Low-frequency Internal Waves Research Articles

Low-frequency Internal Gravity Waves Are Pseudo-incompressible

Starting from the fully compressible fluid equations in a plane-parallel atmosphere, we demonstrate that linear internal gravity waves are naturally pseudo-incompressible in the limit that the wave frequency ω is much less than that of surface gravity waves, i.e., ω≪gkh , where g is the gravitational acceleration and k h is the horizontal wavenumber. We accomplish this by performing a formal expansion of the wave functions and the local dispersion relation in terms of a dimensionless frequency ε=ω/gkh . Further, we show that, in this same low-frequency limit, several forms of the anelastic approximation, including the Lantz–Braginsky–Roberts formulation, poorly reproduce the correct behavior of internal gravity waves. The pseudo-incompressible approximation is achieved by assuming that Eulerian fluctuations of the pressure are small in the continuity equation—whereas, in the anelastic approximation, Eulerian density fluctuations are ignored. In an adiabatic stratification, such as occurs in a convection zone, the two approximations become identical. However, in a stable stratification, the differences between the two approximations are stark and only the pseudo-incompressible approximation remains valid.

Internal Gravity Waves in the Earth’s Ionosphere

The theory of low-frequency internal gravity waves (IGWs) is readdressed in the stable stratified weakly ionized Earth’s ionosphere. The formation of dipolar vortex structures and their dynamical evolution as well as the emergence of chaos in the wave–wave interactions are studied both in the presence and absence of the Pedersen conductivity. The latter is shown to inhibit the formation of solitary vortices and the onset of chaos.

Variability of Internal Wave Strain Over the Western Boundary Region in the North Atlantic

Based on the CTD profiles and mooring measurements of the Line W program in the North Atlantic about 35°N/70°W, we investigated the spatial variability of internal wave-induced strain variance along the Line W section and the associated reasons. The results showed that the internal wave strain variance over the slope could be up to five times larger than that in the interior. We have validated that internal tides and near-inertial waves were not the crucial processes for causing this variability. Meanwhile, we found that the high-frequency internal waves generated by the interaction of anticyclonic eddy and low-frequency internal waves may be a potential cause of strain enhancement. This study implies a potential route of mesoscale eddies to enhance the strain variance induced by internal waves.

Effects of Coriolis force on the nonlinear interactions of acoustic-gravity waves in the atmosphere

The nonlinear theory of acoustic-gravity waves (AGWs) in the atmosphere is revisited with the effects of the Coriolis force. Previous theory in the literature (Mendonça and Stenflo, 2015) is advanced. Starting from a set of fluid equations modified by the Coriolis force, a general linear dispersion relation is derived which manifests the coupling of the high-frequency acoustic-gravity waves (AGWs) and the low-frequency internal gravity waves (IGWs). The frequency of IGWs is enhanced by the Earth’s angular velocity. The latter also significantly modifies the nonlinear coupling of AGWs and IGWs whose evolutions are described by the Zakharov approach as well as the Wigner–Moyal formalism. The consequences of AGWs and the two equivalent evolution equations modified by the Coriolis force are briefly discussed.

Observations of enhanced internal waves in an area of strong mesoscale variability in the southwestern East Sea (Japan Sea)

Oceanic internal waves near the local inertial frequency or near-inertial internal waves and internal waves of tidal origin or internal tides are two types of low-frequency internal waves. Their interactions, interaction with background field, and resulting internal waves at higher frequencies beyond the near-inertial and tidal frequencies have rarely been reported despite its importance on ocean mixing and circulation of energy and materials. Here, we present five episodic enhancements of the high-frequency or continuum frequency waves (CFWs) observed in the southwestern East Sea (Japan Sea) and discuss causes for the enhanced CFWs in relation to near-inertial waves (NIWs), semidiurnal internal tides (SDITs), mesoscale flow fields, and their interactions. The NIWs were amplified due to local surface wind forcing, significantly interacting with mesoscale strain via wave capture. The SDITs were generated in a remote place and propagated into the observational site, largely depending on the mesoscale fields. The observational results suggest that the five episodes of CFWs are results of enhanced NIWs or SDITs, or their wave-wave interaction, rather than locally generated lee-waves. Our study suggests the significant impact of mesoscale circulation on the variability of internal waves from near-inertial to buoyancy frequencies through multiple pathways.

