Disparate Spatial Scales Research Articles

Advanced physics calculations using a multi-fluid plasma model

The multi-fluid plasma model is derived from moments of the Boltzmann equation and typically has two fluids representing electron and ion species. Large mass differences between electrons and ions introduce disparate temporal and spatial scales and require a numerical algorithm with sufficient accuracy to capture the multiple scales. Source terms of the multi-fluid plasma model couple the fluids to themselves (interspecies interactions) and to the electromagnetic fields. The numerical algorithm must treat the inherent stiffness introduced by the multiple physical effects of the model and tightly couple the source terms of the governing equations. A discontinuous Galerkin method is implemented for the spatial representation. Time integration is investigated using explicit, implicit, semi-implicit methods. Semi-implicit treatment is accomplished using a physics-based splitting. The algorithm is applied to study drift turbulence in field reversed configuration plasmas to illustrate the physical accuracy of the model. The algorithm is also applied to plasma sheath formation which demonstrates Langmuir wave propagation.

Robust analysis of soil properties at the national scale: cadmium content of French soils

National‐scale soil datasets exhibit variation over widely disparate spatial scales. Geological variation is an important source at the coarsest scale, and one in which exhaustive information is commonly available, in geological maps. Superimposed on this is continuous spatial variation caused by factors such as relief, vegetation and diffuse ‘background’ pollution. Further variation is caused by locally distinct factors such as point pollution from industrial sites or occasional geological anomalies. In this paper, we propose a single statistical model to encompass all of these effects which we describe as ‘geological variation’, ‘continuous spatial variation’ and ‘local anomalies'. In our model, the geological and continuous spatial variation are described, respectively, by the fixed and random effects of a linear mixed model (LMM) and the local anomalies lead to observations which are spatial outliers with respect to the LMM. We fit the model to a survey of 1887 observations of cadmium concentration in soil (Cd) collected on an incomplete regular grid across the French metropolitan territory (550 000 km2) and use it to predict Cd across France. We find that (i) it is not possible to fit a valid model—in terms of cross‐validation statistics—of Cd variation unless the effects of local anomalies are identified and separated from the larger‐scale processes; (ii) the LMM is not valid if the outliers are merely discarded but a valid model does result if the outliers are winsorized. On the basis of these findings we suggest a practical robust algorithm for national‐scale spatial analysis.

A Dynamic Graph Automata Approach to Modeling Landscape Change in the Andes and the Amazon

A generalization of cellular automata was developed that allows flexible, dynamic updating of variable neighborhood relationships, which in turn allows the integration of interactions at widely disparate spatial and temporal scales. Cells in the landscapes were modeled as vertices of dynamic graph automata that allow temporally variable causal connectivity between spatially nonadjacent cells. A trial was carried out to represent changes in an Amazonian and a tropical Andean landscape modeled as dynamic graph automata with input from a Landsat TM-derived Level 1 classification with the following classes: for the Amazon—forest, nonforest vegetation, water, and urban or bare (soil); for the Andes—forest, scrub (shrub or grassland), agriculture, and bare or exposed ground. Explicit automata transition rules were used to simulate temporal land-cover change. These rules were derived independently from fieldwork in each area, including vegetation plots or transects and informal interviews. Such a generalization of cellular automata was useful for modeling land-use–land-cover change (LULCC), although it potentially increases the computational complexity of an already data intensive process (involving 5–8 million cells, in 1000 stochastic simulations, with each simulation encompassing 15 annual time steps). The interannual predicted LULCC, while more nuanced in the Andean site, poses a serious threat to compositional and configurational stability in both the Andes and the Amazon, with implications for landscape heterogeneity and habitat fragmentation.

