Dispersion Relation For Gravity Waves Research Articles

Inferring neutral winds in the ionospheric transition region from atmospheric-gravity-wave traveling-ionospheric-disturbance (AGW-TID) observations with the EISCAT VHF radar and the Nordic Meteor Radar Cluster

Abstract. Atmospheric gravity waves and traveling ionospheric disturbances can be observed in the neutral atmosphere and the ionosphere at a wide range of spatial and temporal scales. Especially at medium scales, these oscillations are often not resolved in general circulation models and are parameterized. We show that ionospheric disturbances forced by upward-propagating atmospheric gravity waves can be simultaneously observed with the EISCAT very high frequency incoherent scatter radar and the Nordic Meteor Radar Cluster. From combined multi-static measurements, both vertical and horizontal wave parameters can be determined by applying a specially developed Fourier filter analysis method. This method is demonstrated using the example of a strongly pronounced wave mode that occurred during the EISCAT experiment on 7 July 2020. Leveraging the developed technique, we show that the wave characteristics of traveling ionospheric disturbances are notably impacted by the fall transition of the mesosphere and lower thermosphere. We also demonstrate the application of using the determined wave parameters to infer the thermospheric neutral wind velocities. Applying the dissipative anelastic gravity wave dispersion relation, we obtain vertical wind profiles in the lower thermosphere.

Characterization of Gravity Waves in Three Dimensions in the Daytime Thermosphere Using Combined Optical and Radio Measurements and Estimation of Horizontal Neutral Winds

AbstractGravity waves, which are considered omnipresent in the Earth's upper atmosphere, are generally investigated by monitoring the fluctuations in different atmospheric parameters. Here, we report the propagation characteristics of thermospheric gravity waves both in horizontal and vertical directions obtained using collocated optical and radio measurements from Ahmedabad, India for February 2021. The measurements of OI 630.0 nm dayglow emission rates over zenith are used to derive time periods of gravity waves. Wave number analyses of variations in the emission over a large field‐of‐view have been performed to derive gravity wave scale sizes and propagation characteristics in the horizontal direction. Time periods, horizontal scale sizes, and propagation directions are found to be in the range of 31–125 min, 78–243 km, and 203°–248° from east, respectively. Vertical wavelengths of the gravity waves are obtained from radio measurements and are in the range of 26–247 km. As the gravity wave characteristics are influenced by the ambient neutral winds, the measured gravity wave propagation characteristics in three dimensions have been used as inputs into the gravity wave dispersion relation to estimate the magnitudes of thermospheric horizontal neutral winds. These estimated daytime winds in the direction of wave propagation are found to be in the range of 1–105 ms−1, and they compare well with those measured independently from MIGHTI added HWM14 model‐derived winds. The daytime winds estimated by this approach are possibly the first of their kind as obtained from ground‐based measurements.

A simple approximation for the drift rates of rotating polygons on a free fluid surface

We report on laboratory experiments investigating the dynamics of a free fluid surface in a cylindrical tank with a rotating bottom plate. The shear instability in the system creates polygonal structures, which propagate around the domain, along the fixed vertical sidewalls. We analyze the wavelengths and drift rates of these patterns, which are known from previous literature to be created by a complex interplay between centrifugal effects and gravity wave propagation in this unique geometry. We find that with an empirical correction factor, the drift rates of the polygonal vortices can be approximated fairly well by a surprisingly simple formula derived from the dispersion relation of linear gravity waves using easily observable parameters.

