Mode Approximation Research Articles

Thin layer axion dynamo

We study interacting classical magnetic and pseudoscalar fields in frames of the axion electrodynamics. A large scale pseudoscalar field can be the coherent superposition of axions or axion like particles. We consider the evolution of these fields inside a spherical clump. Decomposing the magnetic field into the poloidal and toroidal components, we take into account their symmetry properties. Within a spherical clump, we use a thin layer approximation in the induction and Klein–Gordon equations, where the dependence of the fields on the latitude is accounted for. Then, we derive the dynamo equations in the low mode approximation. The nonlinear evolution equations for the harmonics of the magnetic and pseudoscalar fields are solved numerically. As an application, we consider a dense axion star embedded in solar plasma. The behavior of the harmonics and their typical oscillations frequencies are obtained. We suggest that such small size axionic objects, containing oscillating magnetic fields, can cause electromagnetic flashes, recently observed in the solar corona, contributing to the corona heating.

Spurious Rayleigh-wave apparent anisotropy near major structural boundaries: a numerical and theoretical investigation

SUMMARY The recent developments in array-based surface-wave tomography have made it possible to directly measure apparent phase velocities through wave front tracking. While directionally dependent measurements have been used to infer intrinsic $2\psi $ azimuthal anisotropy (with a 180° periodicity), a few studies have also demonstrated strong but spurious $1\psi $ azimuthal anisotropy (360° periodicity) near major structure boundaries particularly for long period surface waves. In such observations, Rayleigh waves propagating in the direction perpendicular to the boundary from the slow to the fast side persistently show a higher apparent velocity compared to waves propagating in the opposite direction. In this study, we conduct numerical and theoretical investigations to explore the effect of scattering on the apparent Rayleigh-wave phase velocity measurement. Using 2-D spectral-element numerical wavefield simulations, we first reproduce the observation that waves propagating in opposite directions show different apparent phase velocities when passing through a major velocity contrast. Based on mode coupling theory and the locked mode approximation, we then investigate the effect of the scattered fundamental-mode Rayleigh wave and body waves interfering with the incident Rayleigh wave separately. We show that scattered fundamental-mode Rayleigh waves, while dominating the scattered wavefield, mostly cause short wavelength apparent phase velocity variations that could only be studied if the station spacing is less than about one tenth of the surface wave wavelength. Scattered body waves, on the other hand, cause longer wavelength velocity variations that correspond to the existing real data observations. Because of the sensitivity of the $1\psi $ apparent anisotropy to velocity contrasts, incorporating such measurements in surface wave tomography could improve the resolution and sharpen the structural boundaries of the inverted model.

Late time decay of scalar and Dirac fields around an asymptotically de Sitter black hole in the Euler–Heisenberg electrodynamics

We compute the quasinormal modes of massive scalar and Dirac fields within the framework of asymptotically de Sitter black holes in Euler–Heisenberg non-linear electrodynamics. We pay particular attention to the regime μM/mP2≫1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu M/m_{P}^2 \\gg 1,$$\\end{document} where μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} and M denote the masses of the field and the black hole, respectively, and mP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_{P}$$\\end{document} represents the Planck mass, covering a range from primordial to large astrophysical black holes. Through time-domain integration, we demonstrate that, contrary to the asymptotically flat case, the quasinormal modes also dictate the asymptotic decay of fields. Employing the 6th order WKB formula, we derive a precise analytic approximation for quasinormal modes in the regime μM/mP2≫1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu M/m_{P}^2 \\gg 1$$\\end{document} without resorting to expansion in terms of the inverse multipole number. This analytic expression takes on a concise form in the limit of linear electrodynamics, represented by the Reissner–Nordström black holes. Our numerical analysis indicates the stability of the fields under consideration against linear perturbations.

