Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation

The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances. Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I. This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the H\"ormander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, is essential to understand the effect of noise and the phenomenon of phase diffusion. In addition, it is shown that the spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow one to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the RP spectrum at the bifurcation, revealing useful scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical system approach. This approach is not limited to low-dimensional systems and the reduction method presented in part I is applied to a stochastic model relevant to climate dynamics in part III

Coupled processes and the tropical climatology : part III : instabilities of the fully coupled climatology

Coupled processes between the equatorial ocean and atmosphere control the spatial structure of the annual mean state in the Pacific region,in particular the warm-pool/cold- tongue structure.At the same time,coupled processes are known to be responsible for the variability about this mean state,in particular the El-Niño/Southern-Oscillation phe- nomenon.In this paper,we consider the connection between both effects of coupling by investigating the linear stability of fully coupled climatologies in an intermediate coupled model.The new element here is that when parameters-such as the coupling strength-are changed,the potential amplification of disturbances can be greatly influenced by a simul- taneous modification of the mean state.This alters the stability properties of the coupled climatology,relative to the flux-corrected cases that have been previously studied.It ap- pears possible to identify a regime in parameter space where ENSO-like unstable modes coincide with a reasonable warm-pool/cold-tongue structure.These unstable modes are mixed SST/ocean-dynamics modes,that is,they arise through an interaction of oscillatory modes originating from ocean dynamics and oscillatory SST-modes.These effects are qual- itatively similar in this fully coupled problem compared to the ?ux-corrected problem,but the sensitivity of the ENSO mode to parameters and external variations is larger due to feedbacks in the climatology

Imperfections of the thermohaline circulation: latitudinal asymmetry and preferred northern sinking

The present Atlantic thermohaline circulation is dominated by deep water formation in the north despite the fact that surface buoyancy forcing has relatively modest latitudinal asymmetry.Many studies have shown that even with buoyancy forcing that is symmetric about the equator,spontaneous symmetry breaking can produce a single overturning cell with intense sinking in the north.This occurs by salt advection at sufficiently large fresh- water forcing.In this symmetry-breaking case,a southern sinking solution and a symmetric solution are also possible.A simple coupled ocean-atmosphere model of the zonally averaged thermohaline circulation is used to examine the effect of latitudinal asymmetries in the boundary conditions.The greater continental area in the northern hemisphere,combined with the slight asymmetry in the observed fresh-water flux,induce a strong preference for the northern sinking solution.Examining the relation to the solution under symmetric conditions,the salt-advection mechanism still acts to enhance the overturning circulation of the northern sinking branch,but multiple equilibria are much less likely to occur within the realistic parameter range.The most plausible shift between equilibria for paleoclimate applications would be between a strong northern sinking branch and a weak northern sinking branch that is an asymmetric version of the thermally driven solution.However,this is possible only in a very limited range of parameters.There is a substantial parameter range where the northern sinking branch is unique.The role of the fractional region of air-sea interaction at each latitude is substantial in producing north-south asymmetry

Observed Tightening of Tropical Ascent in Recent Decades and Linkage to Regional Precipitation Changes

Climate models predict that the tropical ascending region should tighten under global warming, but observational quantification of the tightening rate is limited. Here we show that the observed spatial extent of the relatively moist, rainy and cloudy regions in the tropics associated with large‐scale ascent has been decreasing at a rate of −1%/decade (−5%/K) from 1979 to 2016, resulting from combined effects of interdecadal variability and anthropogenic forcings, with the former contributing more than the latter. The tightening of tropical ascent is associated with an increase in the occurrence frequency of extremely strong ascent, leading to an increase in the average precipitation rate in the top 1% of monthly rainfall in the tropics. At the margins of the convective zones such as the Southeast Amazonia region, the contraction of large‐scale ascent is related to a long‐term drying trend about −3.2%/decade in the past 38 years

The tropical response to extratropical thermal forcing in an idealized GCM: The importance of radiative feedbacks and convective parameterization

The response of tropical precipitation to extratropical thermal forcing is reexamined using an idealized moist atmospheric GCM that has no water vapor or cloud feedbacks, simplifying the analysis while retaining the aquaplanet configuration coupled to a slab ocean from the authors' previous study. As in earlier studies, tropical precipitation in response to high-latitude forcing is skewed toward the warmed hemisphere. Comparisons with a comprehensive GCM in an identical aquaplanet, mixed-layer framework reveal that the tropical responses tend to be much larger in the comprehensive GCM as a result of positive cloud and water vapor feedbacks that amplify the imposed extratropical thermal forcing. The magnitude of the tropical precipitation response in the idealized model is sensitive to convection scheme parameters. This sensitivity as well as the tropical precipitation response can be understood from a simple theory with two ingredients: the changes in poleward energy fluxes are predicted using a onedimensional energy balance model and a measure of the "total gross moist stability" [??m, which is defined as the total (mean plus eddy) atmospheric energy transport per unit mass transport] of the model tropics converts the energy flux change into a mass flux and a moisture flux change. The idealized model produces a low level of compensation of about 25% between the imposed oceanic flux and the resulting response in the atmospheric energy transport in the tropics regardless of the convection scheme parameter. Because Geophysical Fluid Dynamics Laboratory Atmospheric Model 2 (AM2) with prescribed clouds and water vapor exhibits a similarly low level of compensation, it is argued that roughly 25% of the compensation is dynamically controlled through eddy energy fluxes. The sensitivity of the tropical response to the convection scheme in the idealized model results from different values of ??m: smaller ??m leads to larger tropical precipitation changes for the same response in the energy transport.open624

