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Piotr K. Smolarkiewicz
and
Richard Rotunno

Abstract

The present paper contains a continuation of our study of the flow of a density-stratified fluid past three-dimensional obstacles for Froude number ∼O(1). Linear theory (large Froude number) and potential-flow-type theory (small Froude number) are both invalid in this range, which is of particular relevance to natural, atmospheric flows past large mesoscale mountains. The present study was conceived to provide a systematic investigation of the basic aspects of this flow. Thus, we have excluded the effects of friction, rotation, nonuniform ambient flow, and the complexity of realistic terrain. In Part I of this study we focused on the pair of vertically oriented vortices forming on the lee side when the Froude number decreases below 0.5 (approximately), and argued that their formation may be understood in terms of nonlinear aspects of inviscid gravity waves, i.e., without invoking traditional arguments on the separation of the friction boundary layer. Herein we examine the zone of flow reversal on the windward side of the obstacle, which is also a characteristic feature of the low-Froude number flow. We find that flow stagnation and a tendency for flow reversal upwind of a symmetric bell-shaped obstacle is well predicted by linear inviscid gravity-wave theory. This finding stands in contrast with “horseshoe-vortex” arguments (attributed to frictional boundary layer separation) often invoked in the literature. We also perform experiments on obstacles of varying aspect ratio, β (across-stream length/along-stream length). Here the utility of the linear theory is less clear: considering cases with Fr = 0.33 and, β → ∞, we find that for β ≤ 1 upstream-propagating columnar modes are essentially absent, however, for β increasing beyond unity, they appear with increasing strength. It has been argued in the literature that having a sufficiently strong columnar mode is the means by which the upwind flow is brought to stagnation. The absence of this effect for β ≤ 1 (even though there is upwind-flow stagnation) and its appearance for β > 1 indicate the coexistence of two distinct gravity-wave effects that decelerate the flow upwind of an obstacle at low Froude number.

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David J. Raymond
and
Richard Rotunno

Abstract

The spreading of the low-level cold pool produced by evaporation of precipitation is generally acknowledged to be an important mechanism for the regeneration of moist convection. We show that cooling a stably stratified nocturnal boundary layer produces very different results from the corresponding daytime case in which the boundary layer is neutral. In particular, dynamic behavior is sometimes closer to that of a gravity wave than to a density current. The gravity wave speed defines a minimum propagation speed for a self-regenerating convective system.

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Richard Rotunno
and
Piotr K. Smolarkiewicz

Abstract

No abstract available.

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Piotr K. Smolarkiewicz
and
Richard Rotunno

Abstract

No abstract available.

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Joseph B. Klemp
and
Richard Rotunno

Abstract

The transition of a supercell thunderstorm into its tornadic phase is investigated through high-resolution numerical cloud model simulations initiated within the interior portion of a previously simulated mature supercell storm. With the enhanced grid resolution, the low-level cyclonic vorticity increases dramatically, and the gust front rapidly occludes as small-scale downdrafts develop in the vicinity of the low-level center of circulation. As the occlusion progresses, a ring of high-vorticity air surrounds the circulation center and could be conducive to multiple vortex tornado formation. Numerous features of the simulated transition bear resemblance to those observed in tornadic storms. In the model simulation, the large low-level vorticity is generated through the tilting and intense stretching of air from the inflow side of the storm. This vertical vorticity is derived from the horizontal vorticity of the environmental shear and also from horizontal vorticity generated solenoidally as low-level air approaches the storm along the forward flank cold outflow boundary. Intensification of the rear flank downdraft during the occluding phase is dynamically driven by the strong low-level circulation.

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Richard Rotunno
and
Piotr K. Smolarkiewicz

Abstract

The authors attempt to find a bridge between the vorticity dynamics of a finite cross-stream length hydraulic jump implied by the Navier-Stokes equations and that given by the shallow-water approximation (SWA) with the turbulence of the hydraulic jump parameterized. It is established that, in the actual hydraulic jump, there is horizontal vorticity associated with the time-mean flow in the fluid interior, and that this vorticity has been fluxed down by turbulent eddies from the upper part of the fluid layer. The authors then point out that this vertical flux of cross-stream vorticity component is (minus) the cross-stream flux of vertical vorticity component. The divergence of the latter at the lateral edges of a hydraulic jump of finite cross-stream extent produces time-mean vertical vorticity.) Hence, the line of inquiry devolves to a search for the source of the cross-stream vorticity that is being fluxed downward. For a hydraulic jump in the Ice of a submerged obstacle, the authors argue that that source is the baroclinic production of vorticity at the free surface. It is shown that the SWA version of the flow through the jump requires that the vertical flux of cross-stream vorticity component be independent of depth (but not zero), and that previously only its role as (minus) the cross-stream flux of vertical vorticity has been discussed. On the understanding developed herein of the actual hydraulic-jump vorticity dynamics and the SWA version, the authors describe the relation between the vorticity distributions found in shallow-water models with paramerized turbulence and that in a continuously stratified model of flow past an obstacle.

