Some problems of edge waves and standing waves on beaches

Some problems of edge waves and standing waves on beaches are examined.

The nonlinear interaction of a wave normally incident on a sloping beach with a subharmonic edge wave is studied. A two-timing expansion is used in the full nonlinear theory to obtain the modulation equations which describe the evolution of the waves. It is shown how large amplitude edge waves are produced; and the results of the theory are compared with some recent laboratory experiments.

Traveling edge waves are considered in two situations. First, the full linear theory is examined to find the finite depth effect on the edge waves produced by a moving pressure disturbance. In the second situation, a Stokes' expansion is used to discuss the nonlinear effects in shallow water edge waves traveling over a bottom of arbitrary shape. The results are compared with the ones of the full theory for a uniformly sloping bottom.

The finite amplitude effects for waves incident on a sloping beach, with perfect reflection, are considered. A Stokes' expansion is used in the full nonlinear theory to find the corrections to the dispersion relation for the cases of normal and oblique incidence.

Finally, an abstract formulation of the linear water waves problem is given in terms of a self adjoint but nonlocal operator. The appropriate spectral representations are developed for two particular cases.

Theoretical investigation of turbulent boundary layer over a wavy surface

The important features of the two-dimensional incompressible turbulent flow over a wavy surface of wavelength comparable with the boundary layer thickness are analyzed.

A turbulent field method using model equation for turbulent shear stress similar to the scheme of Bradshaw, Ferriss and Atwell (1967) is employed with suitable modification to cover the viscous sublayer. The governing differential equations are linearized based on the small but finite amplitude to wavelength ratio. An orthogonal wavy coordinate system, accurate to the second order in the amplitude ratio, is adopted to avoid the severe restriction to the validity of linearization due to the large mean velocity gradient near the wall. Analytic solution up to the second order is obtained by using the method of matched-asymptotic-expansion based on the large Reynolds number and hence the small skin friction coefficient.

In the outer part of the layer, the perturbed flow is practically "inviscid." Solutions for the velocity, Reynolds stress and also the wall pressure distributions agree well with the experimental measurement. In the wall region where the perturbed Reynolds stress plays an important role in the process of momentum transport, only a qualitative agreement is obtained. The results also show that the nonlinear second-order effect is negligible for amplitude ratio of 0.03. The discrepancies in the detailed structure of the velocity, shear stress, and skin friction distributions near the wall suggest modifications to the model are required to describe the present problem.

Strongly nonlinear long interfacial waves in uniform basic flows and their dispersion relationship

In this paper, long interfacial waves of finite amplitude in uniform basic flows are considered with the assumption that the aspect ratio between wavelength and water depth is small. A new model is derived using the velocities at arbitrary distances from the still water level as the velocity variables instead of the commonly used depth-averaged velocities. This significantly improves the dispersion properties and makes them applicable to a wider range of water depths. Since its derivation requires no assumption on wave amplitude, the model thus can be used to describe waves with arbitrary amplitude.

The role of Qiongzhou Strait in the seasonal variation of the South China Sea circulation

An analysis of the water level and current data taken in Qiongzhou Strait in the South China Sea (SCS) over the last 37 years (1963 to 1999) was made to examine the characteristics of tidal waves and residual flow through the strait and their roles in the seasonal variation of the SCS circulation. The observations reveal that Qiongzhou Strait is an area where opposing tidal waves interact and a source of water transport to the Gulf of Beibu (Gulf of Tonkin), SCS. A year-round westward mean flow with a maximum speed of 10-40 cm s(-1) is found in Qiongzhou Strait. This accounts for water transport of 0.2-0.4 Sv and 0.1-0.2 Sv into the Gulf of Beibu in winter-spring and summer-autumn, respectively. The outflow from Qiongzhou Strait may cause up to 44% of the gulf water to be refreshed each season, suggesting that it has a significant impact on the seasonal circulation in the Gulf of Beibu. This finding is in contrast to our current understanding that the seasonal circulation patterns in the South China Sea are primarily driven by seasonal winds. Several numerical experiments were conducted to examine the physical mechanisms responsible for the formation of the westward mean flow in Qiongzhou Strait. The model provides a reasonable simulation of semidiurnal and diurnal tidal waves in the strait and the predicted residual flow generally agrees with the observed mean flow. An analysis of the momentum equations indicates that the strong westward flow is driven mainly by tidal rectification over variable bottom topography. Both observations and modeling suggest that the coastal physical processes associated with tidal rectification and buoyancy input must be taken into account when the mass balance of the SCS circulation is investigated, especially for the regional circulation in the Gulf of Beibu.

