Non-dispersive Case Research Articles

Lagrangian approach to nonlinear waves in non-dispersive and dispersive rotating shallow water magnetohydrodynamics

A Lagrangian approach to both hydrostatic non-dispersive in the short-wave range and non-hydrostatic dispersive rotating shallow-water magnetohydrodynamics is developed, and used to analyse weakly and fully nonlinear waves described by the model. Hyperbolic structure in the non-dispersive case is displayed and Riemann invariants are constructed. Characteristic equations are used to establish criteria of breaking and formation of shocks by magneto-gravity waves, and conditions of the appearance of contact discontinuities in Alfvén waves. As in the case of non-magnetic rotating shallow water, rotation cannot prevent breaking. The Lagrangian equations of the model are reduced to a single partial differential ‘master’ equation, which is used to analyse the propagation of weakly nonlinear waves of both families, with or without weak rotation, and with or without weak short-wave dispersion. Corresponding modulation equations are constructed and their main properties sketched. The same master equation is used to obtain fully nonlinear finite-amplitude wave solutions in particular cases of no short-wave dispersion or no rotation.

Dispersive problem and control sets of linear control systems on Lie groups

This paper is dedicated to the study of dispersiveness and controllability of linear control systems on Lie groups. Dispersiveness means absence of recursiveness, that is contrary to the existence of control set. A linear control system on a Lie group associates with a derivation operator. For a linear system with stable derivation, it is shown that the system is dispersive if and only if the trajectories through the neutral element have no limit at infinity. As a consequence, a nonempty limit set at the neutral element is a necessary condition for the linear system to admit a control set or to be controllable. In the nondispersive case, the control sets are described by the Lie subgroup of all recurrent points of the automorphism flow of the system. If the derivation operator is asymptotically stable, the central limit set at the neutral element is the unique control set of the system.

Superradiance in dispersive black hole analogues

Wave equations containing spatial derivatives which are higher than second order arise naturally in the context of condensed matter systems. The solutions of such equations contain more than two modes and consequently, the range of possible interactions between the different modes is significantly enhanced compared to the two mode case. We develop a framework for analysing the different mode interactions based on the classical turning points of the dispersion relation. We then apply this framework to the scattering of deep water gravity waves with a draining bathtub vortex, a system which constitutes the analogue of a rotating black hole in the non-dispersive limit. In particular, we show that the different scattering outcomes are controlled by the light-ring frequencies, a concept routinely applied in black hole physics, and two new frequencies which are related to the strength of dispersion. We find that the frequency range in which the reflected wave is superradiantly amplified appears as a simple modification to the non-dispersive case. However, the condition to observe this amplification is complicated by the fact that a superradiant mode can be reflected back into the system by scattering with one of the additional modes. We provide estimates for the reflection coefficients in the full dispersive regime.

Acoustic and shock wave dispersion in cylindrical ducts

An asymptotic approach is employed to investigate the effects of dispersion on acoustic disturbances propagating through cylindrical ducts, the walls of which may be impermeable or acoustically treated. The disturbances are assumed to be those generated by a rotor-locked pressure field. In the large azimuthal wavenumber (high-frequency) limit, m ≫ 1, linear potential perturbations are found to be weakly dispersive, with the leading order correction occurring at O(m−2/3). When a liner is present, its additional dispersive and dissipative effects appear at O(m−1). The effects of weak nonlinearity, in addition to those of dispersion are then studied. Specifically, on the assumption that the disturbance is comprised of a single radial mode, an evolution equation is derived for the pressure field. Numerical solutions of this equation are carried out for idealized perturbation patterns corresponding to tuned and detuned rotor blades, featuring respectively shocks of uniform and varying strength. In the tuned, untreated case, dispersive effects are found to reduce the dissipation across the shocks without substantially changing the character of the waveform in comparison with the non-dispersive case. The inclusion of the acoustic liner brings about more drastic changes: the periodic shock pattern is transformed into a series of humps with dispersive tails. With the detuned blades, it is found that the untreated duct scenario yields a waveform that differs significantly from the non-dispersive case. In particular, the shock merging that occurs in the latter instance does not transpire. Instead, the shocks exchange amplitudes even as they decay. The presence of the acoustic liner in the detuned case is similar to the tuned case, except with a richer spectrum being generated. Despite the relative simplicity of the models studied, these results exhibit many of the features of previous numerical simulations and point to the need for the inclusion of dispersive effects in designing liners for turbofan applications.

