Height Coordinates Research Articles

Inclusion of Surface Topography into the Vector Vorticity Equation Model (VVM)

[1] Surface topography is implemented into the vector vorticity equation model with a block representation of mountains in the height coordinate. The kinematic boundary conditions at the surface are satisfied with a proper computational boundary condition for vorticity. This approach has been tested in various ways, including idealized 2D mountain waves, the much-studied “Boulder Downslope Windstorm” case, and 3D orographic precipitation over a ridge with and without small-scale irregularities. The model performs reasonably well in all of these cases, with no obvious computational difficulties due to the use of coordinate surfaces intersecting the surface and is ready to be used for arbitrarily prescribed surface topography without any artificial smoothing.

Peeling stress analysis for an inhomogeneous high- T c superconductor with a discontinuous interface at the substrate

This paper presents an analysis of a superconductor–substrate system to calculate the peeling stress of a high temperature superconductor (HTS) when the temperature decreases from ambient to operating conditions (cryogenic temperatures). Firstly, the values for the material properties of the inhomogeneous high temperature superconductor (HTS) were obtained by fitting a second order polynomial to the experimental data. It is assumed that the material properties of the inhomogeneous HTS vary with varying height coordinate and temperature. Then, through the proposed graded finite element method, the coupled thermo-mechanical equations were solved numerically. The numerical results show that the thermal stress generated in the inhomogeneous HTS is larger on a SiTiO 3 substrate than on a MgO substrate. The maximum thermal stresses, i.e., the peeling stresses, occur near the bottom corner of the inhomogeneous HTS and may induce fracture behavior at the bi-material interface. The inhomogeneous HTS cools at a slower pace than the homogeneous HTS from the room temperature to the operating temperature. It is also shown that the magnitude of the peeling stress for a homogeneous HTS is larger than that for an inhomogeneous HTS. It is intended that the model presented here be useful to researchers who are interested in the mechanical properties of an inhomogeneous HTS.

Graded finite element simulation of thermal stress in inhomogeneous high- Tc superconductor

YBa 2Cu 3O y is an orthotropic material with different material properties in a, b and c directions, such as Young’s modulus, coefficient of thermal expansion (CTE), and thermal conductivity. It is assumed that the material properties of inhomogeneous high temperature superconductor (HTS) vary with different height coordinate and temperature. A model is presented in this paper to calculate the thermal stress of inhomogeneous HTS when temperature decreases from ambient to operating conditions (cryogenic temperatures). By fitting a second order polynomial to the experimental data, value of the material properties of inhomogeneous HTS can be obtained. Then, through the proposed graded finite element method, the coupled thermo-mechanical equations were solved numerically. The numerical results show that the temperature profiles distribute the function of time after soaking. It is notable that the temperature profile reaches steady in a very short period of time, so the thermal stress suddenly increases to a very high level for a bulk superconductor. It is also shown that the closer to the sample internal region it is, the larger the heat fluxes are. Besides, the maximum tensile stresses, i.e. the peeling stresses, occur near bottom corner of inhomogeneous HTS. It is intended that the model presented in this paper could be useful to researchers who are interested in mechanical properties of inhomogeneous HTS.

Surface bound adsorption in a microfluidic T-sensor: Numerical comparison and optimization of 2D and 3D models

We compare models in two and three spatial dimensions for describing the interplay of convection, diffusion and adsorption in a microfluidic T-sensor. A distinct section of the channel wall is functionalized with surface bound receptors in order to detect specific molecules from the buffer solution. The geometric shape of this section shall later be optimized with respect to a favorable sensor signal over wide ranges of solute concentrations. However, model-based approaches for solving this inverse problem in 3D would require tremendous computational effort. We have hence simplified the model by eliminating the channel height coordinate. In the 2D model, vertical diffusion of solute molecules is accounted for by a stagnant film over the sensor surface. This contribution addresses how to adjust the flow profile, film mass transfer resistance, and thickness of the stagnant film in the 2D model in order to best approximate the sensor load of the 3D model.

