Large Grid Spacing Research Articles

Towards a new hybrid cumulus parametrization scheme for use in non‐hydrostatic weather prediction models

AbstractClassical mass flux parametrization schemes for cumulus convection generally transport heat and moisture only but do not include a net mass transport. This is well justified for large grid spacings comprising the whole convective circulation in the local grid column, such that all convective mass fluxes locally cancel out. A conceptual problem arises for finer grid spacings as used in contemporary numerical weather prediction (NWP) models, when convection becomes partially resolvable. This problem can be overcome by the hybrid approach presented here. Only updraft and downdraft are parametrized with a net mass transport; the environmental subsidence is treated by the grid‐scale equations. The total mass flux in the continuity equation is split into a grid‐scale and a subgrid‐scale contribution. This parametrization scheme is designed for use in any nonlinear, non‐hydrostatic and fully compressible NWP model. We here have chosen the Lokal–Modell (LM) of Deutscher Wetterdienst.Idealized dry mass lifting experiments (without convective heat transport) demonstrate the feasibility of the hybrid approach. Entrainment causes grid‐scale convergence and the detrained air, if set to the environmental temperature, spreads mainly horizontally on the grid. Gravity waves are generated when convection starts and ends. Whereas their amplitude depends on the details of the switching on and off of convection, the stationary state (after about 30 minutes) does not. Four model runs with different grid spacings (3.5 km to 28 km) confirm that the mass exchange between the model grid and the parametrization scheme is independent of the chosen grid spacing. Total mass in a convective circulation cell is conserved to better than 0.1%, but only if the damping layer at the upper boundary of the LM is shifted to above 20 km.For moist convection (with convective heat transport), a simple cloud model for an updraft has been set up. As the detrained air at the cloud top is colder than the environment, it moves down by about 1 km but then mainly spreads horizontally again over several tens of kilometres as in the dry case without convective heat transport.The hybrid mass flux approach with both a grid‐scale and a subgrid‐scale contribution may fill the gap between coarse‐grid models (grid spacing > 50 km) with classical parametrization schemes, and very highly resolved explicit convection modelling (with a grid spacing of the order of 100 m). Copyright © 2007 Royal Meteorological Society

The use of a square mesh codend and sorting grids to reduce catches of young fish and improve sustainability in a multispecies bottom trawl fishery in the MediterraneanEl uso de copo de malla cuadrada y rejillas separadas para reducir las capturas de pece

In order to improve the capacity of bottom trawl fishing gears to reduce catches of young fish and discards in a highly exploited demersal trawl fisheries in the Mediterranean, the size-selection performance of a 36-mm square-mesh codend and two sorting grids with 20 and 15 mm bar spacing was assessed. Alternate hauls were used to assess the selectivity of 36-mm square-mesh codend. Selectivity of sorting grids was assessed using a double codend in which fish that escaped through the grid were captured in the lower codend while other fish were guided into the upper codend. The mean selection length (L50) for European hake was 18.5 cm and that for the Atlantic horse mackerel was 14.0 cm with the 36-mm square-mesh codend . These values are close to their current minimum landing sizes (20 cm for hake and 12 cm for horse mackerel). The sorting grid with 20-mm bar spacing showed L50 value of 13.3 cm for hake, suggesting that a larger grid-spacing would be needed to obtain sufficient sorting performance. Similarly, for Atlantic mackerel the estimated L50 of 14.3 cm indicates that larger grid spacing is required to attain an L50 that would be close to the current MLS (18 cm). For Atlantic horse mackerel and red mullet, the L50 obtained with the sorting grid with 20-mm bar spacing was close to the MLS of these species (the MLS of red mullet is 11 cm). The selectivity of the sorting grid with 15-mm bar spacing was generally very poor. Size-selection performance of sorting grids was assessed for the first time in the local fisheries. In order to effectively improve size-selection, seasonal and depth-dependent differences between target and by-catch species must be taken into account, which is a good indicator of the difficulty of implementing a single mesh size or grid spacing in the Mediterranean demersal trawl fishery.

