Gravity Wave Dynamics Research Articles

Noisy Lagrangian Tracers for Filtering Random Rotating Compressible Flows

The recovery of a random turbulent velocity field using Lagrangian tracers that move with the fluid flow is a practically important problem. This paper studies the filtering skill of $$L$$ -noisy Lagrangian tracers in recovering random rotating compressible flows that are a linear combination of random incompressible geostrophically balanced (GB) flow and random rotating compressible gravity waves. The idealized random fields are defined through forced damped random amplitudes of Fourier eigenmodes of the rotating shallow-water equations with the rotation rate measured by the Rossby number $$\varepsilon $$ . In many realistic geophysical flows, there is fast rotation so $$\varepsilon $$ satisfies $$\varepsilon \ll 1$$ and the random rotating shallow-water equations become a slow–fast system where often the primary practical objective is the recovery of the GB component from the Lagrangian tracer observations. Unfortunately, the $$L$$ -noisy Lagrangian tracer observations are highly nonlinear and mix the slow GB modes and the fast gravity modes. Despite this inherent nonlinearity, it is shown here that there are closed analytical formulas for the optimal filter for recovering these random rotating compressible flows for any $$\varepsilon $$ involving Ricatti equations with random coefficients. The performance of the optimal filter is compared and contrasted through mathematical theorems and concise numerical experiments with the performance of the optimal filter for the incompressible GB random flow with $$L$$ -noisy Lagrangian tracers involving only the GB part of the flow. In addition, a sub-optimal filter is defined for recovering the GB flow alone through observing the $$L$$ -noisy random compressible Lagrangian trajectories, so the effect of the gravity wave dynamics is unresolved but effects the tracer observations. Rigorous theorems proved below through suitable stochastic fast-wave averaging techniques and explicit formulas rigorously demonstrate that all these filters have comparable skill in recovering the slow GB flow in the limit $$\varepsilon \rightarrow 0$$ for any bounded time interval. Concise numerical experiments confirm the mathematical theory and elucidate various new features of filter performance as the Rossby number $$\varepsilon $$ , the number of tracers $$L$$ and the tracer noise variance change.

On the application of Wentzel–Kramer–Brillouin theory for the simulation of the weakly nonlinear dynamics of gravity waves

The dynamics of internal gravity waves is modelled using Wentzel–Kramer–Brillouin (WKB) theory in position–wave number phase space. A transport equation for the phase‐space wave‐action density is derived for describing one‐dimensional wave fields in a background with height‐dependent stratification and height‐ and time‐dependent horizontal‐mean horizontal wind, where the mean wind is coupled to the waves through the divergence of the mean vertical flux of horizontal momentum associated with the waves. The phase‐space approach bypasses the caustics problem that occurs in WKB ray‐tracing models when the wave number becomes a multivalued function of position, such as in the case of a wave packet encountering a reflecting jet or in the presence of a time‐dependent background flow. Two numerical models were developed to solve the coupled equations for the wave‐action density and horizontal mean wind: an Eulerian model using a finite‐volume method and a Lagrangian ‘phase‐space ray tracer’ that transports wave‐action density along phase‐space paths determined by the classical WKB ray equations for position and wave number. The models are used to simulate the upward propagation of a Gaussian wave packet through a variable stratification, a wind jet and the mean flow induced by the waves. Results from the WKB models are in good agreement with simulations using a weakly nonlinear wave‐resolving model, as well as with a fully nonlinear large‐eddy‐simulation model. The work is a step toward more realistic parametrizations of atmospheric gravity waves in weather and climate models.

Nonlinear dynamics of trapped waves on jet currents and rogue waves.

Nonlinear dynamics of surface gravity waves trapped by an opposing jet current is studied analytically and numerically. For wave fields narrow band in frequency but not necessarily with narrow angular distributions the developed asymptotic weakly nonlinear theory based on the modal approach of Shrira and Slunyaev [J. Fluid. Mech. 738, 65 (2014)] leads to the one-dimensional modified nonlinear Schrödinger equation of self-focusing type for a single mode. Its solutions such as envelope solitons and breathers are considered to be prototypes of rogue waves; these solutions, in contrast to waves in the absence of currents, are robust with respect to transverse perturbations, which suggests a potentially higher probability of rogue waves. Robustness of the long-lived analytical solutions describing modulated trapped waves and solitary wave groups is verified by direct numerical simulations of potential Euler equations.

