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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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Richard Rotunno and Joseph B. Klemp

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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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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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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 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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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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Dale R. Durran and Joseph B. Klemp

Abstract

Numerical mountain wave simulations have documented that intense lee-slope winds frequently arise when wave-overturning occurs above the mountain. Explanations for this amplification process have been proposed by Clark and Peltier in terms of a resonance produced by linear-wave reflections from a self-induced critical layer, and by Smith in terms of solutions to Long's equation for flow beneath a stagnant well-mixed layer. In this paper, we evaluate the predictions of these theories through numerical mountain-wave simulations in which the level of wave-overturning is fixed by a critical layer in the mean flow. The response of the simulated flow to changes in the critical-layer height and the mountain height is in good agreement with Smith's theory. A comparison of Smith's solution with shallow-water theory suggests that the strong lee-slope winds associated with wave-overturning are caused by a continuously stratified analog to the transition from subcritical to supercritical flow in conventional hydraulic theory.

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Joseph B. Klemp and Dale R. Durran

Abstract

A radiative upper boundary condition is proposed for numerical mesoscale models which allows vertically propagating internal gravity waves to pass out of the computational domain with minimal reflection. In this formulation, the pressure along the upper boundary is determined from the Fourier transform of the vertical velocity at that boundary. This boundary condition can easily be incorporated in a wide variety of models and requires little additional computation. The radiation boundary condition is derived from the linear, hydrostatic, Boussinesq equations of motion, neglecting Coriolis effects. However, tests of this radiation boundary condition in the presence of nonhydrostatic, Coriolis, nonlinear and non-Boussinesq effects suggest that it would be effective in many mesoscale modeling applications.

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