Wentzel Kramers Brillouin Research Articles

Global geodesic acoustic mode in an ideal magnetohydrodynamic tokamak plasma

A concise and transparent second order ordinary differential equation (ODE) describing the radial structure of the global geodesic acoustic mode (GAM) is analytically presented in a low-β tokamak plasma. The large-aspect-ratio and circular cross section are assumed to linearize the ideal magnetohydrodynamic equations. We show clearly how finite β-dependent terms affect the global GAM frequency and radial mode structure. A typical Wentzel–Kramers–Brillouin form of solution is found for some reversed shear equilibria. For some other equilibria with lower β, even also in a reversed shear tokamak, the GAM continuum is upraised by the high order β-dependent terms so that its maximum is beyond ωG, where ωG is the classical local frequency of GAM. As a result, no self-consistent solution to the ODE can be found.

Semi-analytic calculations of barrier penetration and alpha particle preformation probabilities

An improved semi-analytic approach for the barrier penetration probability is developed in the framework of the Wentzel–Kramers–Brillouin approximation. It is used to calculate the α-decay half-life, assault frequency and α-preformation probability for many radioactive nuclei in the range Z = 52–99. Calculations are also extended to the super heavy region with Z = 100–120. Results are compared with the experimental data and some other recent studies. This approach achieves a better agreement with the experimental data than many other models. A strong correlation is found between the experimental decay half-life and the calculated penetration probability, assault frequency and preformation probability. Therefore, this approach achieves both reasonable accuracy and good consistency with the expected nuclear physical observations.

Horizontal shear instabilities in rotating stellar radiation zones

Context. Rotational mixing transports angular momentum and chemical elements in stellar radiative zones. It is one of the key processes for modern stellar evolution. In the past two decades, an emphasis has been placed on the turbulent transport induced by the vertical shear instability. However, instabilities arising from horizontal shear and the strength of the anisotropic turbulent transport that they may trigger remain relatively unexplored. The weakest point of this hydrodynamical theory of rotational mixing is the assumption that anisotropic turbulent transport is stronger in horizontal directions than in the vertical one. Aims. This paper investigates the combined effects of stable stratification, rotation, and thermal diffusion on the horizontal shear instabilities that are obtained and discussed in the context of stellar radiative zones. Methods. The eigenvalue problem describing linear instabilities of a flow with a hyperbolic-tangent horizontal shear profile was solved numerically for a wide range of parameters. When possible, the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximation was applied to provide analytical asymptotic dispersion relations in both the nondiffusive and highly diffusive limits. As a first step, we consider a polar f-plane where the gravity and rotation vector are aligned. Results. Two types of instabilities are identified: the inflectional and inertial instabilities. The inflectional instability that arises from the inflection point (i.e., the zero second derivative of the shear flow) is the most unstable when at a zero vertical wavenumber and a finite wavenumber in the streamwise direction along the imposed-flow direction. While the maximum two-dimensional growth rate is independent of the stratification, rotation rate, and thermal diffusivity, the three-dimensional inflectional instability is destabilized by stable stratification, while it is stabilized by thermal diffusion. The inertial instability is rotationally driven, and a WKBJ analysis reveals that its growth rate reaches the maximum value of √f(1 − f) in the inviscid limit as the vertical wavenumber goes to infinity, where f is the dimensionless Coriolis parameter. The inertial instability for a finite vertical wavenumber is stabilized as the stratification increases, whereas it is destabilized by the thermal diffusion. Furthermore, we found a selfsimilarity in both the inflectional and inertial instabilities based on the rescaled parameter PeN2 with the Péclet number Pe and the Brunt–Väisälä frequency N.