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David J. Raymond
and
Richard Rotunno

Abstract

Correction to Volume 46, Issue 18, Article 2830.

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Impact of Vertical Wind Shear on Gravity Wave Propagation in the Land–Sea-Breeze Circulation at the Equator

Yu Du
,
Richard Rotunno
, and
Fuqing Zhang

Abstract

The impact of vertical wind shear on the land–sea-breeze circulation at the equator is explored using idealized 2D numerical simulations and a simple 2D linear analytical model. Both the idealized and linear analytical models indicate Doppler shifting and attenuation effects coexist under the effect of vertical wind shear for the propagation of gravity waves that characterize the land–sea-breeze circulation. Without a background wind, the idealized sea breeze has two ray paths of gravity waves that extend outward and upward from the coast. A uniform background wind causes a tilting of the two ray paths due to Doppler shifting. With vertical shear in the background wind, the downstream ray path of wave propagation can be rapidly attenuated near a certain level, whereas the upstream ray path is not attenuated and the amplitudes even increase with height. The downstream attenuation level is found to descend with increasing linear wind shear. The present analytical model establishes that the attenuation level corresponds to the critical level where the background wind is equal to the horizontal gravity wave phase speed. The upstream gravity wave ray path can propagate upward without attenuation as there is no critical level there.

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Shizuo Fu
,
Richard Rotunno
, and
Huiwen Xue

Abstract

In orographic precipitation events, there are times when subsaturated low-level layers are observed to be below saturated, nearly moist-neutral, upper-level layers. By performing a series of idealized two-dimensional simulations, this study investigates the response of orographic precipitation to subsaturated low-level layers. When the nondimensional parameter N 2 z t /U, where N 2 and z t are, respectively, the dry Brunt–Väisälä frequency and depth of the subsaturated low-level layer, and U the cross-mountain wind speed, exceeds a critical value, the decelerated region on the upwind side of the mountain moves upwind, resulting in weak surface precipitation near the mountain peak. The critical value determined from the simulations is close to that derived from linear theory. When N 2 z t /U is less than the critical value, increasing z t has two competing effects: 1) the vapor-transport effect, meaning that increasing z t decreases the amount of vapor transported to the mountain, and hence tends to decrease surface precipitation; and 2) the updraft-width effect, meaning that increasing z t enhances flow blocking, producing a wider updraft over the upwind slope, and hence tends to increase surface precipitation. When the vapor-transport effect dominates, surface precipitation decreases with z t . When the updraft-width effect dominates, surface precipitation increases with z t . Increasing the maximum mountain height h m or U generally increases surface precipitation. However, for certain combinations of h m and U, the simulations produce lee waves, which substantially reduce surface precipitation. Finally, the response of orographic precipitation in the simulations with both liquid-phase and ice-phase microphysics is similar to that in the simulations with only liquid-phase microphysics.

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Richard Rotunno
and
George H. Bryan

Abstract

This study considers a two-layer fluid with constant density in each layer connected by a layer of continuously varying density for flows past topography in which hydraulic jumps with lee vortices are expected based on shallow-water theory. Numerical integrations of the Navier–Stokes equations at a Reynolds number high enough for a direct numerical simulation of turbulent flow allow an examination of the internal mechanics of the turbulent leeside hydraulic jump and how this mechanics is related to lee vortices. Analysis of the statistically steady state shows that the original source of lee-vortex vertical vorticity is through the leeside descent of baroclinically produced spanwise vorticity associated with the hydraulic jump. This spanwise vorticity is tilted to the vertical at the spanwise extremities of the leeside hydraulic jump. Turbulent energy dissipation in flow through the hydraulic jump allows this leeside vertical vorticity to diffuse and extend downstream. The present simulations also suggest a geometrical interpretation of lee-vortex potential-vorticity creation, a concept central to interpretations of lee vortices based on the shallow-water equations.

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