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A Solution to the Fundamental Linear Fractional Order Differential Equation

This paper provides a solution to the fundamental linear fractional order differential equation, namely, (sub c)d(sup q, sub t) + ax(t) = bu(t). The impulse response solution is shown to be a series, named the F-function, which generalizes the normal exponential function. The F-function provides the basis for a qth order "fractional pole". Complex plane behavior is elucidated and a simple example, the inductor terminated semi- infinite lossy line, is used to demonstrate the theory.

Hartley, Tom T.↗

Legendre-Tau approximation for functional differential equations. Part 3: Eigenvalue approximations and uniform stability

The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.

Ito, K.↗

The Ground Flash Fraction Retrieval Algorithm Employing Differential Evolution: Simulations and Applications

The ability to estimate the fraction of ground flashes in a set of flashes observed by a satellite lightning imager, such as the future GOES-R Geostationary Lightning Mapper (GLM), would likely improve operational and scientific applications (e.g., severe weather warnings, lightning nitrogen oxides studies, and global electric circuit analyses). A Bayesian inversion method, called the Ground Flash Fraction Retrieval Algorithm (GoFFRA), was recently developed for estimating the ground flash fraction. The method uses a constrained mixed exponential distribution model to describe a particular lightning optical measurement called the Maximum Group Area (MGA). To obtain the optimum model parameters (one of which is the desired ground flash fraction), a scalar function must be minimized. This minimization is difficult because of two problems: (1) Label Switching (LS), and (2) Parameter Identity Theft (PIT). The LS problem is well known in the literature on mixed exponential distributions, and the PIT problem was discovered in this study. Each problem occurs when one allows the numerical minimizer to freely roam through the parameter search space; this allows certain solution parameters to interchange roles which leads to fundamental ambiguities, and solution error. A major accomplishment of this study is that we have employed a state-of-the-art genetic-based global optimization algorithm called Differential Evolution (DE) that constrains the parameter search in such a way as to remove both the LS and PIT problems. To test the performance of the GoFFRA when DE is employed, we applied it to analyze simulated MGA datasets that we generated from known mixed exponential distributions. Moreover, we evaluated the GoFFRA/DE method by applying it to analyze actual MGAs derived from low-Earth orbiting lightning imaging sensor data; the actual MGA data were classified as either ground or cloud flash MGAs using National Lightning Detection Network[TM] (NLDN) data. Solution error plots are provided for both the simulations and actual data analyses.

Koshak, William↗

Further Improvement in 3DGRAPE

3DGRAPE/AL:V2 denotes version 2 of the Three-Dimensional Grids About Anything by Poisson's Equation with Upgrades from Ames and Langley computer program. The preceding version, 3DGRAPE/AL, was described in Improved 3DGRAPE (ARC-14069) NASA Tech Briefs, Vol. 21, No. 5 (May 1997), page 66. These programs are so named because they generate volume grids by iteratively solving Poisson's Equation in three dimensions. The grids generated by the various versions of 3DGRAPE have been used in computational fluid dynamics (CFD). The main novel feature of 3DGRAPE/AL:V2 is the incorporation of an optional scheme in which anisotropic Lagrange-based trans-finite interpolation (ALBTFI) is coupled with exponential decay functions to compute and blend interior source terms. In the input to 3DGRAPE/AL:V2 the user can specify whether or not to invoke ALBTFI in combination with exponential-decay controls, angles, and cell size for controlling the character of grid lines. Of the known programs that solve elliptic partial differential equations for generating grids, 3DGRAPE/AL:V2 is the only code that offers a combination of speed and versatility with most options for controlling the densities and other characteristics of grids for CFD.

Alter, Stephen↗

Air-Sea Spray Airborne Radar Profiler Characterizes Energy Fluxes in Hurricanes

A report discusses ASAP (Air-sea Spray Airborne Profiler), a dual-wavelength radar profiler that provides measurement information about the droplet size distribution (DSD) of sea-spray, which can be used to estimate heat and moisture fluxes for hurricane research. Researchers have recently determined that sea spray can have a large effect on the magnitude and distribution of the air-sea energy flux at hurricane -force wind speeds. To obtain information about the DSD, two parameters of the DSD are required; for example, overall DSD amplitude and DSD mean diameter. This requires two measurements. Two frequencies are used, with a large enough separation that the differential frequency provides size information. One frequency is 94 GHz; the other is 220 GHz. These correspond to the Rayleigh and Mie regions. Above a surface wind speed of 10 m/ s, production of sea spray grows exponentially. Both the number of large droplets and the altitude they reach are a function of the surface wind speed.

