scholarly journals Equatorially Bounded Zonally Propagating Linear Waves on a Generalized β Plane

2009 ◽  
Vol 66 (9) ◽  
pp. 2937-2945 ◽  
Author(s):  
Mark D. Fruman
Keyword(s):  
Gravity Waves ◽  
Phase Speed ◽  
Boussinesq Equations ◽  
Buoyancy Frequency ◽  
Spatial Structures ◽  
Restoring Force ◽  
Meridional Component ◽  
Ω Phase ◽  
Linear Waves ◽  
Planetary Rotation

Abstract Meridionally confined zonally propagating wave solutions to the linear hydrostatic Boussinesq equations on a generalized equatorial β plane that includes the “nontraditional” Coriolis force terms associated with the poleward component of planetary rotation are calculated. Kelvin, Rossby, inertia–gravity, and mixed Rossby–gravity modes generalize from the traditional model with the dispersion relation unchanged. The effects of the nontraditional terms on all waves are the curving upward with latitude of the surfaces of constant phase and the equatorial trapping width of the solutions (the equatorial radius of deformation) increasing by order (Ω/N)2 compared to the traditional case, where Ω is the planetary rotation rate and N the buoyancy frequency. In addition, for the Rossby, inertia–gravity, and mixed Rossby–gravity modes, there is a phase shift of O(Ω/N) in the zonal and vertical velocity components relative to the meridional component, and their spatial structures are further modified by differences of O(Ω/N)2. For the Rossby and inertia–gravity waves, the modifications depend also on the phase speed of the wave. In the limit N ≫ Ω, the traditional approximation is justified and lines of constant phase in the y–z plane become horizontal, whereas for N ≪ Ω phase lines become everywhere almost parallel to the planetary rotation vector. In both limits, the phase lines are perpendicular to the dominant restoring force—respectively, gravity and the centrifugal force associated with the solid-body rotation of the atmosphere at rest in the rotating frame.

scholarly journals Linear Nuclear Acoustic Waves in Degenerate Quantum Plasma

2018 ◽  
Vol 5 (1) ◽  
pp. 20-23
Author(s):  
M Hasan ◽  
DMS Zaman
Keyword(s):  
Acoustic Waves ◽  
Mass Density ◽  
Phase Speed ◽  
Reductive Perturbation Method ◽  
Nonlinear Propagation ◽  
Quantum Plasma ◽  
Restoring Force ◽  
Linear Waves ◽  
Basic Features ◽  
Electrostatic Perturbation

A rigorous theoretical investigation has been made on the linear propagation of electrostatic perturbation modes of degenerate pressure driven modified nucleus-acoustic (DPDMNA) ‘waves in a degenerate quantum plasma (DQP) system. It contains cold inertia-less degenerate electron species (DES), cold inertial non-degenerate light nucleus species (LNS) and stationary heavy nucleus species (HNS) which maintains the quasi-neutrality condition at equilibrium only. The mass density of the cold LNS provides the inertia and the cold inertia-less cold LNS provides the inertia and the cold inertia-less DES gives rise to the restoring force. The reductive perturbation method has been used for the study of nonlinear propagation of the DPDMNA waves. The basic features of linear waves are supervised in a theoretical manner. It has been observed that the phase speed of DPDMNA waves changes with the change of charge density of the stationary HNS for both non-relativistic and ultra-relativistic DES; The NA waves with their dispersion properties which are consequential in various astrophysical and laboratory plasmas, have been broadly considered. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 20-23

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

Radiation of internal waves from groups of surface gravity waves

2017 ◽  
Vol 829 ◽  
pp. 280-303 ◽  
Author(s):  
S. Haney ◽  
W. R. Young
Keyword(s):  
Internal Waves ◽  
Energy Flux ◽  
Gravity Waves ◽  
Three Dimensions ◽  
Stokes Drift ◽  
Surface Gravity ◽  
Fourth Power ◽  
Buoyancy Frequency ◽  
Surface Gravity Waves ◽  
Short Period

