Vertical Coordinate Research Articles

ELECTRON BEAMS IN THE GRADIENT MAGNETIC FIELD: CONTROL FOR CONVERTING LONGITUDINAL MOTION INTO TRANSVERSAL

The motion of electrons in a cylindrical magnetic field with the gradient-type potential is considered. The motion of electrons in the cylindrical magnetic field with the gradient-type potential is considered. It is found that in the selected field, the initial motion of electrons along the longitudinal axis is converted into motion along the radius. It is determined that such the transformation is due to the action of a solenoidal magnetic field with large longitudinal gradient. The transformation of the longitudinal direction of motion into the transverse one turned out to be stable in the energy range of 20...55 keV of electrons and in the range of 5...50 mm of radial dimensions of the particle beam. The main dependences of the motion of the electron beam in the given solenoidal magnetic field are studied with the help of the software tool. The results of numerical simulation of electron trajectories in the gradient magnetic field with the circular secondary emission cathode located in the middle of the system are presented. To study the mechanism of stability with respect to magnetic field, two experimentally realized magnetic fields were used. Based on these two fields, arrays of additional 4 fields are numerically synthesized. For the set of 6 named fields, the operation of the gun, in which the particle undergoes the stable transformation of the direction of motion, is numerically studied. It is shown that for the given electron energy and the fixed magnetic field, the parameter that determines the rotation of the particles is the magnetic field gradient at the boundary of the entry region. It is found that the rotation effect takes place for the considered range of radial beam sizes, which leads to particle focusing. The possibility is shown to control the vertical coordinate of the focused beam on the basis of the field adjustment, thereby giving the interpretation of the pore dependence of the registration of electrons on the detector. The dependence of the formation of the final distribution of particles on the amplitude and gradient of the magnetic field along the axis of the system is studied. The results of numerical simulation on the motion of the electron beam are presented. Based on the model of electron flow motion, the characteristics of the resulting electron beam are considered. It is shown that the beam, having radial dimensions of 5...50 mm, is transformed and focused vertically on the area of 1 mm.

Overcoming Bureaucratic Resistance: An Analysis of Barriers to Climate Change Adaptation in South Africa

Climate change is already a reality, and it is affecting the lives and livelihoods of many people globally. Many scientists argue that adaptation is, therefore, necessary to address the impact of climate change on life-supporting systems. Climate change adaptation, however, is a complex process that involves transformations implemented through governance at multiple levels. In this paper, the barriers to climate change adaptation in South Africa are presented and analysed. Semi-structured, in-depth interviews were conducted telephonically and online via Microsoft Teams with 13 government officials working at the Department of Forestry, Fisheries, and the Environment; the KwaZulu-Natal Department of Economic Development and Environmental Affairs; and the uMkhanyakude District Municipality. The findings suggest that the barriers to climate change adaptation in South Africa include inadequate financial resources, a lack of human capacity at the provincial and local levels, limited political will at the local level, limited understanding of climate change adaptation issues by communities, inadequate coordination across government levels and sectors, no legal mandate at the local level, no climate change unit at the district and local levels, a lack of knowledge by some staff members tasked with environmental duties at the local level, not enough climate change plans in place at the local level, and outdated information on climate change used in the IDPs. This paper, therefore, recommends that climate change be a standing item in the Integrated Development Plan for local governments, which will ensure that climate change is budgeted for appropriately. In addition, this paper suggests that a mandate for climate change adaptation be developed for all three government levels. There is also a need for the government to invest in capacity development and improve horizontal and vertical coordination to strengthen the weak climate governance capacity that exists.

Assimilating AMSU-A Radiance Observations with an Ensemble Four-Dimensional Variational (En4DVar) Hybrid Data Assimilation System

Many ensemble-based data assimilation (DA) methods use observation space localization to mitigate the sampling errors due to the insufficient ensemble members. Observation space localization is simpler and more timesaving than model space localization in implementation, but more difficult to directly assimilate satellite radiance observations, a kind of non-local observations. The vertical locations of radiance observations are undetermined and the transmission of observational information is thereby obstructed. To determine the vertical coordinates of radiance observations, a weighted average hypsometry is proposed. Using this hypsometry, AMSU-A radiance observations are directly assimilated with an ensemble four-dimensional variational (En4DVar) DA system. It consists of a four-dimensional ensemble-variational (4DEnVar) system providing ensemble covariance and a 4DVar system. Observing system simulation experiments show that the hypsometry alleviates the degradations in the late period of medium-range forecast in the Northern Extratropics that occur in the traditional peak-based hypsometry. It obviously improves the analysis qualities and forecast skills of the En4DVar system and its two components, especially in the Southern Extratropics, when incorporating AMSU-A radiance observations. The improvement in the En4DVar-initialized forecast is comparable to that in the 4DVar-initialized forecast in the Southern Extratropics and Tropics. It indicates that a proper hypsometry enables efficient extraction of useful information from AMSU-A radiance observations by 4DEnVar with observation space localization. Therefore, the 4DEnVar provides high-quality ensemble covariances for En4DVar.

