共查询到19条相似文献，搜索用时 31 毫秒
1.
The stability criterion of maximum flow efficiency (MFE) has previously been found inherent in typical alluvial channel flow relationships, and this study investigates the general nature of this criterion using a wider range of flow resistance and bedload transport formulae. For straight alluvial channels, in which the effect of sediment sorting is insignificant, our detailed mathematical analysis demonstrates that a flow efficiency factor ε occurs generally as the ratio of sediment (bedload) discharge Qs to stream power Ω (γQS) in the form of . When ε is maximized (i.e. Qs is maximized or Ω is minimized), maximally efficient straight channel geometries derived from most flow resistance and bedload transport formulae are found compatible with observed bankfull hydraulic geometry relations. This study provides support for the use of the criteria of MFE, maximum sediment transporting capacity and minimum stream power for understanding the operation of alluvial rivers, and also addresses limitations to the direct application of its findings. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
2.
Basic flow relationships have previously been seen to be insufficient to explain the self‐adjusting mechanism of alluvial channels and as a consequence extremal hypotheses have been incorporated into the analyses. In contrast, this study finds that by introducing a channel form factor (width/depth ratio), the self‐adjusting mechanism of alluvial channels can be illustrated directly with the basic flow relations of continuity, resistance and sediment transport. Natural channel flow is able to reach an optimum state (Maximum Flow Efficiency (MFE), defined as the maximum sediment transporting capacity per unit available stream power) with regard to the adjustment of channel form such that rivers exhibit regular hydraulic geometry relations at dominant or bankfull stage. Within the context of MFE, this study offers support for the use of the concepts of maximum sediment transporting capacity (MSTC) and minimum stream power (MSP). Furthermore, this study indicates that the principle of least action is able to provide a physical explanation for the existence of MFE, MSTC and MSP. Potential energy is minimized and consequently sediment transport is maximized in alluvial channels. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
3.
A simple analytic model is presented relating local sediment transport capacity to variance in the transverse shear stress distribution in a stream channel. The model is used to develop a physically based conceptual model for the initiation of meandering in straight, bedload‐dominated streams as a result of a feedback mechanism. The feedback maximizes the cross‐sectional shear stress variance and – in order to achieve stability – ultimately minimizes the energy slope at repeated locations along the channel, subject to steady‐state mass flux and the stability of the channel boundary. These locations develop into pools in a fully developed meandering channel; they represent attractor states wherein sediment continuity is satisfied using the least possible energy expenditure per unit length of channel. However, since the cross‐sectional geometry of a pool (and the adjacent bar) is asymmetric, these attractor states are only conditionally stable, requiring strong, curvature‐induced secondary circulation to maintain their asymmetry. Between two successive pools, a stream occupies a metastable, higher energy state (corresponding to a riffle) that requires greater energy expenditure per unit length of channel to transport the same volume of sediment. The model we present links processes at the scale of a channel width to adjustments of the channel sinuosity and slope at the scale of a channel reach. We argue that the reach‐scale extremal hypotheses employed by rational regime models are mathematical formalisms that permit a one‐dimensional theory to describe the three‐dimensional dynamics producing stream morphology. Our model is consistent with the results from stream table experiments, with respect to both the rate of development of meandering and the characteristics of the equilibrium channel morphology. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
4.
Channel adjustments in the North Fork Toutle River and the Toutle River main stem were initiated by deposition of a 2.5 km3 debris avalanche and associated lahars that accompanied the catastrophic eruption of Mount St. Helens, Washington on 18 May 1980. Channel widening was the dominant process. In combination, adjustments caused average boundary shear stress to decrease non-linearly with time and critical shear stress to increase non-linearly with time. At the discharge that is equalled or exceeded 1 per cent of the time, these trends converged by 1991–1992 so that excess shear stress approached minimum values. Extremal hypotheses, such as minimization of unit stream power and minimization of the rate of energy dissipation (minimum stream power), are shown to be applicable to dynamic adjustments of the Toutle River system. Maximization of the Darcy–Weisbach friction factor did not occur, but increases in relative bed roughness, caused by the concomitant reduction in hydraulic depths and bed-material coarsening, were documented. Predictions of stable channel geometries using the minimum stream power approach were unsuccessful when compared to the 1991–1992 geometries and bed-material characteristics measured in the field. It is concluded that the predictions are not applicable because the study reaches are not truly stable and cannot become so until a new floodplain has been formed by renewed channel incision, retreat of stream-side hummocks, and establishment of riparian vegetation to limit the destabilizing effects of large floods. Further, prediction of energy slope (and consequently stream power) by the sediment transport equations is inaccurate because of the inability of the equations to account for significant contributions of finer grained (sand and gravel) bank materials (relative to the coarsened channel bed) from bank retreat and from upstream terrace erosion. 相似文献
5.
