Single Streamflow Research Articles

Quantifying uncertainty in coastal salinity regime for biological application using quantile regression

AbstractSalinity regimes in coastal ecosystems are highly dynamic and driven by complex geomorphic and hydrological processes. Estuarine biota are generally adapted to salinity fluctuation, but are vulnerable to salinity extremes. Characterizing coastal salinity regime for ecological studies therefore requires representing extremes of salinity ranges at time scales relevant to ecology (e.g., daily, monthly, and seasonally). Here, we propose a framework for modeling coastal salinity with these overall goals: (1) quantify uncertainty in salinity associated with important terrestrial and oceanographic drivers, (2) examine time scales of salinity response to river streamflow events, and (3) predict salinity continuously over space at key time scales. Salinity is modeled as quantile surfaces related to river discharge, tidal dynamics, wind, and spatial location, applied to Suwannee Sound estuary, FL, USA, where salinity has been monitored spatially since 1981. Each quantile level is regressed independently, and together they comprise a distribution of salinity uncertainty across space, with upper and lower quantiles describing salinity extremes. Effects of physical drivers on salinity are compared through four base models with various combinations of tide and wind variables, each including spatial coordinates and a single streamflow metric (in cubic meters per second). Multiple time scales of streamflow are considered by taking means across various periods, from 1 to 12 days, and at various lagged intervals prior to salinity sample, totaling 144 streamflow metrics. We found that the Suwannee coastal salinity regime is dynamic at multiple time scales and varies nonlinearly across space from the river effluence outward. Salinity increases nonlinearly with decreasing river flow rates below 200 m3/s, most prominently in the lower quantiles of salinity (τ = 0.05–0.25). Wind appears to have a stronger influence on salinity than astronomic tides for this estuary. The regression approach developed here can be applied to any coastal system that has sufficient spatial and temporal monitoring coverage to capture multiple flood and drought events. It is implemented with a simpleRroutine, and is less computationally‐intensive than finite difference hydrodynamic modeling. The characterizations of salinity uncertainty developed in these analyses can be directly applied to future studies of fish and wildlife responses to changes in watershed management.

Effect of electrodes positions on the performance of microfluidic enzymatic biofuel cell: From two streams to a single-stream flow device

Membraneless microfluidic enzymatic biofuel cells (μEBFCs) operated under the co-laminar flow of fuel and oxidant often face cross-mixing issues, especially at low flow rates. In this work, we propose a new approach to design the single-stream μEBFC based on the investigation of relative positions of the electrodes at the top and bottom of the microchannel. To achieve this, a μEBFC was fabricated with multiwalled carbon nanotube (MWCNT) electrodes via a stencil method. The bioanode and biocathode were modified with glucose dehydrogenase and laccase via direct covalent bonding, respectively. The best results were attained by placing the cathode at the top and anode at the bottom of the microchannel in a Y-shaped two streamflow μEBFC. When a single electrolyte stream was used to achieve more practicality and ease of use, the performance was reduced by 40%. However, with the new design of electrodes in single-stream μEBFC, i.e. cathode at the top and anode at the bottom facing each other in the microchannel, the performance was recovered by 20%. This corresponds to maximum current and power density of 216 ± 12 μA cm−2 and 69.2 ± 9.2 μW cm−2, respectively. The stacking ability of the device was realized by connecting the two cells in series and parallel. The maximum power density delivered by the two single-stream μEBFCs’ stack reached 160 μW cm−2 at 0.3 V. Thus, this study validates the viability of using single-stream μEBFCs in large stacks to power microelectronics more simply and practically.

Considering coasts: Adapting terrestrial models to characterize coastal wetland ecosystems

