Linear Gravity Waves Research Articles

A comparison of gravity wave energy observed by VHF radar and GPS/MET over central North America

Monthly mean values of the kinetic energy at frequencies associated with gravity waves based on wind observations made with the VHF radar at White Sands Missile Range, New Mexico, are compared with potential energies determined using GPS/MET soundings. The monthly mean curves of Ek and Ep are highly correlated, and both show minimum values in the summer season with summer to winter increases of about a factor of 2. The observed ratio Ek/Ep agrees very closely with the prediction of a linear gravity wave model. These observations support previous comparisons of Ek with Ep made using the middle and upper atmosphere radar in Japan.

Resonance phenomena for water waves in channels of arbitrary cross-section

This article studies the evolutionary problem for linear gravity waves on the surface of water in a uniform, symmetric channel which is excited by an antisymmetric pressure force of frequency ω at the free surface. It is shown that there is a countably infinite set of frequencies {ω0, ω1, …} which give rise to resonance phenomena: the amplitude of the wave motion grows like t1/2 as t→∞ in a sense which is precisely specified. Under pressure forcing at any other frequency the solution obeys the principle of limiting amplitude. These results are obtained by combining methods developed for problems in acoustic waveguides with regularity theory for elliptic boundary-value problems in non-smooth domains. Copyright © 1999 John Wiley & Sons, Ltd.

Resonance phenomena for water waves in channels of arbitrary cross‐section

This article studies the evolutionary problem for linear gravity waves on the surface of water in a uniform, symmetric channel which is excited by an antisymmetric pressure force of frequency ω at the free surface. It is shown that there is a countably infinite set of frequencies {ω0, ω1, …} which give rise to resonance phenomena: the amplitude of the wave motion grows like t1/2 as t→∞ in a sense which is precisely specified. Under pressure forcing at any other frequency the solution obeys the principle of limiting amplitude. These results are obtained by combining methods developed for problems in acoustic waveguides with regularity theory for elliptic boundary-value problems in non-smooth domains. Copyright © 1999 John Wiley & Sons, Ltd.

The influence of photochemistry on gravity waves in the middle atmosphere

This paper focuses on the effect of diabatic processes due to photochemical heating on long-period gravity waves in the stratosphere, mesosphere and lower thermosphere. A linear diabatic gravity wave model is established and compared to a model of pure dynamical adiabatic gravity waves. The results indicate that the photochemistry has a damping effect on gravity waves in most regions of the stratosphere and mesosphere. However, the photochemistry has a destabilizing effect on gravity waves in the mesopause region. The photochemical heating process can induce a comparatively strong enhancement of gravity waves at the mesopause for lower temperatures. In the summer polar mesopause region, this growth rate may be greater by about one order of magnitude than the growth rate of gravity waves at other seasons and locations.

A numerical study of nonlinear propagation of a gravity‐wave packet in compressible atmosphere

By using a two‐dimentional full‐implicit‐continuous‐Eulerian scheme, numerical simulation for nonlinear propagation of Gaussian gravity‐wave packets in a compressible and isothermal atmosphere are carried out. The numerical analyses show that for an initially given upgoing gravity‐wave packet whose disturbance velocity is much less than ambient wind velocity, although there exists nonlinear interaction, during the propagation, the whole wave packet and the wave‐associated energy keep moving upward, while the wave front keeps moving downward. Wave‐associated perturbation velocity increases with the increasing height, and the mean flow shows obvious enhancement when the wave packet passes. After a long time propagation (several periods), wave‐associated perturbation and energy can still concentrate in a limited region that is comparable in size to that given initially. The propagation path of wave energy coincides well with the ray path predicted by the linear gravity wave theory, but the magnitude of wave energy propagation velocity is evidently smaller than the group velocity derived from the linear gravity wave theory. This indicates that once gravity waves are generated, they propagate almost freely along their rays, and the nonlinear effect will only lower the propagation velocity of the wave‐associated energy. While gravity‐wave packets propagate in a nonisothermal atmosphere, the nonlinear propagation paths of wave energy depart clearly from the ray paths derived from the linear gravity wave theory under the WKB approximation, which indicates that the linear gravity‐wave theory under the WKB approximation can not predict the nonlinear propagation of gravity‐wave packet in a nonisothermal atmosphere.

