Variational approach to the closure problem of turbulence theory

A new method is proposed to solve the closure problem of turbulence theory and to drive the Kolmogorov law in an Eulerian framework. Instead of using complex Fourier components of velocity field as modal parameters, a complete set of independent real parameters and dynamic equations are worked out to describe the dynamic states of a turbulence. Classical statistical mechanics is used to study the statistical behavior of the turbulence. An approximate stationary solution of the Liouville equation is obtained by a perturbation method based on a Langevin-Fokker-Planck (LFP) model. The dynamic damping coefficient eta of the LFP model is treated as an optimum control parameter to minimize the error of the perturbation solution; this leads to a convergent integral equation for eta to replace the divergent response equation of Kraichnan's direct-interaction (DI) approximation, thereby solving the closure problem without appealing to a Lagrangian formulation. The Kolmogorov constant Ko is evaluated numerically, obtaining Ko = 1.2, which is compatible with the experimental data given by Gibson and Schwartz, (1963).

Modelling the stable, nocturnal boundary layer: different approaches to turbulence closure problem

Il lavoro è dedicato all'analisi fisica e alla modellizzazione dello strato limite atmosferico in condizioni stabili. L'obiettivo principale è quello di migliorare i modelli di parametrizzazione della turbulenza attualmente utilizzati dai modelli meteorologici a grande scala. Questi modelli di parametrizzazione della turbolenza consistono nell' esprimere gli stress di Reynolds come funzioni dei campi medi (componenti orizzontali della velocità e temperatura potenziale) usando delle chiusure. La maggior parte delle chiusure sono state sviluppate per i casi quasi-neutrali, e la difficoltà è trattare l'effetto della stabilità in modo rigoroso. Studieremo in dettaglio due differenti modelli di chiusura della turbolenza per lo strato limite stabile basati su assunzioni diverse: uno schema TKE-l (Mellor-Yamada,1982), che è usato nel modello di previsione BOLAM (Bologna Limited Area Model), e uno schema sviluppato recentemente da Mauritsen et al. (2007). Le assunzioni delle chiusure dei due schemi sono analizzate con dati sperimentali provenienti dalla torre di Cabauw in Olanda e dal sito CIBA in Spagna. Questi schemi di parametrizzazione della turbolenza sono quindi inseriti all'interno di un modello colonnare dello strato limite atmosferico, per testare le loro predizioni senza influenze esterne. Il confronto tra i differenti schemi è effettuato su un caso ben documentato in letteratura, il "GABLS1". Per confermare la validità delle predizioni, un dataset tridimensionale è creato simulando lo stesso caso GABLS1 con una Large Eddy Simulation. ARPS (Advanced Regional Prediction System) è stato usato per questo scopo. La stratificazione stabile vincola il passo di griglia, poichè la LES deve essere ad una risoluzione abbastanza elevata affinchè le tipiche scale verticali di moto siano correttamente risolte. Il confronto di questo dataset tridimensionale con le predizioni degli schemi turbolenti permettono di proporre un insieme di nuove chiusure atte a migliorare il modello di turbolenza di BOLAM. Il lavoro è stato compiuto all' ISAC-CNR di Bologna e al LEGI di Grenoble.

Fluxes of CO(2) above a plantation of Eucalyptus in southeast Brazil

Eddy-covariance measurements of net ecosystem exchange of CO(2) (NEE) and estimates of gross ecosystem productivity (GEP) and ecosystem respiration (R(E)) were obtained in a 2-4 year old Eucalyptus plantation during two years with very different winter rainfall In the first (drier) year the annual NEE GEP and RE were lower than the sums in the second (normal) year and conversely the total respiratory costs of assimilated carbon were higher in the dry year than in the normal year Although the net primary production (NPP) in the first year was 23% lower than that of the second year the decrease in the carbon use efficiency (CUE = NPP/GEP) was 11% and autotrophic respiration utilized more resources in the first dry year than in the second normal year The time variations in NEE were followed by NPP because in these young Eucalyptus plantations NEE is very largely dominated by NPP and heterotrophic respiration plays only a relatively minor role During the dry season a pronounced hysteresis was observed in the relationship between NEE and photosynthetically active radiation and NEE fluxes were inversely proportional to humidity saturation deficit values greater than 0 8 kPa Nighttime fluxes of CO(2) during calm conditions when the friction velocity (u) was below the threshold (0 25 ms(-1)) were estimated based on a Q(10) temperature-dependence relationship adjusted separately for different classes of soil moisture content which regulated the temperature sensitivity of ecosystem respiration (C) 2010 Elsevier B V All rights reserved

