Average Energy Flux Research Articles

Structural and stochastic analysis of a zero‐dimensional climate system

AbstractAn ‘almost trivial’ climate system of geometrical dimension zero is analysed, the complexity of which has been reduced to a minimum. It can be simply described as the globally averaged energy flux balance between infrared emission and solar heat input, expanded by a linear albedo‐temperature feedback. This nonlinear and time‐dependent climate model is formulated as a gradient system of a potential and can be analysed without explicit time integration. It includes many of the results which are also exhibited by one‐dimensional energy balance models. Two equilibrium solutions appear. The stable one is characterized by the interglacial, whereas the unstable equilibrium defines a lower bound for temperature (state variable) changes which the system can absorb. Beyond a threshold of an external parameter combination (fold catastrophe) no equilibria exist so that the system attains a ‘deep freeze’ climate situation. A −2 power law describes the linear response of the (internally stable) system to weather fluctuations.

Structural and stochastic analysis of a zero-dimensional climate system

An ‘almost trivial’ climate system of geometrical dimension zero is analysed, the complexity of which has been reduced to a minimum. It can be simply described as the globally averaged energy flux balance between infrared emission and solar heat input, expanded by a linear albedo-temperature feedback. This nonlinear and time-dependent climate model is formulated as a gradient system of a potential and can be analysed without explicit time integration. It includes many of the results which are also exhibited by one-dimensional energy balance models. Two equilibrium solutions appear. The stable one is characterized by the interglacial, whereas the unstable equilibrium defines a lower bound for temperature (state variable) changes which the system can absorb. Beyond a threshold of an external parameter combination (fold catastrophe) no equilibria exist so that the system attains a ‘deep freeze’ climate situation. A −2 power law describes the linear response of the (internally stable) system to weather fluctuations.

The Effect of a Magnetic Field on the Onset of Thermal Convection when Constant Flux Boundary Conditions Apply

The energy passing through the boundaries of a convective zone in a star, in an outward radial direction per second, must in total be equal to the luminosity of the star, provided of course that this zone does not encompass nuclear energy producing regions. If, in an endeavour to establish the nature of the convective motions in this zone, a section of the zone is modelled by considering steady cellular convection in a horizontal layer of Boussinesq fluid, it is on the basis of an average energy flux being imposed. Moreover, the sum of the conductive and convective heat fluxes is represented by the Nusselt number, which is specified and constant for the zone.

Problem of the average energy flux in a fluctuating medium

Problem of the average energy flux in a fluctuating medium

The ratio of specific heats for postshock plasmas of a detached bow shock: An MHD model

The empirical relationship between the standoff distance of a detached bow shock (generated by the flow of a supersonic gas past an impenetrable obstacle), the size of the obstacle, the Mach number of the gas, and the ratio of specific heats has been generalized to include the magnetic field. The value of the ratio of specific heats (gamma-prime) in the postshock plasma has been calculated in terms of the preshock Alfvenic and sonic Mach numbers and orientation of the magnetic field. The empirical relationship is further generalized by taking into consideration the normal momentum and energy flux due to waves and/or turbulence and/or heat flow in association with high Mach number shocks. The computed value of gamma prime is substantially modified in comparison with that given by the MHD or the gas dynamic model. For this generalized model the computed gamma prime can be considered to be a more precise thermodynamic quantity, since the macroscopic parameters of the plasma have been separated out. Application of this empirical relationship to the earth's bow shock has been given.

Seismic regionalization in North America based on Average Regional Seismic Hazard Index

AbstractCumulative Seismic Hazard Index (CSHI) is the logarithm of the sum of the seismic energies experienced at any location. Average Regional Seismic Hazard Index (ARSHI) is the regional average of Cumulative Hazard Index values normalized to 100 years. Both can be expressed in the same units as Mercalli intensity. ARSHI differs from previously proposed measures of seismicity in that it compensates for variations in the completeness of the historic record and for variations in the rate of attenuation of seismic energy with distance. More rapid attenutation of intensity with distance in the West than in the East reduces the difference in hazard between these regions.In North America between 30° and 50° N latitude, ARSHI varies from 8.44 in the Western Nevada to 5.28 in the Canadian Shield and Plains provinces. Since ARSHI is measured on a logarithmic scale, this is a range of a factor of over 1,000 in average energy flux.A trough of low seismicity between areas of high seismicity roughly coincides with areas of recent volcanic activity and high heat flow in parts of the Great Basin, the Columbia Plateau and adjoining areas. This low seismicity is interpreted as possibly being due to weakening of the crustal rocks by higher than normal temperatures. This interpretation would mean that low seismicity in this area is not evidence of a lack of tectonic activity. It is consistent with either a rising convection cell, a branch of the East Pacific Rise system underlying this area, or with igneous upwelling over an old subduction zone.

