Vertical Vorticity Equation Research Articles

Evaluating Vertical Velocity Retrievals from Vertical Vorticity Equation Constrained Dual-Doppler Analysis of Real, Rapid-Scan Radar Data

Abstract Accurate vertical velocity retrieval from dual-Doppler analysis (DDA) is a long-standing problem of radar meteorology. Typical radar scanning strategies poorly observe the vertical component of motion, leading to large uncertainty in vertical velocity estimates. Using a vertical vorticity equation constraint in addition to a mass conservation constraint in DDA has shown promise in improving vertical velocity retrievals. However, observation system simulation experiments (OSSEs) suggest this technique requires rapid radar volume scans to realize the improvements due to the vorticity tendency term in the vertical vorticity constraint. Here, the vertical vorticity constraint DDA is tested with real, rapid-scan radar data to validate prior OSSEs results. Generally, the vertical vorticity constraint DDA produced more accurate vertical velocities from DDAs than those that did not use the constraint. When the time between volume scans was greater than 30 s, the vertical velocity accuracy was significantly affected by the vorticity tendency estimation method. A technique that uses advection correction on provisional DDA wind fields to shorten the discretization interval for the vorticity tendency calculation improved the vertical velocity retrievals for longer times between volume scans. The skill of these DDAs was similar to those using a shorter time between volume scans. These improvements were due to increased accuracy of the vertical vorticity tendency using the advection correction technique. The real radar data tests also revealed that the vertical vorticity constraint DDAs are more forgiving to radar data errors. These results suggest that vertical vorticity constraint DDA with rapid-scan radars should be prioritized for kinematic analyses.

MPV-Q⁎ view of vorticity development in a saturated atmosphere

Based on the three-dimensional moist potential vorticity (MPV) equation and diabatic heating excluding latent heating (Q*), a Lagrangian vertical vorticity equation is deduced for a saturated atmosphere. The convective stability and water vapor are considered in this equation, and the impacts of sensible heating and radiative heating on the vertical vorticity development are focused. This MPV-Q* theory is then applied to a vortex case over the Yangtze-Huaihe River valley of China in 2016. The contributions of the convective stability, MPV and the horizontal component of MPV (MPV2) to the development of vertical vorticity are investigated, and the influence of nonuniform spatial distribution of Q* on the movement of the vortex is expounded.Theoretical deduction and case analysis show that, the more convectively neutral the atmosphere tends to be, the more favorable it is for the vortex development. Q⁎ affects the development of vertical vorticity by affecting MPV. Considering the convective stability, the effect of Q* on vortices can be divided into four scenarios. When the atmosphere is convectively unstable (stable), the increase (decrease) in MPV2 will lead to an increase in vertical vorticity. Furthermore, the case analysis reveals that the maximum Q* corresponds coincidentally to the high-level clouds, implying the possible effect of cloud radiation.

High-Resolution, Rapid-Scan Dual-Doppler Retrievals of Vertical Velocity in a Simulated Supercell

AbstractObservation system simulation experiments are used to evaluate different dual-Doppler analysis (DDA) methods for retrieving vertical velocity w at grid spacings on the order of 100 m within a simulated tornadic supercell. Variational approaches with and without a vertical vorticity equation constraint are tested, along with a typical (traditional) method involving vertical integration of the mass conservation equation. The analyses employ emulated radar data from dual-Doppler placements 15, 30, and 45 km east of the mesocyclone, with volume scan intervals ranging from 10 to 150 s. The effect of near-surface data loss is examined by denying observations below 1 km in some of the analyses. At the longer radar ranges and when no data denial is imposed, the “traditional” method produces results similar to those of the variational method and is much less expensive to implement. However, at close range and/or with data denial, the variational method is much more accurate, confirming results from previous studies. The vorticity constraint shows the potential to improve the variational analysis substantially, reducing errors in the w retrieval by up to 30% for rapid-scan observations (≤30 s) at close range when the local vorticity tendency is estimated using spatially variable advection correction. However, the vorticity constraint also degrades the analysis for longer scan intervals, and the impact diminishes with increased range. Furthermore, analyses using 30-s data also frequently outperform analyses using 10-s data, suggesting a limit to the benefit of increasing the radar scan rate for variational DDA employing the vorticity constraint.