The Energy Exascale Earth System Model (E3SM) simulates fully coupled processes and interactions among water, energy, carbon and nutrient cycles. E3SM connects vegetation and soil dynamics through nutrient uptake, plant production, litterfall and decomposition as a function of abiotic parameters (e.g. temperature and moisture). However, E3SM is designed to characterize terrestrial ecosystems and connects land and open ocean systems using a single streamflow transport term, ignoring the complex dynamics of energy, water, carbon, and nutrients in coastal systems. The goals of our project were to: (1) Parameterize a point version of E3SM to capture coastal wetland habitats and (2) Determine marsh community responses to increased temperature and elevated CO2. We adapted a version of the E3SM land model, previously configured to represent forested bog hydrology to a coastal ecosystem using datasets from field experiments conducted at the Smithsonian Environmental Research Center's Global Change Research Wetland (GCReW). Tidal forcing in a marsh environment was simulated using a two-column system in which the columns are connected by lateral hydrologic flows. One column simulates interactions between vegetation and soil while a second column simulates variation in water level (both tidal and sea level rise). The updated model captures many aspects of the field experiments, showing that plant community responses to environmental change are non-linear, non-additive and different between plant types. Elevated CO2 treatments increased C3 plant biomass more than C4 (33% vs 17%). Temperature exacerbated CO2 responses in C3 plants (0 °C: 26%, 5.1 °C: 56%). We were more successful at characterizing C3 than C4 responses and simulating above rather than belowground biomass production. Next steps will include updates to key physiological parameters such as root:shoot carbon allocation and the addition of mechanistic feedbacks between vegetation and biogeochemical processes.

Does comprehensive evaluation of hydrological models influence projected changes of mean and high flows in the Godavari River basin?

Understanding the projected changes in the mean and high flows remains a significant challenge due to uncertainty arising from global climate models (GCMs) and hydrological models. Moreover, the calibration approaches used for hydrological models can influence the climate change impact assessment. We use the combination of three hydrological models, four global climate models, and two RCPs (2.6 and 8.5) to analyze the projected changes in mean flow, high flow, and the frequency of high flow under the projected future climate in the Godavari River basin (GRB) until the gauge Tekra. The two evaluation approaches: a simple approach (TASK A) based on the calibration and validation at a single streamflow gauge station and a comprehensive approach (TASK B) based on multi-variable and multisite calibration and validation and trend analysis were employed to evaluate the hydrological models. The differences between the projected changes in mean and high flows calculated using models after TASK A and TASK B were estimated. Our results show that the differences can be up to 10–13% in mean annual flow and high flow, and up to 40% in high flow frequency. The comprehensively evaluated hydrological models were chosen for impact assessment, and they project increases in mean and high flows, and the frequency of high flow at all four gauge stations in the GRB. The projected increases are higher under RCP 8.5 and in the End century (2071–2100). Our results demonstrate the importance of the comprehensive evaluation of hydrological models in advance of climate change impact assessment.

Elastic Turbulence of Aqueous Polymer Solution in Multi-Stream Micro-Channel Flow.

Viscous liquid flow in micro-channels is typically laminar because of the low Reynolds number constraint. However, by introducing elasticity into the fluids, the flow behavior could change drastically to become turbulent; this elasticity can be realized by dissolving small quantities of polymer molecules into an aqueous solvent. Our recent investigation has directly visualized the extension and relaxation of these polymer molecules in an aqueous solution. This elastic-driven phenomenon is known as ‘elastic turbulence’. Hitherto, existing studies on elastic flow instability are mostly limited to single-stream flows, and a comprehensive statistical analysis of a multi-stream elastic turbulent micro-channel flow is needed to provide additional physical understanding. Here, we investigate the flow field characteristics of elastic turbulence in a 3-stream contraction-expansion micro-channel flow. By applying statistical analyses and flow visualization tools, we show that the flow field bares many similarities to that of inertia-driven turbulence. More interestingly, we observed regions with two different types of power-law dependence in the velocity power spectra at high frequencies. This is a typical characteristic of two-dimensional turbulence and has hitherto not been reported for elastic turbulent micro-channel flows.

Single stream inertial focusing in low aspect-ratio triangular microchannels.

A wide range of microfluidic devices for single stream focusing of cells and particles has emerged in recent years, based on both passive and active methods. Inertial microfluidics offers an attractive alternative to these methods, providing efficient and sheathless passive focusing of cells and beads. Nevertheless, in rectangular microchannels, the presence of multiple equilibrium positions necessitates complicated solutions involving manipulation of the 3D structure in order to achieve single stream flows. Here, we present a new approach to single-stream inertial focusing based on a triangular microchannel geometry. Changing the channel cross-sectional shape leads to asymmetry in the velocity profile, resulting in a size-dependent single stable equilibrium position near the channel apex. We demonstrate that soft lithography masters for such microchannels can be fabricated using PMMA through micromilling, and 15 μm diameter beads can be efficiently focused into a single stream. Confocal microscopy was used to confirm the focusing positions in the microchannel cross-section. We further integrated this device with a laser counting system to form a sheathless flow cytometer and demonstrated the counting of beads with an ∼326 s -1 throughput. The use of a triangular cross-section offers a number of benefits, including simplicity of the fundamental principle and geometry, control of design, a small footprint, and ease of integration, as well as high-precision single position focusing.