Groups of surface gravity waves induce horizontally varying Stokes drift that drives convergence of water ahead of the group and divergence behind. The mass flux divergence associated with spatially variable Stokes drift pumps water downwards in front of the group and upwards in the rear. This ‘Stokes pumping’ creates a deep Eulerian return flow that sets the isopycnals below the wave group in motion and generates a trailing wake of internal gravity waves. We compute the energy flux from surface to internal waves by finding solutions of the wave-averaged Boussinesq equations in two and three dimensions forced by Stokes pumping at the surface. The two-dimensional (2-D) case is distinct from the 3-D case in that the stratification must be very strong, or the surface waves very slow for any internal wave (IW) radiation at all. On the other hand, in three dimensions, IW radiation always occurs, but with a larger energy flux as the stratification and surface wave (SW) amplitude increase or as the SW period is shorter. Specifically, the energy flux from SWs to IWs varies as the fourth power of the SW amplitude and of the buoyancy frequency, and is inversely proportional to the fifth power of the SW period. Using parameters typical of short period swell (e.g. 8 s SW period with 1 m amplitude) we find that the energy flux is small compared to both the total energy in a typical SW group and compared to the total IW energy. Therefore this coupling between SWs and IWs is not a significant sink of energy for the SWs nor a source for IWs. In an extreme case (e.g. 4 m amplitude 20 s period SWs) this coupling is a significant source of energy for IWs with frequency close to the buoyancy frequency.

scholarly journals Observation of Kelvin–Helmholtz instabilities and gravity waves in the summer mesopause above Andenes in Northern Norway

2018 ◽  
Vol 18 (9) ◽  
pp. 6721-6732 ◽  
Author(s):  
Gunter Stober ◽  
Svenja Sommer ◽  
Carsten Schult ◽  
Ralph Latteck ◽  
Jorge L. Chau
Keyword(s):  
Gravity Waves ◽  
Middle Atmosphere ◽  
Phase Speed ◽  
Velocity Measurements ◽  
Wind Variability ◽  
Short Period ◽  
Strong Shear ◽  
Summer Echoes ◽  
Period Wave ◽  
Horizontal Wavelength

Abstract. We present observations obtained with the Middle Atmosphere Alomar Radar System (MAARSY) to investigate short-period wave-like features using polar mesospheric summer echoes (PMSEs) as a tracer for the neutral dynamics. We conducted a multibeam experiment including 67 different beam directions during a 9-day campaign in June 2013. We identified two Kelvin–Helmholtz instability (KHI) events from the signal morphology of PMSE. The MAARSY observations are complemented by collocated meteor radar wind data to determine the mesoscale gravity wave activity and the vertical structure of the wind field above the PMSE. The KHIs occurred in a strong shear flow with Richardson numbers Ri < 0.25. In addition, we observed 15 wave-like events in our MAARSY multibeam observations applying a sophisticated decomposition of the radial velocity measurements using volume velocity processing. We retrieved the horizontal wavelength, intrinsic frequency, propagation direction, and phase speed from the horizontally resolved wind variability for 15 events. These events showed horizontal wavelengths between 20 and 40 km, vertical wavelengths between 5 and 10 km, and rather high intrinsic phase speeds between 45 and 85 m s−1 with intrinsic periods of 5–10 min.

The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics

1978 ◽  
Vol 360 (1703) ◽  
pp. 471-488 ◽  
Keyword(s):  
Stream Function ◽  
Gravity Waves ◽  
Velocity Potential ◽  
Travelling Waves ◽  
Normal Modes ◽  
Phase Speed ◽  
Finite Amplitude ◽  
Fundamental Wave ◽  
Wave Steepness ◽  
Horizontal Scale

In this paper we embark on a calculation of all the normal-mode perturbations of nonlinear, irrotational gravity waves as a function of the wave steepness. The method is to use as coordinates the stream-function and velocity potential in the steady, unperturbed wave (seen in a reference frame moving with the phase speed) together with the time t. The dependent quantities are the cartesian displacements and the perturbed stream function at the free surface. To begin we investigate the ‘superharmonics’, i.e. those perturbations having the same horizontal scale as the fundamental wave, or less. When the steepness of the fundamental is small, the normal modes take the form of travelling waves superposed on the basic nonlinear wave. As the steepness increases the frequency of each perturbation tends generally to be diminished. At a steepness ak ≈ 0.436 it appears that the two lowest modes coalesce and an instability arises. There is evidence that this critical steepness corresponds precisely with the steepness at which the phase velocity is a maximum, considered as a function of ak. The calculations are facilitated by the discovery of some new identities between the coefficients in Stokes’s expansion for waves of finite amplitude.

scholarly journals Dynamics of Mechanical Oscillator Mechanism for Stratospheric Gravity Waves Generated by Convection

Atmosphere ◽  
2020 ◽  
Vol 11 (9) ◽  
pp. 942
Author(s):  
Shiwang Yu ◽  
Lifeng Zhang ◽  
Ming Zhang ◽  
Yuan Wang
Keyword(s):  
Gravity Waves ◽  
Boussinesq Approximation ◽  
Generation Mechanism ◽  
Buoyancy Frequency ◽  
Background Wind ◽  
Dynamics Model ◽  
Mechanical Oscillator ◽  
Vortex System ◽  
Wave Resonance ◽  
Intrinsic Characteristic