Instabilities of a Time-Dependent Shear Flow

AbstractThis study offers a systematic stability analysis of unsteady shear flows representing large-scale, low-frequency internal waves in the ocean. The analysis is based on the unbounded time-dependent Couette model. This setup makes it possible to isolate the instabilities caused by uniform shear from those that can be attributed to resonant triad interactions or to the presence of inflection points in vertical velocity profiles. Linear analysis suggests that time-dependent spatially uniform shears are unstable regardless of the Richardson number (Ri). However, the growth rate of instability monotonically decreases with increasing Ri and increases with increasing frequency of oscillations. Therefore, models assuming a steady basic state—which are commonly used to conceptualize shear-induced instability and mixing—can be viewed as singular limits of the corresponding time-dependent systems. The present investigation is focused on the supercritical range of Richardson numbers (Ri > 1/4) where steady parallel flows are stable. An explicit relation is proposed for the growth rate of shear instability as a function of background parameters. For moderately supercritical Richardson numbers (Ri ~ 1), we find that the growth rates obtained are less than, but comparable to, those expected for Kelvin–Helmholtz instabilities of steady shears at Ri < 1/4. Hence, we conclude that the instability of time-dependent flows could represent a viable mixing mechanism in the ocean, particular in regions characterized by relatively weak wave activity and predominantly supercritical large-scale shears.

Microstructure measurements and finescale parameterization assessment of turbulent mixing in the northern South China Sea

Both microscale and finescale measurements were conducted along 20°N and 21°N in the northern South China Sea (SCS) during July 2007. Spatial variability of turbulent kinetic energy (TKE) dissipation rate was examined, and two finescale parameterizations were assessed and compared. TKE dissipation rates along the 21°N section were found to be much higher than those along 20°N; in particular, remarkably high TKE dissipation rates existed near the Luzon Strait and around the Dongsha Plateau, which were likely caused by internal tides and internal solitary waves, respectively. The Gregg–Henyey scaling does not work well in the northern SCS, while the MacKinnon–Gregg scaling with a modified parameter matches the observations in both magnitude and variability. One explanation is that the large-scale/low-mode shear mainly comes from low-frequency internal waves such as internal tides, which are not described well by the Garrett–Munk spectrum.

BUOYANCY-DRIVEN MAGNETOHYDRODYNAMIC WAVES

ABSTRACT Turbulent motions close to the visible solar surface may generate low-frequency internal gravity waves (IGWs) that propagate through the lower solar atmosphere. Magnetic activity is ubiquitous throughout the solar atmosphere, so it is expected that the behavior of IGWs is to be affected. In this article we investigate the role of an equilibrium magnetic field on propagating and standing buoyancy oscillations in a gravitationally stratified medium. We assume that this background magnetic field is parallel to the direction of gravitational stratification. It is known that when the equilibrium magnetic field is weak and the background is isothermal, the frequencies of standing IGWs are sensitive to the presence of magnetism. Here, we generalize this result to the case of a slowly varying temperature. To do this, we make use of the Boussinesq approximation. A comparison between the hydrodynamic and magnetohydrodynamic cases allows us to deduce the effects due to a magnetic field. It is shown that the frequency of IGWs may depart significantly from the Brunt–Väisälä frequency, even for a weak magnetic field. The mathematical techniques applied here give a clearer picture of the wave mode identification, which has previously been misinterpreted. An observational test is urged to validate the theoretical findings.