Evaluation of the impacts of hydrologic model calibration methods on predictability of ecologically-relevant hydrologic indices

Due to the significant costs associated with stream health monitoring, hydrological modeling is widely used to calculate ecologically-relevant hydrologic indices to better understand the overall condition of streams within large and diverse watersheds. However, hydrologic modeling’s ability to replicate these indices is limited, especially when calibrating models by optimizing a single objective function or when selecting a single optimal solution. Hence, this study evaluates the performance of multi-objective model calibration in representing 167 hydrologic indices of ecological interest using the median values of different Pareto-optimal solution sets. For this purpose, two strategies based on three Nash-Sutcliffe Efficiencies (NSE), and root-mean-squared error (RMSE) with explicit hydrograph partitioning, were implemented. Additionally, the k-means clustering technique was employed to define subsets of Pareto-optimal solutions representing high, medium, and low flow conditions. The Soil and Water Assessment Tool (SWAT) was set up for the Honeyoey Creek-Pine Creek Watershed, located in Michigan, USA, and was calibrated using a single streamflow gauging station and the recently proposed multi-objective optimization algorithm – Non-dominated Sorting Genetic Algorithm III (NSGA-III). The results show that the NSE- and RMSE-based calibration strategies can represent 128 and 123 indices within a range of ±30% relative errors, respectively. In addition, the results of this study demonstrated that using different set of solutions, instead of a single optimal solution, introduces more flexibility in the predictability of different hydrologic indices of ecological interest. Moreover, it is suggested that a multi-objective calibration approach allows the systematic identification of groups of poorly represented and closely related indices. Knowing these poorly represented indices would facilitate the formulation of additional objective functions intended to improve model performance or to detect structural inadequacies.

Post-collapse perturbation theory in 1D cosmology – beyond shell-crossing

We develop a new perturbation theory (PT) treatment that can describe gravitational dynamics of large-scale structure after shell-crossing in the one-dimensional cosmological case. Starting with cold initial conditions, the motion of matter distribution follows at early stages the single-stream regime, which can, in one dimension, be described exactly by the first-order Lagrangian perturbation, i.e. the Zel'dovich solution. However, the single-stream flow no longer holds after shell-crossing and a proper account of the multi-stream flow is essential for post-collapse dynamics. In this paper, extending previous work by Colombi (2015, MNRAS 446, 2902), we present a perturbative description for the multi-stream flow after shell-crossing in a cosmological setup. In addition, we introduce an adaptive smoothing scheme to deal with the bulk properties of phase-space structures. The filtering scales in this scheme are linked to the next-crossing time in the post-collapse region, estimated from our PT calculations. Our PT treatment combined with adaptive smoothing is illustrated in several cases. Predictions are compared to simulations and we find that post-collapse PT with adaptive smoothing reproduces the power spectrum and phase-space structures remarkably well even at small scales, where Zel'dovich solution substantially deviates from simulations.

Event-scale power law recession analysis: quantifying methodological uncertainty

Abstract. The study of single streamflow recession events is receiving increasing attention following the presentation of novel theoretical explanations for the emergence of power law forms of the recession relationship, and drivers of its variability. Individually characterizing streamflow recessions often involves describing the similarities and differences between model parameters fitted to each recession time series. Significant methodological sensitivity has been identified in the fitting and parameterization of models that describe populations of many recessions, but the dependence of estimated model parameters on methodological choices has not been evaluated for event-by-event forms of analysis. Here, we use daily streamflow data from 16 catchments in northern California and southern Oregon to investigate how combinations of commonly used streamflow recession definitions and fitting techniques impact parameter estimates of a widely used power law recession model. Results are relevant to watersheds that are relatively steep, forested, and rain-dominated. The highly seasonal mediterranean climate of northern California and southern Oregon ensures study catchments explore a wide range of recession behaviors and wetness states, ideal for a sensitivity analysis. In such catchments, we show the following: (i) methodological decisions, including ones that have received little attention in the literature, can impact parameter value estimates and model goodness of fit; (ii) the central tendencies of event-scale recession parameter probability distributions are largely robust to methodological choices, in the sense that differing methods rank catchments similarly according to the medians of these distributions; (iii) recession parameter distributions are method-dependent, but roughly catchment-independent, such that changing the choices made about a particular method affects a given parameter in similar ways across most catchments; and (iv) the observed correlative relationship between the power-law recession scale parameter and catchment antecedent wetness varies depending on recession definition and fitting choices. Considering study results, we recommend a combination of four key methodological decisions to maximize the quality of fitted recession curves, and to minimize bias in the related populations of fitted recession parameters.

