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## 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.

## 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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## 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.

## 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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## 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.

## 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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## 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.

## 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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## 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.

## 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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## Abstract

The rise of an isolated dry thermal bubble in a quiescent unstratified environment is a prototypical natural convective flow. This study considers the rise of an isolated dry thermal bubble of ellipsoidal shape (elliptical in both horizontal and vertical cross sections). The azimuthal asymmetry of the bubble allows the vorticity tilting mechanism to operate without an environmental wind. The dry Boussinesq equations of motion are solved analytically as a Taylor series in time for the early time behavior of the bubble (involving derivatives of up to the third order in time). The analytic results are supplemented with numerical simulations to examine the longer-time behavior. The first nonzero term in the Taylor expansion for the vertical vorticity is a third-order term, and appears as a four-leaf clover pattern with lobes of alternating sign. The horizontal flow associated with this vorticity pattern first appears as a sheared stagnation point-type flow, but eventually organizes into vertical vortices that fill the bubble. The vortices induce large structural changes to the bubble and eventually reverse the sense of the azimuthal asymmetry.

## Abstract

The rise of an isolated dry thermal bubble in a quiescent unstratified environment is a prototypical natural convective flow. This study considers the rise of an isolated dry thermal bubble of ellipsoidal shape (elliptical in both horizontal and vertical cross sections). The azimuthal asymmetry of the bubble allows the vorticity tilting mechanism to operate without an environmental wind. The dry Boussinesq equations of motion are solved analytically as a Taylor series in time for the early time behavior of the bubble (involving derivatives of up to the third order in time). The analytic results are supplemented with numerical simulations to examine the longer-time behavior. The first nonzero term in the Taylor expansion for the vertical vorticity is a third-order term, and appears as a four-leaf clover pattern with lobes of alternating sign. The horizontal flow associated with this vorticity pattern first appears as a sheared stagnation point-type flow, but eventually organizes into vertical vortices that fill the bubble. The vortices induce large structural changes to the bubble and eventually reverse the sense of the azimuthal asymmetry.

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## Abstract

The Zhang–Gal-Chen single-Doppler velocity retrieval (SDVR) technique is applied to a multicell storm observed by three radars near the Orlando, Florida, airport on 9 August 1991. This dataset is unique in that 3-min volume scans at very high spatial resolution (200 m) are available during a 24-min period. The retrieved (unobserved) wind, determined using only the radial wind and reflectivity from one of the radars, is compared to the (observed) winds obtained from a hybrid three-dimensional wind synthesis.

Error statistics demonstrate that the retrievals perform best when applied in a reference frame moving with the storm; however, the results also show that the specification of this frame is problematic. The findings also indicate that, in an environment where the mean flow has a critical layer, the moving reference frame is best defined as a function of height rather than a volume mean. The benefit of such a reference frame is case dependent and is best realized in regions such as a surface cold pool or upper-level divergence at storm top.

Error statistics demonstrate that the SDVR technique recovers the horizontal wind with greater accuracy than it does the vertical velocity—suggesting that for deep convection, the absence of dynamical constraints is critical. The kinematic and O’Brien techniques and a new variational technique, in which the solution to a second-order ordinary differential equation for the vertical velocity is expressed in terms of Bessel functions, are tested as possible alternatives to the SDVR vertical velocity. Results indicate that this new technique yields vertical velocities significantly better than those using the other three methods.

## Abstract

The Zhang–Gal-Chen single-Doppler velocity retrieval (SDVR) technique is applied to a multicell storm observed by three radars near the Orlando, Florida, airport on 9 August 1991. This dataset is unique in that 3-min volume scans at very high spatial resolution (200 m) are available during a 24-min period. The retrieved (unobserved) wind, determined using only the radial wind and reflectivity from one of the radars, is compared to the (observed) winds obtained from a hybrid three-dimensional wind synthesis.

Error statistics demonstrate that the retrievals perform best when applied in a reference frame moving with the storm; however, the results also show that the specification of this frame is problematic. The findings also indicate that, in an environment where the mean flow has a critical layer, the moving reference frame is best defined as a function of height rather than a volume mean. The benefit of such a reference frame is case dependent and is best realized in regions such as a surface cold pool or upper-level divergence at storm top.

Error statistics demonstrate that the SDVR technique recovers the horizontal wind with greater accuracy than it does the vertical velocity—suggesting that for deep convection, the absence of dynamical constraints is critical. The kinematic and O’Brien techniques and a new variational technique, in which the solution to a second-order ordinary differential equation for the vertical velocity is expressed in terms of Bessel functions, are tested as possible alternatives to the SDVR vertical velocity. Results indicate that this new technique yields vertical velocities significantly better than those using the other three methods.