Capturing features of turbulent Ekman–Stokes boundary layers with a stochastic modeling approach

Abstract. Atmospheric boundary layers (ABLs) exhibit transient processes on various time scales that range from a few days down to seconds, with a scale separation of the large-scale forcing and the small-scale turbulent response. One of the standing challenges in modeling and simulation of ABLs is a physically based representation of complex multiscale boundary layer dynamics. In this study, an idealized time-dependent ABL, the so-called Ekman–Stokes boundary layer (ESBL), is considered as a simple model for the near-surface flow in the mid latitudes and polar regions. The ESBL is driven by a prescribed temporal modulation of the bulk–surface velocity difference. A stochastic one-dimensional turbulence (ODT) model is applied to the ESBL as standalone tool that aims to resolve all relevant scales of the flow along a representative vertical coordinate. It is demonstrated by comparison with reference data that ODT is able to capture relevant features of the time-dependent boundary layer flow. The model predicts a parametric enhancement of the bulk–surface coupling in the event of a boundary layer resonance when the flow is forced with the local Coriolis frequency. The latter reproduces leading order effects of the critical latitudes. The model results suggest that the bulk flow decouples from the surface for high forcing frequencies due to a relative increase in detached residual turbulence.

Stability Properties of the Constant Coefficients Semi-Implicit Time Schemes Solving Nonfiltering Approximation of the Fully Compressible Equations

Abstract A set of control parameters is introduced in the fully elastic nonhydrostatic Euler equations formulated in the mass-based vertical coordinate of Laprise. Contrary to the classical approach, the hydrostatic limit is represented by a subspace of control parameters, instead of a single point. By finding a suitable path from the fully compressible equations to the hydrostatic subspace, we are able to construct a blended system with acoustic modes slowed down and gravity modes nearly unaffected. Numerical stability of the discretized system is thus improved, and the solution remains essentially the fully compressible one. Alternatively, control parameters can be used to redefine the linear model of the constant coefficients semi-implicit time scheme, increasing the numerical stability of the fully compressible system. With a careful choice of the control parameters in both, the linear model used in the semi-implicit temporal scheme, and in the full model, the blended system does not deteriorate the compressible solution while its semi-implicit temporal discretization is more stable. We illustrate the potential of the method in several simple examples and in real case studies using the ALADIN system.

Solution of filtration problems of construction geotechnology in thin water layers

Filtration problems of construction geotechnics during water movement in thin permeable layers of soils and artificial materials are considered. Under such conditions, traditional approaches borrowed from classical hydrogeology give errors. This is due to the fact that under normal conditions of underground aquifers of a sufficiently large depth, in hydrogeological practice, the hydraulic theory of filtration is used, in which the velocities of filtration currents are averaged along the vertical coordinate. The original nonlinear differential equations of filtration flows are subjected to forced linearization by one or another known method. In this case, the method of linearization by introducing the average power (depth) of the filtration flow is widely used. This method was originally developed for aquifers of great depth. In the presented work, an attempt was made to abandon such a linearization and solve some of the problems of unsteady filtration flows in thin aquifers, which are characteristic of building geotechnics, with obtaining analytical solutions of the final form without introducing artificial simplifications. A methodology is proposed for combining analytical and numerical methods in the process of finding solutions to filtration problems of construction geotechnics with obtaining self-similar dependencies. This approach will make it possible to avoid linearization errors and carry out filtration calculations in thin permeable layers of soils and artificial materials more accurately. A review of domestic and foreign scientific publications on the topic of research has shown some stagnation in this direction. The comparison revealed that in thin permeable layers of soils and artificial materials, preference can be given to filtration calculations using the presented method, since it more accurately describes the non-stationary movement of water in construction geotechnics.

