Analytical Parameterization Research Articles

Approaching the asymptotic regime of rapidly rotating convection: boundary layers versus interior dynamics.

Rapidly rotating Rayleigh-Bénard convection is studied by combining results from direct numerical simulations (DNS), laboratory experiments, and asymptotic modeling. The asymptotic theory is shown to provide a good description of the bulk dynamics at low, but finite Rossby number. However, large deviations from the asymptotically predicted heat transfer scaling are found, with laboratory experiments and DNS consistently yielding much larger Nusselt numbers than expected. These deviations are traced down to dynamically active Ekman boundary layers, which are shown to play an integral part in controlling heat transfer even for Ekman numbers as small as 10^{-7}. By adding an analytical parametrization of the Ekman transport to simulations using stress-free boundary conditions, we demonstrate that the heat transfer jumps from values broadly compatible with the asymptotic theory to states of strongly increased heat transfer, in good quantitative agreement with no-slip DNS and compatible with the experimental data. Finally, similarly to nonrotating convection, we find no single scaling behavior, but instead that multiple well-defined dynamical regimes exist in rapidly rotating convection systems.

SPACS: A semi-empirical parameterization for isotopic spallation cross sections

A new semi-empirical parameterization for residue cross sections in spallation reactions is presented. The prescription named SPACS, for spallation cross sections, permits calculating the fragment production in proton- and neutron-induced collisions with light up to heavy non-fissile partners from the Fermi regime to ultra-relativistic energies. The model is fully analytical, based on a new parameterization of the mass yields, accounting for the dependence on bombarding energy. The formalism for the isobaric distribution consists of a commonly used functional form, borrowed from the empirical parameterization of fragmentation cross sections EPAX, with the observed suited adjustments for spallation, and extended to the charge-pickup channel. Structural and even-odd staggering related to the last stage of the primary-residue deexcitation process is additionally explicitly introduced with a new prescription. Calculations are benchmarked with recent data collected at GSI, Darmstadt as well as with previous measurements employing various techniques. The dependences observed experimentally on collision energy, reaction-partner mass, and proton-neutron asymmetry are well described. A fast analytical parameterization, such as SPACS, can be relevant to be implemented in complex simulations as used for practical issues at nuclear facilities and plants. Its predictive power also makes it useful for cross-section estimates in astrophysics and biophysics.

Equation of state in (2+1)-flavor QCD

We present results for the equation of state in (2+1)-flavor QCD using the highly improved staggered quark action and lattices with temporal extent $N_{\tau}=6,~8,~10$, and $12$. We show that these data can be reliably extrapolated to the continuum limit and obtain a number of thermodynamic quantities and the speed of sound in the temperature range $(130-400)$ MeV. We compare our results with previous calculations, and provide an analytic parameterization of the pressure, from which other thermodynamic quantities can be calculated, for use in phenomenology. We show that the energy density in the crossover region, $145~ {\rm MeV} \leq T \leq 163$ MeV, defined by the chiral transition, is $\epsilon_c=(0.18-0.5)~{\rm GeV}/{\rm fm}^3$, $i.e.$, $(1.2-3.1)\ \epsilon_{\rm nuclear}$. At high temperatures, we compare our results with resummed and dimensionally reduced perturbation theory calculations. As a byproduct of our analyses, we obtain the values of the scale parameters $r_0$ from the static quark potential and $w_0$ from the gradient flow.

Output control of a spacecraft motion spectrum

An efficient approach to the output control of the spectrum of multidimensional dynamic systems specified in the state space form is proposed. The approach can be applied for both the continuous and discrete mathematical models. This approach has no limitations on the algebraic and geometric multiplicity of the spectrum elements and allows one to obtain analytical solutions and parameterization (construction) of a set of controllers.

A parameterization for the radio emission of air showers as predicted by CoREAS simulations and applied to LOFAR measurements

Measuring radio emission from air showers provides excellent opportunities to directly measure all air shower properties, including the shower development. To exploit this in large-scale experiments, a simple and analytic parameterization of the distribution of the radio signal at ground level is needed. Data taken with the Low-Frequency Array (LOFAR) show a complex two-dimensional pattern of pulse powers, which is sensitive to the shower geometry. Earlier parameterizations of the lateral signal distribution have proven insufficient to describe these data. In this article, we present a parameterization derived from air-shower simulations. We are able to fit the two-dimensional distribution with a double Gaussian, requiring five fit parameters. All parameters show strong correlations with air shower properties, such as the energy of the shower, the arrival direction, and the shower maximum. We successfully apply the parameterization to data taken with LOFAR and discuss implications for air shower experiments.