The ability to predict the stability of eroding riverbanks is a prerequisite for modelling alluvial channel width adjustment and a requirement for predicting bank erosion rates and sediment yield associated with bank erosion. Mass‐wasting of bank materials under gravity occurs through a variety of specific mechanisms, with a separate analysis required for each type of failure. This paper presents a computer program for the analysis of the stability of steep, cohesive riverbanks with respect to planar‐type failures. Planar‐type failures are common along stream channels destabilized by severe bed degradation. Existing stability analyses for planar‐type failures have a number of limitations that affect their physical basis and predictive ability. The computer program presented here is based on an analysis developed by Darby and Thorne. The software takes account of the geotechnical characteristics of the bank materials, the shape of the bank profile, and the relative elevations of the groundwater and surface water to estimate stability with respect to mass failure along a planar‐type failure surface. Results can be displayed either in terms of a factor of safety (ratio of resisting to driving forces), or probability of failure. The computer analysis is able to determine the relative amounts of bed degradation and bank‐toe erosion required to destabilize an initially stable bank. Data for the analysis are supplied in the form of either HEC‐2 hydrographic survey data files or user‐supplied bank profile data, in conjunction with user‐supplied geotechnical parameter values. Some examples, using data from the Upper Missouri River in Montana, are used to demonstrate potential applications of the software. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
6.
This paper analyses types and rates of change in river meander morphology and the links between mechanisms of change and emergent behaviour of planform morphology. It uses evidence of four dates of aerial photography combined with annual field mapping and ground photography to examine the morphological changes and mechanisms of change in a series of bends on an active meandering river, the River Dane in NW England, over a 25 year period. This unique data set allows insight into the spatial and temporal variability of bank line movement and component processes. Bank lines were mapped photogrametrically from air photos of 1984, 1996, 2001 and 2007 and the digitised courses compared in ArcGIS to produce calculations of erosional and depositional areas and rates. Most bends exhibit morphological change that largely follows the autogenic sequence, identified in qualitative models of meander development, from low sinuosity curves through simple symmetric and asymmetric bends to compound forms with lobe development in the apex region. Rates of erosion and bankline movement increase through this sequence until the compound phase. Relationships of amounts of movement to various curvature measures of bend morphology are complex. Several new loops, distinct from compound bend behaviour, have developed during the study period in formerly straight sections. Mechanisms of morphological change are illustrated for four types of bends: new, rapid growth bend; sharp‐angled bend with mid‐channel bar development; symmetric migrating bend; and simple to compound bend development. The changes take place in phases that are not simply related to discharge but to inherent sequences and feedbacks in development of bars and bend morphology and timescales for these are identified. Overall, emergent behaviour of systematic planform change, moderated by channel confinement and boundary features, is produced from spatially and temporally varied channel processes. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
7.
