Symmetry breaking, mixing, instability, and low frequency variability in a minimal Lorenz-like system

Starting from the classical Saltzman two-dimensional convection equations, we derive via a severe spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled generalized Lorenz system. The consideration of this process breaks an important symmetry and couples the dynamics of fast and slow variables, with the ensuing modifications to the structural properties of the attractor and of the spectral features. When the relevant nondimensional number (Eckert number Ec) is different from zero, an additional time scale of O(Ec−1) is introduced in the system, as shown with standard multiscale analysis and made clear by several numerical evidences. Moreover, the system is ergodic and hyperbolic, the slow variables feature long-term memory with 1/f3/2 power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics, such as the interaction between slow and fast variables, the presence of long-term memory, and the associated extreme value statistics. This analysis shows how neglecting the coupling of slow and fast variables only on the basis of scale analysis can be catastrophic. In fact, this leads to spurious invariances that affect essential dynamical properties (ergodicity, hyperbolicity) and that cause the model losing ability in describing intrinsically multiscale processes.

Symmetry-break in Voronoi tessellations

We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α<0.12. Basically, the same happens in the 3D case, where only the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. In both 2D and 3D cases, already for a moderate amount of Gaussian noise (α>0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomaly in the scaling indicates that large cells preferentially feature large isoperimetric quotients. As the number of faces is strongly correlated with the sphericity (cells with more faces are bulkier), in 3D it is shown that the anomalous scaling is heavily reduced when we perform power law fits separately on cells with a specific number of faces.

Substitutional and orientational disorder in organic crystals: a symmetry-adapted ensemble model

Modelling of disorder in organic crystals is highly desirable since it would allow thermodynamic stabilities and other disorder-sensitive properties to be estimated for such systems. Two disordered organic molecular systems are modeled using a symmetry-adapted ensemble approach, in which the disordered system is treated as an ensemble of the configurations of a supercell with respect to substitution of one disorder component for another. Computation time is kept manageable by performing calculations only on the symmetrically inequivalent configurations. Calculations are presented on a substitutionally disordered system, the dichloro/dibromobenzene solid solution, and on an orientationally disordered system, eniluracil, and the resultant free energies, disorder patterns, and system properties are discussed. The results are found to be in agreement with experiment following manual removal of physically implausible configurations from ensemble averages, highlighting the dangers of a completely automated approach to organic crystal thermodynamics which ignores the barriers to equilibration once the crystal has been formed.

Spontaneous local symmetry breaking: a conformational study of glycine on Cu{311}

Understanding the interplay between intrinsic molecular chirality and chirality of the bonding footprint is crucial in exploiting enantioselectivity at surfaces. As such, achiral glycine and chiral alanine are the most obvious candidates if one is to study this interplay on different surfaces. Here, we have investigated the adsorption of glycine on Cu{311} using reflection-absorption infrared spectroscopy, low-energy electron diffraction, temperature-programmed desorption and first-principles density-functional theory. This combination of techniques has allowed us to accurately identify the molecular conformations present under different conditions, and discuss the overlayer structure in the context of the possible bonding footprints. We have observed coverage-dependent local symmetry breaking, with three-point bonded glycinate moieties forming an achiral arrangement at low coverages, and chirality developing with the presence of two-point bonded moieties at high coverages. Comparison with previous work on the self-assembly of simple amino acids on Cu{311} and the structurally-similar Cu{110} surface has allowed us to rationalise the different conditions necessary for the formation of ordered chiral overlayers.

Effect of ENSO phase on large-scale snow water equivalent distribution in a GCM

Understanding links between the El Nino-Southern Oscillation (ENSO) and snow would be useful for seasonal forecasting, but also for understanding natural variability and interpreting climate change predictions. Here, a 545-year run of the general circulation model HadCM3, with prescribed external forcings and fixed greenhouse gas concentrations, is used to explore the impact of ENSO on snow water equivalent (SWE) anomalies. In North America, positive ENSO events reduce the mean SWE and skew the distribution towards lower values, and vice versa during negative ENSO events. This is associated with a dipole SWE anomaly structure, with anomalies of opposite sign centered in western Canada and the central United States. In Eurasia, warm episodes lead to a more positively skewed distribution and the mean SWE is raised. Again, the opposite effect is seen during cold episodes. In Eurasia the largest anomalies are concentrated in the Himalayas. These correlations with February SWE distribution are seen to exist from the previous June-July-August (JJA) ENSO index onwards, and are weakly detected in 50-year subsections of the control run, but only a shifted North American response can be detected in the anaylsis of 40 years of ERA40 reanalysis data. The ENSO signal in SWE from the long run could still contribute to regional predictions although it would be a weak indicator only

