Spatial scale variability

Weigel BM, Wang LZ, Rasmussen PW, Butcher JT, Stewart PM, Simon TP, Wiley MJ. 2003. Relative influence of variables at multiple spatial scales on sheam macro invertebrates in the Northern Lakes and Forest ecoregion, USA. Freshwat Biol 48 1440-1461. 

Three main patterns of contamination were resolved by MCR-ALS analysis of [SE SO] data matrix (105 samples x 15 variables). Composition profiles (loadings) of the resolved components are shown in Fig. 11 (plots on the left). Variables are identified with a number in the x axis. In the y axis, the relative contribution of every scaled variable to the identified contamination pattern is given. Temporal and spatial sample distribution profiles of the contamination patterns (scores) are represented in Fig. 11 (plots on the right). In the x axis, samples are identified for the two compartments, SE and SO, successively ordered from first to third campaign and, within each campaign, form North-West to South-East. The y axis displays the contribution of every resolved contamination pattern to samples. 

The focus of RANS simulations is on the time-averaged flow behavior of turbulent flows. Yet, all turbulent eddies do contribute to redistributing momentum within the flow domain and by doing so make up the inherently transient character of a turbulent-flow field. In RANS, these effects of the full range of eddies are made visible via the so-called Reynolds decomposition of the NS equations (see, e.g., Tennekes and Lumley, 1972, or Rodi, 1984) of the flow variables into mean and fluctuating components. To this end, a clear distinction is required between the temporal and spatial scales of the mean flow on the one hand and those associated with the turbulent fluctuations on the other hand. 

After determining the underlying factors which affect local precipitation composition at an Individual site, an analysis of the slmlllarlty of factors between different sites can provide valuable Information about the regional character of precipitation and Its sources of variability over that spatial scale. SIMCA ( ) Is a classification method that performs principal component factor analysis for Individual classes (sites) and then classifies samples by calculating the distance from each sample to the PGA model that describes the precipitation character at each site. A score of percent samples which are correctly classified by the PGA models provides an Indication of the separability of the data by sites and, therefore, the uniqueness of the precipitation at a site as modeled by PGA. 

A more refined approach is based on the local description of fluctuations in non-equilibrium systems, which permits us to treat fluctuations of all spatial scales as well as their correlations. The birth-death formalism is applied here to the physically infinitesimal volume vo, which is related to the rest of a system due to the diffusion process. To describe fluctuations in spatially extended systems, the whole volume is divided into blocks having distinctive sizes Ao (vo = Xd, d = 1,2,3 is the space dimension). Enumerating these cells with the discrete variable f and defining the number of particles iVj(f) therein, we can introduce the joint probability of arbitrary particle distribution over cells. Particle diffusion is also considered in terms of particle death in a given cell accompanied with particle birth in the nearest cell. 

New reaction asymptotic law (2.1.78) emerges due to formation during the reaction course of a new spatial scale - the correlation length = Id- Similar to the case of immobile particles, we can expect here that at long times the coordinate r enters into the correlation function in a scaling form rj = r/Io, so that Y(r,t) —> Y(t, t), X (r,t) -> where the second variable... 

Higher order terms can be obtained by writing the inner and outer solutions as expansions in powers of e and solving the sets of equations obtained by comparing coefficients. This enzymatic example is treated extensively in [73] and a connection with the theory of materials with memory is made in [82]. The essence of the singular perturbation analysis, as this method is called, is that there are two (or more in some extensions) time (or spatial) scales involved. If the initial point lies in the domain of attraction of steady states of the fast variables and these are unique and stable, the state of the system will rapidly pass to the stable manifold of the slow variables and, one might... 

Variability may be defined as reflecting fluctuations in the atmosphere, of natural origin, with both temporal and spatial scales examples are diurnal, seasonal, solar activity-related variations impulsive events such as volcano eruptions and solar proton events fluctuations linked to some peculiar meteorological conditions, for example, intense cyclonic activities and jet streams. Variability by itself is a whole program to be conducted ideally on a four-dimensional basis (latitude, longitude, altitude, and time) by space vehicles, for example, satellites or from the space shuttle. This area of research is certainly the most urgent one to be de-... 

Finally, processes operating at larger spatial scales may control the storage of C in soils. Fire, for example, is as important a loss mechanism as decomposition for organic C in thick detrital layers in boreal forests (Flarden et al., 2000). For fire-prone regions, the net status of the land surface as a C sink or source depends as much on the area burned in a given year as on the responses of decomposition rates to weather variability in unbumed areas. 

Emissions of biogenic sulfur compounds to the atmosphere result from an imbalance between metabolic formation processes and biological or physicochemical consumption processes, determined on the spatial scale of the available methods for measuring emission fluxes. Variability in emissions... 

There is an inherent coupling of the behavior of the micro-scale variables to the behavior of macro-scale variables. This in itself presents complications when simrrlating these models. A few researchers have tried to address this problem of couphng of scales in these models. The solid state concentration term defined by the micro scale diffusion equation need to be coupled with the governing equations for the macro-scale to predict electrochemical behavior. Wang and co-workers used volume averaged equations and a parabolic profile approximation for solid-phase concentration. Subramanian et al. developed approximations assuming that the solid-state concentration inside the spherical electrode particle can be expressed as a polynomial in the spatial direction. 

The critical need to achieve high signal-to-noise ratios for spatially resolved measurement of several free radicals has spawned a number of research efforts aimed at improving our ability to observe such radicals as OH, HO2, NO, NO2, etc., with orders of magnitude better sensitivity than was previously available. A major impetus behind this research has been the realization that atmospheric variability on the spatial scale of a hundred meters in the vertical drives fluctuations in several of the key reactiye species, which provide ample concentration variation to carry out covariance studies to establish cause and effect within subsets of free radical reaction sets. 

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