Numerical simulation of solitary gravity waves on deep water with constant vorticity

We present a numerical study of essentially nonlinear dynamics of surface gravity waves on deep water with constant vorticity using governing equations in conformal coordinates. The dispersion relation of surface gravity waves on shear flow is known to have two branches; one of which is weakly dispersive for long waves. Weakly nonlinear evolution of the waves of this branch can be described by the Benjamin–Ono equation, which is integrable and has soliton and multi-soliton solutions. Currently, the extent to which the properties of such solitary waves obtained within the weakly nonlinear model are preserved in the exact Euler equations is unknown. We investigate the behavior of this class of solitary waves without the restrictive assumption of weak nonlinearity by using the exact Euler equations. The evolution of localized initial perturbations leading to the formation of single or multiple solitary waves is modeled, and the properties of finite-amplitude solitary waves are discussed. We show that within the framework of the exact equations, two-soliton collisions are almost elastic, but in contrast to solutions of the Benjamin–Ono equation, the waves receive a phase shift as a result of the interaction.

Imager observation of concentric mesospheric gravity waves over Srinagar, Jammu and Kashmir, India

Using the NARL airglow imager in the Indian subtropical station of Srinagar, Kashmir, India, we report the characteristics of the concentric deformed circular pattern of gravity waves detected at ∼85 km height. OH molecular emission intensities from the height of ∼85 km as observed by the imager in the wide band optical filter of centre wavelength of 840 nm show concentric deformed circular gravity waves drifting northwestward. Using Atmospheric Infrared Sounder (AIRS) image at the centre wavelength of 8.1 µm, deep convective activity around the imager site in the lower atmosphere is recognized as the source mechanism of the observed concentric gravity waves in the mesosphere. The determined phase speed, direction and period of these concentric gravity waves (CGWs) is ∼47 m/s, 150° north of east and ∼10 min (intrinsic period ∼8 min) respectively. At heights of ∼97 km [OI (1S)] and ∼250 km [OI (1D)], these concentric gravity waves are sporadically visible depending on the possible vertical movement of airglow layer height. The dissipated portions of these concentric gravity waves get converted into turbulence at these heights. Using anelastic gravity wave dispersion relation, decreasing horizontal wavelength from the centre of concentric gravity waves is explained. This result is novel in the sense that it disagrees with most of the earlier interpretations of such observations.

A single-mode approximation for gravity waves in the thermosphere

In previous work (Knight et al., 2019) a multilayer Fourier method was introduced for obtaining solutions for a partial differential equation (PDE) associated with a dispersion relation for linear gravity waves in a vertically varying, anelastic, nonhydrostatic, viscous, and thermally diffusive atmosphere over some altitude range, given boundary conditions that allow waves to enter at the lower boundary and exit at the top. The previous work described an exact solution method for the dispersion-relation PDE using all wave modes. The current work describes a type of approximate solution, based on a single upgoing gravity-wave mode for each Fourier component, which is computationally cheaper and simpler to implement. Two different approaches to defining the single-mode solution are presented, one based on the Wentzel–Kramers–Brillouin (WKB) method and the other based on a simplified version of the multilayer approximation. The two approaches are shown to be mathematically equivalent. The accuracy of the single-mode approximation is investigated through comparison with the all-mode solution for a realistic example in which gravity waves generated by a convective storm in the troposphere propagate into the upper thermosphere and for which the single-mode approximation gives reasonably accurate results. It is necessary to include imaginary frequency shifts in some Fourier components in order for the different types of modes (e.g., upgoing vs. downgoing and gravity-wave vs. dissipative) to be well-defined in the presence of molecular viscosity. A systematic approach to determining the smallest possible imaginary frequency shift for any given Fourier component is introduced. The new method efficiently suppresses a specific type of numerical artifact associated with large imaginary frequency shifts.