Durden, Stephen L.↗

Research highlights: June 1990 - May 1991

Linear instability calculations at MSFC have suggested that the Geophysical Fluid Flow Cell (GFFC) should exhibit classic baroclinic instability at accessible parameter settings. Interest was in the mechanisms of transition to temporal chaos and the evolution of spatio-temporal chaos. In order to understand more about such transitions, high resolution numerical experiments for the physically simplest model of two layer baroclinic instability were conducted. This model has the advantage that the numerical code is exponentially convergent and can be efficiently run for very long times, enabling the study of chaotic attractors without the often devastating effects of low-order trunction found in many previous studies. Numerical algorithms for implementing an empirical orthogonal function (EOF) analysis of the high resolution numerical results were completed. Under conditions of rapid rotation and relatively low differential heating, convection in a spherical shell takes place as columnar banana cells wrapped around the annular gap, but with axes oriented along the axis of rotation; these were clearly evident in the GFFC experiments. The results of recent numerical simulations of columnar convection and future research plans are presented.

Source record↗

Discrete approximation methods for parameter identification in delay systems

Approximation schemes for parameter identification problems in which the governing state equation is a linear functional differential equation of retarded type are constructed. The basis of the schemes is the replacement of the parameter identification problem having an infinite dimensional state equation by a sequence of approximating parameter identification problems in which the states are given by finite dimensional discrete difference equations. The difference equations are constructed using linear semigroup theory and rational function approximations to the exponential. Sufficient conditions are given for the convergence of solutions to the approximating problems, which can be obtained using conventional methods, to solutions to the original parameter identification problem. Finite difference and spline based schemes using Paderational function approximations to the exponential are constructed, and shown to satisfy the sufficient conditions for convergence. A discussion and analysis of numerical results obtained through the application of the schemes to several examples is included.

Rosen, I. G.↗

Contamination of liquid oxygen by pressurized gaseous nitrogen

The penetration of pressurized gaseous nitrogen (GN2) into liquid oxygen (LOX) was investigated experimentally in the 7-inch High Temperature Tunnel, the pilot tunnel for the 8-foot High Temperature Tunnel (8'HTT) at Langley Research Center. A preliminary test using a nuclear monitor revealed the extent of the liquid nitrogen (LN2) build-up at the LOX interface as a function of GN2 pressure. Then an adaptation of the differential flash vaporization technique was used to determine the binary diffusivity of the LOX-LN2 system at a temperature of 90.2 K. The measured value D equals 0.000086 sq cm/s + or - 25 percent together with two prior measurements at lower temperatures revealed an excellent fit to the Arrhenius equation, yielding a pre-exponential factor D sub 0 equals 0.0452 sq cm/s and an activation enthalpy H equals 1.08 kcal/mol. At a pressure of 1700 psi and holding time of 15 min, the penetration of LN2 into LOX (to a 1 percent contamination level) was found to be 0.9 cm, indicating but minimal impact upon 8'HTT operations.

Zuckerwar, Allan J.↗

Thermal and Nonthermal Contributions to the Solar Flare X-Ray Flux

The relative thermal and nonthermal contributions to the total energy budget of a solar flare are being determined through analysis of RHESSI X-ray imaging and spectral observations in the energy range from approx. 5 to approx. 50 keV. The classic ways of differentiating between the thermal and nonthermal components - exponential vs. sources - can now be combined for individual flares. In addition, RHESSI's sensitivity down to approx. 4 keV and energy resolution of approx. 1 keV FWHM allow the intensities and equivalent widths of the complex of highly ionized iron lines at approx. 6.7 keV and the complex of highly ionized iron and nickel lines at approx. 8 keV to be measured as a function of time. Using the spectral line and continuum intensities from the Chianti (version 4.2) atomic code, the thermal component of the total flare emission can be more reliably separated from the nonthermal component in the measured X-ray spectrum. The abundance of iron can also be determined from RHESSI line-to-continuum measurements as a function of time during larger flares. Results will be shown of the intensity and equivalent widths of these line complexes for several flares and the temperatures, emission measures, and iron abundances derived from them. Comparisons will be made with 6.7-keV Fe-line fluxes measured with the RESIK bent crystal spectrometer on the Coronas-F spacecraft operating in third order during the peak times of three flares (2002 May 31 at 00:12 UT, 2002 December 2 at 19:26 UT, and 2003 April 26 at 03:OO UT). During the rise and decay of these flares, RESIK was operating in first order allowing the continuum flux to be measured between 2.9 and 3.7 keV for comparison with RHESSI fluxes at its low-energy end.

