Evaluating a stochastic parametrization for a fast--slow system using the Wasserstein distance

Constructing accurate, flexible, and efficient parametrizations is one of the great challenges in the numerical modeling of geophysical fluids. We consider here the simple yet paradigmatic case of a Lorenz 84 model forced by a Lorenz 63 model and derive a parametrization using a recently developed statistical mechanical methodology based on the Ruelle response theory. We derive an expression for the deterministic and the stochastic component of the parametrization and we show that the approach allows for dealing seamlessly with the case of the Lorenz 63 being a fast as well as a slow forcing compared to the characteristic timescales of the Lorenz 84 model. We test our results using both standard metrics based on the moments of the variables of interest as well as Wasserstein distance between the projected measure of the original system on the Lorenz 84 model variables and the measure of the parametrized one. By testing our methods on reduced-phase spaces obtained by projection, we find support for the idea that comparisons based on the Wasserstein distance might be of relevance in many applications despite the curse of dimensionality

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

The exploitation of wild plants in Neolithic North Africa. Use-wear and residue analysis on non-knapped stone tools from the Haua Fteah cave, Cyrenaica, Libya

The North African region offers up essential data for the study of the origins of the earliest forms of plant exploitation. Data available from several Saharan and coastal areas in the region have revealed that the arrival of domestic wheat and barley from the Levant during the Mid Holocene did not replace the exploitation of autochthonous wild plants, especially grasses. The Neolithic layers of the Haua Fteah cave, in Cyrenaica (Northern Libya), have so far produced archaeobotanical assemblages exclusively made up of wild species. This paper investigates production and use of non-knapped stone tools, mainly grinding stones, from the Holocene sequence of the Haua Fteah Cave. The presence of grinding stones may indicate a certain level of behavioural change and the adoption of new economic strategies, relying more strongly on plant exploitation. This assumption has been tested using an integrated approach of use-wear and residue analysis. These methods allowed us to obtain significant new information as to how tools were originally used. Use-wear analysis was carried out adopting a low power approach; such study was complemented by residue analysis, in particular starch granules analysis, only rarely applied in North African contexts. This combined approach was carried out on a selected sample of tools, in order to test survival of plant micro-remains and, where possible, the types of plants processed by the Holocene communities in the cave. Here we confirm the lack of evidence of domesticated crops and the presence of starch granules only belonging to wild plants. The importance of wild plants in the economy of North African prehistoric groups has often been underestimated, on the assumption that they were replaced by the Near Eastern domesticated crops and animals once they were firstly introduced in North Africa. The results of our study show a different picture, confirming the role of the wild species as an important food source during the Holocene.his research was conducted as a part of the Marie Skłodowska-Curie Project FP7-People-2012-IEF ‘AGRINA’, funded by the European Commission. We gratefully acknowledge the permission and support of the Department of Antiquities of Libya to undertake the Cyrenaican Prehistory Project and the financial support from the European Research Grant (grant number 230421) and the Society for Libyan Studies.This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.quaint.2015.11.10

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

Covariant Lyapunov vectors of a quasi-geostrophic baroclinic model: analysis of instabilities and feedbacks

The classical approach for studying atmospheric variability is based on defining a background state and studying the linear stability of the small fluctuations around such a state. Weakly non-linear theories can be constructed using higher order expansions terms. While these methods have undoubtedly great value for elucidating the relevant physical processes, they are unable to follow the dynamics of a turbulent atmosphere. We provide a first example of extension of the classical stability analysis to a non-linearly evolving quasi-geostrophic flow. The so-called covariant Lyapunov vectors (CLVs) provide a covariant basis describing the directions of exponential expansion and decay of perturbations to the non-linear trajectory of the flow. We use such a formalism to re-examine the basic barotropic and baroclinic processes of the atmosphere with a quasi-geostrophic beta-plane two-layer model in a periodic channel driven by a forced meridional temperature gradient ΔT. We explore three settings of ΔT, representative of relatively weak turbulence, well-developed turbulence, and intermediate conditions. We construct the Lorenz energy cycle for each CLV describing the energy exchanges with the background state. A positive baroclinic conversion rate is a necessary but not sufficient condition of instability. Barotropic instability is present only for few very unstable CLVs for large values of ΔT. Slowly growing and decaying hydrodynamic Lyapunov modes closely mirror the properties of the background flow. Following classical necessary conditions for barotropic/baroclinic instability, we find a clear relationship between the properties of the eddy fluxes of a CLV and its instability. CLVs with positive baroclinic conversion seem to form a set of modes for constructing a reduced model of the atmosphere dynamics