Progress, challenges, and perspectives in marine microbial ecology

When observed from above, the surfaces of oceans, estuaries and littoral areas show beautiful and complex patterns and textures. Some of these large-scale surface features are visible from space, and many are in fact generated by the smallest oceanic inhabitants, prokaryotic and eukaryotic unicellular organisms. A similar heterogeneity exists along a depth axis, and a vertical oceanic tour would reveal not only intricate and dynamic physical structures, but also complex biological patterns, particularly at the microbial scale. One of the main challenges of marine microbial ecology has traditionally been to connect these disparate spatial scales that coexist in our aquatic biosphere, and to extrapolate the microbial features and processes typically measured at very small spatial and temporal scales to scales that are relevant to the functioning of oceanic systems. There are logistic, technical and conceptual difficulties associated with this scaling; some of these challenges, such as the spatial and temporal distribution of bacterial production and growth in the oceans, have been the research focus of our community for decades now, and yet we still have much to learn; some challenges are more recent, for example the exploration of spatial and temporal patterns of marine microbial diversity. New paradigms and challenges are constantly emerging, such as the distribution and biogeochemical importance of novel microbial metabolic pathways in the ocean. Many of these existing and emerging challenges and research opportunities in our discipline were presented and discussed during the 10th Symposium on Aquatic Microbial Ecology (SAME 10), held in Faro (Portugal) in September 2007, through a combination of invited plenary talks, targeted thematic workshops and individual contribution of participants. This first special AME issue brings together the highlights of this meeting and presents some of the main topics that were explored. AME Special 1 provides a synthesis of a diverse array of key current topics in marine microbial ecology, including new developments in the measurement of microbial community structure and processes in biofilms, sediments, and the water column, patterns of microbial diversity and their ecological and biogeochemical significance, the current state of our understanding of major oceanic biogeochemical processes, of key biological interactions such as predation, and of specific microbial components, as well as conceptual issues concerning microbial services at the global scale. It is clear that we have a long way to go and that there will be no lack of future challenges, surprises and difficulties, but our community has made great progress in the past decade, and this special AME issue is a reflection of these achievements. I thank all the authors who so effectively contributed to develop this project.

Integrating neo- and paleolimnological approaches to refine interpretations of environmental change

With the development of quantitative transfer functions to relate community structure to physicochemical variables, reconstructions of past environmental conditions have been possible and have enhanced our understanding of various ecosystem processes. There are cases, however, in which this approach is not applicable, or does not provide enough information for the questions being asked. In these cases, some alternatives are to conduct experiments or to examine the distribution of species at a finer spatial resolution. These two approaches have been used as alternatives to or in conjunction with the development of transfer functions. In this review, I discuss the ways in which these two approaches are now being used in paleolimnological studies to enhance our understanding of the ecology of the species found in sediment records and thus refine interpretations of environmental change. My focus is primarily on studies that deal with establishing clearer relationships between environmental variables and the growth or distribution of organisms. I present examples of how these approaches have been integrated in a variety of studies, including those designed to: (1) refine and enhance reconstructions that are based on transfer functions; (2) develop new paleolimnological tools to reconstruct environmental change; (3) explore mechanistic links in the relationships between organisms and commonly reconstructed environmental variables; and (4) pose and test hypotheses based on patterns in the sediment record. These cases demonstrate that the use of these approaches was essential to clarify species-environment relationships as well as lake responses to disturbance. As in all disciplines, however, there are many challenges in this area of research. In particular, the quantitative integration of these approaches with the sediment record is a major challenge, due to disparate spatial and temporal scales. This research can also be quite labor-intensive, and provides information on fewer taxa than in the calibration set approach. It also requires interdisciplinary training and/or collaboration in fields that have historically been less integrated, hence they may require greater effort. These issues may hinder the use of these approaches because of the perceived difficulty. I discuss these challenges and address possible solutions.

From mitochondrial ion channels to arrhythmias in the heart: computational techniques to bridge the spatio-temporal scales.

Computer simulations of electrical behaviour in the whole ventricles have become commonplace during the last few years. The goals of this article are (i) to review the techniques that are currently employed to model cardiac electrical activity in the heart, discussing the strengths and weaknesses of the various approaches, and (ii) to implement a novel modelling approach, based on physiological reasoning, that lifts some of the restrictions imposed by current state-of-the-art ionic models. To illustrate the latter approach, the present study uses a recently developed ionic model of the ventricular myocyte that incorporates an excitation-contraction coupling and mitochondrial energetics model. A paradigm to bridge the vastly disparate spatial and temporal scales, from subcellular processes to the entire organ, and from sub-microseconds to minutes, is presented. Achieving sufficient computational efficiency is the key to success in the quest to develop multiscale realistic models that are expected to lead to better understanding of the mechanisms of arrhythmia induction following failure at the organelle level, and ultimately to the development of novel therapeutic applications.

