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NASA NTRS1990

Recurrence in truncated Boussinesq models for nonlinear waves in shallow water

The rapid spatial recurrence of weakly nonlinear and weakly dispersive progressive shallow-water waves is examined using a numerical integration technique on the discretized and truncated form of the Boussinesq equations. This study primarily examines recurrence in wave fields with Ursell number O(1) and characterizes the sensitivity of recurrence to initial spectral shape and number of allowed frequency modes. It is shown that the rapid spatial recurrence is not an inherent property of the considered Boussinesq systems for evolution distances of 10-50 wavelengths. The main result of the study is that highly truncated Boussinesq models of resonant shallow-water ocean surface gravity waves predict rapid multiple recurrence cycles, but that this is an artifact dependent on the number of allowed modes. For initial conditions consisting of essentially all energy concentrated in a single mode, damping of the recurrence cycles increases as the number of low-power background modes increases. When more than 32 modes are allowed, the recurrence behavior is relatively insensitive to the number of allowed modes.

Elgar, Steve↗
NASA NTRS1993

Breaking Kelvin-Helmholtz waves and cloud-top entrainment as revealed by K-band Doppler radar

Radars have occasionally detected breaking Kelvin-Helmholtz (KH) waves under clear-air conditions in the atmospheric boundary layer and in the free troposphere. However, very few direct measurements of such waves within clouds have previously been reported and those have not clearly documented wave breaking. In this article, we present some of the most detailed and striking radar observations to date of breaking KH waves within clouds and at cloud top and discuss their relevance to the issue of cloud-top entrainment, which is believed to be important in convective and stratiform clouds. Aircraft observations reported by Stith suggest that vortex-like circulations near cloud top are an entrainment mechanism in cumuliform clouds. Laboratory and modeling studies have examined possibility that KH instability may be responsible for mixing at cloud top, but direct observations have not yet been presented. Preliminary analyses shown here may help fill this gap. The data presented in this paper were obtained during two field projects in 1991 that included observations from the NOAA Wave Propagation Laboratory's K-band Doppler radar (wavelength = 8.7 mm) and special rawinsonde ascents. The sensitivity (-30 dBZ at 10 km range), fine spatial resolution (375-m pulse length and 0.5 degrees beamwidth), velocity measurement precision (5-10 cm s-1), scanning capability, and relative immunity to ground clutter make it sensitive to non-precipitating and weakly precipitating clouds, and make it an excellent instrument to study gravity waves in clouds. In particular, the narrow beam width and short pulse length create scattering volumes that are cylinders 37.5 m long and 45 m (90 m) in diameter at 5 km (10 km) range. These characteristics allow the radar to resolve the detailed structure in breaking KH waves such as have been seen in photographic cloud images.

Martner, Brooks E.↗
NASA NTRS1987

Quasi-Lagrangian measurements of density surface fluctuations and power spectra in the stratosphere

Pressure and temperature data from eight superpressure balloon flights at 26 km in the southern hemisphere stratosphere are analyzed. The balloons, which float on a constant density surface, travel steadily westward during summer and eastward during winter, as expected from local climatology. Two types of fluctuations are observed: neutral buoyancy oscillations (NBO) of around 4 min, and 0.1- to 1-hour oscillations that are characterized as small-amplitude density surface fluctuations. Lapse rates and densities are calculated and found to agree well with the expected values. Examples of wave damping and simultaneous fluctuation at two nearby balloons are presented. Spectral analysis is performed clearly showing the NBO and that the majority of the power is in the mesoscale range. Spectral slopes of power versus frequency are measured to be on the average -2.18 + or - 0.24 for pressure and -1.72 + or - 0.24 for temperature. These slopes are compared to the predictions of turbulence theories and the theory of a universal gravity wave spectrum.

Quinn, Elizabeth P.↗
NASA NTRS1985

Turbulence and wave particle interactions in solar-terrestrial plasmas

Activities in the following study areas are reported: (1) particle and wave processes in solar flares; (2) solar convection zone turbulence; and (3) solar radiation emission. To investigate the amplification of cyclotron maser radiation in solar flares, a radio frequency. (RF) heating model was developed for the corona surrounding the energy release site. Then nonlinear simulations of compressible convection display prominent penetration by plumes into regions of stable stratification at the base of the solar convection zone, leading to the excitation of internal gravity waves there. Lastly, linear saturation of electron-beam-driven Langmuir waves by ambient density fluctuations, nonlinear saturation by strong turbulence processes, and radiation emission mechanisms are examined. An additional section discusses solar magnetic fields and hydromagnetic waves in inhomogeneous media, and the effect of magnetic fields on stellar oscillation.

