Nambu representation of an extended Lorenz model with viscous heating

We consider the Nambu and Hamiltonian representations of Rayleigh-Benard convection with a nonlinear thermal heating effect proportional to the Eckert number (Ec). The model we use is an extension of the classical Lorenz-63 model with 4 kinematic and 6 thermal degrees of freedom. The conservative parts of the dynamical equations which include all nonlinearities satisfy Liouville's theorem and permit a conserved Hamiltonian H for arbitrary Ec. For Ec=0 two independent conserved Casimir functions exist, one of these is associated with unavailable potential energy and is also present in the Lorenz-63 truncation. This Casimir C is used to construct a Nambu representation of the conserved part of the dynamical system. The thermal heating effect can be represented either by a second canonical Hamiltonian or as a gradient (metric) system using the time derivative of the Casimir. The results demonstrate the impact of viscous heating in the total energy budget and in the Lorenz energy cycle for kinetic and available potential energy.Comment: 15 pages, no figur

Detection and correction of the misplacement error in THz Spectroscopy by application of singly subtractive Kramers-Kronig relations

In THz reflection spectroscopy the complex permittivity of an opaque medium is determined on the basis of the amplitude and of the phase of the reflected wave. There is usually a problem of phase error due to misplacement of the reference sample. Such experimental error brings inconsistency between phase and amplitude invoked by the causality principle. We propose a rigorous method to solve this relevant experimental problem by using an optimization method based upon singly subtractive Kramers-Kronig relations. The applicability of the method is demonstrated for measured data on an n-type undoped (100) InAs wafer in the spectral range from 0.5 up to 2.5 THz.Comment: 16 pages, 5 figure

Return levels of temperature extremes in southern Pakistan

Southern Pakistan (Sindh) is one of the hottest regions in the world and is highly vulnerable to temperature extremes. In order to improve rural and urban planning, it is useful to gather information about the recurrence of temperature extremes. In this work, return levels of the daily maximum temperature Tmax are estimated, as well as the daily maximum wet-bulb temperature TWmax extremes. We adopt the peaks over threshold (POT) method, which has not yet been used for similar studies in this region. Two main datasets are analyzed: temperatures observed at nine meteorological stations in southern Pakistan from 1980 to 2013, and the ERA-Interim (ECMWF reanalysis) data for the nearest corresponding locations. The analysis provides the 2-, 5-, 10-, 25-, 50-, and 100-year return levels (RLs) of temperature extremes. The 90 % quantile is found to be a suitable threshold for all stations. We find that the RLs of the observed Tmax are above 50 °C at northern stations and above 45 °C at the southern stations. The RLs of the observed TWmax exceed 35 °C in the region, which is considered as a limit of survivability. The RLs estimated from the ERA-Interim data are lower by 3 to 5 °C than the RLs assessed for the nine meteorological stations. A simple bias correction applied to ERA-Interim data improves the RLs remarkably, yet discrepancies are still present. The results have potential implications for the risk assessment of extreme temperatures in Sindh

Effects of stochastic parametrization on extreme value statistics

Extreme geophysical events are of crucial relevance to our daily life: they threaten human lives and cause property damage. To assess the risk and reduce losses, we need to model and probabilistically predict these events. Parametrizations are computational tools used in the Earth system models, which are aimed at reproducing the impact of unresolved scales on resolved scales. The performance of parametrizations has usually been examined on typical events rather than on extreme events. In this paper, we consider a modified version of the two-level Lorenz’96 model and investigate how two parametrizations of the fast degrees of freedom perform in terms of the representation of extreme events. One parametrization is constructed following Wilks [Q. J. R. Meteorol. Soc. 131, 389–407 (2005)] and is constructed through an empirical fitting procedure; the other parametrization is constructed through the statistical mechanical approach proposed by Wouters and Lucarini [J. Stat. Mech. Theory Exp. 2012, P03003 (2012); J. Stat. Phys. 151, 850–860 (2013)]. The two strategies show different advantages and disadvantages. We discover that the agreement between parametrized models and true model is in general worse when looking at extremes rather than at the bulk of the statistics. The results suggest that stochastic parametrizations should be accurately and specifically tested against their performance on extreme events, as usual optimization procedures might neglect them. The provision of accurate parametrizations is a task of paramount importance in many scientific areas and specifically in weather and climate modeling. Parametrizations are needed for representing accurately and efficiently the impact of the scales of motions and of the processes that cannot be explicitly represented by the numerical model. Parametrizations are usually constructed in order to optimize the overall performance of the model, thus aiming at an accurate representation of the bulk of the statistics. Nonetheless, numerical models are key to estimating, anticipating, and predicting extreme events. Here, we analyze critically in a simple yet illustrative example the performance of parametrizations in describing extreme events, and we conclude that good performance on typical conditions cannot be in any way extrapolated for rare conditions, which could, nonetheless, be of great practical relevance

Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach

In this paper we consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system, which are weakly coupled to the remaining variables. In a previous paper we have used the Ruelle response theory on such a weakly coupled system to construct a surrogate dynamics, such that the expectation value of any observable agrees, up to second order in the coupling strength, to its expectation evaluated on the full dynamics. We show here that such surrogate dynamics agree up to second order to an expansion of the Mori-Zwanzig projected dynamics. This implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related, if one takes into account, in addition to the widely adopted stochastic forcing, the often neglected memory effects.Comment: 14 pages, 1 figur

