Amplitude Equations For Waves Research Articles

A multiple time scale approach for anisotropic inertial wave turbulence

Wave turbulence is the study of the long-time statistical behaviour of equations describing a set of weakly nonlinear interacting waves. Such a theory, which has a natural asymptotic closure, allows us to probe the nature of turbulence more deeply than the exact Kolmogorov laws by rigorously proving the direction of the cascade and the existence of an inertial range, predicting stationary spectra for conserved quantities, or evaluating the Kolmogorov constant. An emblematic example is given by fast rotating fluids for which a wave turbulence theory has been derived by Galtier (Phys. Rev. E, vol. 68, issue 1, 2003, p. 015301). This work involves non-trivial analytical developments for a problem that is anisotropic by nature. We propose here a new path for the derivation of the kinetic equation by using the anisotropy at the beginning of the analysis. We show that the helicity basis is not necessary to obtain the wave amplitude equation for the canonical variables that involve a combination of poloidal and toroidal fields. The multiple time scale method adapted to this anisotropic problem is then used to derive the kinetic equation that is the same as the original work when anisotropy is eventually taken into account. This result proves the commutativity between asymptotic closure and anisotropy. In addition, the multiple time scale method informs us that the kinetic equation can be derived without imposing restrictions on the probability distribution of the wave amplitude such as quasi-Gaussianity, or on the phase such as random phase approximation that naturally occurs dynamically.

Resonant and near-resonant internal wave triads for non-uniform stratifications. Part 2. Vertically bounded domain with mild-slope bathymetry

Weakly nonlinear internal wave–wave interaction is a key mechanism that cascades energy from large to small scales, leading to ocean turbulence and mixing. Oceans typically have a non-uniform density stratification profile; moreover, submarine topography leads to a spatially varying bathymetry ( $h$ ). Under these conditions and assuming mild-slope bathymetry, we employ multiple-scale analysis to derive the wave amplitude equations for weakly nonlinear wave–wave interactions. The waves are assumed to have a slowly (rapidly) varying amplitude (phase) in space and time. For uniform stratifications, the horizontal wavenumber ( $k$ ) condition for waves (1, 2, 3), given by ${k}_{(1,a)}+{k}_{(2,b)}\!+\!{k}_{(3,c)}\!=\!0$ , is unaffected as $h$ is varied, where $(a,b,c)$ denote the mode number. Moreover, the nonlinear coupling coefficients (NLC) are proportional to $1/h^2$ , implying that triadic waves grow faster while travelling up a seamount. For non-uniform stratifications, triads that do not satisfy the condition $a=b=c$ may not satisfy the horizontal wavenumber condition as $h$ is varied, and unlike uniform stratification, the NLC may not decrease (increase) monotonically with increasing (decreasing) $h$ . NLC, and hence wave growth rates for weakly nonlinear wave–wave interactions, can also vary rapidly with $h$ . The most unstable daughter wave combination of a triad with a mode-1 parent wave can also change for relatively small changes in $h$ . We also investigate higher-order self-interactions in the presence of a monochromatic, small-amplitude topography; here, the topography behaves as a zero-frequency wave. We derive the amplitude evolution equations and show that higher-order self-interactions might be a viable mechanism of energy cascade.

A vorticity wave packet breaking within a rapidly rotating vortex. Part II: wave/mean flow interaction

AbstractIn Part I, the author analyzed the asymmetric structure of a critical layer embedded in a linearly stable, columnar, and axisymmetric vortex on the ‐plane. This structure is the result of the long time asymptotic and slowly varying interaction between a vorticity wave packet and the vortex. He showed that the presence of a critical layer‐induced mean radial velocity led to the formation of a spiraling helical critical layer. Through matched asymptotic expansions, he found an analytical solution of the leading‐order motion equations inside the critical layer. In this paper, he derives the system of the coupled evolution equations of the wave amplitude and the low‐order CL‐induced mean flow on the critical radius in the quasi‐steady regime. The knowledge of the wave amplitude, the leading‐order mean axial and azimuthal velocity, and axial vorticity evolutions can be simply determined from three first‐order differential equations. The main outcome of the first‐order mean flow truncated system resolution is that the wave packet/vortex interaction leads to a fast vorticity wave breaking. The wave amplitude equation contains a coupling term, involving the vertical gradient of the mean axial vorticity located inside the separatrices, and accelerating the breaking. The leading‐order mean rotational and vertical momenta are conserved within the envelope as time proceeds. Numerical simulations nevertheless show that the wave packet and vortex kinetic energies slightly grow inside the envelope before the breaking onset in most of the cases, whereas the vortex was intensifying at the expense of the wave packet in the previous and unsteady interaction. The vertical wind shear strength has the highest effect on the wave/mean flow interaction. When the shear is moderate, it enhances intensification but when it is very large, it prohibits it in both the unsteady and slowly evolving stages. Including the second‐order mean flow in this system could, however, avoid the breaking and would permit the interaction to generate an asymptotic constant‐speed travelling coherent vortical structure.

