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## Abstract

A three-dimensional numerical simulation is presented for the asymmetric vortex motion which occurs in a Ward-type vortex chamber. The initial state is taken to be one of axisymmetric irrotational flow where the flow enters through the sides at the bottom and exits through the top of the chamber. As tangential velocity is added to the inflowing fluid, the structure of the flow in the meridional plane is modified from a ‘one-celled’ flow(updraft everywhere) to a ‘two-celled’ flow (updraft surrounding a central downdraft). Asymmetric vortices develop in the location of maximum vorticity of the ‘two-celled’ vortex which, it is shown, must be in the gradient between the updraft and the downdraft (but in updraft). Structural features of these asymmetric vortices, such as the tilt with height and propagation rate, are examined. Although the laboratory model upon which the present numerical calculations are based lacks the ability to simulate some important aspects of atmospheric flow, several significant features are shown to resemble the structure of observed tornadoes and mesocyclones.

## Abstract

A three-dimensional numerical simulation is presented for the asymmetric vortex motion which occurs in a Ward-type vortex chamber. The initial state is taken to be one of axisymmetric irrotational flow where the flow enters through the sides at the bottom and exits through the top of the chamber. As tangential velocity is added to the inflowing fluid, the structure of the flow in the meridional plane is modified from a ‘one-celled’ flow(updraft everywhere) to a ‘two-celled’ flow (updraft surrounding a central downdraft). Asymmetric vortices develop in the location of maximum vorticity of the ‘two-celled’ vortex which, it is shown, must be in the gradient between the updraft and the downdraft (but in updraft). Structural features of these asymmetric vortices, such as the tilt with height and propagation rate, are examined. Although the laboratory model upon which the present numerical calculations are based lacks the ability to simulate some important aspects of atmospheric flow, several significant features are shown to resemble the structure of observed tornadoes and mesocyclones.

## Abstract

The influence of weak mean vertical wind shear upon the trapeze instability of Orlanski (1973) is investigated. It is found that the shear limits the growth of unstable waves unless they are propagating at nearly right angles to the mean wind vector, or in other words, the equi-phase lines are parallel to the mean wind direction.

## Abstract

The influence of weak mean vertical wind shear upon the trapeze instability of Orlanski (1973) is investigated. It is found that the shear limits the growth of unstable waves unless they are propagating at nearly right angles to the mean wind vector, or in other words, the equi-phase lines are parallel to the mean wind direction.

## Abstract

Fine-resolution calculations using an axisymmetric numerical model of the flow within a Ward-type vortex chamber are discussed. Particular attention is paid to the vortex-ground interaction. Variations in the swirl ratio *S* from zero to unity lead to radically different vortex structure in the “corner” region (i.e., near *r* = *z* = 0). For *S* Lt; 1, a concentrated vortex forms in the upper chamber but not in the corner. At moderate *S*, we observe vortex breakdown, large-amplitude inertial waves, and very intense swirling motion in the corner. When *S* = 1, the central downdraft penetrates to the lower surface and the vortex breakdown occurs within the boundary layer. These results are consistent with experimental observations and suggest the explanation of a number of observed facets of tornadoes.

## Abstract

Fine-resolution calculations using an axisymmetric numerical model of the flow within a Ward-type vortex chamber are discussed. Particular attention is paid to the vortex-ground interaction. Variations in the swirl ratio *S* from zero to unity lead to radically different vortex structure in the “corner” region (i.e., near *r* = *z* = 0). For *S* Lt; 1, a concentrated vortex forms in the upper chamber but not in the corner. At moderate *S*, we observe vortex breakdown, large-amplitude inertial waves, and very intense swirling motion in the corner. When *S* = 1, the central downdraft penetrates to the lower surface and the vortex breakdown occurs within the boundary layer. These results are consistent with experimental observations and suggest the explanation of a number of observed facets of tornadoes.

## Abstract

An axisymmetric numerical model has been developed to simulate Ward's (1972) laboratory experiments. It was shown by Davies-Jones (1976) that this experiment is more geophysically relevant than all previous experiments in that Ward's experiment exhibits both dynamical and geometrical similarity to actual tornadoes.

Major results are 1) the core size versus inflow angle relationship agrees very nearly with Ward's measurements, 2) the numerical and laboratory surface pressure patterns are in agreement, and 3) it is demonstrated that the core radius is independent of the Reynolds number at high Reynolds number (Ward's data also exhibit this behavior).

Based on this axisymmetric model some speculation concerning the nature of the asymmetric multiple vortex phenomenon is made. Furthermore, the numerical model allows the examination of the interior flow field. As a consequence, an explanation is offered in Section 6 for the double-walled structure sometimes observed in natural vortices.