Toward real-time optical estimation of ocean waves’ space-time fields

Stereo 3D reconstruction is continuously increasing its popularity in the study of mid-to small-scale sea waves. In the recent past, different approaches have been proposed to reconstruct the space-time sea surface elevation field from synchronized stereo frames. Usually, the reconstruction is performed by first recovering a dense and sparse 3D point cloud from stereo pairs and then by interpolating it into a regular grid. Even considering state-of-the-art methods, with typical image resolutions, it's unlikely to perform the first step in less than dozens of seconds (without paralellization), that can easily doubled if we include the subsequent interpolation step. This will limit the applicability of stereo based wave analysis for all the approaches (like monitoring or data assimilation) in which the time between the raw acquisition and the output of processed data is critical.In this paper, we propose a new method to directly estimate the sea surface spectrum over time from a sequence of stereo frames. We exploit the frequency dispersion relation of gravity waves and the (Non-uniform) Fast Fourier Transform to continuously update the 3D surface and 2D wavenumber spectrum estimations given a sparse set of matching features between the two frames. This effectively combines the two aforementioned steps with a performance gain of more than an order of magnitude while keeping a reconstruction accuracy that is extremely close to the current state-of-the-art approaches. Code is available at https://gitlab.com/fibe/wassfast.

Signature and energetics of internal gravity waves in stratified turbulence

Internal gravity waves propagating within homogeneous stratified turbulence are the subject of the present study. A spatiotemporal analysis is carried out on the results of direct numerical simulations including a forcing term, with the aim of showing the energy content of the simulations as a function of frequency, ω, and wave-vector inclination to the horizontal, θ. Clear signatures of the dispersion relation of internal gravity waves, ω = ±N cos θ, where N is the Brunt-Vaisala frequency, are observed in all our simulations, which have low Froude number, Frh 1, and increasing buoyancy Reynolds number up to Reb ≈ 10. Interestingly, we observe the presence of high-frequency waves with ω ∼ N and a corresponding low-frequency vortex mode, both containing a non-negligible amount of energy. These waves are large-scale waves, their energy signature being found at scales larger than the forcing scales. We also observe the growth of energy in the shear modes, constituting a horizontal mean flow, and we show that their continuous growth is due to an upscale energy transfer, from the forcing scales to larger horizontal as well as vertical scales. These shear modes are found to be responsible for Doppler shifting the frequency of the large-scale waves. When considering the wave energy across the simulations at varying Reb, such energy is seen to reduce as Reb is increased and the flow enters the strongly stratified turbulence regime. The classical wave-vortex decomposition, based on a purely spatial decomposition of instantaneous snapshots of the flow, is analyzed within the current framework and is seen to correspond relatively well to the “true” wave signal identified by the spatiotemporal analysis, at least for the large-scale waves with ω ∼ N. Distinct energy peaks in θ-ω space highlight that the waves have preferential directions of propagation, specifically θ = 45◦ and θ ≈ 55◦, similar to observations in studies of wave radiation from localized regions of turbulence. This suggests that the same wave-generation mechanisms may be relevant for homogeneous and inhomogeneous stratified turbulent flows.

Suppression of Wind Ripples and Microwave Backscattering Due to Turbulence Generated by Breaking Surface Waves

The role of wave breaking in microwave backscattering from the sea surface is a problem of great importance for the development of theories and methods on ocean remote sensing, in particular for oil spill remote sensing. Recently it has been shown that microwave radar return is determined by both Bragg and non-Bragg (non-polarized) scattering mechanisms and some evidence has been given that the latter is associated with wave breaking, in particular, with strong breaking such as spilling or plunging. However, our understanding of mechanisms of the action of strong wave breaking on small-scale wind waves (ripples) and thus on the radar return is still insufficient. In this paper an effect of suppression of radar backscattering after strong wave breaking has been revealed experimentally and has been attributed to the wind ripple suppression due to turbulence generated by strong wave breaking. The experiments were carried out in a wind wave tank where a frequency modulated wave train of intense meter-decimeter-scale surface waves was generated by a mechanical wave maker. The wave train was compressed according to the gravity wave dispersion relation (“dispersive focusing”) into a short-wave packet at a given distance from the wave maker. Strong wave breaking with wave crest overturning (spilling) occurred for one or two highest waves in the packet. Short decimeter-centimeter-scale wind waves were generated at gentle winds, simultaneously with the long breaking waves. A Ka-band scatterometer was used to study microwave backscattering from the surface waves in the tank. The scatterometer looking at the area of wave breaking was mounted over the tank at a height of about 1 m above the mean water level, the incidence angle of the microwave radiation was about 50 degrees. It has been obtained that the radar return in the presence of short wind waves is characterized by the radar Doppler spectrum with a peak roughly centered in the vicinity of Bragg wave frequencies. The radar return was strongly enhanced in a wide frequency range of the radar Doppler spectrum when a packet of long breaking waves arrived at the area irradiated by the radar. After the passage of breaking waves, the radar return strongly dropped and then slowly recovered to the initial level. Measurements of velocities in the upper water layer have confirmed that the attenuation of radar backscattering after wave breaking is due to suppression of short wind waves by turbulence generated in the breaking zone. A physical analysis of the effect has been presented.