On the interactions of vorticity, divergence and deformation in a meso-\u03b1-scale vortex

Vorticity, divergence and deformation are three differentiation properties of the air flow. In this paper, by deriving the tendency equations of squared vorticity (SV), squared divergence (SD) and squared deformation (SE), it is found that the three properties can interact with each other through the terms with opposite signs in the equations. These opposite-sign terms permit a quantitative discussion on the interactions between vorticity, divergence and deformation, and the role of these interactions on weather developments. The corresponding framework is first applied to investigate the development of an extratropical meso-α-scale cyclonic vortex in China. Through diagnosing the SV equation and the interactive terms in the vortex event, it is found that the interaction between SV and SD was the controlling factor for the evolution of the vortex. The transition of SD–SV can intensify the vortex, while the transition of SV–SD causes the vortex to die out. This is consistent with the relation between vorticity and divergence that has been obtained from the vertical vorticity equation. Apart from this, a new finding is that deformation also plays a role in influencing the vortex development. On one hand, deformation can interact with vorticity to affect the vortex directly. On the other hand, deformation interacts with divergence and influence the vortex development in an indirect way. These results provide a new insight on the vortex development which is worth further investigating in the future.

Estimates of Gradients in Radar Moments Using a Linear Least Squares Derivative Technique

Abstract The local, linear, least squares derivative (LLSD) approach to radar analysis is a method of quantifying gradients in radar data by fitting a least squares plane to a neighborhood of range bins and finding its slope. When applied to radial velocity fields, for example, LLSD yields part of the azimuthal (rotational) and radial (divergent) components of horizontal shear, which, under certain geometric assumptions, estimate one-half of the two-dimensional vertical vorticity and horizontal divergence equations, respectively. Recent advances in computational capacity as well as increased usage of LLSD products by the meteorological community have motivated an overhaul of the LLSD methodology’s application to radar data. This paper documents the mathematical foundation of the updated LLSD approach, including a complete derivation of its equation set, discussion of its limitations, and considerations for other types of implementation. In addition, updated azimuthal shear calculations are validated against theoretical vorticity using simulated circulations. Applications to nontraditional radar data and new applications to nonvelocity radar data including reflectivity at horizontal polarization, spectrum width, and polarimetric moments are also explored. These LLSD gradient calculations may be leveraged to identify and interrogate a wide variety of severe weather phenomena, either directly by operational forecasters or indirectly as part of future automated algorithms.

Tilting of Horizontal Shear Vorticity and the Development of Updraft Rotation in Supercell Thunderstorms

AbstractThe question of how rotation arises in sheared updrafts is analyzed using the shear and curvature vorticity framework. Local rotation exists where the shear and curvature vorticity have a similar magnitude and the same sign, such that parcels are in near-solid-body rotation. It is shown that the tilting terms of the vertical vorticity equation cannot explain the development of local rotation in the canonical cases where the horizontal vorticity is either purely streamwise or purely crosswise. Rather, vertical shear vorticity develops if crosswise vorticity is tilted, and vertical curvature vorticity develops if streamwise vorticity is tilted. To analyze how local rotation develops, two simulations of updrafts in an environment with crosswise and mostly streamwise vorticity, respectively, are discussed. A trajectory analysis is performed and shear and curvature vorticity budgets are analyzed. It is found that much of the horizontal vorticity near the updraft becomes streamwise, which results from pressure gradient accelerations in the vicinity of the updraft. Consequently, in the analyzed scenarios, the tilting mechanism results primarily in vertical curvature vorticity. Local rotation is achieved via an interchange process that facilitates a partial conversion of vertical curvature vorticity to vertical shear vorticity. Updraft rotation in supercells thus does not result from tilting of horizontal vorticity alone, but partial conversion of curvature to shear vorticity is also required.