Nonlinear Thermal Convection in a Layer with Imposed Energy Flux

Results are reported of an investigation into the effect of the chosen boundary conditions on the steady finite-amplitude convective motions in a layer in which the average energy flux is imposed. The boundary conditions are chosen with a view to the application of the results to solar granulation and supergranulation. It is shown that, at high Rayleigh numbers, solutions do in fact exist for which there is no modulation in the energy flux and little fluctuation in the temperature across the boundaries.

Energy flux for laminar waves of viscous fluid

An expression of the average energy flux to first order approximation for laminar waves of viscous fluid on uniform depth is obtained. This flux, compared with the one given by the conventional approximations, presents an additional term which is significant in the dynamical description of the boundary layer.

Transformation of the wave spectrum in a medium having smooth space-time fluctuations

In conclusion let us sum up concerning the question of the applicability limits of the quasistatic approximation. For u0T≫L the equations for the phase and frequency dispersions, which determine the width of the time spectrum of the waves (specifically, (2.9)), and likewise the equations for the arrival angle (2.6) and the amplitudes of the waves (3.5) are in agreement with the results of the quasistatic calculation with a sufficient degree of accuracy. For u0T≼L there is naturally no such correspondence any more, and here the dispersion of the medium plays an essential role. Let us emphasize the fact that in the presence of dispersion the delay effect (including the Eqs. (2.13) and (2.14) for the phase fluctuations) is determined by the group velocity of the waves. However in considering such questions as the variation of the average energy flux in the wave and the shape of the power spectrum the quasistatic approach is inapplicable in principle even for fulfillment of the condition u0T≫L.

Energy and mass content of high-speed solar-wind streams

The mass and energy flux densities of the solar wind measured by the Vela 3, Vela 4, and Explorer 34 satellites between July 1965 and October 1969 have been examined. Three-hour averages were used to study the properties of large-scale high-speed streams, and 27-day averages were employed in an attempt to identify long-term trends during the rising part of the current 11-year solar cycle. If the cross-sectional area of a stream is estimated by assuming that its extent in heliocentric latitude is the same as its extent in longitude, the total energy (at 1 AU) and mass flux can be calculated by integrating the excess mass and energy flux density above the predisturbance level during corotation of the stream past the observing satellite. The average energy and mass fluxes for dominant high-speed streams are found to be 9 × 1025 erg/sec and 8 × 109 g/sec, respectively. The total mass and energy output of all streams emanating from the sun over a typical 27-day interval, assuming that all last for the entire 27 days, is estimated to be less than 10% (∼7%) of the total solar-wind output from the same range of heliocentric latitudes (generally ±15°). Both large-scale streams and interplanetary shocks are found to contribute relatively little to 27-day average values of either mass or energy flux density. Thus, even though the amount of mass or energy or both per disturbance appeared to increase with increasing solar activity, the solar-wind energy and mass flux density averaged over a solar rotation remained remarkably constant.

Saturation Calculation for Light Propagation in the Turbulent Atmosphere*

Experiments indicate that the variance of irradiance fluctuations of a wave propagating through a turbulent atmosphere saturates with increasing path length or turbulence strength. In addition, many experimental data indicate that the saturation value depends on the path length. In this paper we calculate the variance of the irradiance fluctuations by a method originally proposed by Keller, but we view it in a conceptually different manner. The reason that this method is reconsidered is that it leads to solutions whose validity is essentially independent of distance for the coherent field and average energy flux. Our calculations yield a saturation curve that agrees with the experimental data, with a saturation value that depends on both the wavelength and the length of the propagation path. It also leads to a saturation curve with a pronounced maximum for the case of constant turbulence conditions and variable path length. Finally, the amplitude fluctuations are found to be Rayleigh distributed in the limit of infinite path length.