Adrian Chappell George L. Heritage Ian C. Fuller Andrew R. G. Large David J. Milan 《地球表面变化过程与地形》2003,28(4):349-370
A digital elevation model (DEM) of a fluvial environment represented landform surface variability well and provided a medium for monitoring morphological change over time. Elevation was measured above an arbitrary datum using a ground‐based three‐dimensional tacheometric survey in two reaches of the River Nent, UK, in July 1998, October 1998 (after flood conditions) and June 1999. A detailed geostatistical analysis of the elevation data was used to model the spatial variation of elevation and to produce DEMs in each reach and for each survey period. Maps of the difference in elevation were produced and volumetric change was calculated for each reach and each survey period. The parameters of variogram models were used to describe the morphological character of each reach and to elucidate the linkages between process and the form of channel change operating at different spatial and temporal scales. The analysis of channel change on the River Nent shows the potential of geostatistics for investigating the magnitude and frequency of geomorphic work in other rivers. A flood modified the channel features, but low magnitude and high frequency flows rationalized the morphology. In spite of relatively small amounts of net flux the channel features changed as a consequence of the reworking of existing material. The blocking of chute entrances and redirection of the channel had a considerable effect on the behaviour of the channel. Such small changes suggested that the distributary system was sensitive to variation in sediment regime. Plots of the kriging variances against sampling intervals were used to quantify the temporal variation in sampling redundancy (ranging between ?11 per cent and +93 per cent). These curves illustrated the importance of bespoke sampling designs to reduce sampling effort by incorporating anisotropic variation in space and geomorphic information on flow regime. Variation in the nugget parameter of the variogram models was interpreted as sampling inaccuracy caused by variability in particle size and is believed to be important for future work on surface roughness. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
8.
Wandering rivers are composed of individual anabranches surrounding semi‐permanent islands, linked by single channel reaches. Wandering rivers are important because they provide habitat complexity for aquatic organisms, including salmonids. An anabranch cycle model was developed from previous literature and field observations to illustrate how anabranches within the wandering pattern change from single to multiple channels and vice versa over a number of decades. The model was used to investigate the temporal dynamics of a wandering river through historical case studies and channel characteristics from field data. The wandering Renous River, New Brunswick, was mapped from aerial photographs (1945, 1965, 1983 and 1999) to determine river pattern statistics and for historical analysis of case studies. Five case studies consisting of a stable single channel, newly formed anabranches, anabranches gaining stability following creation, stable anabranches, and an abandoning anabranch were investigated in detail. Long profiles, hydraulic geometry, channel energy, grain size and sediment mobility variables were calculated for each channel. Within the Renous study area, the frequency of channel formation and abandonment were similar over the 54 years of analysis, indicating that the wandering pattern is being maintained. Eight anabranches were formed through avulsions, five were formed through the emergence of islands from channel bars and 11 anabranches were abandoned. The stable anabranch pair displayed similar hydraulic geometry and channel energy characteristics, while unstable anabranch pairs did not. The anabranch pair that gained stability displayed more similar channel energy characteristics than the anabranch pair that was losing stability (abandoning). It appears that anabranch pairs with similar energy characteristics are more stable than anabranches where these characteristics are out of balance. This is consistent with the hypothesis that anabranch pairs of similar length will be more stable than those with dissimilar lengths. Copyright © 2005 John Wiley & Sons, Ltd. 相似文献
9.
We study pairwise interactions of elliptical quasi-geostrophic (QG) vortices as the limiting case of vanishingly thin uniform potential vorticity ellipsoids. In this limit, the product of the vertical extent of the ellipsoid and the potential vorticity within it is held fixed to a finite non-zero constant. Such elliptical “lenses” inherit the property that, in isolation, they steadily rotate without changing shape. Here, we use this property to extend both standard moment models and Hamiltonian ellipsoidal models to approximate the dynamical interaction of such elliptical lenses. By neglecting non-elliptical deformations, the simplified models reduce the dynamics to just four degrees of freedom per vortex. For simplicity, we focus on pairwise interactions between identical elliptical vortices initially separated in both the horizontal and vertical directions. The dynamics of the simplified models are compared with the full QG dynamics of the system, and show good agreement as expected for sufficiently distant lenses. The results reveal the existence of families of steadily rotating equilibria in the initial horizontal and vertical separation parameter space. For sufficiently large vertical separations, equilibria with varying shape exist for all horizontal separations. Below a critical vertical separation (stretched by the constant ratio of buoyancy to Coriolis frequencies N / f), comparable to the mean radius of either vortex, a gap opens in horizontal separation where no equilibria are possible. Solutions near the edge of this gap are unstable. In the full QG system, equilibria at the edge of the gap exhibit corners (infinite curvature) along their boundaries. Comparisons of the model results with the full nonlinear QG evolution show that the early stages of the instability are captured by the Hamiltonian elliptical model but not by the moment model that inaccurately estimates shorter-range interactions. 相似文献
10.
