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Stephen D. Eckermann

Abstract

A straightforward methodology is presented for converting the deterministic multiwave parameterizations of nonorographic gravity wave drag, currently used in general circulation models (GCMs), to stochastic analogs that use fewer waves (in the example herein, a single wave) within each grid box. Deterministic discretizations of source-level momentum flux spectra using a fixed spectrum of many waves with predefined phase speeds are replaced by sampling these source spectra stochastically using waves with randomly assigned phase speeds. Using simple conversion formulas, it is shown that time-mean wave-induced drag, diffusion, and heating-rate profiles identical to those from the deterministic scheme are produced by the stochastic analog. Furthermore, in these examples the need for bulk intermittency factors of small value is largely obviated through the explicit incorporation of stochastic intermittency into the scheme. When implemented in a GCM, the single-wave stochastic analog of an existing deterministic scheme reproduces almost identical time-mean middle-atmosphere climate and drag as its deterministic antecedent but with an order of magnitude reduction in computational expense. The stochastically parameterized drag is also accompanied by inherent variability about the time-mean profile that forces the smallest space–time scales of the GCM. Studies of mean GCM kinetic energy spectra show that this additional stochastic forcing does not lead to excessive increases in dynamical variability at these smallest GCM scales. The results show that the expensive deterministic schemes currently used in GCMs are easily modified and replaced by cheap stochastic analogs without any obvious deleterious impacts on GCM climate or variability, while offering potential advantages of computational savings, reduction of systematic climate biases, and greater and more realistic ensemble spread.

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Stephen D. Eckermann

Abstract

The “Doppler spread” theory of atmospheric gravity waves has developed rapidly in recent years, from an initial theory of wave spectra into a general parameterization of gravity wave effects for use in global models of the middle atmosphere. Yet the theory currently employs certain key approximations that have still to be tested. The author focuses on the omission of the propagation of the other waves in the spectrum when determining the Doppler spreading of a given gravity wave. This approximation is shown to become untenable as waves are refracted to progressively shorter vertical scales, so ray methods are employed to investigate the refraction characteristics of short waves within propagating long-wave fields. Short-wave refraction is reduced compared to the Doppler-spread results. While turning levels are common, critical levels do not occur if all waves propagate upward in the absence of mean wind shear. Consequently, a sharp increase in the probability of wave obliteration beyond the so-called cutoff vertical wavenumber (a central tenet of Doppler-spread theory) no longer occurs. Possible implications of these results for models of wave–field interactions, spectra, and momentum deposition are discussed.

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Stephen D. Eckermann, Jun Ma, and Dave Broutman

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Numerical transform solutions for hydrostatic gravity waves generated by both uniform and sheared flow over elliptical obstacles are used to quantify effects of horizontal geometrical spreading on amplitude evolution with height. Both vertical displacement and steepness amplitudes are considered because of their close connections to drag parameterizations in weather and climate models. Novel diagnostics quantify the location and value of the largest wavefield amplitudes most likely to break at each altitude. These horizontal locations do not stray far from the obstacle peak even at high altitudes. Resulting vertical profiles of wave amplitude are normalized to remove density and refraction effects, thereby quantifying the horizontal geometrical spreading contribution, currently absent from parameterizations. Horizontal geometrical spreading produces monotonic amplitude decreases with height through wave-action conservation as waves propagate into progressively larger horizontal areas. Accumulated amplitude reductions are appreciable for all but the most quasi-two-dimensional obstacles with long axes orthogonal to the flow, and even these are impacted appreciably if the obstacle is rotated by more than 20°–30°. Profiles are insensitive to the obstacle’s functional form but vary strongly in response to changes in its horizontal aspect ratio. Responses to background winds are captured by a vertical coordinate transformation that remaps profiles to a universal form for a given obstacle. These results show that horizontal geometrical spreading has comparable or larger effects on wave amplitudes as the refraction of vertical wavenumbers and thus is important for accurate parameterizations of wave breaking and drag.

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Stephen D. Eckermann, Dave Broutman, and Harold Knight

Abstract

Effects of horizontal geometrical spreading on the amplitude variation with height of linear three-dimensional hydrostatic orographic gravity waves (OGWs) are quantified via relevant simplifications to the governing transform relations, leading to analytical solutions. The analysis is restricted to elliptical Gaussian obstacles with principal axes aligned parallel and perpendicular to unidirectional shear flow and to vertical displacement and steepness amplitudes, given their relevance to OGW drag parameterizations in global models. Two solutions are derived: a “small l” solution in which horizontal wavenumbers l orthogonal to the flow are taken to be much smaller than those parallel to the flow, and a “single k” solution in which horizontal wavenumbers k parallel to the flow have a single mean value. The resulting analytical relations, valid for arbitrary vertical profiles of upstream winds and stability, depend only on the obstacle’s elliptical aspect ratio β and a normalized height coordinate incorporating wind and stability variations. These analytical approximations accurately reproduce the salient features of the exact numerical transform solutions. These include monotonic decreases with height that asymptotically approach z −1/2 forms at large z and strong β dependence in amplitude diminution with height. Steepness singularities close to the surface are shown to be a mathematical consequence of the Hilbert transform approach to deriving complex wavefield solutions. These approximate analytical solutions for horizontal geometrical spreading effects on wave amplitude highlight the importance of this missing physics for current parameterizations of OGW drag and offer an accurate and efficient means of incorporating some of these omitted effects.

