Analytical and computational methods for the study of rare event probabilities in dispersive and dissipative waves

The main focus of this dissertation is the application of importance sampling (IS) to calculate the probabilities associated with rare events in nonlinear, large-dimensional lightwave systems that are driven by noise, including models for fiber-based optical communication system and mode-locked lasers. Throughout the last decade, IS has emerged as a valuable tool for improving the efficiency of simulating rare events in such systems. In particular, it has shown great success in simulating various sources of transmission impairments found in optical communication systems, with examples ranging from large polarization fluctuations resulting from randomly varying fiber birefringence to large pulse-width fluctuations resulting from imperfections in the optical fiber. In many cases, the application of IS is guided by a low-dimensional reduction of the system dynamics. Combining the low-dimensional reduction with Monte Carlo simulations of the original system has been shown to be an extremely effective scheme for computing, for example, the probability with which a pulse deviates significantly from its initial form due to a random forcing. In the context of nonlinear optics, this might represent a transmission error where the propagation model is the nonlinear Schr¨odinger equation (NLSE) with additive or multiplicative noise. A shortcoming of this method is that the efficiency of the IS technique depends strongly on the accuracy of the low-dimensional reduction used to guide the simulations. These low-dimensional reductions are often derived from a formal perturbation theory, referred to as soliton perturbation theory (SPT) for the case of soliton propagation under the forced NLSE. As demonstrated here, such reduction methodsare often inadequate in their description of the pulse\u27s dynamics. In particular, the interaction between a propagating pulse and dispersive radiation leads to a radiation-induced drift in a pulse\u27s phase, which is largely unaccounted for in the reduced systems currently in use. The first part of this dissertation is devoted to understanding the interaction between a pulse and dispersive radiation, leading to the derivation of an improved reduced system based on a variational approach. Once this system is derived and verified numerically, it serves as the basis for an improved IS method that incorporates the dynamics of the radiation, which is subsequently extended to more realistic propagation models. Of particular interest is the case of the NLSE with a periodic modulation of the dispersion constant, referred to as dispersion management (DM), and a related model where this modulation is averaged to give an autonomous, nonlocal equation. Following the nomenclature commonly use in literature, the former (nonautonomous) equation will be referred to as the NLSE+DM and the latter (autonomous) equation as the DMNLSE. A complicating aspect of these more realistic models is that, unlike the NLSE, exact solutions only exist as numerical objects rather than as closed-form solutions, which introduces an addition source of error in the derivation of a reduced system for the pulse dynamics. In the second part of this dissertation, the IS method is extended to the calculation of phase-slip probabilities in mode-locked lasers (MLL). Realistic models for pulse propagation in MLL include the dissipative effects of gain and loss, in addition to nonlocal saturation effects. As a result most of the reduced systems derived for pulse dynamics are extremely complicated, which diminishes their applicability as guides for IS simulations. Therefore, a MLL operating in the soliton propagation regime is considered, where the effects of gain, loss and saturation are treated perturbatively. A simple reduced system for the pulse dynamics is derived for this MLL model, allowing the IS technique to be effectively applied

Highly Efficient Modeling of Dynamic Coronal Loops

Observational and theoretical evidence suggests that coronal heating is impulsive and occurs on very small cross-field spatial scales. A single coronal loop could contain a hundred or more individual strands that are heated quasi-independently by nanoflares. It is therefore an enormous undertaking to model an entire active region or the global corona. Three-dimensional MHD codes have inadequate spatial resolution, and 1D hydro codes are too slow to simulate the many thousands of elemental strands that must be treated in a reasonable representation. Fortunately, thermal conduction and flows tend to smooth out plasma gradients along the magnetic field, so "0D models" are an acceptable alternative. We have developed a highly efficient model called Enthalpy-Based Thermal Evolution of Loops (EBTEL) that accurately describes the evolution of the average temperature, pressure, and density along a coronal strand. It improves significantly upon earlier models of this type--in accuracy, flexibility, and capability. It treats both slowly varying and highly impulsive coronal heating; it provides the differential emission measure distribution, DEM(T), at the transition region footpoints; and there are options for heat flux saturation and nonthermal electron beam heating. EBTEL gives excellent agreement with far more sophisticated 1D hydro simulations despite using four orders of magnitude less computing time. It promises to be a powerful new tool for solar and stellar studies.Comment: 34 pages, 8 figures, accepted by Astrophysical Journal (minor revisions of original submitted version

