Nonlinear effects occurring during the propagation of trapped topographic waves

Baroclinic topographic waves trapped by a sloping bottom in the case of a real stratification are considered. The dispersion properties of these waves are studied. The characteristic scales and amplitudes of trapped topographic waves observed in the Norwegian Sea are determined. The asymptotic method of multi-scale expansions is used to study nonlinear effects occurring during the propagation of these waves. The wave-induced mean flow is determined in the second order of smallness in the wave amplitude. The evolution equation for the envelope—the nonlinear Schrödinger equation—is derived. Modulation instability of these waves is examined. It is shown that trapped topographic waves are modulationally unstable.     

Trapped bottom topographic waves over the inclined bottom

In the Boussinesq approximation, we study baroclinic topographic waves trapped by the flat meridional slope. The existence of these waves is explained by stratification, inclined bottom, and Earth's rotation. We deduce the evolutionary equation for the square of the envelope of a narrow-band wave packet of trapped waves. In the second order of smallness relative to the wave amplitude, we find the mean fields of velocity and density induced by the packet. It is shown that, in the limiting case of weakly nonlinear plane waves, the induced current is zonal. In the Northern hemisphere, depending on the slope of the bottom γ₁, the sign of the phase velocity σ/k (k is the zonal wave number) is either always positive (for γ₁>γ_1cr) or always negative (for γ₁<γ_1cr). If we neglect the vertical component of the Coriolis acceleration, then γ_1cr=0. Translated by Peter V. Malyshev and Dmitry V. Malyshev     

Transmission and reflection of the Rossby waves in a stratified ocean with bottom topography

Transmission and reflection coefficients are calculated for Rossby waves incident on a bottom topography with constant slope in a continuously stratified ocean. The characteristics of the coefficients are interpreted in terms of the quasigeostrophic waves on the slope. In the parameter range where only the barotropic Rossby waves can propagate in the region outside the slope, the bottom trapped wave plays the same role as the topographic Rossby wave in a homogeneous ocean, and hence the transmission is weak unless phase matching takes place. When both of the barotropic and baroclinic Rossby waves can propagate outside the slope, the total transmission can be strong. The bottom trapped wave affects the transmission and reflection, and it leads to the possibility that the Rossby wave is transmitted as a mode different from the incident mode. When the number of the wavy modes on the slope is smaller than that of the Rossby wave modes outside the slope, strong reflection occurs.The results for an ocean with linear distribution of the squared Brunt-Väisälä frequency are compared to those in a uniformly stratified ocean. The weakening of the stratification near the bottom is almost equivalent to reducing the effect of the slope.     

Transport of Sediments Entrapped by Topographic Waves

In the Boussinesq approximation, for topographic waves entrapped by a sloping bottom, we determine mean currents induced by a wave due to nonlinearity with regard for turbulent viscosity and diffusion. We determine the thickness of the bottom boundary layer, the vertical turbulent exchange coefficients, and turbulent stresses on the upper boundary of the boundary layer depending on the parameters of the wave. In the diffusion approximation, we find the vertical distribution of the concentration of sediments suspended by the wave and the flow rates of sediments along and perpendicular to the isobaths. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 5, pp. 13–24, September–October, 2005.     

Ferrobasalts from the Central Indian Ocean Basin

 We report the occurrence of ferrobasalts recovered from the Central Indian Ocean Basin crust generated at the Southeast Indian Ridge during a phase of moderate to fast spreading accretion (∼110–190 mm/yr, full rate).The rocks are rich in plagioclase, FeO^* (13–19%), and TiO₂ (2.27–2.76%), poor in olivine and MgO (3.44–6.20%), and associated with topographic highs and increased amplitude magnetic anomalies corresponding to chrons A25 and A24. We suggest that secon dary eruptions from ancient N-MORB magma, which may have been trapped at a shallow depth in a horizon of neutral buoyancy, could have produced the ferrobasalts. Received: 27 January 1998 / Revision received: 25 May 1998     

