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Tetsuya Takemi
and
Richard Rotunno

Abstract

Using the newly developed Weather Research and Forecasting (WRF) model, this study investigates the effects of subgrid mixing and numerical filtering in mesoscale cloud simulations by examining the sensitivities to the parameters in turbulence-closure schemes as well as the parameters in the numerical filters. Three-dimensional simulations of squall lines in both no-shear and strong-shear environments have been performed. Using the Smagorinsky or 1.5-order turbulent kinetic energy (TKE) subgrid model with standard values for the model constants and no explicit numerical filter, the solution in the no-shear environment is characterized by many poorly resolved grid-scale cells. In the past, such grid-scale noise was avoided by adding a numerical filter which, however, produces excessive damping of the physical small-scale eddies. Without using such a filter, it was found that by increasing the proportionality constant in the eddy viscosity coefficient in the subgrid turbulence models, the cells become well resolved, but that further increases in the constant overly smooth the cells. Such solution sensitivity is also found in the strong-shear cases. The simulations using the subgrid models with viscosity coefficients 1.5 to 2 times larger than those widely used in other cloud models retain more power in short scales, but without an unwanted buildup of energy; with these optimum values, no numerical filters are required to avoid computational noise. These optimum constants do not depend significantly on grid spacings of O(1 km). Therefore, it is concluded that by using the eddy viscosity formulation appropriate for mesoscale cloud simulations, the use of artificial numerical filters is avoided, and the mixing processes are represented by more physically based turbulence-closure models.

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Richard Rotunno
and
Jian-Wen Bao

Abstract

It is universally agreed that cyclogenesis in midlatitudes occurs through baroclinic conversion of the potential energy available from an initial state. The mechanical process by which that conversion takes place is a perennial subject of discussion. At least as far back as the 1950s, it was recognized that in any practical forecast problem, the initial condition is influential. Observational research continues to confirm the prevalence of tropopause-level perturbations preceding surface cyclogenesis. The observations also suggest that the growing disturbances have time-varying vertical structures. Relating these observations to the classical linear theory of baroclinic instability is not immediately obvious since, in the latter, the precise form of the initial condition is not important, and the theory predicts cyclogenesis with a fixed-in-time vertical structure. These differences between theory and observations are but a few of the many that have been recognized and treated in modified theories of baroclinic instability. We attempt herein to draw a closer connection between the modified theories and observations by performing a case study using a hierarchy of models of decreasing complexity.

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Richard Rotunno
and
Joseph B. Klemp

Abstract

In the present investigation we propose a simple theory to explain how a veering environmental wind shear vector can cause an initially symmetric updraft to grow preferentially to the right of the shear vector and acquire cyclonic rotation. The explanation offered is based on linear theory which predicts that interaction of the mean shear with the updraft produces favorable vertical pressure gradients along its right flank. To asses the validity of linear theory for large-amplitude updrafts, the three-dimensional, shallow, anelastic equations are numerically integrated using a simple parameterization for latent heating within a cloud and the linear and nonlinear forcing terms are separately analyzed. These results suggest that although the nonlinear effects strongly promote splitting of the updraft, the linear forcing remains the dominant factor in preferentially enhancing updraft growth on the right flank. We believe this differential forcing is a major contributor to the observed predominance of cyclonically rotating, right moving storms.

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George H. Bryan
and
Richard Rotunno