On Deriving Gravity Wave Characteristics in the Daytime Upper Atmosphere Using Radio Technique

AbstractThe importance of neutral wave dynamics in the understanding of the upper atmospheric processes is well known. Conventionally, optical methods are used to derive information on the neutral wave dynamics by obtaining gravity wave (GW) characteristics. Optical measurement techniques use airglow emissions as tracers to obtain such information that correspond to altitudes from where the emissions emanate. However, in this paper, we describe a method using radio wave measurement technique (digisonde) to obtain information on the neutral GW behavior. It involves monitoring of variations in the heights of isoelectron densities as a function of time, and their phase shifts, if any, to derive vertical propagation speeds and scale sizes of GWs. The daytime values of GW time periods, vertical phase speeds, and vertical scale sizes obtained for the duration of 16–21 May 2015 are in the range of 1.47 ± 0.05 to 2.64 ± 0.07 hr, 30.06 ± 4.35 to 45.69 ± 11.84 m/s, and 183.21 ± 39.23 to 393.07 ± 66.38 km, respectively. Further, we have used the GW dispersion relation to make a first‐order estimation of the horizontal scale sizes. This method of deriving neutral GW characteristics through radio measurement technique is effective for the daytime conditions and opens up new possibilities of investigations of the wave dynamical behavior in the upper atmosphere during all weather conditions.

Nonlinear dispersion for ocean surface waves

Two expressions for the nonlinear dispersion relation for gravity waves on water of constant depth are derived, one for wave fields with discrete amplitude spectra, the other for wave fields with continuous wavenumber energy spectra. Numerical examples for wave quartets and for two-dimensional Pierson–Moskowitz spectra are given, and an important possible application is discussed.

Atmospheric gravity wave ray tracing: Ordinary and extraordinary waves

Calculation of internal gravity-wave ray paths in the atmosphere using a general three-dimensional ray tracing computer program is discussed. To initialize a ray-path calculation, it is necessary to specify the initial values for the components of the wave vector at the source so that the dispersion relation is satisfied. For acoustic waves, or for gravity waves in the absence of wind, there is no ambiguity in determining the magnitude of the wave vector once the frequency and direction of propagation (wave-normal direction) have been specified. For gravity waves with wind, however, it is necessary to solve a quartic equation to specify the magnitude of the wavenumber. Two of the roots of the quartic reduce to the usual solutions in the absence of wind, and we designate these roots as ‘ordinary waves.’ The two new roots (whose values approach infinity as the wind speed approaches zero), we designate as ‘extraordinary waves.’ A section contrasts the properties of the different gravity wave types.Comparison with some previously published examples of ray-path calculations of gravity waves shows that those previous examples were actually extraordinary waves, but their significance was not recognized at that time.In the absence of wind, gravity waves are restricted to a fan of propagation directions centered about horizontal propagation, but asymptotic gravity waves (in the Bousinesq approximation) have a fixed propagation direction for a given frequency. In the presence of a horizontal wind, however, the wave-normal direction is restricted to a fan of directions in the upwind direction, but not in the downwind direction. The ray direction for upwind propagation has no restrictions. A short review of gravity wave theory and the gravity wave dispersion relation is included.