Asymptotical analysis of internal gravity wave dynamics in stratified medium

In paper fundamental problems of internal gravity waves dynamics are considered. Wave dynamics of stratified medium (ocean, atmosphere) is highly dependent on density distribution and bottom topography. The exact analytical solution is obtained only if the water distribution density and bottom shape described by some model functions. The solution of this problem is expressed in terms of the Green’s function and the asymptotic representations of the solutions are considered. The uniform asymptotic forms of the internal gravity waves in horizontally inhomogeneous and non-stationary stratified ocean are obtained. A modified spatio-temporal ray method is proposed, which belongs to the class of geometrical optics methods (WKBJ method). The solution is proposed in terms of wave modes, propagating independently at the adiabatic approximation, and described as a non-integral degree series of a small parameter characterizing the stratified medium properties. Analytical and numerical algorithms of internal gravity wave calculations for the real ocean parameters are presented.

Detection of gravity waves across the USArray: A case study

Barometers deployed as part of the USArray Transportable Array (TA) recorded large (300 Pa peak-to-peak) gravity waves, or gust fronts, that originated near a severe tornadic storm system in the southern United States on April 27, 2011. We present a new method in which the TA is divided into a large number of elemental sub-arrays. Each 3-element sub-array (triad) is sufficiently closely spaced so that the long-period gravity wave signal is coherent, but large enough to provide a robust estimate of the signal's direction and speed. The results from each triad are combined to follow the progress of gravity waves as they propagate across the TA over 2 days in late April, 2011. We observe a large, high-amplitude gravity wave, spanning a region over 200,000 km2, progressing to the NNW away from the tornadic storm region. We also observe gravity waves with lower amplitudes and smaller spatial extent along the gulf coast, propagating southward, away from the storm region. This study demonstrates the functionality of the USArray pressure sensors for analyzing gravity wave dynamics over periods greater than 40 min and across a wide region. In principle the method could be applied to study other long-period wave phenomena recorded by any dense large-scale network and is not limited to gravity waves.

Ocean software for modeling dynamics of gravity waves

This work is aimed at describing the Ocean software package developed for modeling the dynamics of nonlinear gravity waves and the process of breaking on the surface of a liquid of infinite depth. The modeling is carried out only for a simply connected domain. The used algorithms of Longuet-Higgins, Cokelet, and Krasil’nikov are described. The structure of the software package, computational methods, and indicative functions are described. Tactics for conducting the experiments are proposed. The results of modeling and their comparison with the experimental data are given.

Nonlinear gravity-wave interactions in stratified turbulence

To investigate the dynamics of gravity waves in stratified Boussinesq flows, a model is derived that consists of all three-gravity-wave-mode interactions (the GGG model), excluding interactions involving the vortical mode. The GGG model is a natural extension of weak turbulence theory that accounts for exact three-gravity-wave resonances. The model is examined numerically by means of random, large-scale, high-frequency forcing. An immediate observation is a robust growth of the so-called vertically sheared horizontal flow (VSHF). In addition, there is a forward transfer of energy and equilibration of the nonzero-frequency (sometimes called “fast”) gravity-wave modes. These results show that gravity-wave-mode interactions by themselves are capable of systematic interscale energy transfer in a stratified fluid. Comparing numerical simulations of the GGG model and the full Boussinesq system, for the range of Froude numbers (Fr) considered (0.05 ≤ Fr ≤ 1), in both systems the VSHF is hardest to resolve. When adequately resolved, VSHF growth is more vigorous in the GGG model. Furthermore, a VSHF is observed to form in milder stratification scenarios in the GGG model than the full Boussinesq system. Finally, fully three-dimensional nonzero-frequency gravity-wave modes equilibrate in both systems and their scaling with vertical wavenumber follows similar power-laws. The slopes of the power-laws obtained depend on Fr and approach −2 (from above) at Fr = 0.05, which is the strongest stratification that can be properly resolved with our computational resources.

Linear mechanism of generation and intencification of internal gravity waves in the ionosphere at their interaction with a nonuniform zonal wind: 1. Model of the medium and initial dynamic equations

The specific features of the generation and intensification of internal gravity wave structures in different atmospheric-ionospheric regions, caused by zonal local nonuniform winds (shear flows), are studied. The model of the medium has been explained and an initial closed system of equations has been obtained in order to study the linear and nonlinear dynamics of internal gravity waves (IGWs) when they interact with the geomagnetic field in a dissipative ionosphere (for the D, E, and F regions).