The mesoscale balance and imbalance and the corresponding potential vorticity inversion from the view of helicity

The static balance and the geostrophic balance are the common balances in meteorology. All the synoptic systems and most of the mesoscale systems satisfy the above two balances. However, due to the strong convection and non-geostrophic feature, many mesoscale systems usually present as static imbalance, and the quasi-geostrophic approximation is no longer attainable. This paper tried to find out a kind of balance that exists for mesoscale convective system. To do this, the concrete mathematics definitions for balance and imbalance equations were defined. Then, it is proposed that the new balance equation should include the divergence, vorticity, and vertical motion simultaneously, and the helicity equation was a good choice for the basis. Finally, the mesoscale balance and imbalance equations were constructed, as well as a new balance model that was based on the helicity, horizontal divergence, vertical vorticity, continuity, and thermal dynamic equations under same approximations. Moreover, the corresponding potential vorticity (PV) inversion technique was introduced. It was pointed out that by using the PV conservation and the potential temperature conservation, the flows of the mesoscale balance model can be deduced, and their comparison with the real fields would give the degree of the imbalance.

Discussion on the complete-form vorticity equation and slantwise vorticity development

The complete form of the vertical vorticity tendency equation (the complete-form vorticity equation) is derived from the Ertel potential vorticity equation to contain thermodynamic factors. In this study, a new complete-form vorticity equation, which has the same form as the original complete-form vorticity equation, is deduced from the absolute vorticity vector equation combined with the continuity equation and the expression of three-dimensional (3D) entropy gradient. By comparing the complete-form vorticity equation with the classical vertical vorticity equation, it is found that regardless of whether or not the isentropic surface is tilting, the two vorticity equations are in essence the same. The “baroclinic term” of the complete-form vorticity equation is exactly equal to the solenoidal term of the classical one, and there is a significant amount of cancellation between the two baroclinic items (the “slantwise term” and the horizontal vorticity change term) in the complete-form vorticity equation. In operational weather analysis, the tilt of the isentropic surface can be diagnosed according to the density of the isotherm on the upper-level isobaric map. For synoptic-scale motion, the vertical vorticity produced by the tilt of the isentropic surface is due to the contribution of atmospheric baroclinicity, which is measured by the solenoid. The 3D solenoid is parallel to the isentropic surface, so the more tilted the isentropic surface, the bigger the projection of the 3D solenoid in the vertical direction. The baroclinic contribution can be interpreted based on the PV thinking theory, but the relationship between the vorticity field and the potential vorticity field is not immediate.

The impact of deformation on vortex development in a baroclinic moist atmosphere

A mathematical relation between deformation and vertical vorticity tendency is built by introducing the frontogenesis function and the complete vertical vorticity equation, which is derived by virtue of moist potential vorticity. From the mathematical relation, it is shown that properly configured atmospheric conditions can make deformation exert a positive contribution to vortex development at rates comparable to other favorable factors. The effect of deformation on vortex development is not only related to the deformation itself, but also depends on the current thermodynamic and dynamic structures of the atmosphere, such as the convective stability, moist baroclinicity and vertical wind shear (or horizontal vorticity). A diagnostic study of a heavy-rainfall case that occurred during 20–22 July 2012 shows that deformation has the most remarkable effect on the increase in vertical vorticity during the rapid development stage of the low vortex during its whole life cycle. This feature is mainly due to the existence of an approximate neutral layer (about 700 hPa) in the atmosphere where the convective stability tends to be zero. The neutral layer makes the effect of deformation on the vertical vorticity increase significantly during the vortex development stage, and thus drives the vertical vorticity to increase.

An Application of the Immersed Boundary Method for Recovering the Three-Dimensional Wind Fields over Complex Terrain Using Multiple-Doppler Radar Data

