Instability Of Gravity Waves Research Articles

Modulational instability and collapse of internal gravity waves in the atmosphere.

Nonlinear two-dimensional (IGWs) in the atmospheres of the Earth and the Sun are studied. The resulting two-dimensional nonlinear equationhas the form of a generalized nonlinear Schrödinger equationwith nonlocal nonlinearity, that is, when the nonlinear response depends on the wave intensity at some spatial domain. The modulation instability of IGWs is predicted, and specific cases for the Earth's atmosphere are considered. In a number of particular cases, the instability thresholds and instability growth rates are analytically found. Despite the nonlocal nonlinearity, we demonstrate the possibility of critical collapse of IGWs due to the scale homogeneity of the nonlinear term in spatial variables.

Three-Dimensional Static Instability of Gravity Waves and a Possible Parameterization of the Associated Wave Breaking

Abstract Parameterizations of subgrid-scale gravity waves (GWs) in atmospheric models commonly involve the description of the dissipation of GWs. Where they dissipate, GWs have an increased effect on the large-scale flow. Instabilities that trigger wave breaking are an important starting point for the route to dissipation. Possible destabilizing mechanisms are numerous, but the classical vertical static instability is still regarded as a key indicator for the disposition to wave breaking. In this work, we investigate how the horizontal variations associated with a GW could alter the criterion for static instability. To this end, we use an extension of the common parcel displacement method. This three-dimensional static stability analysis predicts a significantly larger range of instability than does the vertical static stability analysis. In this case, the Lindzen-type saturation adjustment to a state of marginal stability is perhaps a less suitable ansatz for the parameterization of the GW breaking. To develop a possible ansatz for the GW dissipation due to three-dimensional instability, we apply the methods of irreversible thermodynamics, which are embedded in the Gibbs formalism of dynamics. In this way, the parameterization does not only satisfy the second law of thermodynamics, but it can also be made consistent with the conservation of energy and further (non-)conservation principles. We develop the parameterization for a discrete spectrum of GW packets. Offline computations of GW drag and dissipative heating rates are performed for two vertical profiles of zonal wind and temperature for summer and winter conditions from CIRA data. The results are compared to benchmarks from the literature.

Two-dimensional stability analysis of finite-amplitude interfacial gravity waves in a two-layer fluid

We explore basic mechanisms for the instability of finite-amplitude interfacial gravity waves through a two-dimensional linear stability analysis of the periodic and irrotational plane motion of the interface between two unbounded homogeneous fluids of different density in the absence of background currents. The flow domains are conformally mapped into two half-planes, where the time-varying interface is always mapped onto the real axis. This unsteady conformal mapping technique with a suitable representation of the interface reduces the linear stability problem to a generalized eigenvalue problem, and allows us to accurately compute the growth rates of unstable disturbances superimposed on steady waves for a wide range of parameters. Numerical results show that the wave-induced Kelvin–Helmholtz (KH) instability due to the tangential velocity jump across the interface produced by the steady waves is the major instability mechanism. Any disturbances whose dominant wavenumbers are greater than a critical value grow exponentially. This critical wavenumber that depends on the steady wave steepness and the density ratio can be approximated by a local KH stability analysis, where the spatial variation of the wave-induced currents is neglected. It is shown, however, that the growth rates need to be found numerically with care and the successive collisions of eigenvalues result in the wave-induced KH instability. In addition, the present study extends the previous results for the small-wavenumber instability, such as modulational instability, of relatively small-amplitude steady waves to finite-amplitude ones.

Instabilities in internal gravity waves

<abstract><p>Internal gravity waves are propagating disturbances in stably stratified fluids, and can transport momentum and energy over large spatial extents. From a fundamental viewpoint, internal waves are interesting due to the nature of their dispersion relation, and their linear dynamics are reasonably well-understood. From an oceanographic viewpoint, a qualitative and quantitative understanding of significant internal wave generation in the ocean is emerging, while their dissipation mechanisms are being debated. This paper reviews the current knowledge on instabilities in internal gravity waves, primarily focusing on the growth of small-amplitude disturbances. Historically, wave-wave interactions based on weakly nonlinear expansions have driven progress in this field, to investigate spontaneous energy transfer to various temporal and spatial scales. Recent advances in numerical/experimental modeling and field observations have further revealed noticeable differences between various internal wave spatial forms in terms of their instability characteristics; this in turn has motivated theoretical calculations on appropriately chosen internal wave fields in various settings. After a brief introduction, we present a pedagogical discussion on linear internal waves and their different two-dimensional spatial forms. The general ideas concerning triadic resonance in internal waves are then introduced, before proceeding towards instability characteristics of plane waves, wave beams and modes. Results from various theoretical, experimental and numerical studies are summarized to provide an overall picture of the gaps in our understanding. An ocean perspective is then given, both in terms of the relevant outstanding questions and the various additional factors at play. While the applications in this review are focused on the ocean, several ideas are relevant to atmospheric and astrophysical systems too.</p></abstract>