Andrew Simon 《地球表面变化过程与地形》1995,20(7):611-628
The management of riverine environments is shown to require a knowledge and awareness of the complex interactions between fluvial and mass-wasting processes, riparian vegetation, and channel form. Identification of the cause of instability rather than the local symptoms, and knowledge of the temporal and spatial aspects of channel adjustment are central to the application of (1) appropriate analyses to estimate future channel changes, (2) appropriate mitigation measures, and (3) the protection of river-crossing structures and adjacent land. Conceptual models of channel evolution and bank-slope development are particularly valuable for interpreting past and present processes, applying appropriate computational techniques to estimate future channel changes, and implementing strategies to mitigate the impacts of processes likely to dominate the channel in the future. Techniques for identification and analysis of channel instability are interdisciplinary and provide a mechanism for estimating changes in channel-bed elevation and channel width with time. Features of channel form and associated riparian vegetation can be used as diagnostic criteria to identify channel processes, the stage of channel evolution and the magnitude and extent of instability. Changes in bed elevation with time can be represented using an exponential function; changes in channel width with time can be calculated using slope stability equations and (or) projection of a temporary angle of stability from a low-angle surface termed the ‘slough line’ that supports re-establishment of woody vegetation. These techniques, in combination with knowledge of the state of channel evolution, can then be used to assess the appropriateness of various mitigation measures to control on-going channel adjustments and to protect river-crossing structures. 相似文献
11.
Occurrence and development of channel bars are major components of the morphodynamics of rivers and their relation to river meandering has been much explored through theory and experimentation. However, field and documentary data of characteristics and evolution over timescales from years to several decades are lacking. Four sets of aerial photographs in the period 1984–2007 were used to map and quantify bar numbers and areas in GIS on an active meandering reach. Bar types were classified. Additional temporal resolution was provided by annual ground photography and mapping for 1981–2010. Analysis was extended backward by use of large scale Ordnance Survey maps from 1873 onwards. As expected, point bars are the most common type but ‘free’ bars of several types are major components of bar deposition. Point bars and attached bars are significantly larger in size than mid‐channel and side bars. Spatial distribution of bars varies down the reach and over time but is related to channel sinuosity, gradient and mobility and to bend evolution. Different types of bar occur in distinctive channel locations, with point and concave‐bend bars in zones of high curvature. Bar activity shows a relation with discharge events and phases and possibly with changing riparian conditions, but superimposed on this is a common sequence of bar evolution from incipient gravel mid‐channel bars to full floodplain integration. This life‐cycle is identified as 7–9 years on average. No evidence for mobility of free bars within the course is found. The results are compared with bar and bend theory; the bars are forced and conform in general to bend theory but detailed variation relates to geomorphic factors and to autogenic sequences of bends and bars. Mid‐channel bars are width induced. Variability of bar occurrence needs to be taken into account in river management and ecological evaluation, including for the EU WFD. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
12.
R. Pepijn van Denderen Ralph M. J. Schielen Astrid Blom Suzanne J. M. H. Hulscher Maarten G. Kleinhans 《地球表面变化过程与地形》2018,43(6):1169-1182
Side channel construction is a common intervention applied to increase a river's conveyance capacity and to increase its ecological value. Past modelling efforts suggest two mechanisms affecting the morphodynamic change of a side channel: (1) a difference in channel slope between the side channel and the main channel and (2) bend flow just upstream of the bifurcation. The objective of this paper was to assess the conditions under which side channels generally aggrade or degrade and to assess the characteristic timescales of the associated morphological change. We use a one‐dimensional bifurcation model to predict the development of side channel systems and the characteristic timescale for a wide range of conditions. We then compare these results to multitemporal aerial images of four side channel systems. We consider the following mechanisms at the bifurcation to be important for side channel development: sediment diversion due to the bifurcation angle, sediment diversion due to the transverse bed slope, partitioning of suspended load, mixed sediment processes such as sorting at the bifurcation, bank erosion, deposition due to vegetation, and floodplain sedimentation. There are limitations to using a one‐dimensional numerical model as it can only account for these mechanisms in a parametrized manner, but the model reproduces general behaviour of the natural side channels until floodplain‐forming processes become important. The main result is a set of stability diagrams with key model parameters that can be used to assess the development of a side channel system and the associated timescale, which will aid in the future design and maintenance of side channel systems. © 2017 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd. 相似文献
13.