Partition of Unity Finite Element Method for the modelling of Acoustic Black Hole wedges

The Acoustic Black Hole (ABH) phenomenon can be exploited to manipulate and mitigate flexural wave propagation in thin-walled structures. ABH structures feature unique space-dependent wavenumber variation and wave celerity reduction in the tapered ABH area, thus posing challenges to the existing modelling techniques. In this work, the Partition of Unity Finite Element Method (PUFEM) is revamped to simulate the structural response of an ABH wedge subject to a harmonic loading. This method allows the incorporation of auxiliary enrichment functions into the finite element framework in order to cope with the ABH-induced wave oscillating behaviour, exemplified by the varying wavenumber and amplitude in space. The PUFEM tapered Timoshenko beam elements are constructed by employing wave enrichment functions with the Wentzel-Kramers-Brillouin (WKB) approximation method. A wavelet enrichment is also investigated as hierarchic refinement. Using these enriched elements, the frequency responses of an ABH wedge and the convergence of numerical solutions are computed and compared with the classical linear FEM and the elements enriched with ‘local’ wave solutions. An adaptive meshing scheme is designed and implemented to further accelerate the solution convergence. It is shown that the PUFEM offers a good computational accuracy and drastic reduction of degrees of freedom for solving the broadband ABH problems, outperforming the classical FEM.

A Introduction to the Classical Spiral Electrodynamics: The” Spiral-Spin”

This paper demonstrates the existence of analytical solutions of the Lorentz equation for charged particles in “uniform pilot time-varying magnetic fields". These analytical solutions represent a temporal generalization of the Larmor's orbits and are expressed through a Schwarz-Christoffel spiral mapping or in spiral coordinates.
 The concepts of "spiral-spin” moment and "polar-spiral" angular momentum are then presented, the existence of a subclass of solutions for which these two angular moments are conserved is demonstrated.
 It is also shown that under the action of the "pilot fields," there exist particular trajectories for which the charged particles have a "spiral-spin" momentum constant proportional to +1/2 (solution named "spiral-spin-up ") and -1/2 (solution named "spiral-spin-down "), respectively.
 The results are in full agreement with the ideas of L.DeBroglie and A. Einstein on the possible existence of pilot fields able to describe the physical reality deterministically.
 Finally, the solution of the Lorentz equation is discussed with the WKB (Wentzel-Kramers-Brillouin) method for a superposition of two uniform magnetic fields with the same direction, the first constant and the second time-varying.

Beyond the Runge-Kutta-Wentzel-Kramers-Brillouin method

We explore higher-dimensional generalizations of the Runge-Kutta-Wentzel-Kramers-Brillouin method for integrating coupled systems of first-order ordinary differential equations with highly oscillatory solutions. Such methods could improve the performance and adaptability of the codes which are used to compute numerical solutions to the Einstein-Boltzmann equations. We test Magnus expansion-based methods on the Einstein-Boltzmann equations for a simple universe model dominated by photons with a small amount of cold dark matter. The Magnus expansion methods achieve an increase in run speed of about 50% compared to a standard Runge-Kutta integration method. A comparison of approximate solutions derived from the Magnus expansion and the Wentzel-Kramers-Brillouin (WKB) method implies the two are distinct mathematical approaches. Simple Magnus expansion solutions show inferior long range accuracy compared to WKB. However we also demonstrate how one can improve on the standard Magnus approach to obtain a new "Jordan-Magnus" method. This has a WKB-like performance on simple two-dimensional systems, although its higher-dimensional generalization remains elusive.

Multisoliton pulse breakup in WKB approximation

The application of the Wentzel-Kramers–Brillouin (WKB) method for Zakharov–Shabat scattering problem with high-amplitude smooth pulses is discussed. For pulses with limited phase variation, the eigenvalue spectrum for high pulse amplitude A lies on a curve in the complex plane to the order of A−1. The eigenvalues here can be accurately estimated using integrals on a real interval, similar to quantum mechanics. For pulses with big phase variation, such features as symmetry breaking can be studied with more involved complex WKB approach. Comparison with numerical and analytical results is reported.

Laser beam self-focusing in collisional plasma with periodical density ripple

AbstractIn the paper, we applied the paraxial region theory and Wentzel–-Kramers–-Brillouin approximation to study laser beam self-focusing in the interaction of laser and collisional plasma with periodical density ripple. The results have shown that, under the influence of collision nonlinear effect, laser presents stable self-focusing, self-defocusing, and oscillational self-focusing in the plasma. Besides, the parameters of plasma with periodical density ripple have a greater impact on the effect of self-defocusing and oscillational self-focusing than stable self-focusing. In certain conditions, beam self-defocusing and oscillational self-focusing would decline and even disappear, and stable self-focusing would further be strengthened. Hence, selecting a suitable periodic plasma system is advantageous for separating self-defocusing and oscillational self-focusing, and for the formation of a more stable collisional self-focusing.