Gauge symmetry of excited states in projected entangled-pair state simulations

While gauge symmetry is a well-established requirement for representing topological orders in projected entangled-pair state (PEPS), its impact on the properties of low-lying excited states remains relatively unexplored. Here we perform PEPS simulations of low-energy dynamics in the Kitaev honeycomb model, which supports fractionalized gauge flux (vison) excitations. We identify gauge symmetry emerging upon optimizing an unbiased PEPS ground state. Using the PEPS adapted local mode approximation, we further classify the low-lying excited states by discerning different vison sectors. Our simulations of spin and spin-dimer dynamical correlations establish close connections with experimental observations. Notably, the selection rule imposed by the locally conserved visons results in nearly flat dispersions in momentum space for excited states belonging to the 2-vison or 4-vison sectors. Published by the American Physical Society 2024

Nonlinear ray tracing in focused fields, part 2. Tracing the flux: tutorial

In this three-part paper series, we develop a method to trace the lines of flux through a three-dimensional wavefield by following a direction that is governed by the derivative of the phase at each point, a process that is best described as flux tracing but which we interchangeably name “nonlinear ray tracing.” In the first part we focused on the high-speed calculation of focused three-dimensional complex wavefields in the paraxial approximation for TEM00 and TEM01 laser modes. The algorithms developed in the first paper are first used to generate the three-dimensional grid of samples of the complex wavefield in the focal region. In this second part, we focus on tracing a flux through this three-dimensional point cloud. For a given “ray” at an arbitrary position in the 3D volume, interpolation of the three-dimensional samples is applied to determine the derivative of the phase (normal to the direction of propagation) at the ray position, which is then used to direct the ray as it “propagates” forward in a straight line over a short distance to a subsequent plane; the process is repeated between consecutive planes. The initial origin of the ray can be chosen arbitrarily at any point, and the ray can then be traced through the volume with appropriate interpolation. Results are demonstrated for focused wavefields in the absence of aberrations, corresponding to the cases highlighted in the first paper. Some of the most interesting results relate to focused Laguerre-Gaussian beams, for which the rays are found to spiral at different rates of curvature. In the third paper we extend the application of this algorithm to the investigation of lens aberration.

An entropy conserving/stable discontinuous Galerkin solver in entropy variables based on the direct enforcement of entropy balance

Numerical schemes that guarantee the preservation of flow properties as prescribed by physical laws, ensure high-fidelity results and improved computational robustness. This work presents a highly accurate entropy conserving/stable solver for the solution of the compressible Euler equations. The method uses a modal Discontinuous Galerkin approximation and takes advantage of both entropy variables and the Direct Enforcement of Entropy Balance (DEEB), proposed by Abgrall [1] to fulfill the second law of thermodynamics at the discrete level. Thanks to the use of an orthonormal modal basis, the cost of adding the DEEB correction to the spatially discretized equations is negligible. Entropy variables are used directly, as unknowns, or to assemble the discrete operators while solving for the conservative variables. In the latter case, the conservative variables are “converted” into the entropy ones through an L2 “entropy projection”. If this strategy is coupled with DEEB, a very efficient alternative to the direct use of the entropy variables is obtained, with a dramatic improvement in robustness over the direct use of the conservative variables. For the time integration both an explicit Strong Stability Preserving Runge-Kutta scheme and the implicit Generalized Crank-Nicolson method are considered. The latter allows for the construction of a discrete entropy conservative/stable scheme both in space and time to be used as a reference. Convergence and conservation properties are demonstrated together with robustness and computational efficiency on a suite of two-dimensional test cases, showing that by coupling DEEB with entropy projection the solver robustness is improved while not affecting the stability limit.

Resonant Modal Approximation of Time-Domain Elastic Scattering from Nano-Bubbles in Elastic Materials

Resonant Modal Approximation of Time-Domain Elastic Scattering from Nano-Bubbles in Elastic Materials

Combined vibration analysis of a top tensioned riser with the geometrical nonlinearity

A coupling model representing the interactions between the riser and vortices is proposed, of which the time-varying top tension and the internal nonlinear axial force due to the bending of the riser are considered. In addition, the van der Pol oscillator is introduced to simulate the time-varying characteristic of the vortices. The motion equations are derived and the first-order mode approximation is obtained with the Galerkin approach. The multiple scale method is applied to study the steady-state solutions of the system. Effects of system parameters (the structural damping, nonlinearity, the amplitude ratio of the varying-tension, the shedding frequency) on the responses are investigated in detail. The synchronous and non-synchronous motions between the structure and wake oscillator are studied. The bifurcation diagrams for the responses in terms of the amplitude ratio and the nonlinear parameter are obtained in the large parameter ranges. The results show that responses with different topological properties including quasi-periodic, periodic and chaotic solutions can occur under different parameter conditions. A pair of chaotic attractors can be found for the large parameter range of the amplitude ratio. The joint effects of the amplitude ratio and the nonlinear parameter on the responses in the large parameter range are studied as well, which are further verified by the phase projections and Poincaré sections. The results show that the topological properties of responses can be controlled by the amplitude ratio. These results can be helpful to the dynamic design of the riser in practice.

