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The present paper examines the northern stratosphere during April 2020, when the polar vortex split into two cyclonic vortices after a winter with the strongest ozone depletion on record. We examin...

In Rayleigh‐Bénard convection, turbulent superstructures are large‐scale patterns of circulation rolls created by hot ascending and cold descending thermal plumes. The evolution of these large‐scale patterns happens on very large time scales τ. Spectral clustering applied to Lagrangian particle trajectories on time intervals smaller than τ can be used to create clusters displaying a structure similar to the patterns detected in the Eulerian frame of reference. However, this technique is unfeasible for the analysis of the evolution of turbulent superstructures due to turbulent dispersion. Therefore, we test the application of concepts of evolutionary spectral clustering on Lagrangian particle trajectories to analyze the long‐term dynamics of turbulent superstructures in the Lagrangian frame of reference.

The present paper examines the northern stratosphere during April 2020, when the polar vortex split into two cyclonic vortices. We examine this split at middle as well as lower stratospheric levels...

In most chemical reactors the homogeneous distribution of reactants is a desired prerequisite for guaranteed product quality and purity. However, oftentimes, especially in new reactor concepts this condition may not be met due to evolving or existing structures in the transport patterns. In this article the usage of Lagrangian analysis techniques in order to analyze chemical and biochemical reactors regarding the behavior of Lagrangian tracers is explored. A Lagrangian analysis on two different datasets is performed trying to show the large versatility of the concepts. The Lagrangian analysis techniques allow new insights into the reactor dynamics and could be a valuable tool to identify and characterize areas of different mixing performance The authors gratefully acknowledge the financial support provided by the German Research Foundation within the Priority Program ‘‘Reactive Bubbly Flows'', SPP 1740 (SCHL 617/12‐2 SCHL 617/13‐2) and by the Federal State of Hamburg, Ministry of Science, Research and Equalities of Hamburg, within the framework of the joint research project “New reactor technologies for chemical and biochemical synthesis processes” (Ref.‐No. LFF‐FV 43) SI

Atmospheric blocks strongly influence surface weather, including extremes such as heat waves and cold spells. Recently, diabatic heating and associated upper‐tropospheric potential vorticity (PV) modification have been identified as important modulators of atmospheric blocking dynamics. Also, robust links between atmospheric blocks and proximate heavy precipitation events have been established. This leads to the question of the extent to which diabatic heating associated with heavy precipitation events influences Northern Hemisphere blocking. This study uses 5 years of 3‐day back trajectories started from objectively identified blocks in the ERA‐Interim dataset to investigate this relationship. A substantial fraction of air parcels in blocks pass through heavy precipitation areas. The exact fraction depends on the choice of heavy precipitation threshold. Roughly 19% of all the trajectories in a block pass a heavy precipitation area (>95th percentile) area while being saturated. Of the air parcels in a block that are heated at least 5 K, 60% pass a heavy precipitation area while saturated. This fraction varies with the season and geographical area. The overall fraction is lowest in summer and highest in winter, higher over oceans than over land, and higher over the Pacific than over the Atlantic. In summer, heating is relevant over the continents and heating over North America influences blocks over the eastern Atlantic. For summer blocks in the North Atlantic and over Scandinavia, heating happens partly over the European continent.

A Lagrangian analysis is applied to identify the main moisture source areas associated with atmospheric rivers (ARs) making landfall along the west coast of South Africa during the extended austral winter months from 1980 to 2014. The results show that areas that provide the anomalous uptake of moisture can be categorized into four regions: (1) the South Atlantic Ocean between 10°S and 30°S, (2) a clear local maximum in the eastern South Atlantic, (3) a continental source of anomalous uptake to the north of the Western Cape, and (4) over South America at a distance of more than 7000 km from the target region. It emerges that the South American moisture source can be linked to a particular phase of the South American low-level jet, known as a no Chaco jet event (NCJE), which transports moisture to the western and central South Atlantic basin. Concisely, we provide strong evidence that the two margins of the South Atlantic Ocean appear connected by two meteorological structures, with the NCJE playing a key role of transporting moisture from South America to the western and central South Atlantic basin, feeding the AR that transports some of the moisture to the west coast of South Africa.