Metrics for Localized Beams and Pulses

Geometric and energy metrics are here proposed to gauge the confinement properties of localized electromagnetic beams (e.g., Bessel beams) and pulses (e.g., X-waves) in both the nondispersive and dispersive cases. As is well known, the frequency superposition of Bessel beams sharing the same axicon angle leads to the generation of tightly bounded wavepackets, namely X-waves, with remarkable spatio–temporal confinement properties. In the nondispersive case, the axicon angle does not change with frequency, and these features can be predicted by closed-form expressions. Remarkably, it is found that the volumetric confinement of such pulses can be optimized for a specific axicon angle. Since the geometric metrics do not account for the unavoidable presence of energy tails outside the main spot, energy metrics are here introduced and compared with the purely geometric definition. Finally, the dispersive case is investigated, accounting for the typical wavenumber dispersion of radial waveguides. As a result, the axicon angle changes with the frequency, and fundamental limits exist for the maximum theoretical fractional bandwidth. Interestingly, design criteria based on approximate analytical expressions are derived to optimize the confinement, even in the dispersive case.

Mass-polariton theory of light in dispersive media

We have recently shown that the electromagnetic pulse in a medium is made of mass-polariton (MP) quasiparticles, which are quantized coupled states of the field and an atomic mass density wave (MDW) [M. Partanen et al., Phys. Rev. A 95, 063850 (2017)]. In this work, we generalize the MP theory of light for dispersive media assuming that absorption and scattering losses are very small. Following our previous work, we present two different approaches to the coupled state of light: (1) the MP quasiparticle theory, which is derived by only using the fundamental conservation laws and the Lorentz transformation; (2) the classical optoelastic continuum dynamics (OCD), which is a generalization of the electrodynamics of continuous media to include the dynamics of the medium under the influence of optical forces. We show that the total momentum and the transferred mass of the light pulse can be determined in a straightforward way if we know the field energy of the pulse and the dispersion relation of the medium. In analogy to the nondispersive case, we also find unambiguous correspondence between the MP and OCD theories. For the coupled MP state of a single photon and the medium, we obtain the total MP momentum ${p}_{\mathrm{MP}}={n}_{\mathrm{p}}\ensuremath{\hbar}\ensuremath{\omega}/c$, where ${n}_{\mathrm{p}}$ is the phase refractive index. The field's share of the MP momentum is equal to ${p}_{\mathrm{field}}=\ensuremath{\hbar}\ensuremath{\omega}/({n}_{\mathrm{g}}c)$, where ${n}_{\mathrm{g}}$ is the group refractive index and the share of the MDW is equal to ${p}_{\mathrm{MDW}}={p}_{\mathrm{MP}}\ensuremath{-}{p}_{\mathrm{field}}$. Thus, as in a nondispersive medium, the total momentum of the MP is equal to the Minkowski momentum and the field's share of the momentum is equal to the Abraham momentum. We also show that the correspondence between the MP and OCD models and the conservation of momentum at interfaces gives an unambiguous formula for the optical force. The dynamics of the light pulse and the related MDW lead to nonequilibrium of the medium and to relaxation of the atomic density by sound waves in the same way as for nondispersive media. We also carry out simulations for optimal measurements of atomic displacements related to the MDW in silicon. In the simulations, we consider different waveguide cross sections and optical pulse widths and account for the breakdown threshold irradiance of materials. We also compare the MP theory to previous theories of the momentum of light in a dispersive medium. We show that our generalized MP theory resolves all the problems related to the Abraham-Minkowski dilemma in a dispersive medium.

