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Katharine M. Kanak
and
Jerry M. Straka

Abstract

Photographic documentation of a rare and enigmatic reticular cloud formation that occurred in conjunction with a thunderstorm outflow anvil on 4 June 1995 at 2230 UTC at Norman, Oklahoma, is presented. A National Weather Service vertical sounding, taken within 1 h of the occurrence of the formation at Norman, is also presented. Possible formation mechanisms for these unusual cloud features are discussed.

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Paul M. Markowski
and
Jerry M. Straka

Abstract

The authors document some of the unusual rotating updrafts (one of which produced a tornado) that developed over central Oklahoma on 28 October 1998 in an environment of strong (1.8 × 10−2 s−1) low-level (0–3 km) mean shear. The maximum convective available potential energy (including virtual temperature effects) a “storm” could have realized was approximately 300 J kg−1; however, most of the storms probably realized less than 100 J kg−1. Average (maximum) parcel virtual temperature excesses were estimated to be 0.4–1.2 K (1.8–2.8 K). Echo tops were measured from less than 5 km to 11.2 km above ground level (AGL), although visual observations and radar data suggested echoes that extended above approximately 5–6 km AGL were not associated with significantly buoyant cloud elements. Radar characteristics of many of the storms were similar to supercell storms (e.g., weak echo regions, echo overhang, velocity couplets, hook echoes), as were some of the visual characteristics near cloud base (e.g., wall clouds, rain-free bases, and striated low-level updrafts); however, visual characteristics in middle to upper portions of the storms were not characteristic of typical severe storms, supercells, or previously documented “minisupercells.” Furthermore, the buoyancy realized by the updrafts was estimated to be considerably less than environments associated with the aforementioned minisupercells.

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Mark A. Askelson
and
Jerry M. Straka

Abstract

The response function is a commonly used measure of analysis scheme properties. Its use in the interpretation of analyses of real-valued data, however, is unnecessarily complicated by the structure of the standard form of the Fourier transform. Specifically, interpretation using this form of the Fourier transform requires knowledge of the relationship between Fourier transform values that are symmetric about the origin. Here, these relationships are used to simplify the application of the response function to the interpretation of analysis scheme properties.

In doing so, Fourier transforms are used because they can be applied to studying effects that both data sampling and weight functions have upon analyses. A complication arises, however, in the treatment of constant and sinusoidal input since they do not have Fourier transforms in the traditional sense. To handle these highly useful forms, distribution theory is used to generalize Fourier transform theory. This extension enables Fourier transform theory to handle both functions that have Fourier transforms in the traditional sense and functions that can be represented using Fourier series.

The key step in simplifying the use of the response function is the expression of the inverse Fourier transform in a magnitude and phase form, which involves folding the integration domain onto itself so that integration is performed over only half of the domain. Once this is accomplished, interpretation of the response function is in terms of amplitude and phase modulations, which indicate how amplitudes and phases of input waves are affected by an analysis scheme. This interpretation is quite elegant since its formulation in terms of properties of input waves results in a one-to-one input-to-output wave interpretation of analysis scheme effects.

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Erik N. Rasmussen
and
Jerry M. Straka

Abstract

It is hypothesized that the precipitation intensity beneath a supercell updraft is strongly influenced by the amount of hydrometeors that are reingested into the updraft after being transported away in the divergent upper-level flow of the anvil. This paper presents the results of a climatological analysis of soundings associated with three types of isolated supercells having distinctive precipitation distributions, the so-called classic, low-precipitation (LP), and high-precipitation (HP) storms. It is shown that storm-relative flow at 9–10 km above the ground is strongest in the environments of LP storms, and relatively weak in the environments of HP storms, with classic storms occurring in environments with intermediate magnitudes of upper storm-relative flow. It is plausible that comparatively strong flow in the anvil-bearing levels of LP storms transports hydrometeors far enough from the updraft that they are relatively unlikely to be reingested into the updraft, leading to greatly diminished precipitation formation in the updraft itself. Conversely, the weak upper flow near HP storms apparently allows a relatively large number of hydrometeors to return to the updraft, leading to the generation of relatively large amounts of precipitation in the updraft. It also is apparent that thermodynamic factors such as convective available potential energy, low-level mixing ratio, and mean relative humidity are of lesser importance in determining storm type from a climatological perspective, although important variations in humidity may not be well sampled in this study. This climatological analysis does not directly evaluate the stated hypothesis; however, the findings do indicate that further modeling and microphysical observations are warranted.

