An analysis of the vertical structure equation for arbitrary thermal profiles

The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. The spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions, both for arbitrary thermal profiles were studied. The results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. In the case of a bounded atmosphere it is shown that the spectrum is always totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigen value, upper and lower bounds which depend only on the surface temperature and the atmosphere height were obtained. All eigenfunctions are bounded, but always have unbounded first derivatives. It was proved that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, is well posed. It was concluded that the vertical structure equation always has a totally discrete spectrum under the assumptions implicit in the primitive equations

Energetic Consistency and Coupling of the Mean and Covariance Dynamics

The dynamical state of the ocean and atmosphere is taken to be a large dimensional random vector in a range of large-scale computational applications, including data assimilation, ensemble prediction, sensitivity analysis, and predictability studies. In each of these applications, numerical evolution of the covariance matrix of the random state plays a central role, because this matrix is used to quantify uncertainty in the state of the dynamical system. Since atmospheric and ocean dynamics are nonlinear, there is no closed evolution equation for the covariance matrix, nor for the mean state. Therefore approximate evolution equations must be used. This article studies theoretical properties of the evolution equations for the mean state and covariance matrix that arise in the second-moment closure approximation (third- and higher-order moment discard). This approximation was introduced by EPSTEIN [1969] in an early effort to introduce a stochastic element into deterministic weather forecasting, and was studied further by FLEMING [1971a,b], EPSTEIN and PITCHER [1972], and PITCHER [1977], also in the context of atmospheric predictability. It has since fallen into disuse, with a simpler one being used in current large-scale applications. The theoretical results of this article make a case that this approximation should be reconsidered for use in large-scale applications, however, because the second moment closure equations possess a property of energetic consistency that the approximate equations now in common use do not possess. A number of properties of solutions of the second-moment closure equations that result from this energetic consistency will be established

Effectiveness of student response systems in terms of learning environment, attitudes and achievement

Most past research on the effectiveness of Student Response Systems (SRS) has focused on higher levels of education and neglected consideration of the learning environment. Therefore, this study is unique in its focus on Grade 7 and Grade 8 students and on the effect of using SRS on students’ perceptions of the learning environment, as well as on the student outcomes of attitudes and achievement. This study also validated a new questionnaire, the How Do You Feel About This Class? (HDYFATC), which incorporates a new learning environment scale (Comfort) developed by the researcher. As schools incorporate technology such as SRS into the classroom, it is important to evaluate its effectiveness in terms of students’ perceptions of the learning environment, attitudes, and achievement.Student perceptions of the learning environment and their attitudes were assessed with the HDYFATC, which combines four learning environment scales (Involvement, Task Orientation, Equity, and Cooperation) from the What Is Happening In this Class? (WIHIC) questionnaire with one created by the researcher (Comfort) to assess how comfortable students are in their science class, and an attitude scale (Enjoyment) from the Test of Science Related Attitudes (TOSRA). Students’ achievement was assessed using the average of their examination scores for the duration of the study.The HDYFATC was administered to a sample of 1097 Grade 7 and Grade 8 students from 47 classes in three schools in New York State. Data analyses supported the HDYFATC’s factorial validity, internal consistency reliability, and ability to differentiate between the perceptions of students in different classrooms. All items had a factor loading of at least 0.40 on their a priori scale and less than 0.40 on all other scales. The total variance was 76.13%, with the largest contribution from the Enjoyment scale. Eigenvalues ranged from 1.29 to 25.62. When the individual was used as the unit of analysis, the internal consistency reliability for different scales of the HDYFATC ranged from 0.94 to 0.95. ANOVA revealed significant differences between students’ perceptions in different classes for each learning environment scale, with eta² values ranging from 0.50 to 0.60 for different scales.To determine the effectiveness of SRS in terms of learning environment, attitudes, and achievement, data obtained from the HDYFATC and achievement scores were subjected to a MANOVA. The dependent variables were the five learning environment scales and two student outcome scales, while use or non-use of SRS was the independent variable. Because the multivariate test using Wilks’ lambda criterion yielded a statistically significant result overall for the whole set of seven dependent variables, the univariate ANOVA results were interpreted separately for each individual dependent variable. The F value for between-group differences was statistically significant for every scale. Very large effect sizes ranged from 1.96 to 2.46 standard deviations for the learning environment scales and were 2.19 and 1.17 standard deviations for attitudes and achievement. For every scale, the SRS group had higher scores than the comparison group.A two-way MANOVA was used to determine if the use of SRS was differentially effective for males and females. The independent variables were the use/non-use of SRS and gender, and the dependent variables were the seven learning environment and student outcome scales. Although both males and females benefited from the use of SRS, Task Orientation was the only scale for which a statistically significant interaction emerged. However, the degree of differential effectiveness found for males and females when using SRS was small and of very little educational importance. Females appeared to benefit slightly more than males from the use of SRS.Simple correlation and multiple regression analyses were used to investigate the relationships between students’ perceptions of the learning environment and the student outcomes of attitudes and achievement. All five learning environment scales correlated positively and significantly with both student attitudes and achievement. The multiple correlation of the five learning environment scales with student attitudes and achievement was, respectively, 0.79 and 0.45. Involvement, Task Orientation, and Comfort were statistically significant independent predictors of student attitudes, while Involvement, Equity, and Comfort were statistically significant independent predictors of achievement