Orbital stability of periodic waves in the class of reduced Ostrovsky equations

Periodic travelling waves are considered in the class of reduced Ostrovsky equations that describe low-frequency internal waves in the presence of rotation. The reduced Ostrovsky equations with either quadratic or cubic nonlinearities can be transformed to integrable equations of the Klein–Gordon type by means of a change of coordinates. By using the conserved momentum and energy as well as an additional conserved quantity due to integrability, we prove that small-amplitude periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. The proof is based on construction of a Lyapunov functional, which is convex at the periodic wave and is conserved in the time evolution. We also show numerically that convexity of the Lyapunov functional holds for periodic waves of arbitrary amplitudes.

Solitons in a forced nonlinear Schrödinger equation with the pseudo-Raman effect.

The dynamics of solitons is considered in the framework of an extended nonlinear Schrödinger equation (NLSE), which is derived from a Zakharov-type model for wind-driven high-frequency surface waves in the ocean, coupled to damped low-frequency internal waves. The drive gives rise to a convective (but not absolute) instability in the system. The resulting NLSE includes a pseudo-stimulated-Raman-scattering (pseudo-SRS) term, which is a spatial-domain counterpart of the SRS term, a well-known ingredient of the temporal-domain NLSE in optics. Analysis of the field-momentum balance and direct simulations demonstrates that wave-number downshift by the pseudo-SRS may be compensated by the upshift induced by the wind traction, thus maintaining robust bright solitons in both stationary and oscillatory forms; in particular, they are not destroyed by the underlying convective instability. Analytical soliton solutions are found in an approximate form and are verified by numerical simulations. Solutions for soliton pairs are obtained in the numerical form.

Fine-scale structure and mixing across the front between the Tsugaru Warm and Oyashio Currents in summer along the Sanriku Coast, east of Japan

High-resolution shipboard observations were made across the front between the Tsugaru Warm Current (TWC) and the Oyashio Current in July 2013. Fine structure in the frontal zones was successfully captured with an underway conductivity–temperature–depth profiler deployed with a typical horizontal interval of 2–3 nautical miles. The front characterized by marked horizontal gradients in temperature and salinity extended from the subsurface onto the shelf. Along this frontal layer, the minimum frequency for internal waves became substantially lower than the local inertial frequency, mainly due to the strong vertical shear of the geostrophic velocity. Turbulent energy dissipation rates e (vertical diffusivity K ρ ) were frequently elevated along the front and its offshore side up to 3 × 10−8 W kg−1 (10−4 m2 s−1), which may have been caused by an “internal tide chimney”, trapping low-frequency internal waves within the band of strong shear. At the onshore side of the TWC on the shelf, strong mixing with e (K ρ ) exceeding 10−6 W kg−1 (10−3 m2 s−1) was also observed. A large portion of the water columns in the frontal area provided suitable conditions for double diffusion; in some layers with moderate turbulence, temperature microstructures indicative of double diffusion were observed. The vigorous mixing processes around the front are likely to modify the properties of the TWC downstream, which could then produce a latitudinal gradient in environments along the coast.

Hypolimnetic turbulence generation associated with superposition of large-scale internal waves in a strongly stratified lake: Lake Biwa, Japan

A 48-h microstructure experiment captured the variation of turbulence in Lake Biwa, Japan, during a strong stratification period, 1 week after a typhoon event. A free-fall microstructure profiler (TurboMAP) and a fine-scale profiler (F-probe) were deployed. An array of five ADCPs positioned close to the experiment site provided current measurements associated with the internal-wave field for a period of one week. Strong winds related to the passage of a typhoon close to the lake generated two low-frequency internal waves: a Kelvin wave and an inertial wave. Both waves were in the first baroclinic mode. This study pictures the superposition of two large-scale internal waves and a stronger current resulting from the two waves being in phase. The synchronization of the waves provoked enhanced shear in the hypolimnion, more than 10 m above the bottom, as well as high dissipation (10−7 W kg−1) and vertical diffusivity reaching 10−4 m2 s−1. The enhanced shear event seems to be related to the current amplitude since it occurred when the current amplitude was increased by the diel wind. Assuming the same turbulence intensity for the enhanced hypolimnetic shear events, this superposition may affect nutrient transfer in the hypolimnion. Also, we witnessed sediment resuspension, consisting of high turbidity and high fluorescence intensity, likely due to a combination of bottom stress and enhanced turbulence.

