Quadratic Quantities Research Articles

𝒫𝒯 and anti-𝒫𝒯 symmetries for astrophysical waves

Context. Discrete symmetries have found numerous applications in photonics and quantum mechanics, but remain little studied in fluid mechanics, particularly in astrophysics. Aims. We aim to show how 𝒫𝒯 and anti-𝒫𝒯 symmetries determine the behaviour of linear perturbations in a wide class of astrophysical problems. They set the location of ‘exceptional points’ in the parameter space and the associated transitions to instability, and are associated with the conservation of quadratic quantities that can be determined explicitly. Methods. We study several classical local problems: the gravitational instability of isothermal spheres and thin discs, the Schwarzschild instability, the Rayleigh-Bénard instability and acoustic waves in dust–gas mixtures. We calculate the locations and the order of the exceptional points using the resultant of two univariate polynomials, as well as the conserved quantities in the different regions of the parameter space using Krein theory. Results. All problems studied here exhibit discrete symmetries, even though Hermiticity is broken by different physical processes (self-gravity, buoyancy, diffusion, and drag). This analysis provides genuine explanations for certain instabilities, and for the existence of regions in the parameter space where waves do not propagate. Those two aspects correspond to regions where 𝒫𝒯 and anti-𝒫𝒯 symmetries are broken respectively. Not all instabilities are associated to symmetry breaking (e.g. the Rayleigh-Benard instability).

Random walk model for dual cascades in wave turbulence.

Dual cascades in turbulent systems with two conserved quadratic quantities famously arise in both two-dimensional hydrodynamic turbulence and also in wave turbulence based on four-wave interactions. Examples for the latter include surface waves and nonlinear Schrödinger equationswith cubic nonlinearity. However, numerical simulations in forced-dissipative equilibrium of two-dimensional turbulence and of a one-dimensional wave system reveal that the physical nature of their cascades is starkly different. This is demonstrated by comparing their spectra in a finite inertial range and by comparing the temporal fluctuations of their spectral fluxes. In particular, the flux fluctuations are much larger in the wave case and frequently lead to instantaneous flux values that have the opposite sign of the mean flux, a phenomenon that is completely absent in the hydrodynamic case. A simple random walk model for the dual cascade in wave turbulence is then formulated that is very successful in explaining these effects. In particular, the model is able to replicate the detailed shape of the observed turbulent spectrum in a finite inertial range, and it also offers a ready explanation for the large flux fluctuations. It is also shown that a nonlinear diffusion model for the wave system cannot explain the observed spectral shapes. Overall, this suggests that in wave turbulence the systematic spectral fluxes observed in a dual cascade do not require an irreversible dynamical mechanism, rather, they arise as the inevitable outcome of blind chance.

Energy‐based model order reduction for linear stochastic Galerkin systems of second order

AbstractWe consider a second‐order linear system of ordinary differential equations (ODEs) including random variables. A stochastic Galerkin method yields a larger deterministic linear system of ODEs. We apply a model order reduction (MOR) of this high‐dimensional linear dynamical system, where its internal energy represents a quadratic quantity of interest. We investigate the properties of this MOR with respect to stability, passivity and energy dissipation. Numerical results are shown for a system modelling a mass‐spring‐damper configuration.

Resonance Expansion of Quadratic Quantities with Regularized Quasinormal Modes

Resonance Expansion of Quadratic Quantities with Regularized Quasinormal Modes

Resonance Expansion of Quadratic Quantities with Regularized Quasinormal Modes

Resonance expansions are an intuitive approach to capture the interaction of an optical resonator with light. Herein, a quasinormal mode expansion approach for quadratic observables is presented. The approach exploits the rigorous Riesz projection method. A numerical implementation of a state‐of‐the‐art quantum light source demonstrates the applicability with emphasis on modal expansions outside the underlying nanophotonic resonator.

Nonconservative Cascades in a Shell Model of Turbulence

Developed turbulent flows in which the intervention of external forces is fundamentally important at scales where the inertial range should exist are quite common. Then the cascade processes are not conservative any more and, therefore, it is necessary to adequately describe the external forces acting in the whole range of scales. If the work of these forces has a power law scaling, then one can assume that the integral of motion changes and the preserving value becomes a quadratic quantity, which includes the dependence on the scale. We develop this idea within the framework of shell models of turbulence. We show that, in terms of nonconservative cascades, one can describe various situations, including (as a particular case) the Obukhov – Bolgiano scaling proposed for turbulence in a stratified medium and for helical turbulence with a helicity injection distributed along the spectrum.