Nonlinear wave dynamics in self-consistent water channels

We study long-wave dynamics in a self-consistent water channel of variable cross-section, taking into account the effects of weak nonlinearity and dispersion. The self-consistency of the water channel is considered within the linear shallow water theory, which implies that the channel depth and width are interrelated, so the wave propagates in such a channel without inner reflection from the bottom even if the water depth changes significantly. In the case of small-amplitude weakly dispersive waves, the reflection from the bottom is also small, which allows the use of a unidirectional approximation. A modified equation for Riemann waves is derived for the nondispersive case. The wave-breaking criterion (gradient catastrophe) for self-consistent channels is defined. If both weak nonlinearity and dispersion are accounted for, the variable-coefficient Korteweg–de Vries (KdV) equation for waves in self-consistent channels is derived. Note that this is the first time that a KdV equation has been derived for waves in strongly inhomogeneous media. Soliton transformation in a channel with an abrupt change in depth is also studied.

Extraordinary optical transmission induced by strong plasmon-phonon coupling: Shape resonance versus non-shape resonance

The study of the localized surface plasmons induced by a rectangular hole has been a basic building block for understanding the shape dependence of extraordinary optical transmission (EOT). A recent discovery has shown that the substrate significantly affects the aperture resonance position. However, the substrate effect on aperture resonance has been limited to the non-dispersive case, in which the refractive index of the substrate is assumed to be constant. Here, by employing strong plasmon-phonon interactions, we study the EOT of a rectangular hole with a dispersive medium. By changing the slot antenna length, various plasmon modes are generated, and they then interact with the phonon of the SiO2 film in the mid-infrared range. We report that the transmission resonance changes from shape resonance to non-shape resonance as the length of the slot increases. The origin of the shape resonance is the destructive interference of the evanescent modes; in contrast, the propagation modes and the material property of the substrate determine the non-shape resonance. Because of the effect of evanescent coupling, shape resonance is much more sensitive than non-shape resonance to the dielectric environment.

Dispersion analysis of Robert–Asselin type filters for linear non-dispersive and dispersive systems

The present paper deals with the dispersion relation preserving (DRP) analysis of the Robert–Asselin type filters coupled with leapfrog time integration method for time dependent non-dispersive and dispersive model systems. As, leapfrog time advancement method is widely used in the numerical modeling of atmosphere and ocean dynamics, despite the fact that in addition to physical mode it also admits a spurious mode. This is the major disadvantage of leapfrog time integration scheme in discrete computing. To suppress the spurious mode associated with leapfrog time integration method, Robert–Asselin type filters are extensively used in the literature. Here, the dispersion analysis is based on the spectral analysis by considering filtered/unfiltered leapfrog time integration method along with spatial discretization schemes. Spatial discretization is performed using second order centered difference and optimized compact schemes. Furthermore, to assess the efficacy of these methods one dimensional convection equation is used as a model equation for the non-dispersive case, while linearized rotating shallow water equations (LRSWE), as a model test problem for the dispersive case.

Resonant-state expansion of dispersive open optical systems: Creating gold from sand

A resonant-state expansion (RSE) for open optical systems with a general frequency dispersion of the relative permittivity, described by a finite number of simple poles, is presented. As in the non-dispersive case, the RSE of dispersive systems converts Maxwell's wave equation into a linear matrix eigenvalue problem in the basis of unperturbed resonant states, in this way numerically exactly determining all relevant eigenmodes of the optical system. This dispersive RSE is verified by application to the analytically solvable system of a sphere in vacuum, with a dispersion of the dielectric constant described by the Drude and Drude-Lorentz models. We calculate the change of the optical modes when converting the sphere material from gold to non-dispersive silica and back to gold, and evaluate the accuracy using the exact solutions.

Fully discrete stability and dispersion analysis for a linear dispersive internal wave model

In the context of modeling internal water waves, some strongly nonlinear reduced models exhibit a nonlocal term involving a Hilbert-type transform. These models accurately account for the physical dispersion regarding the Euler equations. To perform a careful stability analysis and detect whether numerical solutions are reliable or spurious, it is necessary to adapt the classical von Neumann analysis to account for nonlocal dispersive terms. We address this issue by a fully discrete analysis of the one-dimensional linear model, in the flat bottom case. We find a formula for the amplification factor that provides estimates concerned with numerical stability and dispersion (namely, phase errors). Subsequently, we contrast the numerical properties of the original dispersive problem with that of the underlying non-dispersive case, namely a linear hyperbolic system. The stability estimates corroborate the fact that physical dispersion provided by a nonlocal (singular integral) term allows for less restrictive stability conditions.