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## Abstract

A theory is presented for the Great Plains low-level jet in which the jet emerges in the sloping atmospheric boundary layer as the nocturnal phase of an oscillation arising from diurnal variations in turbulent diffusivity (Blackadar mechanism) and surface buoyancy (Holton mechanism). The governing equations are the equations of motion, mass conservation, and thermal energy for a stably stratified fluid in the Boussinesq approximation. Attention is restricted to remote (far above slope) geostrophic winds that blow along the terrain isoheights (southerly for the Great Plains). Diurnally periodic solutions are obtained analytically with diffusivities that vary as piecewise constant functions of time and slope buoyancies that vary as piecewise linear functions of time. The solution is controlled by 11 parameters: slope angle, Coriolis parameter, free-atmosphere Brunt–Väisälä frequency, free-atmosphere geostrophic wind, radiative damping parameter, day and night diffusivities, maximum and minimum surface buoyancies, and times of maximum surface buoyancy and sunset. The Holton mechanism, by itself, results in relatively weak wind maxima but produces strong jets when paired with the Blackadar mechanism. Jets with both Blackadar and Holton mechanisms operating are shown to be broadly consistent with observations and climatological analyses. Jets strengthen with increasing geostrophic wind, maximum surface buoyancy, and day-to-night ratio of the diffusivities and weaken with increasing Brunt–Väisälä frequency and magnitude of minimum slope buoyancy (greater nighttime cooling). Peak winds are maximized for slope angles characteristic of the Great Plains.

## Abstract

A theory is presented for the Great Plains low-level jet in which the jet emerges in the sloping atmospheric boundary layer as the nocturnal phase of an oscillation arising from diurnal variations in turbulent diffusivity (Blackadar mechanism) and surface buoyancy (Holton mechanism). The governing equations are the equations of motion, mass conservation, and thermal energy for a stably stratified fluid in the Boussinesq approximation. Attention is restricted to remote (far above slope) geostrophic winds that blow along the terrain isoheights (southerly for the Great Plains). Diurnally periodic solutions are obtained analytically with diffusivities that vary as piecewise constant functions of time and slope buoyancies that vary as piecewise linear functions of time. The solution is controlled by 11 parameters: slope angle, Coriolis parameter, free-atmosphere Brunt–Väisälä frequency, free-atmosphere geostrophic wind, radiative damping parameter, day and night diffusivities, maximum and minimum surface buoyancies, and times of maximum surface buoyancy and sunset. The Holton mechanism, by itself, results in relatively weak wind maxima but produces strong jets when paired with the Blackadar mechanism. Jets with both Blackadar and Holton mechanisms operating are shown to be broadly consistent with observations and climatological analyses. Jets strengthen with increasing geostrophic wind, maximum surface buoyancy, and day-to-night ratio of the diffusivities and weaken with increasing Brunt–Väisälä frequency and magnitude of minimum slope buoyancy (greater nighttime cooling). Peak winds are maximized for slope angles characteristic of the Great Plains.

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## 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.

## 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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## Abstract

The authors present herein an analysis of a single-Doppler velocity retrieval (SDVR) technique whereby the unobserved wind components are determined from single-Doppler radar data. The analysis is designed to provide information about the behavior and/or sensitivity of the SDVR scheme as a function of various internal and external parameters as well as about observational errors and weights.

Results presented for retrieval of both the mean and local flow indicate that the SDVR breaks down if the reflectivity gradient vanishes or if a reflectivity isoline is locally perpendicular to the radar beam. In the absence of reflectivity or radial velocity errors, the mean flow solution is independent of the integration area, the radar location, the signal wavenumber, and the weights. Given perfect radial wind information, error in the reflectivity field degrades the solution. Contrary to the error-free solution, the solution with error depends on the integration area.

Error statistics indicate that radial wind information alone is not sufficient to retrieve the local wind. Reduced error norms reveal that an optimal (i.e., reduced error norms) integration area exists that is dependent upon the length of time between radar volume scans, suggesting that the velocity field is *not* stationary (as was assumed) over these scans.

## Abstract

The authors present herein an analysis of a single-Doppler velocity retrieval (SDVR) technique whereby the unobserved wind components are determined from single-Doppler radar data. The analysis is designed to provide information about the behavior and/or sensitivity of the SDVR scheme as a function of various internal and external parameters as well as about observational errors and weights.

Results presented for retrieval of both the mean and local flow indicate that the SDVR breaks down if the reflectivity gradient vanishes or if a reflectivity isoline is locally perpendicular to the radar beam. In the absence of reflectivity or radial velocity errors, the mean flow solution is independent of the integration area, the radar location, the signal wavenumber, and the weights. Given perfect radial wind information, error in the reflectivity field degrades the solution. Contrary to the error-free solution, the solution with error depends on the integration area.

Error statistics indicate that radial wind information alone is not sufficient to retrieve the local wind. Reduced error norms reveal that an optimal (i.e., reduced error norms) integration area exists that is dependent upon the length of time between radar volume scans, suggesting that the velocity field is *not* stationary (as was assumed) over these scans.