Stochastic Sea Research Articles

The sea-wave elevation and multistability in a non-autonomous nonlinear Itô’s stochastic differential equation for the rolling angle of a ship

Prediction of the rolling behavior of ships in irregular sea remains one of the most difficult problems in ship engineering. The present work facilitates solution of this problem by derivation of a model which is meaningful from the subject-specific point of view and can efficiently be analyzed with the path-integration method. The model is a single Itô’s stochastic differential equation for the rolling angle of a ship located at a fixed spatial point. The equation appears to be of the third order and nonlinear. It takes into account the elevation of stochastic traveling sea waves. The stochasticity of the elevation is allowed for by stationary stochastic velocity of the waves. The works also notes the picture for the multistability of the derived model. Improvement of capabilities of the methods for multistable nonlinear systems is included in directions for future research.

Simulation-based robust design of mass dampers for response mitigation of tension leg platforms

A robust stochastic design framework is discussed for design of mass dampers. The focus is on applications for the mitigation of the coupled heave and pitch response of Tension Leg Platforms under stochastic sea excitation. The framework presented fully addresses the complex relationship between the coupled dynamics of the platform, the stochastic excitation and the vibration of the dampers. Model parameters that have some level of uncertainty are probabilistically described. In this probabilistic setting, the system reliability is adopted as the design objective. Stochastic simulation is considered for evaluation of the system model response and the overall reliability performance. This way, all nonlinear characteristics of the structural response and environmental excitation are explicitly incorporated into their respective models. An efficient algorithm is discussed for performing the challenging stochastic design optimization. The ideas are illustrated in an application involving a tension leg platform with closely spaced frequencies for the heave and pitch degrees of freedom.

Sea-swell interaction as a mechanism for the generation of freak waves

The probability of freak waves in an inhomogeneous ocean is studied by integration of Alber’s equation. The special phase structure of the inhomogeneous disturbance, required for instability, is provided by bound waves, generated by the quadratic interaction of the stochastic sea with a deterministic, long swell. The probability of freak waves higher than twice the significant wave height increases by a factor of up to 20 compared to the classical value given by Rayleigh’s distribution. The probability of exceptionally high freak waves, with height larger than three times the significant wave height, is shown to increase some 30 000-fold compared to that given by the Rayleigh distribution, which renders their encounter feasible.

Ship Motion Prediction for Maritime Flight Operations

This paper presents a novel and feasible prediction procedure for ship motion in the presence of uncertain tendency of ship motion dynamic variations and stochastic sea state disturbances. An appropriate model aiming to feature the characteristics of the dynamic relationship between an observer and a ship deck is constructed, from which an initial algorithm is implemented. The optimal system order based on Bayes Information Criterion (BIC) is derived, resulting in the development of an accurate adaptive multi-step predictor for estimation of ship motion dynamics. Simulation results demonstrate that the proposed prediction approach substantially reduces the model complexity and exhibits excellent prediction performance, making it suitable for integration into ship-helicopter approaches and landing guidance systems.

Chaotic mixing and transport in a meandering jet flow

Mixing and transport of passive particles are studied in a simple kinematic model of a meandering jet flow motivated by the problem of lateral mixing and transport in the Gulf Stream. We briefly discuss a model stream function, Hamiltonian advection equations, stationary points, and bifurcations. The phase portrait of the chosen model flow in the moving reference frame consists of a central eastward jet, chains of northern and southern circulations, and peripheral westward currents. Under a periodic perturbation of the meander's amplitude, the topology of the phase space is complicated by the presence of chaotic layers and chains of oscillatory and ballistic islands with sticky boundaries immersed into a stochastic sea. Typical chaotic trajectories of advected particles are shown to demonstrate a complicated behavior with long flights in both the directions of motion intermittent with trapping in the circulation cells being stuck to the boundaries of vortex cores and resonant islands. Transport is asymmetric in the sense that mixing between the circulations and the peripheral currents is, in general, different from mixing between the circulations and the jet. The transport properties are characterized by probability distribution functions (PDFs) of durations and lengths of flights. Both the PDFs exhibit at their tails power-law decay with different values of exponents.

