Eigenfunctions Of Modes Research Articles

Sound focusing in shallow water using a vertical phased array

Vertical phased array is an efficient and complex equipment in underwater acoustics. Single-mode excitation and time-reversal sound focusing are two typical techniques based on vertical phased array, but both of them have some shortcomings. In order to realize sound focusing at variable distance and depth using vertical phased array, two techniques are proposed in this paper. One of them is the Green's function construction of target sound field based on normal mode eigen-function and attenuation coefficient extraction, and the other is controllable sound focusing based on optimal target sound field determination. Simulation and experiment results are presented. This technique indicates a method for objective depth detecting in shallow water.

A modified WKB formulation for linear eigenmodes of a collisionless self-gravitating disc in the epicyclic approximation

The short--wave asymptotics (WKB) of spiral density waves in self-gravitating stellar discs is well suited for the study of the dynamics of tightly--wound wavepackets. But the textbook WKB theory is not well adapted to the study of the linear eigenmodes in a collisionless self-gravitating disc because of the transcendental nature of the dispersion relation. We present a modified WKB of spiral density waves, for collisionless discs in the epicyclic limit, in which the perturbed gravitational potential is related to the perturbed surface density by the Poisson integral in Kalnaj's logarithmic spiral form. An integral equation is obtained for the surface density perturbation, which is seen to also reduce to the standard WKB dispersion relation. We specialize to a low mass (or Keplerian) self-gravitating disc around a massive black hole, and derive an integral equation governing the eigenspectra and eigenfunctions of slow precessional modes. For a prograde disc, the integral kernel turns out be real and symmetric, implying that all slow modes are stable. We apply the slow mode integral equation to two unperturbed disc profiles, the Jalali--Tremaine annular discs, and the Kuzmin disc. We determine eigenvalues and eigenfunctions for both $m = 1$ and $m = 2$ slow modes for these profiles and discuss their properties. Our results compare well with those of Jalali--Tremaine.

Diverse wave-particle interactions for energetic ions that traverse Alfvén eigenmodes on their first full orbit

Neutral-beam ions that are deflected onto loss orbits by Alfvén eigenmodes (AE) on their first bounce orbit and are detected by a fast-ion loss detector (FILD) satisfy the “local resonance” condition proposed by Zhang et al. [Nucl. Fusion 55, 22002 (2015)]. This theory qualitatively explains FILD observations for a wide variety of AE-particle interactions. When coherent losses are measured for multiple AE, oscillations at the sum and difference frequencies of the independent modes are often observed in the loss signal. The amplitudes of the sum and difference peaks correlate weakly with the amplitudes of the fundamental loss-signal amplitudes but do not correlate with the measured mode amplitudes. In contrast to a simple uniform-plasma theory of the interaction [Chen et al., Nucl. Fusion 54, 083005 (2014)], the loss-signal amplitude at the sum frequency is often larger than the loss-signal amplitude at the difference frequency, indicating a more detailed computation of the orbital trajectories through the mode eigenfunctions is needed.

Simulating acoustic waves in spotted stars

Acoustic modes of oscillation are affected by stellar activity, however it is unclear how starspots contribute to these changes. Here we investigate the nonmagnetic effects of starspots on global modes with angular degree $\ell \leq 2$ in highly active stars, and characterize the spot seismic signature on synthetic light curves. We perform 3D time-domain simulations of linear acoustic waves to study their interaction with a model starspot. We model the spot as a 3D change in the sound speed stratification with respect to a convectively stable stellar background, built from solar Model S. We perform a parametric study by considering different depths and perturbation amplitudes. Exact numerical simulations allow the investigation of the wavefield-spot interaction beyond first order perturbation theory. The interaction of the axisymmetric modes with the starspot is strongly nonlinear. As mode frequency increases, the frequency shifts for radial modes exceed the value predicted by linear theory, while the shifts for the $\ell=2, m=0$ modes are smaller than predicted by linear theory, with avoided-crossing-like patterns forming between the $m=0$ and $m=1$ mode frequencies. The nonlinear behavior increases with increasing spot amplitude and/or decreasing depth. Linear theory still reproduces the correct shifts for nonaxisymmetric modes. In the nonlinear regime the mode eigenfunctions are not pure spherical harmonics, but rather a mixture of different spherical harmonics. This mode mixing, together with the frequency changes, may lead to misidentification of the modes in the observed acoustic power spectra.