Starch Gel Electrophoresis of Various Fractions of Casein

Summary Preparations of α-, β -, γ-, λ-, and κ-caseins were compared by electrophoresis in starch gel and in free solution. Acid casein separated into at least 17 zones. All fractions of casein contained numerous proteins, but the preparatory procedures brought about considerable enrichment of certain zones while weakening or eliminating others, and a tentative identification of specific zones with each casein fraction was, therefore, possible. α s -Casein appeared to be contaminated with traces of β -, κ-, and γ-casein, but the main contaminants were unidentified fractions travelling between α s - and β -casein. β -Casein was contaminated with traces of γ- and κ-casein, and with the unidentified fractions observed in α s -casein. Our preparations of γ-casein were heavily contaminated with κ- and β -casein, and contained lesser amounts of α s -casein and of unidentified fractions. Identification of the κ-casein zone was confirmed by zone electrophoresis from urea solution into urea-free gel, whereupon the κ-casein formed a characteristic precipitation zone, and by electrophoresis in gels containing calcium. Comparison of three κ-casein preparations indicated differences among them, both in stabilizing power and in electrophoretic properties. The preparation of κ-casein with the least contamination from other casein components appeared to be partially denatured

Fast and slow Kelvin waves in the Madden-Julian Oscillation of a GCM

The structure of the Madden-Julian Oscillation (MJO) in an 1800-day integration of the Hadley Centre Unified Model was analysed, and interpreted within a Kelvin wave framework. The model was forced with constant equinoctial (March) boundary conditions so that a ``clean'' MJO signal could be separated from the effects of the seasonal cycle and forced interannual variability. The simulated MJO was fairly realistic in terms of its large-scale spatial structure and propagation characteristics, although its period of 30 days (corresponding to an average phase speed of 15 \mps) was shorter than that observed. The signal in deep convection was less coherent than in observations, and appeared to move eastward as a sequence of discrete convective anomalies, rather than by a smooth eastward propagation. Both ``fast'' and ``slow'' equatorial Kelvin waves appeared to play an important role in the eastward propagation of the simulated MJO. Enhanced convection over the Indian Ocean was associated with a ``fast'' equatorial Kelvin wave that propagated eastward at 55 m s-1 over the Pacific. On reaching the west coast of South America, a component of this Kelvin wave propagated northward and southward as a trapped wave along the mountain ranges of Central America and the Andes, in agreement with observations. The anomalous surface easterlies over the tropical eastern Pacific associated with this fast Kelvin wave enhanced the climatological mean easterlies and led to positive convective anomalies over the eastern Pacific consistent with the WISHE mechanism. However, WISHE was not able to account for the eastward development of the convective anomalies over the Indian Ocean/western Pacific region. By splitting the equatorial divergence anomalies of the simulated MJO into their du/dx and dv/dy components, the role of Kelvin wave dynamics in the ``slow'' (15 m s-1) average eastward propagation of the simulated MJO was examined. Although the two components were of comparable magnitude, the \dudx\ component exhibited a pronounced eastward propagation which tended to be disrupted by the \dvdy\ component, thus supporting the paradigm of an underlying, but strongly modified, Kelvin wave mechanism

Short Warm Distribution Tails Accelerate the Increase of Humid-Heat Extremes Under Global Warming

Humid-heat extremes threaten human health and are increasing in frequency with global warming, so elucidating factors affecting their rate of change is critical. We investigate the role of wet-bulb temperature (TW) frequency distribution tail shape on the rate of increase in extreme TW threshold exceedances under 2°C global warming. Results indicate that non-Gaussian TW distribution tails are common worldwide across extensive, spatially coherent regions. More rapid increases in the number of days exceeding the historical 95th percentile are projected in locations with shorter-than-Gaussian warm side tails. Asymmetry in the specific humidity distribution, one component of TW, is more closely correlated with TW tail shape than temperature, suggesting that humidity climatology strongly influences the rate of future changes in TW extremes. Short non-Gaussian TW warm tails have notable implications for dangerous humid-heat in regions where current-climate TW extremes approach human safety limits