Viscous Regularization of Breaking Faraday Waves

The effect of the viscosity of a liquid on the parameters of standing surface gravity waves in a vertically oscillating rectangular vessel has been experimentally studied. It has been shown for the first time that a 60-fold increase in the viscosity of a working medium as compared to water fundamentally changes the parameters of the second nonlinear wave mode: waves are regularized in the total absence of their breaking. The effect of viscosity on the resonance dependences and process of damping of waves has been studied. The numerical analysis of the dispersion relation for gravity waves has shown that the effects observed in the experiment are due to the presence of short-range perturbations in the cutoff region, where viscous dissipation becomes a dominant factor and short waves are suppressed.

Determination of gravity wave parameters in the airglow combining photometer and imager data

Abstract. Mesospheric airglow measurements of two or three layers were used to characterize both vertical and horizontal parameters of gravity waves. The data set was acquired coincidentally from a multi-channel filter (Multi-3) photometer and an all-sky imager located at São João do Cariri (7.4∘ S, 36.5∘ W) in the equatorial region from 2001 to 2007. Using a least-square fitting and wavelet analysis technique, the phase and amplitude of each observed wave were determined, as well as the amplitude growth. Using the dispersion relation of gravity waves, the vertical and horizontal wavelengths were estimated and compared to the horizontal wavelength obtained from the keogram analysis of the images observed by an all-sky imager. The results show that both horizontal and vertical wavelengths, obtained from the dispersion relation and keogram analysis, agree very well for the waves observed on the nights of 14 October and 18 December 2006. The determined parameters showed that the observed wave on the night of 18 December 2006 had a period of ∼43.8±2.19 min, with the horizontal wavelength of 235.66 ± 11.78 km having a downward phase propagation, whereas that of 14 October 2006 propagated with a period of ∼36.00±1.80 min with a horizontal wavelength of ∼195±9.80 km, and with an upward phase propagation. The observation of a wave taken by a photometer and an all-sky imager allowed us to conclude that the same wave could be observed by both instruments, permitting the investigation of the two-dimensional wave parameter. Keywords. Atmospheric composition and structure (airglow and aurora) – electromagnetics (wave propagation) – history of geophysics (atmospheric sciences)

Concentric traveling ionospheric disturbances triggered by the launch of a SpaceX Falcon 9 rocket

AbstractWe report the first observation of concentric traveling ionospheric disturbances (CTIDs) triggered by the launch of a SpaceX Falcon 9 rocket on 17 January 2016. The rocket‐triggered ionospheric disturbances show shock acoustic wave signature in the time rate change (time derivative) of total electron content (TEC), followed by CTIDs in the 8–15 min band‐pass filtering of TEC. The CTIDs propagated northward with phase velocity of 241–617 m/s and reached distances more than 1000 km away from the source on the rocket trajectory. The wave characteristics of CTIDs with periods of 10.5–12.7 min and wavelength ~ 200–400 km agree well with the gravity wave dispersion relation. The optimal wave source searching and gravity wave ray tracing technique suggested that the CTIDs have multiple sources which are originated from ~38–120 km altitude before and after the ignition of the second‐stage rocket, ~200 s after the rocket was launched.

A clear link connecting the troposphere and ionosphere: ionospheric reponses to the 2015 Typhoon Dujuan

The global navigation satellite system (GNSS) total electron content (TEC) sequences were used to capture the arrival time and location of the ionosphere disturbances in response to the 2015 Typhoon Dujuan. After removing the de-trended TEC variation, the clear ionosphere disturbances on the typhoon landing day could be distinguished, and these disturbances disappeared from the TEC sequences before and after the typhoon landing day. The foF2 data observed by Xiamen ionosonde station also show ionosphere disturbances. Based on the advantages of GNSS multi-point observations, the disturbances horizontal velocity in the ionosphere were estimated according to the linear theory for a dispersion relation of acoustic gravity waves (AGWs) in an isothermal atmosphere. The average horizontal velocity (\(\sim \)240 m/s) and the radial velocity (\(\sim \)287 m/s) were used in the two-dimensional grid search for the origin point on the Earth’s surface. The origin area was determined to be on the eastern side of Taiwan. Lastly, a possible physical mechanism is discussed in this study. When typhoons land on Taiwan, the severe convective storms and the drag effect from the Central Mountains create an ideal location for development of AGWs. Topographic conditions, like the high lapse rate, contribute to the formation of AGWs, which then propagates into the ionosphere altitude.