Mesoscale Temperature Fluctuations Induced by a Spectrum of Gravity Waves: A Comparison of Parameterizations and Their Impact on Stratospheric Microphysics

Abstract Power spectral densities (PSDs) of mesoscale fluctuations of temperature and rate of change of temperature (heating–cooling rate) due to a spectrum of stratospheric gravity waves are derived using canonical spectral forms based on observations and linear gravity wave theory. The parameterization developed here assumes a continuous distribution of horizontal wave phase speeds, as opposed to a previous spectral parameterization in which all waves were assigned stationary ground-based phase speeds. Significantly different heating–cooling rate PSDs result in each case. The differences are largest at small horizontal scales, where the continuous phase-speed parameterization yields heating–cooling rate PSDs that are several orders of magnitude smaller than in the stationary phase-speed parameterization. A simple Monte Carlo method is used to synthesize randomly phased temperature perturbation time series within tagged air parcels using either spectral parameterization. These time series are incorporate...

Linear Stratospheric Gravity Waves above Convective Thermal Forcing

Abstract The spectra of linear gravity waves generated by a time-varying tropospheric thermal forcing representing organized convection are compared to the spectra of stratospheric gravity waves generated by organized convection in a fully nonlinear two-dimensional squall line simulation. The resemblance between the spectra in the two simulations suggests that stratospheric gravity waves above convection can be understood primarily in terms of the linear response to a time- and space-dependent thermal forcing. In particular, the linear response to thermal forcing accounts for the correlation between the dominant vertical wavelength of the stratospheric waves and the depth of the tropospheric convection as well as the the fact that the dominant frequency of the stratospheric waves is the same as the frequency of oscillation of the main convective updraft.

Observed linear and nonlinear K layer response

Potassium densities and temperatures were measured on board the German Research Vessel ‘Polarstern’ with the containerized IAP (Institute of Atmospheric Physics) K‐lidar. Considerable wave activity was observed in the K density profiles as well as the temperature profiles, in particular during the night of May 23/24, 1996 over the South Atlantic ocean. A strong monochromatic wave was observed between 85 and 100 km. The calculated relative K density variations are in phase with the temperature variations at the bottom of the K layer and 180° out of phase at the top of the K layer as was to be expected from linear gravity wave theory. The height of change (92–93 km) between positive and negative correlation of temperatures and K densities is clearly dominated by the nonlinear layer response, which could be observed largely because of a strong monochromatic gravity wave with vertival wavelength of 30 km and a horizontal period close to 5 hours, quite similar to the predicted nonlinear layer response from theory. These data show that (1) chemical effects are negligible in the response of the K layer to the passage of an atmospheric gravity wave, and (2) that near the peak of the K layer non‐linear effects dominate the response.

Estimating shear velocities in the oceanic crust from compliance measurements by two‐dimensional finite difference modeling

The deep seafloor deforms under the pressure loading of linear ocean surface gravity (water) waves at low frequencies (0.003 to 0.04 Hz). The ratio of seafloor displacement to pressure loading as a function of frequency, known as the seafloor compliance function, depends on the shear velocity structure of the oceanic crust and upper mantle. Compliance measurements are used to estimate oceanic crustal structure, particularly within low shear velocity regions such as sediments, fractured rock, and partial melt. Compliance calculated from laterally homogeneous (one‐dimensional, 1‐D) crustal models shows that a buried low‐velocity zone (LVZ) causes a peak in the compliance function at wavelengths 4 to 6 times longer than the LVZ depth, and that the compliance amplitude depends primarily on the LVZ shear velocity. A new numerical code allows forward modeling of compliance for two‐dimensional oceanic crustal models. The new code demonstrates that the peak in the compliance function directly over a finite width LVZ reaches a maximum value at higher frequency, and is of smaller amplitude, than predicted from 1‐D modeling. The compliance maximum persists outside of the region underlain by the LVZ but diminishes in amplitude and shifts to lower frequencies with increasing distance from the LVZ. The numerical models indicate that significant peaks in the compliance function indicate crustal LVZs, but that multiple compliance measurements are necessary to independently constrain the depth, location, and shear velocity of these features.