Generation of seiches by moving baric formations

We consider a plane problem of generation of barotropic seiches in the case of motion of a region of disturbances of atmospheric pressure over a bounded basin. The nonlinear system of equations of long waves with regard for the quadratic dependence of bottom friction is solved by the finite-difference method. The calculations are carried out for three basins of variable depth corresponding to rectangular sections of the Black Sea and Sea of Azov. It is shown that the passage of the baric anomaly over the basin is accompanied by the generation of the lowest seiches. The oscillations of fluid are especially intense in the shallow-water zones of the basins. Seiches exhibit a trend to intensification as the velocity of motion of the atmospheric disturbance increases. The dependence on the width of the baric anomaly is not monotonic. In the shelf zone, the amplitude of oscillations of the level can be several times higher that the pure hydrostatic response of the sea surface. In the analyzed basins, the influence of nonlinearity on the seiches is weak. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 6, pp. 3–15, November–December, 2007.     

Influence of currents on weakly nonlinear topographic waves

In the Boussinesq approximation, we consider trapped topographic waves in an inhomogeneous current directed along isobaths. The influence of the current on the dispersion properties of trapped topographic waves in the Norwegian Sea is studied. We determine the mean currents and nonoscillatory (on the time scale of the waves) density corrections induced by the waves due to their nonlinearity. It is shown that the influence of currents is significant in the short-wave region. Its influence leads to a decrease in the wavelength for the constant period of waves, whereas the mean current caused by nonlinearity noticeably varies, especially in the bottom layer.     

Thermohydrodynamics of the ocean generation of seiches by moving baric fronts in bounded basins

We consider a plane problem of barotropic seiches generated by a front of atmospheric pressure moving over a bounded basin. A system of nonlinear equations of long waves is solved by the finite-difference method with regard for the bottom friction and Earth's rotation. The numerical analyses are performed for two basins with distributions of depths typical of the Black Sea. It is shown that the passage of a baric front over the basin leads to the generation of lower seiches. The oscillations of level and the corresponding currents are especially intense in the shallow-water zones of the basins. The seiches become more intense as the velocity of transfer of the atmospheric front increases and the width of the front decreases. Earth's rotation leads to the generation of longshore currents and promotes the process of weakening of residual oscillations of the fluid following the passage of the front. The influence of nonlinearity on seiches is small for the analyzed basins. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 2, pp. 3–18, March–April, 2008.     

Spectra of the Black-Sea tsunamis

We perform the analysis of the time spectra of four tsunamis generated in the Black Sea by the earthquakes of 26.07.1927, 11.09.1927, 26.12.1939, and 12.07.1966. For the analysis of the spectra, we used digitized marigrams obtained for 12 points of the Black-Sea coast. The obtained spectra are, as a rule, multimode and have 1–4 spectral maxima. One maximum corresponds to the periods typical of tsunami waves and the other maxima correspond to the oscillations of the sea level with lower frequencies. It seems likely that the events of tsunami are accompanied by low-frequency oscillations of the level caused by the atmospheric forcing, seiches, or other factors. In numerous cases, the oscillations from the predominant energy range lie outside the characteristic range of periods of the tsunami waves. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 5, pp. 21–30, September–October, 2007.     

Capillary-Gravitational Waves of Finite Amplitude on the Surface of a Homogeneous Liquid

The method of multiple scales is used to deduce equations for three nonlinear approximations of the capillary-gravitational disturbances of the free surface of a layer of a homogeneous liquid of constant depth. In these equations, the space-time variations of the wave profile in the expression for the velocity potential on the liquid surface are taken into account. On this basis, we construct asymptotic expansions up to the quantities of the third order of smallness for the velocity potential and elevations of the liquid surface induced by running periodic waves of finite amplitude. Furthermore, we analyze the dependences of the amplitude-phase characteristics of wave disturbances on the surface tension, depth of the liquid, and the length and steepness of waves of the first harmonic. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 5, pp. 25–34, September–October, 2005.     