Nonlinear Acoustics in Underwater and Biomedical Applications: Array Performance Degradation and Time Reversal Invariance

This dissertation describes a model for acoustic propagation in inhomogeneous flu- ids, and explores the focusing by arrays onto targets under various conditions. The work explores the use of arrays, in particular the time reversal array, for underwater and biomedical applications. Aspects of propagation and phasing which can lead to reduced focusing effectiveness are described. An acoustic wave equation was derived for the propagation of finite-amplitude waves in lossy time-varying inhomogeneous fluid media. The equation was solved numerically in both Cartesian and cylindrical geometries using the finite-difference time-domain (FDTD) method. It was found that time reversal arrays are sensitive to several debilitating factors. Focusing ability was determined to be adequate in the presence of temporal jitter in the time reversed signal only up to about one-sixth of a period. Thermoviscous absorption also had a debilitating effect on focal pressure for both linear and nonlinear propagation. It was also found that nonlinearity leads to degradation of focal pressure through amplification of the received signal at the array, and enhanced absorption in the shocked waveforms. This dissertation also examined the heating effects of focused ultrasound in a tissue-like medium. The application considered is therapeutic heating for hyperther- mia. The acoustic model and a thermal model for tissue were coupled to solve for transient and steady temperature profiles in tissue-like media. The Pennes bioheat equation was solved using the FDTD method to calculate the temperature fields in tissue-like media from focused acoustic sources. It was found that the temperature-dependence of the medium's background prop- erties can play an important role in the temperature predictions. Finite-amplitude effects contributed excess heat when source conditions were provided for nonlinear ef- fects to manifest themselves. The effect of medium heterogeneity was also found to be important in redistributing the acoustic and temperature fields, creating regions with hotter and colder temperatures than the mean by local scattering and lensing action. These temperature excursions from the mean were found to increase monotonically with increasing contrast in the medium's properties.

PROPAGATION OF SONIC BOOMS THROUGH A REAL, STRATIFIED ATMOSPHERE

Sonic boom propagation in a quiet) stratified) lossy atmosphere is the subject of this dissertation. Two questions are considered in detail: (1) Does waveform freezing occur? (2) Are sonic booms shocks in steady state? Both assumptions have been invoked in the past to predict sonic boom waveforms at the ground. A very general form of the Burgers equation is derived and used as the model for the problem. The derivation begins with the basic conservation equations. The effects of nonlinearity) attenuation and dispersion due to multiple relaxations) viscosity) and heat conduction) geometrical spreading) and stratification of the medium are included. When the absorption and dispersion terms are neglected) an analytical solution is available. The analytical solution is used to answer the first question. Geometrical spreading and stratification of the medium are found to slow down the nonlinear distortion of finite-amplitude waves. In certain cases the distortion reaches an absolute limit) a phenomenon called waveform freezing. Judging by the maturity of the distortion mechanism, sonic booms generated by aircraft at 18 km altitude are not frozen when they reach the ground. On the other hand, judging by the approach of the waveform to its asymptotic shape, N waves generated by aircraft at 18 km altitude are frozen when they reach the ground. To answer the second question we solve the full Burgers equation and for this purpose develop a new computer code, THOR. The code is based on an algorithm by Lee and Hamilton (J. Acoust. Soc. Am. 97, 906-917, 1995) and has the novel feature that all its calculations are done in the time domain, including absorption and dispersion. Results from the code compare very well with analytical solutions. In a NASA exercise to compare sonic boom computer programs, THOR gave results that agree well with those of other participants and ran faster. We show that sonic booms are not steady state waves because they travel through a varying medium, suffer spreading, and fail to approximate step shocks closely enough. Although developed to predict sonic boom propagation, THOR can solve other problems for which the extended Burgers equation is a good propagation model.

Broadband chaos generated by an optoelectronic oscillator.

We study an optoelectronic time-delay oscillator that displays high-speed chaotic behavior with a flat, broad power spectrum. The chaotic state coexists with a linearly stable fixed point, which, when subjected to a finite-amplitude perturbation, loses stability initially via a periodic train of ultrafast pulses. We derive approximate mappings that do an excellent job of capturing the observed instability. The oscillator provides a simple device for fundamental studies of time-delay dynamical systems and can be used as a building block for ultrawide-band sensor networks.