Reflection of gravity waves by a steep beach with a porous bed

Solutions are determined for normally incident non-breaking linear gravity waves in a perfect fluid over a porous plane bed of arbitrary slope α both with and without bed friction. For simplicity, computations are restricted to α=π/2m, m∈N. Modifications to wave height transformations due to percolation and to friction are determined for a variety of slopes and coefficients. The effect on the reflection coefficient Rf is also studied and excellent qualitative agreement is found with recent work on damping and reflection by permeable structures. In particular, for a choice of parameters, the Rf response is determined in closed form.

An analysis of the structure and forcing of the equatorial semiannual oscillation in zonal wind

The low‐frequency variability in the equatorial middle atmosphere is dominated by the semiannual oscillation (SAO) and quasi‐biennial (QBO) oscillation. Although the general characteristics and forcing mechanisms of these oscillations are thought to be known, these oscillations are still not well reproduced in modeling experiments of the middle atmosphere. The relatively long period (1992–1996) of Upper Atmosphere Research Satellite (UARS) measurements and U.K. Meteorological Office (UKMO) assimilated model data have been used to examine the zonal wind SAO in the upper stratosphere. The SAO in zonal wind is found to be latitudinally asymmetric about the equator with strongest amplitudes in the southern hemisphere subtropics. The latitudinal asymmetry is diagnosed by using the TEM momentum equation and UKMO data. The asymmetry is mostly due to the different phasing of the peak advective forcing by the mean meridional circulation in each hemisphere. The contribution of gravity waves to the zonal wind SAO are also explored by using the forcing from a linear gravity wave model. Gravity waves are found to be the primary source of westerly momentum in each hemisphere during fall and contribute to the latitudinal asymmetry of the SAO by having more easterly forcing in the southern hemisphere during summer compared to in the northern hemisphere during summer.

On the dissipation of internal solitons in coastal seas

Previous descriptions of the turbulent decay of an internal soliton have been based on an adiabatic assumption in which the soliton parameters continuously change in time in response to the dissipation. A complete leading order description of a decaying internal soliton including both horizontal and vertical dissipation is given. It is shown that the adiabatic ansatz fails to fully describe a decaying internal soliton under the influence of weak dissipation. Physically, the breakdown is the consequence of the fact that an adiabatically decaying internal soliton is unable to simultaneously satisfy averaged energy and mass balance relations. An extended region with vertical momentum flux develops behind the decaying soliton in order to compensate for the additional mass lost in the adiabatically decaying internal soliton. The leading order evolution of this extended region is described as is its connection to the stream function field associated with the internal soliton. The magnitude of the vertical flux associated with this region scales linearly with the dissipation parameters. The transition of this extended region back to zero is accomplished through the formation of a spatially decaying packet of linear internal gravity waves.

Vertical wavenumber spectra of wind and temperature from high‐resolution balloon soundings over Illinois

We spectrally analyzed high‐resolution balloon measurements of vertical profiles of temperature and horizontal wind in the troposphere and lower stratosphere over very flat terrain in Illinois. This paper is principally concerned with spectra in the power law range at vertical wavenumbers m ≥10−3 cycle/m. The logarithmic spectral slopes and amplitudes are found to have only insignificant dependencies on meteorological parameters, including time of day, season, wind speed and direction, vertical shear, etc., except that between the troposphere and stratosphere the spectral amplitude scales as (N2)q with q ∼0.3, where N is the buoyancy frequency. The mean slopes are ≈−3 in the stratosphere and ≈ −2.6 in the troposphere. On the average the individual spectra with larger amplitudes have less negative spectral slopes. The wide variation of spectral slopes (σ≈0.5) and amplitudes and the weak dependence on N2 are quite inconsistent with the predictions of theories of saturated spectra. Further, the wind spectra in the troposphere and stratosphere are correlated, which suggests some unsaturated propagation between the regions. The ratio of kinetic to potential energy spectra is constant versus m, consistent with the linear gravity wave polarization relations. The magnitude of the model ratio can be brought into agreement with the observed ratio by assuming a model intrinsic frequency spectrum varying as ω−p with p ∼5/3 to 2 plus an enhancement of energy near the inertial frequency.

