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Register a new userAxisymmetric internal wave transmission and resonance in non-linear stratifications
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To date, the influence of non-linear stratifications and two layer stratifications on internal wave propagation has been studied for two-dimensional wave fields in a cartesian geometry. Here, we use a novel wave generator configuration to investigate transmission in non-linear stratifications of axisymmetric internal wave. Two configurations are studied, both theoretically and experimentally. In the case of a free incident wave, a transmission maximum is found in the vicinity of evanescent frequencies. In the case of a confined incident wave, resonant effects lead to enhanced transmission rates from an upper layer to layer below. We consider the oceanographic relevance of these results by applying them to an example oceanic stratification, finding that there can be real-world implications.
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To date, axisymmetric internal wave fields, which have relevance to atmospheric internal wave fields generated by storm cells and oceanic near-inertial wave fields generated by surface storms, have been experimentally realized using an oscillating sphere or torus as the source. Here, we use a novel wave generator configuration capable of exciting axisymmetric internal wave fields of arbitrary radial form to generate axisymmetric internal wave modes. After establishing the theoretical background for axisymmetric mode propagation, taking into account lateral and vertical confinement, and also accounting for the effects of weak viscosity, we experimentally generate and study modes of different order. We characterize the efficiency of the wave generator through careful measurement of the wave amplitude based upon group velocity arguments. This established, we investigate the ability of vertical confinement to induce resonance, identifying a series of experimental resonant peaks that agree well with theoretical predictions. In the vicinity of resonance, the wave fields undergo a transition to non-linear behaviour that is initiated on the central axis of the domain and proceeds to erode the wave field throughout the domain.
In this paper, we present an experimental study of weakly non-linear interaction of axisymmetric internal gravity waves in a resonant cavity, supported by theoretical considerations. Contrary to plane waves in Cartesian coordinates, for which self-interacting terms are null in a linear stratifiation, the non-linear self-interaction of an internal wave mode in axisymmetric geometry is found to be efficient at producing super-harmonics, i.e. waves whose frequencies are integer multiples of the excitation frequency. Due to the range of frequencies tested in our experiments, the first harmonic frequency is below the cut-off imposed by the stratification so the lowest harmonic created can always propagate. The study shows that the super-harmonic wave field is a sum of standing waves satisfying both the dispersion relation for internal waves and the boundary conditions imposed by the cavity walls, while conserving the axisymmetry.
In the paper taking the assumption of the slowness of the change of the parameters of the vertically stratified medium in the horizontal direction and in time, the evolution of the non-harmonic wave packages of the internal gravity waves has been analyzed. The concrete form of the wave packages can be expressed through some model functions and is defined by the local behavior of the dispersive curves of the separate modes near to the corresponding special points. The solution of this problem is possible with the help of the modified variant of the special-time ray method offered by the authors (the method of geometrical optics), the basic difference of which consists that the asymptotic representation of the solution may be found in the form the series of the non-integer degrees of some small parameter. At that the exponent depends on the concrete form of representation of this package. The obvious kind of the representation is determined from the principle of the localness and the asymptotic behavior of the solution in the stationary and the horizontally-homogeneous case. The phases of the wave packages are determined from the corresponding equations of the eikonal, which can be solved numerically on the characteristics (rays). Amplitudes of the wave packages are determined from the laws of conservation of the some invariants along the characteristics (rays).
We study the flow forced by precession in rigid non-axisymmetric ellipsoidal containers. To do so, we revisit the inviscid and viscous analytical models that have been previously developed for the spheroidal geometry by, respectively, Poincare (Bull. Astronomique, vol. XXVIII, 1910, pp. 1-36) and Busse (J. Fluid Mech., vol. 33, 1968, pp. 739-751), and we report the first numerical simulations of flows in such a geometry. In strong contrast with axisymmetric spheroids, where the forced flow is systematically stationary in the precessing frame, we show that the forced flow is unsteady and periodic. Comparisons of the numerical simulations with the proposed theoretical model show excellent agreement for both axisymmetric and non-axisymmetric containers. Finally, since the studied configuration corresponds to a tidally locked celestial body such as the Earths Moon, we use our model to investigate the challenging but planetary-relevant limit of very small Ekman numbers and the particular case of our Moon.
We report evaluations of a resonant kinetic equation that suggest the slow time evolution of the Garrett and Munk spectrum is {em not}, in fact, slow. Instead nonlinear transfers lead to evolution time scales that are smaller than one wave period at high vertical wavenumber. Such values of the transfer rates are inconsistent with conventional wisdom that regards the Garrett and Munk spectrum as an approximate stationary state and puts the self-consistency of a resonant kinetic equation at a serious risk. We explore possible reasons for and resolutions of this paradox. Inclusion of near-resonant interactions decreases the rate at which the spectrum evolves. This leads to improved self-consistency of the kinetic equation.
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