Mathematical modelling of nonlinear internal waves in a rotating fluid

Large amplitude internal solitary waves in the coastal ocean are commonly modelled with the Korteweg-de Vries (KdV) equation or a closely related evolution equation. The characteristic feature of these models is the solitary wave solution, and it is well documented that these provide the basic paradigm for the interpretation of oceanic observations. However, often internal waves in the ocean survive for several inertial periods, and in that case, the KdV equation is supplemented with a linear non-local term representing the effects of background rotation, commonly called the Ostrovsky equation. This equation does not support solitary wave solutions, and instead a solitary-like initial condition collapses due to radiation of inertia-gravity waves, with instead the long-time outcome typically being an unsteady nonlinear wave packet. The KdV equation and the Ostrovsky equation are formulated on the assumption that only a single vertical mode is used. In this thesis we consider the situation when two vertical modes are used, due to a near-resonance between their respective linear long wave phase speeds. This phenomenon can be described by a pair of coupled Ostrovsky equations, which is derived asymptotically from the full set of Euler equations and solved numerically using a pseudo-spectral method. The derivation of a system of coupled Ostrovsky equations is an important extension of coupled KdV equations on the one hand, and a single Ostrovsky equation on the other hand. The analytic structure and dynamical behaviour of the system have been elucidated in two main cases. The first case is when there is no background shear flow, while the second case is when the background state contains current shear, and both cases lead to new solution types with rich dynamical behaviour. We demonstrate that solitary-like initial conditions typically collapse into two unsteady nonlinear wave packets, propagating with distinct speeds corresponding to the extremum value in the group velocities. However, a background shear flow allows for several types of dynamical behaviour, supporting both unsteady and steady nonlinear wave packets, propagating with the speeds which can be predicted from the linear dispersion relation. In addition, in some cases secondary wave packets are formed associated with certain resonances which also can be identified from the linear dispersion relation. Finally, as a by-product of this study it was shown that a background shear flow can lead to the anomalous version of the single Ostrovsky equation, which supports a steady wave packet

Dynamic modelling for assessing the impact of marine debris on the population of Sea Turtles

Marine debris has significant impacts on marine animals including the sea turtles, which are particularly vulnerable to the presence of waste in the marine environment. We propose a novel mathematical model with three compartments to examine this effect: the sea turtle population, the concentration level of pollution inside sea turtles’ bodies, and the concentration level of pollution in marine environment. We locate the equilibrium points (also known as equilibria) for the suggested model and perform an analytical check on their stability. We also use the bifurcation analysis to examine how changing a model parameter affects the stability of the model's equilibria. Our findings demonstrated the existence of two equilibria: the sea turtles' survival equilibrium and their extinction equilibrium. The eigenvalues of the Jacobian matrix applied to the proposed model have been used to demonstrate the conditions for stability of these equilibria. The resulting bifurcation diagram demonstrates that both equilibrium points undergo transcritical bifurcations when the values of response intensity of toxicity parameter is varied. The findings of this study can help local or national governments make decisions and educate the public about sea turtle conservation in order to sustain sea turtle populations in the future

Temperature control of vacuum dividing wall column – case study on oleochemical fatty acid fractionation

Analysis of oleochemical compositions in distillation column often have large process delays. Inferential control is commonly used by means of stage temperature as the measured variable which provide more responsive in composition control. This work aims to evaluate the performance of temperature control in vacuum dividing wall column (VDWC) for fatty acid oleochemical fractionation. Product purity at 99% used as inferred parameter to determine the temperature. Sensitivity analysis was used to determine the relationship between stage and temperature difference for changes in the manipulated variables. The most sensitive tray was selected and implemented to a Distillate- Side Stream- Boilup (DSV) control configuration in Aspen Dynamics following the work by Othman (2019b). Controller adopted with PI and PID settings using Ziegler-Nichols (ZN) and Internal Model Control (IMC) tuning calculation method. Both methods were compared based on the settling time and overshoot. The best setting was then fine-tuned before tested to set point tracking without any disturbances. From the sensitive analysis, temperature at stage 6, 29 and 34 were selected used as controlled variable which inferred distillate, middle and bottom product purity at 99% respectively. PID controller setting based on ZN method provide the best setup with fastest settling time and smallest overshoot and provide good performance for set point tracking

Formation of wave packets in the Ostrovsky equation for both normal and anomalous dispersion

It is well-known that the Ostrovsky equation with normal dispersion does not support steady solitary waves. An initial Korteweg-de Vries solitary wave decays adiabatically through the radiation of long waves and is eventually replaced by an envelope solitary wave whose carrier wave and envelope move with different velocities (phase and group velocities correspondingly). Here we examine the same issue for the Ostrovsky equation with anomalous dispersion when the wave frequency increases with wavenumber in the limit of very short waves. The essential difference is that now there exists a steady solitary wave solution (Ostrovsky soliton), which in the small-amplitude limit can be described asymptotically through the solitary wave solution of a nonlinear Schrodinger equation, based at that wavenumber where the phase and group velocities coincide. Longtime numerical simulations show that the emergence of this steady envelope solitary wave is a very robust feature. The initial Korteweg-de Vries solitary wave transforms rapidly to this envelope solitary wave in a seemingly non-adiabatic manner. The amplitude of the Ostrovsky soliton strongly correlates with the initial Korteweg-de Vries solitary wave

Adaptation Of Residual Power Series Approach For Solving Time-Fractional Nonlinear Kline-Gordon Equations With Conformable Derivative

In this paper, the time-fractional nonlinear Kline-Gordon equations are considered and solved using the adaptive of residual power series method. The fractional derivative is considered in a conformable sense. Analytical solutions are obtained based on conformable Taylor series expansion by substituting the truncated conformable series solutions to residual error functions. This adaptation can be implemented as a novel alternative technique to handle many nonlinear issues occurring in physics and engineering. Effectiveness, validity, and feasibility of the proposed method are demonstrated by testing some numerical applications. Tabular and graphic results indicate that the method is superior, accurate and appropriate for solving these fractional partial differential models with compatible derivatives

The Propagation of Nonlinear Internal Waves under the Influence of Variable Topography and Earth’s Rotation in a Two-Layer Fluid

A nonlinear equation of the Korteweg–de Vries equation usually describes internal solitary waves in the coastal ocean that lead to an exact solitary wave solution. However, in any real application, there exists the Earth’s rotation. Thus, an additional term is required, and consequently, the Ostrovsky equation is developed. This additional term is believed to destroy the solitary wave solution and form a nonlinear envelope wave packet instead. In addition, an internal solitary wave is commonly disseminated over the variable topography in the ocean. Because of these effects, the Ostrovsky equation is retrieved by a variable-coefficient Ostrovsky equation. In this study, the combined effects of both background rotation and variable topography on a solitary wave in a two-layer fluid is studied since internal waves typically happen here. A numerical simulation for the variable-coefficient Ostrovsky equation with a variable topography is presented. Two basic examples of the depth profile are considered in detail and sustained by numerical results. The first one is the constant-slope bottom, and the second one is the specific bottom profile following the previous studies. These indicate that the combination of variable topography and rotation induces a secondary trailing wave packet