Linear dispersive effect on random lasing modes

A model, by combining Maxwell's equations with all-parameters of Sellmeier's fitting equations and four-level rate equations, is built to investigate linear dispersive effect on the property of random lasing modes. Computed results show that the first excited modes for both dispersive and non-dispersive scattering cases have almost the same resonant frequency but the spectral intensity for dispersive case is lower than that for non-dispersive case, and there exist more modes in the whole spectra for dispersive case. Further analysis demonstrates that threshold of random lasing in dispersive case is higher than that of the non-dispersive case.

Optical and magneto-optical properties of the monoclinic phase of the C70 crystal

Using the Hartree–Fock (HF) single-excitation configuration interaction (CI) model in conjunction with the local field method, the dielectric tensor, birefringence coefficient, circular dichroism, refractive index and effects of spatial dispersion on the dispersion relation for transverse normal polariton waves of the monoclinic phase of the C70 crystal are calculated. Our results indicate that the anisotropy of the C70 molecule remains in C70 solids. Also, our results on the effect of spatial dispersion, compared with the non-dispersive case, indicate that the coupling of the transverse electromagnetic field with the first and second excitonic states is very weak.

Electromagnetic modes in cavities made of negative-index metamaterials

We discuss electromagnetic modes in cavities formed by metamaterials with negative refraction and demonstrate that the straightforward approach to substitute negative values of the electric permittivity and the magnetic permeability leads to quasi-bound states with a negative quality factor. To ensure positive quality factors and a consistent physical interpretation of the quasi-bound states it is essential to include the frequency dispersion of the permittivity and the permeability, as required by positive field energy and causality. The basic mode equation and the boundary conditions including linear frequency dispersion are derived. As an example we consider a disk-like cavity with deformed cross sectional shape. The transition from the unphysical nondispersive case with negative quality factors to the dispersive case with positive quality factors is demonstrated numerically and in an analytical perturbative treatment.

Characteristics of magnetoacoustic sausage modes

Aims: We perform an advanced study of the fast magnetoacoustic sausage oscillations of coronal loops in the context of MHD coronal seismology to establish the dependence of the sausage mode period and cut-off wavenumber on the plasma-beta of the loop-filling plasma. A parametric study of the ratios for different harmonics of the mode is also carried out. Methods: Full magnetohydrodynamic numerical simulations were performed using Lare2d, simulating hot, dense loops in a magnetic slab environment. The symmetric Epstein profile and a simple step-function profile were both used to model the density structure of the simulated loops. Analytical expressions for the cut-off wavenumber and the harmonic ratio between the second longitudinal harmonic and the fundamental were also examined. Results: It was established that the period of the global sausage mode is only very weakly dependent on the value of the plasma-beta inside a coronal loop, which justifies the application of this model to hot flaring loops. The cut-off wavenumber kc for the global mode was found to be dependent on both internal and external values of the plasma-beta, again only weakly. By far the most important factor in this case was the value of the density contrast ratio between the loop and the surroundings. Finally, the deviation of the harmonic ratio P1/2P2 from the ideal non-dispersive case was shown to be considerable at low k, again strongly dependent on plasma density. Quantifying the behaviour of the cut-off wavenumber and the harmonic ratio has significant applications to the field of coronal seismology.

Natural modes and resonances in a dispersive stratified N-layer medium

The properties of the natural modes in a dispersive stratified N-layer medium are investigated. The focus is on the (over)completeness properties of these modes. Also the distribution of the natural frequencies is considered. Both the degree of (over)completeness and the natural frequency distribution turn out to be totally different from what is known for the non-dispersive case.