Traveling fronts in dissipative granular chains and nonlinear lattices

We consider an infinite chain of particles with nonlinear elastic and dissipative nearest neighbors interactions. Assuming the existence of a traveling front between uniformly compressed (or stretched) states, we obtain jump conditions relating the wave speed and limiting particle velocities to the relative displacements at infinity. Using this result, we characterise compression fronts in chains of touching beads, for viscoelastic contact laws that include a nonlinear elastic force (generalised Hertz contact) and viscous dissipation. We compute compression fronts numerically for the generalised Kuwabara–Kono model in which the viscous contact force is proportional to the derivative of the elastic force, without precompression of the chain. Steady fronts are obtained both as the end result of the compression of one end of the chain and using a shooting method which provides numerically exact traveling waves. Depending on the magnitudes of contact damping and strain applied downstream, we obtain either overdamped (monotonic) or underdamped (oscillatory) compression fronts. To explain this transition and approximate the front profiles, we consider a continuum limit valid when the exponent of the contact nonlinearity is close to (and above) unity. Using multiscale expansions, we formally derive two different amplitude equations for long waves, a Burgers equation with logarithmic nonlinearity, and a logarithmic Korteweg–de Vries (KdV)–Burgers equation for small contact damping. Both models possess traveling front solutions that are in good agreement with the front profiles computed numerically in the granular chain. The analysis of the logarithmic KdV–Burgers equation allows one to approximate the critical damping corresponding to the transition from underdamped to overdamped fronts.

Planetary and Gravity Waves in a Polar Basin

AbstractThe eigenfrequencies of freely propagating barotropic, divergent, planetary waves and gravity waves in a spherical polar cap are presented using an approximation in which full spherical geometry is retained in the derivation of the wave amplitude equation. Subsequently, the colatitude angle in the coefficients of the wave amplitude equation is fixed, thereby allowing the eigenvalue problem to be solved using analytical methods. The planetary wave frequencies are compared with published results that adopt the polar-plane approximation to solve the equivalent free-wave problem. Low-order planetary wave frequencies calculated in this study agree well with the polar-plane approximation results. The sensitivity of the wave frequencies to the choice of the fixed colatitude in the coefficients of the wave amplitude equation is discussed.

Amplitude equations for breathing spiral waves in a forced reaction-diffusion system

Based on a multiple scale analysis of a forced reaction-diffusion system leading to amplitude equations, we explain the existence of spiral wave and its photo-induced spatiotemporal behavior in chlorine dioxide-iodine-malonic acid system. When the photo-illumination intensity is modulated, breathing of spiral is observed in which the period of breathing is identical to the period of forcing. We have also derived the condition for breakup and suppression of spiral wave by periodic illumination. The numerical simulations agree well with our analytical treatment.

Impulse Product Parameter in Landslide Generated Impulse Waves

Subaerial landslide generated impulse waves were investigated in a prismatic wave channel based on Froude similitude and granular slide material. The tests included the following seven governing parameters: still water depth, slide impact velocity, slide thickness, bulk slide volume, bulk slide density, slide impact angle, and grain diameter. All governing parameters, except for the grain diameter with a negligible effect, are included in the impulse product parameter P allowing for a simple application. Empirical equations based on 211 experiments for all relevant wave features in practice including the maximum wave height, the maximum wave amplitude with its location and period in the slide impact zone, and both the wave height and amplitude decay and the period increase in the wave propagation zone are a simple function of P. The presented equations were validated with 223 runs resulting in improved goodness of fit. The limitations of the herein derived empirical equations are also highlighted. The wave height and amplitude equations based on a wave channel (2D) agree well with the 1958 Lituya Bay case.