Reexamining the Vertical Structure of Tangential Winds in Tropical Cyclones: Observations and Theory

Abstract A few commonly held beliefs regarding the vertical structure of tropical cyclones drawn from prior studies, both observational and theoretical, are examined in this study. One of these beliefs is that the outward slope of the radius of maximum winds (RMW) is a function of the size of the RMW. Another belief is that the outward slope of the RMW is also a function of the intensity of the storm. Specifically, Shea and Gray found that the RMW becomes increasingly vertical with increasing intensity and decreasing radius. The third belief evaluated here is that the RMW is a surface of constant absolute angular momentum M. These three conventional wisdoms of vertical structure are revisited with a dataset of three-dimensional Doppler wind analyses, comprising seven hurricanes on 17 different days. Azimuthal mean tangential winds are calculated for each storm, and the slopes of the RMW and M surfaces are objectively determined. The outward slope of the RMW is shown to increase with radius, which supports prior studies. In contrast to prior results, no relationship is found between the slope of the RMW and intensity. It is shown that the RMW is indeed closely approximated by an M surface for the majority of storms. However, there is a small but systematic tendency for M to decrease upward along the RMW. Utilizing Emanuel’s analytical hurricane model, a new equation is derived for the slope of the RMW in radius–pressure space. This predicts a linear increase of slope with radius and essentially no dependence of slope on intensity. An exactly analogous equation can be derived in log-pressure height coordinates, and a numerical solution yields the same conclusions in geometric height coordinates. These conclusions are further supported by the results of simulations utilizing Emanuel’s simple, time-dependent, axisymmetric hurricane model. As both the model and the analytical theory are governed by the dual constraints of thermal wind balance and slantwise moist neutrality, it is demonstrated that it is these two assumptions that require the slope of the RMW to be a function of its size but not of the intensity of the storm. Finally, it is shown that within the context of Emanuel’s theory, the RMW must very closely approximate an M surface through most of the depth of the vortex.

Diabatically Induced Secondary Flows in Tropical Cyclones. Part I: Quasi-Steady Forcing

Abstract The Sawyer–Eliassen Equation (SEQ) is here rederived in height coordinates such that the sea surface is also a coordinate surface. Compared with the conventional derivation in mass field coordinates, this formulation adds some complexity, but arguably less than is inherent in terrain-following coordinates or interpolation to the lower physical boundary. Spatial variations of static stability change the vertical structure of the mass flow streamfunction. This effect leads to significant changes in both secondary-circulation structure and intensification of the primary circulation. The SEQ is solved on a piecewise continuous, balanced mean vortex where the shapes of the wind profiles inside and outside the eye and the tilt of the specified heat source can be adjusted independently. A series of sensitivity studies shows that the efficiency with which imposed heating intensifies the vortex is most sensitive to intensity itself as measured by maximum wind and to vortex size as measured by radius of maximum wind. Vortex shape and forcing tilt have impacts 20%–25% as great as intensity and size, suggesting that the aspects of tropical cyclones that predispose them to rapid intensification are environmental or thermodynamic rather than kinematic.

Comment on “Complete Eulerian-mean tracer equation for coarse resolution OGCMs” by M. S. Dubovikov and V. M. Canuto

Dubovikov and Canuto (Dubovikov, M.S. and Canuto, V.M., Complete Eulerian-mean tracer equation for coarse resolution OGCMs. Geophys. Astrophys. Fluid Dyn., 2006, 100, 197–214), after averaging the tracer conservation equation in density coordinates and transforming to height coordinates, concluded that present ocean models are missing important terms in their mean tracer equations. Here we point out some errors made by Dubovikov and Canuto (2006) in their isopycnal averaging procedure. We draw on the temporal-residual-mean (TRM) theory and show that when the isopycnal averaging and coordinate transformation are performed correctly, the tracer equations of present ocean circulation models are recovered. We therefore conclude that present ocean circulation models are not neglecting the leading order terms identified by Dubovikov and Canuto (2006).

Numerical Modelling of Heat Generation and Distribution in the Core and Winding of Power Transformers

The power transformer is a complex and critical component of the power transmission and distribution system. System abnormalities, loading, switching and ambient condition normally contribute to accelerated aging and sudden failure. In the absence of critical components monitoring, the failure risk is always high. For early fault detection and real time condition assessment, an online monitoring system in accordance with the age and conditions of the asset would be an important tool. Power loss, heat generation and heat distribution evaluations in a large-scale oil immersed power transformer are presented here, along with the details of computer implementation and experimental verification.Core power losses are approximately constant with temperature variation or may decrease with that. Over the temperature range of 20 to 100°C the change in hysteresis loss Ph with temperature was negligible. Since the total core loss PT decreased with increasing temperature over this range, almost all the loss reduction was due to a reduction in the eddy current loss component Pe that was inversely proportional to the resistivity. Winding and oil temperature will increase with the load increasing and may create a hot spot. This is caused by degradation insulation and the loss of life in the power transformer.Hottest spot temperature and temperature profiles in radial and height coordinates were found using three different methods in this paper. The finite element method (FEM), finite difference method (FDM) and discrete furrier transform methods (DFT) are used to analyze algorithms in this paper. Computational results based on theoretical considerations and using the DFT method are shown to be in good agreement with FDM and FEM. Two mathematical formulae are proposed for temperature distribution in both radial and horizontal axes of core and windings. COMSOL for FEM, GEMINI for FDM and MATLAB for DFT are used.