The mechanical oscillator mechanism (MOM) for stratospheric gravity waves generated by convection is investigated with a dynamics model using the two-dimensional, nonhydrostatic and linear governing equations based on the Boussinesq approximation. The model is solved analytically with a fixed buoyancy oscillation (BO) at the tropopause as the boundary conditions. Results show that this BO is the source of stratospheric gravity waves and the MOM is the generation mechanism. The characteristics of the stratospheric gravity waves not only depend on the BO, but also rely on the stratospheric state, such as the background wind and the buoyancy frequency. When the vertical wavenumbers of the stratospheric gravity waves are close to those of the intrinsic characteristic waves (ICWs), which are the model solution without BO forcing at the tropopause, resonance occurs. Under the resonance conditions, the amplitudes of the stratospheric gravity waves increase significantly, even for low BO intensity. The background wind in the stratosphere has a large effect on wave resonance. Finally, numerical simulation results of a low-vortex system also verify that the MOM is the generation mechanism of stratospheric gravity waves generated by convection.

scholarly journals Seasonal and intra-diurnal variability of small-scale gravity waves in OH airglow at two Alpine stations

2019 ◽  
Vol 12 (1) ◽  
pp. 457-469 ◽  
Author(s):  
Patrick Hannawald ◽  
Carsten Schmidt ◽  
René Sedlak ◽  
Sabine Wüst ◽  
Michael Bittner
Keyword(s):  
Gravity Waves ◽  
High Frequency ◽  
Phase Speed ◽  
Propagation Direction ◽  
Field Of View ◽  
Small Scale ◽  
Data Sets ◽  
Data Set ◽  
Diurnal Variability ◽  
Fully Automatic

Abstract. Between December 2013 and August 2017 the instrument FAIM (Fast Airglow IMager) observed the OH airglow emission at two Alpine stations. A year of measurements was performed at Oberpfaffenhofen, Germany (48.09∘ N, 11.28∘ E) and 2 years at Sonnblick, Austria (47.05∘ N, 12.96∘ E). Both stations are part of the network for the detection of mesospheric change (NDMC). The temporal resolution is two frames per second and the field-of-view is 55 km × 60 km and 75 km × 90 km at the OH layer altitude of 87 km with a spatial resolution of 200 and 280 m per pixel, respectively. This resulted in two dense data sets allowing precise derivation of horizontal gravity wave parameters. The analysis is based on a two-dimensional fast Fourier transform with fully automatic peak extraction. By combining the information of consecutive images, time-dependent parameters such as the horizontal phase speed are extracted. The instrument is mainly sensitive to high-frequency small- and medium-scale gravity waves. A clear seasonal dependency concerning the meridional propagation direction is found for these waves in summer in the direction to the summer pole. The zonal direction of propagation is eastwards in summer and westwards in winter. Investigations of the data set revealed an intra-diurnal variability, which may be related to tides. The observed horizontal phase speed and the number of wave events per observation hour are higher in summer than in winter.

Stokes-multiplier expansion in an inhomogeneous differential equation with a small parameter

2005 ◽  
Vol 461 (2059) ◽  
pp. 2243-2256 ◽  
Author(s):  
E.I Ólafsdóttir ◽  
A.B Olde Daalhuis ◽  
J Vanneste
Keyword(s):  
Small Parameter ◽  
Gravity Waves ◽  
Homogeneous Solution ◽  
Boussinesq Equations ◽  
Inhomogeneous Equation ◽  
Stokes Phenomenon ◽  
Homogeneous Solutions ◽  
Inhomogeneous Solution ◽  
Type Construction ◽  
Special Solutions

Accurate approximations to the solutions of a second-order inhomogeneous equation with a small parameter ϵ are derived using exponential asymptotics. The subdominant homogeneous solutions that are switched on by an inhomogeneous solution through a Stokes phenomenon are computed. The computation relies on a resurgence relation, and it provides the ϵ -dependent Stokes multiplier in the form of a power series. The ϵ -dependence of the Stokes multiplier is related to constants of integration that can be chosen arbitrarily in the WKB-type construction of the homogeneous solution. The equation under study governs the evolution of special solutions of the Boussinesq equations for rapidly rotating, strongly stratified fluids. In this context, the switching on of subdominant homogeneous solutions is interpreted as the generation of exponentially small inertia–gravity waves.

