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This dissertation includes two parts (designated A and B) that serve two separate but related research purposes. In Part A, coagulation rates between fractal aggregates (200-1000 μm) and small (1.48 μm) particles were studied for collisions induced by differential sedimentation and turbulent fluid shear. The collision frequency functions (beta) between these aggregates and small particles were found to be lower than predicted by a rectilinear collision model but much higher than predicted by a curvilinear collision model for equivalent impermeable spheres. The collision frequencies decreased with the magnitude of aggregate fractal dimensions (D). Based on fractal geometry of aggregates and the comparisons between observed settling velocities and those calculated using Stokes' law, a semi-empirical correlation was derived to describe the permeabilities of settling fractal aggregates. A filtration model was used in conjunction with this fractal permeability correlation to predict capture rates and capture efficiencies of small particles by settling fractal aggregates. In the turbulently sheared fluid, it was demonstrated that the importance of the shear rate (G) on enhancing collision frequencies was dependent on the fractal dimension of aggregates. As D approaches 3, beta became less sensitive to G as predicted by a curvilinear model. It was argued that flow through large pores formed between clusters within fractal aggregates contributed to high aggregate permeabilities and enhanced the coagulation between the aggregates and suspended small particles. In part B, fractal properties of microscopic particles (300 μm) occurring in marine systems were investigated. A new method, called the particle concentration technique (PCT), was developed to calculate the average fractal dimension of all particles in a certain size range by the analysis of particle size distributions in terms of both solid volume and length. During a simulated algae bloom in a mesocosm, as coagulation proceeded the average fractal dimension decreased with time from D = 2.52 to D = 1.68, a value typical of larger marine snow aggregates. Investigations in three eastern Pacific coastal areas suggested that the average fractal dimension indicated the importance of coagulation in determining local particle size distributions. The magnitude of the fractal dimension is likely associated with other factors, such as transparent exopolymer particles (TEP), affecting the coagulation rate of algae during a bloom in seawater.

This is a collection of essays on weakly and strongly nonlinear systems and possible ways of solving/interpreting them. Firstly, we study sand patterns which are often observed on sea (river) beds. One of the most common features looks like straight rolls perpendicular to the water motion. In many cases, the straight rolls are superimposed on a much longer wave so that two vastly different length scales coexist. In general, there are at least two mechanisms responsible for the growth of periodic sand waves. One is linear instability, and the other is nonlinear coupling between long waves and short waves. One novel feature of this work is to suggest that the latter can be much more important than the former one for the generation of long waves. A weakly nonlinear analysis of the corresponding physical system suggests that the nonlinear coupling leads to the growth of the longer features if the amplitude of the shorter waves has a non-zero curvature. For the case of a straight channel and a tidal shallow sea, we derive nonlinear amplitude equations governing the dynamics of the main features. Estimates based on these equations are consistent with measurements. Secondly, we consider strongly nonlinear systems with randomness. The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, nonlinearity and dispersion. The problem is then shown to be isomorphic to a problem of the nonlinear Schroedinger (NLS) type with a random (in space) potential. It is proven that the SIT result continues to hold when the uniform medium of inhomogeneously broadened two-level atoms is replaced by a series of intervals in each of which the frequency mismatch is randomly chosen from some distribution. The exact solution of this problem suggests that nonlinearity can improve the transparency of the medium. Also, the small amplitude, almost monochromatic limit of SIT is taken and results in an envelope equation which is an exactly integrable combination of NLS and a modified SIT equation. Some generalizations are made to describe a broad class of integrable systems which combine randomness, nonlinearity and dispersion.

The highly heterogeneous and anelastic nature of deep ocean seafloor results in complex reverberation as acoustic energy incident from the overlaying water column interacts and scatters from it. To gain a deeper understanding of the mechanisms causing the reverberation in sonar and seafloor scattering experiments, we have developed numerical simulation techniques that are capable of modeling the principal physical properties of complex seafloor structures. A new viscoelastic finite-difference technique for modeling anelastic wave propagation in 2-D and 3-D heterogeneous media, as well as a computationally optimally efficient method for quantifying the anelastic properties in terms of viscoelastic mechanics are presented. A method for reducing numerical dispersion using a Galerkin-wavelet formulation that enables large computational savings is also presented. The widely different regimes of wave propagation occurring in ocean acoustic problems motivate the use of hybrid simulation techniques. HARVEST (Hybrid Adaptive Regime Visco-Elastic Simulation Technique) combines solutions from Gaussian beams, viscoelastic finite-differences, and Kirchhoff extrapolation, to simulate large offset scattering problems. Several scattering hypotheses based on finite-difference simulations of short-range acoustic scattering from realistic seafloor models are presented. Anelastic sediments on the seafloor are found to have a significant impact on the backscattered field from low grazing angle scattering experiments. In addition, small perturbations in the sediment compressional velocity can also dramatically alter the backscattered field due to transitions between pre- and post-critical reflection regimes. The hybrid techniques are employed to simulate deep ocean acoustic reverberation data collected in the vicinity of the northern mid-Atlantic ridge. In general, the simulated data compare well to the real data. Noise partly due to side-lobes in the beam-pattern of the receiver-array is the principal source of reverberation at lower levels. Overall, the employed seafloor models were found to model the real seafloor well. Inaccurately predicted events may partly be attributed to the intrinsic uncertainty in the stochastic seafloor models. For optimal comparison between real and HARVEST simulated data the experimental geometry should be chosen so that 3-D effects may be ignored, and to yield a cross-range resolution in the beam-formed acoustic data that is small relative to the lineation of the seafloor.