The shock-compression responses of PTFE (polytetrafluoroethylene)/Al granular composites subjected to planar shock waves of various pressures are investigated. A 57-mm diameter single-stage gas-gun and 50-mm diameter plane-wave lenses are employed to perform planar shock wave experiments. High frequency manganin piezoresistance stress gauges are used to monitor the stress (regarded as pressure in consideration of the high pressure state) at four Lagrangian positions of the PTFE/Al granular composites specimens. Planar shock wave experiments show characteristics of densification at measured input pressure of 0.5 GPa to 1.27 GPa using single-stage gas-gun and shock-induced reaction (SIR) indicated by growth of shock pressure and specific volume expansion at measured input pressure of 7.29 GPa to 12.25 GPa using plane-wave lenses. The pressure and relative volume states behind the shock wave front are calculated from the experimental recorded pressure profiles using Lagrangian analysis method, which are used to determine the reaction ratios under different shock pressures by comparing with partial reacted Hugoniot calculations. It was shown that the reaction ratios obtained in this research have good agreement with the thermochemical modeling calculations. The corresponding results indicate that the shock-induced reactions of PTFE/Al granular composites occur in the shock wave rising period and the reaction ratios are intimately related to the shock wave pressure.

Foehn flows are typically associated with warm air temperatures. Though several theories for the so-called foehn air warming have been developed over the past century, no conclusion about the most important mechanism has been reached. The development of new methods to calculate accurate air-mass trajectories over complex topography has opened up a new perspective on this question. Air-mass-trajectories derived from wind-field data from COSMO model simulations with 20 s temporal resolution are used in this study to investigate the origin of the foehn air and the contribution of adiabatic and diabatic processes for two foehn events in the Swiss Alps, with a focus on the Rhine valley. The first foehn event investigated has no precipitation on the upstream side of the Alps. The majority of air parcels stem from upstream altitudes above 1.8 km and most of the foehn air warming is due to adiabatic descent (∼79%). In the second event investigated, significant upstream precipitation occurred. For this case, a significantly larger fraction of the foehn air parcels originate within the lowest 2 km of the upstream atmosphere (up to 70%). Adiabatic descent accounts for the largest part of the temperature change (∼70%), while moist diabatic processes explain about 60% of the potential temperature change. The vertical displacement across the Alpine range is correlated with the diabatic temperature change: parcels strongly heated by condensation, deposition and freezing are in general found at high altitudes above the foehn valley, while parcels affected by diabatic cooling through evaporation, sublimation and melting arrive closer to the valley floor. The high-resolution trajectories also indicate a much more complicated vertical and horizontal flow pattern than generally assumed, with several distinct air streams upstream of the mountain range and vertical ‘scrambling’ of air masses.

The precipitation produced by orographic clouds is the result of the interaction between nonlinear dynamical and microphysical processes. Focusing on warm-phase stable clouds, we propose that the precipitation efficiency is essentially controlled by the bulk Damkohler number of the system, i.e. the ratio of the characteristic advective to the microphysical time-scale of the cloud. These time-scales are investigated thoroughly for single air parcels along temporally highly resolved trajectories from quasi-two-dimensional numerical simulations with vertical Froude numbers between 0.38 and 2.31. Based on these results, analytical formulations for the Damkohler number of single parcels and for a bulk Damkohler number for the entire cloud are developed, which depend only on the far upstream flow properties and the cloud droplet number density. Relating the bulk Damkohler numbers to the precipitation efficiencies from the numerical simulations results in a unique scaling relation between the two non-dimensional numbers: no precipitation is observed for very small Damkohler numbers, while for large Damkohler numbers the precipitation efficiency asymptotically approaches a maximum value of about 0.8. For intermediate Damkohler numbers, the precipitation efficiency depends strongly on the microphysical properties of the clouds.

Summary The coupling between two distinct numerical methods presents a major challenge, especially in the case of discrete-continuum coupling for dynamic simulations. This paper presents a general multiscale framework for coupling the discrete element method (DEM) and the finite difference method (FDM). DEM has been shown to be particular suitable for capturing physical phenomena at small length scales where continuum mechanics description no longer applies. Its efficiency, however, is limited because of the computational power required. A multidomain analysis coupling DEM and FDM is thus proposed to reduce the computational cost. To couple overlapping DEM and FDM, a bridging scale term is introduced such that compatibility of dynamic behavior between the DEM-based and the FDM-based models is enforced. This multiscale method couples two commercial packages: the DEM-based code, particle flow code, and the FDM-based code, fast lagrangian analysis of continua. The new method is applied to several reference dynamic tests. Results using the proposed method compare well with benchmark simulations from FDM and DEM. Copyright © 2015 John Wiley & Sons, Ltd.

