Theory of elastic plates in the reference state.
Stress tensor analysis of elastic plates in reference state, noting nonlinear plate theory and variational principle
ELASTIC PLATE↗
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On the equations of motion of shells in the reference state.
Shell motion equations derived in terms of reference state from three-dimensional theory based on Cauchy motion laws for undeformed bodies
STRESS CALCULATION↗
Theory of elastic shells in the reference state.
Nonlinear field equations of elastic shell in terms of reference state subjected to strain displacement
SHELL THEORY↗
Special perturbations employing osculating reference states
The concept of employing osculating reference position and velocity vectors in the numerical integration of the equations of motion of a satellite is examined. The choice of the reference point is shown to have a significant effect upon numerical efficiency and the class of trajectories described by the differential equations of motion. For example, when the position and velocity vectors on the osculating orbit at a fixed reference time are chosen, a universal formulation is yielded. For elliptical orbits, however, this formulation is unattractive for numerical integration purposes due to Poisson terms (mixed secular) appearing in the equations of motion. Other choices for the reference point eliminate this problem but usually at the expense of universality. A number of these formulations, including a universal one, are considered here. Comparisons of the numerical characteristics of these techniques with those of the Encke method are presented.
Born, G. H.↗
Integral Hellmann-Feynman analysis of the binding energy of H2 and LiH using atomic reference states
Integral Hellman-Feynman formula applied to binding energy of molecular H and LiH, using SCF wavefunctions for atomic states
Marron, M. T.↗
The Moist Available Energy of a Conditionally Unstable Atmosphere
Two simple methods for computing the moist available energy for cases in which the total number of parcels is fairly small are discussed. The first is a swapping routine that finds the reference state by exchanging parcels whenever the swap reduces the total enthalpy. However, it can be proved by contrived example that the swapping algorithm does not always find the true reference state. A second brute force routine actually examines all possible arrangements of the parcels and chooses the one with the smallest moist enthalpy. At present, both routines allow only horizontally homogeneous reference states. In an exploratory study, the swapping routine was used to find MAE and reference state for a time sequence of GATE soundings, all of which are conditionally unstable. In spot checks, the brute force routine always agreed with the swapper. Results show that in passing from the given state of the reference state the lowest three hundred mb of the atmosphere typically exchanges places with the 300 mb above it. The reference state then contains a dry warm layer near the surface, but a supersaturated but cool layer in the middle troposphere. The upper portion of the sounding is not modified in passing from the given state to the reference state.
Randall, D.↗
Modelling Greenland Outlet Glaciers
The objective of this project was to develop simple yet realistic models of Greenland outlet glaciers to better understand ongoing changes and to identify possible causes for these changes. Several approaches can be taken to evaluate the interaction between climate forcing and ice dynamics, and the consequent ice-sheet response, which may involve changes in flow style. To evaluate the icesheet response to mass-balance forcing, Van der Veen (Journal of Geophysical Research, in press) makes the assumption that this response can be considered a perturbation on the reference state and may be evaluated separately from how this reference state evolves over time. Mass-balance forcing has an immediate effect on the ice sheet. Initially, the rate of thickness change as compared to the reference state equals the perturbation in snowfall or ablation. If the forcing persists, the ice sheet responds dynamically, adjusting the rate at which ice is evacuated from the interior to the margins, to achieve a new equilibrium. For large ice sheets, this dynamic adjustment may last for thousands of years, with the magnitude of change decreasing steadily over time as a new equilibrium is approached. This response can be described using kinematic wave theory. This theory, modified to pertain to Greenland drainage basins, was used to evaluate possible ice-sheet responses to perturbations in surface mass balance. The reference state is defined based on measurements along the central flowline of Petermann Glacier in north-west Greenland, and perturbations on this state considered. The advantage of this approach is that the particulars of the dynamical flow regime need not be explicitly known but are incorporated through the parameterization of the reference ice flux or longitudinal velocity profile. The results of the kinematic wave model indicate that significant rates of thickness change can occur immediately after the prescribed change in surface mass balance but adjustments in flow rapidly diminish these rates to a few cm/yr at most. The time scale for adjustment is of the order of a thousand years or so.
