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.

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.

Does high-frequency repetitive transcranial magnetic stimulation produce residual and/or cumulative effects within an experimental session?

A common procedure for studying the effects on cognition of repetitive transcranial magnetic stimulation (rTMS) is to deliver rTMS concurrent with task performance, and to compare task performance on these trials versus on trials without rTMS. Recent evidence that TMS can have effects on neural activity that persist longer than the experimental session itself, however, raise questions about the assumption of the transient nature of rTMS that underlies many concurrent (or "online") rTMS designs. To our knowledge, there have been no studies in the cognitive domain examining whether the application of brief trains of rTMS during specific epochs of a complex task may have effects that spill over into subsequent task epochs, and perhaps into subsequent trials. We looked for possible immediate spill-over and longer-term cumulative effects of rTMS in data from two studies of visual short-term delayed recognition. In 54 subjects, 10-Hz rTMS trains were applied to five different brain regions during the 3-s delay period of a spatial task, and in a second group of 15 subjects, electroencephalography (EEG) was recorded while 10-Hz rTMS was applied to two brain areas during the 3-s delay period of both spatial and object tasks. No evidence for immediate effects was found in the comparison of the memory probe-evoked response on trials that were vs. were not preceded by delay-period rTMS. No evidence for cumulative effects was found in analyses of behavioral performance, and of EEG signal, as a function of task block. The implications of these findings, and their relation to the broader literature on acute vs. long-lasting effects of rTMS, are considered.

The response of a uniform horizontal temperature gradient to heating

The response of a uniform horizontal temperature gradient to prescribed fixed heating is calculated in the context of an extended version of surface quasigeostrophic dynamics. It is found that for zero mean surface flow and weak cross-gradient structure the prescribed heating induces a mean temperature anomaly proportional to the spatial Hilbert transform of the heating. The interior potential vorticity generated by the heating enhances this surface response. The time-varying part is independent of the heating and satisfies the usual linearized surface quasigeostrophic dynamics. It is shown that the surface temperature tendency is a spatial Hilbert transform of the temperature anomaly itself. It then follows that the temperature anomaly is periodically modulated with a frequency proportional to the vertical wind shear. A strong local bound on wave energy is also found. Reanalysis diagnostics are presented that indicate consistency with key findings from this theory.

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.

Bilateral high-frequency stimulation of the subthalamic nucleus on attentional performance: transient deleterious effects and enhanced motivation in both intact and parkinsonian rats.

It is now well established that subthalamic nucleus high-frequency stimulation (STN HFS) alleviates motor problems in Parkinson's disease. However, its efficacy for cognitive function remains a matter of debate. The aim of this study was to assess the effects of STN HFS in rats performing a visual attentional task. Bilateral STN HFS was applied in intact and in bilaterally dopamine (DA)-depleted rats. In all animals, STN HFS had a transient debilitating effect on all the variables measured in the task. In DA-depleted rats, STN HFS did not alleviate the deficits induced by the DA lesion such as omissions and latency to make correct responses, but induced perseverative approaches to the food magazine, an indicator of enhanced motivation. In sham-operated controls, STN HFS significantly reduced accuracy and induced perseverative behaviour, mimicking partially the effects of bilateral STN lesions in the same task. These results are in line with the hypothesis that STN HFS only partially mimics inactivation of STN produced by lesioning and confirm the motivational exacerbation induced by STN inactivation.

Prefrontal cortical-ventral striatal interactions involved in affective modulation of attentional performance: implications for corticostriatal circuit function

