Free Internal Waves Research Articles

Generation of a Vertical Fine Structure by Internal Waves on the Sea Shelf

Free internal waves are considered in the Boussinesq approximation with account of horizontal turbulent viscosity and diffusion in a two-dimensional flow in the sea of constant depth. In the linear approximation the boundary-value problem for the amplitude of vertical velocity has complex coefficients and is solved numerically using the third-order implicit Adams method. The vertical component of the Stokes drift velocity and the vertical wave fluxes of mass and salt are found in the second order in the wave amplitude. These fluxes lead to generation of a vertical fine structure that has an irreversible nature.

Vertikal'nyy perenos impul'sa inertsionno-gravitatsionnymi vnutrennimi volnami na dvumernom sdvigovom techenii

Purpose. The paper is aimed at investigating the momentum vertical transfer by inertia-gravity internal waves on a two-dimensional flow with a vertical shear of velocity, and also at studying the Stokes drift of liquid particles and the mean current effect on it. Methods and Results. Free internal waves in an infinite basin of constant depth are considered in the Boussinesq approximation with the regard for the Earth rotation. Two components of the mean current velocity depend on the vertical coordinate. The equation for the vertical velocity amplitude has complex coefficients; therefore the eigenfunction and the wave frequency are complex. The corresponding boundary value problem is solved numerically by the implicit Adams scheme of the third order of accuracy. The wave frequency at a fixed wavenumber was found by the shooting method. It was determined that the frequency imaginary part was small and could be either negative or positive depending on a wave number and a mode number. Thus, both weak attenuation and weak amplification of an internal wave are possible. The vertical wave momentum fluxes are nonzero and can exceed the corresponding turbulent fluxes. The Stokes drift velocity, transverse to the wave direction, is nonzero and less than the longitudinal velocity. The vertical component of the Stokes drift velocity is also nonzero and four orders of magnitude less than the longitudinal component. The signs of the vertical component of the Stokes drift velocity for the waves with the frequencies 10 and 16 cycle/h are opposite, since the signs of their frequency imaginary parts are different; and the vertical component of the Stokes drift velocity is proportional to the wave frequency imaginary part. Conclusions. The vertical momentum wave flux of inertia-gravity internal waves differs from zero in the presence of the current whose velocity component, transverse to the wave propagation direction, depends on the vertical coordinate. The component of the Stokes drift velocity, transverse to the wave propagation direction, is nonzero and less than the longitudinal one. The vertical component of the Stokes drift velocity is also nonzero and can contribute to formation of the vertical fine structure

Vertical Transfer of Momentum by Inertia-Gravity Internal Waves on a Two-Dimensional Shear Flow

Purpose. The paper is aimed at investigating the momentum vertical transfer by inertia-gravity internal waves on a two-dimensional flow with a vertical shear of velocity, and also at studying the Stokes drift of liquid particles and the mean current effect on it. Methods and Results. Free internal waves in an infinite basin of constant depth are considered in the Boussinesq approximation with the regard for the Earth rotation. Two components of the mean current velocity depend on the vertical coordinate. The equation for the vertical velocity amplitude has complex coefficients; therefore the eigenfunction and the wave frequency are complex. The corresponding boundary value problem is solved numerically by the implicit Adams scheme of the third order of accuracy. The wave frequency at a fixed wavenumber was found by the shooting method. It was determined that the frequency imaginary part was small and could be either negative or positive depending on a wave number and a mode number. Thus, both weak attenuation and weak amplification of an internal wave are possible. The vertical wave momentum fluxes are nonzero and can exceed the corresponding turbulent fluxes. The Stokes drift velocity, transverse to the wave direction, is nonzero and less than the longitudinal velocity. The vertical component of the Stokes drift velocity is also nonzero and four orders of magnitude less than the longitudinal component. The signs of the vertical component of the Stokes drift velocity for the waves with the frequencies 10 and 16 cph are opposite, since the signs of their frequency imaginary parts are different; and the vertical component of the Stokes drift velocity is proportional to the wave frequency imaginary part. Conclusions. The vertical momentum wave flux of inertia-gravity internal waves differs from zero in the presence of the current whose velocity component, transverse to the wave propagation direction, depends on the vertical coordinate. The component of the Stokes drift velocity, transverse to the wave propagation direction, is nonzero and less than the longitudinal one. The vertical component of the Stokes drift velocity is also nonzero and can contribute to formation of the vertical fine structure.

