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James W. Rottman and Franco Einaudi

Abstract

The weakly nonlinear theory for internal solitary waves is reviewed and theoretical results of the vertical and horizontal structure of temperature, vertical displacements, and vertical and horizontal perturbations to the wind field associated with steadily propagating solitary waves are presented in two idealized atmospheric configurations. One configuration is representative of solitary waves observed in the lower troposphere and the other of solitary waves that occupy the entire troposphere. The important results of the theory are presented in a form that can be readily used by observationalists. The results obtained are then analyzed using actual rawinsonde data for two well-documented observations of atmospheric solitary waves, which are analogous to the two idealized configurations. The importance and difficulties of properly identifying the waveguide within which the solitary wave is confined are discussed. The fundamental role of a critical level in ducting the disturbances and thus in defining the thickness of the waveguide is illustrated in the example dealing with the solitary wave occupying the entire troposphere. Together, these two examples illustrate the decisions and compromises that must be made in applying the theory to the real atmosphere.

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Chaing Chen, James W. Rottman, and Steven E. Koch

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A two-dimensional, nonhydrostatic, elastic numerical model has been used to study the generation of gravity waves for a stably stratified shear flow over an obstacle. When a low-level wind shear is included in the simulation, we find that the predictions for noticeable upstream effects based on Froude number for a uniform flow are no longer accurate. Upstream effects are encountered in the form of upstream propagating columnar disturbances and internal bores away from the obstacle. The limited parameter space studies conducted in this study suggest that the ratio of the shear depth to the obstacle height (d/H), the obstacle aspect ratio (H/L), and the Froude number (U/NH) are instrumental in determining the strength and the existence of these upstream disturbances. Thus, the present theoretical and empirical understanding of the importance of the Froude number for determining the nature of upstream effects should be modified substantially to include additional nondimensional parameters when shear is present.

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Dave Broutman, James W. Rottman, and Stephen D. Eckermann

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A previously derived approximation to the standard Fourier integral technique for linear mountain waves is extended to include nonhydrostatic effects in a background flow with height-dependent wind and stratification. The approximation involves using ray theory to simplify the vertical eigenfunctions. The generalization to nonhydrostatic waves requires special treatment for resonant modes and caustics. Resonant modes are handled with a small amount of damping, and caustics are handled with a uniformly valid approximation involving the Airy function. This method is developed for both two- and three-dimensional flows, and its results are shown to compare well with an exact analytical result for two-dimensional mountain waves and with a numerical simulation for two- and three-dimensional mountain waves.

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Dave Broutman, Stephen D. Eckermann, and James W. Rottman

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A Fourier method is used to model mountain waves that have nearby turning points in a wind jet. In Fourier space, the propagation equations are solved by ray theory. To correct for the ray singularity at a turning point without time-consuming special-function evaluations, the ray solution is linearly interpolated across the breakdown region. The Fourier solutions for the spatial wavefield are compared with mesoscale model simulations in two cases: two-dimensional flow over idealized topography with uniform stratification and a sech-squared wind profile and three-dimensional flow over the island of Jan Mayen with stratification and wind profiles taken from radiosonde measurements. The latter case reveals the partial transmission of trapped mountain waves into the stratosphere.

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Yue Wu, Stefan G. Llewellyn Smith, James W. Rottman, Dave Broutman, and Jean-Bernard H. Minster

Abstract

Tsunami-generated acoustic–gravity waves have been observed to propagate in the atmosphere up to the ionosphere, where they have an impact on the total electron content. The authors simulate numerically the propagation of two-dimensional linear acoustic–gravity waves in an atmosphere with vertically varying stratification and horizontal background winds. The authors’ goal is to compare the difference in how much energy reaches the lower ionosphere up to an altitude of 180 km, where the atmosphere is assumed to be anelastic or fully compressible. The authors consider three specific atmospheric cases: a uniformly stratified atmosphere without winds, an idealized case with a wind jet, and a realistic case with an atmospheric profile corresponding to the 2004 Sumatra tsunami. Results show that for the last two cases, the number and height of turning points are different for the anelastic and compressible assumptions, and the net result is that compressibility enhances the total transmission of energy through the whole atmosphere.