Shear flow energy redistribution stipulated by the internal-gravity wavy structures in the dissipative ionosphere

The linear mechanism of generation, intensification and further nonlinear dynamics of internal gravity waves (IGW) in stably stratified dissipative ionosphere with non-uniform zonal wind (shear flow) is studied. In case of the shear flows the operators of linear problem are non-selfadjoint, and the corresponding Eigen functions – nonorthogonal. Thus, canonical – modal approach is of less use studying such motions. Non-modal mathematical analysis becomes more adequate for such problems. On the basis of non-modal approach, the equations of dynamics and the energy transfer of IGW disturbances in the ionosphere with a shear flow is obtained. Exact analytical solutions of the linear as well as the nonlinear dynamic equations of the problem are built. The increment of shear instability of IGW is defined. It is revealed that the transient amplification of IGW disturbances due time does not flow exponentially, but in algebraic – power law manner. The effectiveness of the linear amplification mechanism of IGW at interaction with non-uniform zonal wind is analyzed. It is shown that at initial linear stage of evolution IGW effectively temporarily draws energy from the shear flow significantly increasing (by an order of magnitude) own amplitude and energy. With amplitude growth the nonlinear mechanism turns on and the process ends with self-organization of nonlinear solitary, strongly localized IGW vortex structures (the monopole vortex, the transverse vortex chain or the longitudinal vortex street). Accumulation of these vortices in the ionospheric medium can create the strongly turbulent state.

Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test

Under suitable assumptions, the nonlinear dynamics of surface gravity waves can be modeled by the one-dimensional nonlinear Schrödinger equation. Besides traveling wave solutions like solitons, this model admits also breather solutions that are now considered as prototypes of rogue waves in ocean. We propose a novel technique to study the interaction between waves and ships/structures during extreme ocean conditions using such breather solutions. In particular, we discuss a state of the art sea-keeping test in a 90-meter long wave tank by creating a Peregrine breather solution hitting a scaled chemical tanker and we discuss its potential devastating effects on the ship.

Fundamental Problems of Internal Gravity Waves Dynamics in Ocean

In paper fundamental problems of internal gravity waves dynamics are considered. The solution of this problem is expressed in terms of the Green’s function and the asymptotic representations of the solutions are considered. The uniform asymptotic forms of the internal gravity waves in horizontally inhomogeneous and non-stationary stratified ocean are obtained. A modified spatio-temporal ray method is proposed, which belongs to the class of geometrical optics methods (WKBJ method). Analytical and numerical algorithms of internal gravity wave calculations for the real ocean parameters are presented.

Asymptotic solutions of higher approximations of fields of internal gravity waves in variable-depth stratified media

A problem of wave dynamics of internal gravity waves in a variable-depth stratified medium is considered. By using a modified method of geometrical optics (vertical modes—horizontal rays), wave modes of higher approximations of asymptotic solutions are constructed. It is demonstrated that the main contributions to the solution in real stratified media are made by the first terms of the corresponding asymptotic presentations.

Correction to “Gravity wave dynamics and effects in the middle atmosphere”

Correction to “Gravity wave dynamics and effects in the middle atmosphere”

Application of Zakharov equation in three dimensions to deep water gravity waves

A numerical method, based on the Zakharov equation in three dimensions, is developed to study the nonlinear dynamics of deep water gravity waves. We focus primarily on systems describing the evolution of hydrodynamic surface waves, considering a thick incompressible fluid spanning across all space available and subject to gravity. The reference height to the resting state is z = 0. In the case of hydrostatic equilibrium, the bottom of the water is located at a distance H from the water surface. This depth is considered constant. The height of the surface wave is denoted i¨i€¨x,y,t) and the volume occupied by the fluid is restricted by –H < z < h(x,y,t). This study shows that rogue wavesare generated by primary waves whose directions of propagation are nearly parallel (that is0 < qi€ i€¼i€ p/6). This ensures their unexpected character (expressed by thequasi-spontaneous passage from a calm situation to a sea greatly agitated). Key words: Zakharov equation, nonlinear dynamics, deep water gravity waves.