vanderVeen, Cornelis↗
Curve Fits of the NIST-JANNAF Thermochemical Tables Fourth Edition
Piecewise polynomial coefficients providing ideal gas, temperature-dependent molar specific heat, enthalpy, and entropy are tabulated for 1,068 common elemental and molecular species from the Fourth Edition (1998) to the NIST-JANAF Thermochemical Tables. The coefficients provide predictive values covering a wide range of temperatures (in many cases for gas state species from 298.15 to 6,000 K). A total of 1,760 fits are provided covering both single states (for example, gas, liquid, and crystal) and multiple states (crystal and liquid). Appropriate units and reference states are used for the two different fit formats, but the requisite procedure to adjust the coefficients for any other units and reference states is discussed.
high temperature properities↗
Transport Coefficients in Rotating Weakly Compressible Turbulence
Analytical studies of compressible turbulence have found that compressible velocity fluctuations create both effective fluid transport properties and an effective equation of state. This paper investigates the effects of rotation on compressible turbulence. It is shown that rotation modifies the transport properties of compressible turbulence by replacing the turbulence time scale by a rotational time scale, much as rotation modifies the transport properties of incompressible turbulence. But thermal equilibrium properties are modified in a more complex manner. Two regimes are possible: one dominated by incompressible fluctuations, in which the sound speed is modified as it is in non-rotating turbulence, and a rotation dominated regime in which the sound speed enhancement is rotation dependent. The dimensionless parameter which discriminates between regimes is identified. In general, rotation is found to suppress the effects of compressibility. A novel feature of the present analysis is the use of a non-Kolmogorov steady state as the reference state of turbulence. introduction of such steady states expands the power and utility of analytical turbulence closures to a wider range of problems.
Rubinstein, Robert↗
High Energy Boundary Conditions for a Cartesian Mesh Euler Solver
Inlets and exhaust nozzles are common place in the world of flight. Yet, many aerodynamic simulation packages do not provide a method of modelling such high energy boundaries in the flow field. For the purposes of aerodynamic simulation, inlets and exhausts are often fared over and it is assumed that the flow differences resulting from this assumption are minimal. While this is an adequate assumption for the prediction of lift, the lack of a plume behind the aircraft creates an evacuated base region thus effecting both drag and pitching moment values. In addition, the flow in the base region is often mis-predicted resulting in incorrect base drag. In order to accurately predict these quantities, a method for specifying inlet and exhaust conditions needs to be available in aerodynamic simulation packages. A method for a first approximation of a plume without accounting for chemical reactions is added to the Cartesian mesh based aerodynamic simulation package CART3D. The method consists of 3 steps. In the first step, a components approach where each triangle is assigned a component number is used. Here, a method for marking the inlet or exhaust plane triangles as separate components is discussed. In step two, the flow solver is modified to accept a reference state for the components marked inlet or exhaust. In the third step, the flow solver uses these separated components and the reference state to compute the correct flow condition at that triangle. The present method is implemented in the CART3D package which consists of a set of tools for generating a Cartesian volume mesh from a set of component triangulations. The Euler equations are solved on the resulting unstructured Cartesian mesh. The present methods is implemented in this package and its usefulness is demonstrated with two validation cases. A generic missile body is also presented to show the usefulness of the method on a real world geometry.
Pandya, Shishir↗
Evaluation of atomic constants for optical radiation, volume 2
Various atomic constant for 23 elements from helium to mercury were computed and are presented in tables. The data given for each element start with the element name, its atomic number, its ionic state, and the designation and series limit for each parent configuration. This is followed by information on the energy level, parent configuration, and designation for each term available to the program. The matrix elements subtables are ordered by the sequence numbers, which represent the initial and final levels of the transitions. Each subtable gives the following: configuration of the core or parent, designation and energy level for the reference state, effective principal quantum number, energy of the series limit, value of the matrix element for the reference state interacting with itself, and sum of all of the dipole matrix elements listed in the subtable. Dipole and quadrupole interaction data are also given.
Kylstra, C. D.↗
A diagnostic for denitrification in the winter polar stratospheres
The pairwise correlation of NO(y) and N2O data from the Southern and Northern Hemispheres is presented. Both data sets show a linear correlation region, defined as a reference state, and regions of denitrification where the correlation breaks down. Using two-dimensional photochemical model simulations of the atmosphere, a similar linear correlation is found between NO(y) and N2O, thereby establishing a theoretical framework for the reference state. This general approach, which can be extended to other pairs of molecules, should prove to be powerful in further comparisons of aircraft data with numerical models.