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V. Mohan Karyampudi, Steven E. Koch, Chaing Chen, James W. Rottman, and Michael L. Kaplan

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In this paper, Part II of a series, the evolution of a prefrontal bore on the leeside of the Rockies and its subsequent propagation and initiation of convection farther downstream over eastern Colorado and western Nebraska are investigated. The observational evidence for this sequence of events was obtained from combined analyses of high-resolution GOES satellite imagery and Program for Regional Observing and Forecasting Services mesonetwork data over the Colorado region for the severe weather event that occurred during 13–14 April 1986. A 2D nonhydrostatic numerical model is used to further understand the initiation of the bore and its ability to propagate farther downstream and trigger convection.

Analysis of satellite imagery and mesonet data indicated that an internal bore (ahead of a cold front), a moderate downslope windstorm, and a quasi-stationary hydraulic jump were generated within a few hours along the Iceslope as a Pacific cold front and its attendant upper-level jet streak advanced over the Rockies. The bore and the cold front then propagated eastward for several hours and interacted with a Ice cyclone, a dryline, and a warm front, initiating severe weather over Nebraska and Kansas. Wave-ducting analysis showed that favorable wave-trapping mechanisms such as a capping inversion above a neutral layer and wind curvature from a low-level jet, which appeared to he the most dominant ducting mechanism, existed across eastern Colorado and western Nebraska to maintain the bore strength. Numerical simulations of continuously stratified shear flow specified from upstream and downstream soundings suggested that the creation of a density current along the Ice slopes, a downstream inversion height lower than the upstream inversion height, and a strong curvature in the wind profile of the low-level jet are all needed to initiate and sustain the integrity of the propagating bore.

Based on the synthesis of observational analyses and 2D nonhydrostatic model simulations, a schematic illustration of the time evolution of the bore ahead of the Pacific cold front, the hydraulic jump associated with a mountain wave, and the arctic air intrusion from the north to the Ice of the Rockies are presented in the context of severe weather occurrence over western Nebraska and Kansas.

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Yue Wu, Stefan G. Llewellyn Smith, James W. Rottman, Dave Broutman, and Jean-Bernard H. Minster

Abstract

Tsunami-generated linear acoustic–gravity waves in the atmosphere with altitude-dependent vertical stratification and horizontal background winds are studied with the long-term goal of real-time tsunami warning. The initial-value problem is examined using Fourier–Laplace transforms to investigate the time dependence and to compare the cases of anelastic and compressible atmospheres. The approach includes formulating the linear propagation of acoustic–gravity waves in the vertical, solving the vertical displacement of waves and pressure perturbations numerically as a set of coupled ODEs in the Fourier–Laplace domain, and employing den Iseger’s algorithm to carry out a fast and accurate numerical inverse Laplace transform. Results are presented for three cases with different atmospheric and tsunami profiles. Horizontal background winds enhance wave advection in the horizontal but hinder the vertical transmission of internal waves through the whole atmosphere. The effect of compressibility is significant. The rescaled vertical displacement of internal waves at 100-km altitude shows an arrival at the early stage of wave development due to the acoustic branch that is not present in the anelastic case. The long-term displacement also shows an O(1) difference between the compressible and anelastic results for the cases with uniform and realistic stratification. Compressibility hence affects both the speed and amplitude of energy transmitted to the upper atmosphere because of fast acoustic waves.

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John Lindeman, Zafer Boybeyi, Dave Broutman, Jun Ma, Stephen D. Eckermann, and James W. Rottman

Abstract

A Fourier method is combined with a mesoscale model to simulate mountain waves. The mesoscale model describes the nonlinear low-level flow and predicts the emerging wave field above the mountain. This solution serves as the lower boundary condition for the Fourier method, which follows the waves upward to much higher altitudes and downward to the ground to examine parameterizations for the orography and the lower boundary condition. A high-drag case with a Froude number of ⅔ is presented.

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