A glacial systems model configured for large ensemble analysis of Antarctic deglaciation

Abstract. This article describes the Memorial University of Newfoundland/Penn State University (MUN/PSU) glacial systems model (GSM) that has been developed specifically for large-ensemble data-constrained analysis of past Antarctic Ice Sheet evolution. Our approach emphasizes the introduction of a large set of model parameters to explicitly account for the uncertainties inherent in the modelling of such a complex system. At the core of the GSM is a 3-D thermo-mechanically coupled ice sheet model that solves both the shallow ice and shallow shelf approximations. This enables the different stress regimes of ice sheet, ice shelves, and ice streams to be represented. The grounding line is modelled through an analytical sub-grid flux parameterization. To this dynamical core the following have been added: a heavily parameterized basal drag component; a visco-elastic isostatic adjustment solver; a diverse set of climate forcings (to remove any reliance on any single method); tidewater and ice shelf calving functionality; and a new physically motivated, empirically-derived sub-ice-shelf melt (SSM) component. To assess the accuracy of the latter, we compare predicted SSM values against a compilation of published observations. Within parametric and observational uncertainties, computed SSM for the present-day ice sheet is in accord with observations for all but the Filchner ice shelf. The GSM has 31 ensemble parameters that are varied to account (in part) for the uncertainty in the ice physics, the climate forcing, and the ice–ocean interaction. We document the parameters and parametric sensitivity of the model to motivate the choice of ensemble parameters in a quest to approximately bound reality (within the limits of 31 parameters).

Atomistic simulations of stainless steels: a many-body potential for the Fe–Cr–C system

Stainless steels found in real-world applications usually have some C content in the base Fe–Cr alloy, resulting in hard and dislocation-pinning carbides—Fe3C (cementite) and Cr23C6—being present in the finished steel product. The higher complexity of the steel microstructure has implications, for example, for the elastic properties and the evolution of defects such as Frenkel pairs and dislocations. This makes it necessary to re-evaluate the effects of basic radiation phenomena and not simply to rely on results obtained from purely metallic Fe–Cr alloys. In this report, an analytical interatomic potential parameterization in the Abell–Brenner–Tersoff form for the entire Fe–Cr–C system is presented to enable such calculations. The potential reproduces, for example, the lattice parameter(s), formation energies and elastic properties of the principal Fe and Cr carbides (Fe3C, Fe5C2, Fe7C3, Cr3C2, Cr7C3, Cr23C6), the Fe–Cr mixing energy curve, formation energies of simple C point defects in Fe and Cr, and the martensite lattice anisotropy, with fair to excellent agreement with empirical results. Tests of the predictive power of the potential show, for example, that Fe–Cr nanowires and bulk samples become elastically stiffer with increasing Cr and C concentrations. High-concentration nanowires also fracture at shorter relative elongations than wires made of pure Fe. Also, tests with Fe3C inclusions show that these act as obstacles for edge dislocations moving through otherwise pure Fe.

A parameterization method and application in breast tomosynthesis dosimetry

To present a parameterization method based on singular value decomposition (SVD), and to provide analytical parameterization of the mean glandular dose (MGD) conversion factors from eight references for evaluating breast tomosynthesis dose in the Mammography Quality Standards Act (MQSA) protocol and in the UK, European, and IAEA dosimetry protocols. MGD conversion factor is usually listed in lookup tables for the factors such as beam quality, breast thickness, breast glandularity, and projection angle. The authors analyzed multiple sets of MGD conversion factors from the Hologic Selenia Dimensions quality control manual and seven previous papers. Each data set was parameterized using a one- to three-dimensional polynomial function of 2-16 terms. Variable substitution was used to improve accuracy. A least-squares fit was conducted using the SVD. The differences between the originally tabulated MGD conversion factors and the results computed using the parameterization algorithms were (a) 0.08%-0.18% on average and 1.31% maximum for the Selenia Dimensions quality control manual, (b) 0.09%-0.66% on average and 2.97% maximum for the published data by Dance et al. [Phys. Med. Biol. 35, 1211-1219 (1990); ibid. 45, 3225-3240 (2000); ibid. 54, 4361-4372 (2009); ibid. 56, 453-471 (2011)], (c) 0.74%-0.99% on average and 3.94% maximum for the published data by Sechopoulos et al. [Med. Phys. 34, 221-232 (2007); J. Appl. Clin. Med. Phys. 9, 161-171 (2008)], and (d) 0.66%-1.33% on average and 2.72% maximum for the published data by Feng and Sechopoulos [Radiology 263, 35-42 (2012)], excluding one sample in (d) that does not follow the trends in the published data table. A flexible parameterization method is presented in this paper, and was applied to breast tomosynthesis dosimetry. The resultant data offer easy and accurate computations of MGD conversion factors for evaluating mean glandular breast dose in the MQSA protocol and in the UK, European, and IAEA dosimetry protocols. Microsoft Excel™ spreadsheets are provided for the convenience of readers.

