Periodic Internal Waves Research Articles

STEADY INTERNAL WATER WAVES WITH A CRITICAL LAYER BOUNDED BY THE WAVE SURFACE

In this paper we construct small amplitude periodic internal waves traveling at the boundary region between two rotational and homogeneous fluids with different densities. Within a period, the waves we obtain have the property that the gradient of the stream function associated to the fluid beneath the interface vanishes, on the wave surface, at exactly two points. Furthermore, there exists a critical layer which is bounded from above by the wave profile. Besides, we prove, without excluding the presence of stagnation points, that if the vorticity function associated to each fluid in part is real-analytic, bounded, and non-increasing, then capillary-gravity steady internal waves are a priori real-analytic. Our new method provides the real-analyticity of capillary and capillary-gravity waves with stagnation points traveling over a homogeneous rotational fluid under the same restrictions on the vorticity function.

Boundary mixing in lakes: 1. Modeling the effect of shear-induced convection

[1] Recent studies in lakes have shown evidence for a strong influence of shear-induced stratification on mixing in turbulent bottom boundary layers (BBLs) on sloping topography. These observations suggest that the periodic near-bottom shear resulting from internal wave motions may lead to alternating periods of gravitationally stable and unstable stratification in the BBL with relevant implications for turbulence and mixing in the entire basin. The impact of these processes for basin-scale mixing is investigated here in a three-dimensional processes-oriented modeling study, using a well-investigated system (Lake Alpnach, Switzerland) as an example. Consistent with available observations, our results indicate that the BBL becomes gravitationally unstable in areas with upslope flow, covering a substantial fraction of the total bottom area of the lake. While near-bottom convection associated with the unstable stratification in these areas results in strong turbulence, its contribution to net mixing is negligible since the BBL is already well mixed. Conversely, in areas with downslope flow the near-bottom shear generates stable stratification, leading to a suppression of turbulence but also to larger mixing rates due to an enhanced mixing efficiency. Overall, BBL mixing is found to dominate basin-scale mixing. These mechanisms are likely to be important for a large class of stratified natural waters, in which boundary layer mixing is energized by periodic internal waves or basin-scale motions.

Fully nonlinear periodic internal waves in a two-fluid system of finite depth

Periodic travelling wave solutions for a strongly nonlinear model of long internal wave propagation in a two-fluid system are derived and extensively analysed, with the aim of providing structure to the rich parametric space of existence of such waves for the parent Euler system. The waves propagate at the interface between two homogeneous-density incompressible fluids filling the two-dimensional domain between rigid planar boundaries. The class of waves with a prescribed mean elevation, chosen to coincide with the origin of the vertical (parallel to gravity) axis, and prescribed zero period-average momentum and volume-flux is studied in detail. The constraints are selected because of their physical interpretation in terms of possible processes of wave generation in wave-tanks, and give rise to a quadrature formula which is analysed in parameter space with a combination of numerical and analytical tools. The resulting model solutions are validated against those computed numerically from the parent Euler two-layer system with a boundary element method. The parametric domain of existence of model periodic waves is determined in closed form by curves in the amplitude–speed (A, c) parameter plane corresponding to infinite period limiting cases of fronts (conjugate states) and solitary waves. It is found that the existence domain of Euler solutions is a subset of that of the model. A third closed form relation between c and A indicates where the Euler solutions cease to exist within the model's domain, and this is related to appearance of ‘overhanging’ (multiple valued) wave profiles. The model existence domain is further partitioned in regions where the model is expected to provide accurate approximations to Euler solutions based on analytical estimates from the quadrature. The resulting predictions are found to be in good agreement with the numerical Euler solutions, as exhibited by several wave properties, including kinetic and potential energy, over a broad range of parameter values, extending to the limiting cases of critical depth ratio and extreme density ratios. In particular, when the period is sufficiently long, model solutions show that for a given supercritical speed waves of substantially larger amplitude than the limiting amplitude of solitary waves can exist, and are good approximations of the corresponding Euler solutions. This finding can be relevant for modelling field observations of oceanic internal waves, which often occur in wavetrains with multiple peaks.

Influence of Imperfect Internal Waves on Long-Range Underwater Acoustic Propagation

This work presents a numerical analysis of the effect of random fluctuations of internal waves on the chaotic dynamics of ray trajectories in ocean acoustics. The Eikonal equation is considered in a form of the second order, nonlinear ordinary differential equation. Random phase modulations in the form of zero mean Gaussian white noise are considered for modeling an imperfectly periodic single mode internal wave. It is shown that in the presence of random fluctuations the intersection of acoustic rays with the ocean surface occurs sooner and becomes more frequent than predicted by deterministic ocean models.