Some average properties of auroral electron precipitation as determined from satellite observations

The total energy flux of electrons precipitating on the auroral zones has been measured by particle detectors aboard a low-altitude, polar-orbiting satellite for the 5-day period from October 30 to November 4, 1963. The observations were made at local times of approximately 0100 and 1300 hours. Precipitating electrons were observed on almost every transit of the auroral zone. Parameters are defined characterizing the intensity, the angular distribution, and the energy distribution of the precipitating electrons. The relationships among these parameters and their dependence on Kp are examined. The averaged total precipitated energy flux was found to be approximately 5 times more intense on the night side than on the day side of the auroral zone, and the averaged energy distribution was found to be softer on the night side. The averaged angular distributions were found to be generally anisotropic, peaking at the higher pitch angles. The intensity of the precipitation was found to increase, the angular distribution became more isotropic, and the averaged spectral hardness increased as Kp increased. The averaged energy and angular distributions were also found to undergo significant changes that occurred on a global scale and persisted for many hours, thus indicating that large-scale, rather than local, processes were responsible for these changes. The total auroral-zone energy input is computed for the period of the experiment, and by comparison with published measurements for the average solar wind properties, an ‘efficiency’ of the auroral mechanism is calculated. This efficiency varies approximately from 0.01% to 0.2% as Kp varies from 0 to 5.

Solar flares as resulting from the temporary interruption of energy flow to the corona: A case of hydromagnetic resonance

The hypothesis that solar flares may be caused by a choking off of the normal energy flux to the corona by the strong closed magnetic fields of a plage is examined. If the energy flux into a plage from the photosphere is of the order of 108 ergs/cm2 sec, and if a substantial fraction of this energy is carried in the form of Alfven waves, then the rate of dissipation of the waves is slower than the rate at which energy is injected. Since the waves must propagate along the magnetic field and cannot reenter the photosphere, they must remain within the plage; hence, the magnetic and kinetic energy in a small-scale motion (either waves, turbulence, or high-energy particles) must increase with time, eventually causing disruption of the volume when the small-scale energy density exceeds the energy in the mean field. It is believed that the unusually broad wings in the emission lines represent evidence of this phenomenon. The accumulation of waves is manifested as a resonance which occurs initially only at discrete locations in the magnetic field, but later is expected to involve the whole flare volume. The response of a typical volume of flare dimensions due to a trapping of the normal wave supply to the corona is studied through use of the virial equation. For magnetic fields typical of a plage, the region expands in a time scale of 102–103 sec, with a velocity in the neighborhood of 10–20 km/sec. Small-scale velocities within the region, however, have reached 100–300 km/sec, indicating that almost all the energy in the flare resides in small-scale forms. The energy density of the flare region exhibits a behavior much more explosive than the expansion rate. There is a rapid rise to maximum in 102 sec or less, and a slow subsequent decline taking about 103–104 sec due to the dilution of energy caused by expansion of the region. The predicted temporal behavior of the energy density coincides qualitatively with the light curves observed during flares, and it is suggested that the rise and decline of the energy density is to be associated with the optical flare. The total flare is defined as the time required for the energy density of the chromosphere and corona to return to the pre-flare state. During this time (about one hour) a large flare can derive the necessary 1032 ergs from normal photospheric energy output.