A closure theory of intermittency of turbulence

The variational approach to the closure problem of turbulence theory, proposed in an earlier article [Phys. Fluids 26, 2098 (1983); 27, 2229 (1984)], is extended to evaluate the flatness factor, which indicates the degree of intermittency of turbulence. Since the flatness factor is related to the fourth moment of a turbulent velocity field, the corresponding higher-order terms in the perturbation solution of the Liouville equation have to be considered. Most closure methods discard these higher-order terms and fail to explain the intermittency phenomenon. The computed flatness factor of the idealized model of infinite isotropic turbulence ranges from 9 to 15 and has the same order of magnitude as the experimental data of real turbulent flows. The intermittency phenomenon does not necessarily negate the Kolmogorov k−5/3 inertial range spectrum. The Kolmogorov k−5/3 law and the high degree of intermittency can coexist as two consistent consequences of the closure theory of turbulence. The Kolmogorov 1941 theory [J. Fluid Mech. 62, 305 (1974)] cannot be disqualified merely because the energy dissipation rate fluctuates.

An integrated model of soil-canopy spectral radiances, photosynthesis, fluorescence, temperature and energy balance

This paper presents the model SCOPE (Soil Canopy Observation, Photochemistry and Energy fluxes), which is a vertical (1-D) integrated radiative transfer and energy balance model. The model links visible to thermal infrared radiance spectra (0.4 to 50 μm) as observed above the canopy to the fluxes of water, heat and carbon dioxide, as a function of vegetation structure, and the vertical profiles of temperature. Output of the model is the spectrum of outgoing radiation in the viewing direction and the turbulent heat fluxes, photosynthesis and chlorophyll fluorescence. A special routine is dedicated to the calculation of photosynthesis rate and chlorophyll fluorescence at the leaf level as a function of net radiation and leaf temperature. The fluorescence contributions from individual leaves are integrated over the canopy layer to calculate top-of-canopy fluorescence. The calculation of radiative transfer and the energy balance is fully integrated, allowing for feedback between leaf temperatures, leaf chlorophyll fluorescence and radiative fluxes. Leaf temperatures are calculated on the basis of energy balance closure. Model simulations were evaluated against observations reported in the literature and against data collected during field campaigns. These evaluations showed that SCOPE is able to reproduce realistic radiance spectra, directional radiance and energy balance fluxes. The model may be applied for the design of algorithms for the retrieval of evapotranspiration from optical and thermal earth observation data, for validation of existing methods to monitor vegetation functioning, to help interpret canopy fluorescence measurements, and to study the relationships between synoptic observations with diurnally integrated quantities. The model has been implemented in Matlab and has a modular design, thus allowing for great flexibility and scalability.

A MODIS-Based Energy Balance to Estimate Evapotranspiration for Clear-Sky Days in Brazilian Tropical Savannas