Laboratory modeling of the propagation of periodic internal waves over bottom slopes

The experimental investigation of the run-up of periodic internal waves in a two-layer fluid on the coastal slope is performed in an open hydrochannel at the Physical Department of the Lomonosov Moscow State University. The waves are produced by a wave generator. We study the transformation of waves, the vertical structure of the field of velocities of mass transfer, and the behavior of the parameters of internal waves propagating over the sloping bottom. It is shown that the run-up and breaking of internal waves are accompanied by periodic emissions of portions of the heavier fluid from the bottom layer upward along the slope. The Stokes drift velocity changes its sign as a function of depth. Moreover, both the wave length (the horizontal distance between the neighboring crests) and the height of waves over the sloping bottom (the elevation of the crest over the slope along the vertical) decrease as the wave approaches the coast.

Mixing and stratification in lakes of varying horizontal length scales: Scaling arguments and energy partitioning

This work analyzes the effects that changes in the surface area of stratified lakes have on the partitioning of energy among different pools, and, ultimately, on the strength of thermal stratification. Proposed restoration plans for the Salton Sea (USA), the surface area of which could be reduced by 80%, provided the original motivation of this work. Scaling arguments, based on a one‐dimensional model of lake stratification and three‐dimensional simulations, are consistent in predicting weaker mixing for smaller lakes. It is demonstrated that energy partitioning is affected by geometry for lakes that have a quarter internal wave period (Ti/4) that is shorter than the duration of wind forcing events, ∆tw. In these cases, it is demonstrated that mixing is, in general, stronger in larger basins due to greater mechanical energy fluxes across the free surface; larger shear production of turbulent kinetic energy Etk; and a larger fraction of Etk becoming background potential energy through diapycnal mixing. All these variables scale with the length of the basin. In lakes where Ti/4 > ∆tw, the rates of exchange among the different energy pools and the magnitudes of these pools are not sensitive to changes in lake size. Our results suggest that stratification in the reduced Salton Sea will be stronger than in the lake as it exists today, mainly in late summer. It is during that time when, due to high insolation values, stratification becomes strong and Ti/4 in the reduced sea will likely become small in comparison with the diurnal winds.

Exposure of internal waves on the sea surface

We theoretically analyse the impact of subsurface currents induced by internal waves on nonlinear Stokes surface waves. We present analytical and numerical solutions of the modulation equations under conditions that are close to group velocity resonance. Our results show that smoothing of the downcurrent surface waves is accompanied by a relatively high-frequency modulation, while the profile of the opposing current is reproduced by the surface wave's envelope. We confirm the possibility of generating an internal wave forerunner that is a modulated surface wave packet. Long surface waves can create such a wave modulation forerunner ahead of the internal wave, while other relatively short surface waves comprise the trace of the internal wave itself. Modulation of surface waves by a periodic internal wavetrain may exhibit a characteristic period that is less than the internal wave period. This period can be non-uniform while the wave crosses the current zone. Our results confirm that surface wave excitation by means of internal waves, as observed at their group resonance frequencies, is efficient only in the context of opposing currents.

Solitary wave and periodic wave solutions for the thermally forced gravity waves in atmosphere

By introducing a new transformation, a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system, which extends Fan's direct algebraic method to the case when r > 4. The solutions of a first-order nonlinear ordinary differential equation with a higher degree nonlinear term and Fan's direct algebraic method of obtaining exact solutions to nonlinear partial differential equations are applied to the combined KdV–mKdV–GKdV equation, which is derived from a simple incompressible non-hydrostatic Boussinesq equation with the influence of thermal forcing and is applied to investigate internal gravity waves in the atmosphere. As a result, by taking advantage of the new first-order nonlinear ordinary differential equation with a fifth-degree nonlinear term and an eighth-degree nonlinear term, periodic wave solutions associated with the Jacobin elliptic function and the bell and kink profile solitary wave solutions are obtained under the effect of thermal forcing. Most importantly, the mechanism of propagation and generation of the periodic waves and the solitary waves is analysed in detail according to the values of the heating parameter, which show that the effect of heating in atmosphere helps to excite westerly or easterly propagating periodic internal gravity waves and internal solitary waves in atmosphere, which are affected by the local excitation structures in atmosphere. In addition, as an illustrative sample, the properties of the solitary wave solution and Jacobin periodic solution are shown by some figures under the consideration of heating interaction.