Otsenka svyazi dispersnykh svoystv svobodnykh vnutrennikh voln i vertikal'noy struktury polya plotnosti v Barentsevom i Karskom moryakh

Otsenka svyazi dispersnykh svoystv svobodnykh vnutrennikh voln i vertikal'noy struktury polya plotnosti v Barentsevom i Karskom moryakh

Estimation of the Relation between the Dispersion Features of Free Internal Waves and the Density Field Vertical Structure in the Barents and Kara Seas

Estimation of the Relation between the Dispersion Features of Free Internal Waves and the Density Field Vertical Structure in the Barents and Kara Seas

Vertical transport of momentum by internal waves in a shift current

In Boussinesq approximation free internal waves are considered at the account of rotation of the Earth and a plane-parallel current, perpendicular to a direction to wave propagation. For homogeneous stratification and at constant shift of speed of a current the boundary-value problem for amplitude of vertical velocity analytically solved and the dispersive parity is defined. Vertical wave fluxes of momentum and two components of Stokes drift speed are defined. Stokes drift speed, cross-section to a direction a wave propagation, is distinct from zero in the presence of the specified shift current. Similar calculations are executed for two-layer model. The vertical wave flux of momentum can exceed a corresponding turbulent flux.

Vertical Transport of Momentum by Internal Waves in a Shear Current

In Boussinesq approximation free internal waves are considered at the account of rotation of the Earth and a plane-parallel current, perpendicular to a direction to wave propagation. For homogeneous stratification and at constant shift of speed of a current the boundary-value problem for amplitude of vertical velocity analytically solved and the dispersive parity is defined. Vertical wave fluxes of momentum and two components of Stokes drift speed are defined. Stokes drift speed, cross-section to a direction a wave propagation, is distinct from zero in the presence of the specified shift current. Similar calculations are executed for two-layer model. The vertical wave flux of momentum can exceed a corresponding turbulent flux.

Internal waves pump waters in and out of a deep coastal embayment of a large lake

AbstractLarge internal waves are a ubiquitous feature of many thermally stratified lakes, and result in oscillating baroclinic flows that pump water into and out of deep coastal embayments. In the long, narrow, and deep Kempenfelt Bay of Lake Simcoe, we show that stratification and circulation were coupled, so that movements of the thermocline can effectively flush the embayment much faster than hydraulic residence time from river input alone. Internal currents were driven by long‐period internal waves and resulted in large horizontal excursion lengths of several kilometers, which could drive exchange of embayment waters with the main basin. If the embayments are long, wide, and deep, Coriolis forces also deflect the internal wave to follow the coastline on the right‐hand side in the direction of travel as a Kelvin‐type wave, resulting in a net cyclonic circulation in the embayment. This residual counterclockwise flow further facilitated flushing of Kempenfelt Bay. For the summer of 2015, we estimate that forced and free internal wave dynamics alone resulted in a seasonally averaged flushing timescale as short as 17 ± 6 d for the surface mixed layer, and of 13.5 ± 5 d for the hypolimnetic waters of Kempenfelt Bay. Kempenfelt Bay is representative of many long, deep, and narrow embayments found in the Laurentian Great Lakes and arctic fjords. The exchange processes investigated here are relevant for determining the dynamics of water quality parameters used as indicators to evaluate lake health and fish habitat.

Vertical momentum transfer by internal waves when eddy viscosity and diffusion are taken into account

Free internal waves are considered in a Boussinesq approximation in the situation when horizontal eddy viscosity and diffusion in a vertically inhomogeneous flow are taken into account. The dispersion relation and wave damping factor are found in a linear approximation. The Stokes drift velocity is determined in the second order of smallness based on the wave amplitude. It has been indicated that the Stokes drift velocity, transverse with respect to the wave propagation direction, differs from zero if the flow-rate transverse component depends on the vertical coordinate. Vertical momentum fluxes differ from zero and can be comparable with or exceed the corresponding turbulent fluxes if eddy viscosity and diffusion are taken into account.

