Relevance of sampling schemes in light of Ruelle's linear response theory

We reconsider the theory of the linear response of non-equilibrium steady states to perturbations. We �rst show that by using a general functional decomposition for space-time dependent forcings, we can de�ne elementary susceptibilities that allow to construct the response of the system to general perturbations. Starting from the de�nition of SRB measure, we then study the consequence of taking di�erent sampling schemes for analysing the response of the system. We show that only a speci�c choice of the time horizon for evaluating the response of the system to a general time-dependent perturbation allows to obtain the formula �rst presented by Ruelle. We also discuss the special case of periodic perturbations, showing that when they are taken into consideration the sampling can be �ne-tuned to make the de�nition of the correct time horizon immaterial. Finally, we discuss the implications of our results in terms of strategies for analyzing the outputs of numerical experiments by providing a critical review of a formula proposed by Reick.

Vibrational perturbation theory

Expressions are derived for the Jacobian of the coriolis ζ interaction constants and the centrifugal stretching constants (DJ, DJK, etc.) with respect to the force constants in a vibrating-rotating molecule.

The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory

We investigate Fréchet differentiability of the scattered field with respect to variation in the boundary in the case of time–harmonic acoustic scattering by an unbounded, sound–soft, one–dimensional rough surface. We rigorously prove the differentiability of the scattered field and derive a characterization of the Fréchet derivative as the solution to a Dirichlet boundary value problem. As an application of these results we give rigorous error estimates for first–order perturbation theory, justifying small perturbation methods that have a long history in the engineering literature. As an application of our rigorous estimates we show that a plane acoustic wave incident on a sound–soft rough surface can produce an unbounded scattered field.

Towards a general theory of extremes for observables of chaotic dynamical systems

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the cho- sen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan–Yorke dimension of the attractor. Preliminary numer- ical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.

Significance of cross correlations in the stress relaxation of polymer melts

According to linear response theory, all relaxation functions in the linear regime can be obtained using time correlation functions calculated under equilibrium. In this paper, we demonstrate that the cross correlations make a significant contribution to the partial stress relaxation functions in polymer melts. We present two illustrations in the context of polymer rheology using (1) Brownian dynamics simulations of a single chain model for entangled polymers, the slip-spring model, and (2) molecular dynamics simulations of a multichain model. Using the single chain model, we analyze the contribution of the confining potential to the stress relaxation and the plateau modulus. Although the idea is illustrated with a particular model, it applies to any single chain model that uses a potential to confine the motion of the chains. This leads us to question some of the assumptions behind the tube theory, especially the meaning of the entanglement molecular weight obtained from the plateau modulus. To shed some light on this issue, we study the contribution of the nonbonded excluded-volume interactions to the stress relaxation using the multichain model. The proportionality of the bonded/nonbonded contributions to the total stress relaxation (after a density dependent "colloidal" relaxation time) provides some insight into the success of the tube theory in spite of using questionable assumptions. The proportionality indicates that the shape of the relaxation spectrum can indeed be reproduced using the tube theory and the problem is reduced to that of finding the correct prefactor. (c) 2007 American Institute of Physics

Equatorial waves in the lower stratosphere. I: A novel detection method

Wavenumber-frequency spectral analysis and linear wave theory are combined in a novel method to quantitatively estimate equatorial wave activity in the tropical lower stratosphere. The method requires temperature and velocity observations that are regularly spaced in latitude, longitude and time; it is therefore applied to the ECMWF 15-year re-analysis dataset (ERA-15). Signals consistent with idealized Kelvin and Rossby-gravity waves are found at wavenumbers and frequencies in agreement with previous studies. When averaged over 1981-93, the Kelvin wave explains approximately 1 K-2 of temperature variance on the equator at 100 hPa, while the Rossby-gravity wave explains approximately 1 m(2)s(-2) of meridional wind variance. Some inertio-gravity wave and equatorial Rossby wave signals are also found; however the resolution of ERA-15 is not sufficient for the method to provide an accurate climatology of waves with high meridional structure.