A hierarchical multiscale framework for problems with multiscale source terms

This paper presents a hierarchical multiscale framework for problems that involve multiscale source terms. An assumption on the additive decomposition of the source function results in consistent decoupling of the fully coupled system and constitutes the new method. The structure of this decomposition is investigated and its mathematical implications are delineated. This method results in variational embedding of fine-scale information that is derived from the fine-scale equations, in the corresponding coarse-scale equations. It therefore provides a mathematically consistent way of bridging information between disparate spatial scales in the response function that are induced by multiscale forcing functions.

On the Evolution of Thermonuclear Flames on Large Scales

The thermonuclear explosion of a massive white dwarf in a Type Ia supernova explosion is characterized by vastly disparate spatial and temporal scales. The extreme dynamic range inherent to the problem prevents the use of direct numerical simulation and forces modelers to resort to subgrid models to describe physical processes taking place on unresolved scales. We consider the evolution of a model thermonuclear flame in a constant gravitational field on a periodic domain. The gravitational acceleration is aligned with the overall direction of the flame propagation, making the flame surface subject to the Rayleigh-Taylor instability. The flame evolution is followed through an extended initial transient phase well into the steady-state regime. The properties of the evolution of flame surface are examined. We confirm the form of the governing equation of the evolution suggested by Khokhlov (1995). The mechanism of vorticity production and the interaction between vortices and the flame surface are discussed. The results of our investigation provide the bases for revising and extending previous subgrid-scale model.

On Solving the Coronal Heating Problem

The question of what heats the solar corona remains one of the most important problems in astrophysics. Finding a definitive solution involves a number of challenging steps, beginning with an identification of the energy source and ending with a prediction of observable quantities that can be compared directly with actual observations. Critical intermediate steps include realistic modeling of both the energy release process (the conversion of magnetic stress energy or wave energy into heat) and the response of the plasma to the heating. A variety of difficult issues must be addressed: highly disparate spatial scales, physical connections between the corona and lower atmosphere, complex microphysics, and variability and dynamics. Nearly all of the coronal heating mechanisms that have been proposed produce heating that is impulsive from the perspective of elemental magnetic flux strands. It is this perspective that must be adopted to understand how the plasma responds and radiates. In our opinion, the most promising explanation offered so far is Parker's idea of nanoflares occurring in magnetic fields that become tangled by turbulent convection. Exciting new developments include the identification of the “secondary instability” as the likely mechanism of energy release and the demonstration that impulsive heating in sub-resolution strands can explain certain observed properties of coronal loops that are otherwise very difficult to understand. Whatever the detailed mechanism of energy release, it is clear that some form of magnetic reconnection must be occurring at significant altitudes in the corona (above the magnetic carpet), so that the tangling does not increase indefinitely. This article outlines the key elements of a comprehensive strategy for solving the coronal heating problem and warns of obstacles that must be overcome along the way.

The multiscale nature of diesel particulate filter simulation

Widespread market application of diesel particulate filters (DPFs) has been accompanied by the introduction of new filter materials and configurations in terms of cell density, wall thickness, pore size, porosity, catalyst coatings etc. Given this state-of-affairs, materials development, DPF design, system integration, regeneration control strategy optimisation and ash ageing assessment, based on a traditional design of experiments approach, become very time consuming and costly, due to the high number of tests required. This provides a privileged window of opportunity for the application of simulation tools. DPF behaviour depends strongly on the coupling of phenomena occurring over widely disparate spatial and temporal scales and the simulation approach must recognise and exploit these facts.

Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations

The fluid limit of a recently introduced family of nonintegrable (nonlinear) continuous-time random walks is derived in terms of fractional differential equations. In this limit, it is shown that the formalism allows for the modeling of the interaction between multiple transport mechanisms with not only disparate spatial scales but also different temporal scales. For this reason, the resulting fluid equations may find application in the study of a large number of nonlinear multiscale transport problems, ranging from the study of self-organized criticality to the modeling of turbulent transport in fluids and plasmas.

Ultrahigh resolution simulations of mode converted ion cyclotron waves and lower hybrid waves

Full wave studies of mode conversion (MC) processes in toroidal plasmas have required prohibitive amount of computer resources in the past because of the disparate spatial scales involved. The TORIC code [Brambilla, Nucl. Fusion 38 (1998) 1805] solves the linear sixth order reduced wave equation for the ion cyclotron range of frequencies (ICRF), in toroidal geometry using a Fourier representation for the poloidal dimension and finite elements in the flux dimension. The range of problems that TORIC can do has been extended through both new serial algorithms and parallelization of memory and processing. The implementation of out-of-core memory management, FFT convolutions, and improved memory management brought MC studies just into range of the serial version of the code running on a NERSC Cray SV1. Some simple tests and arguments show that more resolution than is possible on a single processor system is needed to fully resolve these scenarios. By distributing the large linear system across many processors in conjunction with the out-of-core technique, the resolution limitations are effectively removed. ScaLAPACK is used to do the linear algebra operations and message passing interface (MPI) is used to distribute the significant amount of post-processing. The new parallel version of the code can easily do the most difficult MC problems on present day tokamaks (Alcator C-Mod and Asdex-Upgrade), with only 32 pc from a local Beowulf cluster. Using 48 or more processors admits us to problems in the lower hybrid range of frequencies.

Advanced computations in plasma physics

Scientific simulation in tandem with theory and experiment is an essential tool for understanding complex plasma behavior. In this paper we review recent progress and future directions for advanced simulations in magnetically confined plasmas with illustrative examples chosen from magnetic confinement research areas such as microturbulence, magnetohydrodynamics, magnetic reconnection, and others. Significant recent progress has been made in both particle and fluid simulations of fine-scale turbulence and large-scale dynamics, giving increasingly good agreement between experimental observations and computational modeling. This was made possible by innovative advances in analytic and computational methods for developing reduced descriptions of physics phenomena spanning widely disparate temporal and spatial scales together with access to powerful new computational resources. In particular, the fusion energy science community has made excellent progress in developing advanced codes for which computer run-time and problem size scale well with the number of processors on massively parallel machines (MPP’s). A good example is the effective usage of the full power of multi-teraflop (multi-trillion floating point computations per second) MPP’s to produce three-dimensional, general geometry, nonlinear particle simulations which have accelerated progress in understanding the nature of turbulence self-regulation by zonal flows. It should be emphasized that these calculations, which typically utilized billions of particles for thousands of time-steps, would not have been possible without access to powerful present generation MPP computers and the associated diagnostic and visualization capabilities. In general, results from advanced simulations provide great encouragement for being able to include increasingly realistic dynamics to enable deeper physics insights into plasmas in both natural and laboratory environments. The associated scientific excitement should serve to stimulate improved cross-cutting collaborations with other fields and also to help attract bright young talent to plasma science.

Globalized Newton-Krylov-Schwarz Algorithms and Software for Parallel Implicit CFD

Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, parallelization is essential. The pseudo-transient matrix-free Newton-Krylov-Schwarz (ψNKS) algorithmic framework is presented as a widely applicable answer. This article shows that for the classical problem of three-dimensional transonic Euler flow about an M6 wing, ψNKS can simultaneously deliver globalized, asymptotically rapid convergence through adaptive pseudo-transient continuation and Newton’s method; reasonable parallelizability for an implicit method through deferred synchronization and favorable communication-to-computation scaling in the Krylov linear solver; and high per processor performance through attention to distributed memory and cache locality, especially through the Schwarz preconditioner. Two discouraging features of ψNKS methods are their sensitivity to the coding of the underlying PDE discretization and the large number of parameters that must be selected to govern convergence. The authors therefore distill several recommendations from their experience and reading of the literature on various algorithmic components of ψNKS, and they describe a freely available MPI-based portable parallel software implementation of the solver employed here.