Multi-stream portrait of the cosmic web

We report the results of the first study of the multi-stream environment of dark matter haloes in cosmological N-body simulations in the LCDM cosmology. The full dynamical state of dark matter can be described as a three-dimensional sub-manifold in six dimensional phase space - the dark matter sheet. In our study we use a Lagrangian sub-manifold x = x(q,t) (where x and q are co-moving Eulerian and Lagrangian coordinates respectively), which is dynamically equivalent to the dark matter sheet but is more convenient for numerical analysis. Our major results can be summarized as follows. At the resolution of the simulation i.e. without additional smoothing, the cosmic web represents a hierarchical structure: each halo is embedded in the filamentary framework of the web predominantly at the filament crossings, and each filament is embedded in the wall like fabric of the web at the wall crossings. Locally, each halo or sub-halo is a peak in the number of streams field. The number of streams in the neighbouring filaments is higher than in the neighbouring walls. The walls are regions where number of streams is equal to three or a few. Voids are uniquely defined by the local condition requiring to be a single-stream flow region. The shells of streams around haloes are quite thin and the closest void region is typically within one and a half FOF radius from the center of the halo.

Systematic uncertainty reduction strategies for developing streamflow forecasts utilizing multiple climate models and hydrologic models

Recent studies show that multimodel combinations improve hydroclimatic predictions by reducing model uncertainty. Given that climate forecasts are available from multiple climate models, which could be ingested with multiple watershed models, what is the best strategy to reduce the uncertainty in streamflow forecasts? To address this question, we consider three possible strategies: (1) reduce the input uncertainty first by combining climate models and then use the multimodel climate forecasts with multiple watershed models (MM-P), (2) ingest the individual climate forecasts (without multimodel combination) with various watershed models and then combine the streamflow predictions that arise from all possible combinations of climate and watershed models (MM-Q), (3) combine the streamflow forecasts obtained from multiple watershed models based on strategy (1) to develop a single streamflow prediction that reduces uncertainty in both climate forecasts and watershed models (MM-PQ). For this purpose, we consider synthetic schemes that generate streamflow and climate forecasts, for comparing the performance of three strategies with the true streamflow generated by a given hydrologic model. Results from the synthetic study show that reducing input uncertainty first (MM-P) by combining climate forecasts results in reduced error in predicting the true streamflow compared to the error of multimodel streamflow forecasts obtained by combining streamflow forecasts from all-possible combination of individual climate model with various hydrologic models (MM-Q). Since the true hydrologic model structure is unknown, it is desirable to consider MM-PQ as an alternate choice that reduces both input uncertainty and hydrologic model uncertainty. Application on two watersheds in NC also indicates that reducing the input uncertainty first is critical before reducing the hydrologic model uncertainty.

Modelling Rainfall Runoff Relations Using HEC-HMS and IHACRES for a Single Rain Event in an Arid Region of Jordan

The HEC-HMS and IHACRES rainfall runoff models were applied to simulate a single streamflow event in Wadi Dhuliel arid catchment that occurred on 30–31/01/2008. Streamflow estimation was performed on the basis of an hourly scale. The aim of this study was to develop a new framework of rainfall-runoff model applications in arid catchments by integrating a re-adjusted satellite-derived rainfall dataset (GSMaP_MVK+) to determine the location of the rainfall storm. The HEC-HMS model was applied using the HEC-GeoHMS extension in ArcView 3.3 while the IHACRES is Java-based version model. The HEC-HMS model input data include soil type, land use/land cover, and slope. By contrast, the lumped model IHACRES was also applied, based on hourly rainfall and temperature data. Both models were calibrated and validated using the observed streamflow data set collected at Al-Za’atari discharge station. The performance of IHACRES showed some weaknesses, while the flow comparison between the calibrated streamflow results fits well with the observed streamflow data in HEC-HMS. The Nash-Sutcliffe efficiency (Ef) for the two models was 0.51 and 0.88 respectively.