Gravity wave instability structures and turbulence from more than 1.5 years of OH* airglow imager observations in Slovenia

Abstract. We analysed 286 nights of data from the OH* airglow imager FAIM 3 (Fast Airglow IMager) acquired at Otlica Observatory (45.93∘ N, 13.91∘ E), Slovenia, between 26 October 2017 and 6 June 2019. Measurements have been performed with a spatial resolution of 24 m per pixel and a temporal resolution of 2.8 s. A two-dimensional fast Fourier transform is applied to the image data to derive horizontal wavelengths between 48 m and 4.5 km in the upper mesosphere/lower thermosphere (UMLT) region. In contrast to the statistics of larger-scale gravity waves (horizontal wavelength up to ca. 50 km; Hannawald et al., 2019), we find a more isotropic distribution of directions of propagation, pointing to the presence of wave structures created above the stratospheric wind fields. A weak seasonal tendency of a majority of waves propagating eastward during winter may be due to instability features from breaking secondary gravity waves that were created in the stratosphere. We also observe an increased southward propagation during summer, which we interpret as an enhanced contribution of secondary gravity waves created as a consequence of primary wave filtering by the meridional mesospheric circulation. We present multiple observations of turbulence episodes captured by our high-resolution airglow imager and estimated the energy dissipation rate in the UMLT from image sequences in 25 cases. Values range around 0.08 and 9.03 W kg−1 and are on average higher than those in recent literature. The values found here would lead to an approximated localized maximum heating of 0.03–3.02 K per turbulence event. These are in the same range as the daily chemical heating rates for the entire atmosphere reported by Marsh (2011), which apparently stresses the importance of dynamical energy conversion in the UMLT.

A parallel cell-centered adaptive level set framework for efficient simulation of two-phase flows with subcycling and non-subcycling

We develop a unified adaptive level set (LS) framework using the multi-level collocated grid for incompressible two-phase flows. This framework allows us to advance all variables level by level using either the subcycling or the non-subcycling method such that the data advancement on each level is fully decoupled. A series of synchronization operations are designed to keep the momentum and mass conserved across all levels. A multi-level re-initialization method for the LS function is also proposed, which greatly improves the mass conservation of the two-phase flows. The collocated grid allows the use of a single set of differential schemes and interpolation operations for all variables, which greatly simplifies the numerical implementation. The capability and robustness of the computational framework are validated by a variety of canonical problems, including the inviscid shear layer, gravity wave, rising bubble, and Rayleigh-Taylor instability. It is shown that the present multi-level scheme can accurately resolve the interfaces of the two-phase flows with gravitational and surface tension effects while having good momentum and energy conservation. At last, a three-dimensional dam breaking problem is simulated to show the efficiency and significant speedup of the proposed framework.

Simulating turbulent mixing caused by local instability of internal gravity waves

Abstract

On the instability of finite-amplitude inertia-gravity waves

The hydrodynamic instability of monochromatic inertia-gravity waves (IGWs) of finite amplitude, propagating at small angles either to the vertical or horizontal, is studied. In both cases, the corresponding angle serves as a small parameter in the problem, and instability is investigated using the Galerkin method. For IGWs that propagate at a small angle to the vertical, it is shown that stable density stratification is a stabilizing factor, and fluid rotation is destabilizing. The opposite is true for IGWs that propagate at a small angle to the horizontal. Previous analysis of the short wave instability of the monochromatic internal gravity waves of finite amplitude that propagate at small angles to the vertical is generalised to the case of the IGW. As an auxiliary but methodologically useful problem, the stability of a monochromatic inertial wave of finite amplitude in a fluid of uniform density is investigated in the same mathematical formulation.