Dennis, Brian R.↗

On the nonlinear three dimensional instability of Stokes layers and other shear layers to pairs of oblique waves

The nonlinear evolution of a pair of initially oblique waves in a high Reynolds Number Stokes layer is studied. Attention is focused on times when disturbances of amplitude epsilon have O(epsilon(exp 1/3)R) growth rates, where R is the Reynolds number. The development of a pair of oblique waves is then controlled by nonlinear critical-layer effects. Viscous effects are included by studying the distinguished scaling epsilon = O(R(exp -1)). This leads to a complicated modification of the kernel function in the integro-differential amplitude equation. When viscosity is not too large, solutions to the amplitude equation develop a finite-time singularity, indicating that an explosive growth can be introduced by nonlinear effects; we suggest that such explosive growth can lead to the bursts observed in experiments. Increasing the importance of viscosity generally delays the occurrence of the finite-time singularity, and sufficiently large viscosity may lead to the disturbance decaying exponentially. For the special case when the streamwise and spanwise wavenumbers are equal, the solution can evolve into a periodic oscillation. A link between the unsteady critical-layer approach to high-Reynolds-number flow instability, and the wave vortex approach is identified.

Wu, Xuesong↗

Finite Control in Underdamped Distributed Parameter Systems

Specifically, a class of distributed parameter systems is defined using Hilbert space methods based on a partial differential equation model of a structure. In this space a simple, easy to check definition of underdamping is constructed based on the well known single degree of freedom concept of critical damping. It is then shown that if a given distributed parameter system satisfies this definition, each mode of the modal expansion of the solution is in fact an underdamped function of time. It is noted that a distributed parameter system which is underdamped is also uniformly exponentially stable. By appealing to the work of Gibson, a finite dimensional model of the system will yield satisfactory control laws. Furthermore, it is shown that the assumption of the underdamping also allows straightforward computation of rough bounds on the magnitude of the unmodeled or residual modes for a modal truncation scheme. Hence, underdamped systems are precisely that class of distributed parameter systems which are not likely to have spillover problems and which will yield convergence of finite dimensional control laws to control laws which are optimal for the full distributed parameter model of the system.

Inman, D. J.↗

Turbulence and the Stabilization Principle

Further results of research, reported in several previous NASA Tech Briefs articles, were obtained on a mathematical formalism for postinstability motions of a dynamical system characterized by exponential divergences of trajectories leading to chaos (including turbulence). To recapitulate: Fictitious control forces are introduced to couple the dynamical equations with a Liouville equation that describes the evolution of the probability density of errors in initial conditions. These forces create a powerful terminal attractor in probability space that corresponds to occurrence of a target trajectory with probability one. The effect in ordinary perceived three-dimensional space is to suppress exponential divergences of neighboring trajectories without affecting the target trajectory. Con sequently, the postinstability motion is represented by a set of functions describing the evolution of such statistical quantities as expectations and higher moments, and this representation is stable. The previously reported findings are analyzed from the perspective of the authors Stabilization Principle, according to which (1) stability is recognized as an attribute of mathematical formalism rather than of underlying physics and (2) a dynamical system that appears unstable when modeled by differentiable functions only can be rendered stable by modifying the dynamical equations to incorporate intrinsic stochasticity.