Since temperature extremes have a strong impact on environment and society, it is crucial to understand their underlying mechanisms. While their relationship to the large-scale atmospheric circulation has been comprehensively investigated, the role of physical processes in the formation of air masses with extreme temperatures is less well understood. This study presents a Lagrangian analysis of the 0.1% most extreme six-hourly hot and cold events in three European regions (UK, Central Europe, Balkans) for the time period 1989–2009. The results provide insight into typical transport patterns and physical processes (adiabatic compression, radiation, surface heat fluxes) occurring along the trajectories of extreme temperature air masses. Cold events in Europe are most frequently induced by advection of cold air masses from the Arctic and Russia. The transport to the target region is characterized by a temperature increase caused by adiabatic compression and, in the maritime setting of the UK, also by diabatic heating due to surface sensible heat fluxes. Despite the warming along the trajectories, the extremeness of the associated 2 m temperature increases, due to the dislocation of the air mass to regions with a milder climate. Hot events are generally associated with weaker horizontal transport, but strong adiabatic warming and local temperature increase caused by enhanced radiation and surface heat fluxes. This in situ warming is particularly strong in Central Europe. Evaluating the temperature evolution along the trajectories reveals that hot and cold extremes develop on a similar time-scale of 2–3 days. This time-scale is mostly set by physical processes for hot extremes and controlled by advective transport for cold extremes. The diagnostics applied in this study lead to an improved process understanding that can provide a basis for more accurate predictions of temperature extremes.

In July and August 2010 floods of unprecedented impact afflicted Pakistan. The floods resulted from a series of intense multi-day precipitation events in July and early August. At the same time a series of blocking anticyclones dominated the upper-level flow over western Russia and breaking waves i.e. equatorward extrusions of stratospheric high potential vorticity (PV) air formed along the downstream flank of the blocks. Previous studies suggested that these extratropical upper-level breaking waves were crucial for instigating the precipitation events in Pakistan. Here a detailed analysis is provided of the extratropical forcing of the precipitation. Piecewise PV inversion is used to quantify the extratropical upper-level forcing associated with the wave breaking and trajectories are calculated to study the pathways and source regions of the moisture that precipitated over Pakistan. Limited-area model simulations are carried out to complement the Lagrangian analysis. The precipitation events over Pakistan resulted from a combination of favourable boundary conditions with strong extratropical and monsoonal forcing factors. Above-normal sea-surface temperatures in the Indian Ocean led to an elevated lower-tropospheric moisture content. Surface monsoonal depressions ensured the transport of moist air from the ocean towards northeastern Pakistan. Along this pathway the air parcel humidity increased substantially (60–90% of precipitated moisture) via evapotranspiration from the land surface. Extratropical breaking waves influenced the surface wind field substantially by enhancing the wind component directed towards the mountains which reinforced the precipitation.

We present the computation of lagrangian-based flow characterization measures for time-dependent, deformable-wall, finite-element blood flow simulations. Applicability of the algorithm is demonstrated in a fluid-structure interaction simulation of blood flow through a total cavopulmonary connection (Fontan procedure), and results are compared with a rigid-vessel simulation. Specifically, we report on several important lagrangian-based measures including flow distributions, finite-time Lyapunov exponent fields, particle residence time, and exposure time calculations. Overall, strong similarity in lagrangian measures of the flow between deformable and rigid-vessel models was observed.

The potential vorticity (PV) in warm conveyor belts (WCBs) is strongly influenced by the latent heating associated with the various microphysical processes occurring during the formation of clouds. The first-order effect is that PV increases below the level of maximum diabatic heating and decreases above. Thus, the WCB reaches the upper troposphere with low PV values and has the potential to influence the large-scale dynamics. In order to quantify the influence of different microphysical processes on the diabatic heating rates (DHRs) and associated PV development during the ascent, a Lagrangian analysis is used and applied to a regional model simulation of a selected WCB event. First, the individual DHRs caused by the various microphysical processes are calculated with the COSMO model. Then, the DHRs and the associated changes in PV are evaluated along the WCB trajectories. The relative role of the different microphysical processes is quantified for the latent heating and the diabatic PV modification, for which the gradient of the latent heating and the absolute vorticity are crucially important. It is shown that condensation of vapour and depositional growth of snow each contribute ∼10 K to the total latent heating. However, the diabatic PV modification due to condensation is stronger since it occurs close to the cold frontal low-level maximum of the z-component of the absolute vorticity, ηz, whereas a similar heating rate gradient caused by depositional growth of snow modifies the PV much less since it occurs in a region with much weaker ηz. This highlights the importance of ηz for determining the PV modification due to a certain diabatic heating rate. Furthermore, cooling processes like the evaporation of rain, also co-occurring with high ηz near the surface cold front, have the potential to strongly modify the PV below the WCB. Copyright © 2011 Royal Meteorological Society