An accelerated time domain finite difference simulation scheme for three‐dimensional transient electromagnetic modeling using geometric multigrid concepts

The fact that the transient electromagnetic (TEM) field is smoothed gradually in space with time allows for a reduced spatial sampling rate of the EM field. On the basis of concepts known from multigrid methods, we have developed a restriction operator in order to map the EM field and the material properties from a fine to a coarser finite difference mesh during a forward field simulation with an explicit time‐stepping scheme. Two advantages follow. First, the grid size can be reduced. Field restriction involves reducing the number of grid nodes by a factor of 2 for each Cartesian direction. Second, as can be seen from the Courant‐Friedrichs‐Levy condition, the larger grid spacing allows for proportionally larger time step sizes. After field restriction, a material averaging scheme is employed in order to calculate the underlying effective medium on the coarse simulation grid. Example results show a factor of up to 5 decrease in solution run time, compared to a scheme that uses a constant grid. Key to the accuracy of the approach is knowledge of the proper time range to restrict the fields. An adequate criterion to decide during run time when to restrict involves an error measure for the locations of interest between the fields on the fine mesh and the restricted fields.

Parallel 3-D simulation of seismic wave propagation in heterogeneous anisotropic media: a grid method approach

SUMMARY This paper presents a parallel numerical technique for modelling wave propagation in 3-D heterogeneous anisotropic media. The scheme is developed by following a so-called 3-D grid method of the elastic-isotropic case. The proposed parallel algorithm needs small data exchanges between subdomains in contrast to that developed based on other numerical techniques; therefore, it is more suitable for a PC-Cluster. The algorithm is implemented on a mesh of mixed tetrahedrons and parallelepipedons, thus providing an accurate description of arbitrary 3-D surface and interface topographies and an easy generation of a non-uniform, unstructured mesh. The unstructured mesh means that the proposed algorithm can reduce the memory requirement by flexibly assigning small grid spacing in regions with low velocities and larger grid spacing in regions with higher velocities. Like the 3-D grid method, the resulting anisotropic scheme naturally satisfies the free-surface boundary conditions of arbitrary surface topography. As a result, the near-surface scattering effects can be more accurately modelled. The proposed scheme can handle a general anisotropy without any interpolations. In this paper, the transversely isotropic medium with a tilted symmetry axis, as typically caused by a system of parallel cracks or fine layers, is discussed in detail. A paraxial absorbing boundary condition in a 3-D general anisotropic case is also proposed. Comparisons with analytical solutions demonstrate the accuracy of the parallel algorithm. Computed 3-D radiation patterns illustrate shear-wave splitting, as predicted by the theory. We show the generality and flexibility of the algorithm by modelling wave propagation in an anisotropic half-space with a hemispherical crater on the surface and in mixed isotropic/anisotropic models with horizontal and inclined interfaces.

Relative Humidity over the West Florida Continental Shelf

Abstract Observed relative humidity variations on the coastal ocean of the West Florida Continental Shelf (WFS) are examined over the 5-yr period 1998–2003. Despite considerable daily variability within seasons, the monthly mean values are nearly constant at about 75%. Summertime specific humidity is twice that during winter, so high air temperatures are responsible for the low summer monthly mean relative humidities. Winter has the greatest relative humidity variability; values range from less than 50% to over 100% as extratropical fronts move over the WFS. Saturation (and fog) occurs as warm moist air passes over colder water. Two different sensors, mounted on multiple moorings, were used to make these observations. Monthly mean values from the Rotronics MP-100F are higher than the Hygrometrix 1020SHT. In addition to sensor differences, a contributing cause to this offset appears to be the locations chosen for sensor deployment. NCEP reanalysis climatology over the WFS and land-based coastal data both show an annual cycle in monthly mean relative humidity, with higher values in summer, suggesting that the reanalysis field is influenced by land. Air–sea fluxes over the WFS are sensitive to small spatial variability in the coastal ocean and atmosphere. The large grid spacing of the NCEP reanalysis does not capture this variability. The lack of coastal ocean data for assimilation biases the NCEP reanalysis fields toward land-based measurements. Increased spatial coverage via evolving Coastal Ocean Observing Systems should remedy this problem by providing required information for describing and understanding the complicated ocean–atmosphere interactions that occur on continental shelves.