This study develops an extension of a variational-based multiple-Doppler radar synthesis method to construct the three-dimensional wind field over complex topography. The immersed boundary method (IBM) is implemented to take into account the influence imposed by a nonflat surface. The IBM has the merit of providing realistic topographic forcing without the need to change the Cartesian grid configuration into a terrain-following coordinate system. Both Dirichlet and Neumann boundary conditions for the wind fields can be incorporated. The wind fields above the terrain are obtained by variationally adjusting the solutions to satisfy a series of weak constraints, which include the multiple-radar radial velocity observations, anelastic continuity equation, vertical vorticity equation, background wind, and spatial smoothness terms. Experiments using model-simulated data reveal that the flow structures over complex orography can be successfully retrieved using radial velocity measurements from multiple Doppler radars. The primary advantages of the original synthesis method are still maintained, that is, the winds along and near the radar baseline are well retrieved, and the resulting three-dimensional flow fields can be used directly for vorticity budget diagnosis. If compared with the traditional wind synthesis algorithm, this method is able to merge data from different sources, and utilize data from any number of radars. This provides more flexibility in designing various scanning strategies, so that the atmosphere may be probed more efficiently using a multiple-radar network. This method is also tested using the radar data collected during the Southwest Monsoon Experiment (SoWMEX), which was conducted in Taiwan from May to June 2008 with reasonable results being obtained.

Impact of a Vertical Vorticity Constraint in Variational Dual-Doppler Wind Analysis: Tests with Real and Simulated Supercell Data

Abstract One of the greatest challenges to dual-Doppler retrieval of the vertical wind is the lack of low-level divergence information available to the mass conservation constraint. This study examines the impact of a vertical vorticity equation constraint on vertical velocity retrievals when radar observations are lacking near the ground. The analysis proceeds in a three-dimensional variational data assimilation (3DVAR) framework with the anelastic form of the vertical vorticity equation imposed along with traditional data, mass conservation, and smoothness constraints. The technique is tested using emulated radial wind observations of a supercell storm simulated by the Advanced Regional Prediction System (ARPS), as well as real dual-Doppler observations of a supercell storm that occurred in Oklahoma on 8 May 2003. Special attention is given to procedures to evaluate the vorticity tendency term, including spatially variable advection correction and estimation of the intrinsic evolution. Volume scan times ranging from 5 min, typical of operational radar networks, down to 30 s, achievable by rapid-scan mobile radars, are considered. The vorticity constraint substantially improves the vertical velocity retrievals in our experiments, particularly for volume scan times smaller than 2 min.

Relationship between the western Pacific subtropical high and the subtropical East Asian diabatic heating during south China heavy rains in June 2005

Based on the daily NCEP/NCAR reanalysis data, the position variation of the western Pacific subtropical high (WPSH) in June 2005 and its relation to the diabatic heating in the subtropical East Asia are analyzed using the complete vertical vorticity equation. The results show that the position variation of the WPSH is indeed associated with the diabatic heating in the subtropical East Asian areas. In comparison with June climatology, stronger heating on the north side of the WPSH and relatively weak ITCZ (intertropical convergence zone) convection on the south side of the WPSH occurred in June 2005. Along with the northward movement of the WPSH, the convective latent heating extended northward from the south side of the WPSH. The heating to the west of the WPSH was generally greater than that inside the WPSH, and each significant enhancement of the heating field corresponded to a subsequent westward extension of the WPSH. In the mid troposphere, the vertical variation of heating on the north of the WPSH was greater than the climatology, which is unfavorable for the northward movement of the WPSH. On the other hand, the vertical variation of heating south of the WPSH was largely smaller than the climatology, which is favorable for the anomalous increase of anticyclonic vorticity, leading to the southward retreat of the WPSH. Before the westward extension of the WPSH in late June 2005, the vertical variation of heating rates to (in) the west (east) of the WPSH was largely higher (lower) than the climatology, which is in favor of the increase of anticyclonic (cyclonic) vorticity to (in) the west (east) of the WPSH, inducing the subsequent westward extension of the WPSH. Similar features appeared in the lower troposphere. In a word, the heating on the north-south, east-west of the WPSH worked together, resulting in the WPSH extending more southward and westward in June 2005, which is favorable to the maintenance of the rainbelt in South China.