Transport Properties of Trapped Topographic Waves

In the Boussinesq approximation, we study weakly nonlinear topographic waves trapped by a flat slope of arbitrary orientation. We compute the mean currents induced by the waves due to the nonlinearity in the quadratic approximation with respect to the wave amplitude in the presence of dissipation of the wave energy into the turbulent motion. In the diffusion approximation, we determine the vertical distribution of the concentration of wave-suspended sediments. It is shown that the consumption of sediments across the isobaths is directed downward along the slope. At the same time, the consumption of sediments along the isobaths has the same direction as the projection of the horizontal wave vector.     

Evaluation of the Parameters of Tsunami Waves along the South Coast of the Crimean Peninsula

Within the framework of a nonlinear model of long waves, we present the estimates of the parameters of tsunami waves along the south coast of the Crimean Peninsula (from Cape Khersones to Cape Meganom) with a space resolution of 2.5 km. The numerical analysis is carried out for four typical positions of the elliptic zones of generation and the range of magnitudes 6.5–7.5. We study the space structure of waves and determine the amplitudes and periods of oscillations of the level at 11 points of the analyzed part of the coastline of the Black Sea. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 3, pp. 3 – 10, May–June, 2005.     

Dynamics of the Crimea shelf waters in summer

In this paper, the main results of field research into the seawater dynamics on the shelf of the Crimea in the summer of 1991 are reported. It is shown that in the weather frequency band, the oscillations with periods of 11–12 days have the greatest amplitude. These represent coastal-trapped waves with a spatial scale of the order of the length of the Black Sea coastline. These waves are generated by distant winds and significantly disturb the response of the Crimea shelf waters to local wind forcing. A transient upwelling-downwelling circulation having a period of 5–7 days is induced by local winds and is not accompanied by the generation of trapped waves at the frequency of forcing. Since the average circulation on the shelf of the Crimea is of cyclonic character, downwelling events are found more frequently and have longer time periods than the upwelling events. The wind-generated trapped waves on the shelves of the Crimea and Kerch Peninsula have an average period of 27 h. The inertial oscillations were well expressed during an experiment resembling long-wave motions.Translated by Mikhail M. Trufanov.     

Barotropic Response of the Sea of Okhotsk to Wind Forcing

We have examined wind-induced circulation in the Sea of Okhotsk using a barotropic model that contains realistic topography with a resolution of 9.25 km. The monthly wind stress field calculated from daily European Centre for Medium-Range Weather Forecasting (ECMWF) Re-Analysis data is used as the forcing, and the integration is carried out for 20 days until the circulation attains an almost steady state. In the case of November (a representative for the winter season from October to March), southward currents of velocity 0.1–0.3 m s⁻¹ occur along the bottom contours off the east of Sakhalin Island. The currents are mostly confined to the shelf (shallower than 200 m) and extend as far south as the Hokkaido coast. In the July case (a representative for the summer season from April to September), significant currents do not occur, even in the shallow shelves. The simulated southward current over the east Sakhalin shelf appears to correspond to the near-shore branch of the East Sakhalin Current (ESC), which was observed with the surface drifters. These seasonal variations simulated in our experiments are consistent with the observations of the ESC. Dynamically, the simulated ESC is interpreted as the arrested topographic wave (ATW), which is the coastally trapped flow driven by steady alongshore wind stress. The volume transport of the simulated ESC over the shelf reaches about 1.0 Sv (1 Sv = 10⁶ m³s⁻¹) in the winter season, which is determined by the integrated onshore Ekman transport in the direction from which shelf waves propagate. This revised version was published online in July 2006 with corrections to the Cover Date.     

Runup of the surface waves of various shapes on a sloping beach

In the long-wave approximation, we perform the numerical analysis of the plane problem of runup of waves of various shapes on a sloping beach. We study transformations of the shape of waves flooding the beach and in the course of their subsequent rundown. The dependence of maximum elevations and lowerings of the sea level on the parameters of the waves approaching the beach, the depth of the shelf, and the slope of the bottom are investigated. It is shown that the shape of waves affects the amplitude characteristics of oscillations of the coastline. The heights of the vertical runup of waves incident on a sloping beach can be several times higher than the amplitude of waves entering the shelf zone.     