TH-C-T-6E-10: The Impact of Calculation Grid Size On the Accuracy of IMRT Dose Distribution

Purpose: To experimentally investigate the accuracy of complex IMRT dose distribution with varying dose grid size. Method and Materials: A Head and Neck compression film phantom was constructed from two semi-hemisphere solid water slabs. Two hypothetical Head and Neck IMRT treatment plans for a 54Gy prescribed dose to a shallow and a deep targets were generated using Philips Pinnacle3 TPS with 1.5, 2, 3, and 4 mm dose computation grid sizes. Radiochromic films were used for dosimetry. The plans were evaluated by computing the dose cumulative histograms. The 1.5 mm grid size plan was used as a benchmark for plan evaluation. Results: Dose differences between 2 mm and 1.5 mm grid size calculations were within 120 cGy (2.2% of prescribed dose) for shallow target and 132 cGy (2.4%) for deep target respectively over 95% of the area of analysis. The dose differences were 273 cGy (5.1%) and 253 cGy (4.7%) for the 4 mm grid size. In gamma function tests, all grid sizes met the criteria of acceptability (i.e., 95% of the region resulted in gamma index less or equal to 1 with a 3% dose difference and a 3 mm DTA criteria) except for deep target and 3 and 4 mm grid sizes. It was observed that larger grid spacing produces higher dose gradient, which contributes to the failure of the gamma test. The analyses of surface dose histograms showed a general trend of larger grid size producing broader histogram for surface dose. All grid spacing overestimate surface dose. Conclusion: Overall, the effect of varying grid size on dose variation appears to be insignificant as long as the grid size is less than 4 mm. However, 2 mm is recommended to ensure acceptable dose calculations, especially in high gradient regions.

Dynamical downscaling: Assessment of value retained and added using the Regional Atmospheric Modeling System (RAMS)

The value restored and added by dynamical downscaling is quantitatively evaluated by considering the spectral behavior of the Regional Atmospheric Modeling System (RAMS) in relation to its domain size and grid spacing. A regional climate model (RCM) simulation is compared with NCEP Reanalysis data regridded to the RAMS grid at each model analysis time for a set of six basic experiments. At large scales, RAMS underestimates atmospheric variability as determined by the column integrated kinetic energy and integrated moisture flux convergence. As the grid spacing increases or domain size increases, the underestimation of atmospheric variability at large scales worsens. The model simulated evolution of the kinetic energy relative to the reanalysis regridded kinetic energy exhibits a decrease with time, which is more pronounced with larger grid spacing. Additional follow‐on experiments confirm that the surface boundary forcing is the dominant factor in generating atmospheric variability for small‐scale features and that it exerts greater control on the RCM solution as the influence of lateral boundary conditions diminish. The sensitivity to surface forcing is also influenced by the model parameterizations, as demonstrated by using a different convection scheme. For the particular case considered, dynamical downscaling with RAMS in RCM mode does not retain value of the large scale which exists in the larger global reanalysis. The utility of the RCM, or value added, is to resolve the smaller‐scale features which have a greater dependence on the surface boundary. This conclusion regarding RAMS is expected to be true for other RCMs as well.