Dulk, G. A.↗
NASA NTRS1985

Midlatitude tropical interactions during winter

Pre-FGGE and FGGE/MONEX data are used to identify short term midlatitude tropical and longitudinal interactions during the winter monsoon. These interactions occur as cold surges, which develop over the East Asian continent and penetrate deep into the tropics with fast gravity wave speed. The observed interactions that occur after a surge include cyclogenesis and enhanced convection in the equatorial region, feedback from equatorial convection to midlatitude circulation systems, tropical east-west (Walker) circulations, and cross-equatorial influence. These interactions are also studied theoretically by analytical solutions of linearized shallow water equations. Response to transient forcing (monsoon surges) are mainly in Rossby and Kelvin modes. When the forcing time scale is short, significant gravity modes are also excited. The responses closely resemble observed winter monsoon flow. Responses to stationary forcing show that deep (barotropic) motions propagate energy away into high latitudes and that shallow (baroclinic) motions are trapped around the equator. It is shown that the barotropic teleconnection-type response to tropical sources found in previous numerical studies was due to the specified vertical wind shear and surface friction.

Chang, C. P.↗
NASA NTRS2000

Accuracy of Modelled Stratospheric Temperatures in the Winter Arctic Vortex from Infra Red Montgolfier Long Duration Balloon Measurements

The temperature of the stratosphere has been measured in the Arctic vortex every 9-10 minutes along the trajectory of four Infra Red Montgolfier long duration balloons flown for 7 to 22 days during the winters of 1997 and 1999. From a number of comparisons to independent sensors, the accuracy of the measurements is demonstrated to be plus or minus 0.5 K during nighttime and at altitude below 28 km (10 hPa). The performances of the analyses of global meteorological models, European Center for Medium Range Weather Forecasts (ECMWF) 31 and 50 levels, United Kingdom Meteorological Office (UKMO), Data Assimilation Office (DAO), National Climatic Prediction Center (NCEP) and NCEP/NCAR reanalysis, used in photochemical simulations of ozone destruction and interpretation of satellite data, are evaluated by comparison to this large (3500 data points) and homogeneous experimental data set. Most of models, except ECMWF31 in 1999, do show a smal1 average warm bias of between 0 and 1.6 K, with deviations particularly large, up to 20 K at high altitude (5hPa) in stratospheric warming conditions in 1999. Particularly wrong was ECMWF 31 levels near its top level at 10 hPa in 1999 where temperature 25 K colder than the real atmosphere were reported. The average dispersion between models and measurements varies from plus or minus 1.0 to plus or minus 3.0 K depending on the model and the year. It is shown to be the result of three contributions. The largest is a long wave modulation likely caused by the displacement of the temperature field in the analyses compared to real atmosphere. The second is the overestimation of the vertical gradient of temperature particularly in warming conditions, which explains the increase of dispersion from 1997 to 1999. Unexpectedly, the third and smallest (plus or minus 0.6-0.7 K) is the contribution of meso and subgrid scale vertical and horizontal features associated to the vertical propagation of orographic or gravity waves. Compared to other models, the newly available ECMWF 50 levels version assimilating the high vertical resolution radiances of the space borne Advanced Microwave Sounding Unit, performs significantly better (0.03 plus or minus 1.12 K on average between 10 and 140 hPa in 1999) than other models.

Pommereau, J.-P.↗
NASA NTRS1989

Scatterometer azimuthal response and wind wave directionality

Azimuthal response of a scatterometer to radiation scattered by the sea surface was studied in a wind-wave tank. The variation of the normalized radar cross section with the azimuth angle is fitted by a three-term series. Results show that the upwind-downwind asymmetry decreases as the wind speed increases. The crosswind modulation depends on the wind velocity. The results show that the evolution of the long-wind-crosswind ratio evolves with wind speed in a manner similar to the evolution of the isotropy of short capillary-gravity waves. The maximum of the isotropy of the short wind waves is obtained for wind velocities close to 4 m/s. For the same value of the velocity, the variations of radar response between long-wind and crosswind directions is minimum. For lower or higher values of wind velocities the directional accuracy of the radar increases, since the wind-wave field tends to align in the wind direction.