The experiments with no-slip boundary conditions reveal a very complicated flow structure in the vicinity of *r* = *z* = 0. The computed flow field is strongly reminiscent of that described by Benjamin (1962).

## Abstract

An axisymmetric numerical model has been developed to simulate Ward's (1972) laboratory experiments. It was shown by Davies-Jones (1976) that this experiment is more geophysically relevant than all previous experiments in that Ward's experiment exhibits both dynamical and geometrical similarity to actual tornadoes.

Major results are 1) the core size versus inflow angle relationship agrees very nearly with Ward's measurements, 2) the numerical and laboratory surface pressure patterns are in agreement, and 3) it is demonstrated that the core radius is independent of the Reynolds number at high Reynolds number (Ward's data also exhibit this behavior).

Based on this axisymmetric model some speculation concerning the nature of the asymmetric multiple vortex phenomenon is made. Furthermore, the numerical model allows the examination of the interior flow field. As a consequence, an explanation is offered in Section 6 for the double-walled structure sometimes observed in natural vortices.

The experiments with no-slip boundary conditions reveal a very complicated flow structure in the vicinity of *r* = *z* = 0. The computed flow field is strongly reminiscent of that described by Benjamin (1962).

## Abstract

A vertical velocity field is chosen which imitates that of the initial stages of cloud development as simulated numerically by Wilhelmson and Klemp (1978). Given this, an approximate version of the equation for the vertical component of the vorticity is solved. The vertical velocity is assumed to vary with height as sin πz/*H* where a is the altitude and *H* is the depth of the domain. At the level of nondivergence (z=*H*/2), the solutions indicate the development of a vortex pair which then splits into two vortex pairs one moving to the right of the mean wind and the other to the left (as observed in the numerical model). At lower levels, owing to the convergence in the updraft and divergence in the downdraft, the cyclonic/anticyclonic member of the vortex pair in the rightward/leftward moving storm is greatly enhanced. The vorticity maximum is initially on the maximum gradient of vertical velocity. At mid-levels the maximum vorticity migrates with time close to the position of maximum vertical velocity. However, at lower levels, the maximum vorticity migrates with time past the position of maximum vertical velocity and thereafter resides on the vertical velocity gradient separating updraft from downdraft, as observed in a number of case studies. Some general comparisons of the present theory with an observational case study are made.

## Abstract

A vertical velocity field is chosen which imitates that of the initial stages of cloud development as simulated numerically by Wilhelmson and Klemp (1978). Given this, an approximate version of the equation for the vertical component of the vorticity is solved. The vertical velocity is assumed to vary with height as sin πz/*H* where a is the altitude and *H* is the depth of the domain. At the level of nondivergence (z=*H*/2), the solutions indicate the development of a vortex pair which then splits into two vortex pairs one moving to the right of the mean wind and the other to the left (as observed in the numerical model). At lower levels, owing to the convergence in the updraft and divergence in the downdraft, the cyclonic/anticyclonic member of the vortex pair in the rightward/leftward moving storm is greatly enhanced. The vorticity maximum is initially on the maximum gradient of vertical velocity. At mid-levels the maximum vorticity migrates with time close to the position of maximum vertical velocity. However, at lower levels, the maximum vorticity migrates with time past the position of maximum vertical velocity and thereafter resides on the vertical velocity gradient separating updraft from downdraft, as observed in a number of case studies. Some general comparisons of the present theory with an observational case study are made.

## Abstract

Given that the earth's atmosphere may be idealized as a rotating, stratified fluid characterized by the Coriolis parameter *f* and the Brunt–V¨is¨l¨ frequency *N*, and that the diurnal cycle of heating and cooling of the land relative to the sea acts as a stationary, oscillatory source of energy of frequency ω (=2π day^{−1}), it follows from the linear theory of motion that where *f* > ω the atmospheric response is confined to within a distance *Nh*(*f*
^{−2} – ω ^{−2})^{−1/2} of the coastline, where *h* is the vertical scale of the heating. When *f* < ω, the atmospheric response is in the form of internal-inertial waves which extend to “Infinity” along ray paths extending upward and outward from the coast. Near the ground, the horizontal extent of the sea breeze is given by the horizontal wale of the dominant wave mode, *Nh*(ω^{2} – *f*
^{−2})^{−1/2}.

Although these concepts are familiar from the linear theory of motion in a rotating, stratified fluid, their relevance with respect to the interpretation of linear models of the land and sea breeze has not been emphasized in the literature. Hence, a critical historical review of extant linear models of the land and sea breeze is presented, and from these varied linear models, a simple model. which allows the above-described conclusions to be reached, is decocted.