Nonlinear coherent dynamics of an atom in an optical lattice

We consider a simple model of the lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. The internal dynamics of the atom is governed by the laser field, which is treated as classical with a large number of photons. The center-of-mass classical atomic motion is governed by the optical potential and the internal atomic degrees of freedom. The resulting Hamilton-Schro dinger equations of motion are a five-dimensional nonlinear dynamical system with two integrals of motion, and the total atomic energy and the Bloch vector length are conserved during the interaction. In our previous papers, the motion of the atom has been shown to be regular or chaotic (in the sense of exponential sensitivity to small variations of the initial conditions and/or the system’s control parameters) depending on the values of the control parameters, atom-field detuning, and recoil frequency. At the exact atom-field resonance, the exact solutions for both the external and internal atomic degrees of freedom can be derived. The center-of-mass motion does not depend in this case on the internal variables, whereas the Rabi oscillations of the atomic inversion is a frequency-modulated signal with the frequency defined by the atomic position in the optical lattice. We study analytically the correlations between the Rabi oscillations and the center-of-mass motion in two limiting cases of a regular motion out of the resonance: (1) far-detuned atoms and (2) rapidly moving atoms. This paper is concentrated on chaotic atomic motion that may be quantified strictly by positive values of the maximal Lyapunov exponent. It is shown that an atom, depending on the value of its total energy, can either oscillate chaotically in a well of the optical potential, or fly ballistically with weak chaotic oscillations of its momentum, or wander in the optical lattice, changing the direction of motion in a chaotic way. In the regime of chaotic wandering, the atomic motion is shown to have fractal properties. We find a useful tool to visualize complicated atomic motion-Poincare mapping of atomic trajectories in an effective three-dimensional phase space onto planes of atomic internal variables and momentum. The Poincare mappings are constructed using the translational invariance of the standing laser wave. We find common features with typical nonhyperbolic Hamiltonian systems-chains of resonant islands of different sizes imbedded in a stochastic sea, stochastic layers, bifurcations, and so on. The phenomenon of the atomic trajectories sticking to boundaries of regular islands, which should have a great influence on atomic transport in optical lattices, is found and demonstrated numerically.

Chaotic Advection and Targeted Mixing

The advection of passive tracers in an oscillating vortex chain is investigated. It is shown that by adding a suitable perturbation to the ideal flow, the induced chaotic advection exhibits two remarkable properties which do not hold in the case of a generic perturbation: Particles remain trapped within a specific domain bounded by two oscillating barriers (suppression of chaotic transport along the channel), and the stochastic sea seems to cover this whole bounded domain (enhancement of mixing within the rolls).

A stochastic sea: The source of plasma sheet boundary layer ion structures observed by Cluster

On 14 February 2001 the Cluster Ion Spectrometry (CIS) experiment onboard three of the Cluster spacecraft observed velocity‐dispersed ion structures (VDIS) as the spacecraft passed from the tail lobes into the plasma sheet boundary layer. These are the first multiple spacecraft observations of the VDIS phenomenon. The Cluster 1 spacecraft (SC1) observed a dispersed ion signature with beamlets and a second structure like that expected to be produced by an echo, while Cluster 3 (SC3) observed much less pronounced structuring a few minutes later. During this same event and over an extended interval the ACE spacecraft observed an interplanetary magnetic field that was directed southward. We have inferred the sources and acceleration mechanisms of the ions in these VDIS observations by following millions of ion trajectories backward and forward in time through time‐dependent electric and magnetic fields obtained from a global MHD simulation. ACE data were used as input for the MHD model. We found that almost all of the particles comprising the first (A1) and second (A2) beamlets observed by SC1 had been nonadiabatic earlier in their history, while particles in the A3 beamlet exhibited a combination of adiabatic and nonadiabatic behavior. Beamlet A4 particles were always adiabatic. Moreover, for all of the beamlets the current sheet crossing that took place prior to their detection occurred between x = −13 RE and x = −16 RE in the tail, well earthward of the permanent stochastic “sea” from which all of the beamlets originated. Our model does not favor the multiple source scenario suggested by A. Keiling et al. Instead, it indicates that the source regions of the structures are spatially correlated. We have carried out a similar analysis of the SC3 observations. In general, SC3 beamlets have higher κ values, partly because of the depolarization of the field lines during these observations. In time forward calculations only a small fraction of ions from SC1 A structures returned to the spacecraft location. “Echoes” were more pronounced on SC3. In addition, in our calculations, some particles from SC1 A structures interacted with the current sheet and returned to the SC3 location, at the time when SC3 observed the A structures. When Cluster observations were organized by latitude instead of time, we found that all three Cluster spacecraft seemed to observe the same primary structure that persisted throughout the interval of observation.