Calculation and measurement of conical beams of three-dimensional periodic internal waves excited by a vertically oscillating piston

The structure of the wave field given by an exact solution of the linearized problem of radi- ation of three-dimensional periodic internal waves in a continuously stratified viscous fluid is analyzed numerically. The waves are generated by a piston, i.e., a disk lying on a fixed horizontal plane and oscil- lating in the vertical direction. The flow fields and the wave displacements are compared with the data of shadow visualization and measurements of the wave amplitudes made using a contact sensor. The calculated and observed wave patterns are in satisfactory agreement and the displacement distributions coincide correct to a fitting coefficient 0.7 < K < 1.1 characterizing the role of the nonlinear effects and other factors neglected in this model. DOI: 10.1134/S0015462807040114

On the parabolic equation method in internal wave propagation

A parabolic equation for the propagation of periodic internal waves over varying bottom topography is derived using the multiple-scale perturbation method. Some computational aspects of the numerical implementation are discussed. The results of numerical experiments on propagation of an incident plane wave over a circular-type shoal are presented in comparison with the analytical result, based on Born approximation.

The generation of three-dimensional internal waves and attendant boundary layers in a viscous continuously stratified fluid. Construction of an analytical solution

An analytical solution of a linearized problem of the emission of periodic internal waves by part of a plane which oscillates with a small amplitude in an arbitrary direction in a viscous exponentially stratified fluid is constructed. Solutions of the dispersion equation are given for all positions of the emitting surface (arbitrary, vertical, horizontal, and critical when one of the beam propagation directions is collinear with the emitting surface). The possibility of transition to the case of a uniform fluid, which is important for applications, is analyzed.

Observations of internal waves and associated mixing phenomena in the Portimao Canyon area

Internal wave activity and induced mixing phenomena in the Portimao Canyon area (southern Portuguese coast) are analysed. Observations of current velocities and CTD measurements were taken during a 24-h period at two fixed stations located along the main axis of the canyon. The station nearest shore was located close to the canyon head and the other station was 5 km seaward. These basic measurements are complemented with vertical sections of temperature, acquired by XBT casts, along the main axis of the canyon, and also with time series of temperatures recorded by a thermistor chain moored on the continental shelf, near the canyon head. The time sequences of velocities and density profiles recorded at the fixed stations were analysed using a procedure based on empirical orthogonal function (EOF) analysis and dynamical modes decomposition (DMD) techniques. By this method, two different sources for internal wave activity are identified: a clear internal tide signal on the one hand, and a shorter than tidal period internal waves which exhibit greater current velocities and density oscillation amplitudes than internal tide, on the other. It is suggested that these shorter period internal waves are responsible for the vertical mixing affecting the water column over the slope and shelf waters in this region. Finally, it is also suggested that the evacuation of these mixed water masses formed around the continental shelf break may be related to a core of relatively cold water flowing over the continental slope toward west.

Generation of beams of three-dimensional periodic internal waves by sources of various types

The energy and force characteristics of periodic internal wave beams in a viscous exponentially stratified fluid are analyzed. The exact solutions of linearized problems of generation obtained by integral transformations describe not only three-dimensional internal waves but also the associated boundary layers of two types. The solutions not containing empirical parameters are brought to a form that allows a direct comparison with experimental data for generators of various types (friction, piston, and combined) of rectangular or elliptic shape. The stress tensor and force components acting on the generator are given in quadratures. In the limiting cases, the solutions are uniformly transformed to the corresponding expressions for the problems in a two-dimensional formulation.

Spatial and Temporal Variability of Internal Wave Forcing on a Coral Reef

Abstract The deployment of a dense spatial array of temperature sensors on a coral reef in the Florida Keys provided a unique view of the interaction of cool water incursions generated by internal waves with the three-dimensional reef bathymetry. Water temperature on the reef surface was sampled every 5 s at 100 points on a 100 m by 150 m grid with concomitant measurements of water column velocity and temperature from mid-May through mid-August 2003. Episodic incursions of cool, subsurface water were driven by periods of strong semidiurnal internal tide and higher-frequency internal wave activity. For every time step in the data record the mean profile of temperature as a function of depth is calculated with a 3-m vertical averaging length scale. Subtracting this mean profile from the raw record yields a within depth, horizontal temperature anomaly. Visualization through time of the anomaly mapped onto the measured reef bathymetry reveals episodic variability of thermal patchiness across the reef as well as persistent features associated with reef bathymetry. Variation in the nature of the cooling and resulting thermal heterogeneity among events and seasons suggests multiple modes of cool water incursion ranging from unbroken, tidal period internal waves to packets of higher-frequency, energetic, broken internal bores.

A general view of the hydrographic and dynamical patterns of the Rías Baixas adjacent sea area

The hydrography and dynamics of the Rias Baixas adjacent shelf region is reviewed in this paper with the aim to serve as a general ‘state-of-the-art’ reference and to help introduce several topic-related articles in this special volume. This introductory article is structured as follows: first, a brief description of the general topographic and bathymetric characteristics and the tidal regime is given; afterwards, the water masses existing in the region, as well as their circulation patterns are reviewed with much more detail. In this context, we focus on the characteristics of the water masses involved in the main oceanographic process that occurs in this area (the upwelling of Eastern North Atlantic Central Water and the Mediterranean Water). To better reflect this purpose, bibliographic data are used, but also 46 CTD new data samplings all around the year from a station off Ría of Vigo are gathered. In this section, we extensively discuss the present dilemmas about the Iberian Poleward Current system, confronting some classical and more recent views about its seasonality. After that, concise but novel and relevant information on very recent observations of short period internal waves in the near shelf area is presented, pointing out about their importance on the thermohaline structure of the water column. This introductory paper is closed with some thoughts on what we consider either interesting or necessary work that should be performed in this area during the next coming years.

