Homogeneous quantum electrodynamic turbulence

The electromagnetic field equations and Dirac equations for oppositely charged wave functions are numerically time-integrated using a spatial Fourier method. The numerical approach used, a spectral transform technique, is based on a continuum representation of physical space. The coupled classical field equations contain a dimensionless parameter which sets the strength of the nonlinear interaction (as the parameter increases, interaction volume decreases). For a parameter value of unity, highly nonlinear behavior in the time-evolution of an individual wave function, analogous to ideal fluid turbulence, is observed. In the truncated Fourier representation which is numerically implemented here, the quantum turbulence is homogeneous but anisotropic and manifests itself in the nonlinear evolution of equilibrium modal spatial spectra for the probability density of each particle and also for the electromagnetic energy density. The results show that nonlinearly interacting fermionic wave functions quickly approach a multi-mode, dynamic equilibrium state, and that this state can be determined by numerical means

A Fourier collocation time domain method for numerically solving Maxwell's equations

A new method for solving Maxwell's equations in the time domain for arbitrary values of permittivity, conductivity, and permeability is presented. Spatial derivatives are found by a Fourier transform method and time integration is performed using a second order, semi-implicit procedure. Electric and magnetic fields are collocated on the same grid points, rather than on interleaved points, as in the Finite Difference Time Domain (FDTD) method. Numerical results are presented for the propagation of a 2-D Transverse Electromagnetic (TEM) mode out of a parallel plate waveguide and into a dielectric and conducting medium

Electromagnetic potential vectors and the Lagrangian of a charged particle

Maxwell's equations can be shown to imply the existence of two independent three-dimensional potential vectors. A comparison between the potential vectors and the electric and magnetic field vectors, using a spatial Fourier transformation, reveals six independent potential components but only four independent electromagnetic field components for each mode. Although the electromagnetic fields determined by Maxwell's equations give a complete description of all possible classical electromagnetic phenomena, potential vectors contains more information and allow for a description of such quantum mechanical phenomena as the Aharonov-Bohm effect. A new result is that a charged particle Lagrangian written in terms of potential vectors automatically contains a 'spontaneous symmetry breaking' potential

Nonlinear dynamics and control of a vibrating rectangular plate

The von Karman equations of nonlinear elasticity are solved for the case of a vibrating rectangular plate by meams of a Fourier spectral transform method. The amplification of a particular Fourier mode by nonlinear transfer of energy is demonstrated for this conservative system. The multi-mode system is reduced to a minimal (two mode) system, retaining the qualitative features of the multi-mode system. The effect of a modal control law on the dynamics of this minimal nonlinear elastic system is examined

On weak and strong magnetohydrodynamic turbulence

Recent numerical and observational studies contain conflicting reports on the spectrum of magnetohydrodynamic turbulence. In an attempt to clarify the issue we investigate anisotropic incompressible magnetohydrodynamic turbulence with a strong guide field $B_0$. We perform numerical simulations of the reduced MHD equations in a special setting that allows us to elucidate the transition between weak and strong turbulent regimes. Denote $k_{\|}$, $k_\perp$ characteristic field-parallel and field-perpendicular wavenumbers of the fluctuations, and $b_{\lambda}$ the fluctuating field at the scale $\lambda\sim 1/k_{\perp}$. We find that when the critical balance condition, $k_{\|}B_0\sim k_{\perp} b_{\lambda}$, is satisfied, the turbulence is strong, and the energy spectrum is $E(k_{\perp})\propto k^{-3/2}_{\perp}$. As the $k_{\|}$ width of the spectrum increases, the turbulence rapidly becomes weaker, and in the limit $k_{\|}B_0\gg k_{\perp} b_{\lambda}$, the spectrum approaches $E(k_{\perp})\propto k_{\perp}^{-2}$. The observed sensitivity of the spectrum to the balance of linear and nonlinear interactions may explain the conflicting numerical and observational findings where this balance condition is not well controlled.Comment: 4 pages, 2 figure

