Search Results

You are looking at 1 - 10 of 37 items for

  • Author or Editor: Stephen D. Eckermann x
  • Refine by Access: Content accessible to me x
Clear All Modify Search
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.

Full access
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.

Full access
Stephen D. Eckermann
and
Robert A. Vincent

Abstract

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.

Full access
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.

Full access
Dong L. Wu
and
Stephen D. Eckermann

Abstract

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.

Full access
Stephen D. Eckermann
,
Jun Ma
, and
Dave Broutman

Abstract

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.

Full access
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.

Full access
Stephen D. Eckermann
,
Dorothy E. Gibson-Wilde
, and
Julio T. Bacmeister

Abstract

Existing analytical models of wave-induced minor constituent fluctuations result from linearized perturbation expansions of a rate equation governing either number density or mixing ratio, whereas many numerical models now use isentropic parcel advection methods to simulate these effects. Exact relationships between the two approaches are not currently clear for gravity waves. Here, the parcel advection method is formalized and applied to derive analytical formulas for the response of vertical constituent profiles of arbitrary shape to adiabatic gravity wave displacements. These relations are compared to corresponding formulas from standard linearized perturbation analyses. Both methods accurately model perturbations produced by nondissipating hydrostatic gravity waves within idealized vertical tracer profiles. Both methods can also model wave-induced perturbations of minor constituents with shorter chemical lifetimes. This is demonstrated by using the parcel method to reproduce previous results for wave-induced fluctuations in upper-stratospheric ozone. The parcel-based approach yields more accurate models of nondissipating hydrostatic gravity wave effects on tracer distibutions with sharp spatial gradients. The authors demonstrate this by using the method to model wave-induced modulations of sporadic sodium layers in the mesosphere and of ozone laminae in the lower stratosphere. Conversely, the parcel method does not accurately model tracer density perturbations produced by nonhydrostatic waves, or situations in which the photochemical response of the constituent leads to significant diabatic feedback on the perturbing wave.

Full access
Stephen D. Eckermann
,
John Lindeman
,
Dave Broutman
,
Jun Ma
, and
Zafer Boybeyi

Abstract

Fully nonlinear mesoscale model simulations are used to investigate the momentum fluxes of gravity waves that emerge at a “far-field” height of 6 km from steady unsheared flow over both an axisymmetric and elliptical obstacle for nondimensional mountain heights ĥm = Fr−1 in the range 0.1–5, where Fr is the surface Froude number. Fourier- and Hilbert-transform diagnostics of model output yield local estimates of phase-averaged momentum flux, while area integrals of momentum flux quantify the amount of surface pressure drag that translates into far-field gravity waves, referred to here as the “wave drag” component. Estimates of surface and wave drag are compared to parameterization predictions and theory. Surface dynamics transition from linear to high-drag (wave breaking) states at critical inverse Froude numbers Fr c −1 predicted to within 10% by transform relations. Wave drag peaks at Fr c −1 < ĥm ≲ 2, where for the elliptical obstacle both surface and wave drag vacillate owing to cyclical buildup and breakdown of waves. For the axisymmetric obstacle, this occurs only at ĥm = 1.2. At ĥm ≳ 2–3 vacillation abates and normalized pressure drag assumes a common normalized form for both obstacles that varies approximately as ĥm −1.3. Wave drag in this range asymptotes to a constant absolute value that, despite its theoretical shortcomings, is predicted to within 10%–40% by an analytical relation based on linear clipped-obstacle drag for a Sheppard-based prediction of dividing streamline height. Constant wave drag at ĥm ∼ 2–5 arises despite large variations with ĥm in the three-dimensional morphology of the local wave momentum fluxes. Specific implications of these results for the parameterization of subgrid-scale orographic drag in global climate and weather models are discussed.

Full access
Dave Broutman
,
Roger H. J. Grimshaw
, and
Stephen D. Eckermann
Full access