Projection-based reduced order modeling and data-driven artificial viscosity closures for incompressible fluid flows

Projection-based reduced order models rely on offline–online model decomposition, where the data-based energetic spatial basis is used in the expensive offline stage to obtain equations of reduced states that evolve in time during the inexpensive online stage. The online stage requires a solution method for the dynamic evolution of the coupled system of pressure and velocity states for incompressible fluid flows. The first contribution of this article is to demonstrate the applicability of the incremental pressure correction scheme for the dynamic evolution of pressure and velocity states. The evolution of a large number of these reduced states in the online stage can be expensive. In contrast, the accuracy significantly decreases if only a few reduced states are considered while not accounting for the interactions between unresolved and resolved states. The second contribution of this article is to compare three closure model forms based on global, modal and tensor artificial viscosity approximation to account for these interactions. The unknown model parameters are determined using two calibration techniques: least squares minimization of error in energy approximation and closure term approximation. This article demonstrates that an appropriate selection of solution methods and data-driven artificial viscosity closure models is essential for consistently accurate dynamics forecasting of incompressible fluid flows.

Automatic detection and 2D localization using a network of unsynchronized passive acoustic sensors in a dispersive waveguide

The time-frequency positions of modal dispersion curves in shallow-water low-frequency impulsive signals are strongly dependent on source-receiver range, making them suitable for range-based localization. Here, we apply a temporal convolutional network (TCN) to spectrograms estimated from individual sensors in an array of unsynchronized hydrophones to simultaneously detect dispersive signals and produce source-range estimates. The TCN is trained on simulated signals generated over a spatial grid and various environmental parameters using the adiabatic approximation for normal modes. Assuming that the number of unique sources is unknown, range measurements from the same source across different sensors are simultaneously associated and used in localization. To accomplish this automatically, the proposed method considers all unique combinations of range measurements from every collection of k sensors. For every range measurement combination, if the location estimate generated using each subcombination of k-1 measurements is within a certain threshold of the remaining measurement, the whole collection is labeled as group-k consistent. All such groups of measurements are represented as neighboring nodes in a graph, and strongly connected components are used to calculate the final source location estimates. The whole detection/localization method is validated using both simulated and experimental marine data. [Work supported by NDSEG and ONR].

Singlet-triplets and excited triplets in trapped medium-body systems with particle number 3 ≤ N ≤ 13 and with spin-4 cold atoms

The spin-textures of trapped medium-body cold systems of spin-4 atoms with the interaction dominated by the attractive λ = 4 channel (the two interacting spins are coupled to 4) have been studied via an exact diagonalization of the spin-dependent Hamiltonian under the single spatial mode approximation. Particle number N (3 ≤ N ≤ 13) is assumed. The eigenstates have been analyzed to find out the details of spin-structures and the inherent excitation modes, and the relation between all of them. The emphasis is to find out how the spectra depend on N. We found that the singlet-triplets (a 3-body subsystem where the three spins are coupled to zero) and various excited triplets are basic constituents and all the eigenstates are composed of them. For low-lying states, only a few particles are involved in the excitation modes, while the other particles would 3-by-3 keep themselves in singlet-triplets.