Angela M. Gurnell 《地球表面变化过程与地形》1997,22(10):967-985
Major hydraulic discontinuities along lowland rivers may be caused by water impoundment behind weirs, by tributary floods, and by tides. An analysis of the geometry of 122 surveyed channel cross-sections located on an 18 km reach of the lower River Dee identifies up to three levels in the bank profile representing minima in the width:mean depth ratio, and distinct changes in the geometric properties of the channel to these three levels in a downstrem direction and within four stretches influenced to varying degrees by hydraulic discontinuities created by a weir and by tidal overtopping of the weir. Simple modelling combined with field observations suggest possible processes that may control the observed changes in channel morphology. © 1997 John Wiley & Sons, Ltd. 相似文献
14.
He Qing Huang Caiyun Deng Gerald C. Nanson Beilin Fan Xiaofang Liu Tonghuan Liu Yuanxu Ma 《地球表面变化过程与地形》2014,39(5):669-675
Taking the width/depth ratio of a river channel as an independent variable, a variational analysis of basic flow relationships shows that alluvial‐channel flow adjusts channel geometry to achieve stationary equilibrium when the condition of maximum flow efficiency (MFE) is satisfied. As a test of the veracity of MFE and to examine if this theory of self‐adjusting channel morphodynamics can be practically applied to large river systems, this study examines the degree of correspondence between theoretically determined equilibrium channel geometries and actual measurements along the middle and lower Yangtze River. Using four different forms of the Meyer‐Peter and Müller bedload relation and relations of flow continuity and resistance we show that the Meyer‐Peter and Müller bedload relation modified on the basis of MFE theory predicts channel dimensions most accurately when applied to the middle and lower Yangtze River. This provides convincing evidence supporting MFE equilibrium theory. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
15.
Leif M. Burge 《地球表面变化过程与地形》2006,31(10):1211-1226
This study presents the first detailed field‐based analysis of the morphology of bifurcations within anabranching cobble–gravel rivers. Bifurcations divide the flow of water and sediment into downstream anabranches, thereby influencing the characteristics of the anabranches and the longevity of river islands. The history, morphology, bed grain size, and flow vectors at five bifurcations on the Renous River, New Brunswick, Canada, were studied in detail. The angles of bifurcations within five anabranching rivers in the Miramichi basin were investigated. The average bifurcation angle was 47°, within the range of values cited for braided river bifurcations. Bifurcation angle decreased when anabranches were of similar length. Shields stresses in channels upstream of bifurcations were lower than reported values for braided rivers. Stable bifurcations displayed lower Shields stresses than unstable bifurcations, contrary to experimental results from braided river bifurcations. Bifurcations in anabranching rivers are stabilized by vegetation that slows channel migration and helps to maintain a uniform upstream flow field. The morphology of stable bifurcations enhances their stability. A large bar, shaped like a shallow ramp that increases in elevation to floodplain level, forms at stable bifurcations. Floodplains at stable bifurcations accrete upstream at rates between 0·9 and 2·5 m a?1. Bars may also form within the entrance of an anabranch downstream of the bifurcation node. These bars are associated with bifurcation instability, forming after a period of stability or an avulsion. Channel abandonment occurs when a bar completely blocks the entrance to one anabranch. The stability of channels upstream of bifurcations and the location of bars at bifurcations influence bifurcation stability and the maintenance of river anabranching in the long term. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
16.
