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When assessing the scattering of radiation belt electrons by fast magnetosonic (MS) waves, it is traditionally assumed that the waves follow the MS/whistler branch of the cold plasma dispersion relation (CPDR) in magnetohydrodynamics. However, MS waves are essentially ion Bernstein modes following a distinct kinetic dispersion relation. This study calculates the MS wave‐induced electron diffusion rates with the kinetic dispersion relation for the first time and compares the results with that obtained with the CPDR. It is found that the kinetic effects lead to a lower minimum resonant energy around 100 eV and a broader resonant pitch angle range. Kinetic effects also result in power spectral density attenuation when transforming wave frequency spectra into wavenumber spectra, so the diffusion rates are overall smaller than the ones obtained using the CPDR. Our results demonstrate that kinetic effects can significantly affect the role that MS waves play in the radiation belt dynamics. Plain Language Summary Magnetosonic (MS) waves belong to the kinetic ion Bernstein modes essentially. But when the cold plasma is dominating, the waves also approximately follow the MS/whistler branch of the cold plasma dispersion relation (CPDR) in magnetohydrodynamics. Subsequently, studies of the electron scattering by MS waves have traditionally assumed the CPDR for simplicity. Motivated by recent studies which involved both satellite observations and kinetic theory revealing that the lower harmonic MS waves clearly follow the kinetic dispersion relation, we assess how the differences between the kinetic and cold plasma dispersion relations affect the MS wave‐induced electron scattering rates. Our results indicate that the kinetic dispersion relation produces relatively lower parallel phase speeds for MS waves, leading to a lower minimum resonant energy and subsequently a broader resonant pitch angle range for electrons (of a given energy). The kinetic effects also decrease the overall diffusion rates by attenuating the wave power spectral density in wavenumber space when mapped from a given frequency spectrum. Key Points Linear kinetic dispersion relation indicates lower phase speeds and a broader range of group speeds for fast magnetosonic (MS) waves The lower phase speeds of MS waves result in a broader range of resonant pitch angles and lower minimum resonant energies of electrons Kinetic effects reduce the wavenumber power spectral density and thus result in smaller electron diffusion rates

Mathematical models of numerous processes in continuum mechanics (CM), plasma physics (PP) and astrophysics (AP) are partial differential equations (PDEs). When developing computer models, these equations are replaced by discrete equations that are solved numerically. In order to investigate mathematical and numerical models of CM, PP and AP, the technique of constructing dispersion relations has been developed. Using dispersion relations allows one to derive particular solutions to systems of PDEs, to investigate the stability of solutions for continuous and discrete models, to estimate the order of approximation and the rate of convergence for discrete models, and to establish of the optimal numerical parameters of a discrete model. Dispersion relations describe wave processes (i.e., processes of perturbation transfer with a velocity different from the velocity of matter) in media. The classical dispersion relation is a nonlinear algebraic equation (relating the wave parameters, namely the wave number and the wave frequency, ), which corresponds to a continuous system of PDEs. There is a technique that allows one to derive a dispersion relation (classical or approximate, respectively) for a continuous or discrete CM, PP, and AP model. This paper presents a symbolic computer algebra library, developed by the authors, which automates this technique. The current version supports the use of nonstationary models with a single spatial variable both for continuum and finite-difference notation. The library is written in Python using the SymPy symbolic computing package and is available at https://pypi.org/project/symdr/ .

Causality: Cause and effect. In classical physics, an effect cannot occur before its cause. In Einstein's theory of special relativity, causality means that an effect cannot occur from a cause that is not in the back (past) light cone of that event.

In this work, the analytical properties of dispersion relations of the equation of internal gravity waves with benchmark and arbitrary distributions of buoyancy frequency are investigated. To solve the problem analytically, the benchmark distribution of the buoyancy frequency is used, which is known from applied oceanological calculations in the presence of seasonal thermocline. Implicit forms of dispersion dependences are obtained; they are expressed through the Bessel function of real index. For nonzero wave numbers, an asymptotic method of studying the dispersion relation is proposed based on constructing uniform asymptotics of the Bessel functions for large values of real index and argument, expressed through the Airy functions. For an arbitrary distribution of buoyancy frequency, the asymptotic representations of dispersions relationships at small wavenumbers are obtained by means of the perturbation method and the WKB method. The solutions constructed in this work allow further computing the amplitude-phase characteristics for the fields of internal gravity waves with benchmark and arbitrary distributions of the buoyancy frequency.