A Variational Multiple–Doppler Radar Three-Dimensional Wind Synthesis Method and Its Impacts on Thermodynamic Retrieval

Abstract A multiple–Doppler radar synthesis method is developed to recover the three-dimensional wind field. In this method the solutions are obtained by variationally adjusting the winds to satisfy a series of constraints in weak formats. Among them, the primary ones are multiple-radar radial velocity observations, the anelastic continuity equation, the vertical vorticity equation, the background wind, and spatial smoothness terms. The retrieved wind products are at two time levels, and can be readily applied to deduce the information about the pressure and temperature through the use of the thermodynamic retrieval algorithm, in which the temporal derivatives of the wind fields are required. Experiments using model-simulated data are conducted, from which two major findings are obtained. First, the wind field along and near the radar baseline can still be recovered. This is a major advantage over the traditional approach. Therefore, the proposed method is capable of providing uninterrupted observations of a weather system as it passes the baseline. This allows for more flexibility when designing the radar deployment in field experiments. Second, if the winds are applied to infer the thermodynamic fields using the traditional dynamic retrieval method, and an extra sounding (e.g., radiosonde or dropsonde) is combined in order to specify the horizontal average of the thermodynamic perturbations, the preferable place to release this sounding is within the region of weak, rather than strong, convection. In additional to the aforementioned findings, with this method there is no need to prescribe the top or bottom boundary conditions for the vertical velocities in the traditional sense. Since the computation is performed without explicit vertical integration of the continuity equation, the problem of error accumulation due to inappropriate boundary conditions for the vertical velocities is prevented. These finding are consistent with some previous publications. Furthermore, the instability that occurs during traditional iterative dual-Doppler wind synthesis based on a Cartesian coordinate can also be avoided. Finally, data from any number of radars can be easily added to the computation. This method is also tested using the radar datasets collected during the Southwest Monsoon Experiment/Terrain-Influenced Monsoon Rainfall Experiment (SoWMEX/TiMREX), which was conducted from May to June 2008 in Taiwan, and reasonable results are obtained.

Use of a Vertical Vorticity Equation in Variational Dual-Doppler Wind Analysis

Abstract The utility of the anelastic vertical vorticity equation in a weak-constraint (least squares error) variational dual-Doppler wind analysis procedure is explored. The analysis winds are obtained by minimizing a cost function accounting for the discrepancies between observed and analyzed radial winds, errors in the mass conservation equation, errors in the anelastic vertical vorticity equation, and spatial smoothness constraints. By using Taylor’s frozen-turbulence hypothesis to shift analysis winds to observation points, discrepancies between radially projected analysis winds and radial wind observations can be calculated at the actual times and locations the data are acquired. The frozen-turbulence hypothesis is also used to evaluate the local derivative term in the vorticity equation. Tests of the analysis procedure are performed with analytical pseudo-observations of an array of translating and temporally decaying counterrotating updrafts and downdrafts generated from a Beltrami flow solution of the Navier–Stokes equations. The experiments explore the value added to the analysis by the vorticity equation constraint in the common scenario of substantial missing low-level data (radial wind observations at heights beneath 1.5 km are withheld from the analysis). Experiments focus on the sensitivity of the most sensitive analysis variable—the vertical velocity component—to values of the weighting coefficients, volume scan period, number of volume scans, and errors in the estimated frozen-turbulence pattern-translation components. Although the vorticity equation constraint is found to add value to many of these analyses, the analysis can become significantly degraded if estimates of the pattern-translation components are largely in error or if the frozen-turbulence hypothesis itself breaks down. However, tests also suggest that these negative impacts can be mitigated if data are available in a rapid-scan mode.

A Phase-Plot Method for Diagnosing Vorticity Concentration Mechanisms in Mesoscale Convective Vortices

