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

Abstract

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

Abstract

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
,
Roger H. J. Grimshaw
, and
Stephen D. Eckermann

Abstract

Several recent studies of internal gravity waves have been expressed in a Lagrangian reference frame, motivated by the observation that in this frame the dispersion relation then excludes the Eulerian Doppler shifting term due to a background flow. Here the dispersion relation in a Lagrangian reference frame is explicitly derived for a background flow and background density that are slowly varying with respect to the waves, but are otherwise arbitrary functions of space and time. Two derivations are given, both yielding the same result. The first derivation involves a transformation of the dispersion relation from Eulerian to Lagrangian coordinates, while the second derivation involves a wave-packet analysis of the equations of motion directly in Lagrangian coordinates. The authors show that, although the Eulerian Doppler shifting term is removed from the dispersion relation by the transformation of the frequency when passing from an Eulerian to a Lagrangian reference frame, a dependence on the background shear is then introduced by the transformation of the wavenumber. This dependence on the background shear is the term that accounts for wave refraction in the Lagrangian frame, and its role has apparently not been fully appreciated in the aforementioned previous studies.

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Dave Broutman
,
Stephen D. Eckermann
, and
Douglas P. Drob

Abstract

A vertical eigenfunction equation is solved to examine the partial reflection and partial transmission of tsunami-generated gravity waves propagating through a height-dependent background atmosphere from the ocean surface into the lower thermosphere. There are multiple turning points for each vertical eigenfunction (at least eight in one example), yet the wave transmission into the thermosphere is significant. Examples are given for gravity wave propagation through an idealized wind jet centered near the mesopause and through a realistic vertical profile of wind and temperature relevant to the tsunami generated by the Sumatra earthquake on 26 December 2004.

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Christopher G. Kruse
,
Ronald B. Smith
, and
Stephen D. Eckermann

Abstract

The vertical propagation and attenuation of mountain waves launched by New Zealand terrain during the Deep Propagating Gravity Wave Experiment (DEEPWAVE) field campaign are investigated. New Zealand mountain waves were frequently attenuated in a lower-stratospheric weak wind layer between z = 15 and 25 km. This layer is termed a “valve layer,” as conditions within this layer (primarily minimum wind speed) control mountain wave momentum flux through it, analogous to a valve controlling mass flux through a pipe. This valve layer is a climatological feature in the wintertime midlatitude lower stratosphere above the subtropical jet.

Mountain wave dynamics within this valve layer are studied using realistic Weather Research and Forecasting (WRF) Model simulations that were extensively validated against research aircraft, radiosonde, and satellite observations. Locally, wave attenuation is horizontally and vertically inhomogeneous, evidenced by numerous regions with wave-induced low Richardson numbers and potential vorticity generation. WRF-simulated gravity wave drag (GWD) is peaked in the valve layer, and momentum flux transmitted through this layer is well approximated by a cubic function of minimum ambient wind speed within it, consistent with linear saturation theory. Valve-layer GWD within the well-validated WRF simulations was 3–6 times larger than that parameterized within MERRA. Previous research suggests increasing parameterized orographic GWD (performed in MERRA2) decreases the stratospheric polar vortex strength by altering planetary wave propagation and drag. The results reported here suggest carefully increasing orographic GWD is warranted, which may help to ameliorate the common cold-pole problem in chemistry–climate models.

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Roger Grimshaw
,
Dave Broutman
,
Brian Laughman
, and
Stephen D. Eckermann

Abstract

Mesospheric bores have been observed and measured in the mesopause region near 100-km altitude, where they propagate horizontally along a duct of relatively strong density stratification. Here, a weakly nonlinear theory is developed for the description of these mesospheric bores. It extends previous theories by allowing internal gravity wave radiation from the duct into the surrounding stratified regions, which are formally assumed to be weakly stratified. The radiation away from the duct is expected to be important for bore energetics. The theory is compared with a numerical simulation of the full Navier–Stokes equations in the Boussinesq approximation. Two initial conditions are considered. The first is a solitary wave solution that would propagate without change of form if the region outside the duct were unstratified. The second is a sinusoid that evolves into an undular bore. The main conclusion is that, while solitary waves and undular bores decay by radiation from the duct, they can survive as significant structures over sufficiently long periods (~100 min) to be observable.