Modeling vertical exchange of heat, salt, and other dissolved substances in the Cariaco Basin

A simple 1.5-dimensional model of vertical exchange for heat, salt, and other dissolved substances has been developed for the Cariaco Basin. The model parameters are derived based on the temperature and salinity data collected monthly at a deepwater station in the eastern part of the basin from 1995 through 2007 during the CARIACO time series program. The model describes the processes of turbulent (eddy) diffusion, which includes the integrated effect of diffusive exchange mechanisms acting in the basin, and of vertical advection, which arises following injection of dense water into deep layers following an inflow from the Caribbean Sea. The model takes into account the changes in the horizontal cross-section area of the basin with respect to depth.Temporal variability is an important feature of the hydrography of the Cariaco Basin. To assess profiles of the vertical eddy diffusion coefficient and vertical advection velocity, we examined a time series of CTD profiles (potential temperature and salinity). Two distinct time intervals were identified as the result of this examination. During the first period, the thermohaline structure of the basin was apparently influenced by one or more inflows. The second period, in contrast, showed little or no influence of an inflow. The data from the second period, where no inflows were observed, were incorporated into corresponding transfer equations to assess the profile of the vertical eddy diffusion coefficient, k(z). Then, the result of this assessment was used with the data from the first period to estimate the profile of the vertical advection velocity, W(z), for a time when the effects of an inflow were evident. For that case, the transfer equations include the terms describing the effect of the inflow.Analysis of the vertical profile of the turbulent diffusion coefficient suggests that, in the upper stratified part of the water column, the diffusion mechanism is mostly associated with transient mixing events, which occur due to shear instability in the field of low frequency internal waves. We speculate that below 400m bottom friction over the sloping bottom and geothermal heat flux play the decisive role in the vertical exchange. Analysis of the W(z) profiles reveals two layers dominated by the entrainment of the ambient fluid into the down flow of dense water from the Caribbean Sea, and two layers where this down flow breaks down through the formation of isopycnal intrusions.

Energy Transfer from High-Shear, Low-Frequency Internal Waves to High-Frequency Waves near Kaena Ridge, Hawaii

Abstract Evidence is presented for the transfer of energy from low-frequency inertial–diurnal internal waves to high-frequency waves in the band between 6 cpd and the buoyancy frequency. This transfer links the most energetic waves in the spectrum, those receiving energy directly from the winds, barotropic tides, and parametric subharmonic instability, with those most directly involved in the breaking process. Transfer estimates are based on month-long records of ocean velocity and temperature obtained continuously over 80–800 m from the research platform (R/P) Floating Instrument Platform (FLIP) in the Hawaii Ocean Mixing Experiment (HOME) Nearfield (2002) and Farfield (2001) experiments, in Hawaiian waters. Triple correlations between low-frequency vertical shears and high-frequency Reynolds stresses, 〈uiw∂Ui/∂z〉, are used to estimate energy transfers. These are supported by bispectral analysis, which show significant energy transfers to pairs of waves with nearly identical frequency. Wavenumber bispectra indicate that the vertical scales of the high-frequency waves are unequal, with one wave of comparable scale to that of the low-frequency parent and the other of much longer scale. The scales of the high-frequency waves contrast with the classical pictures of induced diffusion and elastic scattering interactions and violates the scale-separation assumption of eikonal models of interaction. The possibility that the observed waves are Doppler shifted from intrinsic frequencies near f or N is explored. Peak transfer rates in the Nearfield, an energetic tidal conversion site, are on the order of 2 × 10−7 W kg−1 and are of similar magnitude to estimates of turbulent dissipation that were made near the ridge during HOME. Transfer rates in the Farfield are found to be about half the Nearfield values.