The excitation of spiral density waves through turbulent fluctuations in accretion discs - I. WKBJ theory

We study and elucidate the mechanism of spiral density wave excitation in a differentially rotating turbulent flow. We formulate a set of wave equations with sources that are only non-zero in the presence of turbulent fluctuations. We solve these in a shearing box domain using a WKBJ method. It is found that, for a particular azimuthal wave length, the wave excitation occurs through a sequence of regularly spaced swings during which the wave changes from leading to trailing form. This is a generic process that is expected to occur in shearing discs with turbulence. Trailing waves of equal amplitude propagating in opposite directions are produced, both of which produce an outward angular momentum flux that we give expressions for as functions of the disc parameters and azimuthal wave length. By solving the wave amplitude equations numerically we justify the WKBJ approach for a Keplerian rotation law for all parameter regimes of interest. In order to quantify the wave excitation completely the important wave source terms need to be specified. Assuming conditions of weak nonlinearity, these can be identified and are associated with a quantity related to the potential vorticity, being the only survivors in the linear regime. Under the additional assumption that the source has a flat power spectrum at long azimuthal wave lengths, the optimal azimuthal wave length produced is found to be determined solely by the WKBJ response and is estimated to be 2 pi H, with H being the nominal disc scale height.

Mode conversion in plasmas with two-dimensional inhomogeneities

Most of the mode conversion theories considered so far assume only a plane-layered medium, i.e. a medium where the parameters depend on one spatial coordinate. We generalize the mode-conversion method of Cairns and Lashmore-Davies to plasmas with two-dimensional inhomogeneities. In the method presented here, the frequencies ω 1 and ω 2 of the uncoupled modes belonging to two different dispersion equations are considered as functions of the space variable r and the wave vector k and are coupled together via a small quantity η. We calculate the energy transmission and conversion coefficients analytically by solving two coupled wave amplitude equations in the electron cyclotron range of frequencies. The results are applicable to electron Bernstein wave heating of plasmas with two-dimensional inhomogeneity, e.g. spherical tokamaks.

Beam Profile Analysis of Second Harmonic Generated by KH2PO4

Temperature dependence of second harmonic generation (SHG) was examined theoretically by solving the coupling problem composed of heat conduction equation and complex wave amplitude equations. The latter equations include effect of laser absorption in the crystal. KTiOPO4 (KTP) crystal was supposed as a nonlinear optical crystal, and its SHG characteristics were compared with KH2PO4 (KDP) crystal. Power density distribution in the crystal, time variation of the conversion efficiency and output-beam profile were analyzed quantitatively during repetitive irradiation of pulse laser in Gaussian mode. As a result, it was revealed that there are various mode in output-beam profile, such as ring mode, single mode with one or two rings in the surrounding, depending on incident power density or crystal length. These disorders of transverse mode are caused by the inverse conversion of SHG. It was also concluded that there is an optimum length of crystal to the irradiation conditions, and it can be estimated by the present study.

非線形光学結晶によるレーザ吸収の第二高調波発生効率への影響

Frequency-conversion techniques such as SHG, THG and FOHG with nonlinear optical crystals are indispensable for the applications of all solid-state lasers to precision microfabrication. It is well known that the conversion efficiency is very sensitive to the temperature rise of the crystal, which is induced by laser absorption. In this study, temperature dependence of SHG efficiency has been investigated theoretically, supposing Nd:YAG laser of uniform intensity and KDP (KH 2PO 4) crystal. The coupling problem composed of complex wave amplitude equations and one-dimensional heat conduction equation was analyzed. The complex wave amplitude equations were derived considering the absorption of laser in the crystal. The main results obtained are summarized as follows: (1) Decreasing of SHG efficiency due to laser absorption for the single pulse irradiation to a crystal of 10 mm long, is less than five percent. (2) When SHG efficiency is higher, temperature-rise of the crystal is smaller. Because absorption coefficient for the fundamental wave is larger than that of the second harmonic. (3) When the temperature of crystal rises due to repeated irradiation of pulse laser, SHG efficiency fluctuates and decreases gradually. Remarkable inverse-conversion of SHG appears during laser irradiation with high power density. (4) As either pulse width or irradiation time becomes longer, analytical SHG efficiency agrees with the approximated results obtained under the simple phase-mismatching conditions.