Vibrational intensities in methane

The absorption intensities of the two infra-red active vibrations in methane have been obtained from a perturbation calculation on the equilibrium wave functions derived in the preceding paper. The perturbation field is the change in the potential field due to the nuclei which results from moving the nuclei in the vibrational coordinate concerned, and a simplified form of second order perturbation theory, developed by Pople and Schofield, is used for the calculation. The main approximation involved is the neglect of f and higher harmonics in the spherical harmonic expansion of the nuclear field. The resulting dipole moment derivatives are approximately three times larger than the experimental values, but they show qualitative features and sign relationships which are significant.

Time-resolved gas-phase kinetic and quantum chemical studies of the reaction of silylene with oxygen

Time-resolved kinetic studies of the reaction of silylene, SiH2, generated by laser flash photolysis of phenylsilane, have been carried out to obtain rate constants for its bimolecular reaction with O-2. The reaction was studied in the gas phase over the pressure range 1-100 Torr in SF6 bath gas, at five temperatures in the range 297-600 K. The second order rate constants at 10 Torr were fitted to the Arrhenius equation: log(k/cm(3) molecule(-1) s(-1)) = (-11.08 +/- 0.04) + (1.57 +/- 0.32 kJ mol(-1))/RT ln10 The decrease in rate constant values with increasing temperature, although systematic is very small. The rate constants showed slight increases in value with pressure at each temperature, but this was scarcely beyond experimental uncertainty. From estimates of Lennard-Jones collision rates, this reaction is occurring at ca. 1 in 20 collisions, almost independent of pressure and temperature. Ab initio calculations at the G3 level backed further by multi-configurational (MC) SCF calculations, augmented by second order perturbation theory (MRMP2), support a mechanism in which the initial adduct, H2SiOO, formed in the triplet state (T), undergoes intersystem crossing to the more stable singlet state (S) prior to further low energy isomerisation processes leading, via a sequence of steps, ultimately to dissociation products of which the lowest energy pair are H2O + SiO. The decomposition of the intermediate cyclo-siladioxirane, via O-O bond fission, plays an important role in the overall process. The bottleneck for the overall process appears to be the T -> S process in H2SiOO. This process has a small spin orbit coupling matrix element, consistent with an estimate of its rate constant of 1 x 10(9) s(-1) obtained with the aid of RRKM theory. This interpretation preserves the idea that, as in its reactions in general, SiH2 initially reacts at the encounter rate with O-2. The low values for the secondary reaction barriers on the potential energy surface account for the lack of an observed pressure dependence. Some comparisons are drawn with the reactions of CH2 + O-2 and SiCl2 + O-2.

Gas phase kinetic and quantum chemical studies of the reactions of silylene with the methylsilanes. Absolute rate constants, temperature dependences, RRKM modelling and potential energy surfaces

Time resolved studies of silylene, SiH2, generated by the 193 nm laser. ash photolysis of phenylsilane, have been carried out to obtain rate coefficients for its bimolecular reactions with methyl-, dimethyl- and trimethyl-silanes in the gas phase. The reactions were studied over the pressure range 3 - 100 Torr with SF6 as bath gas and at five temperatures in the range 300 - 625 K. Only slight pressure dependences were found for SiH2 + MeSiH3 ( 485 and 602 K) and for SiH2 + Me2SiH2 ( 600 K). The high pressure rate constants gave the following Arrhenius parameters: [GRAPHICS] These are consistent with fast, near to collision-controlled, association processes. RRKM modelling calculations are consistent with the observed pressure dependences ( and also the lack of them for SiH2 + Me3SiH). Ab initio calculations at both second order perturbation theory (MP2) and coupled cluster (CCSD(T)) levels, showed the presence of weakly-bound complexes along the reaction pathways. In the case of SiH2 + MeSiH3 two complexes, with different geometries, were obtained consistent with earlier studies of SiH2 + SiH4. These complexes were stabilised by methyl substitution in the substrate silane, but all had exceedingly low barriers to rearrangement to product disilanes. Although methyl groups in the substrate silane enhance the intrinsic SiH2 insertion rates, it is doubtful whether the intermediate complexes have a significant effect on the kinetics. A further calculation on the reaction MeSiH + SiH4 shows that the methyl substitution in the silylene should have a much more significant kinetic effect ( as observed in other studies).