Zak, Michail↗

A survey of solar protons and alpha differential spectra between 1 and greater than 400 MeV/nucleon

Attempts to systematically measure and fit a large sample of flare spectra from IMP 7 and 8 satellite data to determine the spectral forms that characterize the data and the range of spectral slopes are reported. Proton spectra are constructed covering the range 0.9 to over 400 MeV, and alpha fluxes from 1.4-80 MeV/nucleon. The event spectra are composed of fluxes measured at different times at each energy, and fitting procedures were employed such as a power law in kinetic energy, an exponential in rigidity, an exponential in energy, and a Bessel function of momentum/nucleon. Best fits were obtained with the exponential in rigidity and the Bessel function, with the Bessel function having a better numerical accuracy for all 21 proton spectra considered.

Mcguire, R. E.↗

Spectral Analyses and Radiation Exposures from Several Ground-Level Enhancement (GLE) Solar Proton Events: A Comparison of Methodologies

Several methods for analyzing the particle spectra from extremely large solar proton events, called Ground-Level Enhancements (GLEs), have been developed and utilized by the scientific community to describe the solar proton energy spectra and have been further applied to ascertain the radiation exposures to humans and radio-sensitive systems, namely electronics. In this paper 12 GLEs dating back to 1956 are discussed, and the three methods for describing the solar proton energy spectra are reviewed. The three spectral fitting methodologies are EXP [an exponential in proton rigidity (R)], WEIB [Weibull fit: an exponential in proton energy], and the Band function (BAND) [a double power law in proton rigidity]. The EXP and WEIB methods use low energy (MeV) GLE solar proton data and make extrapolations out to approx.1 GeV. On the other hand, the BAND method utilizes low- and medium-energy satellite solar proton data combined with high-energy solar proton data deduced from high-latitude neutron monitoring stations. Thus, the BAND method completely describes the entire proton energy spectrum based on actual solar proton observations out to ~10 GeV. Using the differential spectra produced from each of the 12 selected GLEs for each of the three methods, radiation exposures are presented and discussed in detail. These radiation exposures are then compared with the current 30-day and annual crew exposure limits and the radiation effects to electronics.

Atwell, William↗

Rocket measurements of energetic particles in the midlatitude precipitation zone

Measurements of energetic ion and electron properties as a function of altitude in the midlatitude zone of nighttime energetic particle precipitation are reported. The measurements of particle fluxes, energy spectra and pitch angle distributions were obtained by a Langmuir probe, six energetic particle spectrometers and an electrostatic analyzer on board a Nike Apache rocket launched near the center of the midlatitude zone during disturbed conditions. It is found that the incident flux was primarily absorbed rather than backscattered, and consists of mainly energetic hydrogen together with some helium and a small energetic electron component. Observed differential energy spectra of protons having an exponential energy spectrum, and pitch angle distributions at various altitudes indicate that the energetic particle flux decreases rapidly for pitch angles less than 70 deg. An energetic particle energy flux of 0.002 ergs/sq cm per sec is calculated which indicates the significance of energetic particles as a primary nighttime ionization source for altitudes between 120 and 200 km in the midlatitude precipitation zone.

Voss, H. D.↗

Development of a Photon Counting System for Differential Lidar Signal Detection

Photon counting has been chosen as a means to extend the detection range of current airborne DIAL ozone measurements. Lidar backscattered return signals from the on and off-line lasers experience a significant exponential decay. To extract further data from the decaying ozone return signals, photon counting will be used to measure the low light levels, thus extending the detection range. In this application, photon counting will extend signal measurement where the analog return signal is too weak. The current analog measurement range is limited to approximately 25 kilometers from an aircraft flying at 12 kilometers. Photon counting will be able to exceed the current measurement range so as to follow the mid-latitude model of ozone density as a function of height. This report describes the development of a photon counting system. The initial development phase begins with detailed evaluation of individual photomultiplier tubes. The PMT qualities investigated are noise count rates, single electron response peaks, voltage versus gain values, saturation effects, and output signal linearity. These evaluations are followed by analysis of two distinctive tube base gating schemes. The next phase is to construct and operate a photon counting system in a laboratory environment. The laboratory counting simulations are used to determine optimum discriminator setpoints and to continue further evaluations of PMT properties. The final step in the photon counting system evaluation process is the compiling of photon counting measurements on the existing ozone DIAL laser system.