The Lagrangian analysis method is re-analysed. It is shown that when a series of particle velocity profiles (or strain profiles) are measured to determine the dynamic stress–strain relation by the Lagrangian analysis, the crux is attributed to how to simultaneously measure both the stress and particle velocity wave profiles at a Lagrangian position by a combining stress–velocity gauge. The related experimental technique has not been satisfactorily solved yet. Thus, a new method combining the Lagrangian analysis with the Hopkinson pressure bar (HPB) technique is proposed. It is shown that the interface between the HPB and the specimen in fact provides such a combining gauge (called 1sv). Based on this technique, the so-called ‘1sv + nv’ inverse analysis method is developed when particle velocity profiles are measured at n Lagrangian positions, and similarly the so-called ‘1sv + ne’ inverse analysis method are developed when strain profiles are measured at n Lagrangian positions. The feasibility of the new method is evaluated by numerical simulations for a typical rate-dependent viscoelastic material polymethyl methacrylate. The good agreement between the known ‘characteristics solutions’ and the present solutions confirms the validity of the proposed new method.

Analysis of large deformation of geomaterials subjected to time-varying load poses a very difficult problem for the geotechnical profession. Conventional finite element schemes using the updated Lagrangian formulation may suffer from serious numerical difficulties when the deformation of geomaterials is significantly large such that the discretized elements are severely distorted. In this paper, an operator-split arbitrary Lagrangian–Eulerian (ALE) finite element model is proposed for large deformation analysis of a soil mass subjected to either static or dynamic loading, where the soil is modelled as a saturated porous material with solid–fluid coupling and strong material non-linearity. Each time step of the operator-split ALE algorithm consists of a Lagrangian step and an Eulerian step. In the Lagrangian step, the equilibrium equation and continuity equation of the saturated soil are solved by the updated Lagrangian method. In the Eulerian step, mesh smoothing is performed for the deformed body and the state variables obtained in the updated Lagrangian step are then transferred to the new mesh system. The accuracy and efficiency of the proposed ALE method are verified by comparison of its results with the results produced by an analytical solution for one-dimensional finite elastic consolidation of a soil column and with the results from the small strain finite element analysis and the updated Lagrangian analysis. Its performance is further illustrated by simulation of a complex problem involving the transient response of an embankment subjected to earthquake loading. Copyright © 2007 John Wiley & Sons, Ltd.

Facilitating more predictive ore discovery is becoming an increasingly challenging task for the resources geology research and exploration industry. The numerical geodynamic modeling, which uses numerical code to reproduce feedback coupling of geodynamic processes in the computer, is one of endeavors for forming sound predictive exploration strategy by thoroughly understanding the mineralization system and its controlling factors. The FLAC (Fast Lagrangian Analysis of Continua) is a two-dimensional finite-difference code for studying the mechano-thermo-hydrological behavior of a continuous porous medium after it reached equilibrium or steady plastic flow. We used the FLAC to simulate the syn-deformation cooling and fluid flowing after the intrusion was emplaced and solidified in the Fenghuangshan ore field, Tongling Cu district for guiding analysis of ore-controlling factors and selection of the exploration target area. The results of numerical computation demonstrate that the dilatant deformation zones along the contact of the intrusion control the fluid focus and the location and scale of ore bodies. Further more, the simulation results suggest a most prospective exploration area in the south high dilation zone, where the subsequent exploration supported the prediction and the test bore hole disclosed the high quality copper ore bodies in the target. This example has demonstrated the positive role of the numerical geodynamic modeling in facilitating predictive ore discovery.