Quadratic quantity discount contract under price-dependent demand and consumer returns

This study introduces a new type of supply contract, the quadratic quantity discount contract, whose average transfer payment is a quadratic function. Thus, both the total and average quantity discounts increase nonlinearly in order quantity, unlike the typical linear quantity discount contract, whose total discount is nonlinear but average payment is a linear function of order quantity. We reveal the quadratic quantity discount contract’s unique properties by comparing it with linear quantity discount and wholesale price contracts. Our investigation, conducted in basic and closed-loop supply chain situations, reveals the potential of the quadratic quantity discount contract to be considered a serious choice for supply chain managers in various situations. Specifically, we reveal that the quadratic quantity discount contract always coordinates the supply chain, achieving the first-best supply chain profit while always enhancing the profits of all players. This is the most important difference from the typical linear quantity discount contract that entails a profit distribution problem and bargaining power issues. Further, unlike other contracts, its profit improvement performance is predictable; moreover, it offers the acceptable wholesale price with an additional nonlinear average discount. However, when the supplier can exercise its power and only considers its own profit, the typical linear quantity discount contract can be considered the better option, as it produces superior profits for the supplier, but the profits of the retailer and supply chain are sacrificed in this case. If the supplier considers the supply chain’s long-term health and wants voluntary compliance from other players, the quadratic quantity discount contract is the optimal choice.

Helicity distributions and transfer in turbulent channel flows with streamwise rotation

Helicity is a quadratic inviscid conservative quantity in three-dimensional turbulent flows and is crucial for turbulent system evolution. Helicity effects have mainly been highlighted over the past few decades to explore the intrinsic mechanism of turbulent flows, while the statistical characteristics of helicity itself are nearly absent in general anisotropic turbulent flows. In this paper, we investigate the helicity statistics in turbulent channel flows with streamwise rotation at moderate rotation numbers ($Ro_{\tau }=7.5,15$and 30) and Reynolds numbers ($Re_{\tau }=180$and 395), including their spatial and scale distributions, anisotropy and cross-scale transfer. The appearance of a mean secondary flow in the spanwise direction corresponds to a mean streamwise vorticity, which indicates the presence of a high-helicity distribution. Numerical results reveal a regular helicity profile along the wall-normal direction, and a new peak is found in the near-wall region around$y^+=6$of the streamwise or spanwise helicity profiles. The inter-scale helicity transfer is analysed by the filtering method, and the numerical consequences reveal that the second channel of the helicity cascade we proposed previously is dominant in contrast to the first channel. The rotation effects are explored by comparing the numerical results obtained under different rotation numbers. With increasing rotation number, more helical structures in the near-wall regions are present, with peaks of helicity profiles and fluxes coming closer to the wall. With a higher Reynolds number, their amplitudes are larger and scale-space transfer is strengthened. These systematic numerical analyses uncover the helicity distributions and transfer in wall-bounded turbulent flows.

Ramanujan continued fractions of order sixteen

We study the continued fractions [Formula: see text] and [Formula: see text] of order sixteen by adopting the theory of modular functions. These functions are analogues of Rogers–Ramanujan continued fraction [Formula: see text] with modularity and many interesting properties. Here we prove the modularities of [Formula: see text] and [Formula: see text] to find the relation with the generator of the field of modular functions on [Formula: see text]. Moreover we prove that the values [Formula: see text] and [Formula: see text] are algebraic integers for certain imaginary quadratic quantity [Formula: see text].

Quadratic quantities in acoustics: Scattering cross-section and radiation force

Time-harmonic acoustic waves interact with a scatterer: this interaction is calculated using linear, first-order theory. However, there are quadratic, second-order quantities that are of interest. These include the scattering cross-section and the steady radiation force; these quantities can be expressed as integrals of products of first-order quantities over a sphere. These integrals are evaluated exactly. The results are infinite series of products of the coefficients in the spherical multipole expansions of the incident and scattered fields; they do not depend on the radius of the spherical integration surface. For a specific scattering problem, the coefficients can be connected by an appropriate T-matrix. In most previous work, the spherical surface is moved to infinity so that far-field quantities can be introduced: it is shown that this process is not straightforward and it may introduce spurious difficulties.

Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages

The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper, we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving average and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian limit.

Discrete Energy-Conservation Properties in the Numerical Simulation of the Navier–Stokes Equations

Nonlinear convective terms pose the most critical issues when a numerical discretization of the Navier–Stokes equations is performed, especially at high Reynolds numbers. They are indeed responsible for a nonlinear instability arising from the amplification of aliasing errors that come from the evaluation of the products of two or more variables on a finite grid. The classical remedy to this difficulty has been the construction of difference schemes able to reproduce at a discrete level some of the fundamental symmetry properties of the Navier–Stokes equations. The invariant character of quadratic quantities such as global kinetic energy in inviscid incompressible flows is a particular symmetry, whose enforcement typically guarantees a sufficient control of aliasing errors that allows the fulfillment of long-time integration. In this paper, a survey of the most successful approaches developed in this field is presented. The incompressible and compressible cases are both covered, and treated separately, and the topics of spatial and temporal energy conservation are discussed. The theory and the ideas are exposed with full details in classical simplified numerical settings, and the extensions to more complex situations are also reviewed. The effectiveness of the illustrated approaches is documented by numerical simulations of canonical flows and by industrial flow computations taken from the literature.