Giovanangeli, J. P.↗
NASA NTRS2001

A Regional Model Study of Synoptic Features Over West Africa

Synoptic weather features over West Africa were studied in simulations by the regional simulation model (RM) at the NASA/Goddard Institute for Space Studies. These pioneering simulations represent the beginning of an effort to adapt regional models for weather and climate prediction over West Africa. The RM uses a cartesian grid with 50 km horizontal resolution and fifteen vertical levels. An ensemble of four simulations was forced with lateral boundary conditions from ECMWF global analyses for the period 8-22 August 1988. The simulated mid-tropospheric circulation includes the skillful development and movement of several African wave disturbances. Wavelet analysis of mid-tropospheric winds detected a dominant periodicity of about 4 days and a secondary periodicity of 5-8 days. Spatial distributions of RM precipitation and precipitation time series were validated against daily rain gauge measurements and ISCCP satellite infrared cloud imagery. The time-space distribution of simulated precipitation was made more realistic by combining the ECMWR initial conditions with a 24-hr spin-up of the moisture field and also by damping high frequency gravity waves by dynamic initialization. Model precipitation "forecasts" over the Central Sahel were correlated with observations for about three days, but reinitializing with observed data on day 5 resulted in a dramatic improvement in the precipitation validation over the remaining 9 days. Results imply that information via the lateral boundary conditions is not always sufficient to minimize departures between simulated and actual precipitation patterns for more than several days. In addition, there was some evidence that the new initialization may increase the simulations' sensitivity to the quality of lateral boundary conditions.

Druyan, Leonard M.↗
NASA NTRS1982

Turbulence and wave particle interactions in solar-terrestrial plasmas

Theoretical modelling of two dimensional compressible convection in the Sun shows that convective flows can extend over many pressure scale heights without the nonlinear motions becoming supersonic, and that compressional work arising from pressure fluctuations can be comparable to that by buoyancy forces. These results are contrary to what was supposed in prevailing mixing length models for solar convection, and they imply a much greater degree of organized flow extending over the full depth of the convection zone. The nonlinear penetration of motions into the stable region below the convection zone was emphasized. These compressible flows are dominated by downward directed plumes in the unstable zone. Their strong penetration into the region of stable stratification below excites a broad spectrum of internal gravity waves there, and these in turn feed back upon the convection in the unstable zone to produce a rich time dependence.

Dulk, G. A.↗
NASA NTRS1999

Influence of Transport on Two-Dimensional Model Simulation. Tracer Sensitivity to 2-D Model Transport: Long Lived Tracers - 1

In this study, we examine the sensitivity of long lived tracers to changes in the base transport components in our 2-D model. Changes to the strength of the residual circulation in the upper troposphere and stratosphere and changes to the lower stratospheric K(sub zz) had similar effects in that increasing the transport rates decreased the overall stratospheric mean age, and increased the rate of removal of material from the stratosphere. Increasing the stratospheric K(sub yy) increased the mean age due to the greater recycling of air parcels through the middle atmosphere, via the residual circulation, before returning to the troposphere. However, increasing K(sub yy) along with self-consistent increases in the corresponding planetary wave drive, which leads to a stronger residual circulation, more than compensates for the K(sub yy)-effect, and produces significantly younger ages throughout the stratosphere. Simulations with very small tropical stratospheric K(sub yy) decreased the globally averaged age of air by as much as 25% in the middle and upper stratosphere, and resulted in substantially weaker vertical age gradients above 20 km in the extratropics. We found only very small stratospheric tracer sensitivity to the magnitude of the horizontal mixing across the tropopause, and to the strength of the mesospheric gravity wave drag and diffusion used in the model. We also investigated the transport influence on chemically active tracers and found a strong age-tracer correlation, both in concentration and calculated lifetimes. The base model transport gives the most favorable overall comparison with a variety of inert tracer observations, and provides a significant improvement over our previous 1995 model transport. Moderate changes to the base transport were found to provide modest agreement with some of the measurements. Transport scenarios with residence times ranging from moderately shorter to slightly longer relative to the base case simulated N2O lifetimes that were within the observational estimates of Volk et al. [1997]. However, only scenarios with rather fast transport rates were comparable with the Volk et al. estimates of CFCl3 lifetimes. This is inconsistent with model-measurement comparisons of mean age in which the base model or slightly slower transport rates compared the most favorably with balloon SF6 data. For all comparisons shown, large transport changes away from the base case resulted in simulations that were outside the range of measurements, and in many cases, far outside this range.