A free-surface hydrodynamic model for density-stratified flow in the weakly to strongly non-hydrostatic regime

A non-hydrostatic density-stratified hydrodynamic model with a free surface has been developed from the vorticity equations rather than the usual momentum equations. This approach has enabled the model to be obtained in two different forms, weakly non-hydrostatic and fully non-hydrostatic, with the computationally efficient weakly non-hydrostatic form applicable to motions having horizontal scales greater than the local water depth. The hydrodynamic model in both its weakly and fully non-hydrostatic forms is validated numerically using exact nonlinear non-hydrostatic solutions given by the Dubriel–Jacotin–Long equation for periodic internal gravity waves, internal solitary waves, and flow over a ridge. The numerical code is developed based on a semi-Lagrangian scheme and higher order finite-difference spatial differentiation and interpolation. To demonstrate the applicability of the model to coastal ocean situations, the problem of tidal generation of internal solitary waves at a shelf-break is considered. Simulations carried out with the model obtain the evolution of solitary wave generation and propagation consistent with past results. Moreover, the weakly non-hydrostatic simulation is shown to compare favorably with the fully non-hydrostatic simulation. The capability of the present model to simulate efficiently relatively large scale non-hydrostatic motions suggests that the weakly non-hydrostatic form of the model may be suitable for application in a large-area domain while the computationally intensive fully non-hydrostatic form of the model may be used in an embedded sub-domain where higher resolution is needed.

Generation of three-dimensional periodic internal waves by compact sources

Approximate methods [1] for calculating internalwave generation include certain empirical parameters. These parameters are not universal [2] and have to be determined experimentally. In this case, approximate solutions can differ rather considerably from exact ones when the parameters of model sources and sinks are found in the framework of the viscous-fluid theory [3] or of the conventional perfect-fluid theory [1, 4]. In [3], a more exact method was proposed for constructing a solution satisfying both the equations and boundary conditions on a radiating surface. The method allows us to calculate internal waves and complementary boundary layers. In the case of two-dimensional periodic internal waves generated by a strip oscillating in its own plane, calculated results based on the linear radiation theory are in satisfactory agreement with the experimental data of [5].

Generation of Periodic Motions by a Disk Performing Torsional Oscillations in a Viscous, Continuously Stratified Fluid

The solution of the problem of nonlinear generation of periodic internal waves by a boundary flow on a vertical cylinder or a horizontal disk performing torsional oscillations in an exponentially stratified fluid is constructed. The calculations are in satisfactory agreement with the results of experiments in which both horizontal and inclined disks of various diameters and a model propeller performing periodic torsional oscillations, including oscillations against a background of uniform rotation, are used as perturbation sources. The experiments were carried out over a wide range of parameters including the laminar, transition, and turbulent flow regimes. The limits of applicability of the proposed analytic theory of wave radiation are determined.

The seasonal evolution of wind/internal wave resonance in Lake Kinneret

Data from a thermistor chain and wind sensor collected over an annual stratification cycle in Lake Kinneret (Israel) during 2000 were used to investigate the seasonal evolution of wind/internal wave resonance. Internal wave periods determined from an analytical model were compared with observations, and we show that resonance during 2000 occurred during three distinct times of the year—at the onset of stratification (March), during the heating phase (June), and during the cooling phase (November). In all cases, resonance was between the wind and the dominant radial, azimuthal, and vertical mode‐one cyclonic (Kelvin) wave previously observed in Lake Kinneret.

Horizontal transport and dispersion in the surface layer of a medium‐sized lake

Lagrangian GPS drifter experiments, carried out in the surface layer of stratified Lake Kinneret (Israel), are presented. Differential kinematic properties and Lagrangian statistics were calculated and used to estimate the dominant mechanisms for horizontal dispersion. On time scales smaller than a few internal wave periods, internal waves lead to strong divergence and convergence events, causing instantaneous apparent horizontal growth rates that were larger, by up to an order of magnitude, than the actual mean dispersion coefficient. It is shown that the internal wave field modulated the vorticity field so as to satisfy conservation of potential vorticity. On time scales larger than a few internal wave periods, unbounded horizontal shear dispersion was of the same order as the actual mean observed dispersion coefficient (Kxy = 17.1 m2 s−1), while vertical shear dispersion was negligible.