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An alternative hybrid sigma-pressure terrain-following coordinate is presented here that provides smoother coordinate surfaces over terrain by allowing a more rapid decay of the influence of smaller-scale topographic structures with height. This is accomplished by first defining a reference surface pressure that includes the influence of the underlying topography. A smoothed version of this reference surface pressure is then created that represents the larger-scale features of the topography, while the deviations from the smoothed profile contain the smaller-scale terrain structures. In the hybrid-sigma coordinate formulation presented here, the influences of these deviations in the reference surface pressure from their smoothed values are removed more rapidly with increasing height, thereby producing smoother coordinate surfaces. Testing this approach using several idealized simulations demonstrates a significant reduction in the artificial circulations compared to those arising with the basic sigma or the conventional hybrid sigma coordinate, confirming the beneficial aspects of the smoothed hybrid coordinate surfaces. The smoothed hybrid sigma-pressure coordinate proposed here provides flexibility in reducing the influence of the terrain on the coordinate surfaces and can be easily substituted for the basic hybrid sigma-pressure coordinate. Significance Statement Terrain-following vertical coordinates are widely employed in atmospheric numerical models because of their advantages in implementing boundary conditions at the surface. However, it is well known that these coordinates can promote artificial circulations due to numerical errors in computing horizontal pressure gradients and advection along the transformed coordinate surfaces. The purpose of this study is to improve the performance of a pressure-based terrain-following (sigma) hybrid vertical coordinate by selectively removing the influences of smaller-scale terrain features, which results in smoother coordinate surfaces. We demonstrate that this enhancement is beneficial in reducing numerical errors that are inherent in computing with terrain-following vertical coordinates.

A modified hybrid terrain-following vertical coordinate has recently been implemented within the Global Environmental Multiscale atmospheric model that introduces separately controlled height-dependent progressive decaying of the small- and large-scale orography contributions on the vertical coordinate surfaces. The new vertical coordinate allows for a faster decay of the finescale orography imprints on the coordinate surfaces with increasing height while relaxing the compression of the lowest model levels over complex terrain. A number of tests carried out—including experiments involving Environment and Climate Change Canada’s operational regional and global deterministic prediction systems—demonstrate that the new vertical coordinate effectively eliminates terrain-induced spurious generation and amplification of upper-air vertical motion and kinetic energy without increasing the computational cost. Results also show potential improvements in precipitation over complex terrain.

The formation of low stratus cloud over idealized hills is investigated using numerical model simulations. The main driver for the cloud formation is radiative cooling due to outgoing longwave radiation. Despite a purely horizontal flow, the advection terms in the prognostic equations for heat and moisture produce vertical mixing across the upper cloud edge, leading to a loss of cloud water content. This behavior is depicted via a budget analysis. More precisely, this spurious mixing is caused by the diffusive error of the advection scheme in regions where the sloping surfaces of the terrain-following vertical coordinate intersect the cloud top. This study shows that the intensity of the (spurious) numerical diffusion depends strongly on the horizontal resolution, the order of the advection schemes, and the choice of scalar advection scheme. A large-eddy simulation with 4-m horizontal resolution serves as a reference. For horizontal resolutions of a few hundred meters and simulations carried out with a model setup as used in numerical weather prediction, a strong reduction of the simulated liquid-water path is observed. In order to keep the (spurious) numerical diffusion at coarser resolutions small, at least a fifth-order advection scheme should be used. In the present case, a weighted essentially nonoscillatory scalar advection scheme turns out to increase the numerical diffusion along a sharp cloud edge compared with an upwind scheme. Furthermore, the choice of vertical coordinate has a strong impact on the simulated liquid-water path over orography. With a modified definition of the sigma coordinate, it is possible to produce cloud water where the classical sigma coordinate does not allow any cloud formation., Quarterly Journal of the Royal Meteorological Society, 148 (749), ISSN:0035-9009, ISSN:1477-870X

This article presents the results of research concerning the determination of an error of the vertical position of an aircraft during a flight, taking into account, the flight altitude and the values of the ZTD tropospheric product. In particular, this work focuses on the impact of an error in designating the ZTD tropospheric product on the correct determination of a flight altitude. The authors of this work performed a scientific experiment for research data of two flights made by a Cessna 172 around the aerodrome in Dęblin. This article shows the findings of the tropospheric ZTD product specified for the single-frequency SPP positioning method and dual-frequency PPP positioning method. Based on the obtained results, it was discovered that the error of the ZTD tropospheric delay causes an altitude error from 0.08 m at the zenith angle equal to 80o even to 0.79 m at the zenith angle equal to 85o.