Dust ion acoustic solitons in a plasma with kappa-distributed electrons

Dust ion acoustic solitons in an unmagnetized dusty plasma comprising cold dust particles, adiabatic fluid ions, and electrons satisfying a kappa distribution are investigated using both small amplitude and arbitrary amplitude techniques. Their existence domain is discussed in the parameter space of Mach number M and electron density fraction f over a wide range of values of kappa. For all kappa > 3/2, including the Maxwellian distribution, negative dust supports solitons of both polarities over a range in f. In that region of parameter space solitary structures of finite amplitude can be obtained even at the lowest Mach number, the acoustic speed, for all kappa. These cannot be found from small amplitude theories. This surprising behavior is investigated, and it is shown that f(c), the value of f at which the KdV coefficient A vanishes, plays a critical role. In the presence of positive dust, only positive potential solitons are found. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3400229]

Studies On Mixed Layer Depth In The Arabian Sea

This thesis is the result of an elaborate study on the mixed layer depth (MLD) and the various oceanic environmental factors controlling it in the Arabian Sea examining its predictability on annual and short term basis. To accomplish this, the study area between 100 — 250 N latitudes and 600 — 750 E longitudes in the Arabian Sea is divided into 8 subareas of 50 quadrangles. The distribution of monthly means of the surface wind field, net heat exchange mKi868€%WTmN¥tWMWF3UH9 (SST) over each subarea in the annual cycle is examined. The corresponding wind (mechanical) and convective mixing values are computed and presented along with the observed mean MLD for the subareas in the annual cycle. Effects of advection due to surface currents and surface divergence (convergence and divergence) for these subareas are examined for correlating the MLD variations. A representative time series data from typical deep water station under southwest monsoonal forcing is analysed for the spectral components to estimate the amplitude perturbations on the mean MLD variation

Perturbation growth at the convective scale for CSIP IOP18

The Met Office Unified Model is run for a case observed during Intensive Observation Period 18 (IOP18) of the Convective Storms Initiation Project (CSIP). The aims are to identify the physical processes that lead to perturbation growth at the convective scale in response to model-state perturbations and to determine their sensitivity to the character of the perturbations. The case is strongly upper-level forced but with detailed mesoscale/convective-scale evolution that is dependent on smaller-scale processes. Potential temperature is perturbed within the boundary layer. The effects on perturbation growth of both the amplitude and typical scalelength of the perturbations are investigated and perturbations are applied either sequentially (every 30 min throughout the simulation) or at specific times. The direct effects (within one timestep) of the perturbations are to generate propagating Lamb and acoustic waves and produce generally small changes in cloud parameters and convective instability. In exceptional cases a perturbation at a specific gridpoint leads to switching of the diagnosed boundary-layer type or discontinuous changes in convective instability, through the generation or removal of a lid. The indirect effects (during the entire simulation) are changes in the intensity and location of precipitation and in the cloud size distribution. Qualitatively different behaviour is found for strong (1K amplitude) and weak (0.01K amplitude) perturbations, with faster growth after sunrise found only for the weaker perturbations. However, the overall perturbation growth (as measured by the root-mean-square error of accumulated precipitation) reaches similar values at saturation, regardless of the perturbation characterisation.

Orography in a contour dynamics model of large-scale atmospheric flow

The influence of orography on the structure of stationary planetary Rossby waves is studied in the context of a contour dynamics model of the large-scale atmospheric flow. Orography of infinitesimal and finite amplitude is studied using analytical and numerical techniques. Three different types of orography are considered: idealized orography in the form of a global wave, idealized orography in the form of a local table mountain, and the earth's orography. The study confirms the importance of resonances, both in the infinitesimal orography and in the finite orography cases. With finite orography the stationary waves organize themselves into a one-dimensional set of solutions, which due to the resonances, is piecewise connected. It is pointed out that these stationary waves could be relevant for atmospheric regimes.