Width of the equilibrium charge distribution of heavy ions in solid and gaseous media

Based on an analysis of the experimental data on the charge distributions of ions in gases and solids, the analytical parameterization of the width of the charge distribution as a function of the energy, the average equilibrium ion charge, ion nuclear charge, and nuclear charge of target atom has been obtained. The proposed relations are applicable in a broad energy range and describe the existing experimental data satisfactorily.

BINSTAR: a new binary stellar evolution code

We provide a detailed description of a new stellar evolution code, BINSTAR, which has been developed to study interacting binaries. Based on the stellar evolution code STAREVOL, it is specifically designed to study low- and intermediate-mass binaries. We describe the state-of-the-art input physics, which includes treatments of tidal interactions, mass transfer and angular momentum exchange within the system. A generalised Henyey method is used to solve simultaneously the stellar structure equations of each component as well as the separation and eccentricity of the orbit. Test simulations for cases A and B mass transfer are presented and compared with available models. The results of the evolution of Algol systems are in remarkable agreement with the calculations of the Vrije Universiteit Brussel (VUB) group, thus validating our code. We also computed a large grid of models for various masses (2 ≤ M/M⊙ ≤ 20) and seven metallicities (Z = 0.0001, 0.001, 0.004, 0.008, 0.01, 0.02, 0.03) to provide a useful analytical parameterisation of the tidal torque constant E2, which allows the determination of the circularisation and synchronisation timescales for stars with a radiative envelope and convective core. The evolution of E2 during the main sequence shows noticeable differences compared to available models. In particular, our new calculations indicate that the circularisation timescale is constant during core hydrogen burning. We also show that E2 weakly depends on core overshooting but is substantially increased when the metallicity becomes lower.

Empirical estimation of the equilibrium charge distribution of heavy ions in solid and gaseous media

Abstract Based on the analysis of experimental data on equilibrium charge distributions of ions in gases and solid media, the analytical parameterization of the charge distribution width as a function of the average ion charge and nuclear charges of the ion and the target atom and that of the asymmetry as a function of the average ion charge are obtained. The proposed relations are applicable in a broad energy interval and describe the existing experimental data satisfactorily.

Parameterization of homogeneous ice nucleation for cloud and climate models based on classical nucleation theory

Abstract. A new analytical parameterization of homogeneous ice nucleation is developed based on extended classical nucleation theory including new equations for the critical radii of the ice germs, free energies and nucleation rates as simultaneous functions of temperature and water saturation ratio. By representing these quantities as separable products of the analytical functions of temperature and supersaturation, analytical solutions are found for the integral-differential supersaturation equation and concentration of nucleated crystals. Parcel model simulations are used to illustrate the general behavior of various nucleation properties under various conditions, for justifications of the further key analytical simplifications, and for verification of the resulting parameterization. The final parameterization is based upon the values of the supersaturation that determines the current or maximum concentrations of the nucleated ice crystals. The crystal concentration is analytically expressed as a function of time and can be used for parameterization of homogeneous ice nucleation both in the models with small time steps and for substep parameterization in the models with large time steps. The crystal concentration is expressed analytically via the error functions or elementary functions and depends only on the fundamental atmospheric parameters and parameters of classical nucleation theory. The diffusion and kinetic limits of the new parameterization agree with previous semi-empirical parameterizations.