ガウシアンビーム照射による第二高調波発生効率の理論的解析

In recent years, the frequency conversion technique using nonlinear optical crystals are becoming more and more important for the applications of solid-state lasers to precision microfabrication. For the first step of the study, we have analyzed the temperature dependence of SHG conversion efficiency theoretically, supposing uniform beam. The complex wave amplitude equations were derived, considering the absorption of laser in the crystal. The results were presented in the previous paper. In this paper, the analysis was carried out two-dimensionally with an axisymmetrical model based on the previous one-dimensional model, applying the paraxial approximation. Two-dimensional temperature distribution induced by laser absorption and variation of the conversion efficiency of KDP crystal were analyzed quantitatively during repetition irradiation of pulse laser. The influences of pulse width and beam intensity distribution were also investigated Main results are summarized as follows: (1) Depending on SHG and its inverse conversion, temperature of the crystal with Gaussian beam irradiation fluctuates remarkably in the axial direction. (2) During Gaussian beam irradiation, heat conduction in the radial direction prevents temperature from rising in the central part of crystal. (3) The conversion efficiency for a Gaussian beam stays relatively high for a long period, compared with that of uniform beam.

Amplitude equations for non-linear Rayleigh waves

Abstract We investigate asymptotic equations describing small amplitude surface elastic waves in the half-plane (Rayleigh waves). For hyperelastic materials such model equations are Hamiltonian systems, and are seen to lead to the formation of singularities in the surface elastic displacement. At the time of singularity formation the Fourier spectra of the solutions exhibit power law decay, and the observed exponents suggest the existence of both differentiable and non-differentiable singular profiles.

Amplitude equation and pattern selection in Faraday waves.

A nonlinear theory of pattern selection in parametric surface waves (Faraday waves) is presented that is not restricted to small viscous dissipation. By using a multiple scale asymptotic expansion near threshold, a standing wave amplitude equation is derived from the governing equations. The amplitude equation is of gradient form, and the coefficients of the associated Lyapunov function are computed for regular patterns of various symmetries as a function of a viscous damping parameter gamma. For gamma approximately 1, the selected wave pattern comprises a single standing wave (stripe pattern). For gamma<<1, patterns of square symmetry are obtained in the capillary regime (large frequencies). At lower frequencies (the mixed gravity-capillary regime), a sequence of sixfold (hexagonal), eightfold, ...patterns are predicted. For even lower frequencies (gravity waves) a stripe pattern is again selected. Our predictions of the stability regions of the various patterns are in quantitative agreement with recent experiments conducted in large aspect ratio systems.

Amplitude equations for electrostatic waves: Multiple species

The amplitude equation for an unstable electrostatic wave is analyzed using an expansion in the mode amplitude $A(t)$. In the limit of weak instability, i.e. $\gamma\to 0^+$ where $\gamma$ is the linear growth rate, the nonlinear coefficients are singular and their singularities predict the dependence of $A(t)$ on $\gamma$. Generically the scaling $|A(t)|=\gamma^{5/2}r(\gamma t)$ as $\gamma\to 0^+$ is required to cancel the coefficient singularities to all orders. This result predicts the electric field scaling $|E_k|\sim\gamma^{5/2}$ will hold universally for these instabilities (including beam-plasma and two-stream configurations) throughout the dynamical evolution and in the time-asymptotic state. In exceptional cases, such as infinitely massive ions, the coefficients are less singular and the more familiar trapping scaling $|E_k|\sim\gamma^2$ is recovered.

Pattern Selection in Faraday Waves

We present a systematic nonlinear theory of pattern selection for parametric surface waves (Faraday waves), not restricted to fluids of low viscosity. A standing wave amplitude equation is derived from the Navier-Stokes equations that is of gradient form. The associated Lyapunov function is calculated for different regular patterns to determine the selected pattern near threshold. For fluids of large viscosity, the selected wave pattern consists of parallel stripes. At lower viscosity, patterns of square symmetry are obtained in the capillary regime (large frequencies). At lower frequencies (the mixed gravity-capillary regime), a sequence of six-fold (hexagonal), eight-fold, ... patterns are predicted. The regions of stability of the various patterns are in quantitative agreement with recent experiments conducted in large aspect ratio systems.