Effect of small floating disks on the propagation of gravity waves

A dispersion relation for gravity waves in water covered by disk-like impurities embedded in a viscous matrix is derived. The macroscopic equations are obtained by ensemble-averaging the fluid equations at the disk scale in the asymptotic limit of long waves and low disk surface fraction. Various regimes are identified depending on the disk radii and the thickness and viscosity of the top layer. Semi-quantitative analysis in the close-packing regime suggests dramatic modification of the dynamics, with orders of magnitude increase in wave damping and wave dispersion. A simplified model working in this regime is proposed. Possible applications to wave propagation in an ice-covered ocean are discussed and comparison with field data is provided.

Numerical bifurcation for the capillary Whitham equation

The so-called Whitham equation arises in the modeling of free surface water waves, and combines a generic nonlinear quadratic term with the exact linear dispersion relation for gravity waves on the free surface of a fluid with finite depth.In this work, the effect of incorporating capillarity into the Whitham equation is in focus. The capillary Whitham equation is a nonlocal equation similar to the usual Whitham equation, but containing an additional term with a coefficient depending on the Bond number which measures the relative strength of capillary and gravity effects on the wave motion.A spectral collocation scheme for computing approximations to periodic traveling waves for the capillary Whitham equation is put forward. Numerical approximations of periodic traveling waves are computed using a bifurcation approach, and a number of bifurcation curves are found. Our analysis uncovers a rich structure of bifurcation patterns, including subharmonic bifurcations, as well as connecting and crossing branches. Indeed, for some values of the Bond number, the bifurcation diagram features distinct branches of solutions which intersect at a secondary bifurcation point. The same branches may also cross without connecting, and some bifurcation curves feature self-crossings without self-connections.

First three dimensional wave characteristics in the daytime upper atmosphere derived from ground‐based multiwavelength oxygen dayglow emission measurements

AbstractFirst results on the three‐dimensional wave characteristics in the daytime upper atmosphere have been derived using measurements of oxygen dayglow emissions at 557.7, 630.0, and 777.4 nm that originate at around 130, 230, and 300 km (peak of the F region). The horizontal scale sizes of gravity waves (GWs), their time periods, phase propagation angle (counterclockwise from east), and phase speeds are found to vary in the range of 27–227 km, 32–70 min, 207°–253°, and 6–76 ms−1, respectively. Two‐dimensional measurements on the horizontal scale sizes in the daytime have not been reported before. Further, using Hines' (1960) dispersion relation for GWs, vertical scale sizes and phase angles have also been derived. This technique opens up new possibilities in the investigations of daytime wave dynamics in three dimensions in the upper atmosphere.

Observations of wave dispersion and attenuation in landfast ice

AbstractObservations of wave propagation in landfast ice were obtained in Tempelfjorden, Svalbard during March 2015. Wave motion was measured near the ice edge using inertial motion units and consisted of a combination of swell from the North Atlantic and wind‐generated waves. The waves were observed to be unidirectional in the ice with comparable magnitudes in the vertical and horizontal displacements. The dispersion relation was calculated from the measured phase difference between two adjacent sensors separated by a distance of approximately 60 m. Deviations from the gravity wave dispersion relation were observed during the growth phase of the waves and were consistent with the presence of flexural waves. This period of wave growth was accompanied by significant wave attenuation in the high frequency portion of the wave spectrum which persisted for 3–5 h.