Why are some handaxes symmetrical? Testing the influence of handaxe morphology on butchery effectiveness

The morphology of Acheulean handaxes continues to be a subject of debate amongst Lower Palaeolithic archaeologists, with some arguing that many handaxes are over-engineered for a subsistence function alone. This study aims to provide an empirical foundation for these debates by testing the relationship between a range of morphological variables, including symmetry, and the effectiveness of handaxes for butchery. Sixty handaxes were used to butcher 30 fallow deer by both a professional and a non-professional butcher. Regression analysis on the resultant data set indicates that while frontal symmetry may explain a small amount of variance in the effectiveness of handaxes for butchery, a large percentage of variance remains unexplained by symmetry or any of the other morphological variables under consideration.

A critical assessment of unbalanced surface stresses: Some complementary considerations

The wide-ranging survey of twisted growth in polymers by Lotz and Cheng cites extensive evidence consistent with the relief of surface stress being the underlying cause. This complementary note contributes to the discussion by making three main points. First, it is necessary to go further and explain the key issue of how a consistent twist is maintained when, as commonly, this habit has a lower symmetry than the crystallographic lattice. Detailed study has shown that, in polyethylene, this occurs by reorganization of the initial fold surfaces. Second, the suggested explanation by Keith and Padden that. in polyethylene, the asymmetric habit derives from molecules adding to lamellae with inclined fold surfaces is invalid being doubly inconsistent with observation. Third, twisting has now been linked to faster growth by study of row structures in polyethylene. This produces inherently rough fold surfaces in Regime II whose internal stresses drive reorganization and twisting. For slower (Regime I) growth, fold surfaces form with and maintain ordered packing so providing no basis for twisting. These new insights radically alter the context of twisted growth and provide a firm factual basis for further work. (c) 2005 Elsevier Ltd. All rights reserved.

A kinematically distorted flux rope model for magnetic clouds

Constant-α force-free magnetic flux rope models have proven to be a valuable first step toward understanding the global context of in situ observations of magnetic clouds. However, cylindrical symmetry is necessarily assumed when using such models, and it is apparent from both observations and modeling that magnetic clouds have highly noncircular cross sections. A number of approaches have been adopted to relax the circular cross section approximation: frequently, the cross-sectional shape is allowed to take an arbitrarily chosen shape (usually elliptical), increasing the number of free parameters that are fit between data and model. While a better “fit” may be achieved in terms of reducing the mean square error between the model and observed magnetic field time series, it is not always clear that this translates to a more accurate reconstruction of the global structure of the magnetic cloud. We develop a new, noncircular cross section flux rope model that is constrained by observations of CMEs/ICMEs and knowledge of the physical processes acting on the magnetic cloud: The magnetic cloud is assumed to initially take the form of a force-free flux rope in the low corona but to be subsequently deformed by a combination of axis-centered self-expansion and heliocentric radial expansion. The resulting analytical solution is validated by fitting to artificial time series produced by numerical MHD simulations of magnetic clouds and shown to accurately reproduce the global structure.

Harmonic force fields from scaled SCF calculations: Program ASYM40

We report an extended version of our normal coordinate program ASYM40, which may be used to transform Cartesian force constants from ab initio calculations to a force field in nonredundant internal (symmetry) coordinates. When experimental data are available, scale factors for the theoretical force field may then be optimized by least-squares refinement. The alternative of refining an empirical force field to fit a wide variety of data, as with the previous version ASYM20, has been retained. We compare the results of least-squares refinement of the full harmonic force field with least-squares refinement of only the scale factors for an SCF calculated force field and conclude that the latter approach may be useful for large molecules where more sophisticated calculations are impractical. The refinement of scale factors for a theoretical force field is also useful when there are only limited spectroscopic data. The program will accept ab initio calculated force fields from any program that presents Cartesian force constants as output. The program is available through Quantum Chemistry Program Exchange.