Radiatively Damped Density Waves in Optically Thick Protostellar Disks

The redistribution of angular momentum and mass in circumstellar disks by spiral density waves is governed by processes that dissipate the waves. Dissipation is usually attributed to unspecified nonlinear processes or an ad hoc viscosity. In this paper, we examine dissipation by radiative losses associated with the cyclic compression and expansion of disk material by the density waves. We consider linear, discrete modes in optically thick disks and use a simplified treatment of the radiative losses in order to assess the parameter dependences and importance of radiative damping. Wave action conservation principles are generalized to describe the effect of dissipation (by any means) and the consequent coupling of wave and disk energy and momenta. A relation is derived (in the WKBJ limit) between radial mass transport of disk material and the azimuthally averaged dissipation rate. A wave amplitude equation is similarly derived, which describes the radial evolution of wave angular momentum, and therefore the radial deposition of angular momentum in the disk. Associated with the equation is a characteristic damping length, proportional to the cooling time from the base state temperature times the radial wave speed, and a factor that depends weakly on optical depth. The damping length is used to define, in terms of the disk base state temperature and surface density, regimes of strong and weak radiative damping. Waves in warm disks may be damped close to the resonances at which they are excited, while waves in cold disks can propagate to distances at which other dissipative mechanisms may dominate. It is found that the boundary between these regimes cuts across values that are thought to have characterized the primitive solar nebula. Thus both strong and weak radiative damping may be encountered in circumstellar disks in general.

Short-scale convection and long-scale deformationally unstable Rossby wave in a rotating fluid layer heated from below

A rotating fluid layer, heated from below, with a deformable upper and nondeformable lower stress free surfaces is considered in the Boussinesq approximation. The system of the differential equations that governs the long-scale Rossby waves and short-scale convection is obtained in the rapid-rotation approximation. Long-scale flows are unstable due to heating and deformation of the upper surface. The neutral stability curves for Rossby waves and convection are obtained for linearized version of the equations. In a slightly supercritical regime the amplitude equations for convection and Rossby waves are derived by the use of the method of multiscale expansions. The properties of the amplitude equations are discussed. The existence of the two weakly supercritical stationary convection regimes is shown by numerical integration of the equations in the rapid-rotation approximation. In one of them, the amplitude of short-scale convection is modulated due to long-scale deformation of the upper surface associated with the excitation of the Rossby wave. In the other regime, the presence of deformation gives rise to alternating regions with and without convection.

A Three-Dimensional Nonhydrostatic Ray-Tracing Model for Gravity Waves: Formulation and Preliminary Results for the Middle Atmosphere

Abstract The WKB ray-tracing formalism is extended to accommodate internal gravity waves of all frequencies in a rotating, stratified, and compressible three-dimensional atmosphere. This includes the derivation of equations governing the dispersion and refraction of the ray paths, a realistic wave amplitude equation that takes into account both radiative and turbulent damping effects, and extensions of previous wave saturation schemes to accommodate dynamical and convective instabilities along generally slanted axes. These equations have been numerically coded into a global ray-tracing model that the authors have applied to the three-dimensional CIRA 1986 reference atmosphere model in a series of preliminary experiments to investigate the impact of the newly incorporated features on synthesized wave fields in the middle atmosphere. Three main points emerge from these experiments. First, there is a striking reduction in the high-frequency cutoff with decreasing horizontal wavenumber due to a more complete ...

An Eulerian Vlasov-Hilbert Code for the Numerical Simulation of the Interaction of High-Frequency Electromagnetic Waves with Plasma

In order to handle one-dimensional spatial problems in plasmas with very high frequency electromagnetic waves (ω ⪢ ωp), the 1 - 1/2D Eulerian Vlasov code has been modified to interface with the ponderomotive force associated with the complex envelopes of the high frequency electromagnetic fields rather than the fields themselves. The ponderomotive force for the Vlasov code has been computed from the complex amplitude equations and the real Vlasov density perturbations have been converted to complex amplitude form for the mode equations via the Hilbert transform. The resulting Vlasov-Hilbert code allows us to handle high-frequency problems with complex amplitude equations for high frequency electromagnetic waves with the Vlasov code only for the relatively low-frequency plasma wave. Computations are made with this model to situations typical of forward Raman scattering and beatwave; results are in accordance with other computational methods such as full electromagnetic Vlasov-Maxwell code or envelope equations whenever they can be used. This model provides a saving of order (ωpump/ωplasma)2 (i.e., time × space) in computer time, as compared with the direct full electromagnetic Vlasov-Maxwell method with the highest space-time resolution required.