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Jerry M. Straka
and
John R. Anderson

Abstract

The minimal aliasing local spectral (LS) method is a numerical technique that embodies features of both finite-difference (FD) and spectral transform (ST) methods. Anderson first described this method in the context of the one-dimensional advection-diffusion equation. In the current paper, we describe the extension of the LS method to multidimensions. First, we review the one-dimensional version of the LS method from a more rigorous view. In addition, we describe interpolation, differentiation, and dealiasing fitters for the LS method based on Lagrange polynomials. Without the dealiasing filters, this version of the LS method collapses to a standard high-order Taylor series FD scheme. When filter lengths span the integration domain and the dealiasing stage is retained, the LS method becomes an ST method, as described by Anderson. Issues concerning the implementation of the LS method in multidimensions are also discussed. These issues include the form of the high-resolution grid, the implementation of the interpolation stage, and the implementation of the dealiasing stage. Then, we test the LS method with a two-dimensional nonlinear density current problem using idealized boundary conditions. Comparisons are made with a high-resolution reference solution from a reference model, as well as with solutions from a high-order FD model. Results from simulations of the test problem demonstrate that the LS method is more accurate than high-order FD schemes at coarse grid resolutions, and as accurate at finer grid resolutions. Furthermore, the results show that solutions from LS models are more robust than solutions from FD models. After this, we show that dealiasing the nonlinear advection tendencies plays an important role in the success of the LS method, especially for simulations with sharp boundaries that are marginally resolved. For adequately resolved flows, dealiasing does not necessarily improve solutions for the short-term integrations that are presented. However, aliasing errors still must be controlled to prevent a catastrophic buildup of energy at the smallest resolvable wavelengths. Finally, the LS method is tested using open lateral boundary conditions. As the LS method is a higher-order scheme, special treatment of the vertical and lateral boundaries is required. One possibility is to use lower-order versions of the LS method as boundaries are approached, and outflow conditions at the lateral boundaries. This simple treatment results in solutions that compare very favorably to the reference solution of the test problem.

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Mark A. Askelson
,
Patricia M. Pauley
, and
Jerry M. Straka

Abstract

Distance-dependent weighted averaging (DDWA) is a process that is fundamental to most of the objective analysis schemes that are used in meteorology. Despite its ubiquity, aspects of its effects are still poorly understood. This is especially true for the most typical situation of observations that are discrete, bounded, and irregularly distributed.

To facilitate understanding of the effects of DDWA schemes, a framework that enables the determination of response functions for arbitrary weight functions and data distributions is developed. An essential element of this approach is the equivalent analysis, which is a hypothetical analysis that is produced by using, throughout the analysis domain, the same weight function and data distribution that apply at the point where the response function is desired. This artifice enables the derivation of the response function by way of the convolution theorem. Although this approach requires a bit more effort than an alternative one, the reward is additional insight into the impacts of DDWA analyses.

An important insight gained through this approach is the exact nature of the DDWA response function. For DDWA schemes the response function is the complex conjugate of the normalized Fourier transform of the effective weight function. In facilitating this result, this approach affords a better understanding of which elements (weight functions, data distributions, normalization factors, etc.) affect response functions and how they interact to do so.

Tests of the response function for continuous, bounded data and discrete, irregularly distributed data verify the validity of the response functions obtained herein. They also reinforce previous findings regarding the dependence of response functions on analysis location and the impacts of data boundaries and irregular data spacing.