Low-frequency internal waves in magnetized rotating stellar radiation zones

Context. With the progress of observational constraints on dynamical processes in stars, it becomes necessary to understand the angular momentum and the rotation profile history. In this context, internal waves constitute an efficient transport mechanism over long distances in stellar radiation zones. Indeed, they could be one of the mechanisms responsible for the quasi-flat rotation profile of the solar radiative region up to 0.2 R ⊙ .Aims. Angular momentum transport induced by internal waves depends on the properties of their excitation regions and of their dissipation during propagation. Then, the bottom of convective envelopes (the top of convective cores, respectively) are differentially rotating magnetic layers while radiation zones may host fossil magnetic fields. It is therefore necessary to understand the modification of internal wave mechanisms by both rotation and magnetic fields.Methods. We continue our previous work by proceeding step by step. We analytically built a complete formalism that treats the angular momentum transport by internal waves while taking into account both the Coriolis acceleration and the Lorentz force in a non-perturbative way for an axisymmetric toroidal field. We assumed a uniform Alfven frequency and a weak differential rotation to isolate the transport properties as a function of the Rossby and Elsasser numbers.Results. We examined the different possible approximations to describe low-frequency internal waves modified by the Coriolis acceleration and the Lorentz force in a deep spherical shell. The complete structure of these waves, which become magneto-gravito-inertial waves, is given assuming the quasi-linear approximation first in the adiabatic case and then in the dissipative one. Vertical and equatorial trapping phenomena appear that favor retrograde waves. The efficiency of the induced transport as a function of the Rossby and Elsasser numbers is then obtained. Conclusions. A complete study of the transport of angular momentum induced by magneto-gravito-inertial waves in stellar radiative regions is achieved for an axisymmetric toroidal magnetic field for a uniform Alfven frequency and a weak differential rotation. General differential rotation, complex azimuthal magnetic fields, and poloidal and mixed fields will be examined in follow-up studies.

A possible generation mechanism of the strong current over the northwestern shelf of the South China Sea

Continuous observation in late April 2005 on the northwestern shelf of the South China Sea reveals vigorous strong currents, the maximum velocity of which exceeds 3.8 m/s. The strong currents occurred around spring tide period, when the internal tide waves were also expected to be vigorous. Analysis shows that the major peaks of the current power spectrum are in low frequency band. Using a numerical method applied to the actual ocean stratification, we find that the amplitude profiles of the strong current are similar to that of the currents induced by some low-mode internal waves (at diurnal or semi-diurnal frequency). It indicates that the temporal and spatial features of strong currents were possibly induced by low frequency internal waves.

Determination of parameters of low-frequency internal waves in the coastal zone of a fringing sea using field measurements and basing on the nonlinear theory

A method is presented to determine the quadratic nonlinearity parameter and amplitude of low-frequency internal gravity waves in the coastal zone of a fringing sea, based upon their propagation rate dependence on local value of pycnocline vertical displacement produced by the waves. To test the method, the internal wave field observations in the coastal zone of the Sea of Japan are used. The testing results show that the internal wave parameters calculated using the proposed method and the experimental data are in a good agreement with those calculated from theoretical formulas.

Guest editorial to the special issue internal waves in the oceans and estuaries: modeling and observations

Internal waves that owe their existence to density/gradients setup by temperature and/or salinity fields play an important role in ocean dynamics. Whereas the application of internal waves in Naval Warfare is well known, what is not well documented is their role in Natural Hazards. Internal waves can augment surface waves through coupling, and when such a coupling occurs to storm surges and tides, there could be an increasing risk of intensified coastal inundation due to tropical and extra-tropical cyclones and tsunamis. Internal waves, under the right conditions, lead to the so-called Dead Water phenomena and can seriously hamper navigation. Such incidents in the world’s shipping lanes are well documented. We thank Springer Publishers and the editorial staff of Natural Hazards for inviting us to guest edit this special issue, which contains a total of eight papers. The first paper by S. V. Babu and A. D. Rao deals with the dynamics of mixing due to internal waves in the north western part of the Bay of Bengal during winter period. The second paper by Sridevi et al. is about the impact of internal waves on the acoustic field. Paper three is authored by Rao et al. and explains how stratification affects thermal inversions. A more theoretical approach was taken in the fourth paper by Hamdi et al. and examines analytical solutions and conservation laws for long nonlinear internal waves. The theoretical approach is continued in paper five by the same authors, where they employed method of lines simulations of the extended Korteweg–de Vries equation. Craig et al. deal with the coupling of internal waves and surface waves in paper six. Hareesh Kumar et al. describe in paper seven, how low-frequency internal waves influence the transmission loss variability. The final paper, written by Prasad and Rajasekhar,