Fahey, D. W.↗
Feeding Ten Billion People Is Possible Within Four Terrestrial Planetary Boundaries
Global agriculture puts heavy pressure on planetary boundaries, posing the challenge to achieve future food security without compromising Earth system resilience. On the basis of process-detailed, spatially explicit representation of four interlinked planetary boundaries (biosphere integrity, land-system change, freshwater use, nitrogen flows) and agricultural systems in an internally consistent model framework, we here show that almost half of current global food production depends on planetary boundary transgressions. Hotspot regions, mainly in Asia, even face simultaneous transgression of multiple underlying local boundaries. If these boundaries were strictly respected, the present food system could provide a balanced diet (2,355 kcal per capita per day) for 3.4 billion people only. However, as we also demonstrate, transformation towards more sustainable production and consumption patterns could support 10.2 billion people within the planetary boundaries analysed. Key prerequisites are spatially redistributed cropland, improved water–nutrient management, food waste reduction and dietary changes. Adoption of the Sustainable Development Goals by all nations in 2015 is the first ever commitment to a world development path that safeguards the stability of the Earth system as a prerequisite for meeting universal human standards1. The longstanding challenge of achieving food security through sustainable agriculture is particularly acute in this context as world agriculture is a leading cause for the current transgressions of multiple planetary boundaries (PBs) globally and regionally2–5. The PB framework is a comprehensive scientific attempt to synoptically define our planet’s biogeophysical limits to anthropogenic interference. It suggests bounds to nine interacting processes that together delineate a Holocene-like Earth system state. The Holocene is chosen as the reference state as it is the only period known to provide a safe operating space for a world population of several billion people, and according to a precautionary principle, the PBs are set in sufficient distance from processes that may critically undermine Earth system resilience and global sustainability. A challenging question, thus, is whether human development goals such as food security can be met while maintaining multiple PBs along with their subglobal manifestations. Further PB transgressions could jeopardize the chances of providing sufficient food for a world population projected to be wealthier and reach >9 billion by 2050. This conundrum portrays a tradeoff between Earth’s biophysical carrying capacity and humankind’s rising food demand, calling in response for radical rethinking of food production and consumption patterns6–9. Yield gap closures, avoidance of excessive input use, shifts towards less resource-demanding diets, food waste reductions and efficient international trade are crucial options for sustainably increasing the food supply10–15. For example, enhancing water-use efficiency on irrigated and rain-fed farms can triple or quadruple crop yields in low-performing systems, suggesting possible global gains of >20% (ref. 16). Even higher gains appear feasible through globally optimized configurations of the land-use pattern17, and cutting food losses by half could generate food for another billion people18. Thus, collective large-scale implementation of such options could sustain food for a further growing world population19. Yet achieving this within a safe operating space as defined by PBs requires not only a halt to but actually a reversal of existing PB transgressions. Previous studies suggest that such a reconciliation might be possible, but these were based on aggregate representations of PBs (not accounting for the spatial patterns of limits, transgressions and interactions) or considered only one boundary in isolation17,20–23. Here, we systematically quantify to what extent current food production depends on local to global transgressions of the PBs for biosphere integrity, land-system change, freshwater use and nitrogen (N) flows, along with the potential of a range of solutions to avoid these transgressions and still increase food supply (Table 1). To this end, we configured an internally consistent process-based model of the terrestrial biosphere including agriculture (LPJmL) with multiple spatially distributed PBs and their interactions. LPJmL is among the longest-established and best-evaluated biosphere models, showing robust performance regarding simulation of, for example, carbon, water and crop yield dynamics (Supplementary Figs. 1 and 2 and Supplementary Table 1; see ref. 24 for a comprehensive benchmarking and Supplementary Methods for more detail on model evaluations). In principle following established definitions4, we refine the computation of some PBs with respect to their regional patterns and interactions (Methods), providing globally gridded precautionary limits to human interference with the Earth system at a level of great detail. In particular, we account for the evidence that many PBs need to be represented spatially explicitly4 to cover their
Gerten, Dieter↗
Profile-Following Entry Guidance Using Linear Quadratic Regulator Theory
This paper describes one of the entry guidance concepts that is currently being tested as part of Marshall Space Flight Center's Advance Guidance and Control Project. The algorithm is of the reference profile tracking type. The reference profile consists of the reference states, range-to-go, altitude, and flight path angle, and reference controls, bank angle and angle of attack, versus energy. A linear control law using state feedback is used with energy-scheduled gains. The gains are obtained offline using Matlab's steady state linear quadratic regulator function. Lateral trajectory control is effected by performing periodic bank sign reversals based on a heading error corridor. A description and results of the AG&C test cases on which it has been tested are given. Although it is not anticipated that the algorithm will be quite as robust as algorithms with onboard trajectory re-generation capability, the results nevertheless show it to be very robust with respect to varying initial conditions and works satisfactorily even for entries from widely different orbits than that of the reference profile. Moreover, the commanded bank and angle of attack histories are very smooth, making it easier for the attitude control system to implement the guidance commands. Finally, results indicate that the guidance gains are more or less trajectory-independent which is a potentially useful property.