Interpretation of the response function in terms of amplitude and phase modulations is illustrated using examples. Inclusion of phase shift information is important in the evaluation of DDWA schemes when they are applied to situations that may produce significant phase shifts. These situations include those where data boundaries influence the analysis value and where data are irregularly distributed. By illustrating the attendant movement, or shift, of data, phase shift information also provides an elegant interpretation of extrapolation.

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Paul M. Markowski
,
Jerry M. Straka
, and
Erik N. Rasmussen

Abstract

Despite the long-surmised importance of the hook echo and rear-flank downdraft (RFD) in tornadogenesis, only a paucity of direct observations have been obtained at the surface within hook echoes and RFDs. In this paper, in situ surface observations within hook echoes and RFDs are analyzed. These “mobile mesonet” data have unprecedented horizontal spatial resolution and were obtained from the Verifications of the Origins of Rotation in Tornadoes Experiment (VORTEX) and additional field experiments conducted since the conclusion of VORTEX. The surface thermodynamic characteristics of hook echoes and RFDs associated with tornadic and nontornadic supercells are investigated to address whether certain types of hook echoes and RFDs are favorable (or unfavorable) for tornadogenesis.

Tornadogenesis is more likely and tornado intensity and longevity increase as the surface buoyancy, potential buoyancy (as measured by the convective available potential energy), and equivalent potential temperature in the RFD increase, and as the convective inhibition associated with RFD parcels at the surface decreases. It is hypothesized that evaporative cooling and entrainment of midlevel potentially cold air may play smaller roles in the development of RFDs associated with tornadic supercells compared to nontornadic supercells. Furthermore, baroclinity at the surface within the hook echo is not a necessary condition for tornadogenesis. It also will be shown that environments characterized by high boundary layer relative humidity (and low cloud base) may be more conducive to RFDs associated with relatively high buoyancy than environments characterized by low boundary layer relative humidity (and high cloud base).

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Youngsun Jung
,
Ming Xue
,
Guifu Zhang
, and
Jerry M. Straka

Abstract

A data assimilation system based on the ensemble square-root Kalman filter (EnSRF) is extended to include the additional capability of assimilating polarimetric radar variables. It is used to assess the impact of assimilating additional polarimetric observations on convective storm analysis in the Observing System Simulation Experiment (OSSE) framework. The polarimetric variables considered include differential reflectivity Z DR, reflectivity difference Z dp, and specific differential phase K DP. To simulate the observational data more realistically, a new error model is introduced for characterizing the errors of the nonpolarimetric and polarimetric radar variables. The error model includes both correlated and uncorrelated error components for reflectivities at horizontal and vertical polarizations (ZH and ZV , respectively). It is shown that the storm analysis is improved when polarimetric variables are assimilated in addition to ZH or in addition to both ZH and radial velocity Vr . Positive impact is largest when Z DR, Z dp, and K DP are assimilated all together. Improvement is generally larger in vertical velocity, water vapor, and rainwater mixing ratios. The rainwater field benefits the most while the impacts on horizontal wind components and snow mixing ratio are smaller. Improvement is found at all model levels even though the polarimetric data, after the application of thresholds, are mostly limited to the lower levels. Among Z DR, Z dp, and K DP, Z DR is found to produce the largest positive impact on the analysis. It is suggested that Z DR provides more independent information than the other variables. The impact of polarimetric data is also expected to be larger when they are used to retrieve drop size distribution parameters. The polarimetric radar data thresholding prior to assimilation is found to be necessary to minimize the impact of noise. This study is believed to be the first to directly assimilate (simulated) polarimetric data into a numerical model.