Efficient method for solving highly oscillatory ordinary differential equations with applications to physical systems

We present a novel numerical routine (oscode) with a C++ and Python interface for the efficient solution of one-dimensional, second-order, ordinary differential equations with rapidly oscillating solutions. The method is based on a Runge-Kutta-like stepping procedure that makes use of the Wentzel-Kramers-Brillouin (WKB) approximation to skip regions of integration where the characteristic frequency varies slowly. In regions where this is not the case, the method is able to switch to a made-to-measure Runge-Kutta integrator that minimises the total number of function evaluations. We demonstrate the effectiveness of the method with example solutions of the Airy equation and an equation exhibiting a burst of oscillations, discussing the error properties of the method in detail. We then show the method applied to physical systems. First, the one-dimensional, time-independent Schr\"odinger equation is solved as part of a shooting method to search for the energy eigenvalues for a potential with quartic anharmonicity. Then, the method is used to solve the Mukhanov-Sasaki equation describing the evolution of cosmological perturbations, and the primordial power spectrum of the perturbations is computed in different cosmological scenarios. We compare the performance of our solver in calculating a primordial power spectrum of scalar perturbations to that of BINGO, an efficient code specifically designed for such applications.

A new simple model for the α-decay

We analytically find a new simple α-decay formula for a modified harmonic oscillator with the spherical Coulomb potential by employing Wentzel–Kramers–Brillouin approximation in terms of taking into account the Bohr–Sommerfeld quantization condition. We systematically investigate the favored α-decay half-lives of parent nuclei. Our results are in good agreement with the experimental data and other models. We emphasize the importance of shape of the nuclear potential in the examination of α-decay half-lives of the parent nuclei.

The Applicability of WKB Method in Studying Inhomogeneous Dusty Plasma

In this article, the applicability of the Wentzel–Kramer–Brillouin (WKB) method in studying inhomogeneous dusty plasma is discussed. First the power transmission coefficient of electromagnetic waves in inhomogeneous dusty plasma is derived based on the WKB method. Then, the power transmission coefficients of the double exponential and hyperbolic electron number density distribution models are numerically simulated. The numerical results reveal that the power transmission coefficient increases with the increase of incident wave frequency. Furthermore, comparing the power transmission coefficient of the WKB method and the analytical method in common plasma and dusty plasma, it is found that the WKB method is more applicable for common plasma. Finally, the applicable region of the WKB method for dusty plasma is discussed. The results reveal that the electron number density distribution is crucial for the applicable region of the WKB method, as well as for the incident wave frequency.

The nonlinear wave in semiconductor quantum plasma for laser beam in a self-consistent plasma channel

In this paper investigating the current task, our emphasis is on distributed a serious laser beam in a self-made plasma channel. As the intensity of the laser beam increases, the medium displays the neuromuscular electrical conductivity functioning as a result of the mass effectiveness of the effectiveness of the electrons as well as the intensity of the electrons. Based on Wentzel–Kramers–Brillouin and paraxial ray theory, the steady-state solution of an intense, Gaussian electromagnetic beam is studied. The differential equation of the distance with the beam width parameter is derived, including the relativistic effects of self-focusing (SF) and self-channeling ponder motive. Plasma propagation is a radial dynamical force, depending on the width of the beams and σp greater, plasma ratio frequency screws. Once the distribution regimes, the beam power beamwidth is obtained in plane. The σp is a particular value characterized by standard deviation, oscillation and diffusion regimes such as self focusing. The corresponding center parameters are intended for introduction of the plasma density curve and the laser beam is spatially analyzed to the spatial plasma. Performing this margin can lead to a long distance laser beam guide. The disadvantage of numerical predictions is done for laser plasma interaction studies for common factors.