Analyzing the load for the design elements of the working equipment of the loader with a telescopic boom

Problem. The pace of road construction and reconstruction dictates high requirements for modern equipment, and in most cases the requirements relate to work equipment. When performing technological operations, external loads are transmitted through the main working body to all elements of work equipment, components and parts of the machine as a whole. Methodology. The use of software products makes it possible to preanalyze any combinations of design, modeling, as well as strength studies and strength testing. On the basis of the obtained data, it is possible to analyze the load in the nodes of the metal structure of the working equipment without conducting field tests. The work is a continuation of previous studies using computer modeling methods in the engineering industry. Designing or modernization of technical objects at the moment is impossible without creating three-dimensional models. The purpose of this work is to develop a digital model of the telescopic loader equipment in the "Dynamic Simulation" environment of the Autodesk Inventor software package, to perform research on the kinematics of movement and load modes of individual elements. The use of telescopic working equipment expands the capabilities of front loaders on a wheel base, but in turn, leads to an increase in material costs during design and production. Results. The process of separate scooping of bulk material in a stack is considered, which consists in introducing a bucket into the stack with a progressive movement followed by stopping the loader and turning the bucket. Originality. The main forces acting on the working equipment during the rotation of the bucket and the boom during the scooping process are the gravitational forces of the elements of the working equipment, non-conservative forces arising in the cylindrical surface of the section and the normal force from the side of the wall of the stack of material. Practical value. External loads are applied to the edge of the bucket. Their direction depends on the trajectory of the working equipment and in the process of simulation they changed depending on the current vertical coordinate of the cutting edge of the bucket.

Shelf slope, estuarine dynamics and river plumes in a z* vertical coordinate, unstructured grid model

The finite element, unstructured grid model SHYFEM has been used to carry out some fundamental studies on the quality of a specific type of Generalized Vertical Coordinate (GVC) system, the so-called z∗ coordinates versus z−geopotential coordinates. Idealized and realistic test cases are shown to assess the impact of the vertical coordinates on key physical processes, i.e. the internal tide generation at a continental slope, a river plume shape and intensity and the salt water intrusion.The results showed that the z∗ coordinates produce a stronger water column stratification than the z−geopotential ones and together with the unstructured grid modeling they can better solve the coastal sea dynamics.An idealized twin experiment configuration of the model was carried out to verify the proper implementation of the GVC z∗ by simulating the production of internal tides at the continental slope. The results confirmed that the z∗ coordinate model reproduces the expected internal tide fields better than z−geopotential model. A set of realistic experiments in the Po river delta coastal region, demonstrated that the representation of the dynamical processes at the coastal scales is conditioned by the high resolution (25 cm layers), which was only possible by the z∗ formulation in the upper water column.

3D-GPR-RM: A Method for Underground Pipeline Recognition Using 3-Dimensional GPR Images

Ground penetrating radar (GPR), as a non-destructive and rapid detection instrument, has been widely used for underground pipeline detection. However, as the interpretation of 3-dimensional GPR images is still manually performed, the process is inefficient. Aiming at solving the challenges of automatic recognition for underground pipelines, we propose a recognition method based on a deep learning algorithm, which uses 3-dimensional GPR images and the improved 3D depth-wise separable convolution block. In order to expand the number of samples in the dataset, we propose a data augmentation method based on three-dimensional matrix rotation and use a wavelet-based denoising method to filter out the direct wave interference. To prove the effectiveness and efficiency of our method, we compared the classification performance of the improved 3D depth-wise separable convolutional block with the traditional 3D convolutional block and the ordinary 3D depth-wise separable convolutional block under the same conditions. According to the experiment’s results, the number of parameters of the method we proposed is 66.9% less than that of the traditional 3D convolution method, while the classification performance is similar. Furthermore, compared with ordinary 3D depth-wise separable convolution, our method can significantly improve the classification and recognition ability of the neural network, while the number of calculations and the number of parameters remain almost the same. This study demonstrates the effectiveness of 3D-CNN in the field of GPR image interpretation. An improved 3D depth-wise separable convolutional block is also proposed. It greatly reduces the amount of calculation and parameters while ensuring classification performance. It is better than the existing algorithms in performance. At the same time, to obtain the position and direction of the pipeline, in this study, a conic fitting method using the Canny operator is proposed to locate the vertices of B-Scan images and record their horizontal and vertical coordinates. This method can estimate the direction of the pipeline and it lays the foundation for future work such as measuring the pipeline depth.