Photon Elastic Scattering Simulation: Validation and Improvements to Geant4

Several models for the simulation of photon elastic scattering are quantitatively evaluated with respect to a large collection of experimental data retrieved from the literature. They include models based on the form factor approximation, on S-matrix calculations and on analytical parameterizations; they exploit publicly available data libraries and tabulations of theoretical calculations. Some of these models are currently implemented in general purpose Monte Carlo systems; some have been implemented and evaluated for the first time in this paper for possible use in Monte Carlo particle transport. The analysis mainly concerns the energy range between 5 keV and a few MeV. The validation process identifies the newly implemented model based on second order S-matrix calculations as the one best reproducing experimental measurements. The validation results show that, along with Rayleigh scattering, additional processes, not yet implemented in Geant4 nor in other major Monte Carlo systems, should be taken into account to realistically describe photon elastic scattering with matter above 1 MeV. Evaluations of the computational performance of the various simulation algorithms are reported along with the analysis of their physics capabilities.

Simple parameterization for the ground-state energy of the infinite Hubbard chain incorporating Mott physics, spin-dependent phenomena and spatial inhomogeneity

Simple analytical parameterizations for the ground-state energy of the one-dimensional repulsive Hubbard model are developed. The charge dependence of energy is parameterized using exact results extracted from the Bethe-ansatz (BA). The resulting parameterization is shown to be in better agreement with highly precise data obtained from a fully numerical solution to the BA equations than previous expressions (Lima et al 2003 Phys. Rev. Lett.90 146402). Unlike these earlier proposals, the present parameterization correctly predicts a positive Mott gap at half filling for any U > 0. The construction is extended to spin-dependent phenomena by parameterizing the magnetization dependence of the ground-state energy using further exact results and numerical benchmarking. Lastly, the parameterizations developed for the spatially uniform model are extended by means of a simple local-density-type approximation to spatially inhomogeneous models, e.g. in the presence of impurities, external fields or trapping potentials. The results are shown to be in excellent agreement with independent many-body calculations, at a fraction of the computational cost.

Performance Measures Modeling (PMM) in MANET

In this paper We are emphasizing on a Mathematical Model which will be denoted as PMM (Performance Measures Model), for computing routing metrics, with the help of this paper have implemented the mathematical model that will be helpful in calculating various metrics in MANET (Mobile Ad hoc Network) like Packet Delivery Ratio, Normalized Routing Load, Average End To End Delay and many others, In our work we have simulated various routing protocols under different scenarios and after getting some results we use some mathematical expressions for the same and also computed some results and we observed that the some expressions produced closer to the results which are obtained with the simulator, but some expressions produced exactly the same results as the simulator produce. So after repeating the simulation many times and for many routing protocols, we observed the same. Now we can say that using any one of our proposed expression the analytic parameterization of Routing protocols can be calculated such as PDR fraction, Normalized Routing Load and Average End to End Delay. This Approach is much better than the simulation.

Formation of millisecond pulsars with CO white dwarf companions - I. PSR J1614−2230: evidence for a neutron star born massive

The recent discovery of a 2 M_sun binary millisecond pulsar (Demorest et al. 2010) has not only important consequences for the equation-of-state of nuclear matter at high densities but also raises the interesting question if the neutron star PSR J1614-2230 was born massive. The answer is vital for understanding neutron star formation in core collapse supernovae. Furthermore, this system raises interesting issues about the nature of the progenitor binary and how it evolved during its mass exchanging X-ray phase. In this paper we discuss the progenitor evolution of PSR J1614-2230. We have performed detailed stellar evolution modelling of intermediate-mass X-ray binaries undergoing Case A Roche-lobe overflow (RLO) and applied an analytic parameterization for calculating the outcome of either a common envelope evolution or the highly super-Eddington isotropic re-emission mode. We find two viable possibilities for the formation of the PSR J1614-2230 system: either it contained a 2.2-2.6 M_sun giant donor star and evolved through a common envelope and spiral-in phase or, more likely, it descended from a close binary system with a 4.0-5.0 M_sun main sequence donor star via Case A RLO. We conclude that the neutron star must have been born with a mass of ~1.95 M_sun or 1.7+-0.15 M_sun, respectively - which significantly exceeds neutron star birth masses in previously discovered radio pulsar systems. Based on the expected neutron star birth masses from considerations of stellar evolution and explosion models, we find that the progenitor star of PSR J1614-2230 is likely to have been more massive than 20 M_sun.