A Real-Time Model of the Seasonal Temperature of Lunar Polar Region and Data Validation

A small tilt in the spin axis of the moon over the ecliptic plane causes seasonal incidence variation of solar illumination and, especially, causes significant temperature difference at the polar region. In this article, following the position of the subsolar point, the real-time model of solar illumination incidence over the moon polar region is developed. Based on this model with solving the 1-D heat conduction equation, the seasonal temperature of the lunar surface is obtained and is in agreement with the Diviner infrared (IR) data. Meanwhile, using the fluctuation-dissipation theorem and the Wentzel–Kramers–Brillouin (WKB) approach for lunar regolith media, the seasonal microwave (MW) brightness temperature (TB) is also obtained and validated by Chang’e-2 (CE-2) 37-GHz TB data. These data also show that the lunar surface temperature and the MW TB even in the permanent shaded region (PSR) undergo seasonal variation as well. It might be due to the seasonal thermal radiation on the topographic PSR coming from the sunlit crater walls caused by seasonal temperature variation. The Diviner IR data show that the highest temperature in the Hermite-A crater at the north polar PSR can reach 109 K in summer.

Relativistic Self-Focusing of Hermite-cosh-Gaussian Laser Beam in Magnetoplasma with Exponential Plasma Density Ramp

In the present manuscript, we analyse the effect of exponential plasma density ramp for relativistic self-focusing of Hermite-cosh-Gaussian laser pulse in magnetoplasma. The exponential plasma density ramp is found to be more prominent in achieving the stronger self-focusing of Hermite-cosh-Gaussian laser beam in comparison to the tangential plasma density ramp. We propose a theoretical model for propagation of Hermite-cosh-Gaussian laser pulse in magnetoplasma with exponential density ramp. The nonlinearity in the medium arises because of the relativistic motion of electrons, being responsible for relativistic self-focusing. Equation of the beam width parameter is set up by taking the expression for the dielectric function and following Wentzel-Kramers-Brillouin (WKB) with paraxial ray approximations for mode indices m = 0, 1 and 2. Effect of decentered parameter is also analysed, which results in stronger self-focusing of the Hermite-cosh-Gaussian laser beam. Stronger self-focusing of laser beam is more pronounced in high density plasma with higher magnetic field.

Quasi-local method of wave decomposition in a slowly varying medium

The general asymptotic theory for wave propagation in a slowly varying medium, classically known as the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximation, is revisited here with the aim of constructing a new data diagnostic technique useful in atmospheric and oceanic sciences. Using the Wigner transform, a kind of mapping that associates a linear operator with a function, we analytically decompose a flow field into mutually independent wave signals. This method takes account of the variations in the polarisation relations, an eigenvector that represents the kinematic characteristics of each wave component, so as to project the variables onto their eigenspace quasi-locally. The temporal evolution of a specific mode signal obeys a single wave equation characterised by the dispersion relation that also incorporates the effect from the local gradient in the medium. Combining this method with transport theory and applying them to numerical simulation data, we can detect the transfer of energy or other conserved quantities associated with the propagation of each wave signal in a wide variety of situations.

Improved bounds on horizontal convection

For the problem of horizontal convection the Nusselt number based on entropy production is bounded from above by as the horizontal convective Rayleigh number for some constant (Siggers et al., J. Fluid Mech., vol. 517, 2004, pp. 55–70). We re-examine the variational arguments leading to this ‘ultimate regime’ by using the Wentzel–Kramers–Brillouin method to solve the variational problem in the limit and exhibiting solutions that achieve the ultimate scaling. As expected, the optimizing flows have a boundary layer of thickness pressed against the non-uniformly heated surface; but the variational solutions also have rapid oscillatory variation with wavelength along the wall. As a result of the exact solution of the variational problem, the constant is smaller than the previous estimate by a factor of for no-slip and for no-stress boundary conditions. This modest reduction in indicates that the inequalities used by Siggers et al. (J. Fluid Mech., vol. 517, 2004, pp. 55–70) are surprisingly accurate.