A study of the turbulence within a spiralling vortex filament using proper orthogonal decomposition

The stability and turbulence characteristics of a vortex filament emanating from a single-bladed rotor in hover are investigated using proper orthogonal decomposition (POD). The rotor is operated at a tip chord Reynolds number and tip Mach number of 218 000 and 0.23, respectively, and with a blade loading of $C_{T}/{\it\sigma}=0.066$. In-plane components of the velocity field (normal to the axis of the vortex filament) are captured by way of two-dimensional particle image velocimetry with corrections for vortex wander being performed using the ${\it\Gamma}_{1}$ method. The first POD mode alone is found to encompass nearly 75 % of the energy for all vortex ages studied and is determined using a grid of sufficient resolution to avoid numerical integration errors in the decomposition. The findings reveal an equal balance between the axisymmetric and helical modes during vortex roll-up, which immediately transitions to helical mode dominance at all other vortex ages. This helical mode is one of the modes of the elliptic instability. The spatial eigenfunctions of the first few Fourier-azimuthal modes associated with the most energetic POD mode is shown to be sensitive to the choice of the wander correction technique used. Higher Fourier-azimuthal modes are observed in the outer portions of the vortex and appeared not to be affected by the choice of the wander correction technique used.

Localization of a sound source in oceanic waveguides

A method is developed for passive acoustic diagnostics of a mobile sound source in an oceanic waveguide, based on information on the frequency shifts of the maxima of the wave field and mode eigen-functions. An algorithm is formulated for solving the inverse problem. A numerical experiment demonstrates the possibilities of this method for reconstructing the trajectory, velocity, and depth of a source.

Entrainment of Short-wavelength Free-stream Vortical Disturbances into Boundary Layers

This paper is concerned with how free-stream vortical disturbances (FSVD) enter the boundary layer, a process which we refer to as entrainment. A self-consistent description of this physical process is presented, which shows that the non-parallel-flow effect plays a leading-order role even for disturbances with short wavelength comparable with the boundary-layer thickness or shorter. The entrained disturbances are thus drastically different from continuous modes of Orr-Sommerfeld/Squire equations, which neglect non-parallelism at outset. Indeed neglecting non-parallelism is found to cause several nonphysical features of continuous modes, including ‘Fourier entanglement’, abnormal anisotropy of FSVD, and for compressible boundary layers, entanglement of vortical and entropy modes. Eigenfunctions of continuous modes and their superpositions have been used to specify the inlet conditions in direct numerical simulations of bypass transition and boundary-layer receptivity. The findings of present study indicate that this popular practice is inappropriate, and a remedy is suggested.

Helioseismic measurements in the solar envelope using group velocities of surface waves

At intermediate and high degree $l$, solar p- and f modes can be considered as surface waves. Using variational principle, we derive an integral expression for the group velocities of the surface waves in terms of adiabatic eigenfunctions of normal modes, and address the benefits of using group-velocity measurements as a supplementary diagnostic tool in solar seismology. The principal advantage of using group velocities, when compared with direct analysis of the oscillation frequencies, comes from their smaller sensitivity to the uncertainties in the near-photospheric layers. We address some numerical examples where group velocities are used to reveal inconsistencies between the solar models and the seismic data. Further, we implement the group-velocity measurements to the calibration of the specific entropy, helium abundance Y and heavy-element abundance Z in the adiabatically-stratified part of the solar convective envelope, using different recent versions of the equation of state. The results are in close agreement with our earlier measurements based on more sophisticated analysis of the solar oscillation frequencies (Vorontsov et al. 2013, MNRAS 430, 1636). These results bring further support to the downward revision of the solar heavy-element abundances in recent spectroscopic measurements.