Notations and conventions in molecular spectroscopy: part 1. General spectroscopic notation

The field of Molecular Spectroscopy was surveyed in order to determine a set of conventions and symbols which are in common use in the spectroscopic literature. This document, which is Part I in a series, establishes the notations and conventions used for general spectroscopic notations and deals with quantum mechanics, quantum numbers (vibrational states, angular momentum and energy levels), spectroscopic transitions, and miscellaneous notations (e.g. spectroscopic terms). Further parts will follow, dealing inter alia with symmetry notation, permutation and permutation-inversion symmetry notation, vibration-rotation spectroscopy and electronic spectroscopy.

Anharmonic force constant calculations

The relationship of the anharmonic force constants in curvilinear internal coordinates to the observed vibration-rotation spectrum of a molecule is reviewed. A simplified method of setting up the required non-linear coordinate transformations is described: this makes use of an / tensor, which is a straightforward generalization of the / matrix used in the customary description of harmonic force constant calculations. General formulae for the / tensor elements, in terms of the familiar L matrix elements, are presented. The use of non-linear symmetry coordinates and redundancies are described. Sample calculations on the water and ammonia molecules are reported.

Coriolis perturbations in the infrared spectrum of ethylene

Rotational structure has been resolved and analyzed in two of the infrared‐active perpendicular bands of C2H4 vapor: the Type b fundamental band, ν10, at 826 cm—1, and the Type c fundamental band, ν7, at 949 cm—1. Many of the individual PP and RR branch lines have been observed. The analysis has been confined to values of the quantum number K≥3, for which energy levels ethylene shows no detectable deviations from a symmetric‐top rotational structure. The analysis reveals a Coriolis interaction between ν7 and ν10, and between ν4 and ν10, and values of the Coriolis constants ζ7,10z and ζ4,10y are obtained; these are related to normal coordinate calculations for the appropriate symmetry species, and force constants are derived to fit the observed zeta constants. The band center of ν10 has been revised from the original figure of 810 cm—1 to the new value, 826 cm—1, and the inactive frequency ν4 is estimated to lie at 1023±3 cm—1, in good agreement with the previous estimate of 1027 cm—1. The change in the value of ν10 leads to a suggested change in the value of the Raman‐active fundamental ν6 from 1236 to 1222 cm—1. New combination bands have been observed at 2174 cm—1, assigned as ν3+ν10; and at 2252 cm—1, assigned as ν4+ν6; also rotational structure has been resolved and analyzed in the ν6+ν10 band at 2048 cm—1. The new data obtained for the C2H4 molecule are summarized in Table XII, with all of the other data presently available on the vibrational and rotational constants.

Vibrational intensities VI: ethylene and its deutero-isotopes

Absolute intensity measurements have been made on the fundamental vibrations of ethylene and four of its deuteroisotopes. The bands were pressure broadened with nitrogen at 50 atmos, and the intensities were determined by the method of Wilson and Wells except that the observed optical density was integrated against logv rather than v. Normal coordinates have been calculated, and the intensities have been interpreted in terms of quantities (∂p/∂Si) giving the change in dipole moment with respect to each internal symmetry coordinate. Data from the different isotopic species have been used to eliminate ambiguities in the interpretation. Effective bond moments are calculated for each symmetry coordinate.

Rotational spectra induced by vibrations

Mizushima and Venkateswarlu showed in 1953 that certain molecules have the property that excited vibrational states may possess rotational spectra even when the rotational spectrum of the ground vibrational state is forbidden by symmetry. We call such a spectrum a vibrationally induced rotational spectrum, and have made a systematic examination of the point groups which permit such behaviour. We also give formulae for the approximate line frequencies and intensities in these spectra, and discuss some of the problems involved in observing them. The spectra can only arise from degenerate vibrational states, and are of three possible types: i) symmetric top perpendicular spectra, shown by molecules belonging to the point groups Dnh, Dn and Cnh, where n is odd; (ii) symmetric top parallel spectra, shown by molecules belonging to Dnd and S2n, where n is even; and (iii) spherical top spectra, shown by molecules belonging to T or Td. Excited vibrational states of polar molecules of point groups Cnv or Cn, where n is odd, may also possess vibrationally induced perpendicular components of type (i), in addition to their ordinary parallel spectra. In addition to the above limitations on the point groups there are, in general, limitations on the symmetry species of the degenerate vibrational states.