Health care resouce use and stroke outcome

Background and Purpose: Outcome in patients hospitalized for acute stroke varies considerably between populations. Within the framework of the GAIN International trial, a large multicenter trial of a neuroprotective agent (gavestinel, glycine antagonist), stroke outcome in relation to health care resource use has been compared in a large number of countries, allowing for differences in case mix. Methods: This substudy includes 1,422 patients in 19 countries grouped into 10 regions. Data on prognostic variables on admission to hospital, resource use, and outcome were analyzed by regression models. Results: All results were adjusted for differences in prognostic factors on admission (NIH Stroke Scale, age, comorbidity). There were threefold variations in the average number of days in hospital/institutional care (from 20 to 60 days). The proportion of patients who met with professional rehabilitation staff also varied greatly. Three-month case fatality ranged from 11% to 28%, and mean Barthel ADL score at three months varied between 64 and 73. There was no relationship between health care resource use and outcome in terms of survival and ADL function at three months. The proportion of patients living at home at three months did not show any relationship to ADL function across countries. Conclusions: There are wide variations in health care resource use between countries, unexplained by differences in case mix. Across countries, there is no obvious relationship between resource use and clinical outcome after stroke. Differences in health care traditions (treatment pathways) and social We thank the coinvestigators and research staff at the participating centers for their support. Glaxo Wellcome sponsored the GAIN International trial, supported the present analyses and reviewed the final draft of the article

Inference of heating properties from "hot" non-flaring plasmas in active region cores. I. Single nanoflares

The properties that are expected of “hot” non-flaring plasmas due to nanoflare heating in active regions are investigated using hydrodynamic modeling tools, including a two-fluid development of the Enthalpy Based Thermal Evolution of Loops code. Here we study a single nanoflare and show that while simple models predict an emission measure distribution extending well above 10 MK, which is consistent with cooling by thermal conduction, many other effects are likely to limit the existence and detectability of such plasmas. These include: differential heating between electrons and ions, ionization non-equilibrium, and for short nanoflares, the time taken for the coronal density to increase. The most useful temperature range to look for this plasma, often called the “smoking gun” of nanoflare heating, lies between 10 6.6 and 10 7 K. Signatures of the actual heating may be detectable in some instances.Publisher PDFPeer reviewe

Homogenization of the Equations Governing the Flow Between a Slider and a Rough Spinning Disk

We have analyzed the behavior of the flow between a slider bearing and a hard-drive magnetic disk under two types of surface roughness. For both cases the length scale of the roughness along the surface is small as compared to the scale of the slider, so that a homogenization of the governing equations was performed. For the case of longitudinal roughness, we derived a one-dimensional lubrication-type equation for the leading behavior of the pressure in the direction parallel to the velocity of the disk. The coefficients of the equation are determined by solving linear elliptic equations on a domain bounded by the gap height in the vertical direction and the period of the roughness in the span-wise direction. For the case of transverse roughness the unsteady lubrication equations were reduced, following a multiple scale homogenization analysis, to a steady equation for the leading behavior of the pressure in the gap. The reduced equation involves certain averages of the gap height, but retains the same form of the usual steady, compressible lubrication equations. Numerical calculations were performed for both cases, and the solution for the case of transverse roughness was shown be in excellent agreement with a corresponding numerical calculation of the original unsteady equations