Boundary Layer Turbulence and Orographic Precipitation Growth in Cold Clouds: Evidence from Profiling Airborne Radar Data

Abstract Airborne vertically pointing Doppler radar data collected in 10 winter storms over the Medicine Bow Range in Wyoming are used to examine the importance of boundary layer (BL) turbulence for orographic precipitation growth. In all 10 cases, the cloud-base temperature was below 0°C and the bulk Froude number was more than 1.0, implying little or no blocking of the flow by the mountain barrier. Seven of the 10 storms sampled were postfrontal, with weak static stability and relatively shallow cloud tops. Doppler vertical velocity transects depict an approximately 1-km-deep turbulent layer draped over the terrain, sometimes clearly distinct from the stratified flow in the free troposphere aloft, where vertical motion is largely controlled by gravity wave dynamics. Spectral analysis of near-surface Doppler vertical velocity data in terrain-following coordinates reveals an inertial subrange with decreasing power with height toward the BL top. The composite of radar data profiles from the 10 flights is analyzed in frequency-by-altitude diagrams, with altitude expressed above ground level. These diagrams indicate a wide range of vertical velocities in the BL, and rapid snow growth within the BL as air rises through the cloud base, especially when BL turbulence is more intense. This snow growth is concentrated on the windward side of mountains, above the terrain–cloud base intersection. The dominant snow growth mechanism in the BL (i.e., by accretion or vapor deposition) cannot be established because of restrictions in aircraft flight level over complex terrain. Snow aggregation may have contributed to the observed rapid increase in reflectivity in the BL along the windward slope.

The instability of periodic surface gravity waves

Euler's equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this paper, we discuss the stability of periodic travelling wave solutions to the full set of nonlinear equations via a non-local formulation of the water wave problem, modified from that of Ablowitz, Fokas &amp; Musslimani (J. Fluid Mech., vol. 562, 2006, p. 313), restricted to a one-dimensional surface. Transforming the non-local formulation to a travelling coordinate frame, we obtain a new formulation for the stationary solutions in the travelling reference frame as a single equation for the surface in physical coordinates. We demonstrate that this equation can be used to numerically determine non-trivial travelling wave solutions by exploiting the bifurcation structure of this new equation. Specifically, we use the continuous dependence of the amplitude of the solutions on their propagation speed. Finally, we numerically examine the spectral stability of the periodic travelling wave solutions by extending Fourier–Floquet analysis to apply to the associated linear non-local problem. In addition to presenting the full spectrum of this linear stability problem, we recover past well-known results such as the Benjamin–Feir instability for waves in deep water. In shallow water, we find different instabilities. These shallow water instabilities are critically related to the wavelength of the perturbation and are difficult to find numerically. To address this problem, we propose a strategy to estimate a priori the location in the complex plane of the eigenvalues associated with the instability.

On the generation and evolution of numerically simulated large-amplitude internal gravity wave packets

Numerical simulations of internal gravity wave (IGW) dynamics typically rely on wave velocity and density fields which are either generated through forcing terms in the governing equations or are explicitly introduced as initial conditions. Both approaches are based on the associated solution to the inviscid linear internal wave equations and, thus, assume weak-amplitude, space-filling waves. Using spectral multidomain-based numerical simulations of the two-dimensional Navier–Stokes equations and focusing on the forcing-driven approach, this study examines the generation and subsequent evolution of large-amplitude IGW packets which are strongly localized in the vertical in a linearly stratified fluid. When the vertical envelope of the forcing terms varies relatively rapid when compared to the vertical wavelength, the associated large vertical gradients in the Reynolds stress field drive a nonpropagating negative horizontal mean flow component in the source region. The highly nonlinear interaction of this mean current with the propagating IGW packet leads to amplification of the wave, a significant distortion of its rear flank, and a substantial decay of its amplitude. Scaling arguments show that the mean flow is enhanced with a stronger degree of localization of the forcing, larger degree of hydrostaticity, and increasing wave packet steepness. Horizontal localization results in a pronounced reduction in mean flow strength mainly on account of the reduced vertical gradient of the wave Reynolds stress. Finally, two techniques are proposed toward the efficient containment of the mean flow at minimal computational cost. The findings of this study are of particular value in overcoming challenges in the design of robust computational process studies of IGW packet (or continuously forced wave train) interactions with a sloping boundary, critical layer, or caustic, where large wave amplitudes are required for any instabilities to develop. In addition, the detailed description of the dynamics of the wave generation region may offer a first probe into the underlying physics of disruptive near-source nonlinearities observed in laboratory experiments of persistently forced IGW beams. Finally, the question arises as to whether a highly vertically compact IGW packet, which has propagated far from its source but still maintains its original structure and amplitude, can indeed occur in nature.