Conservative Split-Explicit Time Integration Methods for the Compressible Nonhydrostatic Equations

Abstract Historically, time-split schemes for numerically integrating the nonhydrostatic compressible equations of motion have not formally conserved mass and other first-order flux quantities. In this paper, split-explicit integration techniques are developed that numerically conserve these properties by integrating prognostic equations for conserved quantities represented in flux form. These procedures are presented for both terrain-following height and hydrostatic pressure (mass) vertical coordinates, two potentially attractive frameworks for which the equation sets and integration techniques differ significantly. For each set of equations, the linear dispersion equation for acoustic/gravity waves is derived and analyzed to determine which terms must be solved in the small (acoustic) time steps and how these terms are represented in the time integration to achieve stability. Efficient techniques for including numerical filters for acoustic and external modes are also presented. Simulations for several idealized test cases in both the height and mass coordinates are presented to demonstrate that these integration techniques appear robust over a wide range of scales, from subcloud to synoptic.

Design considerations in projection phase-shift moiré topography based on theoretical analysis of fringe formation

Moiré topography is a well-established optical technique to measure the shape of three-dimensional surfaces, based on the geometric interference between an optical grid and its image deformed by an object surface. The technique produces fringes that represent contours of equal height, and from the recordings of several phase-shifted topograms surface height coordinates can be calculated. To perform these calculations, it is assumed that object height variation is small in comparison with the measurement setup dimensions, and this approximation leads to systematic errors in measurement accuracy. We present the mathematical description of the fringe formation process in projection moiré topography, and on the basis of these equations we establish the relation between setup geometry and upper limits of the systematic measurement errors. We derive the equations that determine design specifications needed to reduce the effects of approximations to be below the measurement resolution of the setup. It is shown that setup geometry should be adapted to the gray-scale measurement resolution of the imaging system. We show that, using an iterative correction from one fringe order to the next, measurement accuracy can be maintained over the entire object depth.

Influence of helix angle and density variation on the cutting force in wood-based products machining

Abstract After noticing influences of the helix angle on cutting forces and consequently on tool life for wood machining in previous studies [1] , it was decided to deepen the analysis because tool makers need to understand such effects. In this paper, the oblique cutting edge is divided into small differential segments. For each oblique element, cutting forces are formulated to introduce the density variation of wood based materials: cutting constants, obtained by measuring cutting forces for various orthogonal millings on MDF and particleboard, are written in function of height coordinate of the board thickness. To validate the three orthogonal cutting forces calculated from measured constants, a comparison with new experimental values is necessary. The predictions obtained indicate the effects of helix angle on the tool life.

Vertical discretizations giving optimal representation of normal modes: Sensitivity to the form of the pressure‐gradient term

AbstractThe normal‐mode dispersion properties and structures of some vertical discretizations of the compressible Euler equations are re‐examined. It is shown that the dispersion properties can be sensitive to the form in which the pressure‐gradient term is expressed. For a height coordinate and for an isentropic vertical coordinate, discretizations are identified that have optimal dispersion properties and, at the same time, lend themselves to mass conservation by predicting the relevant density variable. Copyright © 2006 Royal Meteorological Society

Kinematic Analysis of the Cervical Spine Following Implantation of an Artificial Cervical Disc

