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Register a new userComplete Hamiltonian formalism for inertial waves in rotating fluids
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Complete Hamiltonian formalism is suggested for inertial waves in rotating incompressible fluid. Resonance three-wave interaction processes -- decay instability and confluence of two waves -- are shown to play a key role in the weakly nonlinear dynamics and statistics of inertial waves in the rapid rotation case. Future applications of the Hamiltonian approach in inertial wave theory are investigated and discussed.
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We quantify the strength of the waves and their impact on the energy cascade in rotating turbulence by studying the wave number and frequency energy spectrum, and the time correlation functions of individual Fourier modes in numerical simulations in three dimensions in periodic boxes. From the spectrum, we find that a significant fraction of the energy is concentrated in modes with wave frequency $omega approx 0$, even when the external forcing injects no energy directly into these modes. However, for modes for which the period of the inertial waves $tau_omega$ is faster than the turnover time $tau_textrm{NL}$, a significant fraction of the remaining energy is concentrated in the modes that satisfy the dispersion relation of the waves. No evidence of accumulation of energy in the modes with $tau_omega = tau_textrm{NL}$ is observed, unlike what critical balance arguments predict. From the time correlation functions, we find that for modes with $tau_omega < tau_textrm{sw}$ (with $tau_textrm{sw}$ the sweeping time) the dominant decorrelation time is the wave period, and that these modes also show a slower modulation on the timescale $tau_textrm{NL}$ as assumed in wave turbulence theories. The rest of the modes are decorrelated with the sweeping time, including the very energetic modes modes with $omega approx 0$.
(abbreviated) In this paper we develop a consistent WKBJ formalism, together with a formal first order perturbation theory for calculating the properties of the inertial modes of a uniformly rotating coreless body (modelled as a polytrope and referred hereafter to as a planet) under the assumption of a spherically symmetric structure. The eigenfrequencies, spatial form of the associated eigenfunctions and other properties we obtained analytically using the WKBJ eigenfunctions are in good agreement with corresponding results obtained by numerical means for a variety of planet models even for global modes with a large scale distribution of perturbed quantities. This indicates that even though they are embedded in a dense spectrum, such modes can be identified and followed as model parameters changed and that first order perturbation theory can be applied. This is used to estimate corrections to the eigenfrequencies as a consequence of the anelastic approximation, which we argue here to be small when the rotation frequency is small. These are compared with simulation results in an accompanying paper with a good agreement between theoretical and numerical results. The results reported here may provide a basis of theoretical investigations of inertial waves in many astrophysical and other applications, where a rotating body can be modelled as a uniformly rotating barotropic object, for which the density has, close to its surface, an approximately power law dependence on distance from the surface.
The so-called Landau-Levich-Deryaguin problem treats the coating flow dynamics of a thin viscous liquid film entrained by a moving solid surface. In this context, we use a simple experimental set-up consisting of a partially-immersed rotating disc in a liquid tank to study the role of inertia, and also curvature, on liquid entrainment. Using water and UCON$^{mbox{{TM}}}$ mixtures, we point out a rich phenomenology in the presence of strong inertia : ejection of multiple liquid sheets on the emerging side of the disc, sheet fragmentation, ligament formation and atomization of the liquid flux entrained over the discs rim. We focus our study on a single liquid sheet and the related average liquid flow rate entrained over a thin disc for various depth-to-radius ratio $h/R < 1$. We show that the liquid sheet is created via a ballistic mechanism as liquid is lifted out of the pool by the rotating disc. We then show that the flow rate in the entrained liquid film is controlled by both viscous and surface tension forces as in the classical Landau-Levich-Deryaguin problem despite the three dimensional, non-uniform and unsteady nature of the flow, and also despite the large values of the film thickness based flow Reynolds number. When the characteristic Froude and Weber numbers become significant, strong inertial effects influence the entrained liquid flux over the disc at large radius-to-immersion-depth ratio, namely via entrainment by the discs lateral walls and via a contribution to the flow rate extracted from the 3D liquid sheet itself, respectively.
We examine a two dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface. The two media are separated by a free common interface. The gravity driven surface and internal water waves (at the common interface between the media) in the presence of a depth-dependent current are studied under certain physical assumptions. Both media are considered incompressible and with prescribed vorticities. Using the Hamiltonian approach the Hamiltonian of the system is constructed in terms of wave variables and the equations of motion are calculated. The resultant equations of motion are then analysed to show that wave-current interaction is influenced only by the current profile in the strips adjacent to the surface and the interface. Small amplitude and long-wave approximations are also presented.
Laboratory experimental results are presented for nonlinear Internal Solitary Waves (ISW) propagation in deep water configuration with miscible fluids. The results are validated against direct numerical simulations and traveling wave exact solutions where the effect of the diffused interface is taken into account. The waves are generated by means of a dam break and their evolution is recorded with Laser Induced Fluorescence (LIF) and Particle Image Velocimetry (PIV). In particular, data collected in a frame moving with the waves are presented here for the first time. Our results are representative of geophysical applications in the deep ocean where weakly nonlinear theories fail to capture the characteristics of large amplitude ISWs from field observations.
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