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Alan Shapiro

Abstract

An exact analytic solution of the Navier–Stokes equations is used to validate a three-dimensional nonhydrostatic numerical flow model, the Advanced Regional Prediction System developed at the Center for Analysis and Prediction of Storms. The exact solution is a viscously decaying extension of a Beltrami flow used in previous studies of thunderstorm rotation, and consists of a periodic array of counterrotating updrafts and downdrafts. This flow is noteworthy in that it is three-dimensional, free of singularities, and satisfies the Navier–Stokes equations with nontrivial (i.e., nonvanishing) inertial terms. The simple form of the analytic solution and its provision for arbitrarily large spatial gradients suggest its potential utility in validating numerical flow models and in testing the relative merits of various numerical solution algorithms.

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Alan Shapiro

Abstract

Vertically sheared airflow over semi-infinite barriers is investigated with a simple hydrodynamical model. The idealized flow is steady, two-dimensional, neutrally buoyant, and inviscid, bounded on the bottom by a semi-infinite impermeable barrier and on the top by a rigid tropopause lid. With attention further restricted to an exponentially decreasing wind shear, the equations of motion (Euler's equations) reduce, without approximation, to a modified Poisson equation for a pseudo streamfunction and a formula for the Exner function. The free parameters characterizing the model's environment are the tropopause height, the density scale height, the wind speed at ground level, and the wind speed at tropopause level. Additional parameters characterize the barrier geometry.

Exact solutions of the equations of motion are obtained for semi-infinite plateau barriers and for a barrier qualitatively resembling the shallow density current associated with some thunderstorm outflows. These solutions are noteworthy in that the reduction of a certain nondimensional shear parameter (through negative values) results in greater vertical parcel displacements over the barrier despite a corresponding reduction in the vertical velocity. This steepening tendency culminates in overturning motions associated with both upstream and downstream steering levels. In this latter case the low-level inflow impinging on the barrier participates in a mixed jump and overturning updraft reminiscent of updrafts simulated in numerical convective models. Conversely, for large values of the nondimensional shear parameter, parcels undergo small vertical parcel displacements over the barrier despite large vertical velocities. This latter behavior may account for the finding that strong convergence along the leading edge of storm outflows does not always trigger deep convection even in unstable environments.

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Alan Shapiro

Abstract

A theoretical model for unsteady drag-induced transfer of horizontal momentum between air and raindrops in moderate to heavy rainfall is presented. The model accounts for a two-way coupling in which the relative horizontal motion between air and raindrops appears as a drag forcing in both the air and raindrop equations of motion. Analytical solutions of these coupled equations are obtained for the case of rain falling through (i) an initial step change in environmental wind, (ii) a uniform shear profile, and (iii) periodically varying vertical shears of various wavenumbers (a crude proxy for turbulent eddies). Formulas for the propagation (descent) speeds of the shear zones are obtained for (ii), (iii), and for the later stage of (i). However, these speeds are generally quite small—on the order of a few centimeters per second even for heavy rainfall. More importantly, the solutions of (i) and (iii) indicate that the drag interaction leads to a decay of the velocity gradients. A formula for the e-folding decay time of the periodically varying shear profiles indicates that at small wavelengths, the smallest decay times are found for the smaller drops, but at large wavelengths, the smallest decay times are found for the larger drops. The decay times decrease with decreasing wavelength, and approach a value equal to the reciprocal of the product of the rainwater mixing ratio and a drag parameter in the limit of vanishing wavelength. For parameters typical of moderate to heavy rainfall, the small-scale decay times are on the order of a few minutes.

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Alan Shapiro and Paul Markowski

Abstract

Theoretical hydrodynamic models for the behavior of vortices with axially varying rotation rates are presented. The flows are inviscid, axisymmetric, and incompressible. Two flow classes are considered: (i) radially unbounded solid body–type vortices and (ii) vortex cores of finite radius embedded within radially decaying vortex profiles.

