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NASA NTRS1991

Inverse problems in diffraction

A two-dimensional problem of diffraction of a plane electromagnetic wave on a smooth 2 pi-periodic surface is considered. A numerical algorithm solving this problem is developed. An inverse problem of determination of the shape of 2 pi-periodic surface using the performance data of reverse scattering is considered. The inverse problem was solved by means of minimization of the residual functional with the help of the gradient descent method. The initial data were calculated with the help of the numerical method. On each step of the iterative method of minimization, the residual functional was calculated approximately with the help of the small slope method. The examples of the shape determination are considered.

Mikheev, Andrew G.↗

DOE OSTI2024

Liquid Crystal Orientation and Shape Optimization for the Active Response of Liquid Crystal Elastomers

Liquid crystal elastomers (LCEs) are responsive materials that can undergo large reversible deformations upon exposure to external stimuli, such as electrical and thermal fields. Controlling the alignment of their liquid crystals mesogens to achieve desired shape changes unlocks a new design paradigm that is unavailable when using traditional materials. While experimental measurements can provide valuable insights into their behavior, computational analysis is essential to exploit their full potential. Accurate simulation is not, however, the end goal; rather, it is the means to achieve their optimal design. Such design optimization problems are best solved with algorithms that require gradients, i.e., sensitivities, of the cost and constraint functions with respect to the design parameters, to efficiently traverse the design space. In this work, a nonlinear LCE model and adjoint sensitivity analysis are implemented in a scalable and flexible finite element-based open source framework and integrated into a gradient-based design optimization tool. To display the versatility of the computational framework, LCE design problems that optimize both the material, i.e., liquid crystal orientation, and structural shape to reach a target actuated shapes or maximize energy absorption are solved. Multiple parameterizations, customized to address fabrication limitations, are investigated in both 2D and 3D. The case studies are followed by a discussion on the simulation and design optimization hurdles, as well as potential avenues for improving the robustness of similar computational frameworks for applications of interest.

42 ENGINEERING↗

NASA NTRS1998

Incompressible Turbulent Wing-Body Junction Flow

The overall objective of this study is to contribute to the optimized design of fan bypass systems in advanced turbofan engines. Increasing the engine bypass ratios have provided a major boost in engine performance improvement over the last fifty years. An engine with high bypass ratio (11-16:1) such as the Advanced Ducted Propulsion (ADP) is being developed and is expected to provide an additional 25% improvement in overall efficiency over the early turbofans. Such significant improvements in overall efficiency would reduce the cost per seat mile, which is a major government and Industry challenge for the 21th century. The research is part of the Advanced Subsonic Technology (AST) program that involves a NASA, U.S. Industry and FAA partnership with the goal of a safe and highly productive global air transportation system. The immediate objective of the study is to perform numerical simulation of duct-strut interactions to elucidate the loss mechanisms associated with this configuration that is typical of advanced turbofan engines such as ADP. However, at present experimental data for a duct-strut configuration are not available. Thus, as a first step a wing-body junction flow would be studied and is the specific objective of the present study. At the outset it is to be recognized that while duct-strut interaction flow is similar to that of wing-body junction flows, there are some differences owing to the presence of a wall at both ends of the strut. Likewise, some differences are due to the sheared inflow (as opposed to a uniform inflow) velocity profile. It is however expected that some features of a wing-body junction flow would persist. Next, some of the salient aspects of the complex flow near a wing-body junction, as revealed by various studies reported in the literature will be reviewed. One of the principle characteristics of the juncture flow, is the presence of the mean flow components in a plane perpendicular to the direction of the oncoming free-stream flow. The lateral curvature of the wing/strat causes the oncoming turbulent layer to skew about am axis (x-axis) parallel to the plane (xz-plane) of the mean shear. This is the principle mechanism for the generation of secondary flow. Such skew-induced secondary flows are slow to be attenuated by Reynolds stresses. Additional contribution to the generation of secondary flow comes from anisotropies in Reynolds stresses. Upstream of the strut, the mean-vorticity is directed span wise (along the y-direction). The presence of secondary flow in the vicinity of the strut causes the vorticity to stretch around the obstacle in a horse-shoe shape, with each leg having a vorticity of the opposite sense. The blockage effect of the strut imposes a severe adverse pressure gradient on the oncoming turbulent shear layer, causing boundary layer separation ahead of the leading edge, resulting in a vortex that rolls up and flows downstream into the juncture region. The separation vortices trailing in the wake of the wing can alter the lift or drag characteristics of the surfaces downstream of the wing-body juncture. Likewise, on submarines, the wake flow behind the appendage can degrade the performance of the propeller located downstream. The complex nature of this flow is caused by the presence of all six components of Reynolds stresses. Devenport and Simpson report that in the vicinity of the horse-shoe vortex there is intense recirculation with turbulent stresses being much larger than those normally observed in turbulent flows. These features contribute to making this flow a challenge to predict numerically. Some of the past studies provide useful insights into this flow that would guide our numerical efforts. In measurements reported by Shabaka and Bradshaw, the eddy viscosity tensor is seen to be non-isotropic and has negative components in certain regions. In an effort to evaluate the closure assumptions of various turbulence models, Devenport and Simpson used their own extensive measurements in juncture flows around the nose of a wing-body junction. Measured values of mean-velocity and/or turbulence kinetic energy was used to predict the magnitude of the shear stress vector. Algebraic stress models performed the best followed by Cebeci-Smith eddy viscosity model. The flow is reported to be dominated by a pressure field produced by the wing and the velocity field generated by the horseshoe vortex that is wrapped around the junction between the wing and wall. Kubendran et al. conclude from an experimental study that the shape of leading edge of the wing as characterized by its slenderness ratio is a major factor in determining the flow fields in the juncture region. The more thinner the leading edge of the juncture, the weaker the horseshoe vortex is. Also, with a slender leading edge, the secondary flow in the juncture would be due mainly to the cross-stream gradients of Reynolds stresses rather than due to a lateral skewing of the shear layer.