Different responses of the Rayleigh–Taylor type and resistive drift wave instabilities to the velocity shear

The effects of velocity shear on the unstable modes driven by the effective gravity (Rayleigh–Taylor and interchange) and resistive drift wave instabilities for inhomogeneous equilibrium fluid/plasma density are analyzed for the localized eigenmode problems. It is shown that the effect of the velocity shear drastically depends on the type of instability. Whereas the velocity shear can significantly suppress both Rayleigh–Taylor and interchange instabilities, it has only a weak impact on the growth rate of the resistive drift wave. This is directly related to the physical nature of these instabilities. For the Rayleigh–Taylor and interchange instabilities, the shear flow tilts the eddies of the stream functions, while for the resistive drift wave instability the shear flow simply shifts the eddies in the radial direction with no tilting. However, for a large velocity shear, the eigenmode solutions for resistive drift waves cease to exist.

Experimental study of dispersion and modulational instability of surface gravity waves on constant vorticity currents

This paper examines experimentally the dispersion and stability of weakly nonlinear waves on opposing linearly vertically sheared current profiles (with constant vorticity). Measurements are compared against predictions from the unidirectional constant vorticity nonlinear Schrodinger equation (the vor-NLSE) derived by Thomas et al. (Phys. Fluids, vol. 24, no. 12, 2012, 127102). The shear rate is negative in opposing currents when the magnitude of the current in the laboratory reference frame is negative (i.e. opposing the direction of wave propagation) and reduces with depth, as is most commonly encountered in nature. Compared to a uniform current with the same surface velocity, negative shear has the effect of increasing wavelength and enhancing stability. In experiments with a regular low-steepness wave, the dispersion relationship between wavelength and frequency is examined on five opposing current profiles with shear rates from to . For all current profiles, the linear constant vorticity dispersion relation predicts the wavenumber to within the confidence bounds associated with estimates of shear rate and surface current velocity. The effect of shear on modulational instability was determined by the spectral evolution of a carrier wave seeded with spectral sidebands on opposing current profiles with shear rates between and . Numerical solutions of the vor-NLSE are consistently found to predict sideband growth to within two standard deviations across repeated experiments, performing considerably better than its uniform-current NLSE counterpart. Similarly, the amplification of experimental wave envelopes is predicted well by numerical solutions of the vor-NLSE, and significantly over-predicted by the uniform-current NLSE.

Large‐Amplitude Mountain Waves in the Mesosphere Observed on 21 June 2014 During DEEPWAVE: 2. Nonlinear Dynamics, Wave Breaking, and Instabilities

AbstractWeak cross‐mountain flow over the New Zealand South Island on 21 June 2014 during the Deep Propagating Gravity Wave Experiment (DEEPWAVE) led to large‐amplitude mountain waves in the mesosphere and lower thermosphere. The mesosphere and lower thermosphere responses were observed by ground‐based instruments in the lee of the Southern Alps supporting DEEPWAVE, including an Advanced Mesosphere Temperature Mapper, a Rayleigh lidar, an All‐Sky Imager, and a Fabry‐Perot Interferometer. The character of the mountain wave responses at horizontal scales of ~30–90 km reveals strong “sawtooth” variations in the temperature field suggesting large vertical and horizontal displacements leading to mountain wave overturning. The observations also reveal multiple examples of apparent instability structures within the mountain wave field that arose accompanying large amplitudes and exhibited various forms, scales, and evolutions. This paper employs detailed data analyses and results of numerical modeling of gravity wave instability dynamics to interpret these mountain wave dynamics, their instability forms, scales, and expected environmental influences. Results demonstrate apparently general instability pathways for breaking of large‐amplitude gravity waves in environments without and with mean shear. A close link is also found between large‐amplitude gravity waves and the dominant instability scales that may yield additional abilities to quantify gravity wave characteristics and effects.

Numerical Simulations of Modulated Waves in a Higher-Order Dysthe Equation

The nonlinear stage of the modulational (Benjamin–Feir) instability of unidirectional deep-water surface gravity waves is simulated numerically by the fifth-order nonlinear envelope equations. The conditions of steep and breaking waves are concerned. The results are compared with the solution of the full potential Euler equations and with the lower-order envelope models (the 3-order nonlinear Schrodinger equation and the standard 4-order Dysthe equations). The generalized Dysthe model is shown to exhibit the tendency to re-stabilization of steep waves with respect to long perturbations.