Identifying step changes in single streamflow and evaporation records due to forest cover change

AbstractTechniques that identify forestry‐induced changes to streamflow or evaporation are needed to assess available water resources. Equally, there is a growing appreciation that climate cycles may be having a profound impact on the land‐surface hydrology. The ability to see forestry‐induced change above the effects of climate dynamics, therefore, becomes a critical issue. Paired‐catchment analyses have proved very valuable in identifying change, but cannot quantify the relative impacts of climate and land‐cover change, and data from adjacent reference basins are not always available. Within this study, we examined whether step changes within single time‐series of streamflow or evaporation (P‐Q) could be identified without reference to those of a control catchment. The UC‐DHR method was used for this analysis, and included a special routine to allow a known change‐point (e.g. start of logging) to be specified or alternatively identified by the model. Data from three experimental catchments important for their seminal forestry impact studies were selected for the analyses.The study demonstrated that clear‐cutting 29% of the Hore catchment and 40% selective felling of the Berembun basin produced a step change in the discharge trend that was clearly observable above the climate‐related dynamics and uncertainty. In contrast, step changes in P‐Q following the same selective felling event or following 22% afforestation of the Upper Hodder basin were not larger than the uncertainty bands or magnitude of the inter‐annual cycles produced by the climate dynamics, respectively. This demonstrates that while step changes can be observed in single hydrological time‐series, errors within the observations can sometimes mask the identification of change. This masking of change is also possible where the longer‐term cyclical behaviour in Q or P‐Q from natural climate dynamics is large, while the spatial extent of forestry change is small. Copyright © 2011 John Wiley & Sons, Ltd.

Improved jet noise modeling using a new time-scale

To calculate the noise emanating from a turbulent flow using an acoustic analogy knowledge concerning the unsteady characteristics of the turbulence is required. Specifically, the form of the turbulent correlation tensor together with various time and length-scales are needed. However, if a Reynolds Averaged Navier-Stores calculation is used as the starting point then one can only obtain steady characteristics of the flow and it is necessary to model the unsteady behavior in some way. While there has been considerable attention given to the correct way to model the form of the correlation tensor less attention has been given to the underlying physics that dictate the proper choice of time-scale. In this paper the authors recognize that there are several time dependent processes occurring within a turbulent flow and propose a new way of obtaining the time-scale. Isothermal single-stream flow jets with Mach numbers 0.75 and 0.90 have been chosen for the present study. The Mani-Gliebe-Balsa-Khavaran method has been used for prediction of noise at different angles, and there is good agreement between the noise predictions and observations. Furthermore, the new time-scale has an inherent frequency dependency that arises naturally from the underlying physics, thus avoiding supplementary mathematical enhancements needed in previous modeling.

Vascularized materials: Tree-shaped flow architectures matched canopy to canopy

In this paper we develop flow architectures for “vascularizing” smart materials that have self-healing capabilities. The flow architectures are configured as two trees matched canopy to canopy. A single stream flows through both trees and bathes every subvolume (crack site) of the material. Several types of tree-tree configurations are optimized. Trees that have only one level of branching and bathe a rectangular domain have optimal external shapes that are nearly square. They also have optimal ratios of channel sizes before and after branching. Trees optimized on square domains perform nearly as well as trees on freely morphing rectangular domains. The minimized global flow resistance decreases slowly as the number of subvolumes increases. It is more beneficial to bathe the entire volume with a single (optimized) one-stream architecture than to bathe it with several streams that serve small clusters of volume elements. These conclusions are reinforced by an analytical optimization of the same class of architectures in the limit of a large number of assembled subvolumes. We also show that the freedom to morph the design and to increase its performance can be enhanced by using tree-tree architectures with more than one level of branching.