Anatomically segregated systems linking the frontal cortex and the striatum are involved in various aspects of cognitive, affective, and motor processing. In this study, we examined the effects of combined unilateral lesions of the medial prefrontal cortex (mPFC) and the core subregion of the nucleus accumbens (AcbC) in opposite hemispheres (disconnection) on a continuous performance, visual attention test [five-choice serial reaction-time task (5CSRTT)]. The disconnection lesion produced a set of specific changes in performance of the 5CSRTT, resembling changes that followed bilateral AcbC lesions while, in addition, comprising a subset of the behavioral changes after bilateral mPFC lesions previously reported using the same task. Specifically, both mPFC/AcbC disconnection and bilateral AcbC lesions markedly affected aspects of response control related to affective feedback, as indexed by perseverative responding in the 5CSRTT. These effects were comparable, although not identical, to those in animals with either bilateral AcbC or mPFC/AcbC disconnection lesions. The mPFC/AcbC disconnection resulted in a behavioral profile largely distinct from that produced by disconnection of a similar circuit described previously, between the mPFC and the dorsomedial striatum, which were shown to form a functional network underlying aspects of visual attention and attention to action. This distinction provides an insight into the functional specialization of corticostriatal circuits in similar behavioral contexts.

Response of the lower St. Lawrence Estuary to external forcing in winter

Mostly because of a lack of observations, fundamental aspects of the St. Lawrence Estuary's wintertime response to forcing remain poorly understood. The results of a field campaign over the winter of 2002/03 in the estuary are presented. The response of the system to tidal forcing is assessed through the use of harmonic analyses of temperature, salinity, sea level, and current observations. The analyses confirm previous evidence for the presence of semidiurnal internal tides, albeit at greater depths than previously observed for ice-free months. The low-frequency tidal streams were found to be mostly baroclinic in character and to produce an important neap tide intensification of the estuarine circulation. Despite stronger atmospheric momentum forcing in winter, the response is found to be less coherent with the winds than seen in previous studies of ice-free months. The tidal residuals show the cold intermediate layer in the estuary is renewed rapidly ( 14 days) in late March by the advection of a wedge of near-freezing waters from the Gulf of St. Lawrence. In situ processes appeared to play a lesser role in the renewal of this layer. In particular, significant wintertime deepening of the estuarine surface mixed layer was prevented by surface stability, which remained high throughout the winter. The observations also suggest that the bottom circulation was intensified during winter, with the intrusion in the deep layer of relatively warm Atlantic waters, such that the 3 C isotherm rose from below 150 m to near 60 m.

Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test

In this article, we use the no-response test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient gamma of the equation del (.) gamma(x)del + c(x) with piecewise regular gamma and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the no-response test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of gamma(x), x is an element of partial derivative D at the boundary. A second consequence of this formula is that the blow-up rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.

The significance of volcanic eruption strength and frequency for climate

A simple physical model of the atmospheric effects of large explosive volcanic eruptions is developed. Using only one input parameter - the initial amount of sulphur dioxide injected into the stratosphere - the global-average stratospheric optical-depth perturbation and surface temperature response are modelled. The simplicity of this model avoids issues of incomplete data (applicable to more comprehensive models), making it a powerful and useful tool for atmospheric diagnostics of this climate forcing mechanism. It may also provide a computationally inexpensive and accurate way of introducing volcanic activity into larger climate models. The modelled surface temperature response for an initial sulphur-dioxide injection, coupled with emission-history statistics, is used to demonstrate that the most climatically significant volcanic eruptions are those of sufficient explosivity to just reach into the stratosphere (and achieve longevity). This study also highlights the fact that this measure of significance is highly sensitive to the representation of the climatic response and the frequency data used, and that we are far from producing a definitive history of explosive volcanism for at least the past 1000 years. Given this high degree of uncertainty, these results suggest that eruptions that release around and above 0.1 Mt SO2 into the stratosphere have the maximum climatic impact.

A link between low-frequency mesoscale eddy variability around Madagascar and the large-scale Indian Ocean variability

A connection is shown to exist between the mesoscale eddy activity around Madagascar and the large-scale interannual variability in the Indian Ocean. We use the combined TOPEX/Poseidon-ERS sea surface height (SSH) data for the period 1993–2003. The SSH-fields in the Mozambique Channel and east of Madagascar exhibit a significant interannual oscillation. This is related to the arrival of large-scale anomalies that propagate westward along 10°–15°S in response to the Indian Ocean dipole (IOD) events. Positive (negative) SSH anomalies associated to a positive (negative) IOD phase induce a shift in the intensity and position of the tropical and subtropical gyres. A weakening (strengthening) results in the intensity of the South Equatorial Current and its branches along east Madagascar. In addition, the flow through the narrows of the Mozambique Channel around 17°S increases (decreases) during periods of a stronger and northward (southward) extension of the subtropical (tropical) gyre. Interaction between the currents in the narrows and southward propagating eddies from the northern Channel leads to interannual variability in the eddy kinetic energy of the central Channel in phase with the one in the SSH-field.