Krishnamurthy, R.↗

DOE OSTI2015

Predicting the intensity mapping signal for multi-J CO lines

We present a novel approach to estimating the intensity mapping signal of any CO rotational line emitted during the Epoch of Reionization (EoR). Our approach is based on large velocity gradient (LVG) modeling, a radiative transfer modeling technique that generates the full CO spectral line energy distribution (SLED) for a specified gas kinetic temperature, volume density, velocity gradient, molecular abundance, and column density. These parameters, which drive the physics of CO transitions and ultimately dictate the shape and amplitude of the CO SLED, can be linked to the global properties of the host galaxy, mainly the star formation rate (SFR) and the SFR surface density. By further employing an empirically derived SFR−M relation for high redshift galaxies, we can express the LVG parameters, and thus the specific intensity of any CO rotational transition, as functions of the host halo mass M and redshift z. Integrating over the range of halo masses expected to host CO-luminous galaxies, we predict a mean CO(1-0) brightness temperature ranging from ∼ 0.6 μK at z = 6 to ∼ 0.03 μK at z = 10 with brightness temperature fluctuations of Δ{sub CO}{sup 2} ∼ 0.1 and 0.005 μK respectively, at k = 0.1 Mpc{sup −1}. In this model, the CO emission signal remains strong for higher rotational levels at z = 6, with ( T{sub CO} ) ∼ 0.3 and 0.05 μK for the CO J = 6arrow5 and CO J = 10arrow9 transitions respectively. Including the effects of CO photodissociation in these molecular clouds, especially at low metallicities, results in the overall reduction in the amplitude of the CO signal, with the low- and high-J lines weakening by 2–20% and 10–45%, respectively, over the redshift range 4 < z < 10.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2015

SU-E-J-248: Contributions of Tumor and Stroma Phenotyping in Computer-Aided Diagnosis