Forecasting daily streamflow using hybrid ANN models

In this paper, the classic ‘divide and conquer (DAC)’ paradigm is applied as a top-down black-box technique for the forecasting of daily streamflows from the streamflow records alone, i.e. without employing exogenous variables of the runoff generating process such as rainfall. To this end, three forms of hybrid artificial neural networks (ANNs) are used as univariate time series models, namely, the threshold-based ANN (TANN), the cluster-based ANN (CANN), and the periodic ANN (PANN). For the purpose of comparison of forecasting efficiency, the normal multi-layer perceptron form of ANN (MLP–ANN) is selected as the baseline ANN model. Having first applied the MLP–ANN models without any data-grouping procedure, the influence of various data preprocessing procedures on the MLP–ANN model forecasting performance is then investigated. The preprocessing procedures considered are: standardization, log-transformation, rescaling, deseasonalization, and combinations of these. In the context of the single streamflow series considered, deseasonalization without rescaling was found to be the most effective preprocessing procedure. Some discussions are presented (i) on data preprocessing and (ii) on selection of the best ANN model. Overall, among the three variations of hybrid ANNs tested, the PANN model performed best. Compared with the MLP–ANN fitted to the deseasonalized data, the PANN based on the soft seasonal partitioning performed better for short lead times (≤3 days), but the advantage vanishes for longer lead times.

Immobilized 2-(4-hydrazinocarbonylphenyl)-4,5-diphenylimidazole as solid phase luminophore in peroxyoxalate chemiluminescence.

A new luminophore for application in peroxyoxalate chemiluminescence is presented. An analogue of the well-known chemiluminescence compound lophine, i.e. 2-(4-hydrazinocarbonylphenyl)-4,5-diphenylimidazole (HCPI), has been covalently immobilized to controlled pore glass and a porous methacrylate resin. By using this reagent in a solid phase detection reactor, sensitive determinations of hydrogen peroxide have been demonstrated. In homogeneous solution HCPI emits poorly as a result of 1,1'-oxalyldiimidazole excitation, but when immobilized its efficiency is almost comparable to highly efficient luminophores such as 3-aminofluoranthene. Linearity extends in the single stream flow system over several orders of magnitude with both materials. The limit of detection was 1 nmol/l (10 fmole injected), when using the porous methacrylate support.

Initial time-varying flow in the smooth-bore magnetron

A computer program has been written to study the flow pattern at the switching on of a 2-dimensional smooth-bore magnetron. Time-variation and multistream flow are allowed. The results suggest that a simple single-stream flow cannot be established, but that either a double-stream or a multistream flow is established.

Velocity and Potential Distribution in a Linear, Smooth Anode Magnetron (Magnetic Diode). II

We present the results of numerical calculations of the potential, space-charge, and current distributions in linear magnetrons, taking into account the three-dimensional spread of the initial electron velocities. For most practical configurations, the potential distribution is nearly parabolic within the electron cloud, with a slight dip in front of the cathode and the space-charge distribution is remarkably constant and approximately equal to that predicted by the single stream flow, although the electron trajectories are now anything but linear. For other choices of the parameters, the space-charge distribution can be ``triangular'' or may resemble that associated with the so-called double-stream flow. In all such cases the potential, space charge, and current distributions cannot be predicted on the basis of a Brillouin flow and have to be calculated separately. We have also shown that in most practical cases the systems seem to operate in what is called the magnetic-field-limited, rather than space-charge-limited regime, so that the anode current depends on the values of the external parameters only and is independent of the amount of space charge or the potential distribution between the electrodes. In ballistic terms this means that the force due to the magnetic field acting on an electron at the peak of its trajectory is greater than the corresponding force due to the electric field. Finally, we have shown that contrary to our expectations, there seems to be a great deal of agreement between the operations of systems using finite and infinite cathodes, even when the initial electron velocities are taken into account.

General Steady-state Theory of Cylindrical Magnetrons

ABSTRACT This paper presents a general steady-state theory of cylindrical magnetrons. Space charge and the Maxwellian velocity distribution of the emitted electrons are taken into account, but the collisions between individual electrons are neglected. The paper explains the mathematical basis of the calculations and gives expressions for the volume density of the electrons and for the two components of the electron current density. No calculations of the actual potential distributions are given but the problem is briefly discussed at the end of the paper. It is shown however that for certain important ranges of potential distributions the following conclusions concerning the steady-state theory of cylindrical magnetrons can be drawn (1)Brillouin's single-stream flow is not possible when the emission velocities are taken into account (2)In a well ‘ cut-off ’ magnetron the azimuthal component of the current density may be several hundred times larger than the radial component. Thus, even slight imperfection...