Do government knowledge production and use systems matter for global climate change adaptation tracking? Insights from Eastern Africa

National contexts play a critical role in shaping the transposition of international laws and agreements, such as the Paris Agreement. However, the relevance of national contexts when assessing global progress in adaptation to climate change has received little theoretical and empirical attention. To bridge this gap, we conduct a comparative study of government systems for producing and using policy knowledge on the livestock sectors of three Eastern Africa countries. We find distinct features within and between countries, which may explain variations in how adaptation progress is tracked. In particular, our study shows that prevailing administrative structures influence horizontal and vertical coordination, with implications for the flow of knowledge within government. The extent of coordination and the establishment of knowledge production procedures and accountability mechanisms affect the compatibility of the various knowledge streams in each country which, in turn, determines the potential for integrating adaptation tracking across the various administrative units. Our findings suggest that the effectiveness and feasibility of tracking adaptation progress over time and space will depend on the adequacy and successful linkage of tracking programs with existing systems of knowledge production and use. These findings underscore the relevance of a fit-for-context approach that examines how adaptation tracking can effectively be integrated into existing structures and processes while developing strategies for improving knowledge production and use.

TESTING THE EXCAVATOR PERFORMANCE (USING TOPCON 3D EXCAVATOR X63 SYSTEM) ESPECIALLY FOR NAVIGATION AND EARTHWORK

Construction sites commonly utilize bulldozers, wheel loaders, excavators, scrapers, and graders. Among these, excavators are versatile hydraulic heavy-duty equipment operated by humans. They are employed for various tasks like digging, levelling the ground, transporting and dumping loads, as well as providing straight traction. However, certain hazardous environments, such as nuclear disasters or earthquakes, are not suitable for human on-site work. To enhance productivity, accuracy, and profitability in excavation projects, the adoption of 3D machine control is recommended. The Topcon 3D Excavator X63 System offers advanced and precise GNSS positioning technology, coupled with Hidromek with Assist and an intuitive software interface, to significantly improve excavation operations. In this study, the accuracy of the coordinates of the route followed by the Excavator was checked by using RTK GNSS method by using P1 reference point. While the differences obtained in horizontal coordinates are 2–2.5 cm and 4–6 cm in vertical coordinates. In addition, excavation calculations of the earthwork area were performed and checked with the number of bucket of the excavator. The differences obtained from the earthwork were calculated as 0.8 cubic meters for each bucket.

Urban Morphology and Functional Hierarchy of the Jewish Quarter of Zaragoza (Spain) at the End of the Fifteenth Century

This article analyzes the morphology and functionality of the Jewish quarter of Zaragoza at the end of the fifteenth century, shortly before the expulsion of the Jews in 1492, from a multidisciplinary perspective. It deals with the specific nature and understanding of one of the most important Jewish neighborhoods in Sepharad on the internal levels of Hebrew law and the confessional community, where urbanism is traced out from the privacy of the home and not from the public space and where the neighborhood is linked to the synagogues. Over two thousand documents from the 1492 to 1500 period, from the Archive of the Crown of Aragon, notarial protocols, municipal sources, and inquisitorial trials, have been consulted. A method of vertical cartographic coordination based on geographic information systems is applied as a way of geo-referencing the oldest surviving plans to reconstruct the physical and experiential space of the city in the Late Middle Ages.

The European art of governing. The tension between collegiality and national interests in the Cabinets of the European Commissioners

ABSTRACT To explore what we call the ‘European art of governing’, this article examines the Cabinets of the members of the European Commission. They function at the interface between levels and modes of governance, between the Commission and Member State governments, between national interests (of capitals) and sectoral interests (their portfolios). A new interpretation of the ‘denationalization’ process is given, focusing more on profiles than on practices. Cabinets have been remarkably stable since the end of the 1960s in their role as conduits of collegiality. However they also reflect a new form of ‘intergovernmentalization’ of the Commission. Using a historical and sociological approach, the article shows how collegiality at the Commission has been exercised since the Delors period of presidentialization and Juncker’s introduction of real Vice-Presidents, generating complex forms of competition between horizontal and vertical coordination processes.