Cloud-resolving simulations of deep convection over a heated mountain

Cloud-resolving numerical simulations of airflow over a diurnally heated mountain ridge are conducted to explore the mechanisms and sensitivities of convective initiation under high pressure conditions. The simulations are based on a well-observed convection event from the Convective and Orographically Induced Precipitation Study (COPS) during summer 2007, where an isolated afternoon thunderstorm developed over the Black Forest mountains of central Europe, but they are idealized to facilitate understanding and reduce computational expense. In the conditionally unstable but strongly inhibited flow under consideration, sharp horizontal convergence over the mountain acts to locally weaken the inhibition and moisten the dry midtroposphere through shallow cumulus detrainment. The onset of deep convection occurs not through the deep ascent of a single updraft but rather through a rapid succession of thermals that are vented through the mountain convergence zone into the deepening cloud mass. Emerging thermals rise through the saturated wakes of their predecessors, which diminishes the suppressive effects of entrainment and allows for rapid glaciation above the freezing level as supercooled cloud drops rime onto preexisting ice particles. These effects strongly enhance the midlevel cloud buoyancy and enable rapid ascent to the tropopause. The existence and vigor of the convection is highly sensitive to small changes in background wind speed U0, which controls the strength of the mountain convergence and the ability of midlevel moisture to accumulate above the mountain. Whereas vigorous deep convection develops for U0 = 0 m s−1, deep convection is completely eliminated for U0 = 3 m s−1. Although deep convection is able to develop under intermediate winds (U0 = 1.5 m s−1), its formation is highly sensitive to small-amplitude perturbations in the initial flow.

A linear model for the structure of turbulence beneath surface water waves

The structure of turbulence in the ocean surface layer is investigated using a simplified semi-analytical model based on rapid-distortion theory. In this model, which is linear with respect to the turbulence, the flow comprises a mean Eulerian shear current, the Stokes drift of an irrotational surface wave, which accounts for the irreversible effect of the waves on the turbulence, and the turbulence itself, whose time evolution is calculated. By analysing the equations of motion used in the model, which are linearised versions of the Craik–Leibovich equations containing a ‘vortex force’, it is found that a flow including mean shear and a Stokes drift is formally equivalent to a flow including mean shear and rotation. In particular, Craik and Leibovich’s condition for the linear instability of the first kind of flow is equivalent to Bradshaw’s condition for the linear instability of the second. However, the present study goes beyond linear stability analyses by considering flow disturbances of finite amplitude, which allows calculating turbulence statistics and addressing cases where the linear stability is neutral. Results from the model show that the turbulence displays a structure with a continuous variation of the anisotropy and elongation, ranging from streaky structures, for distortion by shear only, to streamwise vortices resembling Langmuir circulations, for distortion by Stokes drift only. The TKE grows faster for distortion by a shear and a Stokes drift gradient with the same sign (a situation relevant to wind waves), but the turbulence is more isotropic in that case (which is linearly unstable to Langmuir circulations).

Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow

Wave-activity conservation laws are key to understanding wave propagation in inhomogeneous environments. Their most general formulation follows from the Hamiltonian structure of geophysical fluid dynamics. For large-scale atmospheric dynamics, the Eliassen–Palm wave activity is a well-known example and is central to theoretical analysis. On the mesoscale, while such conservation laws have been worked out in two dimensions, their application to a horizontally homogeneous background flow in three dimensions fails because of a degeneracy created by the absence of a background potential vorticity gradient. Earlier three-dimensional results based on linear WKB theory considered only Doppler-shifted gravity waves, not waves in a stratified shear flow. Consideration of a background flow depending only on altitude is motivated by the parameterization of subgrid-scales in climate models where there is an imposed separation of horizontal length and time scales, but vertical coupling within each column. Here we show how this degeneracy can be overcome and wave-activity conservation laws derived for three-dimensional disturbances to a horizontally homogeneous background flow. Explicit expressions for pseudoenergy and pseudomomentum in the anelastic and Boussinesq models are derived, and it is shown how the previously derived relations for the two-dimensional problem can be treated as a limiting case of the three-dimensional problem. The results also generalize earlier three-dimensional results in that there is no slowly varying WKB-type requirement on the background flow, and the results are extendable to finite amplitude. The relationship A E =cA P between pseudoenergy A E and pseudomomentum A P, where c is the horizontal phase speed in the direction of symmetry associated with A P, has important applications to gravity-wave parameterization and provides a generalized statement of the first Eliassen–Palm theorem.

On the available energy of an axisymmetric vortex

A theory of available energy for axisymmetric circulations is presented. The theory is a generalization of the classical theory of available potential energy, in that it accounts for both thermal and angular momentum constraints on the circulation. The generalization relies on the Hamiltonian structure of the (conservative) dynamics, is exact at finite amplitude, and has a local form. Application of the theory is presented for the case of an axisymmetric vortex on an f -plane in the context of the Boussinesq equations.