Prospective cohort study. To assess the biomechanical profile of the cervical spine following cervical arthroplasty. Spinal arthroplasty offers the promise of maintaining functional spinal motion, thereby potentially avoiding adjacent segment disease. Disc replacement may become the next gold standard for the treatment of degenerative cervical spine disease, and must be studied rigorously to ensure in vivo efficacy and safety. A total of 20 patients underwent single or 2-level implantation of the Bryan artificial cervical disc (Medtronic Sofamor Danek, Memphis TN) for treatment of cervical degenerative disc disease producing radiculopathy and/or myelopathy. Lateral neutral, flexion, and extension cervical radiographs were obtained before surgery and at intervals up to 24 months after surgery. Kinematic parameters, including sagittal rotation, horizontal translation, change in disc height, and center of rotation (COR), were assessed for each spinal level using quantitative motion analysis software. Motion was preserved in the operated spinal segments (mean range of motion 7.8 degrees) up to 24 months following surgery. The relative contribution of each spinal segment to overall spinal sagittal rotation differed depending on whether the disc was placed at C5-C6 or C6-C7. Overall cervical motion (C2-C7) was moderately but significantly increased during late follow-up. Sagittal rotation, anterior and posterior disc height, translation, and COR coordinates did not change significantly following surgery. The COR was most frequently located posterior and inferior to the center of the disc space. The Bryan artificial cervical disc provided in vivo functional spinal motion at the operated level, reproducing the preoperative kinematics of the spondylotic disc.

Model modulation to add smaller scale structures to large scale electron density models

Abstract. Usually regional and global electron density models provide large scale spatial structures only and smooth out the smaller scale features of the electron density distribution. We present a method to modulate existing electron density models by multiplication: M(h, φ, λ, t) = L(h, φ, λ, t) × S1(h, φ, λ, t) × S2(h, φ, λ, t) × ... Sn(h, φ, λ, t) M: resulting electron density distribution, L: large scale model, S1...Sn: modulating models for n the smaller scale structures; h: height; φ, λ: geographic coordinates, t: Universal Time. There are no restrictions to the nature of the large scale model provided it takes height and horizontal coordinates as input. Examples are models of the "profiler" type which use large scale "maps" for profile anchor points (e.g., E, F1, F2 peak properties) like the International Reference Ionosphere (IRI). Typical examples for smaller scale structures are ridges, troughs and wavelike disturbances. The advantage of modulation by multiplication is that there is no danger to get zero or negative values of electron density as long as the background and modulations are >0 everywhere. For each modulation model, unity means "undisturbed".

The Annual Cycle of the Axial Angular Momentum of the Atmosphere

Abstract Earlier analyses of the annual cycle of the axial angular momentum (AAM) are extended to include mass flows and vertical transports as observed, and to establish angular momentum budgets for various control volumes, using the European Centre for Medium-Range Forecasts (ECMWF) Re-Analyses (ERA) for the years 1979–92, transformed to height coordinates. In particular, the role of the torques is examined. The annual cycle of the zonally averaged angular momentum is large in the latitude belt 20° ⩽ |ϕ| ⩽ 45°, with little attenuation in the vertical up to a height of ∼12 km. The oscillation of the mass term (AAM due to the earth’s rotation) dominates in the lower troposphere, but that of the wind term (relative AAM) is more important elsewhere. The cycle of the friction torque as related to the trade winds prevails in the Tropics. Mountain torque and friction torque are equally important in the extratropical latitudes of the Northern Hemisphere. The annual and the semiannual cycle of the global angular momentum are in good balance with the global mountain and friction torques. The addition of the global gravity wave torque destroys this agreement. The transports must be adjusted if budgets of domains of less than global extent are to be considered. Both a streamfunction, representing the nondivergent part of the fluxes, and a flux potential, describing the divergences/convergences, are determined. The streamfunction pattern mainly reflects the seasonal shift of the Hadley cell. The flux potential links the annual oscillations of the angular momentum with the torques. It is concluded that the interaction of the torques with the angular momentum is restricted to the lower troposphere, in particular, in the Tropics. The range of influence is deeper in the Northern Hemisphere than in the Southern Hemisphere, presumably because of the mountains. The angular momentum cycle in the upper troposphere and stratosphere is not affected by the torques and reflects interhemispheric flux patterns. Budgets for the polar as well as for the midlatitude domains show that fluxes in the stratosphere are important.

An empirical calibration for 4He quantification in minerals and rocks by laser fusion and noble gas mass spectrometry using Cerro de Mercado (Durango, Mexico) fluorapatite as a standard

An empirical calibration with a natural mineral standard (fluorapatite from Cerro de Mercado, Mexico) is proposed as a method to determine the 4He concentration of mineral and rock samples. The procedure is based on the fusion of several aliquots of the fluorapatite standard with a well-spaced weight distribution in order to obtain a good correlation in coordinates of 4He peak height versus fluorapatite weight. The weight is then converted to moles using the accepted mineral age (31.4 Ma) and appropriate formula. Experimental peak height of 4He for the unknown samples are converted to moles with the regression determined for fluorapatite. The procedure is fast and inexpensive, and both precision and accuracy are always below 10% and usually about 3–5%.

