A guiding centre Vlasov equation and its application to Alfven waves.
Guiding center Vlasov equation derived dielectric tensor of collisionless plasma, obtaining dispersion relation of Alfven waves in warm plasma
Fejer, J. A.↗
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Wave propagation in plasmas with very strong magnetic field.
Third moment equation derived from Boltzmann- Vlasov equation for wave propagation in plasmas with very strong magnetic field used instead of state equation
STATE EQUATION↗
Resonance, particle trapping, and Landau damping in finite amplitude obliquely propagating waves
The equations of motion for a particle in resonance with a small finite amplitude wave are solved approximately, using secularity free perturbation theory. The wave propagates at an arbitrary angle to a uniform background magnetic field in an infinite collisionless plasma. The wave fields include a longitudinal electrostatic component and elliptically polarized transverse electric and magnetic components. The trajectories of trapped and resonant untrapped particles are described, for each of the possible wave-particle resonances. These trajectories are used to construct an estimate of the nonlinear time dependent Landau damping rate of the wave.
Palmadesso, P. J.↗
Finite-Difference Algorithms For Computing Sound Waves
Governing equations considered as matrix system. Method variant of method described in "Scheme for Finite-Difference Computations of Waves" (ARC-12970). Present method begins with matrix-vector formulation of fundamental equations, involving first-order partial derivatives of primitive variables with respect to space and time. Particular matrix formulation places time and spatial coordinates on equal footing, so governing equations considered as matrix system and treated as unit. Spatial and temporal discretizations not treated separately as in other finite-difference methods, instead treated together by linking spatial-grid interval and time step via common scale factor related to speed of sound.
Davis, Sanford↗
Wave speeds in the macroscopic extended model for ultrarelativistic gases
Equations determining wave speeds for a model of ultrarelativistic gases are investigated. This model is already present in literature; it deals with an arbitrary number of moments and it was proposed in the context of exact macroscopic approaches in Extended Thermodynamics. We find these results: the whole system for the determination of the wave speeds can be divided into independent subsystems which are expressed by linear combinations, through scalar coefficients, of tensors all of the same order; some wave speeds, but not all of them, are expressed by square roots of rational numbers; finally, we prove that these wave speeds for the macroscopic model are the same of those furnished by the kinetic model.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Progressive wave expansions and open boundary problems
In this paper we construct progressive wave expansions and asymptotic boundary conditions for wave-like equations in exterior domains, including applications to electromagnetics, compressible flows and aero-acoustics. The development of the conditions will be discussed in two parts. The first part will include derivations of asymptotic conditions based on the well-known progressive wave expansions for the two-dimensional wave equations. A key feature in the derivations is that the resulting family of boundary conditions involves a single derivative in the direction normal to the open boundary. These conditions are easy to implement and an application in electromagnetics will be presented. The second part of the paper will discuss the theory for hyperbolic systems in two dimensions. Here, the focus will be to obtain the expansions in a general way and to use them to derive a class of boundary conditions that involve only time derivatives or time and tangential derivatives. Maxwell's equations and the compressible Euler equations are used as examples. Simulations with the linearized Euler equations are presented to validate the theory.
Hagstrom, T.↗
Study of electromagnetic waves in plasmas since the boltzmann equation <etude des ondes electromagnetiques dans les plasmas a partir de l'equation de boltzmann<
Electromagnetic waves in plasmas - landau absorption in plasma having no magnetic field
MAGNETIC FIELD↗
Magneto-atmospheric waves
A theoretical treatment of magneto-atmospheric waves is presented and applied to the modelling of waves in the solar atmosphere. The waves arise in compressible, stratified, electrically conductive atmospheres within gravitational fields when permeated by a magnetic field. Compression, buoyancy, and distortion of the magnetic field all contribute to the existence of the waves. Basic linearized equations are introduced to describe the waves and attention is given to plane-stratified atmospheres and their stability. A dispersion relation is defined for wave propagation in a plane-stratified atmosphere when there are no plane-wave solutions. Solutions are found for the full wave equation in the presence of either a vertical or a horizontal magnetic field. The theory is applied to describing waves in sunspots, in penumbrae, and flare-induced coronal disturbances.