Purpose: To gain insight into the role of parenchyma stroma in the characterization of breast tumors by incorporating computerized mammographic parenchyma assessment into breast CADx in the task of distinguishing between malignant and benign lesions. Methods: This study was performed on 182 biopsy-proven breast mass lesions, including 76 benign and 106 malignant lesions. For each full-field digital mammogram (FFDM) case, our quantitative imaging analysis was performed on both the tumor and a region-of-interest (ROI) from the normal contralateral breast. The lesion characterization includes automatic lesion segmentation and feature extraction. Radiographic texture analysis (RTA) was applied on the normal ROIs to assess the mammographic parenchymal patterns of these contralateral normal breasts. Classification performance of both individual computer extracted features and the output from a Bayesian artificial neural network (BANN) were evaluated with a leave-one-lesion-out method using receiver operating characteristic (ROC) analysis with area under the curve (AUC) as the figure of merit. Results: Lesion characterization included computer-extracted phenotypes of spiculation, size, shape, and margin. For parenchymal pattern characterization, five texture features were selected, including power law beta, contrast, and edge gradient. Merging of these computer-selected features using BANN classifiers yielded AUC values of 0.79 (SE=0.03) and 0.67 (SE=0.04) in the task of distinguishing between malignant and benign lesions using only tumor phenotypes and texture features from the contralateral breasts, respectively. Incorporation of tumor phenotypes with parenchyma texture features into the BANN yielded improved classification performance with an AUC value of 0.83 (SE=0.03) in the task of differentiating malignant from benign lesions. Conclusion: Combining computerized tumor and parenchyma phenotyping was found to significantly improve breast cancer diagnostic accuracy highlighting the need to consider both tumor and stroma in decision making. Funding: University of Chicago Dean Bridge Fund, NCI U24-CA143848-05, P50-CA58223 Breast SPORE program, and Breast Cancer Research Foundation. COI: MLG is a stockholder in R2 technology/Hologic and receives royalties from Hologic, GE Medical Systems, MEDIAN Technologies, Riverain Medical, Mitsubishi, and Toshiba. MLG is a cofounder and stockholder in Quantitative Insights.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2024

Integrated Turbine Component Cooling Designs Facilitated by Additive Manufacturing and Optimization

This research program involved the development of improved gas turbine cooling by optimizing the combined internal and film cooling configurations used for turbine blades and vanes. Improved cooling of turbine airfoils allows the gas turbines to be operated at higher temperatures which improves efficiency for gas turbine-based power generation. This also facilitates operation with hydrogen-based fuels which inherently generate higher operating temperatures. Designs of the new cooling configurations were based on the added geometric flexibility that is available when using metal additive manufacturing techniques. When making these designs, adjustments were made to account for imperfections occurring in the metal additive manufacturing process. New designs for film cooling holes were developed using computational adjoint gradient optimization techniques which generated a unique film cooling hole with 70% greater film cooling effectiveness compared to conventional shaped film cooling holes. A major factor in improving the film cooling effectiveness were small external protrusions on either side of the new hole which generated vortices that pushed coolant towards the surface and increased the lateral spreading of the coolant. Our studies also included optimization of film cooling holes that can be built with conventional manufacturing techniques and optimizing the internal coolant channels that supply coolant to the film cooling holes. Various internal coolant channel shapes were investigated and designs that maximized the cooling of the airfoil while minimizing pressure losses in the channel flows were identified. Testing and verification of the performance of the final configurations was done with a transonic wind tunnel facility which allowed testing at engine realistic high velocities. Final testing of the combined internal and film cooling configurations was done by incorporating the configurations in Penn State’s National Experimental Turbine (NExT) vane. These tests confirmed the enhanced overall cooling effectiveness provided by the new cooling configuration designs.

03 NATURAL GAS↗

NASA NTRS1975

Solar-wind and solar-flare maturation of the lunar regolith

Detailed predictions concerning the depth and time-dependent accumulation of solar-wind effects and solar-flare tracks in lunar dust grains were obtained by use of an adaptation of a Monte Carlo soil-mixing computer code described by Duraud et al. (1975). The predictions are compared to experimental measurements obtained by analyzing lunar dust grains as well as artificially irradiated minerals by a variety of techniques. A study of amorphous coatings of solar-wind radiation-damaged material on certain lunar grains sets limits on the integrated residence time of these grains in the ancient solar wind. Other topics discussed include solar wind maturation, the peculiar shape of the experimental distribution of central track densities in 50-micron grains, and the interpretation of both track gradients in 50-micron feldspars and the relatively 'low' concentration of solar-wind species implanted in ilmenite grains.

Bibring, J. P.↗

NASA NTRS1987

Distributed computer system enhances productivity for SRB joint optimization

The Programming System for Structural Synthesis software package, which couples structural analysis and optimization, has been distributed over a network of work stations for use in Space Shuttle Solid Rocket Booster joint redesign optimization. Finite difference computing techniques were applied to the optimization gradients in parallel execution, allowing several work stations to simultaneously contribute to the problem's solution. An optimal joint shape was obtained which achieves minimum weight while keeping the gap between joints well closed and limiting structural stresses. The optimization cycle was reduced from two hours to one-half hour.