A class of problems of wave propagation in waveguides consisting of one or several layers that are characterized by linear variation of the squared refractive index along the normal to the interfaces between them is considered in this paper. In various problems arising in practical applications, it is necessary to efficiently solve the dispersion relations for such waveguides in order to compute horizontal wavenumbers for different frequencies. Such relations are transcendental equations written in terms of Airy functions, and their numerical solutions may require significant computational effort. A procedure that allows one to reduce a dispersion relation to an ordinary differential equation written in terms of elementary functions exclusively is proposed. The proposed technique is illustrated on two cases of waveguides with both compact and non-compact cross-sections. The developed reduction method can be used in applications such as geoacoustic inversion.

In numerical simulations of complex flows with discontinuities, it is necessary to use nonlinear schemes. The spectrum of the scheme used has a significant impact on the resolution and stability of the computation. Based on the approximate dispersion relation method, we combine the corresponding spectral property with the dispersion relation preservation proposed by De and Eswaran (J Comput Phys 218:398–416, 2006) and propose a quasi-linear dispersion relation preservation (QL-GRP) analysis method, through which the group velocity of the nonlinear scheme can be determined. In particular, we derive the group velocity property when a high-order Runge–Kutta scheme is used and compare the performance of different time schemes with QL-GRP. The rationality of the QL-GRP method is verified by a numerical simulation and the discrete Fourier transform method. To further evaluate the performance of a nonlinear scheme in finding the group velocity, new hyperbolic equations are designed. The validity of QL-GRP and the group velocity preservation of several schemes are investigated using two examples of the equation for one-dimensional wave propagation and the new hyperbolic equations. The results show that the QL-GRP method integrated with high-order time schemes can determine the group velocity for nonlinear schemes and evaluate their performance reasonably and efficiently.

We use the formalism of tensor network states to investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low-energy excitations. In particular, we show that the matrix product state transfer matrix (MPS-TM)-a central object in the computation of static correlation functions-provides important information about the location and magnitude of the minima of the low-energy dispersion relation(s), and we present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low-energy spectrum of the system and the form of the static correlation functions. Finally, we discuss how the MPS-TM connects to the exact quantum transfer matrix of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of the MPS, which allows one to reinterpret variational MPS techniques (such as the density matrix renormalization group) as an application of Wilson's numerical renormalization group along the virtual (imaginary time) dimension of the system.

Low-energy partial-wave πN scattering data is reexamined with the help of the production representation of partial-wave S matrix, where branch cuts and poles are thoroughly under consideration. The left-hand cut contribution to the phase shift is determined, with controlled systematic error estimates, by using the results of O( p 3) chiral perturbative amplitudes obtained in the extended-onmass- shell scheme. In S 11 and P 11 channels, severe discrepancies are observed between the phase shift data and the sum of all known contributions. Statistically satisfactory fits to the data can only be achieved by adding extra poles in the two channels. We find that a S 11 resonance pole locates at z r = (0.895 0.081)−(0.164 0.023)i GeV, on the complex s-plane. On the other hand, a P 11 virtual pole, as an accompanying partner of the nucleon bound-state pole, locates at z v = (0.966 0.018) GeV, slightly above the nucleon pole on the real axis below threshold. Physical origin of the two newly established poles is explored to the best of our knowledge. It is emphasized that the O( p 3) calculation greatly improves the fit quality comparing with the previous O( p 2) one.

In this work we establish two wave modes for the integrable fifth-order Korteweg-de Vries (TfKdV) equations. We determine necessary conditions of the nonlinearity and dispersion parameters of the equation for multiple-soliton solutions to exist. We apply the simplified Hirota method to derive multiple-soliton solutions under these conditions. We also examine the dispersion relations and the phase shifts of the developed models.

The addition of clay minerals in drilling fluids modifies the dispersion's viscosity. In this article, scientific advances related to the use of clays and clay minerals (bentonite, palygorskite, sepiolite and mixtures of clay minerals) in drilling fluids are summarized and discussed based on their specific structure, rheological properties, applications, prevailing challenges and future directions. The rheological properties of drilling fluids are affected by the temperature, type of electrolytes, pH and concentration of clay minerals. Bentonites are smectite-rich clays often used in drilling fluids, and their composition varies from deposit to deposit. Such variations significantly affect the behaviour of bentonite-based drilling fluids. Palygorskite is suitable for use in oil-based drilling fluids, but the gelation and gel structures of palygorskite-added drilling fluids have not received much attention. Sepiolite is often used in water-based drilling fluids as a rheological additive. Dispersions containing mixtures of clays including bentonite, kaolin, palygorskite and sepiolite are used in drilling fluids requiring specific features such as high-density drilling fluids or those used in impermeable slurry walls. In these cases, the surface chemistry–microstructure–property relationships of mixed-clay dispersions need to be understood fully. The prevailing challenges and future directions in drilling fluids research include safety, ‘green’ processes and high-temperature and high-pressure-resistant clay minerals.