Thermohydrodynamics of the ocean decay of nonlinear oscillations in a bounded basin of variable depth

We study the time decay of surges of a liquid in a round shallow-water basin of variable depth. The dependence of the logarithmic decrement of oscillations on the bottom topography and wind velocity is analyzed. The role of convective acceleration and bottom friction in the formation of both the level of vertical displacement of the surface of the basin and the velocity field of horizontal wave currents is estimated. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 2, pp. 3–11, March–April, 2006.     

Simulation of the quasi-biennial oscillations of the zonal wind in the equatorial stratosphere: Part I. Low-parameter models

The paper focuses on the simulation of the quasi-biennial oscillations (QBOs) of zonal velocity in the equatorial stratosphere. Low-parameter models are used to examine two mechanisms for excitation of the QBO: one through the interaction of planetary waves with the mean flow at critical levels and another through gravity-wave obliteration. The possible use of each of these mechanisms for generating the QBO is shown, the ranges of parameter values where this generation is possible are determined, and the dependences of the period and amplitude of the limit cycle on the model parameters are analyzed. A relative role of waves of different scales in the formation of the period of the oscillations of zonal wind is studied with a coupled model combining both mechanisms. The conditions that are required to reproduce the QBO in general circulation models are discussed.     

Study on Influences of Fringing Reef on Harbor Oscillations Triggered by N-Waves

Influences of topographic variations of the offshore fringing reef on the harbor oscillations excited by incident Nwaves with different amplitudes and waveform types are studied for the first time. Both the propagation of the Nwaves over the reef and the subsequently-induced harbor oscillations are simulated by a Boussinesq-type numerical model, FUNWAVE-TVD. The present study concentrates on revealing the influences of the plane reef-face slope,the reef-face profile shape and the lagoon width on the maximum runup, the wave energy distribution and the total wave energy within the harbor. It shows that both the wave energy distribution uniformity and the total wave energy gradually increase with decreasing reef-face slope. The profile shape of the reef face suffering leading-elevation Nwaves(LEN waves) has a negligible impact on the wave energy distribution uniformity, while for leading-depression N-waves(LDN waves), the latter gradually decreases with the mean water depth over the reef face. The total wave energy always first increases and then decreases with the mean water depth over the reef face. In general, the total wave energy first sharply decreases and then slightly increases with the lagoon width, regardless of the reef-face width and the incident waveform type. The maximum runup subjected to the LEN waves decreases monotonously with the lagoon width. However, for the LDN waves, its changing trend with the lagoon width relies on the incident wave amplitude.     

Modeling wave�Ccurrent bottom boundary layers beneath shoaling and breaking waves

The boundary layer characteristics beneath waves transforming on a natural beach are affected by both waves and wave-induced currents, and their predictability is more difficult and challenging than for those observed over a seabed of uniform depth. In this research, a first-order boundary layer model is developed to investigate the characteristics of bottom boundary layers in a wave–current coexisting environment beneath shoaling and breaking waves. The main difference between the present modeling approach and previous methods is in the mathematical formulation for the mean horizontal pressure gradient term in the governing equations for the cross-shore wave-induced currents. This term is obtained from the wave-averaged momentum equation, and its magnitude depends on the balance between the wave excess momentum flux gradient and the hydrostatic pressure gradient due to spatial variations in the wave field of propagating waves and mean water level fluctuations. A turbulence closure scheme is used with a modified low Reynolds number k-ε model. The model was validated with two published experimental datasets for normally incident shoaling and breaking waves over a sloping seabed. For shoaling waves, model results agree well with data for the instantaneous velocity profiles, oscillatory wave amplitudes, and mean velocity profiles. For breaking waves, a good agreement is obtained between model and data for the vertical distribution of mean shear stress. In particular, the model reproduced the local onshore mean flow near the bottom beneath shoaling waves, and the vertically decreasing pattern of mean shear stress beneath breaking waves. These successful demonstrations for wave–current bottom boundary layers are attributed to a novel formulation of the mean pressure gradient incorporated in the present model. The proposed new formulation plays an important role in modeling the boundary layer characteristics beneath shoaling and breaking waves, and ensuring that the present model is applicable to nearshore sediment transport and morphology evolution.