A statistical mechanical approach for the computation of the climatic response to general forcings

The climate belongs to the class of non-equilibrium forced and dissipative systems, for which most results of quasi-equilibrium statistical mechanics, including the fluctuation-dissipation theorem, do not apply. In this paper we show for the first time how the Ruelle linear response theory, developed for studying rigorously the impact of perturbations on general observables of non-equilibrium statistical mechanical systems, can be applied with great success to analyze the climatic response to general forcings. The crucial value of the Ruelle theory lies in the fact that it allows to compute the response of the system in terms of expectation values of explicit and computable functions of the phase space averaged over the invariant measure of the unperturbed state. We choose as test bed a classical version of the Lorenz 96 model, which, in spite of its simplicity, has a well-recognized prototypical value as it is a spatially extended one-dimensional model and presents the basic ingredients, such as dissipation, advection and the presence of an external forcing, of the actual atmosphere. We recapitulate the main aspects of the general response theory and propose some new general results. We then analyze the frequency dependence of the response of both local and global observables to perturbations having localized as well as global spatial patterns. We derive analytically several properties of the corresponding susceptibilities, such as asymptotic behavior, validity of Kramers-Kronig relations, and sum rules, whose main ingredient is the causality principle. We show that all the coefficients of the leading asymptotic expansions as well as the integral constraints can be written as linear function of parameters that describe the unperturbed properties of the system, such as its average energy. Some newly obtained empirical closure equations for such parameters allow to define such properties as an explicit function of the unperturbed forcing parameter alone for a general class of chaotic Lorenz 96 models. We then verify the theoretical predictions from the outputs of the simulations up to a high degree of precision. The theory is used to explain differences in the response of local and global observables, to define the intensive properties of the system, which do not depend on the spatial resolution of the Lorenz 96 model, and to generalize the concept of climate sensitivity to all time scales. We also show how to reconstruct the linear Green function, which maps perturbations of general time patterns into changes in the expectation value of the considered observable for finite as well as infinite time. Finally, we propose a simple yet general methodology to study general Climate Change problems on virtually any time scale by resorting to only well selected simulations, and by taking full advantage of ensemble methods. The specific case of globally averaged surface temperature response to a general pattern of change of the CO2 concentration is discussed. We believe that the proposed approach may constitute a mathematically rigorous and practically very effective way to approach the problem of climate sensitivity, climate prediction, and climate change from a radically new perspective.

Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow

Wave-activity conservation laws are key to understanding wave propagation in inhomogeneous environments. Their most general formulation follows from the Hamiltonian structure of geophysical fluid dynamics. For large-scale atmospheric dynamics, the Eliassen–Palm wave activity is a well-known example and is central to theoretical analysis. On the mesoscale, while such conservation laws have been worked out in two dimensions, their application to a horizontally homogeneous background flow in three dimensions fails because of a degeneracy created by the absence of a background potential vorticity gradient. Earlier three-dimensional results based on linear WKB theory considered only Doppler-shifted gravity waves, not waves in a stratified shear flow. Consideration of a background flow depending only on altitude is motivated by the parameterization of subgrid-scales in climate models where there is an imposed separation of horizontal length and time scales, but vertical coupling within each column. Here we show how this degeneracy can be overcome and wave-activity conservation laws derived for three-dimensional disturbances to a horizontally homogeneous background flow. Explicit expressions for pseudoenergy and pseudomomentum in the anelastic and Boussinesq models are derived, and it is shown how the previously derived relations for the two-dimensional problem can be treated as a limiting case of the three-dimensional problem. The results also generalize earlier three-dimensional results in that there is no slowly varying WKB-type requirement on the background flow, and the results are extendable to finite amplitude. The relationship A E =cA P between pseudoenergy A E and pseudomomentum A P, where c is the horizontal phase speed in the direction of symmetry associated with A P, has important applications to gravity-wave parameterization and provides a generalized statement of the first Eliassen–Palm theorem.

