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Joseph B. Klemp

Abstract

Horizontally diffusive computational damping terms are frequently employed in 3D atmospheric simulation models to enhance stability and to suppress small-scale noise. In configuring these filters, it is desirable that damping effects are concentrated on the smaller-scale disturbances close to the grid scale and that the dissipation is spatially isotropic. On Cartesian meshes, the isotropy of the damping can vary greatly depending on the numerical formulation of the horizontal filter. The most isotropic behavior appears to result from recursive application of a 2D Laplacian that combines both along-axis and diagonal contributions. Also, the recursive application of 1D Laplacians in each coordinate direction provides better isotropy than the recursive application of the 2D Laplacian represented with a five-point operator. Increased isotropy also permits a larger maximum diffusivity, which may be beneficial in certain filter applications. On hexagonal and triangular meshes, Laplacian operators exhibit excellent isotropy, owing to the more isotropic nature of the meshes. However, previous research has established that straightforward application of the Laplacian may yield a diffusion operator that damps both resolved physical modes and unresolved high-wavenumber (aliased) modes, but it does not converge to the proper analytic behavior. Special averaging is then required to recover an accurate representation for the Laplacian. A consequence of this averaging is that the resulting filters do not act on the aliased modes (the checkerboard mode in particular) and thus employing the unaveraged diffusion operators may be preferable. The damping characteristics and stability constraints are derived for both the unaveraged and averaged Laplacian filters for C-grid staggering on these meshes.

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Vim Toutenhoofd and Joseph B. Klemp

Abstract

Observations are described of a small, isolated cumulonimbus developing in a wind field with relatively little directional shear. The storm displayed a high degree of symmetry about a vertical plane through the center of the storm oriented parallel to the wind shear vector. Single-Doppler observations of this storm reveal a region in which the horizontal component of the wind vector was opposite to that of the mid-level environmental wind, suggesting the presence of a vortex pair circulation. The storm was simulated with a three-dimensional cloud model which reproduced these and some of the other observed storm characteristics. The environmental wind shear in which the storm developed is similar to that of the composite sounding documented by Fankhauser and Mohr (1977) for weak, isolated or scattered storms in northeast Colorado. Therefore, this symmetric structure, involving two counter-rotating vortices, may be a common feature of isolated storms in this area.

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Joseph B. Klemp and Richard Rotunno

Abstract

The transition of a supercell thunderstorm into its tornadic phase is investigated through high-resolution numerical cloud model simulations initiated within the interior portion of a previously simulated mature supercell storm. With the enhanced grid resolution, the low-level cyclonic vorticity increases dramatically, and the gust front rapidly occludes as small-scale downdrafts develop in the vicinity of the low-level center of circulation. As the occlusion progresses, a ring of high-vorticity air surrounds the circulation center and could be conducive to multiple vortex tornado formation. Numerous features of the simulated transition bear resemblance to those observed in tornadic storms. In the model simulation, the large low-level vorticity is generated through the tilting and intense stretching of air from the inflow side of the storm. This vertical vorticity is derived from the horizontal vorticity of the environmental shear and also from horizontal vorticity generated solenoidally as low-level air approaches the storm along the forward flank cold outflow boundary. Intensification of the rear flank downdraft during the occluding phase is dynamically driven by the strong low-level circulation.

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Richard Rotunno and Joseph B. Klemp

Abstract

In the present investigation we propose a simple theory to explain how a veering environmental wind shear vector can cause an initially symmetric updraft to grow preferentially to the right of the shear vector and acquire cyclonic rotation. The explanation offered is based on linear theory which predicts that interaction of the mean shear with the updraft produces favorable vertical pressure gradients along its right flank. To asses the validity of linear theory for large-amplitude updrafts, the three-dimensional, shallow, anelastic equations are numerically integrated using a simple parameterization for latent heating within a cloud and the linear and nonlinear forcing terms are separately analyzed. These results suggest that although the nonlinear effects strongly promote splitting of the updraft, the linear forcing remains the dominant factor in preferentially enhancing updraft growth on the right flank. We believe this differential forcing is a major contributor to the observed predominance of cyclonically rotating, right moving storms.

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Robert B. Wilhelmson and Joseph B. Klemp

Abstract

We have used a three-dimensional cloud model to investigate the splitting of an initially isolated storm in a one-directional east-west shear. The simulated evolution of storm splitting in some cases follows all four stages suggested by Achtemeier (1969) after analysis of radar data, including the development of two self-sustaining storm. One of these storms moves to the right of the mean wind vector and the other to the left. In the right-moving storm the updraft rotates cyclonically and the downdraft anticyclonically, forming a vortex pair, as depicted in the schematic model of Fankhauser (1971). The vortex pair structure is also similar to that observed with Doppler radar and analyzed by Ray (1976). The downdraft-induced gust front interacts with the low-level environmental wind to produce the convergence necessary to sustain the storm. This convergence extends to the south and west of the storm, and if enough low-level moisture is available a flanking line develops. The distribution of rainwater within the updraft suggests the existence of an over-hang and book typically observed in severe storms.

To understand when splitting might occur the strength and distribution of the vertical wind shear were varied. The various simulations suggest that strong shear at and just above cloud base is important for the splitting process to be successful. For splitting to occur the low-level inflow from the cast in our simulations must be sufficiently strong to inhibit the propagation of the gust front toward the cast. If the gust front (or wind shift line) can propagate away from the storm toward the cast, the region of low-level convergence moves away from the storm and initial splitting in the lower updraft cannot he sustained. Further, without the precipitation-induced downdraft and associated low-level outflow splitting does not occur.