The HTF-coordinate is a feature that has been imported into the Weather Research and Forecasting (WRF) model version 4, and it is important to understand how it can be used efficiently. Indicators are needed to describe the characteristics of the distribution of the vertical coordinate and to evaluate the rationality of the configuration of the HTF-coordinate. Such indicators, including the maximum factor of compression (MFC), eta corresponding to the MFC, and the critical point of compression (CPC), were defined and calculated in this study. The indicators were also validated to efficiently and accurately describe the characteristics of the distribution of the vertical coordinate of the WRF model. They constituted the integrated system to resolve the problem of the configuration of the HTF-coordinate of WRF. The nonmonotonicity, compression, and stretching that were caused by the nonlinearity of the HTF-coordinate were also illustrated and validated.

A new hybrid, sigma-pressure vertical coordinate was recently added to the Weather Research and Forecasting (WRF) Model in an effort to reduce numerical noise in the model equations near complex terrain. Testing of this hybrid, terrain-following coordinate was undertaken in the WRF-based Rapid Refresh (RAP) and High-Resolution Rapid Refresh (HRRR) models to assess impacts on retrospective and real-time simulations. Initial cold-start simulations indicated that the majority of differences between the hybrid and traditional sigma coordinate were confined to regions downstream of mountainous terrain and focused in the upper levels. Week-long retrospective simulations generally resulted in small improvements for the RAP, and a neutral impact in the HRRR when the hybrid coordinate was used. However, one possibility is that the inclusion of data assimilation in the experiments may have minimized differences between the vertical coordinates. Finally, analysis of turbulence forecasts with the new hybrid coordinate indicate a significant reduction in spurious vertical motion over the full length of the Rocky Mountains. Overall, the results indicate a potential to improve forecast metrics through implementation of the hybrid coordinate, particularly at upper levels, and downstream of complex terrain.

The correct simulation of fog and low stratus (FLS) is a difficult task for numerical weather prediction (NWP) models. The Swiss Plateau experiences many days with FLS in winter. Most NWP models employ terrain-following vertical coordinates. As a consequence, the typically flat cloud top is intersected by sloping coordinate surfaces above hilly terrain such as the Swiss Plateau. Horizontal advection across the sloping coordinate surfaces leads to spurious numerical diffusion which promotes erroneous FLS dissipation. To address this problem, we propose a new vertical coordinate formulation which features a local smoothing of the model levels. We demonstrate the positive impact of the new vertical coordinate formulation on a case study in detail and for a full month using the COSMO model. The improved vertical coordinate formulation is not yet sufficient to obtain perfect FLS forecasts, it is however a crucial aspect to consider on the way thereto., Journal of Advances in Modeling Earth Systems, 13 (8), ISSN:1942-2466

Active attacking in physical-layer security has not been significantly studied while potentially causing serious consequences for the legitimate networks. In this paper, we propose a novel method to estimate and remove the jamming signals from multiple multi-antenna jammers in a two-way relay network with multi-antenna legitimate and relay nodes. We carefully consider the signals in the time slots in order to exploit the repetition of the signals and design the transmitted signals which can work in different cases. The numerical results show that the secrecy maximum achievable sum-rate (MASR) at the legitimate nodes is higher than that of the conventional method when considering the affect of transmit SNR; the number antennas at the legitimate and relay nodes; normalized distance between one legitimate node and the relay; and the vertical coordinate of the relay.

Although a terrain-following vertical coordinate is well suited for the application of surface boundary conditions, it is well known that the influences of the terrain on the coordinate surfaces can contribute to increase numerical errors, particularly over steep topography. To reduce these errors, a hybrid sigma–pressure coordinate is formulated in the Weather Research and Forecasting (WRF) Model, and its effects are illustrated for both an idealized test case and a real-data forecast for upper-level turbulence. The idealized test case confirms that with the basic sigma coordinate, significant upper-level disturbances can be produced due to numerical errors that arise as the advection of strong horizontal flow is computed along coordinate surfaces that are perturbed by smaller-scale terrain influences. With the hybrid coordinate, this artificial noise is largely eliminated as the mid- and upper-level coordinate surfaces correspond much more closely to constant pressure surfaces. In real-data simulations for upper-level turbulence forecasting, the WRF Model using the basic sigma coordinate tends to overpredict the strength of upper-air turbulence over mountainous regions because of numerical errors arising as a strong upper-level jet is advected along irregular coordinate surfaces. With the hybrid coordinate, these errors are reduced, resulting in an improved forecast of upper-level turbulence. Analysis of kinetic energy spectra for these simulations confirms that artificial amplitudes in the smaller scales at upper levels that arise with the basic sigma coordinate are effectively removed when the hybrid coordinate is used.

