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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 Klemp

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We examine the rotation and propagation of the supercell-like convection produced by our three-dimensional cloud model. The rotation in the supercell is studied in terms of the conservation of equivalent potential vorticity and V. Bjerknes' first circulation theorem; neither of these have been used previously in this connection, and we find that they significantly contribute to the current level of understanding in this area. Using these we amplify the findings of our previous work in which we found that the source of midlevel rotation is the horizontally oriented vorticity associated with the environmental shear, while the low-level rotation derives from the baroclinic generation of horizontally oriented vorticity along the low-level cold-air boundary. We further demonstrate that these same processes that amplify the low-level rotation also produce the distinctive cloud feature known as the “wall cloud.”

We find that the thunderstorm propagates rightward primarily because of the favorable dynamic vertical pressure gradient that, owing to storm rotation, is always present on the right flank of the updraft. Simulations without precipitation physics demonstrate that this rightward propagation occurs even in the absence of a cold outflow and gust front near the surface.

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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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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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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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Robert M. Rauber, Ken Heideman, and Joseph Klemp
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William C. Skamarock and Joseph B. Klemp

Abstract

Although atmospheric phenomena tend to be localized in both time and space, numerical models generally employ only uniform discretizations or fixed nested grids. An adaptive grid technique implemented in 2D and 3D nonhydrostatic elastic atmospheric models is described. The adaptive technique makes use of separate rectangular refinements to increase resolution where truncation error estimates are large. Multiple, rotated, overlapping grids are used along with an arbitrary number of discrete grid-refinement levels. Refinements are placed and removed automatically during the integration based an estimates of the truncation error in the evolving solution. The technique can be viewed as an extension of the nesting technique often used in atmospheric models.

The adaptive model integrates the compressible, nonhydrostatic equations of motion. Although sound waves are not significant in the solution, they do constrain the time step. A splitting technique is used to accommodate the sound waves by advancing certain terms with a separate smaller time step. The terms responsible for gravity waves are also integrated with the smaller time step, and with the acoustic modes filtered through the use of divergence damping, the resulting model can be run as efficiently as hydrostatic models. Boundary conditions developed for the splitting technique in the adaptive framework are described and tested in the 2D and 3D models. The adaptive technique is shown to be efficient when compared to single fixed-grid simulations. Two new features are included in the basic solver.

Also considered are additional complications that arise because of the necessary use of parameterized physics. The dependence of many parameterizations on grid scale creates difficulties in evaluating truncation error and raises more general questions concerning solution error in nested and adaptive models.

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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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William A. Gallus Jr. and Joseph B. Klemp

Abstract

A two-dimensional nonhydrostatic version of the NCEP regional Eta Model together with analytic theory are used to examine flow over isolated mountains in numerical simulations using a step-terrain vertical coordinate. Linear theory indicates that a singularity arises in the steady flow over the step corners for hydrostatic waves and that this discontinuity is independent of height. Analytic solutions for both hydrostatic and nonhydrostatic waves reveal a complex behavior that varies with both horizontal and vertical resolution.

Witch of Agnessi experiments are performed with a 2D version of the Eta Model over a range of mountain half-widths. The simulations reveal that for inviscid flow over a mountain using the step-terrain coordinate, flow will not properly descend along the lee slope. Rather, the flow separates downstream of the mountain and creates a zone of artificially weak flow along the lee slope. This behavior arises due to artificial vorticity production at the corner of each step and can be remedied by altering the finite differencing adjacent to the step to minimize spurious vorticity production.

In numerical simulations with the step-terrain coordinate for narrow mountains where nonhydrostatic effects are important, the disturbances that arise at step corners may be of the same horizontal scale as those produced by the overall mountain, and the superposition of these disturbances may reasonably approximate the structure of the continuous mountain wave. For wider mountains, where perturbations are nearly hydrostatic, the disturbances above the step corners have horizontal scales that are much smaller than the overall scale of the mountain and appear as sharp spikes in the flow field. The deviations from the “classic” Witch of Agnesi solution are significant unless the vertical resolution is very small compared to the height of the mountain. In contrast, simulations with the terrain-following vertical coordinate produce accurate solutions provided the vertical grid interval is small compared to the vertical wavelength of the mountain waves (typically at least an order of magnitude larger than the mountain height).

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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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