Thomas, J. H.↗
Numerical methods for high-dimensional probability density function equations
In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker–Planck and Dostupov–Pugachev equations), random wave theory (Malakhov–Saichev equations) and coarse-grained stochastic systems (Mori–Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Quantitative study of the trapped particle bunching instability in Langmuir waves
The bunching instability of particles trapped in Langmuir waves is studied using Vlasov simulations. A measure of particle bunching is defined and used to extract the growth rate from numerical simulations, which are compared with theory [Dodin et al., Phys. Rev. Lett. 110, 215006 (2013)]. In addition, the general theory of trapped particle instability in 1D is revisited and a more accurate description of the dispersion relation is obtained. Excellent agreement between numerical and theoretical predictions of growth rates of the bunching instability is shown over a range of parameters.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Interaction of ocean waves with a soft bottom
Soft muddy bottoms have significant effects on properties of water waves which propagate over them. The wave dispersion equation is modified and wave energy is dissipated by the coupling between the waves in water and those induced in the mud layer. These effects are theoretically determined by assuming a viscoelastic mud layer. A boundary-value problem is solved for the water-mud system with sinusoidal waves. The theoretical dissipation rates are compared favorably with field measurements.
Hsiao, S. V.↗
The response of stationary planetary waves to tropospheric forcing
The lower boundary forcing of airflow over topography, and the internal forcing that results from the geographical distribution of diabatic heating, are studied in light of a steady state, linear, quasi-geostrophic model of stationary waves on a sphere. The lower boundary vertical motions forced by airflow over topography depend on whether the horizontal deflection of airflow around topographic features is taken into account, the level of the wind profile at which flow over topography is assumed to take place, and the topographic data set that was used in the forcing formulation. The lower boundary forcing is taken to be given by the observed stationary planetary wave in lower boundary geopotential height, and the internal forcing is computed using the planetary wave propagation equation on the observed wave structure.
Alpert, J. C.↗
Numerical solution of equations governing longitudinal suspension line wave motion during the parachute unfurling process
Equations are presented which govern the dynamics of the lines-first parachute unfurling process, including wave motion in the parachute suspension lines. Techniques are developed for obtaining numerical solutions to the governing equations. Histories of tension at test data, and generally good agreement is observed. Errors in computed results are attributed to several areas of uncertainty, the most significant being a poorly defined boundary condition on the wave motion at the vehicle-suspension line boundary.
Poole, L. R.↗
Wave induced density modification in RF sheaths and close to wave launchers
With the return to full metal walls - a necessary step towards viable fusion machines - and due to the high power densities of current-day ICRH (Ion Cyclotron Resonance Heating) or RF (radio frequency) antennas, there is ample renewed interest in exploring the reasons for wave-induced sputtering and formation of hot spots. Moreover, there is experimental evidence on various machines that RF waves influence the density profile close to the wave launchers so that waves indirectly influence their own coupling efficiency. The present study presents a return to first principles and describes the wave-particle interaction using a 2-time scale model involving the equation of motion, the continuity equation and the wave equation on each of the time scales. Through the changing density pattern, the fast time scale dynamics is affected by the slow time scale events. In turn, the slow time scale density and flows are modified by the presence of the RF waves through quasilinear terms. Although finite zero order flows are identified, the usual cold plasma dielectric tensor - ignoring such flows - is adopted as a first approximation to describe the wave response to the RF driver. The resulting set of equations is composed of linear and nonlinear equations and is tackled in 1D in the present paper. Whereas the former can be solved using standard numerical techniques, the latter require special handling. At the price of multiple iterations, a simple ’derivative switch-on’ procedure allows to reformulate the nonlinear problem as a sequence of linear problems. Analytical expressions allow a first crude assessment - revealing that the ponderomotive potential plays a role similar to that of the electrostatic potential arising from charge separation - but numerical implementation is required to get a feeling of the full dynamics. A few tentative examples are provided to illustrate the phenomena involved.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Alternative formulation of weak magnetohydrodynamic turbulence theory