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Matthew S. Gilmore
,
Jerry M. Straka
, and
Erik N. Rasmussen

Abstract

Weisman and Klemp suggested that their liquid-only, deep convective storm experiments should be repeated with a liquid-ice microphysics scheme to determine if the solutions are qualitatively the same. Using a three-dimensional, nonhydrostatic cloud model, such results are compared between three microphysics schemes: the “Kessler” liquid-only scheme (used by Weisman and Klemp), a Lin–Farley–Orville-like scheme with liquid and ice parameterization (Li), and the same Lin–Farley–Orville-like microphysics scheme but with only liquid processes turned on (Lr). Convection is simulated using a single thermodynamic profile and a variety of shear profiles. The shear profiles are represented by five idealized half-circle wind hodographs with arc lengths (U s ) of 20, 25, 30, 40, and 50 m s−1. The precipitation, cold pool characteristics, and storm evolution produced by the different schemes are compared.

The Kessler scheme produces similar accumulated precipitation over 2 h compared to Lr for all shear regimes. Although Kessler's rain evaporation rate is 1.5–1.8 times faster in the lower troposphere, rain production is also faster via accretion and autoconversion of cloud water. In addition, nearly ∼40% more accumulated precipitation occurs in Li compared to Lr. This can be attributed primarily to increased precipitation production rates and enhanced low-level precipitation fluxes in Li for all shear regimes. Differences in the amount of precipitation reaching ground and the low-level cooling rates also cause differences in storm cold pools.

For the U s = 25 shear regime, microphysics cases with colder low-level outflow are shown to be associated with temporarily weaker (Li) or shorter-lived (Kessler) supercells as compared to cases with warmer outflow (Lr). This is consistent with a previous study showing that the cold pool has a greater relative impact on the storm updraft compared to dynamic forcing for weaker shear.

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Mark A. Askelson
,
Jean-Pierre Aubagnac
, and
Jerry M. Straka

Abstract

Spatial objective analysis is routinely performed in several applications that utilize radar data. Because of their relative simplicity and computational efficiency, one-pass distance-dependent weighted-average (DDWA) schemes that utilize either the Cressman or the Barnes filter are often used in these applications. The DDWA schemes that have traditionally been used do not, however, directly account for two fundamental characteristics of radar data. These are 1) the spacing of radar data depends on direction and 2) radar data density systematically decreases with increasing range.

A DDWA scheme based on an adaptation of the Barnes filter is proposed. This scheme, termed the adaptive Barnes (A-B) scheme, explicitly takes into account radar data properties 1 and 2 above. Both theoretical and experimental investigations indicate that two attributes of the A-B scheme, direction-splitting and automatic adaptation to data density, may facilitate the preservation of the maximum amount of meaningful information possible within the confines of one-pass DDWA schemes.

It is shown that in the idealized situation of infinite, continuous data and for an analysis in rectangular-Cartesian coordinates, a direction-splitting scheme does not induce phase shifts if the weight function is even in each direction. Moreover, for radar data that are infinite, collected at regular radial, azimuthal, and elevational increments, and collocated with analysis points, the direction-splitting design of the A-B filter removes gradients in the analysis weights. This is a beneficial attribute when considering the treatment of gradient information of rectangular Cartesian data by an analysis system because then postanalysis gradients equal the analysis of gradients. The direction-splitting design of the A-B filter is unable, however, to circumvent the impact of the varying physical distances between adjacent measurements that are inherent to the spherical coordinate system of ground-based weather radars. Because of this, even with the direction-splitting design of the A-B filter postanalysis gradients do not equal the analysis of gradients.

Ringing in the response function of a one-dimensional Barnes filter is illustrated. The negative impact of data windows on the main lobe of the response function is found to decrease rapidly as the window is widened relative to the weight function. Unless an analysis point is near a data boundary, in which case both ringing and phase shifting will adversely affect the analysis, window effects are unlikely to be significant in applications of the A-B filter to radar data.

The A-B filter has potential drawbacks, the most significant of which is misinterpretations owing to the use of the A-B filter without comprehension of its direction- and range-dependent response function. Despite its drawbacks, the A-B filter has the potential to improve analyses owing to the aforementioned attributes and thus to aid research efforts in areas such as multiple-Doppler wind analyses, pseudo-dual-Doppler analyses, and retrieval studies.

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