Elsayed-Ali, Hani↗

Spiking Neurons for Analysis of Patterns

Artificial neural networks comprising spiking neurons of a novel type have been conceived as improved pattern-analysis and pattern-recognition computational systems. These neurons are represented by a mathematical model denoted the state-variable model (SVM), which among other things, exploits a computational parallelism inherent in spiking-neuron geometry. Networks of SVM neurons offer advantages of speed and computational efficiency, relative to traditional artificial neural networks. The SVM also overcomes some of the limitations of prior spiking-neuron models. There are numerous potential pattern-recognition, tracking, and data-reduction (data preprocessing) applications for these SVM neural networks on Earth and in exploration of remote planets. Spiking neurons imitate biological neurons more closely than do the neurons of traditional artificial neural networks. A spiking neuron includes a central cell body (soma) surrounded by a tree-like interconnection network (dendrites). Spiking neurons are so named because they generate trains of output pulses (spikes) in response to inputs received from sensors or from other neurons. They gain their speed advantage over traditional neural networks by using the timing of individual spikes for computation, whereas traditional artificial neurons use averages of activity levels over time. Moreover, spiking neurons use the delays inherent in dendritic processing in order to efficiently encode the information content of incoming signals. Because traditional artificial neurons fail to capture this encoding, they have less processing capability, and so it is necessary to use more gates when implementing traditional artificial neurons in electronic circuitry. Such higher-order functions as dynamic tasking are effected by use of pools (collections) of spiking neurons interconnected by spike-transmitting fibers. The SVM includes adaptive thresholds and submodels of transport of ions (in imitation of such transport in biological neurons). These features enable the neurons to adapt their responses to high-rate inputs from sensors, and to adapt their firing thresholds to mitigate noise or effects of potential sensor failure. The mathematical derivation of the SVM starts from a prior model, known in the art as the point soma model, which captures all of the salient properties of neuronal response while keeping the computational cost low. The point-soma latency time is modified to be an exponentially decaying function of the strength of the applied potential. Choosing computational efficiency over biological fidelity, the dendrites surrounding a neuron are represented by simplified compartmental submodels and there are no dendritic spines. Updates to the dendritic potential, calcium-ion concentrations and conductances, and potassium-ion conductances are done by use of equations similar to those of the point soma. Diffusion processes in dendrites are modeled by averaging among nearest-neighbor compartments. Inputs to each of the dendritic compartments come from sensors. Alternatively or in addition, when an affected neuron is part of a pool, inputs can come from other spiking neurons. At present, SVM neural networks are implemented by computational simulation, using algorithms that encode the SVM and its submodels. However, it should be possible to implement these neural networks in hardware: The differential equations for the dendritic and cellular processes in the SVM model of spiking neurons map to equivalent circuits that can be implemented directly in analog very-large-scale integrated (VLSI) circuits.

Huntsberger, Terrance↗

VLA stereoscopy of solar active regions. 2: Altitude, relative motion, and center-to-limb darkening of 20 centimeter emission

A newly developed method of stereoscopic correlation is applied to solar radio maps at a wavelength of 20 cm. On the basis of stereoscopic correlations between radio maps obtained on 6 different days we establish the presence of 66 radio source components associated with 22 active regions. We find the following statistical results for active region source structures at 20 cm: 1. The observed lifetime is consistent with an exponential distribution having an e-folding time scale of greater than 18 days. 2. The average altitude of 20 cm sources is 25 +/- 15 Mm; 90% of the sources are found in heights less than 40 Mm. 3. The average diameter of discrete source structures is 48 +/- 15 Mm, implying a vertical/horizontal aspect ratio of q(sub A) approx. = 0.5. 4. No significant source motion has been found with respect to the standard differential rotation rate of Omega = 13.45 deg - ((3 deg (sin(exp 2)B)). 5. We find a statistical limb darkening, which can be described by the relation (T(sub B)(alpha))/T(sub B)(0)) = 0.4 + ((0.6(cos(exp 2) alpha)) for sources with T(sub B) greater than 0.5 MK. 6. The degree of source polarization is 15% +/- 10% and is independent of source location. 7. Bright sources (greater than or approximately equal to 0.5 MK), or equivalently, long-lived sources (greater than or approximately equal to 5 days) show a systematic variation of their altitude as function of the center-limb distance.

Aschwanden, Markus J.↗