Polynomial dispersion of trajectories in sticky dynamics

Hamiltonian chaotic dynamics is, in general, not ergodic and the boundaries of the ergodic or quasiergodic area (stochastic sea, stochastic layers, stochastic webs, etc.) are sticky, i.e., trajectories can spend an arbitrarily long time in the vicinity of the boundaries with a nonexponentially small probability. Segments of trajectories imposed by the stickiness are called flights. The flights have polynomial dispersion that can lead to non-Gaussian statistics of displacements and to anomalous transport in phase space. In particular, the presence of flights influences the distribution of Poincaré recurrences. We use the distribution function of (l,t;epsilon, epsilon0) -separation of trajectories that at time instant t and trajectory length l are separated for the first time by epsilon<<1, being initially at a distance epsilon0 <<epsilon. The connection of this function, called the complexity function [Afraimovich and Zaslavsky, Chaos 13, 519 (2003)], with the distribution of Poincaré recurrences is established for three cases: (i) for the case of superdiffusion in standard and web maps for which the stickiness is defined by the boundaries of hierarchical sets of islands; (ii) for the case of the Sinai billiard with infinite horizon, where the stickiness is defined by zero-measure slits in the phase space; (iii) for the square billiard with a slit (bar-in-square billiard) where the Lyapunov exponent is zero and the stickiness is defined by the vicinity of the trajectory to the closest periodic trajectories obtained from the Diophantine approximation. Finally, the powerwise asymptotics of the Poincaré recurrences can be connected, in some cases, to the anomalous transport exponent.

Calculation of a capsizing rate of a ship in stochastic beam seas

This study introduces a method of calculating a capsizing rate of a ship. The phenomenon ‘capsizing’ is described as a jump of local equilibrium point from that near the upright position of a ship to what describes the upside-down attitude of the capsized ship; the rate of occurrence of such jumps was calculated. The potential function corresponding to the roll restoring moment have two potential wells located at the roll displacement angle 0 and 180°, respectively. A nonlinear Fokker–Planck equation for the joint probability density function of roll angle and velocity was solved. The excitation to the ship was assumed to be a combination of a regular harmonic wave and a white noise process.

Stochastic ion heating at the magnetopause due to kinetic Alfvén waves

The magnetopause and boundary layer are typically characterized by large amplitude transverse wave activity with frequencies below the ion cyclotron frequency. The signatures of the transverse waves suggest that they are kinetic Alfvén waves with wavelength on the order of the ion gyroradius [Johnson and Cheng, 1997a; Johnson et al., 2001]. We investigate ion motion in the presence of large amplitude kinetic Alfvén waves with wavelength the order of ρi and demonstrate that for sufficiently large wave amplitude (δB⟂/B0 &gt; 0.05) the particle orbits become stochastic. As a result, low energy particles in the core of the ion distribution can migrate to higher energy through the stochastic sea leading to an increase in T⟂ and a broadening of the distribution. This process can explain transverse ion energization and formation of conies which have been observed near the magnetopause in both the magnetosheath and low‐latitude boundary layer.