Evapotranspiration (ET) plays an important role in global climate dynamics and in primary production of terrestrial ecosystems; it represents the mass and energy transfer from the land to atmosphere. Limitations to measuring ET at large scales using ground-based methods have motivated the development of satellite remote sensing techniques. The purpose of this work is to evaluate the accuracy of the SEBAL algorithm for estimating surface turbulent heat fluxes at regional scale, using 28 images from MODIS. SEBAL estimates are compared with eddy-covariance (EC) measurements and results from the hydrological model MGB-IPH. SEBAL instantaneous estimates of latent heat flux (LE) yielded r(2) = 0.64 and r(2) = 0.62 over sugarcane croplands and savannas when compared against in situ EC estimates. At the same sites, daily aggregated estimates of LE were r(2) = 0.76 and r(2) = 0.66, respectively. Energy balance closure showed that turbulent fluxes over sugarcane croplands were underestimated by 7% and 9% over savannas. Average daily ET from SEBAL is in close agreement with estimates from the hydrological model for an overlay of 38,100 km(2) (r(2) = 0.88). Inputs to which the algorithm is most sensitive are vegetation index (NDVI), gradient of temperature (dT) to compute sensible heat flux (H) and net radiation (Re). It was verified that SEBAL has a tendency to overestimate results both at local and regional scales probably because of low sensitivity to soil moisture and water stress. Nevertheless the results confirm the potential of the SEBAL algorithm, when used with MODIS images for estimating instantaneous LE and daily ET from large areas.

Optimum Trajectory of Dynamic Input-Output Balance Model for Open Economy

When export and import is connected with output of basic production, and criterion functional represents a final state of economy, the generalization of classical qualitative results of the main-line theory on a case of dynamic input-output balance optimization model for open economy is given.

Summertime Influences of Tidal Energy Advection on the Surface Energy Balance in a Mangrove Forest

Mangrove forests are ecosystems susceptible to changing water levels and temperatures due to climate change as well as perturbations resulting from tropical storms. Numerical models can be used to project mangrove forest responses to regional and global environmental changes, and the reliability of these models depends on surface energy balance closure. However, for tidal ecosystems, the surface energy balance is complex because the energy transport associated with tidal activity remains poorly understood. This study aimed to quantify impacts of tidal flows on energy dynamics within a mangrove ecosystem. To address the research objective, an intensive 10-day study was conducted in a mangrove forest located along the Shark River in the Everglades National Park, FL, USA. Forest–atmosphere turbulent exchanges of energy were quantified with an eddy covariance system installed on a 30-m-tall flux tower. Energy transport associated with tidal activity was calculated based on a coupled mass and energy balance approach. The mass balance included tidal flows and accumulation of water on the forest floor. The energy balance included temporal changes in enthalpy, resulting from tidal flows and temperature changes in the water column. By serving as a net sink or a source of available energy, flood waters reduced the impact of high radiational loads on the mangrove forest. Also, the regression slope of available energy versus sink terms increased from 0.730 to 0.754 and from 0.798 to 0.857, including total enthalpy change in the water column in the surface energy balance for 30-min periods and daily daytime sums, respectively. Results indicated that tidal inundation provides an important mechanism for heat removal and that tidal exchange should be considered in surface energy budgets of coastal ecosystems. Results also demonstrated the importance of including tidal energy advection in mangrove biophysical models that are used for predicting ecosystem response to changing climate and regional freshwater management practices.

The issue of macroeconomic closure revisited and extended

Léon Walras (1874) already had realized that his neo-classical general equilibrium model could not accommodate autonomous investment. Sen analysed the same issue in a simple, one-sector macroeconomic model of a closed economy. He showed that fixing investment in the model, built strictly on neo-classical assumptions, would make the system overdetermined, thus, one should loosen some neo-classical condition of competitive equilibrium. He analysed three not neo-classical “closure options”, which could make the model well determined in the case of fixed investment. Others later extended his list and it showed that the closure dilemma arises in the more complex computable general equilibrium (CGE) models as well, as does the choice of adjustment mechanism assumed to bring about equilibrium at the macro level. By means of numerical models, it was also illustrated that the adopted closure rule can significantly affect the results of policy simulations based on a CGE model. Despite these warnings, the issue of macro closure is often neglected in policy simulations. It is, therefore, worth revisiting the issue and demonstrating by further examples its importance, as well as pointing out that the closure problem in the CGE models extends well beyond the problem of how to incorporate autonomous investment into a CGE model. Several closure rules are discussed in this paper and their diverse outcomes are illustrated by numerical models calibrated on statistical data. First, the analyses is done in a one-sector model, similar to Sen’s, but extended into a model of an open economy. Next, the same analyses are repeated using a fully-fledged multisectoral CGE model, calibrated on the same statistical data. Comparing the results obtained by the two models it is shown that although, using the same closure option, they generate quite similar results in terms of the direction and – to a somewhat lesser extent – of the magnitude of change in the main macro variables, the predictions of the multi-sectoral CGE model are clearly more realistic and balanced.