Low-frequency internal waves in magnetized rotating stellar radiation zones

Context. The study of helioseismology, asteroseismology, and powerful ground-based instrumentation dedicated to stellar physics is developing strongly (cf. CoRoT, KEPLER, and ESPaDOnS). This generates tight constraints on the stellar internal structure and dynamical processes. In this context, it is thus necessary to go beyond the non-rotating and the non-magnetic picture of stellar interiors, particularly for large-scale transport mechanisms and waves. Aims. We focus on low-frequency internal waves in magnetic, rotating, stably stratified stellar radiation zones. For frequencies, which can be close to the Alfven and the inertial frequencies, we go beyond the non-magnetic and non-rotating description of wave dynamics with taking the Coriolis acceleration and the Lorentz force into account. Then, we have to couple wave dynamics with fossil magnetic fields, which must have mixed configurations (both poloidal and toroidal) to survive in stellar radiation zones. Methods. We chose to study such coupling step by step, first with purely toroidal fields and then with purely poloidal fields, to unravel their modification by each corresponding component of a realistic mixed-field. Thus, we analytically built a complete formalism, which describes both effects of the Coriolis acceleration and of the Lorentz force in a non-perturbative way in the case of an axisymmetric toroidal field. We consider here the case where both Alfven frequency and angular velocity are chosen to be uniform, to isolate wave properties. Results. The different approximations possible for low-frequency internal waves in this model are examined and discussed. In this way, the traditional approximation used to describe the dynamics of low-frequency regular elliptic gravito-inertial waves in the purely hydrodynamical case is generalized to the magnetic one. The complete structure of internal waves, which become magneto-gravitoinertial waves, is then derived and compared to the non-magnetic case. The asymptotic behaviour of such waves is obtained. Conclusions. A global study of magneto-gravito-inertial waves in stellar radiation zones is achieved in the case of an axisymmetric toroidal magnetic field. In the near future, consequences for angular momentum transport and the case of general differential rotation and azimuthal magnetic field have to be studied. Moreover, the same methodology must be applied to the case of poloidal fields, and the hyperbolic regime has to be carefully studied.

Low-frequency internal waves and their influence on transmission loss variability

Analysis of the time series data collected from a stationary location in the continental shelf of the southeastern Arabian Sea during different month indicated prominent internal wave (IW) activity. Time evolution of temperature, resolved using Morlet wavelet technique, revealed that maximum energy was concentrated in the diurnal band at the density interface, whereas within the interior of thermocline, the dominant energy concentration shifted to semi-diurnal band. Both these harmonics have maximum amplitude (>15 m) during the pre-monsoon and monsoon season when the water column was highly stratified (>0.05 kg/m4), but they were not discernable in the temperature record when the stratification was weak (i.e., especially during winter). An acoustic propagation model based on ray theory, Bellhop (http:/oalib.hlsresearch.com/modes/acoustictoolbox/at.zip) was utilized to compute the transmission loss (TL) associated with the passage of low-frequency IWs. The TL was computed using the model considering (1) range-dependent and range-independent environmental scenario and (2) for different source and receiver depth configurations. Intermittent fading of acoustic signals was observed in the presence of IW. It was also observed that fading of signals very much depends on the source–receiver configuration.