Fleming, Eric L.↗
NASA NTRS2012

View-Angle Dependent AIRS Cloud Radiances and Fluctuations: Implications of Organized Cloud Structures for Tropical Circulations

Interactions between wave dynamics and moisture generate clouds in a wide range of scales. Organized cloud structures produce statistically asymmetric radiances and perturbations in AIRS and AMSU-B measurements. With high resolution (approx.14 km beamwidth) and high-sensitivity instruments, these wave-modulated cloud structures can be readily detected from calibrated Levell radiance data. In this study we analyzed eight-year (2003 - 2010) statistics of AIRS cloud-induced radiances and found that in tropical convective regions the ascending (13:30 LST) measurements reveal higher view-angle asymmetry in cloud radiance than the descending (1:30 LST). The daytime asymmetry suggests 10% more cloudiness when the instrument views east, implying tilted and banded structures in most of the anvil clouds to which AIRS is sensitive. Such banded cloud structures are likely a manifestation of embedded westward propagating gravity waves in tropical convective systems. More importantly, organized cloud structures carry asymmetric momentum fluxes in addition to energy fluxes, which must be taken into account for modeling wave-wave and wave-mean flow interactions in tropical circulations.

Wu, Dong L.↗
NASA NTRS1986

Transfer function analysis of thermospheric perturbations

Applying perturbation theory, a spectral model in terms of vectors spherical harmonics (Legendre polynomials) is used to describe the short term thermospheric perturbations originating in the auroral regions. The source may be Joule heating, particle precipitation or ExB ion drift-momentum coupling. A multiconstituent atmosphere is considered, allowing for the collisional momentum exchange between species including Ar, O2, N2, O, He and H. The coupled equations of energy, mass and momentum conservation are solved simultaneously for the major species N2 and O. Applying homogeneous boundary conditions, the integration is carred out from the Earth's surface up to 700 km. In the analysis, the spherical harmonics are treated as eigenfunctions, assuming that the Earth's rotation (and prevailing circulation) do not significantly affect perturbations with periods which are typically much less than one day. Under these simplifying assumptions, and given a particular source distribution in the vertical, a two dimensional transfer function is constructed to describe the three dimensional response of the atmosphere. In the order of increasing horizontal wave numbers (order of polynomials), this transfer function reveals five components. To compile the transfer function, the numerical computations are very time consuming (about 100 hours on a VAX for one particular vertical source distribution). However, given the transfer function, the atmospheric response in space and time (using Fourier integral representation) can be constructed with a few seconds of a central processing unit. This model is applied in a case study of wind and temperature measurements on the Dynamics Explorer B, which show features characteristic of a ringlike excitation source in the auroral oval. The data can be interpreted as gravity waves which are focused (and amplified) in the polar region and then are reflected to propagate toward lower latitudes.

Mayr, H. G.↗
NASA NTRS1994

Investigation of atmospheric waves on Neptune

This document constitutes the final report for grant NAGW-2442 of the Neptune Data Analysis Program, which supported research concerning atmospheric dynamics on Neptune. Professor Von R. Eshleman was the principal investigator. David P. Hinson was a Co-Investigator. The grant covered the period 1 March 1991 through 31 August 1994, including a six month no-cost extension. Funding from this grant resulted in publication of one journal article and one book chapter as well as presentation of results at two conferences and in numerous seminars. A complete bibliography is given below. A copy of the journal article is attached along with abstracts from the book chapter and the conference presentations. With support from this grant we extended our analysis and interpretation of the Voyager Project. This research contributed to an improvement in our basic understanding of atmospheric dynamics on Neptune. The highlight was the discovery and characterization of inertio-gravity waves in the troposphere and stratosphere. Results include measures of basic wave properties, such as amplitudes and vertical wavelengths, as well as estimates of the effect of the waves on the photochemistry and momentum balance of the stratosphere. This investigation also yielded a better understanding of the potential of radio occultation experiments for studies of atmospheric waves. At the same time we developed new methods of data analysis for exploiting these capabilities. These are currently being applied to radio occultation data obtained with the Magellan spacecraft to study waves in the atmosphere of Venus. Future planetary missions, such as Mars Global Surveyor and Cassini, will benefit from these accomplishments.