Chaos in dissipative relativistic standard maps

The relativistic generalization of the dissipative standard map is introduced, based on the problem of acceleration and heating (or cooling) of charged particles in the electric field of an electromagnetic wave packet. The question arises as to how the relativistic effects change the nonlinear dynamics described by a dissipative standard map. It is shown that the dissipation modifies the positions of the fixed points, but the origin (the central point) remains identical with that of the corresponding Hamiltonian system. However, the phase-space structure around the origin is drastically modified even if a small dissipation is present. The formation of an “ordered” stochastic structure which is not washed out (in the stochastic sea) for longer times shows that the phase mixing is weak and the nonuniformity of the stochastic acceleration increases because of the dissipation. A new type of stochastic attractor of a higher order is found by numerical simulations. In the context of a scaling-law hypothesis (or renormalization group approach), the transition stochastic sea (high acceleration of relativistic particles)–stochastic attractor (low acceleration) is similar to a Bose–Einstein condensation (or, simply, a condensation gas–liquid) at low temperatures, the dissipative parameter being the control parameter for such a transition. The dissipation parameter can also be considered as a time (aging) parameter of the system, and this may have some applications in biological systems. A Frenkel–Kontorova model of the dissipative relativistic standard map (DRSM) and possible applications to “incommensurate fractals” and lattice dynamics of thermoelectric materials are also considered.

Chaotic advection near a three-vortex collapse.

Dynamical and statistical properties of tracer advection are studied in a family of flows produced by three point-vortices of different signs. Tracer dynamics is analyzed by numerical construction of Poincaré sections, and is found to be strongly chaotic: advection pattern in the region around the center of vorticity is dominated by a well developed stochastic sea, which grows as the vortex system's initial conditions are set closer to those leading to the collapse of the vortices; at the same time, the islands of regular motion around vortices, known as vortex cores, shrink. An estimation of the core's radii from the minimum distance of vortex approach to each other is obtained. Tracer transport was found to be anomalous: for all of the three numerically investigated cases, the variance of the tracer distribution grows faster than a linear function of time, corresponding to a superdiffusive regime. The transport exponent varies with time decades, implying the presence of multi-fractal transport features. Yet, its value is never too far from 3/2, indicating some kind of universality. Statistics of Poincaré recurrences is non-Poissonian: distributions have long power-law tails. The anomalous properties of tracer statistics are the result of the complex structure of the advection phase space, in particular, of strong stickiness on the boundaries between the regions of chaotic and regular motion. The role of the different phase space structures involved in this phenomenon is analyzed. Based on this analysis, a kinetic description is constructed, which takes into account different time and space scalings by using a fractional equation.

Relation between stable orbits and quantum transmission resonance in ballistic cavities

Classical and quantum-mechanical transport properties in chaotic cavities are investigated to establish a link between them. Because of the stickiness at the boundary between stochastic seas and islands of regular orbits in phase space, classical trajectories spend a long time in the vicinity of a few regular orbits. The trapping results in an exclusive excitation of these stable orbits even when the cavity is terminated by classical leads. The wave-function pattern at quantum transmission resonances is found to be identical with one of the stable orbits. The correspondence implies that the transmission resonance takes place when the stable orbit satisfies Bohr and Sommerfeld's quantization rule, and hence explains why conductance fluctuations in ballistic cavities contain only several frequency components.

Semi-classical orbits in a caterpillar like Sinaı̈ billiard

The transport properties of a caterpillar-like Sinai billiard are investigated in the ballistic regime. The experimental kinetic coefficients exhibit pronounced features, characteristic of this kind of billiard. A Monte-Carlo style simulation of the semi-classical electron motion is performed in order to model the transmission probabilities Tij of an ideal caterpillar structure. By visualising the spatial charge density distributions, the main features in the transmission probabilities are related to special electron orbits which dominate the electron dynamics at a given B. Finally, Poincaré sections are calculated close to the commensurability condition. The appearance of stability islands surrounded by the stochastic sea characterises the presence of regular electron skipping orbits involved in the ballistic transport phenomena.

