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

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

High dimensional decision dilemmas in climate models

An important source of uncertainty in climate models is linked to the calibration of model parameters. Interest in systematic and automated parameter optimization procedures stems from the desire to improve the model climatology and to quantify the average sensitivity associated with potential changes in the climate system. Building upon on the smoothness of the response of an atmospheric circulation model (AGCM) to changes of four adjustable parameters, Neelin et al. (2010) used a quadratic metamodel to objectively calibrate the AGCM. The metamodel accurately estimates global spatial averages of common fields of climatic interest, from precipitation, to low and high level winds, from temperature at various levels to sea level pressure and geopotential height, while providing a computationally cheap strategy to explore the influence of parameter settings. Here, guided by the metamodel, the ambiguities or dilemmas related to the decision making process in relation to model sensitivity and optimization are examined. Simulations of current climate are subject to considerable regional-scale biases. Those biases may vary substantially depending on the climate variable considered, and/or on the performance metric adopted. Common dilemmas are associated with model revisions yielding improvement in one field or regional pattern or season, but degradation in another, or improvement in the model climatology but degradation in the interannual variability representation. Challenges are posed to the modeler by the high dimensionality of the model output fields and by the large number of adjustable parameters. The use of the metamodel in the optimization strategy helps visualize trade-offs at a regional level, e.g., how mismatches between sensitivity and error spatial fields yield regional errors under minimization of global objective functions

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

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.Comment: 16 pages, 10 figure

Recent Advances Concerning Certain Class of Geophysical Flows

This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for large-scale oceanic and atmospheric dynamics are derived from the Navier-Stokes equations coupled to the heat convection by adopting the Boussinesq and hydrostatic approximations, while the tropical atmosphere model considered here is a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture. We are mainly concerned with the global well-posedness of strong solutions to these systems, with full or partial viscosity, as well as certain singular perturbation small parameter limits related to these systems, including the small aspect ratio limit from the Navier-Stokes equations to the PEs, and a small relaxation-parameter in the tropical atmosphere model. These limits provide a rigorous justification to the hydrostatic balance in the PEs, and to the relaxation limit of the tropical atmosphere model, respectively. Some conditional uniqueness of weak solutions, and the global well-posedness of weak solutions with certain class of discontinuous initial data, to the PEs are also presented.Comment: arXiv admin note: text overlap with arXiv:1507.0523

Climate dynamics and fluid mechanics: Natural variability and related uncertainties

The purpose of this review-and-research paper is twofold: (i) to review the role played in climate dynamics by fluid-dynamical models; and (ii) to contribute to the understanding and reduction of the uncertainties in future climate-change projections. To illustrate the first point, we focus on the large-scale, wind-driven flow of the mid-latitude oceans which contribute in a crucial way to Earth's climate, and to changes therein. We study the low-frequency variability (LFV) of the wind-driven, double-gyre circulation in mid-latitude ocean basins, via the bifurcation sequence that leads from steady states through periodic solutions and on to the chaotic, irregular flows documented in the observations. This sequence involves local, pitchfork and Hopf bifurcations, as well as global, homoclinic ones. The natural climate variability induced by the LFV of the ocean circulation is but one of the causes of uncertainties in climate projections. Another major cause of such uncertainties could reside in the structural instability in the topological sense, of the equations governing climate dynamics, including but not restricted to those of atmospheric and ocean dynamics. We propose a novel approach to understand, and possibly reduce, these uncertainties, based on the concepts and methods of random dynamical systems theory. As a very first step, we study the effect of noise on the topological classes of the Arnol'd family of circle maps, a paradigmatic model of frequency locking as occurring in the nonlinear interactions between the El Nino-Southern Oscillations (ENSO) and the seasonal cycle. It is shown that the maps' fine-grained resonant landscape is smoothed by the noise, thus permitting their coarse-grained classification. This result is consistent with stabilizing effects of stochastic parametrization obtained in modeling of ENSO phenomenon via some general circulation models.Comment: Invited survey paper for Special Issue on The Euler Equations: 250 Years On, in Physica D: Nonlinear phenomen

Tropical air-sea interaction in general circulation models

An intercomparison is undertaken of the tropical behavior of 17 coupled ocean-atmosphere models in which at least one component may be termed a general circulation model (GCM). The aim is to provide a taxonomy—a description and rough classification—of behavior across the ensemble of models, focusing on interannual variability. The temporal behavior of the sea surface temperature (SST) field along the equator is presented for each model, SST being chosen as the primary variable for intercomparison due to its crucial role in mediating the coupling and because it is a sensitive indicator of climate drift. A wide variety of possible types of behavior are noted among the models. Models with substantial interannual tropical variability may be roughly classified into cases with propagating SST anomalies and cases in which the SST anomalies develop in place. A number of the models also exhibit significant drift with respect to SST climatology. However, there is not a clear relationship between climate drift and the presence or absence of interannual oscillations. In several cases, the mode of climate drift within the tropical Pacific appears to involve coupled feedback mechanisms similar to those responsible for El Niño variability. Implications for coupled-model development and for climate prediction on seasonal to interannual time scales are discussed. Overall, the results indicate considerable sensitivity of the tropical coupled ocean-atmosphere system and suggest that the simulation of the warm-pool/cold-tongue configuration in the equatorial Pacific represents a challenging test for climate model parameterizations

Pacific climate variability and the possible impact on global surface CO2 flux

<p>Abstract</p> <p>Background</p> <p>Climate variability modifies both oceanic and terrestrial surface CO2 flux. Using observed/assimilated data sets, earlier studies have shown that tropical oceanic climate variability has strong impacts on the land surface temperature and soil moisture, and that there is a negative correlation between the oceanic and terrestrial CO2 fluxes. However, these data sets only cover less than the most recent 20 years and are insufficient for identifying decadal and longer periodic variabilities. To investigate possible impacts of interannual to interdecadal climate variability on CO2 flux exchange, the last 125 years of an earth system model (ESM) control run are examined.</p> <p>Results</p> <p>Global integration of the terrestrial CO2 flux anomaly shows variation much greater in amplitude and longer in periodic timescale than the oceanic flux. The terrestrial CO2 flux anomaly correlates negatively with the oceanic flux in some periods, but positively in others, as the periodic timescale is different between the two variables. To determine the spatial pattern of the variability, a series of composite analyses are performed. The results show that the oceanic CO2 flux variability peaks when the eastern tropical Pacific has a large sea surface temperature anomaly (SSTA). By contrast, the terrestrial CO2 flux variability peaks when the SSTA appears in the central tropical Pacific. The former pattern of variability resembles the ENSO-mode and the latter the ENSO-modoki¹.</p> <p>Conclusions</p> <p>Our results imply that the oceanic and terrestrial CO2 flux anomalies may correlate either positively or negatively depending on the relative phase of these two modes in the tropical Pacific.</p