## Abstract

Given that the earth's atmosphere may be idealized as a rotating, stratified fluid characterized by the Coriolis parameter *f* and the Brunt–V¨is¨l¨ frequency *N*, and that the diurnal cycle of heating and cooling of the land relative to the sea acts as a stationary, oscillatory source of energy of frequency ω (=2π day^{−1}), it follows from the linear theory of motion that where *f* > ω the atmospheric response is confined to within a distance *Nh*(*f*
^{−2} – ω ^{−2})^{−1/2} of the coastline, where *h* is the vertical scale of the heating. When *f* < ω, the atmospheric response is in the form of internal-inertial waves which extend to “Infinity” along ray paths extending upward and outward from the coast. Near the ground, the horizontal extent of the sea breeze is given by the horizontal wale of the dominant wave mode, *Nh*(ω^{2} – *f*
^{−2})^{−1/2}.

Although these concepts are familiar from the linear theory of motion in a rotating, stratified fluid, their relevance with respect to the interpretation of linear models of the land and sea breeze has not been emphasized in the literature. Hence, a critical historical review of extant linear models of the land and sea breeze is presented, and from these varied linear models, a simple model. which allows the above-described conclusions to be reached, is decocted.

## Abstract

In a previous paper a formula was derived for the maximum potential intensity of the tangential wind in a tropical cyclone called PI^{+}. The formula, PI^{+2} = EPI^{2} + *αr _{m}w_{m}η_{m}*, where EPI is the maximum potential intensity of the gradient wind and

*αr*represents the supergradient winds. The latter term is the product of the radius

_{m}w_{m}η_{m}*r*, the vertical velocity

_{m}*w*, the azimuthal vorticity

_{m}*η*at the radius and height of the maximum tangential wind (

_{m}*r*) and the (nearly constant)

_{m}, z_{m}*α*. Examination of a series of simulations of idealized tropical cyclones indicate an increasing contribution from the supergradient-wind term to PI

^{+}as the radius of maximum wind increases. In the present paper, the physical content of the supergradient-wind term is developed showing how it is directly related to tropical-cyclone boundary-layer dynamics. It is found that

*r*where −

_{m}w_{m}η_{m}∝ u^{2}_{min}Z_{m}(r_{m})/l_{v}(z_{m}) ∝ r_{m}*u*is the maximum boundary-layer radial inflow velocity and

_{min}*l*(

_{v}*z*) is the vertical mixing length.

## Abstract

In a previous paper a formula was derived for the maximum potential intensity of the tangential wind in a tropical cyclone called PI^{+}. The formula, PI^{+2} = EPI^{2} + *αr _{m}w_{m}η_{m}*, where EPI is the maximum potential intensity of the gradient wind and

*αr*represents the supergradient winds. The latter term is the product of the radius

_{m}w_{m}η_{m}*r*, the vertical velocity

_{m}*w*, the azimuthal vorticity

_{m}*η*at the radius and height of the maximum tangential wind (

_{m}*r*) and the (nearly constant)

_{m}, z_{m}*α*. Examination of a series of simulations of idealized tropical cyclones indicate an increasing contribution from the supergradient-wind term to PI

^{+}as the radius of maximum wind increases. In the present paper, the physical content of the supergradient-wind term is developed showing how it is directly related to tropical-cyclone boundary-layer dynamics. It is found that

*r*where −

_{m}w_{m}η_{m}∝ u^{2}_{min}Z_{m}(r_{m})/l_{v}(z_{m}) ∝ r_{m}*u*is the maximum boundary-layer radial inflow velocity and

_{min}*l*(

_{v}*z*) is the vertical mixing length.

## Abstract

In a recent study, the authors investigated the mechanisms leading to the formation of diurnal along-valley winds in a valley formed by two isolated mountain ridges on a horizontal plain. The main focus was on the relation between the valley heat budget and the valley–plain pressure difference. The present work investigates the influence of the valley surroundings on the evolution of the valley winds. Three valley–plain configurations with identical valley volumes are studied: a periodic valley, an isolated valley on a plain (the former case), and an isolated valley entrenched in an elevated plateau. According to the valley volume argument (topographic amplification factor), these three cases should develop identical temperature perturbations and thus similar along-valley winds. However, substantial differences are found between the three cases, in particular a much stronger daytime up-valley wind and nighttime down-valley wind for the plateau configuration. The analysis demonstrates the importance of the exchange of along-valley momentum between the valley atmosphere and its surroundings and of the upper-level pressure gradient in explaining the differences among the cases. Furthermore, differences in the upper-level pressure gradient are shown to be related to the heat exchange of the air above the valley atmosphere with the surroundings, which is related to larger-scale cross-valley circulations.

