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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

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

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

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

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 and Maurizio Fantini

Abstract

We investigate B. Farrell's hypothesis that the development of a surface cyclone with the passage of an upper trough, as observed by S. Pettessen and coworkers, may be understood in terms of an initial-value problem on the Eady model. We consider the response of the Eady model to perturbations whose horizontal wavelengths are short enough to ensure their stability, and whose perturbation potential vorticity is zero. We depart from Farrell with the latter condition as it eliminates the continuous spectrum and allows the evolution of the perturbation to be understood solely in terms of the two normal modes of the Eady model—one with maximum amplitude at the upper lid, which propagates eastward with respect to the midlevel flow, and one westward propagating, with maximum amplitude at the lower surface. Imagine an initial upper-level disturbance with no surface perturbation; this is represented by the two Eady modes in combination such that the initial surface perturbation pressure is zero. As the flow evolves out of this initial condition, a pressure disturbance appears at the surface as the two modes propagate past one another. That is, a surface cyclone forms, deepens, and then weakens, as the upper trough passes. This amplification of the surface trough is not due to mere geometrical interference, but rather is the consequence of an energy-exchanging interplay between waves and mean flow. This distinction is emphasized by comparison with a model in which a superficially similar phenomenon occurs, but without such an interplay.

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Tetsuya Takemi and Richard Rotunno
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Yu Du and Richard Rotunno

Abstract

The characteristics and mechanisms of diurnal rainfall and winds near the south coast of China are explored using satellite data (CMORPH), long-term hourly WRF Model data (Du model data), a simple 2D linear model, and 2D idealized simulations. Both the CMORPH and Du model data indicate that the diurnal cycle of rainfall has two propagation modes near the coast: onshore and offshore. The diurnally periodic winds (vertical motions) also show a similar propagation feature. Analysis of the rainfall budget indicates that vertically integrated vertical vapor advection plays a key role in the diurnal cycle of rainfall and thus provides a physical connection between winds and rainfall in the diurnal cycle. It was found that a simple 2D linear land–sea breeze model with a background wind can well capture the two propagation modes, which are associated with inertia–gravity waves, in terms of speed and phase. The background wind changes the pattern of the inertia–gravity waves and further affects the diurnal propagation. The effect of the background wind on the diurnal propagation was verified through idealized simulations using a simplified version of the WRF Model that can also capture the diurnal features.

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Kerry Emanuel and Richard Rotunno
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