Scalar QNM spectra of Kerr and Reissner-Nordström revealed by eigenvalue repulsions in Kerr-Newman

Recent studies of the gravito-electromagnetic frequency spectra of Kerr-Newman (KN) black holes have revealed two families of quasinormal modes (QNMs), namely photon sphere modes and near-horizon modes. However, they can only be unambiguously distinguished in the Reissner-Nordström (RN) limit, due to a phenomenon called eigenvalue repulsion (also known as level repulsion, avoided crossing or the Wigner-Teller effect), whereby the two families can interact strongly near extremality. We find that these features are also present in the QNM spectra of a scalar field in KN, where the perturbation modes are described by ODEs and thus easier to explore. Starting from the RN limit, we study how the scalar QNM spectra of KN dramatically changes as we vary the ratio of charge to angular momentum, all the way until the Kerr limit, while staying at a fixed distance from extremality. This scalar field case clarifies the (so far puzzling) relationship between the QNM spectra of RN and Kerr black holes and the nature of the eigenvalue repulsions in KN, that ultimately settle the fate of the QNM spectra in Kerr. We study not just the slowest-decaying QNMs (both for ℓ = m = 0 and ℓ = m = 2), but several sub-dominant overtones as well, as these turn out to play a crucial role understanding the KN QNM spectra. We also give a new high-order WKB expansion of KN QNMs that typically describes the photon sphere modes beyond the eikonal limit, and use a matched asymptotic expansion to get a very good approximation of the near-horizon modes near extremality.

Stability of Kuramoto networks subject to large and small fluctuations from heterogeneous and spatially correlated noise.

Oscillatory networks subjected to noise are broadly used to model physical and technological systems. Due to their nonlinear coupling, such networks typically have multiple stable and unstable states that a network might visit due to noise. In this article, we focus on the assessment of fluctuations resulting from heterogeneous and spatially correlated noise inputs on Kuramoto model networks. We evaluate the typical, small fluctuations near synchronized states and connect the network variance to the overlap between stable modes of synchronization and the input noise covariance. Going beyond small to large fluctuations, we introduce the indicator mode approximation that projects the dynamics onto a single amplitude dimension. Such an approximation allows for estimating rates of fluctuations to saddle instabilities, resulting in phase slips between connected oscillators. Statistics for both regimes are quantified in terms of effective noise amplitudes that are compared and contrasted for several noise models. Bridging the gap between small and large fluctuations, we show that a larger network variance does not necessarily lead to higher rates of large fluctuations.

Optimization and Dispersion Tailoring of Chalcogenide M-Type Fibers Using a Modified Genetic Algorithm

M-type optical fibers in which a core is surrounded by a thin ring layer with a higher refractive index have attracted increasing attention in recent years. One of their advantageous features is the ability to operate a non-fundamental LP02 mode possessing unusual dispersion properties, namely, a zero-dispersion wavelength (ZDW) shifted to the short wavelength region relative to the material ZDW. The LP02 mode can be selectively excited since it is predominantly localized near the core, while the fundamental LP01 and other higher modes are localized near the ring (for proper fiber parameters). In this paper, we present a comprehensive theoretical analysis of effective dispersion tailoring for the HE12 mode of highly nonlinear chalcogenide glass fibers (for which the LP mode approximation fails due to large refractive index contrasts). We demonstrate fiber designs for which ZDWs can be shifted to the spectral region < 2 μm, which is of great interest for the development of mid-IR supercontinuum sources and frequency-tunable pulse sources with standard near-IR pumping. We obtained the characteristic equation and solved it numerically to find mode fields and dispersion characteristics. We show the possibility of achieving dispersion characteristics of the HE12 mode with one, two, three, and four ZDWs in the wavelength range of 1.5–5.5 μm. We used a modified genetic algorithm (MGA) to design fibers with desired dispersion parameters. In particular, by applying an MGA, we optimized four fiber parameters and constructed a fiber for which HE12 mode dispersion is anomalous in the 1.735–5.155 μm range.

Cavity-modified Fermi's golden rule rate constants: Beyond the single mode approximation.

We extend our recently proposed theoretical framework for estimating cavity-modified equilibrium Fermi's golden rule (FGR) rate constants beyond the single cavity mode case to cases where the molecular system is coupled to multiple cavity modes. We show that the cumulative effect of simultaneous coupling to multiple modes can enhance FGR rate constants by orders of magnitude relative to the single mode case. We also present an analysis of the conditions necessary for maximizing this effect in the Marcus limit of FGR-based rate theory.