Dukeman, Greg A.↗
Realistic error bounds for a reduced-state model-reference controller
Realistic error bounds for reduced-state model-reference controller
Taylor, T. M.↗
Operational Approach to Generalized Coherent States
Generalized coherent states for general potentials, constructed through a controlling mechanism, can also be obtained applying on a reference state suitable operators. An explicit example is supplied.
DeMartino, Salvatore↗
The PMIP4 Contribution to CMIP6 - Part 1: Overview and Over-Arching Analysis Plan
This paper is the first of a series of four GMD (Geoscientific Model Development) papers on the PMIP4-CMIP6 (Paleoclimate Modelling Intercomparison Project - Phase 4 -- Coupled Model Intercomparison Project - Phase 6) experiments. Part 2 (Otto-Bliesner et al., 2017) gives details about the two PMIP4-CMIP6 interglacial experiments, Part 3 (Jungclaus et al., 2017) about the last millennium experiment, and Part 4 (Kageyama et al., 2017) about the Last Glacial Maximum experiment. The mid-Pliocene Warm Period experiment is part of the Pliocene Model Intercomparison Project (PlioMIP) - Phase 2, detailed in Haywood et al. (2016). The goal of the Paleoclimate Modelling Intercomparison Project (PMIP) is to understand the response of the climate system to different climate forcings for documented climatic states very different from the present and historical climates. Through comparison with observations of the environmental impact of these climate changes, or with climate reconstructions based on physical, chemical, or biological records, PMIP also addresses the issue of how well state-of-the-art numerical models simulate climate change. Climate models are usually developed using the present and historical climates as references, but climate projections show that future climates will lie well outside these conditions. Palaeoclimates very different from these reference states therefore provide stringent tests for state-of-the-art models and a way to assess whether their sensitivity to forcings is compatible with palaeoclimatic evidence. Simulations of five different periods have been designed to address the objectives of the sixth phase of the Coupled Model Intercomparison Project (CMIP6): the millennium prior to the industrial epoch (CMIP6 name: past1000); the mid-Holocene, 6000 years ago (midHolocene); the Last Glacial Maximum, 21,000 years ago (lgm); the Last Interglacial, 127,000 years ago (lig127k); and the mid-Pliocene Warm Period, 3.2 million years ago (midPliocene-eoi400). These climatic periods are well documented by palaeoclimatic and palaeoenvironmental records, with climate and environmental changes relevant for the study and projection of future climate changes. This paper describes the motivation for the choice of these periods and the design of the numerical experiments and database requests, with a focus on their novel features compared to the experiments performed in previous phases of PMIP and CMIP. It also outlines the analysis plan that takes advantage of the comparisons of the results across periods and across CMIP6 in collaboration with other MIPs.
climate↗
Equations of state and constitutive equations
Equations-of state information in the otherwise undeformed state as a starting point for the development of constitutive equations is considered in addition to thermodynamics, free energy functions, and conceptual difficulties including the definition of reference states for strain. V-T effects in the form of the Simha-Somcynsky (1969) equation of state are explicitly discussed, and it is shown how this model can be modified to produce a constitutive equation. Continuum mechanics approaches are considered, and examples are given of developments based on linear viscoelastic theory which directly incorporate stress-induced volume changes, and on large-strain elastic theory.
Landel, Robert F.↗