Rogers, James L., Jr.↗

NASA NTRS1994

Thermocapillary motion of deformable drops

The thermocapillary motion of initially spherical drops/bubbles driven by a constant temperature gradient in an unbounded liquid medium is simulated numerically. Effects of convection of momentum and energy, as well as shape deformations, are addressed. The method used is based on interface tracking on a base cartesian grid, and uses a smeared color or indicator function for the determination of the surface topology. Quad-tree adaptive refinement of the cartesian grid is implemented to enhance the fidelity of the surface tracking. It is shown that convection of energy results in a slowing of the drop, as the isotherms get wrapped around the front of the drop. Shape deformation resulting from inertial effects affect the migration velocity. The physical results obtained are in agreement with the existing literature. Furthermore, remarks are made on the sensitivity of the calculated solutions to the smearing of the fluid properties. Analysis and simulations show that the migration velocity depends very strongly on the smearing of the interfacial force whereas it is rather insensitive to the smearing of other properties, hence the adaptive grid.

Haj-Hariri, Hossein↗

NASA NTRS1997

Current Collection in a Magnetic Field

It is found that the upper-bound limit for current collection in the case of strong magnetic field from the current is close to that given by the Parker-Murphy formula. This conclusion is consistent with the results obtained in laboratory experiments. This limit weakly depends on the shape of the wire. The adiabatic limit in this case will be easily surpassed due to strong magnetic field gradients near the separatrix. The calculations can be done using the kinetic equation in the drift approximation. Analytical results are obtained for the region where the Earth's magnetic field is dominant. The current collection can be calculated (neglecting scattering) using a particle simulation code. Dr. Singh has agreed to collaborate, allowing the use of his particle code. The code can be adapted for the case when the current magnetic field is strong. The needed dm for these modifications is 3-4 months. The analytical description and essential part of the program is prepared for the calculation of the current in the region where the adiabatic description can be used. This was completed with the collaboration of Drs. Khazanov and Liemohn. A scheme of measuring the end body position is also proposed. The scheme was discussed in the laboratory (with Dr. Stone) and it was concluded that it can be proposed for engineering analysis.

Krivorutsky, E. N.↗

NASA NTRS2001

Feasibility of Actively Cooled Silicon Nitride Airfoil for Turbine Applications Demonstrated

Nickel-base superalloys currently limit gas turbine engine performance. Active cooling has extended the temperature range of service of nickel-base superalloys in current gas turbine engines, but the margin for further improvement appears modest. Therefore, significant advancements in materials technology are needed to raise turbine inlet temperatures above 2400 F to increase engine specific thrust and operating efficiency. Because of their low density and high-temperature strength and thermal conductivity, in situ toughened silicon nitride ceramics have received a great deal of attention for cooled structures. However, the high processing costs and low impact resistance of silicon nitride ceramics have proven to be major obstacles for widespread applications. Advanced rapid prototyping technology in combination with conventional gel casting and sintering can reduce high processing costs and may offer an affordable manufacturing approach. Researchers at the NASA Glenn Research Center, in cooperation with a local university and an aerospace company, are developing actively cooled and functionally graded ceramic structures. The objective of this program is to develop cost-effective manufacturing technology and experimental and analytical capabilities for environmentally stable, aerodynamically efficient, foreign-object-damage-resistant, in situ toughened silicon nitride turbine nozzle vanes, and to test these vanes under simulated engine conditions. Starting with computer aided design (CAD) files of an airfoil and a flat plate with internal cooling passages, the permanent and removable mold components for gel casting ceramic slips were made by stereolithography and Sanders machines, respectively. The gel-cast part was dried and sintered to final shape. Several in situ toughened silicon nitride generic airfoils with internal cooling passages have been fabricated. The uncoated and thermal barrier coated airfoils and flat plates were burner rig tested for 30 min without and with air cooling. Without cooling, the surface temperature of the flat plate reached approximately 2350 F. Starting with computer aided design (CAD) files of an airfoil and a flat plate with internal cooling passages, the permanent and removable mold components for gel casting ceramic slips were made by stereolithography and Sanders machines, respectively. The gel-cast part was dried and sintered to final shape. Several in situ toughened silicon nitride generic airfoils with internal cooling passages have been fabricated. The uncoated and thermal barrier coated airfoils and flat plates were burner rig tested for 30 min without and with air cooling. Without cooling, the surface temperature of the flat plate reached approximately 2350 F. With cooling, the surface temperature decreased to approximately 1910 F--a drop of approximately 440 F. This preliminary study demonstrates that a near-net-shape silicon nitride airfoil can be fabricated and that silicon nitride can sustain severe thermal shock and the thermal gradients induced by cooling and, thus, is a viable candidate for cooled components.