Coalescence of metastable states in chemical reactions: double poles of the scattering matrix and exceptional points

Accidental degeneracy of two scattering resonances in chemical reactions is discussed. A novel analytical parameterization for the scattering matrix elements valid in the presence of two interacting resonances is derived and applied to investigate the coalescence of metastable states leading to double poles in the partial-wave amplitude. We show how double poles can manifest themselves in reaction cross-sections and discuss their relationship with exceptional points. The presence of two degenerate resonances interacting with each other can create a peculiar double-peak structure in the reaction probability. At the energy where the two decaying states coalesce the scattering matrix has a double pole and correspondingly the reaction probability reaches its minimum value. This novel formula has been tested with the well-established data obtained for the F + H2 reaction, for which previous studies have revealed the existence of two interfering resonance pathways. The good agreement of model predictions and quantum scattering calculations shows that the present approach is reliable. Hence, we claim that it provides a way for the observation of interacting resonances in molecular collisions. Resonance positions and partial widths can be obtained by fitting the numerically calculated state-to-state reaction probabilities to the square modulus of the novel expression for the scattering matrix elements derived in the paper.

Application of an analytical formula for UV Index reconstructions for two locations in Southwestern Spain

This paper focuses on the application of a simple analytical parameterization to the filling of the Ultraviolet Index (UVI) data gaps, and the reconstruction of past UVI values at Badajoz and Caceres (Southwestern Spain). The empirical model involves three independent variables: the solar zenith angle, the total ozone column and the clearness index. Regarding the first application, daily UVI was estimated for more than 30 days when UV measurements were not available in 2007. For these cases, the missing UVI data were replaced by estimated values, thus affecting the UVI annual mean and median. Regarding the second application, the reconstruction of past UVI time-series (1950–2000) is performed only for clear-sky cases (cloud and aerosol free conditions) using the COST 726 total ozone climatology. The linear UVI trends for two periods (1957–1978 and 1979–2000) are calculated for summer months using linear least squares fits. Both locations show statistically significant UVI trends for the most recent period 1979–2000, with values of +4.4 ± 1.6% per decade for Badajoz, and +4.9 ± 1.8% per decade for Caceres. This result is mainly driven by the ozone decline at northern mid-latitudes during this period. No significant trend is found for the other analysed period.

Sensitivity of the global distribution of cirrus ice crystal concentration to heterogeneous freezing

This study presents the sensitivity of global ice crystal number concentration, Nc, to the parameterization of heterogeneous ice nuclei (IN). Simulations are carried out with the NASA Global Modeling Initiative chemical and transport model coupled to an analytical ice microphysics parameterization. Heterogeneous freezing is described using nucleation spectra derived from theoretical considerations and empirical data for dust, black carbon, ammonium sulfate, and glassy aerosol as IN precursors. When competition between homogeneous and heterogeneous freezing is considered, global mean Nc vary by up to a factor of twenty depending on the heterogeneous freezing spectrum used. IN effects on Nc strongly depend on dust and black carbon concentrations and are strongest under conditions of weak updraft and high temperature. Regardless of the heterogeneous spectrum used, dust is an important contributor of IN over large regions of the Northern Hemisphere. Black carbon however exhibits appreciable effects on Nc when the freezing fraction is greater than 1%. Compared to in situ observations, Nc is overpredicted at temperatures below 205 K, even if a fraction of liquid aerosol is allowed to act as glassy IN. Assuming that cirrus formation is forced by weak updraft addressed this overprediction but promoted heterogeneous freezing effects to the point where homogeneous freezing is inhibited for IN concentrations as low as 1 L−1. Chemistry and dynamics must be considered to explain cirrus characteristics at low temperature. Only cloud formation scenarios where competition between homogeneous and heterogeneous freezing is the dominant feature would result in maximum supersaturation levels consistent with observations.

Empirical Evaluation of a Simple Analytical Formula for the Ultraviolet Index

This paper focuses on the estimation of the UV Index (UVI) for all sky conditions using a simple analytical parameterization involving three independent variables: the solar zenith angle, the total ozone column and the clearness index. Measurements of the UVI made at Badajoz and Cáceres (Southwestern Spain) from January 2006 to December 2007 are used to estimate optimal fitting parameters for the model formula, while measurements from January to December 2008 are used to show that the formula-based estimations have mean absolute errors lower than 6% and R(2) ca 0.99.