Asymptotic model of electroporation

Electroporation is described mathematically by a partial differential equation (PDE) that governs the distribution of pores as a function of their radius and time. This PDE does not have an analytical solution and, because of the presence of disparate spatial and temporal scales, numerical solutions are hard to obtain. These difficulties limit the application of the PDE only to experimental setups with a uniformly polarized membrane. This study performs a rigorous, asymptotic reduction of the PDE to an ordinary differential equation (ODE) that describes the dynamics of the pore density $N(t).$ Given $N(t),$ the precise distribution of the pores in the space of their radii can be determined by an asymptotic approximation. Thus, the asymptotic ODE represents most of the phenomenology contained in the PDE. It is easy to solve numerically, which makes it a powerful tool to study electroporation in experimental setups with significant spatial dependence, such vesicles or cells in an external field.

Statistical mechanics of neocortical interactions: Training and testing canonical momenta indicators of EEG

A series of papers has developed a statistical mechanics of neocortical interactions (SMNI), deriving aggregate behavior of experimentally observed columns of neurons from statistical electrical-chemical properties of synaptic interactions. While not useful to yield insights at the single neuron level, SMNI has demonstrated its capability in describing large-scale properties of short-term memory and electroencephalographic (EEG) systematics. The necessity of including nonlinear and stochastic structures in this development has been stressed. Sets of EEG and evoked potential data were fit, collected to investigate genetic predispositions to alcoholism and to extract brain “signatures” of short-term memory. Adaptive Simulated Annealing (ASA), a global optimization algorithm, was used to perform maximum likelihood fits of Lagrangians defined by path integrals of multivariate conditional probabilities. Canonical momenta indicators (CMI) are thereby derived for individual's EEG data. The CMI give better signal recognition than the raw data, and can be used to advantage as correlates of behavioral states. These results give strong quantitative support for an accurate intuitive picture, portraying neocortical interactions as having common algebraic or physics mechanisms that scale across quite disparate spatial scales and functional or behavioral phenomena, i.e., describing interactions among neurons, columns of neurons, and regional masses of neurons. This paper adds to these previous investigations two important aspects, a description of how the CMI may be used in source localization, and calculations using previously ASA-fitted parameters in out-of-sample data.

An Adaptive Mesh Refinement Algorithm for the Radiative Transport Equation

The discrete ordinates form of the radiative transport equation (RTE) is spatially discretized and solved using an adaptive mesh refinement (AMR) algorithm. This technique permits local grid refinement to minimize spatial discretization error of the RTE. An error estimator is applied to define regions for local grid refinement; overlapping refined grids are recursively placed in these regions; and the RTE is then solved over the entire domain. The procedure continues until the spatial discretization error has been reduced to a sufficient level. The following aspects of the algorithm are discussed: error estimation, grid generation, communication between refined levels, and solution sequencing. This initial formulation employs the step scheme and is valid for absorbing and isotropically scattering media in two-dimensional enclosures. The utility of the algorithm is tested by comparing the convergence characteristics and accuracy to those of the standard single-grid algorithm. For two simple benchmark problems, the AMR algorithm maintains the convergence characteristics of the standard single-grid algorithm, but it does not provide any efficiency gains due to a lack of disparate spatial scales. In a third, more localized problem, however, the AMR algorithm demonstrates significant memory and CPU time reductions.