On the ultraviolet signatures of small scale heating in coronal loops

Studying the statistical properties of solar ultraviolet emission lines could provide information about the nature of small scale coronal heating. We expand on previous work to investigate these properties. We study whether the predicted statistical distribution of ion emission line intensities produced by a specified heating function is affected by the isoelectronic sequence to which the ion belongs, as well as the characteristic temperature at which it was formed. Particular emphasis is placed on the strong resonance lines belonging to the lithium isoelectronic sequence. Predictions for emission lines observed by existing space-based UV spectrometers are given. The effects on the statistics of a line when observed with a wide-band imaging instrument rather than a spectrometer are also investigated. We use a hydrodynamic model to simulate the UV emission of a loop system heated by nanoflares on small, spatially unresolved scales. We select lines emitted at similar temperatures but belonging to different isoelectronic groups: Fe IX and Ne VIII, Fe XII and Mg X, Fe XVII, Fe XIX and Fe XXIV. Our simulations confirm previous results that almost all lines have an intensity distribution that follows a power-law, in a similar way to the heating function. However, only the high temperature lines best preserve the heating function's power law index (Fe XIX being the best ion in the case presented here). The Li isoelectronic lines have different statistical properties with respect to the lines from other sequences, due to the extended high temperature tail of their contribution functions. However, this is not the case for Fe XXIV which may be used as a diagnostic of the coronal heating function. We also show that the power-law index of the heating function is effectively preserved when a line is observed by a wide-band imaging instrument rather than a spectromenter

Coronal loop widths and pressure scale heights

The scale heights of stratification and the widths of steady solar coronal loops exhibit properties unexplained by standard theory: observed scale heights are often much greater than static theory predicts, while the nearly-constant widths of loop emission signatures defy theoretical expectations for large flux tubes in stratified media. In this work we relate the cross-sectional profile of a coronal flux tube to its density scale height in steady-state plasma flow regimes. Steady flows may shorten or lengthen the scale height according to how the tube cross-sectional area varies with arclength. In a near-potential corona the flux tubes are expected to be sufficiently expansive in many active regions for scale heights to be increased by steady flows. On the other hand, cases where scale lengths are actually increased to observed sizes form a small part of the solution space, close to regimes where density profiles reverse. Therefore, although steady flows are the only steady process known to be capable of extending scale heights significantly, they are not expected to be not responsible for the majority of extended active region scale heights

The flaring and quiescent components of the solar corona

The solar corona is a template to understand stellar activity. The Sun is a moderately active star, and its corona differs from active stars: active stellar coronae have a double-peaked EM(T) with the hot peak at 8-20 MK, while the non flaring solar corona has one peak at 1-2 MK. We study the average contribution of flares to the solar EM(T) to investigate indirectly the hypothesis that the hot peak of the EM(T) of active stellar coronae is due to a large number of unresolved solar-like flares, and to infer properties on the flare distribution from nano- to macro-flares. We measure the disk-integrated time-averaged emission measure, EM_F(T), of an unbiased sample of solar flares analyzing uninterrupted GOES/XRS light curves over time intervals of one month. We obtain the EM_Q(T) of quiescent corona for the same time intervals from the Yohkoh/SXT data. To investigate how EM_F(T) and EM_Q(T) vary with the solar cycle, we evaluate them at different phases of the cycle (from Dec. 1991 to Apr. 1998). Irrespective of the solar cycle phase, EM_F(T) appears like a peak of the distribution significantly larger than the values of EM_Q(T) for T~5-10 MK. As a result the time-averaged EM(T) of the whole solar corona is double-peaked, with the hot peak, due to time-averaged flares, located at temperature similar of that of active stars, but less enhanced. The EM_F(T) shape supports the hypothesis that the hot EM(T) peak of active coronae is due to unresolved solar-like flares. If this is the case, quiescent and flare components should follow different scaling laws for increasing stellar activity. In the assumption that the heating of the corona is entirely due to flares, from nano- to macro-flares, then either the flare distribution or the confined plasma response to flares, or both, are bimodal.Comment: 8 pages, 7 postscript figures, accepted for publication in Astronomy and Astrophysic