Seismic wavefield calculation for laterally heterogeneous whole earth models using the pseudospectral method

SUMMARY A method for simulating seismic wave propagation in a laterally heterogeneous whole earth model is presented by solving the elastodynamic equations in 2-D cylindrical coordinates using the pseudospectral method (PSM). The PSM is an attractive timedomain technique that uses the fast Fourier transform for an accurate diierentiation of ¢eld variables in the equations. Since no dispersion error arises in Fourier diierentiation, even when using a large grid spacing, computer memoryand time are reduced by several orders of magnitude compared to traditional ¢nite-diierence methods. In order to examine body-wave phases with current computing resources, a slice through the sphere is approximated with a 2-D cylindrical model. An irregular grid spacing is used in the vertical coordinate to improve the treatment of the various structural boundaries appearing in the earth model by matching the heterogeneity in the model. Synthetic seismograms obtained by the PSM calculation are compared with those calculated from an exact simulation method for a spherically homogeneous (1-D) earth model and achieve good agreement. The PSM method is illustrated by constructing the seismic P^SV wave¢eld for strongly heterogeneous earth models including a shield structure near the free surface and velocity perturbations just above the core^mantle boundary. The visualization of the evolution of seismic P and S waves in time and space is tracked using a sequence of snapshots and synthetic seismograms. These displays allow direct insight into the nature of the complex seismic wave behaviour in the Earth’s interior.

Accuracy of numerical methods for solving the advection–diffusion equation as applied to spore and insect dispersal

Three algorithms for solving a simplified 3-D advection–diffusion equation were compared as to their accuracy and speed in the context of insect and spore dispersal. The algorithms tested were the explicit central difference (ECD) method, the implicit Crank–Nicholson (ICN) method, and the implicit Chapeau function (ICF) method. The three algorithms were used only to simulate the diffusion process. A hold-release wind shifting method was developed to simulate the wind advection process, which shifts the concentration an integer number of grids and accumulates the remaining wind travel distance (which is less than the grid spacing) to the next time step. The test problem was the dispersal of a cloud of particles (originally in only one grid cell) in a 3-D space. The major criterion for testing the accuracy was R 2, which represents the proportion of the total variation in particle distribution in all grid cells that is accounted for by the particle distribution through numerical solutions. Other criteria included total remaining mass, peak positive density, and largest negative density. High R 2 values were obtained for the ECD method with (Δ t K z)/(Δ z) 2≤0.5 (Δ t=time step; K z=vertical eddy diffusion coefficient; Δ z=vertical grid spacing), and for the two implicit methods with Δ t K z/(Δ z) 2≤5. The ICN method gave higher R 2 values than the ICF method when the concentration gradients were high, but its accuracy decreased more rapidly with the progress of time than the ICF method with a combination of a large grid spacing and a large time step. With very steep concentration gradients, the ICF method generated huge negative values, the ICN method generated negative values to a lesser extent, and the ECD method generated only small negative values. It was also found that good mass and/or peak preservation did not necessarily correspond to a higher R 2 value. Based on the R 2 value and the requirement for concentration positivity, for simulations with steep concentration gradients, the ECD method would be most appropriate, followed by the ICN method, and the ICF method would be least appropriate due to large negative values. For simulations with low concentration gradients, the ECD or ICF or ICN method could be used, but the ICN method would not be appropriate for use in a combination of a large time step and a large grid spacing. The results from this study could help selection and use of appropriate numerical methods in studying the spatial dynamics of spores and insects.

Block Spins for Partial Differential Equations

We investigate the use of renormalisation group methods to solve partial differential equations (PDEs) numerically. Our approach focuses on coarse-graining the underlying continuum process as opposed to the conventional numerical analysis method of sampling it. We calculate exactly the coarse-grained or `perfect' Laplacian operator and investigate the numerical effectiveness of the technique on a series of 1+1-dimensional PDEs with varying levels of smoothness in the dynamics: the diffusion equation, the time-dependent Ginzburg-Landau equation, the Swift-Hohenberg equation and the damped Kuramoto-Sivashinsky equation. We find that the renormalisation group is superior to conventional sampling-based discretisations in representing faithfully the dynamics with a large grid spacing, introducing no detectable lattice artifacts as long as there is a natural ultra-violet cut off in the problem. We discuss limitations and open problems of this approach.