Eshleman, Von R.↗
NASA NTRS2002

Are Solar Activity Variations Amplified by the QBO: A Modeling Study

Solar cycle activity effects (SCAE) in the lower and middle atmosphere, reported in several studies, are difficult to explain on the basis of the small changes in solar radiation that accompany the 11-year cycle. It is therefore natural to speculate that dynamical processes may come into play to produce a leverage. Such a leverage may be provided by the Quasi-Biennial Oscillation (QBO) in the zonal circulation of the stratosphere, which has been linked to solar activity variations. Driven primarily by wave mean flow interaction, the QBO period and its amplitude are variable but are also strongly influenced by the seasonal cycle in the solar radiation. This influence extends to low altitudes and is referred to as 'downward control'. Small changes in the solar radiative forcing may produce small changes in the period and phase of the QBO, but these in turn may produce measurable differences in the wind field. Thus, the QBO may be an amplifier of solar activity variations and a natural conduit of these variations to lower altitudes. To test this hypothesis, we conducted experiments with a 2D version of our Numerical Spectral Model that incorporates Hines' Doppler Spread Parameterization for small-scale gravity waves (GW). Solar cycle radiance variations (SCRV) are accounted for by changing the radiative heating rate on a logarithmic scale from 0.1% at the surface to 1% at 50 km to 10% at 100 km. With and without SCRV, but with the same GW flux, we then conduct numerical experiments to evaluate the magnitude of the SCAE in the zonal circulation. The numerical results indicate that, under certain conditions, the SCAE is significant and can extend to lower altitudes where the SCRV is small. For a modeled QBO period of 30 months, we find that the seasonal cycle in the solar forcing acts as a strong pacemaker to lock up the phase and period of the QBO. The SCAE then shows up primarily as a distinct but relatively weak amplitude modulation. But with a different QBO period between 30 and 34 (or less than 30, presumably) months, the seasonal phase lock is weak. Solar flux variations in the seasonal cycle then cause variations in the QBO period and phase. These amplify the SCAE to produce relatively large variations in the wind field. The SCAE in this case extends to mid-latitudes.

Mengel, J. G.↗
NASA NTRS2001

Solar Cycle Variations and Equatorial Oscillations: Modeling Study

Solar cycle activity effects (SCAE) in the lower and middle atmosphere, reported in several studies, are difficult to explain on the basis of the small changes in solar radiation that accompany the 11-year cycle, It is therefore natural to speculate that dynamical processes may come into play to produce a leverage. Such a leverage may be provided by the Quasi-Biennial Oscillation (QBO) in the zonal circulation of the stratosphere, which has been linked to solar activity variations. Driven primarily by wave mean flow interaction, the QBO period and its amplitude are variable but are also strongly influenced by the seasonal cycle in the solar radiation. This influence extends to low altitudes referred to as "downward control". Relatively small changes in solar radiative forcing can produce small changes in the period and phase of the QBO, but this in turn can produce measurable differences in the wind field. Thus, the QBO may be an amplifier of solar activity variations and a natural conduit of these variations to lower altitudes. To test this hypothesis, we conducted experiments with a 2D (two-dimensional) version of our Numerical Spectral Model that incorporates Hines' Doppler Spread Parameterization for small-scale gravity waves (GW). Solar cycle radiance variations (SCRV) are accounted for by changing the radiative heating rate on a logarithmic scale from 0.1 % at the surface to 1 % at 50 km to 10% at 100 km. With and without SCRV, but with the same GW flux, we then conduct numerical experiments to evaluate the magnitude of the SCAE in the zonal circulation. The numerical results indicate that, under certain conditions, the SCAE is significant and can extend to lower altitudes where the SCRV is inconsequential. At 20-km the differences in the modeled wind velocities are as large as 5 m/s. For a modeled QBO period of 30 months, we find that the seasonal cycle in the solar forcing (through the Semi-annual Oscillation (SAO)) acts as a strong pacemaker to lockup the phase and period of the QBO. The SCAE then shows up primarily as a distinct but relatively weak amplitude modulation. But with the QBO period between 30 and 34 (or less than 30, presumably) months, the seasonal phase lock is weak. Solar flux radiance variations in the seasonal cycle then cause variations in the QBO period and phase that amplify the SCAE to produce relatively large variations in the wind field. These variations also extend to mid latitudes.