## Abstract

In a recent study, the authors investigated the mechanisms leading to the formation of diurnal along-valley winds in a valley formed by two isolated mountain ridges on a horizontal plain. The main focus was on the relation between the valley heat budget and the valley–plain pressure difference. The present work investigates the influence of the valley surroundings on the evolution of the valley winds. Three valley–plain configurations with identical valley volumes are studied: a periodic valley, an isolated valley on a plain (the former case), and an isolated valley entrenched in an elevated plateau. According to the valley volume argument (topographic amplification factor), these three cases should develop identical temperature perturbations and thus similar along-valley winds. However, substantial differences are found between the three cases, in particular a much stronger daytime up-valley wind and nighttime down-valley wind for the plateau configuration. The analysis demonstrates the importance of the exchange of along-valley momentum between the valley atmosphere and its surroundings and of the upper-level pressure gradient in explaining the differences among the cases. Furthermore, differences in the upper-level pressure gradient are shown to be related to the heat exchange of the air above the valley atmosphere with the surroundings, which is related to larger-scale cross-valley circulations.

## Abstract

Observations and models of nocturnal katabatic winds indicate strong low-level stability with much weaker stability aloft. When such winds encounter an embedded depression in an otherwise smooth sloping plane, the flow responds in a manner that is largely describable by the inviscid fluid dynamics of stratified flow. Building on earlier work, the present study presents a series of numerical simulations based on the simplest nontrivial idealization relevant to the observations: the height-independent flow of a two-layer stratified fluid past a two-dimensional valley. Stratified flow past a valley has received much less attention than the related problem of stratified flow past a hill. Hence, the present paper gives a detailed review of existing theory and fills a few gaps along the way. The theory is used as an interpretive guide to an extensive set of numerical simulations. The solutions exhibit a variety of behaviors that depend on the nondimensional input parameters. These behaviors range from complete flow through the valley to valley-flow stagnation to situations involving internal wave breaking, lee waves, and quasi-stationary waves in the valley. A diagram is presented that organizes the solutions into flow regimes as a function of the nondimensional input parameters.

## Abstract

Observations and models of nocturnal katabatic winds indicate strong low-level stability with much weaker stability aloft. When such winds encounter an embedded depression in an otherwise smooth sloping plane, the flow responds in a manner that is largely describable by the inviscid fluid dynamics of stratified flow. Building on earlier work, the present study presents a series of numerical simulations based on the simplest nontrivial idealization relevant to the observations: the height-independent flow of a two-layer stratified fluid past a two-dimensional valley. Stratified flow past a valley has received much less attention than the related problem of stratified flow past a hill. Hence, the present paper gives a detailed review of existing theory and fills a few gaps along the way. The theory is used as an interpretive guide to an extensive set of numerical simulations. The solutions exhibit a variety of behaviors that depend on the nondimensional input parameters. These behaviors range from complete flow through the valley to valley-flow stagnation to situations involving internal wave breaking, lee waves, and quasi-stationary waves in the valley. A diagram is presented that organizes the solutions into flow regimes as a function of the nondimensional input parameters.

## Abstract

We examine the rotation and propagation of the supercell-like convection produced by our three-dimensional cloud model. The rotation in the supercell is studied in terms of the conservation of equivalent potential vorticity and V. Bjerknes' first circulation theorem; neither of these have been used previously in this connection, and we find that they significantly contribute to the current level of understanding in this area. Using these we amplify the findings of our previous work in which we found that the source of midlevel rotation is the horizontally oriented vorticity associated with the environmental shear, while the low-level rotation derives from the baroclinic generation of horizontally oriented vorticity along the low-level cold-air boundary. We further demonstrate that these same processes that amplify the low-level rotation also produce the distinctive cloud feature known as the “wall cloud.”

We find that the thunderstorm propagates rightward primarily because of the favorable dynamic vertical pressure gradient that, owing to storm rotation, is always present on the right flank of the updraft. Simulations without precipitation physics demonstrate that this rightward propagation occurs even in the absence of a cold outflow and gust front near the surface.

## Abstract

We examine the rotation and propagation of the supercell-like convection produced by our three-dimensional cloud model. The rotation in the supercell is studied in terms of the conservation of equivalent potential vorticity and V. Bjerknes' first circulation theorem; neither of these have been used previously in this connection, and we find that they significantly contribute to the current level of understanding in this area. Using these we amplify the findings of our previous work in which we found that the source of midlevel rotation is the horizontally oriented vorticity associated with the environmental shear, while the low-level rotation derives from the baroclinic generation of horizontally oriented vorticity along the low-level cold-air boundary. We further demonstrate that these same processes that amplify the low-level rotation also produce the distinctive cloud feature known as the “wall cloud.”

We find that the thunderstorm propagates rightward primarily because of the favorable dynamic vertical pressure gradient that, owing to storm rotation, is always present on the right flank of the updraft. Simulations without precipitation physics demonstrate that this rightward propagation occurs even in the absence of a cold outflow and gust front near the surface.