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Richard Rotunno

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.

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Richard Rotunno

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.

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Richard Rotunno

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, Nh2f −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.

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Richard Rotunno

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.

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Richard Rotunno

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.

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Richard Rotunno

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).

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Richard Rotunno

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 = EPI2 + αrmwmηm , where EPI is the maximum potential intensity of the gradient wind and αrmwmηm represents the supergradient winds. The latter term is the product of the radius rm , the vertical velocity wm , the azimuthal vorticity ηm at the radius and height of the maximum tangential wind (rm , zm ), and the (nearly constant) 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 m w m η m u min 2 z m ( r m ) / l υ ( z m ) r m , where −u min is the maximum boundary layer radial inflow velocity and lυ (z) is the vertical mixing length.

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Daniel Keyser
and
Richard Rotunno

Abstract

We review and discuss a difference in interpretation of the role of turbulence in modifying the potential-vorticity distribution in the vicinity of upper-level jet-front systems. In the late 1970s, M. A. Shapiro presented observational evidence that turbulent mixing of heat can result in a positive anomaly of the Ertel potential vorticity on the cyclonic-shear side of upper-level jets near the level of maximum wind. E. F. Danielsen and collaborators disputed this evidence and the accompanying interpretation. They argued that the turbulent mixing of potential vorticity can be described in terms of downgradient diffusion, in the same sense as for a passive chemical tracer. Accordingly, turbulent mixing cannot produce anomalies from initially smooth distributions of potential vorticity. In our view, this dispute stems from differences in the averaging procedures used to analyze turbulent flows, which lead to fundamentally different definitions of potential vorticity. Shapiro defined potential vorticity as the scalar product of the averaged absolute vorticity and the averaged potential-temperature gradient, whereas Danielson et al. defined it, in their analytical framework, as the average of the scalar product of these quantities. We conclude that the positive anomaly of potential vorticity identified by Shapiro is plausible if one accepts the definition of potential vorticity used in his studies. Moreover. we believe Shapiro's alternative to be the only practical option when working with observed or simulated data. Even if Danielsen's alternative could be adopted in practice, we suggest that its utility as a tracer is problematic in view of the questionable validity of the downgradient diffusion of potential vorticity.

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Tetsuya Takemi
and
Richard Rotunno
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Richard Rotunno
and
Joseph Klemp

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.

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