Mayr, H. G.↗
NASA NTRSDate not supplied

The Response of the Quasi-Biennial Oscillation to Increased CO and its Modulation by Composition Feedbacks

The Quasi-Biennial Oscillation (QBO) impacts the large-scale circulation by altering lower stratospheric meridional temperature gradients which alter the propagation of upward propagating planetary waves. Proposed teleconnections of the QBO include impacts on the stratospheric polar vortices, extratropical surface winter climate, and the Madden-Julian Oscillation. However, long-term projections of the QBO remain highly uncertain. While recent multi-model investigations (Richter et al. 2019, Butchart et al. 2020) show that the amplitude of the QBO weakens robustly among models, changes in QBO period and stability remain highly uncertain. Here we examine the QBO response to increased greenhouse gases using the NASA Goddard Institute for Space Studies Middle Atmosphere Model E2.2 (Rind et al., 2020; Orbe et al. 2020). Compared to lower vertical resolution versions of ModelE, E2.2 also has a higher model top (0.002 hPa) and employs additional interactive non-orographic gravity wave drag sources from convection and shear, which produce a sufficiently realistic QBO, thus rendering it suitable for use in climate change studies. Overall, we find that both the QBO period and amplitude decrease in response to increased CO , the former related to increased lower stratospheric momentum fluxes (associated with convection) and the latter associated partly with a stronger residual mean circulation. Experiments integrated using fully interactive chemistry also reveal that ozone feedbacks significantly impact the magnitude of the QBO amplitude response. Finally, integrations using fixed (pre-industrial) sea surface temperatures (SST) show that the QBO amplitude responds differently to rapid adjustments versus SST feedbacks, as compared to the QBO period.

Quasi-Biennial Oscillation↗
NASA NTRS1999

Cassini Radio Science Experiments on Saturn and Titan Preserved Because of Lewis Analysis

The Cassini mission to Saturn is an international venture with participation from NASA, the European Space Agency, and the Italian Space Agency. The Cassini spacecraft was launched from Cape Canaveral in October 1997 and is scheduled to arrive at Saturn in July 2004. After arrival, the spacecraft will orbit Saturn about 60 times over a period of 4 years. During this time, the Cassini Radio Science Subsystem will be used to investigate the atmosphere and rings of Saturn and the atmosphere of its largest moon, Titan--which is larger than Mercury and is the only moon in our solar system with a dense atmosphere. A critical component in Cassini s Radio Science Subsystem is a traveling-wave tube (TWT) that was designed at the NASA Lewis Research Center and built by Hughes Electronic Dynamics Division (ref. 1). This TWT will amplify downlink microwave signals at a frequency of 32 GHz for the Deep Space Network and will be involved in a number of experiments. These include occultation experiments in which the microwave signal will be beamed through rings and atmospheres toward Earth. Researchers will analyze the received signals to determine the sizes and distributions of the particles in the rings and the structure and composition of the atmospheres. The Radio Science Subsystem also will also be used to more accurately determine the mass and size of Saturn and its moons, to investigate the solar corona, and to search for gravity waves from outside the solar system.

Wilson, Jeffrey D.↗
NASA NTRS1987

Global excitation of wave phenomena in a dissipative multiconstituent medium. III - Response characteristics for different sources in the earth's thermosphere

A linear trasnfer function model of the earth's thermosphere which includes the electric field momentum source is used to study the differences in the response characteristics for Joule heating and momentum coupling in the thermosphere. It is found that, for Joule/particle heating, the temperature and density perturbations contain a relatively large trapped component which has the property of a low-pass filter, with slow decay after the source is turned off. The decay time is sensitive to the altitude of energy deposition and is significantly reduced as the source peak moves from 125 to 150 km. For electric field momentum coupling, the trapped components in the temperature and density perturbations are relatively small. In the curl field of the velocity, however, the trapped component dominates, but compared with the temperature and density its decay time is much shorter. Outside the source region the form of excitation is of secondary importance for the generation of the various propagating gravity wave modes.

Mayr, H. G.↗