Vertical fluxes induced by weakly nonlinear internal waves on a shelf

Free internal waves in a vertically nonuniform stratified flow are considered with account of the coefficients of horizontal turbulent transport. The dispersion equation and the wave decay rate are derived in the linear approximation. The vertical component of the Stokes drift velocity and the wave fluxes of heat and salt are determined in the second order of the wave amplitude. It is shown that it is the vertical component of the Stokes drift velocity which is nonzero, when turbulent viscosity and diffusion are taken into account, that mainly contribute to the wave transport. The wave flux of salt is greater than the turbulent flux. Taking the flow into account leads to a decrease in the vertical wave fluxes but the wave flux of salt remains greater than the turbulent flux.

Interharmonics in internal gravity waves generated by tide-topography interaction

The dynamics and spectrum of internal gravity waves generated in a linearly stratified fluid by tidal flow over a flat-topped ridge are investigated at five different latitudes using an inviscid two-dimensional numerical model. The resulting wave field includes progressive freely propagating waves which satisfy the dispersion relation, and forced waves which are trapped non-propagating oscillations with frequencies outside the internal wave band. The flow is largely stable with respect to shear instabilities, and, throughout the runs, there is a negligibly small amount of overturning which is confined to the highly nonlinear regions along the sloping topography and where tidal beams reflect from the boundaries. The wave spectrum exhibits a self-similar structure with prominent peaks at tidal harmonics and interharmonics, whose magnitudes decay exponentially with frequency. Two strong subharmonics are generated by an instability of tidal beams which is particularly strong for near-critical latitudes where the Coriolis frequency is half the tidal frequency. When both subharmonics are within the free internal wave range (as in cases 0°–20° N), they form a resonant triad with the tidal harmonic. When at least one of the two subharmonics is outside of the range (as in cases 30°–40° N) the observed instability is no longer a resonant triad interaction. We argue that the two subharmonics are generated by parametric subharmonic instability that can produce both progressive and forced waves. Other interharmonics are produced through wave–wave interactions and are not an artefact of Doppler shifting as assumed by previous authors. As the two subharmonics are, in general, not proper fractions of the tidal frequency, the wave–wave interactions are capable of transferring energy to a continuum of frequencies.

A comparison between vertical motions measured by ADCP and inferred from temperature data

Abstract. Combined vertical current (w) and thermistor string data demonstrate that high-, near-buoyancy frequency internal "wave" trains along a pycnocline in a flat-bottom shelf sea consist for 2 periods of a dominant mode-1 non-linear part, while thereafter mainly of linear [mode-2, quadrupled frequency] waves, to first order. In a simple [linear] heat budget the use of unfiltered temperature gradient or its time mean changes results by only 10%. The observations also demonstrate that temperature is not always adequate to estimate vertical motions using the linear 1-D heat equation. In shallow seas, tidal-w estimated from temperature data can be an order of magnitude weaker than directly observed w, and thus do not represent free internal waves. In the ocean, not too far from the main internal wave topography source, tidal motions represent linear waves and are well described by temperature-inferred w. There however, temperature-inferred w and directly observed w differ strongly near the buoyancy frequency, at which w is dominated by non-linear waves, and near [sub]inertial frequencies, at which w is dominated by eddies and gyroscopic waves.

Investigation of the Dependence of the Space Structure and Phase Characteristics of the First-Mode Internal Waves in the Atlantic Ocean on the Hydrological Conditions

By using the linear theory of internal waves in a continuously stratified ideal incompressible liquid of finite depth and hydrological data, we study the dispersion properties and space structure of the vertical velocity of the first mode of free internal waves in the Atlantic Ocean. The dependence of the characteristics of waves on the hydrological structure of waters is analyzed.

On the polarization of oscillatory currents in the Bay of Biscay

Rotary spectral properties were examined using ∼20 records of yearlong current observations in the Bay of Biscay. Here data are discussed from two representative records from 1000 m above the bottom at two sites, one above the continental slope and another above the abyssal plain. Nearly raw spectra revealed a strong prevalence for clockwise over anti‐clockwise circular motions at frequencies (σ) between ∼0.85f < σ < N (f and N denoting the local inertial and buoyancy frequencies, respectively). The observations showed rotary properties that were previously associated with free internal waves (for f < σ < N, N ≫ f), but only for motions in bands of ∼0.9 cpd width in frequency around inertial, tidal, and their nonlinear sum‐interaction frequencies. This width was larger than the enhanced kinetic energy bands of ∼0.15 cpd around inertial and tidal constituent frequencies. Nevertheless, it was suggested that both bands were associated with the same nonlinear interactions due to advection between subinertial and peak‐frequency motions that also resulted in internal wave intermittency.