Numerical Approximation of PGD Modes to Parameterized Elliptic Problems

Numerical Approximation of PGD Modes to Parameterized Elliptic Problems

Linear and Nonlinear Modes and Data Signatures in Dynamic Systems Biology Models

The particulars of stimulus–response experiments performed on dynamic biosystems clearly limit what one can learn and validate about their structural interconnectivity (topology), even when collected kinetic output data are perfect (noise-free). As always, available access ports and other data limitations rule. For linear systems, exponential modes, visible and hidden, play an important role in understanding data limitations, embodied in what we call dynamical signatures in the data. We show here how to circumscribe and analyze modal response data in compartmentalizing model structures—so that modal analysis can be used constructively in systems biology mechanistic model building—for some nonlinear (NL) as well as linear biosystems. We do this by developing and exploiting the modal basis for dynamical signatures in hypothetical (perfect) input–output (I-O) data associated with a (mechanistic) structural model—one that includes inputs and outputs explicitly. The methodology establishes model dimensionality (size and complexity) from particular I-O datasets; helps select among multiple candidate models (model distinguishability); helps in designing new I-O experiments to extract “hidden” structure; and helps to simplify (reduce) models to their essentials. These modal analysis tools are introduced to NL enzyme-regulated and protein–protein interaction biosystems via nonlinear normal mode (NNM) and quasi-steady state approximation (QSSA) analyses and unified with linear models on invariant 2-dimensional manifolds in phase space, with properties similarly informative about their dominant dynamical properties. Some automation of these highly technical aspects of biomodeling is also introduced.

A Simple, Correct Pedagogical Presentation of Airplane Longitudinal Dynamics

Various shortcomings and inaccuracies in the existing approach and presentation of airplane flight dynamics in pedagogy have been identified. An improved pedagogical presentation is offered which is clean - uses the two timescales in longitudinal dynamics to derive short period and phugoid mode parameters, simple - starts with the longitudinal dynamics equations rather than the complete six degree of freedom formulation, and correct - uses a physically correct model of the dynamic (rate) derivatives and employs the concept of a static residual in deriving the slower (phugoid) mode approximations. The redundant and confusing concepts of dimensional derivatives and static stability are junked. A companion paper presents a similar approach to the lateral-directional dynamics.

Modal analysis of concrete filled steel tubular structure under lateral impact loading

In this paper, deformation mode and the large deflection of the concrete filled steel tubular (CFST) structural member with fixed ends under lateral impact loading was studied by rigid-plastic modal approximation method and finite element analysis by ABAQUS. Using the yield criterion and beam-on-foundation model, the modal analytical solution for plastic behavior of CFST member under lateral impact is obtained considering the interaction of bending and axial stretching. The rigid-plastic modal solution captures the FE results and the deformation mode is accordance with the available experimental results. It shows that lateral deformation of CFST member can be effective predicted by the modal solution consisting of geometric and physical parameters variations. The influence of steel tube thickness, cross-section diameter, effective length, strength and density of concrete and steel, and initial impulsive are discussed in detail. The results show that deflection of the CFST member increase with the increase of the length to diameter ratio, initial impulse and smaller thickness to diameter ratio. Finally, the relationship between lateral deformation and geometric variations are established by optimal sets of modal solution.

Compact and Accurate Phugoid Mode Approximation with Residualization

Literal approximations to the phugoid mode are systematically derived by a process of residualization with fast-slow modal decomposition. Approximations with a static residual alone and another including a dynamic residual as well are obtained. The two sets of approximations are shown to be analytically identical to each other, therefore previous attempts by some researchers to obtain "improved" literal approximations by introducing a dynamic residual are seen to be redundant. Our static residualization yields new expressions for the phugoid mode parameters which include an "inertia" term and a "phase damping" term not generally observed in previous approximations. In fact, the "phase damping" term is seen to be the second-most numerically significant damping effect after the drag and is always found to detract from phugoid damping. A third term involving the pitch damping derivative Cmq is relatively insignificant but interestingly is found in many cases to undamp the phugoid motion. Approximations are derived with and without compressibility effects in the form of the derivative CmMa to highlight the contribution of this derivative in obtaining accurate estimates of the phugoid frequency, even at low speeds. Finally, recommended literal approximations to the phugoid frequency and damping are presented which are compact and whose numerical accuracy is established for ten different combinations of airplanes / flight conditions.