In a recent paper, the weak turbulence theory for incompressible magnetohydrodynamics is formulated by employing the method customarily applied in the context of kinetic weak plasma turbulence theory. Such an approach simplified certain mathematical procedures including achieving the closure relationship. The formulation in the above-cited paper starts from the equations of incompressible magnetohydrodynamic (MHD) theory expressed via Elsasser variables. The derivation of nonlinear wave kinetic equation therein is obtained via a truncated solution at the second-order of iteration following the standard practice. In the present paper, the weak MHD turbulence theory is alternatively formulated by employing the pristine form of incompressible MHD equation rather than that expressed in terms of Elsasser fields. The perturbative expansion of the nonlinear momentum equation is carried out up to the third-order iteration rather than imposing the truncation at the second order. Furthermore, it is found that while the resulting wave kinetic equation is identical to that obtained in the previous paper cited above, the third-order nonlinear correction plays an essential role for properly calculating derived quantities such as the total and residual energies.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Zonal-flow dynamics from a phase-space perspective
The wave kinetic equation (WKE) describing drift-wave (DW) turbulence is widely used in the studies of zonal flows (ZFs) emerging from DW turbulence. But, this formulation neglects the exchange of enstrophy between DWs and ZFs and also ignores effects beyond the geometrical-optics limit. Furthermore, we derive a modified theory that takes both of these effects into account, while still treating DW quanta (“driftons”) as particles in phase space. The drifton dynamics is described by an equation of the Wigner–Moyal type, which is commonly known in the phase-space formulation of quantum mechanics. In the geometrical-optics limit, this formulation features additional terms missing in the traditional WKE that ensure exact conservation of the total enstrophy of the system, in addition to the total energy, which is the only conserved invariant in previous theories based on the WKE. We present numerical simulations to illustrate the importance of these additional terms. The proposed formulation can be considered as a phase-space representation of the second-order cumulant expansion, or CE2.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Generation of Highly Oblique Lower Band Chorus Via Nonlinear Three-Wave Resonance
Chorus in the inner magnetosphere has been observed frequently at geomagnetically active times, typically exhibiting a two-band structure with a quasi-parallel lower band and an upper band with a broad range of wave normal angles. But recent observations by Van Allen Probes confirm another type of lower band chorus, which has a large wave normal angle close to the resonance cone angle. It has been proposed that these waves could be generated by a low-energy beam-like electron component or by temperature anisotropy of keV electrons in the presence of a low-energy plateau-like electron component. This paper, however, presents an alternative mechanism for generation of this highly oblique lower band chorus. Through a nonlinear three-wave resonance, a quasi-parallel lower band chorus wave can interact with a mildly oblique upper band chorus wave, producing a highly oblique quasi-electrostatic lower band chorus wave. This theoretical analysis is confirmed by 2-D electromagnetic particle-in-cell simulations. Furthermore, as the newly generated waves propagate away from the equator, their wave normal angle can further increase and they are able to scatter low-energy electrons to form a plateau-like structure in the parallel velocity distribution. As a result, the three-wave resonance mechanism may also explain the generation of quasi-parallel upper band chorus which has also been observed in the magnetosphere.
58 GEOSCIENCES↗
Aeroacoustics
A review of the acoustics of moving media is provided, taking into account a derivation of basic equations, wave-like solutions of acoustic equations, the solutions of acoustic equations by superposition of elementary sources, the source distribution in unbounded regions, the radiation field, energy relations, and moving sound sources. The characteristics of aerodynamic sound are considered along with the effect of solid boundaries, the effect of uniform flow, acoustic field equations, and the effects of nonuniform mean flow on the generation of sound. Attention is given to Lighthill's acoustic analogy, the solution of Lighthill's equation when no solid boundaries are present, the application of Lighthill's theory to turbulent flows, and the physics of jet noise.
Goldstein, M. E.↗