Space scales

The first step is to define the objectives of the flow model, and to identify those flow aspects that are relevant for the performance of the reactor. Then, the engineer must identify and quantify the various times and space scales involved, as well as the geometry of the system. These actions allow the problem to be represented by a mathematical model. Creating this model accurately is the most crucial task in the flow modeling project. 

Valentini, R., Scarascia Mugnozza, G.E., De Agnelis, P. and Matteucci, G. 1995 Coupling water and carbon metabolism of natural vegetation at integrated time and space scales. Agricultural and Forest Meteorology 73 297-306. 

The appropriate time and space scales are imposed by estimated health effects functions, source and population patterns, data quality and availability, and by the user s information needs. These constraints have led to a wide range of analytical approaches. 

Input Errors. Errors in model input often constitute one of the most significant causes of discrepancies between observed data and model predictions. As shown in Figure 2, the natural system receives the "true" input (usually as a "driving function") whereas the model receives the "observed" input as detected by some measurement method or device. Whenever a measurement is made possible source of error is introduced. System inputs usually vary continuously both in space and time, whereas measurements are usually point values, or averages of multiple point values, and for a particular time or accumulated over a time period. Although continuous measurement devices are in common use, errors are still possible, and essentially all models require transformation of a continuous record into discrete time and space scales acceptable to the model formulation and structure. 

System Representation Errors. System representation errors refer to differences in the processes and the time and space scales represented in the model, versus those that determine the response of the natural system. In essence, these errors are the major ones of concern when one asks "How good is the model ". Whenever comparing model output with observed data in an attempt to evaluate model capabilities, the analyst must have an understanding of the major natural processes, and human impacts, that influence the observed data. Differences between model output and observed data can then be analyzed in light of the limitations of the model algorithm used to represent a particularly critical process, and to insure that all such critical processes are modeled to some appropriate level of detail. For example, a... 

C. Space-Scale Analysis Based on Continuous Wavelet Transform Low-Frequency Rhythms in Human DNA Sequences... 

The continuous wavelet transform (WT) is a space-scale analysis that consists in expanding signals in terms of wavelets that are constructed from a single function, the analyzing wavelet /, by means of dilations and translations [13, 27-29]. When using the successive derivatives of the Gaussian function as analyzing wavelets, namely... 

Figure 2 reports the results [34, 35] of a space-scale decomposition of the GC content fluctuations of a 10-Mbp-long fragment of human chromosome 22, when using the Gaussian [Eq. (6)] as smoothing filter. This... 

Another classification of model is related to the time and space scales of interest. Ambient air quality standards are stated for measurement averaging periods varying from an hour to a year. However, for computational purposes, it is often necessary to use periods of less than an hour for a typical resolution-cell size in a model. Spatial scales of interest vary from a few tenths of a meter (e.g., for the area immediately adjacent to a roadway) up to hundreds of kilometers (e.g., in simulations that will elucidate urban-rural interactions). Large spatial scales are also warranted when multiday simulations are necessary for even a moderate-sized urban area. Under some climatologic conditions, recirculations can cause interaction of today s pollution with tomorrow s. Typical resolution specifications couple spatial scales with temporal sc es. Therefore, the full matrix of time scales and space scales is not needed, because of the dependence of time scales on space scales. Some typical categories by scale are as follows ... 

The atmospheric chemical processes undergone by most pollutants are not readily describable by first-order kinetics. Hence, the simple Gaussian plume solution in Equation 5-3 is inapplicable in most cases where physiochemical transformations significantly alter concentrations on a time scale or space scale appropriate to an urban airshed. 

The fundamental elements of deterministic models involve a combination of chemical and meteorologic input, preprocessing with data transmission, logic that describes atmospheric processes, and concentration-field output tables or displays. In addition to deterministic models, there are statistical schemes that relate precursors (or emission) to photo-chemical-oxidant concentrations. Models may be classified according to time and space scales, depending on the purposes for which th are designed. 

Figure 1. Time and space scales of algae, patches of algae, zooplankton and fish, A, and various means of remote sensing of biological properties in the oceans, B. Note that ships and buoys give no synopticity. Adapted from Esaias (36) ...

Minimization of the total free energy (sum of eqns 2.2 and 2.3) with respect to d leads to a predicted domain spacing scaling as (Semenov 1985)... 

A systematic study of the domain spacing scaling in two nearly symmetric PS-PI diblocks in neutral solvents was also undertaken by Hashimoto et al. (1983b). Results of S AXS experiments on these polymers dissolved in toluene and dioctyl phthalate (DOP) were summarized in a scaling relationship for the domain spacing in the ordered phase in the semidilute and concentrated regimes (0.15 < 4> < 0.6)... 

Crystallization in poly(ethy ene)-poly(ethylethylene) (PE-PEE) semicrystalline diblock copolymers has been investigated using SAXS and WAXS on oriented specimens. Microphase separation was found to precede crystallization for all samples, with 37-90wt% PE (Douzinas and Cohen 1992). The scaling of the lamellar domain spacing in the crystalline phase for the same samples was determined from measurements of the principal SAXS peak position (Douzinas et al. 1991). It was found that the domain spacing scales in agreement with the predictions of the theory of Whitmore and Noolandi (1988) (Section 5.3.5), i.e. 

---