Equilibrium Energy Spectrum Research Articles

Statistics of Point Vortex Turbulence in Non-neutral Flows and in Flows with Translational and Rotational Symmetries

A theory (Esler and Ashbee in J Fluid Mech 779:275–308, 2015) describing the statistics of N freely-evolving point vortices in a bounded two-dimensional domain is extended. First, the case of a non-neutral vortex gas is addressed, and it is shown that the density of states function can be identified with the probability density function of an infinite sum of independent non-central chi-squared random variables, the details of which depend only on the shape of the domain. Equations for the equilibrium energy spectrum and other statistical quantities follow, the validity of which are verified against direct numerical simulations of the equations of motion. Second, domains with additional conserved quantities associated with a symmetry (e.g., circle, periodic channel) are investigated, and it is shown that the treatment of the non-neutral case can be modified to account for the additional constraint.

Equilibrium energy spectrum of point vortex motion with remarks on ensemble choice and ergodicity

The equilibrium energy spectrum of the point vortex system is shown to be predictable as a function of the vortex interaction (Hamiltonian) energy.

The energy spectrum in a barotropic atmosphere

Abstract. In a forced-dissipative barotropic model of the atmosphere on a spherical planet, by following mathematical techniques in (Thompson, P. D.: The equilibrium energy spectrum of randomly forced two-dimensional turbulence, Journal of the Atmospheric Sciences, 30, 1593–1598, 1973) but applying them in a novel context of the discrete spectrum on a rotating sphere, the "minus 2" energy spectrum for wavenumbers much greater than a characteristic wavenumber of the baroclinic forcing has been obtained if the forcing is taken in the simplest and most fundamental form. Some observation-based atmospheric kinetic energy spectra, with their slopes lying between "minus 2" and "minus 3" laws, are discussed from the perspective of the deduced "minus 2" energy spectrum.

Cold neutron production in solid and liquid CH 4 moderators

A numerical study of cold neutron production in cryogenic moderators composed of solid or liquid CH 4 at temperatures ranging from 20.4 to 111.7 K is made. The study is performed by a multigroup neutron transport analysis with a set of energy-averaged cross-sections (group constants) generated by the authors. Particular attention is paid to the total and angular properties of a space–energy cold-neutron flux, especially in terms of an equilibrium energy spectrum at each temperature, a storage property of high-density cold neutrons and an anisotropic emission of cold neutrons from the moderator surface.

A Theory for the Statistical Equilibrium Energy Spectrum and Heat Flux Produced by Transient Baroclinic Waves

Obtaining a physically based understanding of the variations with spatial scale of the amplitude and dispersive properties of midlatitude transient baroclinic waves and the heat flux associated with these waves is a central goal of dynamic meteorology and climate studies. Recently, stochastic forcing of highly nonnormal dynamical systems, such as arise from analysis of the equations governing perturbations to the midlatitude westerly jet, has been shown to induce large transfers of energy from the mean to the perturbation scale. In the case of a baroclinic atmospheric jet, this energy transfer to the synoptic scale produces dispersive properties, distributions of wave energy with wavenumber, and heat fluxes that are intrinsically associated with the nonnormal dynamics underlying baroclinic wave development. In this work a method for calculating the spectrum and heat flux arising from stochastic forcing is described and predictions of this theory for a model atmosphere are compared with observations. The calculated energy spectrum is found to be in remarkable agreement with observations, in contrast with the predictions of modal instability theory. The calculated heat flux exhibits a realistic distribution with height and its associated energetic cycle agrees with observed seasonal mean energetics.

Temporal Variations in Predictability

A large ensemble of predictability runs made during the course of a long equilibrium run in a two-level nonlinear quasi-geostrophic model with orography is examined in order to elucidate characteristics contributing to temporal variations in error growth. After the initial dissipation of the small-scale error, an error spectrum is developed wherein all scales grow with about the same doubling time until saturation is reached first at the smallest scales. Toward the end of the predictability runs, the error spectrum steepens toward the equilibrium energy spectrum. This error growth is largest during times of large equilibrium kinetic energy. Because of a lag relationship between the equilibrium kinetic energy and the available potential energy, it is possible to marginally predict times of large and small error growth. Removal of the orography during a forecast produces much larger and more linear growth rates characteristic of present operational forecast model errors.