Semi‐diurnal and diurnal tidal dissipation from TOPEX/Poseidon altimetry

Tidal energy dissipation is estimated for eight semi‐diurnal and diurnal constituents using a global inverse solution constrained by TOPEX/Poseidon altimeter data. Very similar spatial patterns are obtained for all semi‐diurnal constituents, with about one third of the total dissipation occurring in the deep ocean over rough topography. Maps for diurnal constituents are also similar amongst themselves, but quite different from the semi‐diurnal results. For diurnals a smaller fraction of dissipation, roughly 10%, occurs in the deep ocean. Much of the difference can be explained by the very different spatial pattern of diurnal and semi‐diurnal tidal currents. The lack of free internal waves at frequencies poleward of 30° at diurnal frequencies also probably plays a role, limiting the effectiveness of baroclinic conversion as an energy sink for barotropic diurnal tides.

Island-generated internal waves at Scott Reef, Western Australia

Oceanographic field studies were carried out, in August–October 1993, of the water circulation around Scott Reef, a 39 km wide island rising nearly vertically in 500 m depth on the outer continental shelf slope of Western Australia. Macro-tide prevailed and generated 0.6 m s -1 tidal currents. Internal waves 60 m peak-to-trough occurred around Scott Reef mainly at the semi-diurnal frequencies. A numerical model and the field data suggested that the internal waves were locally generated by the interaction of the tidal currents and the bathymetry. Also these waves were predicted to rotate counterclockwise around the island with maximum amplitude along the island slopes, while radiating energy as free internal waves propagating freely away in the open ocean. They were generated apparently without eddies forming in the lee of the island. The wave frequencies were not restricted to the diurnal and semi-diurnal tidal frequencies which dominated the sea surface fluctuations, indeed higher frequencies were important. The energy at the inertial frequency was negligible. It appears that oceanic islands are internal wave generators and may contribute to the oceanic variability even far away.

An influence of vertical structure of density and flows on free internal waves

In the framework of general linear theory we study the influence of a thin stratification of fields of density and velocity of plane-parallel shear flow on dispersion curves and vertical profiles of modes of free internal gravitational waves. It is shown that an accounting of thin structural singularities can essentially change kinematic characteristics of such waves.

The effect of the vertical density structure on internal waves in the neighbourhood of the intertial period

With the latitudinal variability of the Coriolis parameter considered, the influence of the vertical density structure on the parameters of free internal waves is stuided. Numerical calculations are performed for Brunt-Vaisala frequency profiles characteristic of the four seasons of the year in the deep section of the Black Sea. Results of the analysis of the dependences of the wavenumber and group velocity on the frequency are presented, and the domain of application of thef-plane approximation is defined.

Radiative instability of stratified shear flows in the Drazin model

The results of analytical stability calculations of an ideal liquid stratified shear flow described by the Drazin model /1/ are presented. It is shown that the previously found solution /1/, which gives the boundaries of the region of stability in a plane with parameters αJ, where α is a dimensionless wave number and J is the minimum value of the Richardson number are generally incomplete. The presence of constant stratification in the whole flow region enables free internal waves to exist which, emanating from the shear layer, give rise to additional radiative instability, previously examined in /2/ for a flow with tangential velocity discontinuity. The influence of the outer boundaries on the stability of the shear flow is clarified and the analytical results obtained are compared with numerical calculations carried out in /3, 4/ for certain models with smooth velocity and density profiles.

An Analytic Theory on How Friction Affects Free Internal Waves in the Equatorial Waveguide

Abstract We study theoretically the effects of linear (Rayleigh) friction on free internal waves in the equatorial wayeguide. The waves may be vertically propagating or in standing vertical modes. Analytic solutions on a beta-plane show that the meridional scale of the wayeguide becomes significantly larger than the inviscid value only when the frequency is much less than the friction coefficient. In the limit of zero frequency, the meridional scale grows without bound. The amount of zonal damping depends on the wave type. Kelvin, high-frequency Yanai (mixed Rossby-gravity), inertial-gravity and low-wavenumber (non-dispersive) Rossby waves decay relatively slowly while low-frequency Yanai and high-wavenumber (short) Rossby waves are much more strongly attenuated. With friction, transitions to evanescence are spread over frequency bands; waves that had zero group velocity without friction are zonally evanescent with friction. Off the equator, friction changes amplitude nodes into non-zero minima and smooth...