Note on Maxwell's field equations and the constitutive equations.
Maxwell field equations and constitutive equations, discussing best pedagogical formulation for electrodynamics
VELOCITY DISTRIBUTION↗
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Directional finite difference method for directly solving 3D gyrokinetic field equations with enhanced accuracy
The gyrokinetic (GK) field equation is a three-dimensional (3D) elliptic equation, but it is often simplified to a set of two-dimensional (2D) equations by assuming that the field does not vary along a specific direction. However, this simplification can introduce inevitable 0th-order numerical errors, as nonlinear mode coupling in toroidal geometry can produce undesirable harmonic modes that violate the assumption. In this work, we propose a novel directional finite difference method (FDM) with a local coordinate transformation to better resolve the target field of interest. The directional FDM can accurately solve 3D GK field equations without simplifications, which can overcome the limitations of conventional methods. The accuracy and efficiency of different FDMs are analyzed in great detail for a variety of geometries, from simple 2D Cartesian coordinates to realistic 3D curvilinear coordinates. The 0th-order numerical errors of simplified 2D GK equations were found to be more problematic for low-harmonic modes and low aspect ratio geometries such as spherical tokamaks. On the other hand, the directional 3D FDM can accurately resolve a much wider range of harmonic modes aligned to the direction of interest, including the low-harmonic modes. In conclusion, we demonstrate that the directional 3D FDM is a highly effective algorithm for solving the 3D GK field equations, achieving accuracy improvements of 10 to 100 times or more, particularly for low-harmonic modes in spherical tokamaks.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
All homogeneous solutions of einstein's field equations with incoherent matter and electromagnetic radiation.
All homogeneous solutions of Einstein field equations with incoherent matter and null electromagnetic field
EINSTEIN EQUATION↗
A $C^1$-Conforming Arbitrary-Order Two-Dimensional Virtual Element Method for the Fourth-Order Phase-Field Equation
We present a two-dimensional conforming virtual element method for the fourth-order phase-field equation. Our proposed numerical approach to the solution of this high-order phase-field (HOPF) equation relies on the design of an arbitrary-order accurate, virtual element space with $C^1$ global regularity. Such regularity is guaranteed by taking the values of the virtual element functions and their full gradient at the mesh vertices as degrees of freedom. Attaining high-order accuracy requires also edge polynomial moments of the trace of the virtual element functions and their normal derivatives. In this work, we detail the scheme construction, and prove its convergence by deriving error estimates in different norms. A set of representative test cases allows us to assess the behavior of the method.
97 MATHEMATICS AND COMPUTING↗
Mach's principle as boundary condition for Einstein's field equations
Mach principle as boundary condition for Einstein field equations
Wheeler, J. A.↗
New homogeneous solutions of einstein's field equations with incoherent matter obtained by a spinor technique.
Solution of Einstein field equations with incoherent matter for homogeneous space-time case using spinor technique, including Goedel theorem proof
EINSTEIN EQUATION↗
Mean-field equation for phase-modulated optical parametric oscillator
The widely established techniques for the generation of ultrashort optical pulses rely on passive mode locking of lasers, with the output pulse duration and emission spectrum determined by the intrinsic lifetime of laser transition in the gain medium. Due to the instantaneous nature of nonlinear gain, optical parametric oscillators (OPOs) are capable of generating optical radiation in all timescales from continuous-wave (cw) to ultrashort femtosecond regime, if driven by laser pump sources in the corresponding time domain. In the ultrashort timescale, operation of OPOs conventionally relies on mode-locked pump lasers, with the concomitant disadvantages of large footprint and high cost. At the same time, the lack of gain storage mandates the use of synchronous pumping, resulting in increased complexity. In this paper, we present the concept of phase-modulated OPO driven by cw pump laser. The approach overcomes the traditional drawbacks of ultrafast OPOs, enabling femtosecond pulse generation without the need for synchronous pumping, resulting in a simplified, compact, and cost-effective architecture using cw input pump lasers. We derive a mean-field equation for a degenerate χ ( 2 ) OPO driven by a cw laser with intracavity electro-optic modulator (EOM), and also including dispersion compensation. The equation predicts the formation of stable femtosecond pulses ( < 200 fs ) , in both normal and anomalous dispersion regimes, with a controllable repetition rate determined by the frequency of the EOM. The remarkable functionality of the proposed scheme paves the way for the development of an alternative class of widely tunable coherent femtosecond light sources in both bulk and integrated format based on χ ( 2 ) OPOs using cw pump lasers. Published by the American Physical Society 2024
Sanchez, A. D. (ORCID:0000000327630339)↗
A study of the force-field equation for the propagation of galactic cosmic rays
A new development is given of the solution of the equation of the force-field approximation for the propagation of galactic cosmic rays in the interplanetary region. It leads to simpler methods for determining the force-field parameters. A method is given for determining the separable diffusion coefficient from observations of galactic electron spectrum and near-earth electron spectra; it is shown that this diffusion coefficient is not unique but may have a periodic-like dependence upon rigidity; and the method is used to obtain diffusion coefficients for 1965 and 1968. Approximate formulae relating small changes in intensity and diffusion coefficient are developed and some applications of these noted; in one it is shown that the form of, and changes in, diffusion coefficient deduced previously for a neutron monitor event during June-September 1969 are unnecessarily constrained and therefore probably not correct.