scholarly journals Boundary Layer Dynamics in a Simple Model for Convectively Coupled Gravity Waves

2009 ◽  
Vol 66 (9) ◽  
pp. 2780-2795 ◽  
Author(s):  
Michael L. Waite ◽  
Boualem Khouider
Keyword(s):  
Boundary Layer ◽  
Gravity Waves ◽  
Deep Convection ◽  
Potential Temperature ◽  
Phase Speed ◽  
Free Troposphere ◽  
Synoptic Scale ◽  
Basic States ◽  
Boundary Layer Dynamics ◽  
Baroclinic Modes

Abstract A simplified model of intermediate complexity for convectively coupled gravity waves that incorporates the bulk dynamics of the atmospheric boundary layer is developed and analyzed. The model comprises equations for velocity, potential temperature, and moist entropy in the boundary layer as well as equations for the free tropospheric barotropic (vertically uniform) velocity and first two baroclinic modes of vertical structure. It is based on the multicloud model of Khouider and Majda coupled to the bulk boundary layer–shallow cumulus model of Stevens. The original multicloud model has a purely thermodynamic boundary layer and no barotropic velocity mode. Here, boundary layer horizontal velocity divergence is matched with barotropic convergence in the free troposphere and yields environmental downdrafts. Both environmental and convective downdrafts act to transport dry midtropospheric air into the boundary layer. Basic states in radiative–convective equilibrium are found and are shown to be consistent with observations of boundary layer and free troposphere climatology. The linear stability of these basic states, in the case without rotation, is then analyzed for a variety of tropospheric regimes. The inclusion of boundary layer dynamics—specifically, environmental downdrafts and entrainment of free tropospheric air—enhances the instability of both the synoptic-scale moist gravity waves and nonpropagating congestus modes in the multicloud model. The congestus mode has a preferred synoptic-scale wavelength, which is absent when a purely thermodynamic boundary layer is employed. The weak destabilization of a fast mesoscale wave, with a phase speed of 26 m s−1 and coupling to deep convection, is also discussed.

Inertia-gravity waves beyond the inertial latitude. Part 1. Inviscid singular focusing

2010 ◽  
Vol 664 ◽  
pp. 478-509 ◽  
Author(s):  
VICTOR I. SHRIRA ◽  
WILLIAM A. TOWNSEND
Keyword(s):  
Wave Field ◽  
Internal Waves ◽  
Gravity Waves ◽  
Deep Ocean ◽  
Vertical Shear ◽  
Buoyancy Frequency ◽  
Local Minima ◽  
Wave Evolution ◽  
Vertical Scales ◽  
Inertia Gravity Waves

The paper is concerned with analytical study of inertia-gravity waves in rotating density-stratified ideal fluid confined in a spherical shell. It primarily aims at clarifying the possible role of these motions in deep ocean mixing. Recently, it was found that on the ‘non-traditional’ β-plane inertia-gravity internal waves can propagate polewards beyond their inertial latitude, where the wave frequency equals the local Coriolis parameter, by turning into subinertial modes trapped in the narrowing waveguides around the local minima of buoyancy frequency N. The behaviour of characteristics was established: wave horizontal and vertical scales decrease as the wave advances polewards and tend to zero at a latitude corresponding to an attractor of characteristics. However, the basic questions about wave evolution, its quantitative description and the possibility of its reflection from the critical latitude remain open. The present work addresses these issues by studying the linear inviscid evolution of finite bandwidth wavepackets on the ‘non-traditional’ β-plane past the inertial latitude for generic oceanic stratification. Beyond the inertial latitude, the wave field is confined in narrowing waveguides of three distinct generic types around different local minima of the buoyancy frequency. In the oceanic context, the widest is adjacent to the flat bottom, the thinnest is the upper mixed layer, and the middle one is located between the seasonal and main thermocline. We find explicit asymptotic solutions describing the wave field in the WKB approximation. As a byproduct, the conservation of wave action principle is explicitly formulated for all types of internal waves on the ‘non-traditional’ β-plane. The wave velocities and vertical shear tend to infinity and become singular at the attractor latitude or its vicinity for both monochromatic and finite bandwidth packets. We call this phenomenon singular focusing. These WKB solutions are shown to remain valid up to singularity for the bottom and mid-ocean waveguides. The main conclusion is that even in the inviscid setting the wave evolution towards smaller and smaller horizontal and vertical scales is irreversible: there is no reflection. For situations typical of deep ocean, a simultaneous increase in wave amplitude and decrease of vertical scale causes a sharp increase of vertical shear, which may lead to wave breaking and increased mixing.

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