Bhatt, Ramakrishna T.↗

NASA NTRS2011

Use of Dual-Polarization Radar Variables to Assess Low-Level Wind Shear in Severe Thunderstorm Near-storm Environments in the Tennessee Valley

The upgrade of the National Weather Service (NWS) network of S ]band dual-polarization radars is currently underway, and the incorporation of polarimetric information into the real ]time forecasting process will enhance the forecaster fs ability to assess thunderstorms and their near ]storm environments. Recent research has suggested that the combination of polarimetric variables differential reflectivity (ZDR) and specific differential phase (KDP) can be useful in the assessment of low level wind shear within a thunderstorm. In an environment with strong low ]level veering of the wind, ZDR values will be largest along the right inflow edge of the thunderstorm near a large gradient in horizontal reflectivity (indicative of large raindrops falling with a relative lack of smaller drops), and take the shape of an arc. Meanwhile, KDP values, which are proportional to liquid water content and indicative of a large number of smaller drops, are maximized deeper into the forward flank precipitation shield than the ZDR arc as the smaller drops are being advected further from the updraft core by the low level winds than the larger raindrops. Using findings from previous work, three severe weather events that occurred in North Alabama were examined in order to assess the utility of these signatures in determining the potential for tornadic activity. The first case is from October 26, 2010, where a large number of storms indicated tornadic potential from a standard reflectivity and velocity analysis but very few storms actually produced tornadoes. The second event is from February 28, 2011, where tornadic storms were present early on in the event, but as the day progressed, the tornado threat transitioned to a high wind threat. The third case is from April 27, 2011, where multiple rounds of tornadic storms ransacked the Tennessee Valley. This event provides a dataset including multiple modes of tornadic development, including QLCS and supercell structures. The overarching goal of examining these three events is to compare dual ]polarization features from this larger dataset to previous work and to determine if these signatures can be a useful indication of the potential for tornadic activity associated with the amount of low ]level wind shear in the near ]storm environment.

Crowe, Christina C.↗

NASA NTRS2014

Mean Flow Augmented Acoustics in Rocket Systems

Combustion instability in solid rocket motors and liquid engines has long been a subject of concern. Many rockets display violent fluctuations in pressure, velocity, and temperature originating from the complex interactions between the combustion process and gas dynamics. Recent advances in energy based modeling of combustion instabilities require accurate determination of acoustic frequencies and mode shapes. Of particular interest is the acoustic mean flow interactions within the converging section of a rocket nozzle, where gradients of pressure, density, and velocity become large. The expulsion of unsteady energy through the nozzle of a rocket is identified as the predominate source of acoustic damping for most rocket systems. Recently, an approach to address nozzle damping with mean flow effects was implemented by French [1]. This new approach extends the work originated by Sigman and Zinn [2] by solving the acoustic velocity potential equation (AVPE) formulated by perturbing the Euler equations [3]. The present study aims to implement the French model within the COMSOL Multiphysiscs framework and analyzes one of the author's presented test cases.

Fischbach, Sean↗

DOE OSTI2023

High absorption photovoltaic material and methods of making the same

A high absorption photovoltaic material and method of making the material for use in a solar cell are disclosed. The photovoltaic material includes a surface modified with a layer of repeating photonic crystal structures. The photonic crystal structures are approximately inverse conically shaped and have a curved sidewall that has an approximately Gaussian shape. The photonic crystal structures generally have a high vertical depth and sidewall angle. The structures also have a gradient refractive index profile and exhibit the parallel-to-interface refraction light trapping effect. An anti-reflective coating is disposed over the photonic crystal structure layer. The photovoltaic material exhibits near unity light absorption over a broad range of visible and near infrared wavelengths and incidence angles, even at reduced thicknesses. The photovoltaic structures are formed via a combined photolithography and reactive-ion etching method at low power with a gas mixture having a high ratio of an etchant component to a passivation component.