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Robert B. Wilhelmson and Joseph B. Klemp

Abstract

A three-dimensional numerical storm model is used to investigate the observed splitting of several reflectivity echoes on 3 April 1964 in Oklahoma. Representative soundings from this day exhibit a nearly one-directional environmental wind shear vector and the presence of strong low-level wind shear. In the numerical simulation an initial cloud splits into two long-lived rotating storms, one that moves to the left of the mean winds and the other to the right. The left-moving storm develops more slowly than the right-moving one due to the deviation of the environmental wind hodograph from a straight line below 1 km. Further, the left mover eventually splits. Convergence induced by the cold, low-level storm outflow plays a major role in the development of both the first and second splits. However, the second split appears to be dynamically different than the first as the left-moving updraft remains essentially unchanged while a new updraft forms immediately adjacent to it. Because of the different propagational characteristics of the new storm it separates from the left mover. As the left-and right-moving storms move apart, new clouds develop in between them along an expanding cold outflow boundary. In this manner the evolving storm configuration becomes similar to that of a squall line, but has evolved from a single convective cell in the absence of imposed convergence. A comparison of the simulation with observed reflectivity and surface data reveals sufficient similarity to suggest that the explanations for the model storm development also may apply to some of the observed events.

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Joseph B. Klemp and Robert B. Wilhelmson

Abstract

A new three-dimensional cloud model has been developed for investigating the dynamic character of convective storms. This model solves the compressible equations of motion using a splitting procedure which provides numerical efficiency by treating the sound wave modes separately. For the subgrid turbulence processes, a time-dependent turbulence energy equation is solved which depends on local buoyancy, shear and dissipation. First-order closure is applied to nearly conservative variables with eddy coefficients based on the computed turbulence energy. Open lateral boundaries are incorporated in the model that respond to internal forcing and permit gravity waves to propagate out of the integration domain with little apparent reflection. Microphysical processes are included in the model using a Kessler-type parameterization. Simulations conducted for an unsheared environment reveal that the updraft temperatures follow a moist adiabatic lapse rate and that the convection is dissipated by water loading of the updraft. The influence of a one-directional shear on the storm development is also investigated. A simulation with a veering and backing wind profile exhibits interesting features which include a double vortex circulation, cell splitting and, secondary cell formation.

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Joseph B. Klemp and Robert B. Wilhelmson

Abstract

Using a three-dimensional numerical cloud model, self-sustaining right- and left-moving storms are simulated which arise through splitting of the original storm. The right-moving storm develops a structure which bears strong resemblance to Browning's (1964) conceptual model, while the left-moving storm has mirror image characteristics. By altering the direction of the environmental shear at low and middle levels, either the right- or the left-moving storm can be selectively enhanced. Specifically, if the wind hodograph turns clockwise with height, a single right-moving storm envolves from the splitting process. Conversely, counterclockwise turning of the hodograph favors development of the left-moving storm.

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Dale R. Durran and Joseph B. Klemp

Abstract

The effects of latent heat release on the dynamics of mountain lee waves are examined with the aid of two-dimensional numerical simulations, for several situations in which the Scorer parameter has a nearly two-layer vertical structure. Changes in the moisture in the lowest layer are found to produce three fundamentally different behaviors: 1) resonant waves in an absolutely stable environment are distorted and untrapped by an increase in moisture; 2) resonant waves in a conditionally unstable layer are destroyed by an increase in moisture; and 3) resonant waves in a moist environment are detuned by a decrease in moisture. Changes in the humidity in the upper layer are found to amplify or damp the wave response, depending on the depth of the lower layer. In most situations, the wave response is significantly more complicated than that predicted by simply replacing the dry stability with an equivalent moist stability in the saturated layer.

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William C. Skamarock and Joseph B. Klemp

Abstract

The mathematical equivalence of the linearized two-dimensional (2D) shallow-water system and the 2D acoustic-advection system strongly suggests that time-split schemes designed for the hydrostatic equations can be employed in nonhydrostatic models and vice versa. Stability analyses are presented for several time-split numerical methods for integrating the two systems. The primary interest is in the nonhydrostatic system and in explicit numerical schemes where no multidimensional elliptic equations arise; thus, a detailed analysis of the Klemp and Wilhelmson (KW) explicit technique for integrating the time-split nonhydrostatic system is undertaken. It is found that the interaction between propagating and advecting acoustic modes can introduce severe constraints on the maximum allowable time steps. Proper filtering can remove these constraints. Other explicit time-split schemes are analysed, and, of all the explicit schemes considered, it is believed that the KW time-split method offers the best combination of stability, minimal filtering, simplicity, and freedom from spurious noise for integrating the nonhydrostatic or hydrostatic equations.

Schemes wherein the fast modes are integrated implicitly and the slow modes explicitly are also analyzed. These semi-implicit schemes can be used with a greater variety of advection schemes than the explicit time-split approaches and generally require less filtering than the split-explicit schemes for stability. However, a multidimensional elliptic equation must be solved with each time step.

For nonhydrostatic elastic models using the KW time-split method, an acoustic filter is presented that allows a reduction of previously necessary filtering in the KW scheme, and a method for integrating the buoyancy equation is discussed that results in the large time step being limited by a Courant condition based on the advection velocity and not on the fastest gravity-wave speed.

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