Turbulent diffusion in rapidly rotating flows with and without stable stratification

Three different approaches are used for evaluating some Lagrangian properties of homogeneous turbulence containing anisotropy due to the application of a stable stratification and a solid-body rotation. The two external frequencies are the magnitude of the system vorticity 2Q, chosen vertical here, and the Brunt-Vaisala frequency N, which gives the strength of the vertical stratification. Analytical results are derived using linear theory for the Eulerian velocity correlations (single-point, two-time) in the vertical and the horizontal directions, and Lagrangian ones are assumed to be equivalent, in agreement with an additional Corrsin assumption used by Kaneda. They are compared with results from the kinematic simulation model (KS) by Nicolleau & Vassilicos, which also incorporates the wave-vortex dynamics inherited from linear theory, and directly yields Lagrangian correlations as well as Eulerian ones. Finally, results from direct numerical simulations (DNS) are obtained and compared for the rotation-dominant case B = 2Ω/N = 10, the stratification-dominant case B = 1/10, the non-dispersive case B = 1, and pure stratification B = 0 and pure rotation N = 0

Integrated Analytic Radionuclide Transport Model for a Spent Nuclear Fuel Repository in Saturated Fractured Rock

Simple analytic expressions are presented for radionuclide transport from a KBS 3-type repository, where spent nuclear fuel is placed in copper canisters surrounded by bentonite clay and deposited at a depth of 500 m in fractured granitic rock.Dissolution of readily accessible and fuel matrix embedded nuclides, chain decay, and nuclide precipitation is treated within the canister. Transport in the canister void and buffer is modeled with a dual stirred tank analogy, where transport resistances represent an assumed small initial damage in the canister and transport features of the buffer-geosphere interface. Initial, transient diffusion in the buffer is treated with a simple correction term. Chain decay is not included in the buffer.Geosphere transport expressions handle advection, longitudinal dispersion, matrix diffusion, sorption, and radioactive decay, but not chain decay. The treatment is based on earlier results for an instantaneous inlet and for a constant inlet to the geosphere in the nondispersive case. A correction is added so that longitudinal dispersion is taken approximately into account. The correction utilizes analytical expressions for the temporal moments of the geosphere release curve in the dispersive case.The near-field/geosphere integration is treated in a simplified manner avoiding numerical convolutions. The instantaneous inlet expression for the geosphere release is used when the near-field release decreases rapidly in comparison to a typical response time in the geosphere; the constant inlet expression is used in the opposite case.Twenty-seven calculation cases from a safety assessment of a KBS 3 repository using borehole data from three different field investigation sites were repeated with the analytic expressions. The agreement in both near-field and geosphere releases is in general well within an order of magnitude for the variety of long- and short-lived, sorbing, nonsorbing, solubility limited, immediately accessible, and fuel matrix embedded single and ingrowing species dominating the releases and doses in the safety assessment calculations. Also, probabilistic dose calculation results are in good agreement, making the analytic model a versatile complement to the various tools used in long-term safety evaluations of a KBS 3 type of repository in saturated fractured rock.

Optical and magneto-opticalproperties of the simple cubic phase of the C60 crystal

Using the Hartree-Fock and single configuration interactionmodel in conjunction with the local field method, the dielectrictensor, refractive index, circular dichroism, birefringencecoefficient and the effects of spatial dispersion on thespectrum of the transverse normal polariton waves of the simplecubic phase of the C60 crystal are calculated. Our resultson magneto-optical circular dichroism in the energy range3-4 eV almost reproduces the reported experimental results.Also, our results on the effect of spatial dispersion, comparedwith the non-dispersive case, indicate that the first branch ofthe transverse normal polariton in the C60 crystal is dueto the coupling of the second excitonic state of the C60molecule with the electromagnetic field and the first excitonicstate has very weak coupling.

The Bremmer series for a multi-dimensional acoustic scattering problem

The Bremmer series is used to reduce a complex scattering problem to a sequence of simpler single-scattering problems. In the Bremmer series, the wave equation is first decomposed into a coupled system of one-way wave equations. The system is then decoupled into a sequence of one-way wave equations with a fixed-point iteration. In this paper, a left-symbol representation of the decomposition operator and the vertical-propagation operator are used. Time-domain convergence of the Bremmer series is shown for a set of dispersive medium models. The non-dispersive case is treated with an approximation procedure.