Groundwater waves in aquifers of intermediate depths

In order to model recent observations of groundwater dynamics in beaches, a system of equations is derived for the propagation of periodic watertable waves in uncofined aquifers of intermediate depths, i.e. for finite values of the dimensionless aquifer depth nwd K which is assumed small under the Dupuit-Forchheimer approach that leads to the Boussinesq equation. Detailed consideration is given to equations of second- and infinite-order in this parameter. In each case, small amplitude ( η d ⪡ 1 ) as well as finite amplitude versions are discussed. The small amplitude equations have solutions of the form η( x, t) = η 0 e − kx e iwt in analogy with the linearized Boussinesq equation but the complex wave numbers k are different. These new wave numbers compare well with observations from a Hele-Shaw cell which were previously unexplained. The “exact” velocity potential for small amplitude watertable waves, the equivalent of Airy waves, is presented. These waves show a number of remarkable features. They become non-dispersive in the short-wave limit with a finite and quite slow decay rate affording an explanation for observed behaviour of wave-induced porewater pressure fluctuations in beaches. They also show an increasing amplitude of pressure fluctuations towards the base, in analogy with the evanescent modes of linear surface gravity waves.

Observed wavenumber‐frequency properties of microwave backscatter from the ocean surface at near‐grazing angles

This paper presents microwave observations of the ocean surface obtained with a high‐resolution X band imaging Doppler radar during March 1994. Besides providing time‐series imagery of the ocean surface, the radar has sufficient spatial resolution to produce wave vector‐frequency (three‐dimensional) spectra of observed power and Doppler frequency modulations due to the wind waves. The wave vector‐frequency spectra provide a complete representation of the mean spatio‐temporal properties of vertically polarized X band ocean backscatter at near‐grazing incidence. Evident in the spectra are the expected contributions from linear gravity waves exhibiting high coherences between power and Doppler (orbital) velocity and consistent with the composite‐surface‐model view of sea surface microwave backscatter. Occasionally, gravity wave harmonics possibly indicative of wave nonlinearities are observed. Also readily identified in the spectra are spatially broadband modulations due to localized scattering sources which are influenced by young gravity waves and the wind. Contributions from these apparently “non‐Bragg” scattering sources account for a significant part of the observed coherent modulations. The propagation velocity of this modulation is consistent with scattering from short gravity waves well beyond the spectral peak.

The Diffraction of Kelvin Waves and Bores at Coastal Bends

Abstract Bends in coastal mountain ranges may diffract propagating atmospheric Kelvin waves and trapped coastal currents. Analytic solutions exist for the diffraction of both linear Kelvin waves and linear nonrotating gravity waves. Within the context of the single-layer shallow-water equations, we examine the diffraction of nonlinear gravity waves and bores in a nonrotating reference frame and nonlinear Kelvin waves and coastally trapped bores in a rotating reference frame. The diffraction process can significantly decrease the amplitude of linear and nonlinear waves and bores in the nonrotating reference frame. Unlike for their linear counterpart, however, the diffraction-related amplitude decay for the nonrotating nonlinear waves takes place entirely within the region of the bend and does not produce a continuous decay after the bend. Moreover, theory predicts a critical bend angle at which bore amplitudes will be zero at the wall after propagation around the bend, but shallow-water model simulations d...

Gravity‐wave drag on two mountains

AbstractThe generation of linear internal gravity waves by the flow over two idealized mountains, whose summits are aligned in the same direction as the mean wind, is studied. In particular, the sensitivity of the wave drag to the downwind distance between the mountains is examined by using a three‐dimensional numerical model based on the linearized steady‐state Boussinesq equations of motion. Both two‐ and three‐dimensional orography is considered. When the upstream conditions are such that the wind and buoyancy‐frequency do not vary with height, and the mountains are wide enough for non‐hydrostatic effects to be unimportant, the model predicts the same drag dependence on two‐dimensional ridge spacing as was derived analytically by Grisogono et al. The corresponding three‐dimensional numerical result predicted by the model is qualitatively very similar to the two‐dimensional result. When the mountains are narrow and the waves are non‐hydrostatic the drag dependence on mountain spacing is more complicated; maxima and minima in the drag can occur when the distance between the two mountains is such that waves generated by the mountain furthest upstream are in phase (maxima in the drag), or out of phase (minima in the drag), with the waves generated by the downstream mountain. It is shown that this is an important effect when the waves are trapped by upstream profiles whose wind speed varies with height.