Interface depth used in a two-layer model of nonlinear internal waves

A two-layer model includes three parameters: interface depth h 1, upper layer density $$\rho_{1}$$ , and lower layer density $$\rho_{2}$$ . Many theoretical and laboratorial studies of internal waves, as well as most numerical models, are based on the two-layer assumption. However, these three parameters cannot be directly measured because a pycnocline in the real ocean has finite thickness, and the densities in both the mixed layer and the deep ocean are not constant. In the present study, seven different methods are used to determine the interface depth of the two-layer model and compared with the depth of maximum vertical displacement: the depth of maximum buoyancy frequency (Ν max), the depth where the first mode eigenfunction has its maximum (Φmax), the depth where the lowest mode temperature empirical orthogonal function has its maximum, the depth where either the two-layer Korteweg–de Vries (KdV) or Benjamin–Ono equation has closest coefficients with their continuously stratified counterparts, and the same KdV approach with stratification replaced by two idealized distributions. The multi-ship measurement conducted near the Luzon Strait is used for deep ocean comparison, and two measurements conducted in the east of Dongsha Atoll are used for shallow water comparison. The results show that in the deep ocean, the KdV approach with idealized type I stratification gives the interface closest to the depth of maximum vertical displacement. In shallow waters, the KdV approach agrees with the measurement best.

Pulsations of rapidly rotating stars with compositional discontinuities

AbstractRecent observations of rapidly rotating stars have revealed the presence of regular patterns in their pulsation spectra. This has raised the question as to their physical origin, and, in particular, whether they can be explained by an asymptotic frequency formula for low-degree acoustic modes, as recently discovered through numerical calculations and theoretical considerations. In this context, a key question is whether compositional/density gradients can adversely affect such patterns to the point of hindering their identification. To answer this question, we calculate frequency spectra using two-dimensional ESTER stellar models. These models use a multi-domain spectral approach, allowing us to easily insert a compositional discontinuity while retaining a high numerical accuracy. We analyse the effects of such discontinuities on both the frequencies and eigenfunctions of pulsation modes in the asymptotic regime. We find that although there is more scatter around the asymptotic frequency formula, the semi-large frequency separation can still be clearly identified in a spectrum of low-degree acoustic modes.

Destabilization of low-n peeling modes by trapped energetic particles

The kinetic effect of trapped energetic particles (EPs), arising from perpendicular neutral beam injection, on the stable low-n peeling modes in tokamak plasmas is investigated, through numerical solution of the mode's dispersion relation derived from an energy principle. A resistive-wall peeling mode with m/n=6/1, with m and n being the poloidal and toroidal mode numbers, respectively, is destabilized by trapped EPs as the EPs' pressure exceeds a critical value βc*, which is sensitive to the pitch angle of trapped EPs. The dependence of βc* on the particle pitch angle is eventually determined by the bounce average of the mode eigenfunction. Peeling modes with higher m and n numbers can also be destabilized by trapped EPs. Depending on the wall distance, either a resistive-wall peeling mode or an ideal-kink peeling mode can be destabilized by EPs.

Optical solitons in the parity-time-symmetric Bessel complex potential

Optical solitons in the parity-time (PT)-symmetric Bessel complex potential are studied, including the linear case, and self-focusing and self-defocusing nonlinear cases. For the linear case, the PT-symmetric breaking points, eigenvalues and the eigenfunction for different modulated depths of the PT-symmetric Bessel complex potential are obtained numerically. The PT-symmetric breaking points increase linearly with increasing the real part of the modulated depths of the PT potential. Below the PT-symmetric breaking points, the eigenfunctions of linear modes are symmetrical; however, the symmetries of the eigenfunction break above the PT-symmetric breaking points. For nonlinear cases, the existence and stability of fundamental and multipole solitons are studied in self-focusing and self-defocusing media. The eigenvalue for the linear case is equal to the critical propagation constant bc of the existing soliton. Fundamental solitons are stable in the whole region and multipole solitons are stable with the propagation constants being close to bc both for self-focusing and self-defocusing nonlinearities. The range of solitons’ stability decreases with an increase of the number of the intensity peaks of the solitons.

Geologically constrained morphological variability and boundary effects on embayed beaches