Abstract

An axisymmetric numerical model is used to evaluate the maximum possible intensity of tropical cyclones. As compared with traditionally formulated nonhydrostatic models, this new model has improved mass and energy conservation in saturated conditions. In comparison with the axisymmetric model developed by Rotunno and Emanuel, the new model produces weaker cyclones (by ∼10%, in terms of maximum azimuthal velocity); the difference is attributable to several approximations in the Rotunno–Emanuel model. Then, using a single specification for initial conditions (with a sea surface temperature of 26°C), the authors conduct model sensitivity tests to determine the sensitivity of maximum azimuthal velocity (υ max) to uncertain aspects of the modeling system. For fixed mixing lengths in the turbulence parameterization, a converged value of υ max is achieved for radial grid spacing of order 1 km and vertical grid spacing of order 250 m. The fall velocity of condensate (Vt ) changes υ max by up to 60%, and the largest υ max occurs for pseudoadiabatic thermodynamics (i.e., for Vt > 10 m s−1). The sensitivity of υ max to the ratio of surface exchange coefficients for entropy and momentum (CE /CD ) matches the theoretical result, υ max ∼ (CE /CD )1/2, for nearly inviscid flow, but simulations with increasing turbulence intensity show less dependence on CE /CD ; this result suggests that the effect of CE /CD is less important than has been argued previously. The authors find that υ max is most sensitive to the intensity of turbulence in the radial direction. However, some settings, such as inviscid flow, yield clearly unnatural structures; for example, υ max exceeds 110 m s−1, despite a maximum observed intensity of ∼70 m s−1 for this environment. The authors show that turbulence in the radial direction limits maximum axisymmetric intensity by weakening the radial gradients of angular momentum (which prevents environmental air from being drawn to small radius) and of entropy (which is consistent with weaker intensity by consideration of thermal wind balance). It is also argued that future studies should consider parameterized turbulence as an important factor in simulated tropical cyclone intensity.

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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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Piotr K. Smolarkiewicz
and
Richard Rotunno

Abstract

No abstract available.

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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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Chin-Hoh Moeng
and
Richard Rotunno

Abstract

A number of puzzling features of the skewness of the vertical velocity field, Sw (z), are found in observations and large-eddy simulations (LES) of the buoyancy-driven planetary boundary layer (PBL). For example, observations of Sw (z) in cases where the air is heated from below indicate that Sw (z) > 0 and remains relatively constant for z ≳ 0.3zi , whereas all large-eddy simulations of these cases show a continuing increase of Sw (z) with height. In cases where the air is both heated from below and cooled from above, as in some of the stratus-topped PBL cases, large-eddy simulations show a rather curious feature: Sw is positive in the upper layer and negative in the lower layer. In considering these features, it occurred to us that a theoretical model of what one should expect of the skewness distribution, even in simple situations, did not exist. Hence in the present paper we examine the skewness distributions from direct numerical simulations of several simple archetypes of buoyancy-driven turbulent flow. While these simulations do not resolve the discrepancy between LES and observations, they help in understanding the LES results, and suggest avenues for future research.

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Piotr K. Smolarkiewicz
and
Richard Rotunno

Abstract

We study the flow of a density-stratified fluid past a three-dimensional obstacle, using a numerical model. Our special concern is the response of the fluid when the Froude number is near or less than unity. Linear theory is inapplicable in this range of Froude number, and the present numerical solutions show the rich variety of phenomena that emerge in this essentially nonlinear flow regime. Two such phenomena, which occupy Parts I and II of this study, are the formation of a pair of vertically oriented vortices on the lee side and a zone of flow reversal on the windward side of the obstacle. The Ice vortices have been explained as a consequence of the separation of the viscous boundary layer from the obstacle however, this boundary layer is absent (by design) in the present experiments and lee vortices still occur. We argue that a vertical component of vorticity develops on the lee side owing to the tilting of horizontally oriented vorticity produced baroclinically as the isentropes deform in response to the flow over the obstacle. This deformation is adequately predicted by linear gravity-wave, which allows one to deduce, using the next-order correction to linen theory, the existence of a vortex pair of the proper sense in the lee of the obstacle. Thus, the lee vortices are closely associated with the dynamics of gravity waves. The generation of the lee vortices may also be understood as a consequence of Ertel's theorem which in the present circumstance demands that vortex lines adhere to isentropic surfaces— since the isentropes are depressed behind the hill, the vortex lines must run upward and downward along the depression implying vertically oriented vorticity.

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