Scattering cross-section of a cylindrical conducting particle illuminated by electromagnetic plane waves near a conducting quarter-space

The study of the electromagnetic scattering by a particle located near a perfectly conducting corner is a fundamental topic in radar imaging, communications, remote sensing and other applications. In earlier works, the analytical modeling considered the linear field and its scattering properties. Nonetheless, energy efficiencies (or cross-sections) are meaningful fundamental physical observables connected with quadratic quantities of the linear field, which reveal additional properties of the scatterer not obtained from the linear scattered field. The aim of this work is therefore directed toward developing an improved analytical formalism for the energy efficiency factors (or cross-sections) of a cylindrical conducting particle, illuminated by electromagnetic plane waves, and located near a perfectly conducting corner-space. Incident plane progressive waves with an arbitrary angle of incidence are considered in a homogeneous, non-dissipative and non-magnetic fluid. The multiple scattering effects occurring between the electrically-conducting cylindrical particle and corner are described rigorously using the modal expansion method in cylindrical coordinates, the classical method of images and the translational addition theorem. Exact closed-form series expansions are derived and numerical examples illustrate the analysis with particular emphases on the distances from the corner-space, particle size and polarization of the incident field. The results are useful to predict the energetic scattering for various applications, and can serve to validate the results of purely numerical methods in experimental design. Some analogies with the acoustical cross-section theory are also noted.

Three Ramanujan continued fractions with modularity

We study three Ramanujan continued fractions c(τ),W(τ) and T(τ). In fact, c(τ) and W(τ) are modular functions of level 16, and T(τ) is a modular function of level 32. We first prove that the values of c(τ) and W(τ) can generate the ray class field modulo 4 over an imaginary quadratic field K. We also prove that 2/(1−c(τ)),1/W(τ),T(τ)+1/T(τ) are algebraic integers for any imaginary quadratic quantity τ. Furthermore, we find the modular equations of c(τ),T(τ) and W(τ) for any level, and we show that c(τ) and W(τ) satisfy the Kronecker's congruence. We can express the value c(rτ) (respectively, T(rτ),W(rτ)) in terms of radicals for any positive rational number r when the value c(τ) (respectively, T(τ),W(τ)) can be written as radicals.

Extrinsic extinction cross-section in the multiple acoustic scattering by fluid particles

Cross-sections (and their related energy efficiency factors) are physical parameters used in the quantitative analysis of different phenomena arising from the interaction of waves with a particle (or multiple particles). Earlier works with the acoustic scattering theory considered such quadratic (i.e., nonlinear) quantities for a single scatterer, although a few extended the formalism for a pair of scatterers but were limited to the scattering cross-section only. Therefore, the standard formalism applied to viscous particles is not suitable for the complete description of the cross-sections and energy balance of the multiple-particle system because both absorption and extinction phenomena arise during the multiple scattering process. Based upon the law of the conservation of energy, this work provides a complete comprehensive analysis for the extrinsic scattering, absorption, and extinction cross-sections (i.e., in the far-field) of a pair of viscous scatterers of arbitrary shape, immersed in a nonviscous isotropic fluid. A law of acoustic extinction taking into consideration interparticle effects in wave propagation is established, which constitutes a generalized form of the optical theorem in multiple scattering. Analytical expressions for the scattering, absorption, and extinction cross-sections are derived for plane progressive waves with arbitrary incidence. The mathematical expressions are formulated in partial-wave series expansions in cylindrical coordinates involving the angle of incidence, the addition theorem for the cylindrical wave functions, and the expansion coefficients of the scatterers. The analysis shows that the multiple scattering cross-section depends upon the expansion coefficients of both scatterers in addition to an interference factor that depends on the interparticle distance. However, the extinction cross-section depends on the expansion coefficients of the scatterer located in a particular system of coordinates, in addition to the interference term. Numerical examples illustrate the analysis for two viscous fluid circular cylindrical cross-sections immersed in a non-viscous fluid. Computations for the (non-dimensional) scattering, absorption, and extinction cross-section factors are performed with particular emphasis on varying the angle of incidence, the interparticle distance, and the sizes, and the physical properties of the particles. A symmetric behavior is observed for the dimensionless multiple scattering cross-section, while asymmetries arise for both the dimensionless absorption and extinction cross-sections with respect to the angle of incidence. The present analysis provides a complete analytical and computational method for the prediction of cross-section and energy efficiency factors in multiple acoustic scattering of plane waves of arbitrary incidence by a pair of scatterers. The results can be used as a priori information in the direct or inverse characterization of multiple scattering systems such as acoustically engineered fluid metamaterials with reconfigurable periodicities, cloaking devices, liquid crystals, and other applications.