Cut-cell Eta Model: History and Challenges Overcome

Incentive for writing a limited area weather prediction model stemmed from the author’s several years stay at the University of California in Los Angeles, at the end of the sixties. Exposed to what he refers to as the Akio Arakawa approach, having had an idea for a scheme that was an improvement to what Arakawa was using, and being aware of the importance of topography for the weather of the country he was to continue his career in, led in 1973 to his first limited area 3D code, the forerunner of what was to become the Eta model. Refinements and enhancements introduced by the author in subsequent years and of the collaborator he acquired, Zaviša Janjić, resulted in the code that when installed at the then U.S. National Meteorological Center, attracted attention. Hallmarks of the model were Mesinger’s eta vertical coordinate, and Janjić’s transformation of the Arakawa horizontal advection scheme to the model’s semi-staggered B/E grid. In 1993 the Eta became the primary regional forecasting model of the U.S. Weather Bureau, and in 1998 its precipitation accuracy of 24-48 h forecasts became higher across all intensity thresholds than that of its predecessor, the Nested Grid Model (NGM) for its 00-24 h forecasts. Lately, the Eta is extensively used also as a regional climate model (RCM), mostly over the South American domain, and in near-real time as a tool for the North American Regional Reanalysis (NARR), run by the U.S. National Centers for Environmental Prediction/Climate Prediction Center. Several later unique numerical refinements of the Eta addressing problems noticed are summarized in a “before and after” fashion, and results are mentioned of its ensemble skill compared to that of its highly acclaimed driver European Centre for Medium-Range Weather Forecasts (ECMWF) model.

Gravity currents of viscous fluids in a vertically widening and converging fracture

We are investigating flows in the viscous-buoyancy balance regime in a converging channel with an upward increase in the width, with the gap of the channel varying according to a xkzr power function, being x and z the horizontal and vertical coordinate, respectively, and with 0<k<1 and 0<r<1 in order to be consistent with the model. The fluid rheology is described according to the Ostwald–de Waele model, with a power-law relationship between shear stress and shear rate and with application for shear-thinning, shear-thickening and, as a special case, Newtonian fluids. While for the case of flow in the direction of widening of the horizontal channel, a self-similar solution of the first kind can be expected, for flow toward the origin, with channel narrowing horizontally, the solution is self-similar of the second kind, with the space and a reduced time coupled in a self-similar independent variable but with an unknown parameter of the transformation group that makes the differential problem invariant. The solution is found in the phase plane by numerical integration of the paths connecting pairs of singular points, separately for the pre-closure phase, in which the current front advances invading the channel, and for the post-closure or leveling phase, in which the fluid has reached the origin and the front no longer propagates, while the level progressively increases canceling the pressure gradient. Integration is performed with a trial and error procedure by modifying the unknown parameter, generally named eigenvalue and specifically critical eigenvalue when the path has been successfully integrated. The overall effect of an increasing permeability upward is that of an increase in the front speed, with the current profile also becoming locally steeper. The effect of an increase in the fluid-behavior index is mixed, as it reduces the speed of the front but still increases the steepness of the local current profile. In any case, the model implies that the eigenvalue tends to infinity for k→1 even in the presence of an increase in the vertical permeability (r > 0).

The Effect of the Microbiota Metabolite – Butyric Acid on Motor Coordination, Muscle Strength and the Level of Oxidative Stress in Skeletal Muscles in Mice with Dysbiosis

According to modern concepts, the composition and diversity of the intestinal microbiota play an essential role in maintaining immunity, homeostasis, and, in general, the physiological functions of the host organism. Recently the positive role of the microbiota and its metabolites especially short-chain fatty acids, in the metabolism and functional activity of skeletal muscles was reported. The aim of our work was to analyze muscle strength and motor coordination in mice after injection of broad–spectrum antibiotics with simultaneous administration of a microbiota metabolite – one of the representatives of short-chain fatty acids – butyric acid. In addition, we determined the level of malondialdehyde, the concentration of total glutathione and the activity of glutathione peroxidases in the muscles of the hind limbs in mice with administration of antibiotics and butyric acid. The administration of antibiotics to adolescent mice for two weeks induced higher mortality and decrease of weight, and also caused significant changes in motor behavior, including an increase in horizontal motor activity, decrease in vertical motor activity, muscle strength, and motor coordination. A higher level of oxidative stress was found in the muscle tissues of the hind limbs of mice treated with antibiotics. At the same time, oral administration of butyric acid prevented the observed changes and improved not only behavioral disorders, but also partially reduced the level of oxidative stress. In conclusion, metabolite of normal microbiota has a positive effect on the functional and biochemical parameters of skeletal muscles in dysbiosis, which can be used to prevent loss of muscle function in various pathological conditions.