Stratospheric gravity waves revealed in NWP model forecasts

AbstractCurrent operational numerical weather prediction models potentially have sufficient spatial and temporal resolution to resolve much of the inertia‐gravity wave spectrum (e.g. horizontal wavelengths in the region of 100–1000 km typically, and even longer zonal wavelengths in the Tropics). The effects of gravity wave dynamics are usually considered to be subgrid‐scale and parametrizable, either as stationary, mountain waves or travelling, non‐orographic waves. Simple techniques for isolating these waves in both the Met Office and ECMWF forecast models are presented and their associated wave fluxes (including three‐dimensional Eliassen–Palm fluxes) are computed for August 2006. In addition, a limited‐area configuration of the Met Office Unified model with 4 km horizontal resolution is run over a region that includes the southern half of the Andes mountain range. Spectral decompositions of the wave energy and momentum fluxes show a distinct peak in power at horizontal wavelengths of about 400 km, irrespective of whether a 4 or 40 km grid is used, although about one half of the power resides in wavelengths shorter than 200 km. The computed Eliassen–Palm fluxes are shown to be similar to the net parametrized gravity wave drag, although the latter are too widespread and uniform over the oceans, reflecting the absence of geographical source variation. Comparison of the time mean temperature amplitude of the waves in the mid‐stratosphere with estimates derived from limb‐sounder satellite measurements from the HIRDLS project suggests that forecast models are capable of capturing the correct overall strength and distribution of gravity wave activity. Copyright © 2011 British Crown copyright, the Met Office. Published by John Wiley &amp; Sons Ltd.

Simultaneous airglow, lidar, and radar measurements of mesospheric gravity waves over Japan

To investigate gravity wave dynamics in the mesosphere and lower thermosphere (MLT) region, we conducted coordinated observations of mesospheric gravity waves over Japan during the Aeronomy and Dynamics Observation campaign. Two all‐sky airglow imagers were used in this campaign to derive a two‐dimensional structure of the gravity waves; these imagers were installed at the middle and upper atmosphere (MU) observatory in Shigaraki (34.9°N, 136.1°E) and at the Dynic Astropark Observatory in Taga (35.2°N, 136.3°E). Simultaneous measurements of the horizontal winds and the temperature in the MLT region were provided by the meteor‐mode observations of the MU radar at Shigaraki and by a sodium temperature lidar at Uji (34.9°N, 135.8°E), respectively. On 2 October 2008, gravity waves having a horizontal wavelength of ∼170 km, wave period of ∼1 h, and propagating northeastward at ∼50 m s−1 were observed in the airglow keograms. Similar wave structures were observed in the time series of the meteor wind and lidar temperature data; the polarity of these waves varied consistently with the airglow intensity variations according to the linear theory of gravity waves. The phase speeds and momentum fluxes of the gravity waves, estimated from the wind and temperature observations, were also in good agreement with those obtained from the airglow measurements. These results demonstrate, both qualitatively and quantitatively, that an identical gravity wave structure was detected in all the airglow intensities, radar winds, and lidar temperature.

Influence of a future climate on the microphysical and optical properties of orographic cirrus clouds in ECHAM5

The European Centre/Hamburg 5 (ECHAM5) general circulation model is used in order to investigate the influence of a warmer climate on the microphysical and optical properties of orographic cirrus clouds. The main goal of this study is to highlight the variety of processes influencing the formation of orographic cirrus and to emphasize the importance of coupling dynamics and cloud microphysics in order to provide realistic predictions of the influence of a future climate on cloud microphysical and radiative properties. Therefore, a coupling of gravity wave dynamics and cloud microphysics is implemented in the model. The influence of additional moisture on the propagation of gravity waves is investigated by using the dry and moist Brunt‐Väisäla frequency in the calculation of the gravity wave‐induced vertical velocity in two different simulations. In both simulations, the vertical velocities increase in the warmer climate. The additional moisture decreases the Brunt‐Väisäla frequencies leading to less flow blocking and thus higher effective mountain heights. As this effect dominates over the decrease of the gravity wave amplitude due to more moisture, higher vertical velocities occur in a future climate. The opposite effect of a decreased vertical velocity in a future climate can be seen over the dry regions. From the present to the future climate, the ice crystal number concentration decreases despite the increased vertical velocities. Higher temperatures lead to a faster growth of the ice crystals, and the supersaturation is depleted faster such that no new crystals can be formed. The ice water content increases as more water vapor is available in a warmer climate. The net effect of a decreased ice crystal number concentration and an increased ice water content is an increased optical depth in a future climate. This result is in good agreement with recent cloud resolving studies. The effect of orographic cirrus clouds on the radiation is given by an increased short‐ and long‐wave cloud forcing, whereas the latter dominates. However, from the present to the future climate, no changes in orographic cloud cover and cloud forcing over mountains can be seen.