the radius, the first harmonic becomes slightly smaller than its linear theoretical prediction and the second and third harmonics become detectable. Two distinct cases emerge depending on the ratio Ω between the oscillation frequency and the buoyancy frequency. When Ω > 0.5, the second harmonic is evanescent and localized near the sphere in the plane through its centre perpendicular to the direction of oscillation, while the third harmonic is negligible. When Ω < 0.5, the second harmonic is propagative and appears to have an amplitude that exceeds the amplitude of the first harmonic, while the third harmonic is evanescent and localized near the sphere on either side of the plane through its centre perpendicular to the direction of oscillation. Moreover, the propagative first and second harmonics have radically different horizontal radiation patterns and are of dipole and quadrupole types, respectively.