For radially unbounded solid body–type vortices with axially varying rotation rates, the von Kármán–Bödewadt similarity principle is applicable and leads to exact nonlinear solutions of the Euler equations. A vortex overlying nonrotating fluid, a vortex overlying a vortex of different strength, and more generally, a vortex with N horizontal layers of different rotation rate are considered. These vortices cannot exist in a steady state because continuity of pressure across the horizontal interface between the vortex layers demands that a secondary (meridional) circulation be generated. These similarity solutions are characterized by radial and azimuthal velocity fields that increase with radius and a vertical velocity field that is independent of radius. These solutions describe nonlinear interactions between the vortex circulations and the vortex-induced secondary circulations, and may play a role in the dynamics of the interior regions of broad mesoscale vortices. Decaying, amplifying, and oscillatory solutions are found for different vertical boundary conditions and axial distributions of vorticity. The oscillatory solutions are characterized by pulsations of vortex strength in lower and upper levels associated with periodic reversals in the sense of the secondary circulation. These solutions provide simple illustrations of the “vortex valve effect,” sometimes used to explain cyclic changes in updraft and rotation strength in tornadic storms.

A linear analysis of the Euler equations is used to describe the short-time behavior of an elevated vortex of finite radius embedded within a radially decaying vortex profile (i.e., elevated Rankine-type vortices). The linear solution describes the formation of a central updraft (as in the similarity solution) and an annular downdraft ringing the periphery of the vortex core (not accounted for in the similarity solution). Downdraft strength is sensitive to both the vortex core aspect ratio and outer vortex decay rate, being stronger and narrower for broader vortices and larger decay rates. It is hypothesized that this dynamically induced downdraft may facilitate the transport of mesocyclone vorticity down to low levels in supercell thunderstorms.

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William J. Martin and Alan Shapiro

Abstract

From geometrical considerations, the optimum tilt angle for a meteorological radar at which the best possible vertical resolution results is derived. This optimum angle is a compromise between the effects of beam divergence and range gate spacing. For typical S-band radar parameters, this optimum tilt angle is found to be about 7°. However, wind analyses at this tilt angle were found not to be accurate in practice because of ground clutter contamination, and suboptimal angles need to be used. Most of the ground clutter was found to be sensed in the radar beam sidelobes. The data presented here imply that ground clutter is a serious contaminant at tilt angles as high as 45°. For clear-air wind profiling in the boundary layer, the impact of ground clutter contamination increased as the tilt angle was increased.

Data presented from four radars [the Goodland, Kansas, Weather Surveillance Radar-1988 Doppler (WSR-88D); the University of Oklahoma’s Doppler on Wheels; NCAR’s S-band dual-polarization Doppler radar (S-Pol); and NSSL’s Cimarron] suggest that a fairly narrow range of tilt angles from 1° to 2° is generally acceptable for wind profiling of the boundary layer in clear-air conditions. Tilt angles outside this range lead to significant systematic errors, primarily from ground clutter contamination.

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Alan Shapiro and John J. Mewes

Abstract

New formulations of three-dimensional dual-Doppler wind analysis are presented. The new formulations are conceptually simple, preserve the radial nature of the wind observations, involve only one analysis step (i.e., all constraints are imposed in one functional), and are in a form in which the well-posed condition can most readily be checked. These techniques minimize functionals that incorporate the anelastic mass conservation equation and the radial wind observations as strong or weak constraints. The minimizations are accomplished by appealing directly to the Euler–Lagrange equations and proceed most naturally in the “coplane” cylindrical polar coordinate system. In one method, the anelastic mass conservation equation is applied as a weak constraint, while the radial wind observations are imposed as strong constraints. This results in an algorithm similar to the Armijo wind analysis but with provision for vertical velocity data specification on both upper and lower boundaries (as in the O’Brien adjustment). In another method, the anelastic mass conservation equation is imposed as a strong constraint, while the radial wind observations are used as weak constraints. In a third method, both mass conservation and the radial wind observations are used as weak constraints. In each of the latter two formulations, the analysis reduces to solving a second-order linear partial differential equation, the solution of which is unique. As in Armijo’s research, the shape of the analysis domains must be suitably restricted if the problems are to be well posed.

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John J. Mewes and Alan Shapiro

Abstract

The anelastic vertical vorticity equation is imposed as a flow constraint in dual-Doppler wind analysis techniques, with emphasis on improving retrievals of the vertical velocity field. Several new techniques in which the vorticity equation is imposed as a weak constraint are presented. In one method, the vorticity equation is used to obtain the vertical velocity boundary condition field required when mass continuity is used as a strong constraint. In a second method, the vorticity equation is used to solve for the set of two boundary condition fields required when the vertical velocity is expressed in a “weak constraint form.” A final correction step to the horizontal wind field ensures that mass continuity is satisfied exactly in the final analysis. This latter method can be easily adapted to retrieve a single boundary condition when an opposing boundary condition can be accurately assessed using another method.