Behavior of classical particles immersed in the classical electromagnetic zero-point field

This article presents a general analysis of some aspects of the interaction of classical particles with the classical electromagnetic zero-point field (cemzpf). The analysis provides a possible observational test for stochastic electrodynamics (SED). A convergence form factor derived semiclassically supports the narrow linewidth and related approximations of SED by introducing a typically sharp frequency cutoff. An extended classical charge monopole can then be shown to perform a simple jiggling motion under the influence of the cemzpf. Besides this motion (same as polarizable particles), monopolar particles also display a random walk in velocity space which leads them to ever-increasing translational kinetic energies. Hence, classical particles under the influence of the cemzpf display a conspicuous behavior because of the following well-known interrelated results: First, no velocity-dependent forces exist for classical particles moving exclusively through the cemzpf. Second, both monopolar and polarizable particles in SED are predicted to perform a random walk in velocity space due to the action of this field. Only collisions may provide a stopping mechanism. An analysis of the work of Boyer and others concerning particle collisions with walls, suggests the idea that collisions transfer energy from an unconfined gas of mutually colliding particles to the random field. Using this, a Fokker-Planck model for an unconfined gas of mutually colliding classical particles is constructed. It displays a universal equilibrium energy spectrum ${E}^{\ensuremath{-}\mathrm{const}}$ for the gas particles under the cemzpf as seen from any point fixed to co-moving coordinates. Primary cosmic rays have such an energy distribution. This motivated the proposal of a zero-point field (zpf) cosmic-ray acceleration mechanism in a previous work. Such a proposal requires a careful examination. However, methodologically speaking, one should first examine the alternative possibility that the behavior predicted in SED for classical particles does not occur in nature. If that would happen to be the case, then SED and the cemzpf concept should be critically revised. That the cemzpf concept may apparently lead to difficulties, is seen by presenting a paradoxical example where a monopolar particle moving through the cemzpf is predicted to suffer an enormous frictional force due to the surrounding zpf. The prediction obviously violates the Lorentz invariance of the cemzpf energy density spectrum. But the paradox is easily resolved by realizing the improper ultrarelativistic behavior of the Lorentz-Dirac equation which is used in the example. Extreme care must then be exerted in the use of the equations of motion of classical charged particles when moving under the influence of a zpf. The search for internal contradictions in SED, not related with the well-known renormalization and other difficulties of classical electrodynamics, has so far been unsuccessful. This and several points of rigor here and elsewhere included, are enough to indicate that the conspicuous behavior of classical particles discussed here is correctly predicted from the assumptions of SED. It is therefore proposed that this predicted behavior may serve as an observational test for the validity of SED.

Thermodynamics of two-dimensional plasmas or discrete line vortex fluids

The equilibrium properties of a two-dimensional plasma are examined theoretically within the framework of the random phase approximation. The guiding-center case is of most interest because of the formal equivalence to two-dimensional discrete line vortex fluids and because of the formal admissibility of negative temperatures. Using a judicious treatment of the periodic boundary conditions, extensive thermodynamics for nonnegative temperatures and a new value for the energy at which the temperature becomes negative are found. An explicit form for the structure function is derived and compared with the result of a Monte Carlo calculation. Good agreement is found for the negative temperature threshold prediction and for the general shape of the structure function. An equilibrium energy spectrum is calculated, which for positive temperatures, is the Debye–Hückel result; for negative temperatures it predicts an enhanced excitation of the lowest mode, corresponding to macroscopic charge separation.

The equilibrium statistical mechanics of simple quasi-geostrophic models

We have applied the methods of classical statistical mechanics to derive the inviscid equilibrium states for one- and two-layer nonlinear quasi-geostrophic flows, with and without bottom topography and variable rotation rate. In the one-layer case without topography we recover the equilibrium energy spectrum given by Kraichnan (1967). In the two-layer case, we find that the internal radius of deformation constitutes an important dividing scale: at scales of motion larger than the radius of deformation the equilibrium flow is nearly barotropic, while at smaller scales the stream functions in the two layers are statistically uncorrelated. The equilibrium lower-layer flow is positively correlated with bottom topography (anticyclonic flow over seamounts) and the correlation extends to the upper layer at scales larger than the radius of deformation. We suggest that some of the statistical trends observed in non-equilibrium flows may be looked on as manifestations of the tendency for turbulent interactions to maximize the entropy of the system.

The Equilibrium Energy Spectrum of Randomly Forced Two-Dimensional Turbulence

Abstract From a statistical mechanical treatment of an ensemble of randomly forced two-dimensional flows of a viscous fluid, we derive two independent integral constraints on the form of the equilibrium energy spectrum. With a single hypothesis about the shape-preserving properties of the spectrum, those constraints determine the spectrum to within the value of a single universal dimensionless constant. In all other respects the argument is deductive and does not depend on closure approximations or hypotheses about the process of nonlinear energy transfer. The spectrum exhibits minus third-power dependence for, small scales, but minus first-power dependence for large scales. It is in good agreement with the results of detailed numerical integrations of the Navier-Stokes equations.

Energy spectrum of electrons in the outer radiation belt

The equilibrium energy spectrum of electrons in the outer radiation belt is determined by the injection spectrum and the loss processes that operate to remove the electrons or change their energy. The loss processes considered here are ionization energy loss, multiple scattering, and electron-electron scattering; the injection spectra considered are neutron β-decay electrons and monoenergetic electrons of 780 and 20 kev. The problem is treated numerically. The results of the numerical calculation are compared with recent measurements of the outerbelt electron spectrum; it appears that neutron decays produce a reasonable fraction of the outer-belt electrons, but other processes such as acceleration may be important.