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Andreas Dörnbrack
,
Stephen D. Eckermann
,
Bifford P. Williams
, and
Julie Haggerty

Abstract

Stratospheric gravity waves observed during the DEEPWAVE research flight RF25 over the Southern Ocean are analyzed and compared with numerical weather prediction (NWP) model results. The quantitative agreement of the NWP model output and the tropospheric and lower-stratospheric observations is remarkable. The high-resolution NWP models are even able to reproduce qualitatively the observed upper-stratospheric gravity waves detected by an airborne Rayleigh lidar. The usage of high-resolution ERA5 data—partially capturing the long internal gravity waves—enabled a thorough interpretation of the particular event. Here, the observed and modeled gravity waves are excited by the stratospheric flow past a deep tropopause depression belonging to an eastward-propagating Rossby wave train. In the reference frame of the propagating Rossby wave, vertically propagating hydrostatic gravity waves appear stationary; in reality, of course, they are transient and propagate horizontally at the phase speed of the Rossby wave. The subsequent refraction of these transient gravity waves into the polar night jet explains their observed and modeled patchy stratospheric occurrence near 60°S. The combination of both unique airborne observations and high-resolution NWP output provides evidence for the one case investigated in this paper. As the excitation of such gravity waves persists during the quasi-linear propagation phase of the Rossby wave’s life cycle, a hypothesis is formulated that parts of the stratospheric gravity wave belt over the Southern Ocean might be generated by such Rossby wave trains propagating along the midlatitude waveguide.

Open access
Stephen D. Eckermann
,
Cory A. Barton
, and
James F. Kelly

Abstract

The virtual temperature used to model moisture-modified tropospheric dynamics is generalized to include a new thermospheric component. The resulting hybrid virtual potential temperature (HVPT) transitions seamlessly with height, from moist virtual potential temperature (MVPT) in the troposphere, to potential temperature in the stratosphere and mesosphere, to thermospheric virtual potential temperature thereafter. For numerical weather prediction (NWP) models looking to extend into the thermosphere, but still heavily invested in retaining MVPT-based dynamical cores for tropospheric prediction, upgrading to HVPT allows the core to capture critical new aspects of variable composition thermospheric dynamics, while leaving the original MVPT-based tropospheric equations and numerics essentially untouched. In this way, HVPT augmentation can both simplify and streamline extension into the thermosphere at little computational cost beyond the inevitable need for more vertical layers and somewhat smaller time steps. To demonstrate, we upgrade the MVPT-based dynamical core of the Navy global NWP model to HVPT, then test its performance in forecasting analytical globally balanced states containing hot or rapidly heated thermospheres and height-varying gas constants. These tests confirm that HVPT augmentation offers an efficient and effective means of extending MVPT-based NWP models into the thermosphere to accelerate development of future ground-to-space NWP models supporting space weather applications. The related issues of variable gravitational acceleration and shallow-atmosphere approximations are also briefly discussed.

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Stephen D. Eckermann
,
Dave Broutman
,
Jun Ma
, and
John Lindeman

Abstract

A time-dependent generalization of a Fourier-ray method is presented and tested for fast numerical computation of high-resolution nonhydrostatic mountain-wave fields. The method is used to model mountain waves from Jan Mayen on 25 January 2000, a period when wavelike cloud banding was observed long distances downstream of the island by the Advanced Very High Resolution Radiometer Version 3 (AVHRR-3). Surface weather patterns show intensifying surface geostrophic winds over the island at 1200 UTC caused by rapid eastward passage of a compact low pressure system. The 1200 UTC wind profiles over the island increase with height to a jet maximum of ∼60–70 m s−1, yielding Scorer parameters that indicate vertical trapping of any short wavelength mountain waves. Separate Fourier-ray solutions were computed using high-resolution Jan Mayen orography and 1200 UTC vertical profiles of winds and temperatures over the island from a radiosonde sounding and an analysis system. The radiosonde-based simulations produce a purely diverging trapped wave solution that reproduces the salient features in the AVHRR-3 imagery. Differences in simulated wave patterns governed by the radiosonde and analysis profiles are explained in terms of resonant modes and are corroborated by spatial ray-group trajectories computed for wavenumbers along the resonant mode curves. Output from a nonlinear Lipps–Hemler orographic flow model also compares well with the Fourier-ray solution horizontally. Differences in vertical cross sections are ascribed to the Fourier-ray model’s current omission of tunneling of trapped wave energy through evanescent layers.

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Dave Broutman
,
Jun Ma
,
Stephen D. Eckermann
, and
John Lindeman

Abstract

The Fourier-ray method involves ray tracing in a Fourier-transform domain. The ray solutions are then Fourier synthesized to produce a spatial solution. Here previous steady-state developments of the Fourier-ray method are extended to include a transient source of mountain waves. The method is illustrated with an initial value problem in which the background flow is started abruptly from rest and then maintained at steady velocity. The resulting wave transience is modeled in a simple way. All rays that radiate from the mountain, including the initial rays, are assigned the full amplitude of the longtime steady-state solution. Time dependence comes in through the changing position of the initial rays. This is sufficient to account for wave transience in a test case, as demonstrated by comparison with simulations from a mesoscale numerical model.

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