In recent years, there has been much interest in the numerical solution of shallow water equations. The numerical procedure used to solve the shallow water equations must resolve the physics of the problem without introducing spurious oscillations or excessive numerical diffusion. Staggered-grid finite difference methods have been used extensively in modeling surface flow without introducing spurious oscillations. Finite element methods, permitting a high degree of grid flexibility for complex geometries and facilitating grid refinement near land boundaries to resolve important processes, have become much more prevalent. However, early finite element simulations of shallow water systems were plagued by spurious oscillations and the various methods introduced to eliminate these oscillations through artificial diffusion were generally unsuccessful due to excessive damping of physical components of the solution. Here, we give a brief overview on some finite element models of the shallow water equations, with particular attention given to the wave and characteristic formulations. In the literature, standard analysis, based on Fourier decompositions of these methods, has always neglected contributions from the nonlinear terms. We derive ${\cal L}\sp{\infty} ((0,T); {\cal L}\sp2(\Omega))$ and ${\cal L}\sp2((0,T); {\cal H}\sp1(\Omega))$ a priori error estimates for both the continuous-time and discrete-time Galerkin approximation to the nonlinear wave model, finding these to be optimal in ${\cal H}\sp1(\Omega).$ Finally, we derive ${\cal L}\sp{\infty}((0,T); {\cal L}\sp2(\Omega))$ and ${\cal L}\sp2((0,T); {\cal H}\sp1(\Omega))$ a priori error estimates for our proposed Characteristic-Galerkin approximation to the nonlinear primitive model. We find these estimates to be optimal in ${\cal H}\sp1(\Omega)$ but with less restrictive time-step constraints when compared to the Galerkin estimates for the wave model.

The dynamics of offshore structures subjected to random waves is investigated. The load modeling and response analysis are conducted by using a new approach. The problems of the wave kinematics simulation and of the dynamic response determination are treated in an unified approach using filter techniques and the state space analysis. Some interesting results on water depth characterization in connection with the two parameter spectra are derived. These permit the application of the linear wave theory from monochromatic waves to random seas. Throughout this investigation filters which can generate the relevant wave kinematics or represent the random wave forces are sought. This is achieved by relying on developments in the field of signal processing and filter design. Novel platform concepts which hold promise for offshore oil production in deep waters are adopted for the dynamic analysis. In this context, attention is focussed on the efficacy of the method of state space analysis in the time domain, using the well known Liapunov equation. The advantages of this technique, particularly when motion control is of interest, is emphasized. The dynamic response of an idealized guyed tower is obtained in a closed form by using the Liapunov equation. A complete numerical example is included for the guyed tower. An alternate solution in the frequency domain is also obtained for comparison. Finally, the applicability of the state space method for modeling the wave excitation on an idealized 3 degree of freedom tension leg platform is investigated. The tension leg platform dynamics is examined conceptually, and further research recommended. Some key problems in the use of this powerful method are identified.

The exploration and production of oil at offshore locations is extending into ever increasing water depths. To exploit this resource in water depths in excess of 300 meters, a new class of offshore structures called compliant platforms are being developed. The compliant nature of these platforms introduces nonlinear behavior which can not be neglected as in conventional offshore platforms. Consequently, new analytical methods to estimate response statistics are needed. Stochastic linearization is perhaps the most frequently used analytical approximation for analyzing the response of nonlinear systems. Linearization, however, can not predict responses at frequencies outside the excitation frequencies. Therefore, some response statistics may be significantly unconservative. In addition for a gaussian excitation, the linearized solution leads to a gaussian probability distribution, whereas the true response is non-gaussian. In this study, a higher order method termed equivalent stochastic "quadratization" is proposed to circumvent these shortcomings. The nonlinearity is replaced by a polynomial expansion up to quadratic order. The Volterra series method is used to approximate the response of the resulting nonlinear system. The excitation is assumed to be gaussian, however, the response is described by a nongaussian probability distribution. The method is developed for analyzing the stationary response of single and multi-degree-of-freedom systems. A useful application of the proposed method is for analyzing the stochastic response of compliant offshore platforms due to nonlinear drag forces. The method is applied for analyzing a three-degree-of-freedom model of a Tension Leg Platform (TLP) subject to wave and current forces. The proposed method predicts the low frequency response induced by the drag force at the surge natural frequency which linearization can not account for. In addition to nonlinear drag forces, nonlinear potential forces significantly affect the TLP response. These forces are derived in the form of second order Volterra series. A stochastic response analysis of the TLP system due to combined nonlinear drag and nonlinear potential forces is performed to evaluate the relative significance of these forces. The focus of the study is on the nonlinear low frequency and high frequency responses.

Drilling and cutting operations are required in many industrial applications. For example an advanced flexible manufacturing system using robotics technologies may be required to perform such operations. Variation in the mechanical properties of a material to be processed, and compliance add complexity to the already nonlinear robotic system. Thus, these tasks are difficult to perform. For cutting and drilling processes, it is often desirable to maintain a constant cutting force to maintain productivity and also to maintain the quality of the finished workpiece. To counter the effect of the material variations while maintaining a constant cutting force, a controller mechanism is necessary which will react to these variations, and adjust the robot controller's parameters to minimize the effects of the variations on the system's performance. / A single-input/single-output (SISO) model-reference adaptive control (MRAC) scheme to cope with varying material hardness, sensor compliance, and non-rigid body dynamics in the control of cutting forces is presented. The controller is formulated to maintain stable, damped, force control when rigid body and rigid contact assumptions are not valid, and when material hardness is variable, in discrete-time and Cartesian-space. It is suited to super-position on a Cartesian-space hybrid control scheme, and implementation on a multiprocessor control system.