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Crispin J. Marks and Stephen D. Eckermann

Abstract

The WKB ray-tracing formalism is extended to accommodate internal gravity waves of all frequencies in a rotating, stratified, and compressible three-dimensional atmosphere. This includes the derivation of equations governing the dispersion and refraction of the ray paths, a realistic wave amplitude equation that takes into account both radiative and turbulent damping effects, and extensions of previous wave saturation schemes to accommodate dynamical and convective instabilities along generally slanted axes.

These equations have been numerically coded into a global ray-tracing model that the authors have applied to the three-dimensional CIRA 1986 reference atmosphere model in a series of preliminary experiments to investigate the impact of the newly incorporated features on synthesized wave fields in the middle atmosphere.

Three main points emerge from these experiments. First, there is a striking reduction in the high-frequency cutoff with decreasing horizontal wavenumber due to a more complete dispersion relation. Second, adoption of a climatological, height-varying turbulent diffusivity profile based on measurements indicates that turbulent damping is more important than scale-dependent infrared radiative damping over a wide range of wavelengths and frequencies in all but the lower levels of the middle atmosphere. Last, the authors demonstrate that the presence of climatological planetary waves during the northern winter produces greatly varied ray paths for waves of fixed characteristics launched from different longitudes. The implications of these findings for future ray-tracing studies are discussed.

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Stephen D. Eckermann and Robert A. Vincent

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Four extended observational campaigns were conducted during August and November 1988 with an ST (stratosphere–troposphere) radar in southern Australia during the passage of cold fronts over the system, giving around 30 days of three-dimensional wind measurements with 15-min time and 0.5-km height resolution over the 2–11.5-km height range. Order of magnitude increases in the variance of time-fluctuating wind velocities were measured during frontal passages, which are definitively ascribed to gravity waves. The time–height morphology of the horizontal- and vertical-velocity fluctuations differed. Bursts of horizontal-velocity variance u2 + υ2 ∼ 10–100 m2 s−2 arose at upper levels about a day before the frontal boundary arrived, and this activity gradually extended to lower heights as the front neared. The arrival of the frontal boundary marked a sudden reduction in this activity. After the frontal boundary passed, reduced activity persisted for ∼ 12 hours, after which bursts in u2 + υ2 returned at upper levels and persisted typically for about a day. These bursts arose in regions of high mean wind speeds (∼20–50 m s−1), and analysis associates this activity with a spectrum of many saturating inertia–gravity waves with long horizontal wavelengths and large ground-based phase speeds. Strong interaction between the waves and the mean flow is likely. In contrast, bursts in vertical-velocity fluctuations, w′, were confined almost entirely to the troposphere and were quasi-sinusoidal in appearance. These fluctuations are ascribed to gravity waves with high intrinsic frequencies. Significant w′ amplitudes were evident both after and prior to frontal passage, but the largest amplitudes (w′ ∼ 0.5 m s−1) occurred with the onset of strong vertical circulation when the frontal boundary arrived. The smaller w′ amplitudes observed in the stratosphere are due in part to the more oblique propagation of wave energy in this more stable environment, but may also reflect vertical ducting of this activity at altitudes of small static stability just below the tropopause. Two clear cases of ducted w′ oscillations are identified with the aid of radiosonde temperature data from a nearby site. Comparisons between these measurements and the limited numerical modeling of frontal gravity waves show some similarities in wave characteristics.

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Dong L. Wu and Stephen D. Eckermann

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The gravity wave (GW)–resolving capabilities of 118-GHz saturated thermal radiances acquired throughout the stratosphere by the Microwave Limb Sounder (MLS) on the Aura satellite are investigated and initial results presented. Because the saturated (optically thick) radiances resolve GW perturbations from a given altitude at different horizontal locations, variances are evaluated at 12 pressure altitudes between ∼21 and 51 km using the 40 saturated radiances found at the bottom of each limb scan. Forward modeling simulations show that these variances are controlled mostly by GWs with vertical wavelengths λz > 5 km and horizontal along-track wavelengths of λy ∼ 100–200 km. The tilted cigar-shaped three-dimensional weighting functions yield highly selective responses to GWs of high intrinsic frequency that propagate toward the instrument. The latter property is used to infer the net meridional component of GW propagation by differencing the variances acquired from ascending (A) and descending (D) orbits. Because of improved vertical resolution and sensitivity, Aura MLS GW variances are ∼5–8 times larger than those from the Upper Atmosphere Research Satellite (UARS) MLS. Like UARS MLS variances, monthly-mean Aura MLS variances in January and July 2005 are enhanced when local background wind speeds are large, due largely to GW visibility effects. Zonal asymmetries in variance maps reveal enhanced GW activity at high latitudes due to forcing by flow over major mountain ranges and at tropical and subtropical latitudes due to enhanced deep convective generation as inferred from contemporaneous MLS cloud-ice data. At 21–28-km altitude (heights not measured by the UARS MLS), GW variance in the tropics is systematically enhanced and shows clear variations with the phase of the quasi-biennial oscillation, in general agreement with GW temperature variances derived from radiosonde, rocketsonde, and limb-scan vertical profiles. GW-induced temperature variances at ∼44-km altitude derived from operational global analysis fields of the ECMWF Integrated Forecast System in August 2006 reveal latitudinal bands of enhanced GW variance and preferred GW meridional propagation directions that are similar to those inferred from the MLS variances, highlighting the potential of MLS GW data for validating the stratospheric GWs simulated and/or parameterized in global models.

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

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

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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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