Comprehensive empirical data of the response of unstable streams over a range of environmental conditions are unavailable. In this study, as a substitute for empirical data, a physically based numerical model of channel evolution is used in a range of numerical simulation experiments designed to predict the sensitivity of channel response to changes in control variables. The scope of the study is limited by the scope of the numerical model which applies to straight, sand-bed streams with cohesive bank materials that have been destabilized by sediment starvation and evolve towards equilibrium through bed degradation followed by channel widening. Results are presented for stable and unstable channel conditions. Stable channel depths are most sensitive to channel discharge, though the critical threshold shear stress for the entrainment of cohesive bank materials and discharge are both significant in determining the width. The sediment load, channel gradient, bank material cohesion, size of failed bank material aggregates and the initial bank height have sensitivities an order of magnitude smaller than discharge for both width and depth. Variations in bed material characteristics within the sand-size range are found to have little impact on simulated stable channel morphology. For unstable channels, the relative dominance of parameter sensitivities is examined in the context of an empirical-conceptual model of channel evolution proposed by Thorne and Osman (1988), to highlight the relationships between parameter dominance, time, and the processes and forms characterizing individual stages of channel evolution. Rates of change with time of width and depth sensitivity parameters for five tested independent variables (discharge, sediment supply, channel gradient, bank material cohesion and bed material size) are found to vary as a function of time, such that different stages of channel evolution are characterized by variations in the relative dominance of tested variables. The results support the hypothesis proposed by Thorne and Osman (1988) that the critical bank height required to initiate mass-wasting and widening may be regarded as a geomorphic threshold. 相似文献
17.
In the first decades of the 20th century, the Ebro River was the Iberian channel with the most active fluvial dynamics and the most remarkable spatial‐temporal evolution. Its meandering typology, the dimensions of its floodplain, and the singularities of its flow regime produced an especially interesting set of river functions. The largest dynamics of the Ebro River are concentrated along the meandering profile of the central sector. During the 20th century, this sector experienced a large alteration of its geomorphological structure. We present here an analysis of this evolution through the cartographic study of a long segment of the river (~250 km) in 1927, 1956 and 2003. The results show a large reduction in bank sinuosity, a progressive loss of fluvial territory, and a large decrease in channel width. These changes are especially clear in the areas previously most ecologically connected with the active channel. The fluvial territory of the river in 2003 was approximately half that found during the first decades of the 20th century. Forest plantations, which were non‐existent in 1927, occupied more than 1500 ha of the study area in the last decade. This intense geomorphological transformation becomes ecologically visible in (i) a 35% reduction of the area occupied by riparian vegetation; (ii) a loss of the heterogeneity of riparian forest spots, which were formerly structured in an irregular mosaic far from the river thalweg; and (iii) a modification of the riparian forest structure, which is currently linear, uniform, thin and very close to the river axis. The ecomorphological alteration was intensified by the remarkable reduction in bank length (13%) and the reduced dynamism of the present river system, indicated by an increase in the percentage of fluvial territory occupied by riparian forests and a reduction in the area occupied by the active channel. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
18.
Assessing water resources is an important issue, especially in the context of climatic changes. Although numerous hydrological models exist, new approaches are still under investigation. In this context, we propose a modelling approach based on the physical principle of least action. We present new hypotheses to develop the model further, to widen its application. The improved version of the model MODHYPMA was applied on 20 sub-catchments in Africa and the USA. Its performance was compared with two well-known lumped conceptual models, GR4J and HBV. The model could be successfully calibrated and validated. In calibration, GR4J performed better, while other models had similar performance. In validation, MODHYPMA and GR4J performed similarly and better than HBV. The parameter λ has medium sensitivity while parameters λ and TX have low sensitivity. The parameter uncertainty for MODHYPMA, analysed using the GLUE methodology, was higher during high flows but with good p and r factors.
EDITOR D. Koutsoyiannis ASSOCIATE EDITOR not assigned 相似文献
19.
ABSTRACTThe objective of this paper is to understand how the natural dynamics of a time-varying catchment, i.e. the rainfall pattern, transforms the random component of rainfall and how this transformation influences the river discharge. To this end, this paper develops a rainfall–runoff modelling approach that aims to capture the multiple sources and types of uncertainty in a single framework. The main assumption is that hydrological systems are nonlinear dynamical systems which can be described by stochastic differential equations (SDE). The dynamics of the system is based on the least action principle (LAP) as derived from Noether’s theorem. The inflow process is considered as a sum of deterministic and random components. Using data from the Ouémé River basin (Benin, West Africa), the basic properties for the random component are considered and the triple relationship between the structure of the inflowing rainfall, the corresponding SDE that describes the river basin and the associated Fokker-Planck equations (FPE) is analysed.
EDITOR D. Koutsoyiannis; ASSOCIATE EDITOR D. Gerten 相似文献