Avalanches, breathers, and flow reversal in a continuous Lorenz-96 model

For the discrete model suggested by Lorenz in 1996, a one-dimensional long-wave approximation with nonlinear excitation and diffusion is derived. The model is energy conserving but non-Hamiltonian. In a low-order truncation, weak external forcing of the zonal mean flow induces avalanchelike breather solutions which cause reversal of the mean flow by a wave-mean flow interaction. The mechanism is an outburst-recharge process similar to avalanches in a sandpile model

Universal Properties of Nonlinear Response Functions of Nonequilibrium Steady States

We derive universal properties of nonlinear response functions of nonequilibrium steady states. In particular, sum rules and asymptotic behaviors are derived. Their consequences are illustrated for nonlinear optical materials and nonlinear electrical conductors.Comment: 10 pages, 1 figure; added a few sentences and references to explain detail

Early 21st century snow cover state over the western river basins of the Indus River system

In this paper we assess the snow cover and its dynamics for the western river basins of the Indus River system (IRS) and their sub-basins located in Afghanistan, China, India and Pakistan for the period 2001–2012. First, we validate the Moderate Resolution Imaging Spectroradiometer (MODIS) daily snow products from Terra (MOD10A1) and Aqua (MYD10A1) against the Landsat Thematic Mapper/Enhanced Thematic Mapper plus (TM/ETM+) data set, and then improve them for clouds by applying a validated non-spectral cloud removal technique. The improved snow product has been analysed on a seasonal and annual basis against different topographic parameters (aspect, elevation and slope). Our results show a decreasing tendency for the annual average snow cover for the westerlies-influenced basins (upper Indus basin (UIB), Astore, Hunza, Shigar and Shyok) and an increasing tendency for the monsoon-influenced basins (Jhelum, Kabul, Swat and Gilgit). Seasonal average snow cover decreases during winter and autumn, and increases during spring and summer, which is consistent with the observed cooling and warming trends during the respective seasons. Sub-basins at relatively higher latitudes/altitudes show higher variability than basins at lower latitudes/middle altitudes. Northeastern and northwestern aspects feature greater snow cover. The mean end-of-summer regional snow line altitude (SLA) zones range from 3000 to 5000 m a.s.l. for all basins. Our analysis provides an indication of a descending end-of-summer regional SLA zone for most of the studied basins, which is significant for the Shyok and Kabul basins, thus indicating a change in their water resources. Such results are consistent with the observed hydro-climatic data, recently collected local perceptions and glacier mass balances for the investigated period within the UIB. Moreover, our analysis shows a significant correlation between winter season snow cover and the North Atlantic Oscillation (NAO) index of the previous autumn. Similarly, the inter-annual variability of spring season snow cover and spring season precipitation explains well the inter-annual variability of the summer season discharge from most of the basins. These findings indicate some potential for the seasonal stream flow forecast in the region, suggesting snow cover as a possible predictor

Predicting climate change using response theory: global averages and spatial patterns

The provision of accurate methods for predicting the climate response to anthropogenic and natural forcings is a key contemporary scientific challenge. Using a simplified and efficient open-source general circulation model of the atmosphere featuring O(105105) degrees of freedom, we show how it is possible to approach such a problem using nonequilibrium statistical mechanics. Response theory allows one to practically compute the time-dependent measure supported on the pullback attractor of the climate system, whose dynamics is non-autonomous as a result of time-dependent forcings. We propose a simple yet efficient method for predicting—at any lead time and in an ensemble sense—the change in climate properties resulting from increase in the concentration of CO22 using test perturbation model runs. We assess strengths and limitations of the response theory in predicting the changes in the globally averaged values of surface temperature and of the yearly total precipitation, as well as in their spatial patterns. The quality of the predictions obtained for the surface temperature fields is rather good, while in the case of precipitation a good skill is observed only for the global average. We also show how it is possible to define accurately concepts like the inertia of the climate system or to predict when climate change is detectable given a scenario of forcing. Our analysis can be extended for dealing with more complex portfolios of forcings and can be adapted to treat, in principle, any climate observable. Our conclusion is that climate change is indeed a problem that can be effectively seen through a statistical mechanical lens, and that there is great potential for optimizing the current coordinated modelling exercises run for the preparation of the subsequent reports of the Intergovernmental Panel for Climate Change

Stochastic resonance for nonequilibrium systems

Stochastic resonance (SR) is a prominent phenomenon in many natural and engineered noisy systems, whereby the response to a periodic forcing is greatly amplified when the intensity of the noise is tuned to within a specific range of values. We propose here a general mathematical framework based on large deviation theory and, specifically, on the theory of quasipotentials, for describing SR in noisy N -dimensional nonequilibrium systems possessing two metastable states and undergoing a periodically modulated forcing. The drift and the volatility fields of the equations of motion can be fairly general, and the competing attractors of the deterministic dynamics and the edge state living on the basin boundary can, in principle, feature chaotic dynamics. Similarly, the perturbation field of the forcing can be fairly general. Our approach is able to recover as special cases the classical results previously presented in the literature for systems obeying detailed balance and allows for expressing the parameters describing SR and the statistics of residence times in the two-state approximation in terms of the unperturbed drift field, the volatility field, and the perturbation field. We clarify which specific properties of the forcing are relevant for amplifying or suppressing SR in a system and classify forcings according to classes of equivalence. Our results indicate a route for a detailed understanding of SR in rather general systems