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

Global-scale climate impact functions: the relationship between climate forcing and impact

Although there is a strong policy interest in the impacts of climate change corresponding to different degrees of climate change, there is so far little consistent empirical evidence of the relationship between climate forcing and impact. This is because the vast majority of impact assessments use emissions-based scenarios with associated socio-economic assumptions, and it is not feasible to infer impacts at other temperature changes by interpolation. This paper presents an assessment of the global-scale impacts of climate change in 2050 corresponding to defined increases in global mean temperature, using spatially-explicit impacts models representing impacts in the water resources, river flooding, coastal, agriculture, ecosystem and built environment sectors. Pattern-scaling is used to construct climate scenarios associated with specific changes in global mean surface temperature, and a relationship between temperature and sea level used to construct sea level rise scenarios. Climate scenarios are constructed from 21 climate models to give an indication of the uncertainty between forcing and response. The analysis shows that there is considerable uncertainty in the impacts associated with a given increase in global mean temperature, due largely to uncertainty in the projected regional change in precipitation. This has important policy implications. There is evidence for some sectors of a non-linear relationship between global mean temperature change and impact, due to the changing relative importance of temperature and precipitation change. In the socio-economic sectors considered here, the relationships are reasonably consistent between socio-economic scenarios if impacts are expressed in proportional terms, but there can be large differences in absolute terms. There are a number of caveats with the approach, including the use of pattern-scaling to construct scenarios, the use of one impacts model per sector, and the sensitivity of the shape of the relationships between forcing and response to the definition of the impact indicator

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