Three-dimensional small-scale instabilities of plane internal gravity waves

We study the evolution of three-dimensional (3-D), small-scale, small-amplitude perturbations on a plane internal gravity wave using the local stability approach. The plane internal wave is characterised by its non-dimensional amplitude, $A$, and the angle the group velocity vector makes with gravity, $\unicode[STIX]{x1D6F7}$. For a given $(A,\unicode[STIX]{x1D6F7})$, the local stability equations are solved on the periodic fluid particle trajectories to obtain growth rates for all two-dimensional (2-D) and 3-D perturbation wave vectors. For small $A$, the local stability approach recovers previous results of 2-D parametric subharmonic instability (PSI) while offering new insights into 3-D PSI. Higher-order triadic resonances, and associated deviations from them, are also observed at small $A$. Moreover, for small $A$, purely transverse instabilities resulting from parametric resonance are shown to occur at select values of $\unicode[STIX]{x1D6F7}$. The possibility of a non-resonant instability mechanism for transverse perturbations at finite $A$ allows us to derive a heuristic, modified gravitational instability criterion. We then study the extension of small $A$ to finite $A$ internal wave instabilities, where we recover and build upon existing knowledge of small-scale, small-amplitude internal wave instabilities. Four distinct regions of the $(A,\unicode[STIX]{x1D6F7})$-plane based on the dominant instability modes are identified: 2-D PSI, 3-D oblique, quasi-2-D shear-aligned, and 3-D transverse. Our study demonstrates the local stability approach as a physically insightful and computationally efficient tool, with potentially broad utility for studies that are based on other theoretical approaches and numerical simulations of small-scale instabilities of internal waves in various settings.

Erratum to: On shear instability and propagation of surface gravity waves in sea straits

Erratum to: On shear instability and propagation of surface gravity waves in sea straits

On shear instability and propagation of surface gravity waves in sea straits

Normal mode analysis of stability of shear flows of inviscid, incompressible fluids in channels with variable cross section and with a free upper surface is undertaken. Some general analytical results are obtained for basic flows with smooth velocity profiles in channels with smooth topography function. A Kelvin-Helmholtz instability study in a channel with constant topography function is also analyzed.

A nonlinear Schrödinger equation for gravity–capillary water waves on arbitrary depth with constant vorticity. Part 1

A nonlinear Schrödinger equation for the envelope of two-dimensional gravity–capillary waves propagating at the free surface of a vertically sheared current of constant vorticity is derived. In this paper we extend to gravity–capillary wave trains the results of Thomas et al. (Phys. Fluids, 2012, 127102) and complete the stability analysis and stability diagram of Djordjevic & Redekopp (J. Fluid Mech., vol. 79, 1977, pp. 703–714) in the presence of vorticity. The vorticity effect on the modulational instability of weakly nonlinear gravity–capillary wave packets is investigated. It is shown that the vorticity modifies significantly the modulational instability of gravity–capillary wave trains, namely the growth rate and instability bandwidth. It is found that the rate of growth of modulational instability of short gravity waves influenced by surface tension behaves like pure gravity waves: (i) in infinite depth, the growth rate is reduced in the presence of positive vorticity and amplified in the presence of negative vorticity; (ii) in finite depth, it is reduced when the vorticity is positive and amplified and finally reduced when the vorticity is negative. The combined effect of vorticity and surface tension is to increase the rate of growth of modulational instability of short gravity waves influenced by surface tension, namely when the vorticity is negative. The rate of growth of modulational instability of capillary waves is amplified by negative vorticity and attenuated by positive vorticity. Stability diagrams are plotted and it is shown that they are significantly modified by the introduction of the vorticity.

On the instability of internal gravity waves propagating at small but finite angles to the vertical

The question of the instability of internal gravity waves (IGWs) propagating at small angles to the vertical is re-visited. The case of an IGW of finite-amplitude propagating at a very small but finite angle to the vertical is considered. This angle serves as a small parameter in the problem, and the instability of such an IGW is investigated by using the Fourier method and the Sivashinsky integral relations. The analysis undertaken confirms the existence of short-wave instability for small IGW amplitudes and for an arbitrarily small value of their propagation angle to the vertical. For small viscosity and thermal conductivity of the fluid medium, the growth rate of the most unstable mode is proportional to the square of the amplitude of the IGW. The results obtained may be of interest for interpreting the results of observations and confirming the existence of turbulence in the middle atmosphere.

Linear-shear-current modified Schrödinger equation for gravity waves in finite water depth.