Born-Oppenheimer failure in the separation of low frequency vibrations

Changes in the effective potential function of a low-frequency large-amplitude molecular vibration, resulting from excitation of a high-frequency vibration, are discussed. It is shown that in some situations a significant contribution to such changes may arise from failure of the Born-Oppenheimer separation of the low-frequency mode. In the particular example of the HF dimer, recent evidence that the tunneling barrier increases on exciting either of the H-stretching vibrations is probably due to this effect.

Nitrous acid: vibrational frequency shifts due to 15N substitution, and the harmonic force field

Vibration rotation spectra of HO15 NO and DO15 NO have been measured at a resolution of 0•04 cm-1 to determine the isotopic shifts in the vibrational band origins. These have been used together with recently determined data on the vibrational band origins, Coriolis constants, and centrifugal distorition constants, to determine the harmonic force field of both cis and trans nitrous acid in least squares refinement calculations. The results are discussed in relation to recent ab initio calculations, the inertia defects, and the torsional potential function.

Germin and germin-like proteins: evolution, structure, and function

Germin and germin-like proteins (GLPs) are encoded by a family of genes found in all plants. They are part of the cupin superfamily of biochemically diverse proteins, a superfamily that has a conserved tertiary structure, though with limited similarity in primary sequence. The subgroups of GLPs have different enzyme functions that include the two hydrogen peroxide-generating enzymes, oxalate oxidase (OxO) and superoxide dismutase. This review summarizes the sequence and structural details of GLPs and also discusses their evolutionary progression, particularly their amplification in gene number during the evolution of the land plants. In terms of function, the GLPs are known to be differentially expressed during specific periods of plant growth and development, a pattern of evolutionary subfunctionalization. They are also implicated in the response of plants to biotic (viruses, bacteria, mycorrhizae, fungi, insects, nematodes, and parasitic plants) and abiotic (salt, heat/cold, drought, nutrient, and metal) stress. Most detailed data come from studies of fungal pathogenesis in cereals. This involvement with the protection of plants from environmental stress of various types has led to numerous plant breeding studies that have found links between GLPs and QTLs for disease and stress resistance. In addition the OxO enzyme has considerable commercial significance, based principally on its use in the medical diagnosis of oxalate concentration in plasma and urine. Finally, this review provides information on the nutritional importance of these proteins in the human diet, as several members are known to be allergenic, a feature related to their thermal stability and evolutionary connection to the seed storage proteins, also members of the cupin superfamily.

Very short delays prior to escape from potential predators may function efficiently as adaptive risk-assessment periods

Periods between predator detection and an escape response (escape delays) by prey upon attack by a predator often arise because animals trade-off the benefits such a delay gives for assessing risk accurately with the costs of not escaping as quickly as possible. We tested whether freezing behaviour (complete immobility in a previously foraging bird) observed in chaffinches before escaping from an approaching potential threat functions as a period of risk-assessment, and whether information on predator identity is gained even when time available is very short. We flew either a model of a sparrowhawk (predator) or a woodpigeon (no threat) at single chaffinches. Escape delays were significantly shorter with the hawk, except when a model first appeared close to the chaffinch. Chaffinches were significantly more vigilant when they resumed feeding after exposure to the sparrowhawk compared to the woodpigeon showing that they were able to distinguish between threats, and this applied even when time available for assessment was short (an average of 0.29 s). Our results show freezing in chaffinches functions as an effective economic risk assessment period, and that threat information is gained even when very short periods of time are available during an attack.