Abstract Mesoscale convective vortex (MCV) analysis results show that these vortices form by way of different evolutionary paths. Rewriting the traditional form of the relative vertical vorticity equation in terms of momentum advection curl produces an alternative form of the equation containing two terms. When the terms are normalized and plotted on orthogonal axes, a phase-plot path depicting MCV evolutionary growth is created. Thermodynamics is included in the phase plot by correlating the path to the heating characteristics of the troposphere. The application of the phase-plot scheme to several cases shows that for MCV formation events, there are two interconnected regions that combine to produce the vortex. The upper-middle- and upper-troposphere vorticity growth is governed primarily by vertical motion, with heating driving the vorticity growth in the upper-middle region. The lower-middle and lower-troposphere vorticity growth is governed primarily by horizontal motion, with the vertical heating gradient driving the vorticity growth in the lower-middle region. Which regime leads the vorticity growth is found to be case dependent. In the middle troposphere, evolutionary paths are governed by the relative strengths of heating and heating gradient. Additional phase-plot and mesoscale analyses clarify the characteristics of two MCV formation modes. In some cases, heating drives the complete formation of the MCV, whereas in cases with lesser heating, tipping is vital to the MCV formation process. In total, these results help synthesize many of the various discoveries regarding the origin and formation of the MCV.

Comparing the Dynamical Development of Two Mesoscale Convective Vortices

Abstract Examination of much of the literature regarding mesoscale convective vortices (MCVs) exposes significant incongruity over their origin and development. A vorticity budget analysis can conclusively demonstrate that incongruous results are due, at least in part, to different MCV developmental paths. A study of the dynamics of two numerically generated MCVs for the purpose of comparing their evolution from the initial stages to maturity produced the result that the final MCV structure was achieved from different developmental paths. Each MCV was diagnosed using model output from the fifth-generation Pennsylvania State University–NCAR Mesoscale Model (MM5). The analysis consisted of detailing the dynamics of the MCV formation through a diagnosis of the vertical vorticity equation. The results of the study, combined with a comparison with other studies, led to the conclusion that the same mature vertical MCV structure is produced from a variety of dynamical paths. Such a conclusion impacts the nature ...

Moist Potential Vorticity and Up-Sliding Slantwise VorticityDevelopment

By using the moist potential vorticity equation derived from completeatmospheric equations including the effect of mass forcing, the theoryof up-sliding slantwise vorticity development (USVD) is proposed basedon the theory of slantwise vorticity development. When an air parcelslides up along a slantwise isentropic surface, its vertical componentof relative vorticity is developed. Based on the theory of USVD, acomplete vertical vorticity equation is expected with mass forcing,which explicitly includes the effect of both internal forcings andexternal forcings.

Early Evolution of Vertical Vorticity in a Numerically Simulated Idealized Convective Line

The generation and organization of mesoscale convective vortices (MCVs) is a recurring theme in midlatitude and tropical meteorology during the warm season. In this work a simulation of a finite-length idealized convective line in a westerly shear environment is investigated in the absence of ambient vertical vorticity. An asymmetry in average vertical vorticity forms rapidly at early times in the present simulation. This study focuses on the formation and organization of vertical vorticity at these early simulation times. Previous simulations suggest that tilting of either ambient or storm-generated horizontal vorticity is the primary mechanism responsible for the formation, organization, and maintenance of MCVs. This study confirms recent work regarding the generation of vertical vorticity at early times in the simulation. A Lagrangian budget analysis of the vertical vorticity equation, however, shows that vorticity convergence becomes a comparable, and at times dominant, mechanism for the enhancement and long-term organization of vertical vorticity early in the simulation. Despite differences in the initial ambient horizontal vorticity, hodograph, and convective available potential energy, the Lagrangian budget analysis in the present midlatitude case is consistent with the Lagrangian budget results of a previous tropical squall line simulation. The study of idealized convective lines in midlatitude environmental conditions therefore provide valuable insight into understanding vertical vorticity production in tropical squall lines and their potential relevance to tropical cyclogenesis.

Use of the Vorticity Equation in Dual-Doppler Analysis of the Vertical Velocity Field

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...

Global numerical prediction of bursting frequency in turbulent boundary layers

The frequencies of the bursting events associated with the streamwise coherent structures of spatially developing incompressible turbulent boundary layers were predicted. The structures were modeled as wavelike disturbances associated with the turbulent mean flow using a direct‐resonance theory. Global numerical solutions for the resonant eigenmodes of the Orr‐Sommerfeld and the vertical vorticity equations were developed. The global method involves the use of second and fourth order accurate finite difference formulae for the differential equations as well as the boundary conditions. The predicted resonance frequencies were found to agree very well with previous results using a local shooting technique and measured data.