The Tunnel Sealing Experiment (TSX) was a full-scale in situ demonstration of technology for constructing nearly water tight-seals in excavations through crystalline rock deep below the surface of the earth. The experiment has been carried out at Atomic Energy of Canada Limited's (AECL's) Underground Research Laboratory near Lac du Bonnet, Canada, in support of international programs for geologic disposal of radioactive waste. The TSX, with partners from Canada, Japan, France and the United States, was carried under conditions of high pressure (up to 4 MPa) and elevated temperature (up to 85°C). Comparing numerical model predictions with eight years of data collected from approximately 900 sensors was an important component of this experiment. Model of Transport In Fractured/porous Media (MOTIF), a finite element computer program developed by AECL for simulating fully coupled or uncoupled fluid flow, solute transport and heat transport, was used to model both the ambient temperature and heated phases of the TSX. The plan to heat the water in the TSX to 85°C was developed using model predictions and a comparison of simulated results with measurements during heating of the water in the TSX to about 50°C. The three-dimensional MOTIF simulations were conducted in parallel with axisymmetric modelling using Fast Lagrangian Analysis of Continua (FLAC), which computed the heat loss from pipes that carried the heated water through the rock to and from the experiment. The numerical model was initially used to develop a plan for operation of the experiment heaters, and then subsequently used to predict temperatures and hydraulic heads in the TSX bulkhead seals and surrounding rock. Copyright © 2005 John Wiley & Sons, Ltd.

Monitoring of the progressive convergence of a tunnel shows that deformations occurring in the soil surrounding a tunnel exhibit a strong evolution with time. This time-dependent behaviour can be linked to three essential factors: the distance from the point of interest to the working face over time, the distance of unsupported tunnel to the working face and the viscous properties of the soil. The objective of this paper is to propose a constitutive model of the time-dependent behaviour of soil which has been developed within the framework of elastoplasticity–viscoplasticity and critical state soil mechanics. The consideration of viscoplastic characteristic sets the current model apart from the CJS (Cambou, Jafari and Sidoroff) model as the basic elastoplastic model, and introduces an additional viscous mechanism. The evolution of the viscous yield surface is governed by a particular hardening called ‘viscous hardening’ with a bounding surface. The proposed constitutive model has been applied in the analysis of tunnelling. Two kinds of numerical calculations have been used in the analysis, axisymmetric analysis and plane strain analysis. Monitoring of the progressive convergence of a tunnel conducted in the railway tunnel of Tartaiguille (France), has been used to describe the calculation procedure proposed and the capability of the model. The finite difference software, fast Lagrangian analysis of continua (FLAC), has been used for the numerical simulation of the problems. The comparison of results shows that the observed deformations could have been reasonably predicted by using the constitutive model and calculation strategy proposed. Copyright © 2004 John Wiley & Sons, Ltd.

A detailed Lagrangian analysis of partially mixed structures in an SMX static mixer is presented, with emphasis on laminar flows with Reynolds numbers between 1 and 100. The distribution of a small amount of equal-viscosity additive is examined based on computing the trajectories of passive tracer particles through the mixer. Three radial positions (one centerline and two off-center) are chosen for injection and the mixing patterns are compared in order to investigate the effect of injection location on mixing. Mixing measures, such as the decrease in the variation coefficient with axial distance, and the increase in the average rate of stretching, are discussed in order to quantify mixing performance. It is found that mixing rates for the centerline injections are larger than for the off-center injection positions investigated. The presence of self-similarity in the stretching field and the mixing patterns, and the exponential decrease in variation coefficient with increasing axial distance provides evidence for chaotic mixing behavior in this device.

A continuum sensitivity analysis is presented for large inelastic deformations and metal forming processes. The formulation is based on the differentiation of the governing field equations of the direct problem and development of weak forms for the corresponding field sensitivity equations. Special attention is given to modelling of the sensitivity boundary conditions that result due to frictional contact between the die and the workpiece. The contact problem in the direct deformation analysis is modelled using an augmented Lagrangian formulation. To avoid issues of non-differentiability of the contact conditions, appropriate regularizing assumptions are introduced for the calculation of the sensitivity of the contact tractions. The proposed analysis is used for the calculation of sensitivity fields with respect to various process parameters including the die surface. The accuracy and effectiveness of the proposed method are demonstrated with a number of representative example problems. In the die design applications, a Bezier representation of the die curve is introduced. The control points of the Bezier curve are used as the design parameters. Comparison of the computed sensitivity results with those obtained using the direct analysis for two nearby dies and a finite difference approximation indicate a very high accuracy of the proposed analysis. The method is applied to the design of extrusion dies that minimize the standard deviation of the material state in the final product or minimize the required extrusion force for a given reduction ratio. An open-forging die is also designed which for a specified stroke and initial workpiece produces a final product of desired shape.