Simultaneous observations of mesospheric gravity waves with the MU radar and a sodium lidar

Simultaneous observations of mesospheric gravity waves have been carried out using meteor wind measurements with the middle and upper atmosphere (MU) radar at Shigaraki, Japan (34.9°N, 136.1°E), and density perturbations of the sodium lidar at Hachioji, Tokyo, Japan (35.6°N, 139.4°E). The study utilizes 7 hours of data collected on the night of December 15–16, 1993, during a time period when a fairly monochromatic gravity wave was dominant. Using hodograph analysis, the dominant gravity wave was found to exhibit a vertical wavelength of 16 km, an intrinsic period of 9.1 hours, and a horizontal wavelength of about 1900 km. The horizontal propagation direction of the gravity wave was determined from the phase relations between the horizontal wind components and the temperature perturbations at Shigaraki. The wave propagated southward, being almost orthogonal to the baseline between Shigaraki and Hachioji. Employing the dispersion and polarization relations for linear gravity waves, the wave‐induced neutral density perturbations from the MU radar observations were estimated. A comparison with the corresponding density perturbations derived from the sodium density measurements showed good agreement. The amplitudes of the neutral density perturbations observed at both locations, which are separated by 310 km, were similar, with a maximum perturbation of ∼7% and a good correlation of phase. Time variations of the hourly variance of the density perturbations also agreed quite well between the two independent determinations, which again supports the view that the radar and the lidar detected the same gravity wave.

The dispersion relation at large frequencies for linear surface gravity waves on a Maxwell fluid exhibiting a spatial decay

The phase velocity of linear surface gravity waves on Maxwell fluids exhibiting a temporal decay and waves exhibiting a spatial decay have similar phase velocity in the limit of long waves where both the dimensionless wave number and frequency is small. For dimensionless wave numbers and frequencies of the order of unity in magnitude the phase velocity curves of these two types of waves is completely different. This is the case even for waves on Newtonian fluids. It is shown that both of these two phase velocity curves for any Maxwell fluids with non-zero relaxation time resume the value of the phase velocity of a Rayleigh wave on an incompressible Hookean solid in the limit of infinite wave number and frequency.

On Hamiltonian Balanced Dynamics and the Slowest Invariant Manifold

The concept of a slowest invariant manifold is investigated for the five-component model of Lorenz under conservative dynamics. It is shown that Lorenz's model is a two-degree-of-freedom canonical Hamiltonian system, consisting of a nonlinear vorticity-triad oscillator coupled to a linear gravity wave oscillator, whose solutions consist of regular and chaotic orbits. When either the Rossby number or the rotational Froude number is small, there is a formal separation of timescales, and one can speak of fast and slow motion. In the same regime, the coupling is weak, and the Kolmogorov–Arnold-Moser theorem is shown to apply. The chaotic orbits are inherently unbalanced and are confined to regions sandwiched between invariant tori consisting of quasi-periodic regular orbits. The regular orbits generally contain free fast motion, but a slowest invariant manifold may be geometrically defined as the set of all slow cores of invariant tori (defined by zero fast action) that are smoothly related to such cores in the uncoupled system. This slowest invariant manifold is not global; in fact, its structure is fractal; but it is of nearly full measure in the limit of weak coupling. It is also nonlinearly stable. As the coupling increases, the slowest invariant manifold shrinks until it disappears altogether. The results clarify previous definitions of a slowest invariant manifold and highlight the ambiguity in the definition of “slowness.” An asymptotic procedure, analogous to standard initialization techniques, is found to yield nonzero free fast motion even when the core solutions contain none. A hierarchy of Hamiltonian balanced models preserving the symmetries in the original low-order model is formulated; these models are compared with classic balanced models, asymptotically initialized solutions of the full system and the slowest invariant manifold defined by the core solutions. The analysis suggests that for sufficiently small Rossby or rotational Froude numbers, a stable slowest invariant manifold can be defined for this system, which has zero free gravity wave activity, but it cannot be defined everywhere. The implications of the results for more complex systems are discussed.