Gleeson, L. J.↗
Projective embedding of dynamical systems: Uniform mean field equations
Herein we study embeddings of continuous dynamical systems in larger dimensions via projector operators. We call this technique PEDS, projective embedding of dynamical systems, as the stable fixed point of the original system dynamics are recovered via projection from the higher dimensional space. In this paper we provide a general definition and prove that for a particular type of rank-1 projector operator, the uniform mean field projector, the equations of motion become a mean field approximation of the dynamical system. While in general the embedding depends on a specified variable ordering, the same is not true for the uniform mean field projector. We prove a variety of results on the relationship between the spectrum of the Jacobian for fixed points in the original and in the embedded system. Direct applications of PEDS can be non-convex optimization and machine learning.
97 MATHEMATICS AND COMPUTING↗
The dynamics of fibril magnetic fields. I - Effect of flux tubes on convection. II - The mean field equations
Observations have established the general fibril state of the magnetic field at the visible surface of the sun. This extraordinary state of the field, in separate intense flux tubes, implies that the convection has a relatively permanent, closed topology, in spite of the large Reynolds number. This paper explores some of the effects of the separate flux tubes on the convective motions, generally pushing the convective cells toward aligning their downdrafts with the flux tubes so as to minimize the dissipation.
Parker, E. N.↗
Tusas: A fully implicit parallel approach for coupled phase-field equations
In this study, we develop a fully-coupled, fully-implicit approach for phase-field modeling of solidification in metals and alloys. Predictive simulation of solidification in pure metals and metal alloys remains a significant challenge in the field of materials science, as microstructure formation during the solidification process plays a critical role in the properties and performance of the solid material. Our simulation approach consists of a finite element spatial discretization of the fully-coupled nonlinear system of partial differential equations at the microscale, which is treated implicitly in time with a preconditioned Jacobian-free Newton-Krylov method. The approach is algorithmically scalable as well as efficient due to an effective preconditioning strategy based on algebraic multigrid and block factorization. We implement this approach in the open-source Tusas framework, which is a general, flexible tool developed in C++ for solving coupled systems of nonlinear partial differential equations. The performance of our approach is analyzed in terms of algorithmic scalability and efficiency, while the computational performance of Tusas is presented in terms of parallel scalability and efficiency on emerging heterogeneous architectures. We demonstrate that modern algorithms, discretizations, and computational science, and heterogeneous hardware provide a robust route for predictive phase-field simulation of microstructure evolution during additive manufacturing.
97 MATHEMATICS AND COMPUTING↗
Gravitational field equations for sources with axial symmetry and angular momentum.
Parametrization of reduced problem of stationary axially symmetric gravity fields
ANGULAR MOMENTUM↗
Can Malin's gravitational-field equations be modified to obtain a viable theory of gravity
Malin's (1975) gravitational theory, which was recently shown by Lindblom and Nester (1975) to be incorrect, is modified by means of a recently proposed method for obtaining viable gravitational theories. The resulting self-consistent theory, which is in effect a Rastall-type modification of the Einstein theory, exhibits nonconservation of momentum, yet agrees with all experimental limits known to date within the post-Newtonian approximation framework.