These new analysis techniques are tested and compared with each other and to common traditional approaches in which the vorticity equation is not used. These tests use simulated data sampled from Advanced Regional Prediction System (ARPS) simulations of a supercell thunderstorm and of a thunderstorm evolving in a typical dry microburst environment. The results suggest that the vorticity equation holds significant potential for improving dual-Doppler analyses of the vertical velocity field, especially under less than optimal data coverage.

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Yefim L. Kogan and Alan Shapiro

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The development and merger of pairs of convective clouds in a shear-free environment were simulated in an explicit microphysical cloud model. The occurrence or nonoccurrence of updraft merger and the timing of merger depended critically on the initial spacing of the thermal perturbations imposed in the model's initialization. In the unmerged cases the presence of a neighbor cloud was detrimental to cloud development at all times. In the merged cases this negative interaction was still operating but only until the onset of updraft merger.

Based on the visual form of the updraft merger, it was hypothesized that low-level merger was a consequence of mutual advection, that is, that each cloud caught its neighbor in its radial inflow and advected it inward. This low-level advection hypothesis was quantified by considering a potential flow induced by two line sinks whose strengths were set equal to the low-level mass flux into the numerically simulated clouds. The merger times obtained from the advection hypothesis were in good agreement with the merger times observed in the simulations. Moreover, if merger did not occur, the advection hypothesis suggested that merger should not have occurred.

The merger process was accompanied by the presence of trimodal drop spectra at the upper levels of the cloud. It was shown that the drop size distribution depends not only on the autoconversion and accretion rates, but also on the nonlinear interaction between various source and sink terms affecting rain formation, particularly on the rates of condensation-evaporation, sedimentation, and breakup processes. The analysis of raindrop trajectories showed the details of rain formation in different cloud regions and the effect of dynamical conditions on the growth of rain particles.

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William J. Martin and Alan Shapiro

Abstract

The source of clear-air reflectivity from operational and research meteorological radars has been a subject of much debate and study over the entire history of radar meteorology. Recent studies have suggested that bird migrations routinely contaminate wind profiles obtained at night, while historical studies have suggested insects as the main source of such nocturnal clear-air echoes. This study analyzes two cases of nocturnal clear-air return using data from operational Weather Surveillance Radar-1988 Doppler (WSR-88D) and X- and W-band research radars. The research radars have sufficient resolution to resolve the echo as point targets in some cases. By examining the radar cross section of the resolved point targets, and by determining the target density, it is found for both cases of nocturnal clear-air echoes that the targets are almost certainly insects. The analysis of the dependence of the echo strength on radar wavelength also supports this conclusion.

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Joshua G. Gebauer and Alan Shapiro

Abstract

The frequency and intensity of the Great Plains nocturnal low-level jet (LLJ) are enhanced by baroclinicity over the sloped terrain of the region. A classical description of baroclinic-induced diurnal wind oscillations over the Great Plains considers differential heating of the slope with respect to air at the same elevation far removed from the slope, but with buoyancy constant along the slope (Holton mechanism). Baroclinicity can also occur due to differential heating of the slope itself, which creates a gradient in buoyancy along the slope. The relative prevalence of the two types of baroclinicity in this region has received scant attention in the literature. The present study uses 19 years of data from the Oklahoma Mesonet to evaluate the characteristics of along-slope buoyancy gradients over the region. A mean negative afternoon along-slope buoyancy gradient (east–west gradient) is found over Oklahoma. The sign of this afternoon buoyancy gradient is favorable for LLJ formation, as it results in the strongest southerly geostrophic wind near the ground around sunset, which is conducive to nocturnal jet formation via the inertial oscillation mechanism. The negative afternoon buoyancy gradient is at least partially created by an east–west gradient in diurnal heating and is stronger and more consistent in the summer months, which is when LLJs are most frequent. The contribution of the along-slope buoyancy gradient to the low-level geostrophic wind was found to be as important as the contribution of the Holton mechanism. Overall, these results indicate that along-slope buoyancy gradients should be accounted for in studies of LLJ dynamics over the Great Plains.

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