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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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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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Mario Marcello Miglietta
and
Richard Rotunno

Abstract

In two recent papers, the authors reported on numerical simulations of conditionally unstable flows past an idealized mesoscale mountain ridge. These idealized simulations, which were performed with a three-dimensional, explicitly cloud-resolving model, allowed the investigation of simulated precipitation characteristics as a function of the prescribed environment. The numerical solutions were carried out for a uniform wind flowing past a bell-shaped ridge and using an idealized unstable sounding with prescribed values of the relevant parameters.

In the present work the application of these theoretical results to observed cases of orographically forced convective rainfall including the Big Thompson flood (1976, Colorado), the Oahu flood (1974, Hawaii), and the Gard flood (2002, France) is reported. Specifically, numerical simulations have been carried out using observed and idealized soundings relevant to these cases but with idealized topography. It is found that using the observed soundings, but with idealized constant-wind profiles, the simulated rain rates fit reasonably well within the previous theoretically derived parameter space for intense orographic convective rainfall. However, in order to reproduce larger rainfall rates, in closer agreement with observations, in the first two cases it was necessary to initialize the sounding with a wind profile characterized by low-level flow toward the mountain with weak flow aloft (as observed for the across-mountain wind component). For the Gard case, the situation was more complex and it is found unlikely that the situation can be reduced to a simple two-dimensional problem.

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F. Zhang
,
Chris Snyder
, and
Richard Rotunno

Abstract

A mesoscale model is used here to investigate the possible sources of forecast error for the 24–25 January 2000 snowstorm along the east coast of the United States. The primary focus is the quantitative precipitation forecast out to lead times of 36 h. The success of the present high-resolution control forecast shows that the storm could have been well forecasted with conventional data in real time. Various experiments suggest that insufficient model grid resolution and errors in the initial conditions both contributed significantly to problems in the forecast. Other experiments, motivated by the possibility that the forecast errors arose from the operational analysis poorly fitting one or two key soundings, test the effects of withholding single soundings from the control initial conditions. While no single sounding results in forecast changes that are more than a small fraction of the error in the operational forecast, these experiments do reveal that the detailed mesoscale distribution of precipitation in the 24- or 36-h forecast can be significantly altered even by such small changes in the initial conditions. The experiments also reveal that the forecast changes arise from the rapid growth of error at scales below 500 km in association with moist processes. The results presented emphasize the difficulty of forecasting precipitation relative to, say, surface pressure and suggest that the predictability of mesoscale precipitation features in cases of the type studied here may be limited to less than 2–3 days.

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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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George H. Bryan
and
Richard Rotunno

Abstract

This study presents analytic results for steady gravity currents in a channel using the deep anelastic equations. Results are cast in terms of a nondimensional parameter H/H 0 that relates the channel depth H to a scale depth H 0 (the depth at which density goes to zero in an isentropic atmosphere). The classic results based on the incompressible equations correspond to H/H 0 = 0. For cold gravity currents (at the bottom of a channel), assuming energy-conserving flow, the nondimensional current depth h/H is much smaller, and nondimensional propagation speed C/(gH)1/2 is slightly smaller as H/H 0 increases. For flows with energy dissipation, C/(gH)1/2 decreases as H/H 0 increases, even for fixed h/H. The authors conclude that as H/H 0 increases the normalized hydrostatic pressure rise in the cold pool increases near the bottom of the channel, whereas drag decreases near the top of the channel; these changes require gravity currents to propagate slower for steady flow to be maintained. From these results, the authors find that steady cold pools have a likely maximum depth of 4 km in the atmosphere (in the absence of shear). For warm gravity currents (at the top of a channel), h/H is slightly larger and C/(gH)1/2 is much larger as H/H 0 increases. The authors also conduct two-dimensional numerical simulations of “lock-exchange flow” to provide an independent evaluation of the analytic results. For cold gravity currents the simulations support the analytic results. However, for warm gravity currents the simulations show unsteady behavior that cannot be captured by the analytic theory and which appears to have no analog in incompressible flow.

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Heather Dawn Reeves
and
Richard Rotunno

Abstract

The effects of upstream relative humidity (RH) on low-level wind and precipitation patterns for low-speed, statically stable flows over a mountain are investigated using idealized two- and three-dimensional numerical-simulation experiments in which RH is increased from 0% to 100%. For RH less than some critical threshold, the flow upstream becomes less decelerated as RH is increased; for RH greater than this threshold, the flow upstream becomes more decelerated as RH is increased. This increasing deceleration with RH is due to locally enhanced static stability resulting from enhanced condensation near the freezing level. Analyses from the simulations indicate that the lifted condensation level and the height of the freezing level are significant control parameters for the upstream-flow deceleration in the steady-state solutions. Dimensional analysis using these control parameters (as well as others) brings forth new nondimensional parameters that are shown to enter into analytic formulas for the orographic upstream-flow deceleration in a moist atmosphere.