Vertical discretizations for compressible Euler equation atmospheric models giving optimal representation of normal modes

Accurate representation of different kinds of wave motion is essential for numerical models of the atmosphere, but is sensitive to details of the discretization. In this paper, numerical dispersion relations are computed for different vertical discretizations of the compressible Euler equations and compared with the analytical dispersion relation. A height coordinate, an isentropic coordinate, and a terrain-following mass-based coordinate are considered, and, for each of these, different choices of prognostic variables and grid staggerings are considered. The discretizations are categorized according to whether their dispersion relations are optimal, are near optimal, have a single zero-frequency computational mode, or are problematic in other ways. Some general understanding of the factors that affect the numerical dispersion properties is obtained: heuristic arguments concerning the normal mode structures, and the amount of averaging and coarse differencing in the finite difference scheme, are shown to be useful guides to which configurations will be optimal; the number of degrees of freedom in the discretization is shown to be an accurate guide to the existence of computational modes; there is only minor sensitivity to whether the equations for thermodynamic variables are discretized in advective form or flux form; and an accurate representation of acoustic modes is found to be a prerequisite for accurate representation of inertia-gravity modes, which, in turn, is found to be a prerequisite for accurate representation of Rossby modes.

How Sensitive Are Coarse General Circulation Models to Fundamental Approximations in the Equations of Motion?

The advent of high-precision gravity missions presents the opportunity to accurately measure variations in the distribution of mass in the ocean. Such a data source will prove valuable in state estimation and constraining general circulation models (GCMs) in general. However, conventional GCMs make the Boussinesq approximations, a consequence of which is that mass is not conserved. By use of the height‐pressure coordinate isomorphism implemented in the Massachusetts Institute of Technology general circulation model (MITGCM), the impact of non-Boussinesq effects can be evaluated. Although implementing a non-Boussinesq model in pressure coordinates is relatively straightforward, making a direct comparison between height and pressure coordinate (i.e., Boussinesq and non-Boussinesq) models is not simple. However, a careful comparison of the height coordinate and the pressure coordinate solutions ensures that only non-Boussinesq effects can be responsible for the observed differences. As a yardstick, these differences are also compared with those between the Boussinesq hydrostatic and models in which the hydrostatic approximation has been relaxed, another approximation commonly made in GCMs. Model errors (differences) caused by the Boussinesq and hydrostatic approximations are demonstrated to be of comparable magnitude. Differences induced by small changes in subgrid-scale parameterizations are at least as large. Therefore, non-Boussinesq and nonhydrostatic effects are most likely negligible with respect to other model uncertainties. However, because there is no additional cost incurred in using a pressure coordinate model, it is argued that non-Boussinesq modeling is preferable simply for tidiness. It is also concluded that even coarse-resolution GCMs can be sensitive to small perturbations in the dynamical equations.

Rescaled height coordinates for accurate representation of free-surface flows in ocean circulation models

Conventional height coordinate models have previously represented free-surface variations by a variable thickness upper layer. We present a rescaled height coordinate, which we call z ∗ , that treats the time-dependent free surface as a coordinate surface. This coordinate is isomorphic with the atmospheric η coordinate, sometimes known as the step-mountain coordinate. The z ∗ coordinate has also been used in a coastal ocean model. However, unlike both these implementations, here we use the finite volume method within the z ∗ coordinate framework, allowing an accurate representation of topography. The resulting scheme provides a very accurate representation of motions over steep topography in a three-dimensional general circulation model, even with fast and large amplitude free-surface variations.

Numerical simulation of tides, Rossby and Kelvin waves with the COMMA-LIM model

A 48-layer version of the COMMA-LIM (Cologne Model of the Middle Atmosphere at Leipzig Institute for Meteorology) three-dimensional global mechanistic model of the Earth's atmosphere from 0 km to 135 km with logarithmic pressure height coordinates was developed. The model is capable of reproducing the global structures and propagation of different planetary waves in the middle atmosphere. The contribution of gravity waves, tides, Rossby and Kelvin waves on the zonally averaged momentum budget of the mesosphere / lower thermosphere region is investigated.