Fog and low stratus pose a major challenge for numerical weather prediction (NWP) models. Despite high resolution in the horizontal (~1 km) and vertical (~20 m), operational NWP models often fail to accurately predict fog and low stratus. This is a major issue at airports which require visibility predictions, or for energy agencies estimating day-ahead input into the electrical grid from photovoltaic power.Most studies dedicated to fog and low stratus forecasts have focused on the physical parameterisations or grid resolutions. We illustrate how horizontal advection at the cloud top of fog and low stratus in a grid with sloping vertical coordinates leads to spurious numerical diffusion and subsequent erroneous dissipation of the clouds. This cannot be prevented by employing a higher-order advection scheme. After all, the formulation of the terrain-following vertical coordinate plays a crucial role in regions which do not exhibit perfectly flat orography. We suggest a new vertical coordinate formulation which allows for a faster decay of the orographic signal with altitude and present its positive impact on fog and low stratus forecasts. Our experiments indicate that smoothing of the vertical coordinates at low altitudes is a crucial measure to prevent premature dissipation of fog and low stratus in high-resolution NWP models.

The coordinate and resolution dependence of three second moment turbulent closure models are studied using one-dimensional NCOM experiments and large eddy simulations at Ocean Station Papa. Our res...

The results of MultiGNSS experimental research in precise positioning mode with TerraStar corrections are presented. The potential possibilities to ensure (provide) safe landing operation of Unmanned Aerial Systems of all categories, including CAT III meteorological minimum, are demonstrated. The errors of vertical coordinate determination are proposed to be estimated in two vertical planes (East Up and North Up correspondently). Experimental data approve accuracy of coordinates determination to be 10 cm (99%) for horizontal plane and 18 cm (99%) for vertical planes.

The SL-AV atmospheric model version using hybrid vertical coordinates combies the advantages of sigma and isobaric coordinates. The formulation and discretization of model equations maintain the equivalency of the new model version to the basic sigma version in the special case, when hybrid coordinates coincide with sigma coordinates. The SL-AV model version with hybrid vertical coordinate is verified with medium-range weather forecasts. The decrease in the errors of predicted geopotential height and wind as compared to the sigma model version is demonstrated. The use of hybrid coordinates also leads to a certain increase in forecast skill scores for some meteorological parameters characterizing aviation significant weather.

Thin space constructions on curved plans there are used at building of public structures, trade centers. Sport constructions. Working-out the methods of forming of the surfaces on the curved plans that is one of the modern task of the architecture and town building. The orthogonal coordinate system that is formed at the plane with some plane directrix curve and the system of the right lines orthogonal to the directrix curve there is regarded at the stat. The coordinate system forms some trapezium-curved segment. Taking some function of the vertical coordinate there is possible to receive different surfaces at the curved plans. Conjugating different directrix curves, it’s possible to receive the combined surfaces. The article presents a system of orthogonal coordinates of curvilinear-trapezoidal planes and methods for forming surfaces on these planes. Surfaces with a function of the vertical coordinate of a general view are considered, and surfaces on combined plans of segments of the same type are shown.

An improved model is developed to analyze the horizontal dynamic response of piles in saturated soil. The proposed model retains the simplicity of the plane strain model while accounting for 3-D effects. The vertical coordinate is suppressed by integrating the governing equations over the length of the pile. The three-dimensional model is reduced to a simpler model that is similar to the plane strain model. Explicit closed-form solutions are obtained by introducing potential functions. The proposed model is verified by comparison with more rigorous solutions. Finally, the influence of pile slenderness, permeability of soil, and other parameters on dynamic soil reaction and dynamic response of pile is investigated. It is revealed that the proposed model avoids the drawbacks of the earlier model and is effective in predicting the horizontal dynamic response of the pile.