Headlands, rock outcrops and engineering structures impact beach and nearshore dynamics of coastal embayments, inducing boundary effects that constrain the lateral and vertical beach variability. This study analyses morphological change in six embayed beaches with diverse levels of exposure to wave action and various degrees of geological control in the mesotidal coast of southwestern Portugal. The aim is to identify whether geological boundaries constrain the morphological behaviour of embayed beaches and assess whether their effects can be decoupled from datasets of morphological change. Topographic data, obtained over a two-year period on each of the six embayments, were analysed using empirical orthogonal functions (EOF) to decompose temporal and spatial variability in the datasets. First and second mode eigenfunctions were explored using time-variable linear correlation analysis with several nearshore parameters that include hydrodynamic variables, sedimentary and geometric characteristics of each embayment in order to derive forcing–response relationships.Our results demonstrate that natural geological boundaries constrain the morphological behaviour of embayed beaches, producing conspicuous alongshore variability in all embayments. Localised responses induced by lateral and vertical boundary interference with beach and nearshore dynamics include beach rotation, topographically-controlled rip circulation and restrained profile fluctuation. Spatial decoupling in cross- and longshore responses is accompanied by a temporal decoupling in response times, both of which are slower in more constrained embayments (from 1day in exposed embayments to 1week or more in the most sheltered ones). Normalised wave power was correlated at the 99% confidence level with the primary mode of morphological variability at most embayments, which represent 67% to 94% of the variance in the datasets. This correlation stresses the importance of combined parameterization of wave and tide forcing in process–response relations between hydrodynamics and morphological change for mesotidal coastal environments. Lateral and vertical geological boundaries exert their effects fundamentally by restraining longshore sediment transport, inducing cellular surf zone circulation and by impacting cross-shore sediment transport. It is postulated that decreasing sediment abundance and substrate depth intensify vertical boundary effects, while higher indentation and wave obliquity enhance the effects of lateral boundaries.

Supersonic regime of the Hall-magnetohydrodynamics resistive tearing instability

An earlier analysis of the Hall-magnetohydrodynamics (MHD) tearing instability [E. Ahedo and J. J. Ramos, Plasma Phys. Controlled Fusion 51, 055018 (2009)] is extended to cover the regime where the growth rate becomes comparable or exceeds the sound frequency. Like in the previous subsonic work, a resistive, two-fluid Hall-MHD model with massless electrons and zero-Larmor-radius ions is adopted and a linear stability analysis about a force-free equilibrium in slab geometry is carried out. A salient feature of this supersonic regime is that the mode eigenfunctions become intrinsically complex, but the growth rate remains purely real. Even more interestingly, the dispersion relation remains of the same form as in the subsonic regime for any value of the instability Mach number, provided only that the ion skin depth is sufficiently small for the mode ion inertial layer width to be smaller than the macroscopic lengths, a generous bound that scales like a positive power of the Lundquist number.

Non-linear viscous damping and gravitational wave detectability of the f-mode instability in neutron stars

We study the damping of the gravitational radiation-driven f-mode instability in rotating neutron stars by nonlinear bulk viscosity in the so-called supra-thermal regime. In this regime the dissipative action of bulk viscosity is known to be enhanced as a result of nonlinear contributions with respect to the oscillation amplitude. Our analysis of the f-mode instability is based on a time-domain code that evolves linear perturbations of rapidly rotating polytropic neutron star models. The extracted mode frequency and eigenfunctions are subsequently used in standard energy integrals for the gravitational wave growth and viscous damping. We find that nonlinear bulk viscosity has a moderate impact on the size of the f-mode instability window, becoming an important factor and saturating the mode's growth at a relatively large oscillation amplitude. We show similarly that nonlinear bulk viscosity leads to a rather high saturation amplitude even for the r-mode instability. In addition, we show that the action of bulk viscosity can be significantly mitigated by the presence of superfluidity in neutron star matter. Apart from revising the f-mode's instability window we provide results on the mode's gravitational wave detectability. Considering an f-mode-unstable neutron star located in the Virgo cluster and assuming a mode amplitude at the level allowed by bulk viscosity, we find that the emitted gravitational wave signal could be detectable by advanced ground-based detectors such as Advanced LIGO/Virgo and the Einstein Telescope.

Three-dimensional hydrodynamic instabilities in stellar core collapses

A spherically symmetric hydrodynamic stellar core collapse under gravity is time-dependent and may become unstable once disturbed. Specifically for a homologously collapse of stellar core characterized by a polytropic exponent \Gamma=4/3, we examine oscillations and/or instabilities of three dimensional (3D) general polytropic perturbations. For compressible 3D perturbations, the polytropic index \gamma of perturbations can differ from \Gamma=4/3 of the general polytropic hydrodynamic background flow. Our model formulation here is more general and allows the existence of internal gravity g(-)-modes and/or g(+)-modes. Eigenvalues and eigen-functions of various oscillatory and unstable perturbation modes are computed. As studied in several specialized cases of Goldreich & Weber and of Cao & Lou, we further confirm that acoustic p-modes and surface f-modes remain stable in the current more general situations. In comparison, g(-)-modes and sufficiently high radial order g(+)-modes are unstable, leading to inevitable convective motions within the collapsing stellar interior. Unstable growths of 3D g-mode disturbances are governed dominantly by the angular momentum conservation and modified by the gas pressure restoring force. We note in particular that unstable temporal growths of 3D vortical perturbations exist even when the specific entropy distribution becomes uniform and \gamma=\Gamma=4/3. Conceptually, unstable g-modes might bear conceivable physical consequences on supernova explosions, the initial kicks of nascent proto-neutron stars (PNSs) and break-ups of the collapsing core, while unstable growths of vortical perturbations can lead to fast spins of compact objects and other relevant results.