Statistical mechanics of neocortical interactions: Canonical momenta indicatorsof electroencephalography

A series of papers has developed a statistical mechanics of neocortical interactions (SMNI), deriving aggregate behavior of experimentally observed columns of neurons from statistical electrical-chemical properties of synaptic interactions. While not useful to yield insights at the single neuron level, SMNI has demonstrated its capability in describing large-scale properties of short-term memory and electroencephalographic (EEG) systematics. The necessity of including nonlinear and stochastic structures in this development has been stressed. Sets of EEG and evoked potential data were fit, collected to investigate genetic predispositions to alcoholism and to extract brain ``signatures'' of short-term memory. Adaptive simulated annealing (ASA), a global optimization algorithm, was used to perform maximum likelihood fits of Lagrangians defined by path integrals of multivariate conditional probabilities. Canonical momenta indicators (CMI) are thereby derived for an individual's EEG data. The CMI give better signal recognition than the raw data, and can be used to advantage as correlates of behavioral states. These results give strong quantitative support for an accurate intuitive picture, portraying neocortical interactions as having common algebraic or physics mechanisms that scale across quite disparate spatial scales and functional or behavioral phenomena, i.e., describing interactions among neurons, columns of neurons, and regional masses of neurons.

Adaptive numerical advection. The coordinate transformation equation method

In many hydrodynamical applications, advection is the most important dynamical process. Advected quantities typically develop greatly disparate spatial scales because of the inherent non-linearity of advection. A method is outlined how to tackle this process numerically. The basic idea of the coordinate transformation equation (CTE) method is to introduce a moving coordinate system in physical space before applying any discretisation. This keeps the problem of discretisation separate from the problem of adaptation. In the new coordinate system, which changes in time, the effects on the numerical representation of the advection terms are twofold. Firstly, the advection is with respect to a relative velocity between coordinates and flow, which is arranged to be less than the absolute velocity, leading to a less restrictive stability criterion for the explicit time stepping compared to the Courant condition. Since the relative velocity is smaller than the absolute velocity, this also leads to a decrease in the importance of the non-linear advection term in comparison with other terms in the equation being solved, and therefore to a decrease in numerical errors from advection. Secondly, a smoothing of the spatial gradients of the quantities to be advected decreases the diffusion due to the discretisation, especially if monotonic (but therefore diffusive) discretisation schemes are used. Illustrations of the applicability and accuracy of the method are given, using comparisons with the results of other advection techniques and analytical solutions of linear advection problems, i.e., problems in which the velocity field is given, in one and two dimensions, including rarefaction and compression waves. The remaining illustrations are of non-linear problems, in which the velocity field is solved for as part of the problem. In such problems the CTE is coupled non-linearly to the equations of motion. However, the main shortcoming of the method is the large computing time required to solve the CTE system.

Statistical mechanics of neocortical interactions: A scaling paradigm applied to electroencephalography.

A series of papers has developed a statistical mechanics of neocortical interactions (SMNI), deriving aggregate behavior of experimentally observed columns of neurons from statistical electrical-chemical properties of synaptic interactions. While not useful to yield insights at the single neuron level, SMNI has demonstrated its capability in describing large-scale properties of short-term memory and electroencephalographic (EEG) systematics. The necessity of including nonlinear and stochastic structures in this development has been stressed. In this paper, a more stringent test is placed on SMNI: The algebraic and numerical algorithms previously developed in this and similar systems are brought to bear to fit large sets of EEG and evoked potential data being collected to investigate genetic predispositions to alcoholism and to extract brain signatures of short-term memory. Using the numerical algorithm of Very Fast Simulated Re-Annealing, it is demonstrated that SMNI can indeed fit this data within experimentally observed ranges of its underlying neuronal-synaptic parameters, and use the quantitative modeling results to examine physical neocortical mechanisms to discriminate between high-risk and low-risk populations genetically predisposed to alcoholism. Since this first study is a control to span relatively long time epochs, similar to earlier attempts to establish such correlations, this discrimination is inconclusive because of other neuronal activity which can mask such effects. However, the SMNI model is shown to be consistent with EEG data during selective attention tasks and with neocortical mechanisms describing short-term memory previously published using this approach. This paper explicitly identifies similar nonlinear stochastic mechanisms of interaction at the microscopic-neuronal, mesoscopic-columnar and macroscopic-regional scales of neocortical interactions. These results give strong quantitative support for an accurate intuitive picture, portraying neocortical interactions as having common algebraic or physics mechanisms that scale across quite disparate spatial scales and functional or behavioral phenomena, i.e., describing interactions among neurons, columns of neurons, and regional masses of neurons.