The Principle of Energetic Consistency: Application to the Shallow-Water Equations

If the complete state of the earth's atmosphere (e.g., pressure, temperature, winds and humidity, everywhere throughout the atmosphere) were known at any particular initial time, then solving the equations that govern the dynamical behavior of the atmosphere would give the complete state at all subsequent times. Part of the difficulty of weather prediction is that the governing equations can only be solved approximately, which is what weather prediction models do. But weather forecasts would still be far from perfect even if the equations could be solved exactly, because the atmospheric state is not and cannot be known completely at any initial forecast time. Rather, the initial state for a weather forecast can only be estimated from incomplete observations taken near the initial time, through a process known as data assimilation. Weather prediction models carry out their computations on a grid of points covering the earth's atmosphere. The formulation of these models is guided by a mathematical convergence theory which guarantees that, given the exact initial state, the model solution approaches the exact solution of the governing equations as the computational grid is made more fine. For the data assimilation process, however, there does not yet exist a convergence theory. This book chapter represents an effort to begin establishing a convergence theory for data assimilation methods. The main result, which is called the principle of energetic consistency, provides a necessary condition that a convergent method must satisfy. Current methods violate this principle, as shown in earlier work of the author, and therefore are not convergent. The principle is illustrated by showing how to apply it as a simple test of convergence for proposed methods

Creating Havoc: Havoc Development Program

One area where use of the computer is essential is in the modern scientific laboratory. High speed computation, data storage and data analysis enable scientists to perform experiments that would otherwise be impractical. A problem inherent to the effective use of special purpose laboratory computers, however, is the fact that this equipment has generally been developed for highly specific uses, and has either tried to cope with existing high-level languages or has abandoned the attempt and required the user to program in a low-level assembly or machine language. Our idea was to design, develop and implement a programming language that is suited to the needs of a laboratory scientist. Our results have led us to believe that the best way to achieve our goals was using an interpretive/compiled programming environment (similar in spirit to FORTH) in which large programs could be built in small, coherent pieces, that could easily be tested on as high or low a level as the programmer desired. Our language, Havoc, adheres to these principles while providing many of the more widespread and useful language features not found in FORTH. Besides giving it motivation, this preliminary report describes the current design and implementation status of the HAVOC system. The current version of the HAVOC system is available for the Macintosh

Universal optimality of the E8E_8 and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers $n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory.Comment: 95 pages, 6 figure

Dynamic Tensile Strength of Lunar Rock Types

The dynamic tensile strengths of four rocks have been determined. A flat plate impact experiment is used to generate ∼1-μs-duration tensile stress pulses in rock samples by superposing rarefaction waves to induce fracture. A gabbroic anorthosite and a basalt were selected because they are the same rock types as occur on the lunar highlands and mare, respectively. Although these have dynamic tensile strengths which lie within the ranges 153–174 MPa and 157–179 –MPa, whereas Arkansas novaculite and Westerly granite exhibit dynamic tensile strengths of 67–88 MPa and 95–116 MPa, respectively, the effect of chemical weathering and other factors, which may affect application of the present results to the moon, have not been explicitly studied. The reported tensile strengths are based on a series of experiments on each rock where determination of incipient spallation is made by terminal microscopic examination. These data are generally consistent with previous determinations, at least one of which was for a significantly chemically altered (hydroxylated) but physically coherent rock. The tensile failure data do not bear a simple relation to compressive results and imply that any modeling involving rock fracture consider the tensile strength of igneous rocks under impulse loads distinct from the values for static tensile strength. Generally, the dynamic tensile strengths of nonporous igneous rocks range from ∼ 100 to 180 MPa, with the more basic, and even amphibole-bearing samples, yielding the higher values

Adjoints and Low-rank Covariance Representation

Quantitative measures of the uncertainty of Earth System estimates can be as important as the estimates themselves. Second moments of estimation errors are described by the covariance matrix, whose direct calculation is impractical when the number of degrees of freedom of the system state is large. Ensemble and reduced-state approaches to prediction and data assimilation replace full estimation error covariance matrices by low-rank approximations. The appropriateness of such approximations depends on the spectrum of the full error covariance matrix, whose calculation is also often impractical. Here we examine the situation where the error covariance is a linear transformation of a forcing error covariance. We use operator norms and adjoints to relate the appropriateness of low-rank representations to the conditioning of this transformation. The analysis is used to investigate low-rank representations of the steady-state response to random forcing of an idealized discrete-time dynamical system