According to Sverdrup's (1953) model of the spring bloom, phytoplankton biomass decreases in winter when the mixed layer depth exceeds the critical depth. We have used a one-dimensional mathematical model integrated by the Lagrangian Ensemble method to simulate a population of diatoms during the winter between two growing seasons off the Azores. The model allows us to diagnose the demographic changes in the simulated diatom population from a variety of perspectives. The total population falls to a minimum of 70 million diatoms m-2 at the end of February. The vertical distribution of the population dynamics is first analysed in terms of daily Eulerian averages over 1 m depth intervals. Growth starts in February when the diurnal thermocline becomes shallower than 50 m, but while the mixed layer is still 200 m deep. The natural mortality has a minimum in winter because it is reduced (in the model) with temperature and population density. Eulerian analysis suggests that in winter, diatoms have a life expectancy of more than 3 months, so a significant number will survive the months of December, January and February when there is very little growth. Losses to grazing are negligible in winter. Lagrangian analysis shows how an individual diatom responds to its changing ambient environment caused by variation in depth (due to turbulent mixing) and the diurnal and seasonal changes in the photosynthetically active radiance. The different trajectories followed by the thousands of plankton particles simulated by the model produce diversity in growth rate ranging over several orders of magnitude, so care has to be taken in statistical analysis. The paper ends with a re-assessment of the value of the critical depth and compensation depth as predictors for onset of the spring bloom. The compensation depth was computed by Eulerian averaging over 1 m depth inter-vals each day. For 1 month after the vernal equinox the compensation depth follows the ascent of the mixed layer as it rises from a depth of 100 m to 40 m. Lagrangian analysis reveals that this is due to the photo-adaptation better matching the ambient irradiance experienced by diatoms in the mixed layer compared with those at the same depth in the seasonal thermo-cline. By mid-April the spring bloom has already ad-vanced so far that self shading influences the compensation depth, which then rises into the mixed layer. We conclude that Sverdrup's criterion is not useful for predicting changes in the diatom population simulated by our model.

An evaporation model based on Lagrangian turbulent diffusion principles is developed and compared with simpler single- and dual-source K-theory models of evaporation. The performance of the Lagrangian model is assessed against observations of total evaporation by an eddy-correlation instrument and found to be satisfactory for a millet crop in west Africa. Three versions of existing simpler models containing K-theory descriptions of within-canopy turbulence are described and their results also compared. The two K-theory models that explicitly take into acccount the soil source perform better than the single-source Penman-Monteith model. A Lagrangian analysis does not seem to be necessary for this kind of crop because the low source-density profile of the crop, associated with a low leaf-area index, caused the near-field effect to be very small. The overall difference between the evaporation estimates of the dual-source and Lagrangian models is therefore small. It is concluded that, for practical purposes, K-theory remains an adequate approximation of turbulent transport in sparse-crop evaporation models. Significant anomalies in the surface heat fluxes, particularly those of sensible heat, accompanied the decrease in the sea ice concentration. Substantial atmospheric warming was simulated over and in the vicinity of areas in which leads were considered. In all but one experiment there were anomalous easterlies between about 40 and 60°S with westerly anomalies further to the south. The surface pressure at high latitudes appears to change in a consistent fashion with the fraction of open water, with the largest changes occurring in the Weddell and near the Ross Seas. Some of the feedbacks which may enhance the responses here, but which are not included in our model, are discussed.

A novel one-dimensional Lagrangian simulation of beam–wave interaction in helix traveling-wave tubes is introduced in this paper. Unlike the traditional transmission-line coupled-mode Lagrangian analysis, direct electromagnetic coupling of the electric field to the RF signal from the Fourier harmonic components of RF current in a modulated beam is discussed here in terms of the interaction impedance of the RF structure by considering the modulated stream of moving charges as a radiating source. The model presented is very simple and fast, as all of the expressions and equations developed are in closed form and the simulation involves only one numerical integration at each step. The results obtained using the present model show good agreement (deviation

To study the dispersion of scalar trace constituents such as heat, water vapour or CO2 from extensive sources within vegetation canopies, a Lagrangian approach is essential because of the need to account for the influence of travel time (as a fraction of the Lagrangian time scale) on the dispersion properties of marked fluid particles. A simple, analytic Lagrangian theory is developed for a canopy in which the turbulence is homogeneous. For a given scalar source distribution, predictions are obtained for mean scalar concentration, vertical flux, terms in the balance equation for vertical flux, and vertical diffusivities at first, second and third order. the predictions agree remarkably well with measurements of dispersion from an elevated plane source within a model plant canopy in a wind tunnel. the theory also explains the countergradient heat and CO2 fluxes observed in real forest canopies.