Unstructured strategic decision-making processes : CRE decision-making in the Italian consulting industry

This thesis aims at developing a better understanding of unstructured strategic decision making processes and the conditions for achieving successful decision outcomes. Specifically it focuses on the processes used to make CRE (Corporate Real Estate) decisions. The starting point for this thesis is that our knowledge of such processes is incomplete. A comprehensive study of the most recent CRE literature together with Behavioural Organization Theory has provided a research framework for the exploration of CRE recommended =best practice‘, and of how organizational variables impact on and shape these practices. To reveal the fundamental differences between CRE decision-making in practice and the prescriptive =best practice‘ advocated in the CRE literature, a study of seven Italian management consulting firms was undertaken addressing the aspects of content and process of decisions. This thesis makes its primary contribution by identifying the importance and difficulty of finding the right balance between problem complexity, process richness and cohesion to ensure a decision-making process that is sufficiently rich and yet quick enough to deliver a prompt outcome. While doing so, this research also provides more empirical evidence to some of the most established theories of decision-making while reinterpreting their mono-dimensional arguments in a multi-dimensional model of successful decision-making.

Unstructured processes of strategic decision-making

The author aims at developing a better understanding of unstructured strategic decision making processes and the conditions for achieving successful decision outcomes. Specifically he investigates the processes used to make CRE (Corporate Real Estate) decisions. To reveal the fundamental differences between CRE decision-making in practice and the prescriptive ‘best practice’ advocated in the CRE literature, a study of seven leading Italian management consulting firms is undertaken addressing the aspects of content and process of decisions. This research makes its primary contribution by identifying the importance and difficulty of finding the right balance between problem complexity, process richness and cohesion to ensure a decision-making process that is sufficiently rich and yet quick enough to deliver a prompt outcome. While doing so, the study also provides more empirical evidence to some of the most established theories of decision-making, while reinterpreting their mono-dimensional arguments in a multi-dimensional model of successful decision-making.

A passive scalar field convected by turbulence

Classical statistical mechanics is applied to the study of a passive scalar field convected by isotropic turbulence. A complete set of independent real parameters and dynamic equations are worked out to describe the dynamic state of the passive scalar field. The corresponding Liouville equation is solved by a perturbation method based upon a Langevin–Fokker–Planck model. The closure problem is treated by a variational approach reported in earlier papers. Two integral equations are obtained for two unknown functions: the scalar variance spectrum F(k) and the effective damping coefficient (k). The appearance of the energy spectrum of the velocity field in the two integral equations represents the coupling of the scalar field with the velocity field. As an application of the theory, the two integral equations are solved to derive the inertial-convective-range spectrum, obtaining F(k)=0.61 −1/3 k−5/3. Here is the dissipation rate of the scalar variance and is the dissipation rate of the energy of the velocity field. This theoretical value of the scalar Kolmogorov constant, 0.61, is in good agreement with experiments.

Nonequilibrium statistical mechanics of one-dimensional turbulence

The method of statistical mechanics is applied to the study of the one-dimensional model of turbulence proposed in an earlier paper. The closure problem is solved by the variational approach which has been developed for the three-dimensional case, yielding two integral equations for two unknown functions. By solving the two integral equations, the Kolmogorov k−5/3 law is derived and the (one-dimensional) Kolmogorov constant Ko is evaluated, obtaining Ko=0.55, which is in good agreement with the result of numerical experiments on one-dimensional turbulence.

Universal equilibrium range of turbulence

An attempt is made to determine the form of F(x), the dimensionless function of universal nature which occurs in the energy spectrum for the universal equilibrium range of fully developed turbulence, by the method of statistical mechanics without introducing any parameter of semiempirical nature. Then, the validity of the variational approach to the closure problem of turbulence theory is tested by applying it to the study of the universal equilbrium range of turbulence.