Response operators for Markov processes in a finite state space: radius of convergence and link to the response theory for Axiom A systems

Using straightforward linear algebra we derive response operators describing the impact of small perturbations to finite state Markov processes. The results can be used for studying empirically constructed—e.g. from observations or through coarse graining of model simulations—finite state approximation of statistical mechanical systems. Recent results concerning the convergence of the statistical properties of finite state Markov approximation of the full asymptotic dynamics on the SRB measure in the limit of finer and finer partitions of the phase space are suggestive of some degree of robustness of the obtained results in the case of Axiom A system. Our findings give closed formulas for the linear and nonlinear response theory at all orders of perturbation and provide matrix expressions that can be directly implemented in any coding language, plus providing bounds on the radius of convergence of the perturbative theory. In particular, we relate the convergence of the response theory to the rate of mixing of the unperturbed system. One can use the formulas derived for finite state Markov processes to recover previous findings obtained on the response of continuous time Axiom A dynamical systems to perturbations, by considering the generator of time evolution for the measure and for the observables. A very basic, low-tech, and computationally cheap analysis of the response of the Lorenz ’63 model to perturbations provides rather encouraging results regarding the possibility of using the approximate representation given by finite state Markov processes to compute the system’s response

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

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

Regional climate models' performance in representing precipitation and temperature over selected Mediterranean areas

This paper discusses the relative performance of several climate models in providing reliable forcing for hydrological modeling in six representative catchments in the Mediterranean region. We consider 14 Regional Climate Models (RCMs), from the EU-FP6 ENSEMBLES project, run for the A1B emission scenario on a common 0.22° (about 24 km) rotated grid over Europe and the Mediterranean region. In the validation period (1951 to 2010) we consider daily precipitation and surface temperatures from the observed data fields (E-OBS) data set, available from the ENSEMBLES project and the data providers in the ECA&D project. Our primary objective is to rank the 14 RCMs for each catchment and select the four best-performing ones to use as common forcing for hydrological models in the six Mediterranean basins considered in the EU-FP7 CLIMB project. Using a common suite of four RCMs for all studied catchments reduces the (epistemic) uncertainty when evaluating trends and climate change impacts in the 21st century. We present and discuss the validation setting, as well as the obtained results and, in some detail, the difficulties we experienced when processing the data. In doing so we also provide useful information and advice for researchers not directly involved in climate modeling, but interested in the use of climate model outputs for hydrological modeling and, more generally, climate change impact studies in the Mediterranean region

Three-dimensional random Voronoi tessellations: From cubic crystal lattices to Poisson point processes

We perturb the SC, BCC, and FCC crystal structures with a spatial Gaussian noise whose adimensional strength is controlled by the parameter a, and analyze the topological and metrical properties of the resulting Voronoi Tessellations (VT). The topological properties of the VT of the SC and FCC crystals are unstable with respect to the introduction of noise, because the corresponding polyhedra are geometrically degenerate, whereas the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. For weak noise, the mean area of the perturbed BCC and FCC crystals VT increases quadratically with a. In the case of perturbed SCC crystals, there is an optimal amount of noise that minimizes the mean area of the cells. Already for a moderate noise (a>0.5), the properties of the three perturbed VT are indistinguishable, and for intense noise (a>2), results converge to the Poisson-VT limit. Notably, 2-parameter gamma distributions are an excellent model for the empirical of of all considered properties. The VT of the perturbed BCC and FCC structures are local maxima for the isoperimetric quotient, which measures the degre of sphericity of the cells, among space filling VT. In the BCC case, this suggests a weaker form of the recentluy disproved Kelvin conjecture. Due to the fluctuations of the shape of the cells, anomalous scalings with exponents >3/2 is observed between the area and the volumes of the cells, and, except for the FCC case, also for a->0. In the Poisson-VT limit, the exponent is about 1.67. As the number of faces is positively correlated with the sphericity of the cells, the anomalous scaling is heavily reduced when we perform powerlaw fits separately on cells with a specific number of faces