A nonlinear Schrödinger equation for the propagation of two-dimensional surface gravity waves on linear shear currents in finite water depth is derived. In the derivation, linear shear currents are assumed to be a linear combination of depth-uniform currents and constant vorticity. Therefore, the equation includes the combined effects of depth-uniform currents and constant vorticity. Next, using the equation, the properties of the modulational instability of gravity waves on linear shear currents are investigated. It is showed that shear currents significantly modify the modulational instability properties of weakly nonlinear waves. Furthermore, the influence of linear shear currents on Peregrine breather which can be seen as a prototype of freak waves is also studied. It is demonstrated that depth-uniform opposing currents can reduce the breather extension in both the time and spatial domain in intermediate water depth, but following currents has the adverse impact, indicating that a wave packets with freak waves formed on following currents contain more hazardous waves in finite water depth. However, the corresponding and coexisting vorticity can counteract the influence of currents. Additionally, if the water depth is deep enough, shear currents have negligible effect on the characteristics of Peregrine breathers.

Sixteen year variation of horizontal phase velocity and propagation direction of mesospheric and thermospheric waves in airglow images at Shigaraki, Japan

AbstractWe analyzed the horizontal phase velocity of gravity waves and medium‐scale traveling ionospheric disturbances (MSTIDs) by using the three‐dimensional fast Fourier transform method developed by Matsuda et al. (2014) for 557.7 nm (altitude: 90–100 km) and 630.0 nm (altitude: 200–300 km) airglow images obtained at Shigaraki MU Observatory (34.8°N, 136.1°E, dip angle: 49°) over ∼16 years from 16 March 1999 to 20 February 2015. The analysis of 557.7 nm airglow images shows clear seasonal variation of the propagation direction of gravity waves in the mesopause region. In spring, summer, fall, and winter, the peak directions are northeastward, northeastward, northwestward, and southwestward, respectively. The difference in east‐west propagation direction between summer and winter is probably caused by the wind filtering effect due to the zonal mesospheric jet. Comparison with tropospheric reanalysis data shows that the difference in north‐south propagation direction between summer and winter is caused by differences in the latitudinal location of wave sources due to convective activity in the troposphere relative to Shigaraki. The analysis of 630.0 nm airglow images shows that the propagation direction of MSTIDs is mainly southwestward with a minor northeastward component throughout the 16 years. A clear negative correlation is seen between the yearly power spectral density of MSTIDs and F10.7 solar flux. This negative correlation with solar activity may be explained by the linear growth rate of the Perkins instability and secondary wave generation of gravity waves in the thermosphere.

High-resolution observations of small-scale gravity waves and turbulence features in the OH airglow layer

Abstract. A new version of the Fast Airglow Imager (FAIM) for the detection of atmospheric waves in the OH airglow layer has been set up at the German Remote Sensing Data Center (DFD) of the German Aerospace Center (DLR) at Oberpfaffenhofen (48.09° N, 11.28° E), Germany. The spatial resolution of the instrument is 17 m pixel−1 in zenith direction with a field of view (FOV) of 11.1 km × 9.0 km at the OH layer height of ca. 87 km. Since November 2015, the system has been in operation in two different setups (zenith angles 46 and 0°) with a temporal resolution of 2.5 to 2.8 s. In a first case study we present observations of two small wave-like features that might be attributed to gravity wave instabilities. In order to spectrally analyse harmonic structures even on small spatial scales down to 550 m horizontal wavelength, we made use of the maximum entropy method (MEM) since this method exhibits an excellent wavelength resolution. MEM further allows analysing relatively short data series, which considerably helps to reduce problems such as stationarity of the underlying data series from a statistical point of view. We present an observation of the subsequent decay of well-organized wave fronts into eddies, which we tentatively interpret in terms of an indication for the onset of turbulence. Another remarkable event which demonstrates the technical capabilities of the instrument was observed during the night of 4–5 April 2016. It reveals the disintegration of a rather homogenous brightness variation into several filaments moving in different directions and with different speeds. It resembles the formation of a vortex with a horizontal axis of rotation likely related to a vertical wind shear. This case shows a notable similarity to what is expected from theoretical modelling of Kelvin–Helmholtz instabilities (KHIs). The comparatively high spatial resolution of the presented new version of the FAIM provides new insights into the structure of atmospheric wave instability and turbulent processes. Infrared imaging of wave dynamics on the sub-kilometre scale in the airglow layer supports the findings of theoretical simulations and modellings.