Representing Eddy Diffusion in the Surface Boundary Layer of Ocean Models With General Vertical Coordinates

AbstractThe mixing of tracers by mesoscale eddies, parameterized in many ocean general circulation models (OGCMs) as a diffusive‐advective process, contributes significantly to the distribution of tracers in the ocean. In the ocean interior, diffusive contribution occurs mostly along the direction parallel to local neutral density surfaces. However, near the surface of the ocean, small‐scale turbulence and the presence of the boundary itself break this constraint and the mesoscale transport occurs mostly along a plane parallel to the ocean surface (horizontal). Although this process is easily represented in OGCMs with geopotential vertical coordinates, the representation is more challenging in OGCMs that use a general vertical coordinate, where surfaces can be tilted with respect to the horizontal. We propose a method for representing the diffusive horizontal mesoscale fluxes within the surface boundary layer of general vertical coordinate OGCMs. The method relies on regridding/remapping techniques to represent tracers in a geopotential grid. Horizontal fluxes are calculated on this grid and then remapped back to the native grid, where fluxes are applied. The algorithm is implemented in an ocean model and tested in idealized and realistic settings. Horizontal diffusion can account for up to 10% of the total northward heat transport in the Southern Ocean and Western boundary current regions of the Northern Hemisphere. It also reduces the vertical stratification of the upper ocean, which results in an overall deepening of the surface boundary layer depth. Finally, enabling horizontal diffusion leads to meaningful reductions in the near‐surface global bias of potential temperature and salinity.

Mechanical properties and shape-control abilities of a cable dome under asymmetrical loads

This paper focuses on an adaptive cable dome and investigates its mechanical properties and shape control abilities under asymmetrical loads. The analysis includes studying its force and displacement responses as well as the feasibility of shape control under these conditions. All the units of the structure have been specially designed to allow for length adjustment. First, an intelligent algorithm is used to determine the optimal adjustment amounts of the established structural shape control mathematical model. Then the mechanical properties and shape active control performance of the structure are studied through tests and simulations. Results show that, the actual internal forces of the test model show small discrepancies with the theoretical values, with a maximum error of only 3.52%, after assembly and tensioning. The changes in internal forces and displacements of the test model under 1/4, 1/2, and 3/4 span asymmetric loads are obtained and found to be in good agreement with the theoretical values, which verifies the design and construction of the test model are reasonable. Moreover, under asymmetric loads, the test model can approximately restore the upper free joints' vertical coordinates to their initial state through active control, and the average decrease in internal forces of the units is at a minimum of 14.56%. These results demonstrate the shape control feasibility of adaptive cable dome structure under asymmetric loads and validate the accuracy of the simulated results.

Horticulture producer's willingness to participate in contract-based supply chain coordination: A case study from Queensland (Australia).

Product export provides an option for horticulture producers in Queensland (Australia) to diversify their market and increase returns from production. Vertical supply chain coordination in the form of contract-based marketing agreements between producers and export agents/distributors could facilitate increased export. The aim of this study was to investigate the willingness of horticulture producers to participate in export focused contract-based marketing agreements. To achieve this aim, a survey including a discrete choice experiment was conducted. The results from a mixed logit model and a latent class model suggest that there are three clusters of producers: a) the export interested, b) the likely risk averse, and c) those well established in the domestic markets. Only producers in group a) expressed a preference for contract-based export marketing agreements. These producers appear to be younger, already have some export experience, and have a relatively high level of collaboration in their product supply chains. Producers in groups b) and c) expressed an interest in stronger coordination within the domestic retail sector, potentially in the form of contract farming. Prices of produce and potential higher production costs are determinants identified by all producer groups as important for their decision-making about changes to their supply chain.

On the superposition of solutions of the (3 + 1) dimensional Charney–Obukhov equation for the ocean

New exact solutions describing Rossby waves and vortices in ocean propagating along the zonal direction at a constant velocity are found for the (3 + 1)-dimensional nonlinear Charney–Obukhov equation. These solutions are a partial superposition of previously discovered exact solutions of the Charney–Obukhov equation. Partial superposition is found for that part of the solution of the Charney–Obukhov equation, which complements the zonal stream. The presence of such a superposition in the solutions of a nonlinear equation with two nonlinear boundary conditions is a remarkable property of the Charney–Obukhov equation for the ocean and allows one to simulate a wide class of fluid flows based on exact solutions. As an example, we discuss solutions that include superposition of trigonometric functions and the functions of Bessel in a horizontal plane, and superposition of spherically symmetrical solutions in the vertical coordinate. Visualization of the solutions found shows that, depending on the values of the parameters included in the solutions, they can describe both a flow with a large number of vortices and a periodic structure with alternating high and low pressure fronts.