Self-focusing of cosh-Gaussian laser beam and its effect on the excitation of ion-acoustic wave and stimulated Brillouin backscattering in collisionless plasma

An analytical and numerical study has been carried out for self-focusing of an intense cosh-Gaussian laser beam in collisionless plasma and its impact on the excitation of ion-acoustic wave and stimulated Brillouin backscattering process. The analytical model has been developed under Wentzel–Kramers–Brillouin and paraxial ray approximations. The nonlinearities of ponderomotive force on electron and the relativistic oscillation of the electron mass have been used in this study. The nonlinear differential equations have been set up for the beam width parameters of the main beam, ion-acoustic wave, backscattered wave and back reflectivity of stimulated Brillouin scattering (SBS). These equations have been solved numerically for different values of decentred parameter (b), relative plasma density (ωp0/ω0) and incident laser intensity (a). The results have been compared with only relativistic nonlinearity and Gaussian profile of laser beam. The focusing of laser beam, ion-acoustic wave and scattered wave are found to be strong under relativistic-ponderomotive regime compared to only relativistic regime. Further, it is observed that focusing/intensity of main laser beam, ion acoustic wave and SBS back reflectivity increases with increasing the values of b and ωp0/ω0. Itis also found that back reflectivity of SBS process gets suppressed with the increase in the value of a. This study may be useful in laser induced fusion scheme where back scattering of SBS plays very important role.

Quantum-corrected rotating acoustic black holes in Lorentz-violating background

In this paper we explore the effect of the generalized uncertainty principle and modified dispersion relation to compute Hawking radiation from a rotating acoustic black hole in the tunneling formalism by using the Wentzel-Kramers-Brillouin (WKB) approximation applied to the Hamilton-Jacobi method. The starting point is to consider the planar acoustic black hole metric found in a Lorentz-violating Abelian Higgs model. In our analyzes we investigate quantum corrections for the Hawking temperature and entropy. A logarithmic correction and an extra term that depends on a conserved charge were obtained. We also have found that the changing in the Hawking temperature ${\cal T}_H$ for a dispersive medium due to a Lorentz-violating background accounts for supersonic velocities in the general form $(v_g-v_p)/v_p = \Delta {\cal T}_H/{\cal T}_H\sim10^{-5}$ in Bose-Einstein-Condensate (BEC) systems.

Heat kernel for higher-order differential operators and generalized exponential functions

We consider the heat kernel for higher-derivative and nonlocal operators in $d$-dimensional Euclidean space-time and its asymptotic behavior. As a building block for operators of such type, we consider the heat kernel of the minimal operator - generic power of the Laplacian - and show that it is given by the expression essentially different from the conventional exponential Wentzel-Kramers-Brillouin (WKB) ansatz. Rather it is represented by the generalized exponential function (GEF) directly related to what is known in mathematics as the Fox-Wright $\varPsi$-functions and Fox $H$-functions. The structure of its essential singularity in the proper time parameter is different from that of the usual exponential ansatz, which invalidated previous attempts to directly generalize the Schwinger-DeWitt heat kernel technique to higher-derivative operators. In particular, contrary to the conventional exponential decay of the heat kernel in space, we show the oscillatory behavior of GEF for higher-derivative operators. We give several integral representations for the generalized exponential function, find its asymptotics and semiclassical expansion, which turns out to be essentially different for local operators and nonlocal operators of noninteger order. Finally, we briefly discuss further applications of the GEF technique to generic higher-derivative and pseudodifferential operators in curved space-time, which might be critically important for applications of Horava-Lifshitz and other UV renormalizable quantum gravity models.

A numerical analysis of motion in symmetric double-well harmonic potentials using pseudospectral methods

I use pseudospectral methods defined with a set of Hermite polynomials and with a set of half-range, nonclassical Hermite polynomials to test their applicability in analyzing the motion of a particle in symmetric double-well harmonic potentials. The results of these two kinds of pseudospectral method and those of the Wentzel-Kramers-Brillouin (WKB) approximation are compared with each other. The pseudospectral method with half-range Hermite polynomials provides more accurate results of the eigenvalues of the five lowest states in symmetric double-well harmonic potentials than that with classical polynomials. It is shown that the former method with a set of the nonclassical polynomials exhibits a faster convergence than the latter and has a feasibility to study motion in a variety of double-well potentials.