A self-adaptive time integration algorithm for solving partial differential equations

Non-uniform spatial grids are commonly used to resolve locally fast-changing physical phenomena in space. If a traditional explicit time integration scheme is used to advance solutions in the temporal dimension, the step size is restricted by the stability criterion, which is in turn dictated by the smallest grid spacing in the spatial dimensions. It turns out that the excessively small time step enforced by the smallest spatial grid spacing in a domain is usually unnecessary for most of the grid points with larger grid spacing. A new adaptive time integration method is introduced in this paper to improve the computational efficiency. The basic idea is to use different time step sizes at different spatial grid points. The stability criterion is still satisfied at all grid points by using different time step sizes. In this way, the time step size adjusts automatically based on the local spatial grid spacing. Complexity analysis and implementation details are also discussed in the paper. The non-linear Burger's equation is used in the numerical experiment. Both the complexity analysis and numerical computations demostrate significant improvement of computational efficiency.

Improved Cartesian coordinate finite difference simulations of small cylindrical objects.

An analytical, cylindrical coordinate formula for the thermal resistance around small cylindrical objects has been incorporated into finite difference equations in Cartesian coordinates to improve the accuracy of the numerical simulations of hyperthermia cancer treatments. This is done by introducing a circular finite difference control volume which is centered on the cylindrical object. The temperature distributions calculated from this new formulation and from the formulation obtained from a conventional, rectangular control volume are compared to the predictions from an analytical solution. The results show that for a given Cartesian grid spacing, the new method is superior to the conventional one. This will allow more accurate numerical solutions to be obtained at larger grid spacings when the effect of blood vessels or other cylindrical objects, such as ferromagnetic implants, hot water tubes, etc., are being investigated.

Portable Laser Scanner for Measuring Soil Surface Roughness

AbstractSoil surface elevations are commonly measured along a profile at relatively large grid spacings between 5 and 50 mm. This limitation in measurement technique allows the description of surface roughness due to tillage marks and large clods only on the order of ≥20 mm. A portable scanner was designed to measure soil topography at variable grid spacings down to submillimeter scales. The scanner consisted of an optical transducer for measuring soil surface elevation, a motor‐driven X‐Y traversing frame, a set of processing and control electronics, and a personal computer. The optical transducer was mounted on the carriage of the X‐Y traversing frame and was programmed to scan a maximum area of 1 by 1 m. The personal computer controlled the scanning and recorded elevation data. The system had no static noise and the combined frame vibration and spot size produced a positional and elevational resolution on the order 0.1 to 0.3 mm. This scanner can be used to digitize surfaces at fine grid spacings and the detailed surface topography can then be used to study surface boundary processes such as soil erosion.

The influence of the size of the grid used for dose calculation on the accuracy of dose estimation

The standard presentation of a dose distribution as an isodose map is based on interpolation between dose values calculated on a matrix of equally spaced points. We explored the question of how the spacing of the grid used for the dose matrix affects the error due to interpolating the dose at any point. We defined two types of errors: the dose error, which is the difference between the interpolated and true dose at a given point; and the position error, which is the distance between the point of interest and the nearest point which has, in fact, the dose value estimated for the point of interest. We examine the problem using both an analytical beam profile (a Fermi function) and measured 60Co, x-ray and proton beam profiles. Our analysis showed that the interpolation errors are proportional to the curvature of the dose distribution and are relatively high in regions on either side of, but not including, the steepest part of the penumbra. Our results showed how big an interpolation error one should expect for a given size of the calculation grid. The specification of accuracy should be cast in the form of a pair of requirements, one for dose and the other for position. At a given point, only one of the two requirements needs to be satisfied. The position requirement is almost always the less demanding in clinical practice and permits the use of a larger grid spacing than if only a dose requirement is applied.(ABSTRACT TRUNCATED AT 250 WORDS)

A discretization scheme that allows coarse grid-spacing in finite-difference process simulation