A new dynamical core of Environment and Climate Change Canada’s Global Environmental Multiscale (GEM) atmospheric model is presented. Unlike the existing log-hydrostatic-pressure-type terrain-following vertical coordinate, the proposed core adopts a height-based approach. The move to a height-based vertical coordinate is motivated by its potential for improving model stability over steep terrain, which is expected to become more prevalent with the increasing demand for very high-resolution forecasting systems. A dynamical core with height-based vertical coordinate generally requires an iterative solution approach. In addition to a three-dimensional iterative solver, a simplified approach has been devised allowing the use of a direct solver for the new dynamical core that separates a three-dimensional elliptic boundary value problem into a set of two-dimensional independent Helmholtz problems. The issue of dynamics–physics coupling has also been studied, and incorporating the physics tendencies within the discretized dynamical equations is found to be the most acceptable approach for the height-based vertical coordinate. The new dynamical core is evaluated using numerical experiments that include two-dimensional nonhydrostatic theoretical cases as well as 25-km resolution global forecasts. For a wide range of horizontal grid resolutions—from a few meters to up to 25 km—the results from the direct solution approach are found to be equivalent to the iterative approach for the new dynamical core. Furthermore, results from the different numerical experiments confirm that the new height-based dynamical core is equivalent to the existing pressure-based core in terms of solution accuracy.

We present the results of the verification of the regional climate model (RCM) RegCM 4.5 for the territory of Southern Russia using NCEP/NCAR Reanalysis and GHCN + CAMS data. In this paper we aimed to explore the complexity of choosing of a computational domain, numerical grid parameters and their impact on the climate model prediction for models with hydrostatic and non-hydrostatic cores for the vertical coordinate. We upgraded the RegCM base code with a new module that allows splitting a territory into an arbitrary number of zones, each of which is characterized by its own set of physical parameters. In order to study the sensitivity of the simulations results we performed \(>100\) simulations varying the climatic conditions with terrain topography, surface roughness, soil, and vegetation properties, and the numerical grids. The features of numerical regional climate algorithm were discussed in case of the shared research facilities of high-performance computing at Lomonosov Moscow State University.

In this paper we consider the discrete Allen-Cahn equation posed on a two-dimensional rectangular lattice. We analyze the large-time behaviour of solutions that start as bounded perturbations to the well-known planar front solution that travels in the horizontal direction. In particular, we construct an asymptotic phase function \begin{document}$ \gamma_j(t) $\end{document} and show that for each vertical coordinate \begin{document}$ j $\end{document} the corresponding horizontal slice of the solution converges to the planar front shifted by \begin{document}$ \gamma_j(t) $\end{document} . We exploit the comparison principle to show that the evolution of these phase variables can be approximated by an appropriate discretization of the mean curvature flow with a direction-dependent drift term. This generalizes the results obtained in [ 47 ] for the spatially continuous setting. Finally, we prove that the horizontal planar wave is nonlinearly stable with respect to perturbations that are asymptotically periodic in the vertical direction.

To model free-surface flows, a three-dimensional model is developed using a new approach where the horizontal velocity components are projected onto a basis of functions depending on the vertical coordinate z. The choice of this basis and the number of its terms depend on the physical problem and the desired precision. The vertical velocity component and the non-hydrostatic pressure are analytically determined. The integration of the variational form of main flow equations with an adequate choice of test functions transforms the three-dimensional model in a two-dimensional model. The finite element method using a two-dimensional mesh is used to solve the problem. This model is stable and robust because it does not use a mesh in the vertical direction. It is also accurate as the functions that best fit the physical nature of the flow can be directly introduced. The application of the proposed model to flows in curved channels is given.

In the non-linear in-plane elastic buckling and postbuckling analyses of shallow parabolic arches, to overcome the difficulty in deriving an accurate expression for the non-linear normal strain, an approximation assumption that the derivative of the vertical coordinate with respect to the horizontal coordinate satisfies ( d y / d z ) 1 ≪ 1 has been adopted in many investigations. The merit of the assumption is that it leads to the same differential equations of equilibrium and the same solutions as those for shallow circular arches. However, the accuracy of the assumption and the limitation of the analytical solutions have not been examined and because of the approximation, the analytical solutions may lead to significant errors for the buckling loads of shallow parabolic arches in some cases. This paper investigates the effects of the approximation assumption on the accuracy of in-plane buckling and postbuckling analyses of pin-ended and fixed shallow parabolic arches by comparing the analytical solutions with their finite element counterparts. It is found that the analytical solutions based on the assumption have some limitations because the assumption holds approximately only for extremely shallow parabolic arches, but is not valid for most shallow parabolic arches. The analytical solutions for the buckling loads based on the assumption are larger than the corresponding finite element results for parabolic arches with a rise-to-span ratio greater than 0.08, the error of the analytical solution increases with an increase of the rise-to-span ratio of the arch, and the sources for the errors are identified and discussed. Hence, caution should be exercised when using the analytical solutions to predict the buckling load of shallow parabolic arches, particularly of those with a rise-to-span ratio greater than 0.08.

