Two time-splitting methods for integrating the elastic equations are presented. The methods are based on a third-order Runge-Kutta time scheme and the Crowley advection schemes. The schemes are combined with a forward-backward scheme for integrating high-frequency acoustic and gravity modes to create stable splitexplicit schemes for integrating the compressible Navier-Stokes equations. The time-split methods facilitate the use of both centered and upwind-biased discretizations for the advection terms, allow for larger time steps, and produce more accurate solutions than existing approaches. The time-split Crowley scheme illustrates a methodology for combining any pure forward-in-time advection schemes with an explicit time-splitting method. Based on both linear and nonlinear tests, the third-order Runge-Kutta-based time-splitting scheme appears to offer the best combination of efficiency and simplicity for integrating compressible nonhydrostatic atmospheric models.

SUMMARYA comparison between solutions from simulations of a non-linear density current test problem was made in order to study the behaviour of a variety of numerical methods. The test problem was diffusion-limited so that a grid-converged reference solution could be generated using high spatial resolution. Solutions of the test problem using several different resolutions were computed by the participants of the 'Workshop on Numerical Methods for Solving Nonlinear Flow Problems', which was held on 11-13 September 1990 at the National Center for Supercomputing Applications (NCSA). In general, it was found that when the flow was adequately resolved, all of the numerical schemes produced solutions that contained the basic physics as well as most of the flow detail of the reference solution. However, when the flow was marginally resolved, there were significant differences between the solutions produced by the various models. Finally, when the flow was poorly resolved, none of the models performed very well. While higher-order and spectral-type schemes performed best for adequately and marginally resolved flow, solutions made with these schemes were virtually unusable for poorly resolved flow. In contrast, the monotonic schemes provided the most coherent and smooth solutions for poorly resolved flow, however with noticeable amplitude and phase speed errors, even at finer resolutions.

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Warnings about convective-scale hazards are currently based on observations, but the time has come to develop warning methods in which numerical model forecasts play a much larger role.

An ''additive noise'' method for initializing ensemble forecasts of convective storms and maintaining ensemble spread during data assimilation is developed and tested for a simplified numerical cloud model (no radiation, terrain, or surface fluxes) and radar observations of the 8 May 2003 Oklahoma City supercell. Every 5 min during a 90-min data-assimilation window, local perturbations in the wind, temperature, and water-vapor fields are added to each ensemble member where the reflectivity observations indicate precipitation. These perturbations are random but have been smoothed so that they have correlation length scales of a few kilometers. An ensemble Kalman filter technique is used to assimilate Doppler velocity observations into the cloud model. The supercell and other nearby cells that develop in the model are qualitatively similar to those that were observed. Relative to previous storm-scale ensemble methods, the additive-noise technique reduces the number of spurious cells and their negative consequences during the data assimilation. The additive-noise method is designed to maintain ensemble spread within convective storms during long periods of data assimilation, and it adapts to changing storm configurations. It would be straightforward to use this method in a mesoscale model with explicit convection and inhomogeneous storm environments.

Polynomials are used together with polyharmonic spline (PHS) radial basis functions (RBFs) to create local RBF-finite-difference (RBF-FD) weights on different node layouts for spatial discretizations that can be viewed as enhancements of the classical finite differences (FD). The presented method replicates the convergence properties of FD but for arbitrary node layouts. It is tested on the 2D compressible Navier-Stokes equations at low Mach number, relevant to atmospheric flows. Test cases are taken from the numerical weather prediction community and solved on bounded domains. Thus, attention is given on how to handle boundaries with the RBF-FD method, as well as a novel implementation for hyperviscosity. Comparisons are done on Cartesian, hexagonal, and quasi-uniform node layouts. Consideration and guidelines are given on PHS order, polynomial degree and stencil size. The main advantages of the present method are: 1) capturing the basic physics of the problem surprisingly well, even at very coarse resolutions, 2) high-order accuracy without the need of tuning a shape parameter, and 3) the inclusion of polynomials eliminates stagnation (saturation) errors. A MATLAB code is given to calculate the differentiation weights for this novel approach.

Recent observational studies have shown that strong midlatitude mesoscale convective systems (MCSs) tend to decay as they move into environments with less instability and smaller deep-layer vertical wind shear. These observed shear profiles that contain significant upper-level shear are often different from the shear profiles considered to be the most favorable for the maintenance of strong, long-lived convective systems in some past idealized simulations. Thus, to explore the role of upper-level shear in strong MCS environments, a set of two-dimensional (2D) simulations of density currents within a dry, statically neutral environment is used to quantify the dependence of lifting along an idealized cold pool on the upper-level shear. A set of three-dimensional (3D) simulations of MCSs is produced to gauge the effects of the upper-level shear in a more realistic framework.Results from the 2D experiments show that the addition of upper-level shear to a wind profile with weak to moderate low-level shear increases the vertical displacement of parcels despite a decrease in the vertical velocity along the cold pool interface. Parcels that are elevated above the surface (1-2 km) overturn and are responsible for the deep lifting in the deep-shear environments, while the surface-based parcels typically are lifted through the cold pool region in a rearward-sloping path. This deep overturning helps to maintain the leading convection and greatly increases the size and total precipitation output of the convective systems in more complex 3D simulations, even in the presence of 3D structures. These results show that the shear profile throughout the entire troposphere must be considered to gain a more complete understanding of the structure and maintenance of strong midlatitude MCSs.

High-resolution Doppler radar observations of tornadoes reveal a distinctive tornado-scale signature with the following properties: a reflectivity minimum aloft inside the tornado core (described previously as an "eye"), a high-reflectivity tube aloft that is slightly wider than the tornado core, and a tapering of this high-reflectivity tube near the ground. The results of simple one-dimensional and two-dimensional models demonstrate how these characteristics develop. Important processes in the models include centrifugal ejection of hydrometeors and/or debris by the rotating flow and recycling of some objects by the nearsurface inflow and updraft.Doppler radars sample the motion of objects within the tornado rather than the actual airflow. Since objects move at different speeds and along different trajectories than the air, error is introduced into kinematic analyses of tornadoes based on radar observations. In a steady, axisymmetric tornado, objects move outward relative to the air and move more slowly than the air in the tangential direction; in addition, the vertical air-relative speed of an object is less than it is in still air. The differences between air motion and object motion are greater for objects with greater characteristic fall speeds (i.e., larger, denser objects) and can have magnitudes of tens of meters per second. Estimates of these differences for specified object and tornado characteristics can be obtained from an approximation of the one-dimensional model.Doppler On Wheels observations of the 30 May 1998 Spencer, South Dakota, tornado demonstrate how the apparent tornado structure can change when the radar-scatterer type changes. When the Spencer tornado entered the town and started lofting debris, changes occurred in the Doppler velocity and reflectivity fields that are consistent with an increase in mean scatterer size.