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F. Zhang
,
Chris Snyder
, and
Richard Rotunno

Abstract

In a previous study by the authors, it was shown that the problematic numerical prediction of the 24–25 January 2000 snowstorm along the east coast of the United States was in some measure due to rapid error growth at scales below 500 km. In particular they found that moist processes were responsible for this strong initial-condition sensitivity of the 1–2-day prediction of mesoscale forecast aspects. In the present study they take a more systematic look at the processes by which small initial differences (“errors”) grow in those numerical forecasts. For initial errors restricted to scales below 100 km, results show that errors first grow as small-scale differences associated with moist convection, then spread upscale as their growth begins to slow. In the context of mesoscale numerical predictions with 30-km resolution, the initial growth is associated with nonlinearities in the convective parameterization (or in the explicit microphysical parameterizations, if no convective parameterization is used) and proceeds at a rate that increases as the initial error amplitude decreases. In higher-resolution (3.3 km) simulations, errors first grow as differences in the timing and position of individual convective cells. Amplification at that stage occurs on a timescale on the order of 1 h, comparable to that of moist convection. The errors in the convective-scale motions subsequently influence the development of meso- and larger-scale forecast aspects such as the position of the surface low and the distribution of precipitation, thus providing evidence that growth of initial errors from convective scales places an intrinsic limit on the predictability of larger scales.

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Morris L. Weisman
and
Richard Rotunno

Abstract

Based on the analysis of idealized two- and three-dimensional cloud model simulations, Rotunno et al. (hereafter RKW) and Weisman et al. (hereafter WKR) put forth a theory that squall-line strength and longevity was most sensitive to the strength of the component of low-level (0–3 km AGL) ambient vertical wind shear perpendicular to squall-line orientation. An “optimal” state was proposed by , based on the relative strength of the circulation associated with the storm-generated cold pool and the circulation associated with the ambient shear, whereby the deepest leading edge lifting and most effective convective retriggering occurred when these circulations were in near balance. Since this work, subsequent studies have brought into question the basic validity of the proposed optimal state, based on concerns as to the appropriate distribution of shear relative to the cold pool for optimal lifting, as well as the relevance of such concepts to fully complex squall lines, especially considering the potential role of deeper-layer shears in promoting system strength and longevity. In the following, the basic interpretations of the theory are reconfirmed and clarified through both the analysis of a simplified two-dimensional vorticity–streamfunction model that allows for a more direct interpretation of the role of the shear in controlling the circulation around the cold pool, and through an analysis of an extensive set of 3D squall-line simulations, run at higher resolution and covering a larger range of environmental shear conditions than presented by .

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Mario Marcello Miglietta
and
Richard Rotunno

Abstract

In two recent papers, the authors performed numerical simulations with a three-dimensional, explicitly cloud-resolving model for a uniform wind flowing past a bell-shaped ridge and using an idealized unstable (Weisman–Klemp) sounding with prescribed values of the relevant parameters. More recently, some observed cases of orographically forced wind profiles were analyzed, showing that, in order to reproduce larger rainfall rates, it was necessary to initialize the sounding with low-level flow toward the mountain with weak flow aloft (as observed). Additional experiments using the Weisman–Klemp sounding, but with nonuniform wind profiles, are performed here to identify the conditions in which the presence of a low-level cross-mountain flow together with calm flow aloft may increase the rain rates in conditionally unstable flows over the orography. The sensitivity of the solutions to the wind speed at the bottom and the top of a shear layer and the effect of different mountain widths and heights are systematically analyzed herein.

Large rainfall rates are obtained when the cold pool, caused by the evaporative cooling of rain from precipitating convective clouds, remains quasi stationary upstream of the mountain peak. This condition occurs when the cold-pool propagation is approximately countered by the environmental wind. The large precipitation amounts can be attributed to weak upper-level flow, which favors stronger updrafts and upright convective cells, and to the ground-relative stationarity of the cells. This solution feature is produced with ambient wind shear within a narrow region of the parameter space explored here and does not occur in the numerical solutions obtained in the authors’ previous studies with uniform wind profiles.

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