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Register a new userWeak and strong wave turbulence spectra for elastic thin plate
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Variety of statistically steady energy spectra in elastic wave turbulence have been reported in numerical simulations, experiments, and theoretical studies. Focusing on the energy levels of the system, we have performed direct numerical simulations according to the F{o}ppl--von K{a}rm{a}n equation, and successfully reproduced the variability of the energy spectra by changing the magnitude of external force systematically. When the total energies in wave fields are small, the energy spectra are close to a statistically steady solution of the kinetic equation in the weak turbulence theory. On the other hand, in large-energy wave fields, another self-similar spectrum is found. Coexistence of the weakly nonlinear spectrum in large wavenumbers and the strongly nonlinear spectrum in small wavenumbers are also found in moderate energy wave fields.
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A weakly nonlinear spectrum and a strongly nonlinear spectrum coexist in a statistically steady state of elastic wave turbulence. The analytical representation of the nonlinear frequency is obtained by evaluating the extended self-nonlinear interactions. The {em critical/} wavenumbers at which the nonlinear frequencies are comparable with the linear frequencies agree with the {em separation/} wavenumbers between the weak and strong turbulence spectra. We also confirm the validity of our analytical representation of the separation wavenumbers through comparison with the results of direct numerical simulations by changing the material parameters of a vibrating plate.
In wave turbulence, it has been believed that statistical properties are well described by the weak turbulence theory, in which nonlinear interactions among wavenumbers are assumed to be small. In the weak turbulence theory, separation of linear and nonlinear time scales derived from the weak nonlinearity is also assumed. However, the separation of the time scales is often violated even in weak turbulent systems where the nonlinear interactions are actually weak. To get rid of this inconsistency, closed equations are derived without assuming the separation of the time scales in accordance with Direct-Interaction Approximation (DIA), which has been successfully applied to Navier--Stokes turbulence. The kinetic equation of the weak turbulence theory is recovered from the DIA equations if the weak nonlinearity is assumed as an additional assumption. It suggests that the DIA equations is a natural extension of the conventional kinetic equation to not-necessarily-weak wave turbulence.
We consider a Hamiltonian description of the vibrations of a clamped, elastic circular plate. The Hamiltonian of this system features a potential energy with two distinct contributions: one that depends on the local mean curvature of the plate, and a second that depends on its Gaussian curvature. We quantize this model using a complete, orthonormal set of eigenfunctions for the clamped, vibrating plate. The resulting quanta are the flexural phonons of the thin circular plate. As an application, we use this quantized description to calculate the fluctuations in displacement of the plate for arbitrary temperature. We compare the fluctuation profile with that from an elastic membrane under tension. At low temperature, we find that while both profiles have a circular ring of local maxima, the ring in the membrane profile is much more pronounced and sharper. We also note that with increasing temperature the plate profile develops two additional rings of extrema.
By performing two parallel numerical experiments -- solving the dynamical Hamiltonian equations and solving the Hasselmann kinetic equation -- we examined the applicability of the theory of weak turbulence to the description of the time evolution of an ensemble of free surface waves (a swell) on deep water. We observed qualitative coincidence of the results. To achieve quantitative coincidence, we augmented the kinetic equation by an empirical dissipation term modelling the strongly nonlinear process of white-capping. Fitting the two experiments, we determined the dissipation function due to wave breaking and found that it depends very sharply on the parameter of nonlinearity (the surface steepness). The onset of white-capping can be compared to a second-order phase transition. This result corroborates with experimental observations by Banner, Babanin, Young.
A two-field Hamiltonian gyrofluid model for kinetic Alfven waves retaining ion finite Larmor radius corrections, parallel magnetic field fluctuations and electron inertia, is used to study turbulent cascades from the MHD to the sub-ion scales. Special attention is paid to the case of imbalance between waves propagating along or opposite to the ambient magnetic field. For weak turbulence in the absence of electron inertia, kinetic equations for the spectral density of the conserved quantities (total energy and generalized cross-helicity) are obtained. They provide a global description, matching between the regimes of reduced MHD at large scales and electron reduced MHD at small scales, previously considered in the literature. In the limit of ultra-local interactions, Leith-type nonlinear diffusion equations in the Fourier space are derived and heuristically extended to the strong turbulence regime by modifying the transfer time appropriately. Relations with existing phenomenological models for imbalanced MHD and balanced sub-ion turbulence are discussed. It turns out that in the presence of dispersive effects, the dynamics is sensitive on the way turbulence is maintained in a steady state. Furthermore, the total energy spectrum at sub-ion scales becomes steeper as the generalized cross-helicity flux is increased.
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