So far, we have only used the Cartesian geopotential coordinate system consisting of three orthogonal spatial coordinates x, y, z. Relaxing the orthogonality between the vertical coordinate z and the two horizontal coordinates x, y can make it much easier to analyze phenomena in atmospheres and oceans, and devise compelling models of them. The development of such non-orthogonal coordinate systems remains at the forefront of research in numerical modeling. The purpose of this chapter is, therefore, to present the salient issues relating to these generalized vertical coordinates.

We characterize those mappings from a compact subset of $\mathbb{R}$ into the Heisenberg group $\mathbb{H}^{n}$ which can be extended to a $C^{m}$ horizontal curve in $\mathbb{H}^{n}$. The characterization combines the classical Whitney conditions with an estimate comparing changes in the vertical coordinate with those predicted by the Taylor series of the horizontal coordinates., 22 pages

This study considered zenith total delay (ZTD) estimation from precise point positioning (PPP) based on GPS only (PPP GPS), GLONASS only (PPP GLO), and GPS+GLONASS (PPP GPS+GLO) using both a conventional strategy when applying a model for the hydrostatic component with an estimation of the wet component and an alternative strategy. The proposed alternative strategy is to estimate both the hydrostatic and the wet components of the tropospheric delay using different process noises with different mapping functions for both components in an extended Kalman filter (EKF). It was found that the receiver clock offsets and the estimated ambiguities would absorb some errors in the ZTD when using the conventional strategy. The RMS values of the differences between the double differenced (DD) GPS ZTD and the PPP ZTD, using the alternative strategy, were 6.5, 7.3, and 6.7 mm for PPP GPS, PPP GLO, and PPP GPS+GLO, respectively. The results were validated over one continuous week and then over one year. Validation was also performed through comparison with the IGS ZTD values, for 12 weeks, with an overall RMS of 5.9 mm and against IGS real-time products with an overall RMS of 8.1 mm. Furthermore, the alternative strategy also provided significant improvements in the 5 cm convergence time in the vertical coordinate component of the float ambiguity solutions to be on average, 51, 36 and 27 minutes for PPP GPS, PPP GLO and PPP GPS+GLO solutions respectively.

Two-dimensional steady-state flows of a viscous incompressible fluid over a fixed flat boundary, generated by a periodic near-wall volume force, are studied. The corresponding boundary-value problems for the Navier - Stokes equations are calculated using the expansion of the required solutions in a Fourier series in the longitudinal coordinate and a second-order finite-difference approximation in the vertical coordinate. The dependence of the far flow field on the force parameters is investigated.

The wave speed of acoustic emission in the medium is uncertain, which is influenced by the source characteristics and the relative position between the acoustic emission source and the sensors. Due to this difference, the results of TDOA location method determination of wave speed in advance are very discrete. As to liner location ,the more farther the distance between two acoustic emission source sensor are, the more serious the discrete error are. Any of the two sensors, a location line can be obtained by setting the wave speed as a horizontal coordinate and the location as the vertical coordinate. The horizontal coordinate of location line of the different sensors is the real wave speed of acoustic emission events. This method has lower computational complexity, which can overcome the influence on acoustic emission location which wave speed setting error brings, having some practical value in Engineering.

Past investigations have documented large divergent wind anomalies in stratospheric reanalyses over steep terrain, which were attributed to discretization errors produced by the terrain-following (sigma) vertical coordinate in the forecast model. However, forecasting experiments have reported negligible differences in skill between sigma- and hybrid-coordinate models. This leads to the paradoxical conclusion that discretization errors in the forecast model yield significant stratospheric analysis errors, but insignificant stratospheric forecast errors. The authors reexamine this issue by performing two forecast-assimilation experiments that are identical except for the vertical coordinate: one uses a sigma coordinate and the other uses a hybrid coordinate. The sigma-coordinate analyses exhibit large divergent wind anomalies over terrain that extend from the surface to the model top and distort explicitly resolved orographic gravity waves. Above the tropopause, divergent wind errors are suppressed by an order of magnitude or more in the hybrid-coordinate analyses. Over a 3-month period, stratospheric skill scores in the hybrid experiment show statistically significant improvements relative to the sigma experiment. Previous studies, which found no such differences, all used forecasts initialized from a common archived analysis. The results show that the dominant pathway for error growth and net skill impacts is via 0–9-h forecast backgrounds cycling successively through the data assimilation phase without significant observational correction. The skill impacts noted here should further motivate weather and climate models to adopt a hybrid coordinate with the best error suppression characteristics for a given modeling application.