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

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

El Niño Dynamics

Bringer of storms and droughts, the El Niño∕Southern Oscillation results from the complex, sometimes chaotic interplay of ocean and atmosphere

El Niño Dynamics

Bringer of storms and droughts, the El Niño∕Southern Oscillation results from the complex, sometimes chaotic interplay of ocean and atmosphere

Sensitivity of terrestrial precipitation trends to the structural evolution of sea surface temperatures

Pronounced intermodel differences in the projected response of land surface precipitation (LSP) to future anthropogenic forcing remain in the Coupled Model Intercomparison Project Phase 5 model integrations. A large fraction of the intermodel spread in projected LSP trends is demonstrated here to be associated with systematic differences in simulated sea surface temperature (SST) trends, especially the representation of changes in (i) the interhemispheric SST gradient and (ii) the tropical Pacific SSTs. By contrast, intermodel differences in global mean SST, representative of differing global climate sensitivities, exert limited systematic influence on LSP patterns. These results highlight the importance to regional terrestrial precipitation changes of properly simulating the spatial distribution of large-scale, remote changes as reflected in the SST response to increasing greenhouse gases. Moreover, they provide guidance regarding which region-specific precipitation projections may be potentially better constrained for use in climate change impact assessments

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

Collaborative Research: Robust Climate Projections and Stochastic Stability of Dynamical Systems

The project was completed along the lines of the original proposal, with additional elements arising as new results were obtained. The originally proposed three thrusts were expanded to include an additional, fourth one. (i) The e#11;ffects of stochastic perturbations on climate models have been examined at the fundamental level by using the theory of deterministic and random dynamical systems, in both #12;nite and in#12;nite dimensions. (ii) The theoretical results have been implemented #12;first on a delay-diff#11;erential equation (DDE) model of the El-Nino/Southern-Oscillation (ENSO) phenomenon. (iii) More detailed, physical aspects of model robustness have been considered, as proposed, within the stripped-down ICTP-AGCM (formerly SPEEDY) climate model. This aspect of the research has been complemented by both observational and intermediate-model aspects of mid-latitude and tropical climate. (iv) An additional thrust of the research relied on new and unexpected results of (i) and involved reduced-modeling strategies and associated prediction aspects have been tested within the team's empirical model reduction (EMR) framework. Finally, more detailed, physical aspects have been considered within the stripped-down SPEEDY climate model. The results of each of these four complementary e#11;fforts are presented in the next four sections, organized by topic and by the team members concentrating on the topic under discussion

Weakening and strengthening structures in the Hadley Circulation change under global warming and implications for cloud response and climate sensitivity

It has long been recognized that differences in climate model-simulated cloud feedbacks are a primary source of uncertainties for the model-predicted surface temperature change induced by increasing greenhouse gases such as CO_2. Large-scale circulation broadly determines when and where clouds form and how they evolve. However, the linkage between large-scale circulation change and cloud radiative effect (CRE) change under global warming has not been thoroughly studied. By analyzing 15 climate models, we show that the change of the Hadley Circulation exhibits meridionally varying weakening and strengthening structures, physically consistent with the cloud changes in distinct cloud regimes. The regions that experience a weakening (strengthening) of the zonal-mean circulation account for 54% (46%) of the multimodel-mean top-of-atmosphere (TOA) CRE change integrated over 45°S–40°N. The simulated Hadley Circulation structure changes per degree of surface warming differ greatly between the models, and the intermodel spread in the Hadley Circulation change is well correlated with the intermodel spread in the TOA CRE change. This correlation underscores the close interactions between large-scale circulation and clouds and suggests that the uncertainties of cloud feedbacks and climate sensitivity reside in the intimate coupling between large-scale circulation and clouds. New model performance metrics proposed in this work, which emphasize how models reproduce satellite-observed spatial variations of zonal-mean cloud fraction and relative humidity associated with the Hadley Circulation, indicate that the models closer to the satellite observations tend to have equilibrium climate sensitivity higher than the multimodel mean

Universality of rain event size distributions

We compare rain event size distributions derived from measurements in climatically different regions, which we find to be well approximated by power laws of similar exponents over broad ranges. Differences can be seen in the large-scale cutoffs of the distributions. Event duration distributions suggest that the scale-free aspects are related to the absence of characteristic scales in the meteorological mesoscale