Kuang, Ping↗

NASA NTRSDate not supplied

Aerothermal Shape Optimization of Actively-Cooled Battery Packs using Conjugate Heat Transfer

Thermal management for battery is important for electric aircraft because battery temperature is critically important to vehicle safety, and it also has direct impact on the efficiency of the battery system. Because ambient air is a readily available resource for aircraft, this paper considers an active cooling concept with forced convection of ambient air through the battery pack. Conjugate heat transfer analysis is used to solve the coupled aero-thermal problem, which consists of a finite-volume computational fluid dynamics solver for the fluid domain, and a conduction heat transfer solver for the solid domain. A mixed Neumann and Dirichlet boundary condition is developed for the fluid-solid interface, which allows the solid domain to completely submerge in the fluid domain. A gradient-based optimization method is adopted, and the discrete adjoint approach implemented in DAFoam is used to efficiently compute the gradients. The aero-thermal coupling for primal analysis and gradient computation is handled using the OpenMDAO-based MPhys framework. A constant heat source is prescribed for the battery cells, and the battery shape (design variable) is optimized to minimize cooling pump power and battery weight (composite objective function) while keeping the battery temperature below a threshold (constraint). The optimized design achieves a 44.6% and 1.5% reduction in the cooling pump power and battery weight, respectively, and the maximal temperature constraint is satisfied. This work has the potential to reduce battery-pack weight, improve performance, and reduce the weight of thermal management systems for electric vertical take-off and landing aircraft.

thermal management↗

NASA NTRSDate not supplied

Aerothermal Shape Optimization of Actively-Cooled Battery Packs Using Conjugate Heat Transfer

heat transfer↗

NASA NTRS1996

Towards and FVE-FAC Method for Determining Thermocapillary Effects on Weld Pool Shape

Several practical materials processes, e.g., welding, float-zone purification, and Czochralski crystal growth, involve a pool of molten metal with a free surface, with strong temperature gradients along the surface. In some cases, the resulting thermocapillary flow is vigorous enough to convect heat toward the edges of the pool, increasing the driving force in a sort of positive feedback. In this work we examine this mechanism and its effect on the solid-liquid interface through a model problem: a half space of pure substance with concentrated axisymmetric surface heating, where surface tension is strong enough to keep the liquid free surface flat. The numerical method proposed for this problem utilizes a finite volume element (FVE) discretization in cylindrical coordinates. Because of the axisymmetric nature of the model problem, the control volumes used are torroidal prisms, formed by taking a polygonal cross-section in the (r, z) plane and sweeping it completely around the z-axis. Conservation of energy (in the solid), and conservation of energy, momentum, and mass (in the liquid) are enforced globally by integrating these quantities and enforcing conservation over each control volume. Judicious application of the Divergence Theorem and Stokes' Theorem, combined with a Crank-Nicolson time-stepping scheme leads to an implicit algebraic system to be solved at each time step. It is known that near the boundary of the pool, that is, near the solid-liquid interface, the full conduction-convection solution will require extremely fine length scales to resolve the physical behavior of the system. Furthermore, this boundary moves as a function of time. Accordingly, we develop the foundation of an adaptive refinement scheme based on the principles of Fast Adaptive Composite Grid methods (FAC). Implementation of the method and numerical results will appear in a later report.

Canright, David↗

NASA NTRS1972

Idealized model for the radial gradient of modulated cosmic rays.

Development of a conceptually idealized model for interpreting existing and future measurements of the radial gradient of modulated cosmic rays in the limit that deceleration and convection effects become comparable. The proposed model assumes that the modulated spectrum consists of two idealized regions: at low energy, below some critical transition energy T sub c, a region where the shape of the spectrum is assumed to be determined wholly by a balance between convection and deceleration effects; and, at higher energies, above T sub c, a region where the magnitude and shape of the spectrum are taken to be those given by the usual convection-diffusion modulation. The application of this model, invoking some straightforward assumptions about the behavior of T sub c, suggests that the radial gradient in the regime where deceleration processes are dominant may be related to the spectral index of the unmodulated spectrum at the energy T sub c and to the absolute level of modulation. This is in contrast to the gradient at high energies, above T sub c, which is related only to local interplanetary characteristics through convection and diffusion.

O'Gallagher, J. J.↗