Hierarchical structures in the phase space and fractional kinetics: I. Classical systems.

Hamiltonian chaotic dynamics is not ergodic due to the infinite number of islands imbedded in the stochastic sea. Stickiness of the islands' boundaries makes the wandering process very erratic with multifractal space-time structure. This complication of the chaotic process can be described on the basis of fractional kinetics. Anomalous properties of the chaotic transport become more transparent when there exists a set of islands with a hierarchical structure. Different consequences of the described phenomenon are discussed: a distribution of Poincare recurrences, characteristic exponents of transport, nonuniversality of transport, log periodicity, and chaos erasing. (c) 2000 American Institute of Physics.

Transport Properties of a Classical One-Dimensional Kicked Billiard Model

We study a classical 1-dimensional kicked billiard model and investigate its transport behavior. The roles played by the two system parameters α and K, governing the direction and strength of the kick, respectively, are found to be quite crucial. For the perturbations which are not strong, i.e. K < 1, we find that as the phase parameter α changes within its range of interest from -π/2 to π/2, the phase space is in turn characterized by the structure of a prevalently connected stochastic web (-π/2 ⩽ α < 0), local stochastic webs surrounded by a stochastic sea (0 < α < π/2) and the global stochastic sea (α = π/2). Extensive numerical investigations also indicate that the system's transport behavior in the irregular regions of the phase space for K < 1 has a dependence on the system parameters and the transport coefficient D can be expressed as D ≈ D0(α)Kf(α). For strong kicks, i.e. K ≫ 1, the phase space is occupied by the stochastic sea, and the transport behavior of the system seems to be similar to that of the kicked rotor and independent of α.

Explorations in monetary cliometrics: the Reichsbank: 1876-1920

The seasonal unit root tests make it possible to determine the nature of the deterministic and stochastic sea sonal fluctuations. In this paper, we apply this method to the original monthly series of the Reichsbank monetary stock (constructed in weekly data with 2160 observations) and emphasize deterministic seasonal fluctuations with notably a strong seasonality at the beginning and at the end of the year. This statistical result is closely related to the turning points detected by the historical analysis.

Monte Carlo simulations of the influence of particle nonsphericity on remote sensing of ocean water

The paper reports Monte Carlo modeling of backscatter returns of a laser beam from ocean water. The Monte Carlo code used for simulations employs the Stokes vector formalism to account for polarization effects at each scattering event, from either water or suspended particles, and at reflection from or transmission through a stochastic sea surface. Scattering from water and suspended matter are considered separately. The Rayleigh‐scattering phase function and Mueller matrix are used for modeling scattering from water. Several forms of phase function and Mueller matrices are used to model scattering from the particles. Ensembles of spheres and spheroids with effectively the same size distribution are considered. Mueller matrices and phase functions computed by the T‐matrix method for spheroids and Mie theory for spheres are used in the Monte Carlo simulations. The polarization state of the returns are compared for two cases of water containing spherical or spheroidal scattering particles. These are further compared with results derived from experimental data of Voss and Fry for the Mueller matrix of ocean water, which has a form characteristic of nonspherical particles, in combination with the Henyey‐Greenstein phase function.

The limiting wave height in wind-induced wave trains

The vertical acceleration threshold concept has been applied to evaluate the limiting wave height in the train of wind-induced waves propagated over a horizontal bottom. This concept yields very simple computation of the probability of breaking for stochastic sea in deep and finite water depths. The computations confirmed the available field and laboratory observations that the limiting wave steepness in the deep water is lower than the steepness predicted by Stokes. For shallow water depth, the limiting wave height is smaller than 0.55 h. This conclusion is consistent with field as well as wave tank observations.