Theory of Optical Harmonic Generation at a Metal Surface

It is shown that a light wave of the high intensity obtainable from lasers produces a sufficiently strong nonlinear polarization on a reflecting metal surface to result in an observable amount of second harmonic generation. The analysis is based upon a self-consistent set of Maxwell's equations and the classical Boltzmann equation, respectively, for the electromagnetic fields and the distribution function of the conduction electrons. The conduction electrons are considered to be completely free except for a potential barrier at the metal surface, and the equations are solved for the fields varying with the frequency $\ensuremath{\omega}$ of the incident wave, and also for the fields varying with the frequency $2\ensuremath{\omega}$ in the approximation where the surface barrier can be taken as a step potential. The effect of the incident light wave is treated as a perturbation to the motion of the electrons and the frequency $\ensuremath{\omega}$ is assumed to be less than half the plasma frequency ${\ensuremath{\omega}}_{p}$ so that neither the fundamental nor the second harmonic wave can lead to plasma resonance. The part of the polarization varying as ${e}^{\ensuremath{-}2i\ensuremath{\omega}t}$ which is quadratic in the incident field is found to have the form ${\mathrm{P}}_{2}(\mathrm{NL})=\ensuremath{\alpha}({\mathrm{E}}_{1}\ifmmode\times\else\texttimes\fi{}{\mathrm{H}}_{1})+\ensuremath{\beta}{\mathrm{E}}_{1}\mathrm{div}{\mathrm{E}}_{1},$ where ${\mathbf{E}}_{1}$ and ${\mathbf{H}}_{1}$ are, respectively, the electric and magnetic fields varying as ${e}^{\ensuremath{-}i\ensuremath{\omega}t}$ and where the magnitudes of the coefficients $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ have been determined. Since div ${\mathbf{E}}_{1}$ differs from zero only near the surface of the metal, the second term in ${\mathbf{P}}_{2}$ (NL) can be considered as a surface contribution in contrast to the volume contribution of the first term. It is shown that these two terms give rise to comparable effects of second harmonic generation. The ratio of the average energy flux reflected with frequency $2\ensuremath{\omega}$ from the surface to the incident flux is found to be of the order of magnitude ${(\frac{e|{E}_{\mathrm{inc}}|}{\mathrm{mc}{\ensuremath{\omega}}_{p}})}^{2}$, where ${E}_{\mathrm{inc}}$ is the amplitude of the incident electric vector.

PERIODIC GRAVITY WAVES OVER A GENTLE SLOPE AT A THIRD ORDER OF APPROXIMATION

The concepts of energy flux and group velocity for nonlinear periodic gravity waves are discussed The average energy flux, average energy per wavelength and "group velocity" are calculated for irrotational periodic gravity waves at a third order of approximation. Then the principle of conservation of transmitted energy between wave orthogonals is applied to the same order of approximation for determining the variation of wave height m decreasing depth. The results are presented as nomographs. It is seen that the "shoaling coefficient" needs to be expressed at least at a third order of approximation to account for experimental results.

Energy-Flux Operator for a Lattice

A systematic derivation of the energy-flux operator for a three-dimensional lattice is given. The treatment is based on the general expressions for the energy flux which are valid for all phases of matter; a short derivation of these expressions, making no restrictions to two-body forces, is presented. The average energy flux is transformed to the phonon representation, and it is shown that the diagonal contribution from the harmonic forces has the familiar form $\ensuremath{\Sigma}{\mathrm{k}s}^{}{N}_{\mathrm{k}s}\ensuremath{\hbar}{\ensuremath{\omega}}_{\mathrm{k}s}{\mathrm{v}}_{\mathrm{k}s}$. There are, in addition, nondiagonal contributions to the energy flux, even in the harmonic approximation. The significance of these corrections is discussed. The contributions to the average flux from the anharmonic forces and from lattice imperfections are also treated. Finally, the problem of forming wave packets of the plane-wave normal modes to obtain an expression for the local energy flux is considered.

On Scattering and Reflection of Sound by Rough Surfaces

We consider random distributions of arbitrary identical protuberances on free or rigid base planes, and derive the approximate reflection coefficients R = |1 + Z1 − Z|2 and differential scattering cross sections per unit area σ(s,ki) = ρk2|f(s,ki)1 − Z|2. Here Z = πρf(k0,ki)k2n ⋅ k0, where f(s,ki) is the scattering amplitude of an isolated protuberance. ki, k0, and s are the directions of incidence, specular reflection (with respect to the base plane), and observation; n is the normal of the base; ρ is the average number of scatterers in unit area, and k = 2π/λ. (For a two-dimensional “striated” surface we divide Z by π/k, and σ by π/2k.) We find for example, that if the horizon angle β approaches zero (i.e., near grazing incidence), then the reflection coefficients approach unity minus terms of the order β, while the back scattering vanishes like β1, β2 for a free, rigid base. Explicit forms are given for arbitrary hemispheres and circular semicylinders, and for the limiting cases of free and rigid surfaces with scatterer radius very small and very large compared to wavelength. The analysis is based on a Green's function formulation of the problem of a single configuration; R and σ follow from an approximation of the ensemble averaged energy flux which takes account of multiple coherent scattering. (The transmission problem of a “random screen” is treated simultaneously.) An elementary derivation is also given, and extensions to distributions of nonidentical scatterers are made.