Smalley, L. L.↗
Application of internal gravitational field equations to geophysical measurement of G
The spherical harmonic equation for the gravitational potential inside the earth is presented. The equation satisfies Poisson's equation and converges uniformly. It obviates the need for downward continuation of the exterior potential with its attendant convergence difficulties but of course requires some knowledge of the earth's density distribution. The equation is used to derive the general expression for the geophysical measurement of the gravitational constant G made inside the earth, such as in boreholes and mine shafts. Numerical evidence is also presented to show that the long- to intermediate-wavelength gravity anomalies can masquerade as the 'fifth force' if not properly corrected for.
Rubincam, David Parry↗
Overview of Methods for Deriving the Radiative Transfer Theory from the Maxwell Equations. I: Approach Based on the Far-Field Foldy Equations
In this paper, we revisit, with further enhancements and clarifications, the self-consistent first-principles approach developed previously for deriving the vector radiative transfer theory for a discrete random medium with a sparse concentration of particles. We specifically consider the case of a plane-parallel particulate layer embedded in an otherwise homogeneous unbounded medium. The solution method is based on the far-field Foldy equations, an order-of-scattering expansion for the total field derived under the Twersky approximation, the computation of the coherent field by assuming that the positions of the particles are uncorrelated, and the ladder approximation for the coherency dyadic. The latter yields an integral equation for the diffuse specific coherency dyadic, defined through an angular spectrum repre- sentation for the coherency dyadic, which in turn, gives the vector radiative transfer equation for the diffuse specific intensity column vector. We analyze specifically the computation of the coherent field for inhomogeneous particulate media and multiple species of particles, the continuous extension of the far-field representation to the near field, the Foldy approximation, and the Foldy integral equation for the coherent field. Finally, we discuss the transition from the vector to the scalar radiative transfer equation.
Doicu, Adrian↗
A Semi-Analytical Method for Determining the Energy Release Rate of Cracks in Adhesively-Bonded Single-Lap Composite Joints
A semi-analytical method for determining the strain energy release rate due to a prescribed interface crack in an adhesively-bonded, single-lap composite joint subjected to axial tension is presented. The field equations in terms of displacements within the joint are formulated by using first-order shear deformable, laminated plate theory together with kinematic relations and force equilibrium conditions. The stress distributions for the adherends and adhesive are determined after the appropriate boundary and loading conditions are applied and the equations for the field displacements are solved. Based on the adhesive stress distributions, the forces at the crack tip are obtained and the strain energy release rate of the crack is determined by using the virtual crack closure technique (VCCT). Additionally, the test specimen geometry from both the ASTM D3165 and D1002 test standards are utilized during the derivation of the field equations in order to correlate analytical models with future test results. The system of second-order differential field equations is solved to provide the adherend and adhesive stress response using the symbolic computation tool, Maple 9. Finite element analyses using J-integral as well as VCCT were performed to verify the developed analytical model. The finite element analyses were conducted using the commercial finite element analysis software ABAQUS. The results determined using the analytical method correlated well with the results from the finite element analyses.
Yang, Charles↗
Spatial power spectra of the crustal geomagnetic field and core geomagnetic field
Equations providing numerical values of the geomagnetic field spherical harmonic spatial power spectrum as defined by Lowes (1966, 1974) are obtained and this power spectrum is related to various other power spectra. Equations relating the spherical harmonic spatial power spectrum to average great circle power spectra for components of the vector magnetic field in the radial direction, along the great circle track and perpendicular to the first two directions are derived under the assumption that the sources of the field are internal. A statistical model for the crustal and core geomagnetic fields is proposed and used to derive equations for the expected main and crustal spherical harmonic power spectra. The model equations are then compared with observations to determine a scale factor which is then used to obtain an estimate for the core radius and a great circle power spectrum for the field component perpendicular to the great circle and radial directions which are in good agreement with observations. The predicted spherical harmonic power spectrum for the crustal field is found to be consistent with POGO satellite and aircraft data. Other possible models for the crustal and core geomagnetic fields are also briefly considered.
Mcleod, M. G.↗
Which electromagnetic equations apply in rotating coordinates
It was discovered some years ago by Schiff that two equations for fields in vacuum do not carry over without change from an inertial frame to a frame with rotating axes of space coordinates, even for a region with all velocities much lower than the speed of light. However, the belief that all four of the field equations are invariant under such conditions is still prevalent and causes misconceptions in physical applications, including astrophysical and geophysical ones. The purpose of the present paper is therefore to call attention to Schiff's discovery, discussing its basis and its extension to fields in material media, and to interpret the additional terms that must be added to the equations in order to obtain valid transformations to rotating axes of coordinates.
Webster, D. L.↗