A backscattering model based on corrector theory of homogenization for the random Helmholtz equation

This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the energy is specularly reflected and the backscattered signal vanishes in a first order approximation. What is measured at the detector consists therefore of corrections to leading order terms, and we focus in this work on the homogenization corrector. We describe the propagation by a random Helmholtz equation, and derive an expression of the corrector in this layered framework. We moreover obtain a transport model for quadratic quantities in the random wavefield in a high frequency limit.

Linear stability of ideal MHD configurations. II. Results for stationary equilibrium configurations

In this paper, we continue exploring the consequences of the general equation of motion (EOM) governing all Lagrangian perturbations ξ about a time-dependent, ideal magnetohydrodynamic (MHD) configuration, which includes self-gravity, external gravity, pressure gradients, compressibility, inertial effects, and anisotropic Lorentz force. We here address the specific case of MHD stability for 3D stationary equilibria, where the perturbed EOM features a symmetric operator F and an antisymmetric Doppler-Coriolis operator v·∇. For this case, we state and prove the general properties for the solutions ξ of the governing dynamical system. For axisymmetric perturbations about axisymmetric equilibria with purely toroidal, or purely poloidal magnetic fields, specific stability theorems can be formulated. We derive a useful integral expression for the quadratic quantity given by the inner product ⟨ξ,F[ξ]⟩. For deriving stability statements on MHD states where self-gravity is involved as well, we provide an upper bound on the perturbed self-gravitational energy associated with the displacement ξ. The resulting expression elucidates the role of potentially stabilizing versus destabilizing contributions and shows the role of gravity, entropy gradients, velocity shear, currents, Lorentz forces, inertia, and pressure gradients in offering many routes to unstable behavior in flowing gases and plasmas. These have historically mostly been studied for static v=0 configurations, looking at stability of exactly force-balanced states, or by assuming stationarity similar to our approach here (i.e., ∂t≡0 for the state we perturb), but typically in combination with some reduced dimensionality on the configuration of interest (translational or axisymmetry). We show that in these limits, we find and generalize expressions well-known from, e.g., the study of ideal MHD stability of tokamak plasmas or from Schwarzschild's criteria controlling convection in hydrodynamic, (external) gravitating systems. When applied to stationary, axisymmetric configurations in motion, we can use our upper bound to derive a sufficient stability criterion for perturbations of arbitrary azimuthal mode number m used in ξ(r)=η(r,z) exp(imφ).

On a new application of the path integrals in polymer statistical physics

We propose a new approach based on the path integral formalism to the calculation of the probability distribution functions of quadratic quantities of the Gaussian polymer chain in d-dimensional space, such as the radius of gyration and potential energy in the parabolic well. In both cases we obtain the exact relations for the characteristic function and cumulants. Using the standard steepest-descent method, we evaluate the probability distribution functions in two limiting cases of the large and small values of corresponding variables.

Gravity Wave Diagnostics and Characteristics in Mesoscale Fields

Abstract As numerical models of complex atmospheric flows increase their quality and resolution, it becomes valuable to isolate and quantify the embedded resolved gravity waves. The authors propose a spatial filtering method combined with a selection of quadratic diagnostic quantities such as heat, momentum, and energy fluxes to do this. These covariant quantities were found to be insensitive to filter cutoff length scales between 300 and 700 km, suggesting the existence of a “cospectral gap.” The gravity waves identified with the proposed method display known properties from idealized studies, including vertical propagation, upwind propagation, the relationship between momentum and energy flux, and agreement with fluxes derived from an alternative method involving simulations with and without terrain. The proposed method is applied to 2- and 6-km-resolution realistic WRF simulations of orographic and nonorographic gravity waves over and around New Zealand within complex frontal cyclones. Deep mountain wave, shallow mountain wave, jet-generated gravity wave, and convection-generated gravity wave events were chosen for analysis. The four wave events shared the characteristics of positive vertical energy flux, negative zonal momentum flux, and upwind horizontal energy flux. Two of the gravity wave events were dissipated nonlinearly.

New Stability Analysis for Linear Systems with Time-Varying Delay Based on Combined Convex Technique

A novel combined convex method is developed for the stability of linear systems with a time-varying delay. A new delay-dependent stability condition expressed in terms of linear matrix inequalities (LMIs) is derived by employing a dedicated constructed Lyapunov-Krasovskii functional (LKF), utilizing the Wirtinger inequality and the reciprocally convex approach to handle the integral term of quadratic quantities. Different from the previous convex techniques which only tackle the time-varying delay, our method adopts the idea of combined convex technique which can tackle not only the delay but also the delay variation. Four well-known examples are illustrated to show the effectiveness of the proposed results.