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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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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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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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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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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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William C. Skamarock
and
Joseph B. Klemp

Abstract

No abstract available.

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Morris L. Weisman
and
Joseph B. Klemp

Abstract

Using a three-dimensional numerical cloud model, we investigate the effects of directionally varying wind shear on convective storm structure and evolution over a wide range of shear magnitudes. As with a previous series of experiments using unidirectional wind shear profiles (Weisman and Klemp), the current results evince a spectrum of storm types ranging from short lived single cells at low shears, multicells at intermediate shears, to supercells at high shears. With a clockwise curved hodograph, the supercellular growth is confined to the right flank of the storm system while multicellular growth is favored on the left flank. An analysis of the dynamic structure of the various cells reveals that the quasi-steady supercell updrafts are strongly enhanced by dynamically induced pressure gradients on the right flank of the storm system. We use this feature along with other related storm characteristics (such as updraft rotation) to propose a dynamically based storm classification scheme. Following Browning, this scheme includes two basic storm types: ordinary cells and supercells. Multicell storm systems and squall lines would then be made up of a combination of supercells and ordinary cells. As in the unidirectional shear experiments, a convective bulk Richardson numbercharacterizes the environment conducive to producing particular storm types.

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Dale R. Durran
and
Joseph B. Klemp

Abstract

A two-dimensional, nonlinear, nonhydrostatic model is described which allows the calculation of moist airflow in mountainous terrain. The model is compressible, uses a terrain-following coordinate system, and employs lateral and upper boundary conditions which minimize wave reflections.

The model's accuracy and sensitivity are examined. These tests suggest that in numerical simulations of vertically propagating, highly nonlinear mountain waves, a wave absorbing layer does not accurately mimic the effects of wave breakdown and dissipation at high levels in the atmosphere. In order to obtain a correct simulation, the region in which the waves are physically absorbed must generally be included in the computational domain (a nonreflective upper boundary condition should be used as well).

The utility of the model is demonstrated in two examples (linear waves in a uniform atmosphere and the 11 January 1972 Boulder windstorm) which illustrate how the presence of moisture can influence propagating waves. In both cases, the addition of moisture to the upstream flow greatly reduces the wave response.

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