Spectral asymmetry of the massless Dirac operator on a 3-torus

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.

The ocean's role in polar climate change: asymmetric Arctic and Antarctic responses to greenhouse gas and ozone forcing

In recent decades, the Arctic has been warming and sea ice disappearing. By contrast, the Southern Ocean around Antarctica has been (mainly) cooling and sea-ice extent growing. We argue here that interhemispheric asymmetries in the mean ocean circulation, with sinking in the northern North Atlantic and upwelling around Antarctica, strongly influence the sea-surface temperature (SST) response to anthropogenic greenhouse gas (GHG) forcing, accelerating warming in the Arctic while delaying it in the Antarctic. Furthermore, while the amplitude of GHG forcing has been similar at the poles, significant ozone depletion only occurs over Antarctica. We suggest that the initial response of SST around Antarctica to ozone depletion is one of cooling and only later adds to the GHG-induced warming trend as upwelling of sub-surface warm water associated with stronger surface westerlies impacts surface properties. We organize our discussion around ‘climate response functions’ (CRFs), i.e. the response of the climate to ‘step’ changes in anthropogenic forcing in which GHG and/or ozone-hole forcing is abruptly turned on and the transient response of the climate revealed and studied. Convolutions of known or postulated GHG and ozone-hole forcing functions with their respective CRFs then yield the transient forced SST response (implied by linear response theory), providing a context for discussion of the differing warming/cooling trends in the Arctic and Antarctic. We speculate that the period through which we are now passing may be one in which the delayed warming of SST associated with GHG forcing around Antarctica is largely cancelled by the cooling effects associated with the ozone hole. By mid-century, however, ozone-hole effects may instead be adding to GHG warming around Antarctica but with diminished amplitude as the ozone hole heals. The Arctic, meanwhile, responding to GHG forcing but in a manner amplified by ocean heat transport, may continue to warm at an accelerating rate.

Gas-phase rate coefficients for the reactions of O(3P), S(3P), Se(3P), and Te(3P) with alkenes: application of perturbation frontier molecular orbital theory, correlations, and structure-activity relations (SARs)

The kinetics of the reactions of the atoms O(P-3), S(P-3), Se(P-3), and Te((3)p) with a series of alkenes are examined for correlations relating the logarithms of the rate coefficients to the energies of the highest occupied molecular orbitals (HOMOs) of the alkenes. These correlations may be employed to predict rate coefficients from the calculated HOMO energy of any other alkene of interest. The rate coefficients obtained from the correlations were used to formulate structure-activity relations (SARs) for reactions of O((3)p), S(P-3), Se (P-3), and Te((3)p) with alkenes. A comparison of the values predicted by both the correlations and the SARs with experimental data where they exist allowed us to assess the reliability of our method. We demonstrate the applicability of perturbation frontier molecular orbital theory to gas-phase reactions of these atoms with alkenes. The correlations are apparently not applicable to reactions of C(P-3), Si(P-3), N(S-4), and Al(P-2) atoms with alkenes, a conclusion that could be explained in terms of a different mechanism for reaction of these atoms.

Stochastic perturbations to dynamical systems: a response theory approach

Using the formalism of the Ruelle response theory, we study how the invariant measure of an Axiom A dynamical system changes as a result of adding noise, and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics. We first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system, both in the additive and in the multiplicative case. We also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency-dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing. This provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings. Using Kramers-Kronig theory, it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the Green function of the system for any desired observable. We then extend our results to rather general patterns of random forcing, from the case of several white noise forcings, to noise terms with memory, up to the case of a space-time random field. Explicit formulas are provided for each relevant case analysed. As a general result, we find, using an argument of positive-definiteness, that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system. We provide an example of application of our results by considering the spatially extended chaotic Lorenz 96 model. These results clarify the property of stochastic stability of SRB measures in Axiom A flows, provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies, and introduce new ways to study the response of a system to external perturbations. Taking into account the chaotic hypothesis, we expect that our results have practical relevance for a more general class of system than those belonging to Axiom A.