Anisotrophy in MHD turbulence due to a mean magnetic field

The development of anisotropy in an initially isotropic spectrum is studied numerically for two-dimensional magnetohydrodynamic (MHD) turbulence. The anisotropy develops due to the combined effects of an externally imposed dc magnetic field and viscous and resistive dissipation at high wave numbers. The effect is most pronounced at high mechanical and magnetic Reynolds numbers. The anisotropy is greater at the higher wave numbers.;The statistical structure of two-dimensional MHD turbulence is also considered. It is shown that the three known rugged invariants of the isotropic case reduce to two for the anisotropic case. Randomness and ergodicity are also briefly discussed

The Statistical Mechanics of Ideal MHD Turbulence

Turbulence is a universal, nonlinear phenomenon found in all energetic fluid and plasma motion. In particular. understanding magneto hydrodynamic (MHD) turbulence and incorporating its effects in the computation and prediction of the flow of ionized gases in space, for example, are great challenges that must be met if such computations and predictions are to be meaningful. Although a general solution to the "problem of turbulence" does not exist in closed form, numerical integrations allow us to explore the phase space of solutions for both ideal and dissipative flows. For homogeneous, incompressible turbulence, Fourier methods are appropriate, and phase space is defined by the Fourier coefficients of the physical fields. In the case of ideal MHD flows, a fairly robust statistical mechanics has been developed, in which the symmetry and ergodic properties of phase space is understood. A discussion of these properties will illuminate our principal discovery: Coherent structure and randomness co-exist in ideal MHD turbulence. For dissipative flows, as opposed to ideal flows, progress beyond the dimensional analysis of Kolmogorov has been difficult. Here, some possible future directions that draw on the ideal results will also be discussed. Our conclusion will be that while ideal turbulence is now well understood, real turbulence still presents great challenges

Statistical Theory of the Ideal MHD Geodynamo

A statistical theory of geodynamo action is developed, using a mathematical model of the geodynamo as a rotating outer core containing an ideal (i.e., no dissipation), incompressible, turbulent, convecting magnetofluid. On the concentric inner and outer spherical bounding surfaces the normal components of the velocity, magnetic field, vorticity and electric current are zero, as is the temperature fluctuation. This allows the use of a set of Galerkin expansion functions that are common to both velocity and magnetic field, as well as vorticity, current and the temperature fluctuation. The resulting dynamical system, based on the Boussinesq form of the magnetohydrodynamic (MHD) equations, represents MHD turbulence in a spherical domain. These basic equations (minus the temperature equation) and boundary conditions have been used previously in numerical simulations of forced, decaying MHD turbulence inside a sphere [1,2]. Here, the ideal case is studied through statistical analysis and leads to a prediction that an ideal coherent structure will be found in the form of a large-scale quasistationary magnetic field that results from broken ergodicity, an effect that has been previously studied both analytically and numerically for homogeneous MHD turbulence [3,4]. The axial dipole component becomes prominent when there is a relatively large magnetic helicity (proportional to the global correlation of magnetic vector potential and magnetic field) and a stationary, nonzero cross helicity (proportional to the global correlation of velocity and magnetic field). The expected angle of the dipole moment vector with respect to the rotation axis is found to decrease to a minimum as the average cross helicity increases for a fixed value of magnetic helicity and then to increase again when average cross helicity approaches its maximum possible value. Only a relatively small value of cross helicity is needed to produce a dipole moment vector that is aligned at approx.10deg with the rotation axis

Magnetic Helicity and Planetary Dynamos

A model planetary dynamo based on the Boussinesq approximation along with homogeneous boundary conditions is considered. A statistical theory describing a large-scale MHD dynamo is found, in which magnetic helicity is the critical paramete

Broken Symmetries and Magnetic Dynamos

Phase space symmetries inherent in the statistical theory of ideal magnetohydrodynamic (MHD) turbulence are known to be broken dynamically to produce large-scale coherent magnetic structure. Here, results of a numerical study of decaying MHD turbulence are presented that show large-scale coherent structure also arises and persists in the presence of dissipation. Dynamically broken symmetries in MHD turbulence may thus play a fundamental role in the dynamo process