The central processing unit requirements of two-dimensional numerical process simulation require much larger typical grid spacings than those used in one-dimensional simulations. With these larger grid spacings, the discretization error in diffusion simulation can have a large effect on the final simulated profile. This discretization error is analyzed and a first-order correction is applied and compared to the standard discretization method. Results show that this correction allows for a significant increase in the allowed grid spacing. As a typical example, implantation and subsequent diffusion of arsenic into silicon was modeled with different grid spacings using the SUPREM-IV program (and all its default diffusion parameters) in a one-dimensional mode. In all cases, the total dose of the implant was a constant, and differences in the implantation profile width (due to the different grid spacings) were small enough to have no significant effect on the final diffused profiles. It is seen that the standard method overestimates the amount of diffusion. The proposed method shows much better agreement between the coarse and fine grids. >

COMPOSITE NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS Part II. Local Combinations

Abstract In Part II local combinations of PDE solutions are considered, where the grid spacing is reduced in only the marching or evolving direction. Via a local combination of difference solutions, extended ranges of stability for the classical explicit (CE) and, analogously, Euler methods are found. The range of stability is extended (n + 1)2 times the original limit where “n” is the number of combinations. Accounting for the computing effort for this multistep composite, the computing effort can, at best, be cut in half by the composite over the original CE method. The TE of these methods is analyzed and local extrapolations are considered. Local composites of different analogs, such as the Saul'yev average (SA) method, are also considered. The composite of the Crank-Nicholson and CE methods is shown to be equivalent to the optimum implicit method but stable only for p < 1.5. A new composite of the SA and CE is presented and shown to be fourth order in Ax spacing and superior to the other methods considered at large grid spacings.

Methods for Numerical Simulation of Water and Gas Coning

Abstract This paper describes and evaluates three numerical methods for the simulation of well coning behavior. The first method employs the implicit pressure-explicit saturation (IMPES) analysis with pressure-explicit saturation (IMPES) analysis with the production terms treated implicitly. The second technique is similar to the first model except that the interblock transmissibilities are also treated implicitly in the saturation equation. The third model is fully implicitly with respect to all variables in a manner qualified in the introduction and utilizes simultaneous solution of the difference equations describing the multiphase flow. The use of implicit transmissibilities in the IMPES model results in a several-fold increase in the allowable time-increment size over that attainable with the implicit production IMPES scheme, while the computing time per step is increased by less than 10 percent. The fully implicit model accepts larger time-increment sizes than possible with the first two methods but requires 3.3 times the computing time per time step needed by the second model. The fully implicit model is substantially more efficient for problems involving high capillary forces (treated explicitly in the IMPES methods) and small computing grid blocks at the wellbore. In problems involving moderate capillary forces and larger grid spacings, the fully implicit method and the implicit transmissibility IMPES technique are comparable in computing efficiency. The results of three coning studies are presented: a water-oil problem, a three-phase coning presented: a water-oil problem, a three-phase coning example, and a comparison of simulation results with a laboratory coning experiment. Also presented is an analysis of truncation error and a comparison of computational work requirements. Introduction This study was performed to evaluate three finite-difference schemes for simulating well coning behavior. The basis for this evaluation was the IMPES (implicit pressure-explicit saturation) model with explicit transmissibilities and implicit production terms. This model is referred to hereafter production terms. This model is referred to hereafter as Model 1. The next model evaluated in this work is an IMPES model similar to Model 1, except that the saturation-dependent interblock transmissibilities are treated implicitly rather than explicitly in the saturation equation. The third model is fully implicitly with respect to all variables and terms transmissibilities, pressure, saturation an capillary pressure - and utilizes simultaneous solution of the difference equations describing the multiphase flow. These two models are referred to hereafter simply as Model 2 and Model 3. For the purpose of clarity all models are described in purpose of clarity all models are described in reference to the problem of incompressible, two-phase flow. The techniques are equally applicable, however, to compressible, three-phase flow models. The examples chosen for illustration employ both incompressible and compressible simulation models. SPEJ p. 425