Nonlinear decay ofrmodes in rapidly rotating neutron stars

We investigate the dynamics of $r$ modes at amplitudes in the nonlinear regime for rapidly but uniformly rotating neutron stars with a polytropic equation of state. For this, we perform three-dimensional relativistic hydrodynamical simulations, making the simplifying assumption of a fixed spacetime. To excite $r$ modes, we linearly scale exact eigenfunctions to large amplitudes. We find that for initial dimensionless amplitudes around three, $r$ modes decay on time scales around ten oscillation periods, while at amplitudes of order unity, they are stable over the evolution time scale. Together with the decay, a strong differential rotation develops, conserving the total angular momentum, with angular velocities in the range $0.5\dots{}1.2$ of the initial one. We evolved two models with different rotation rates and found slower decay for the more rapidly rotating one. We present $r$-mode eigenfunctions and frequencies, and compare them to known analytic results for slowly rotating Newtonian stars. As a diagnostic tool, we discuss conserved energy and angular momentum for the case of a fixed axisymmetric background metric and introduce a measure for the energy of nonaxisymmetric fluid oscillation modes.

Robust computation of acoustic normal modes in attenuating ocean waveguides

The normal mode representation [Jensen etal., Computational Ocean Acoustics, AIP Press, 1997] provides for the mathematical analysis and numerical computation of acoustic fields in a range-independent ocean waveguide. Existing algorithms [e.g., M. B. Porter and E. L. Reiss, J. Acoust. Soc. Am. 77, 1760] provide for efficient computation of the mode eigenvalues and eigenfunctions for narrow-band acoustic fields. However, for extensive computations involving a large set of environmental parameter values or acoustic frequencies, reliability issues, such as eigenvalues omitted in the calculation, have been noted in the case of attenuating media. In this work, a computational method is presented that computes the normal modes by approximate solution of the mode depth-dependence equation on a discrete computational grid, using a selected discrete basis. Empirical evidence of the robustness of the method is provided by comparisons with established numerical benchmarks and by examining the acoustic parameter dependence of the mode spectra over thousands of parameter values. Theoretical support for the reliability of the computation is then discussed. [Work supported by the Office of Naval Research.]

A geometric framework for black hole perturbations

Black hole perturbation theory is typically studied on time surfaces that extend between the bifurcation sphere and spatial infinity. From a physical point of view, however, it may be favorable to employ time surfaces that extend between the future event horizon and future null infinity. This framework resolves problems regarding the representation of quasinormal mode eigenfunctions and the construction of short-ranged potentials for the perturbation equations in frequency domain.

Higher order mode analyses in Feynman- α method

Using a generalized formula for the space and energy dependent Feynman- α method, which was originally derived by Endo et al. and Muñoz-Cobo et al., the effect of higher order modes of the α-mode eigenvalue problem on the Feynman Y function has been investigated. To deal with a large number of higher order modes, the diffusion approximation is adopted instead of the transport theory for a one-dimensional homogeneous infinite slab. By making a transport correction to low order mode eigenvalues and eigenfunctions, the formula can accurately reproduce the Monte Carlo simulation results of the Feynman- α method. By virtue of these efforts, an accurate numerical application of the generalized formula, which has not been performed due to the difficulty in solving the higher order α-mode eigenvalue problem, has been made possible. Sample numerical examples for a near-critical system and a deeply-subcritical system quantitatively demonstrate how the Feynman Y functions are decomposed into the higher order mode components. While the higher order mode components in the Feynman Y function can be negligible in a near-critical system, the Feynman Y function in a deeply-subcritical system is found to be severely contaminated by the higher order modes.