Nonlinear Analysis of Tropical Waves and Cyclogenesis Excited by Pressure Disturbance in Atmosphere

The horizontal equations of motion for an inviscid homogeneous fluid under the influence of pressure disturbance and waves are applied to investigate the nonlinear process of solitary waves and cyclone genesis forced by a moving pressure disturbance in atmosphere. Based on the reductive perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies the Korteweg–de Vries equation with a forcing term (fKdV equation for short), which describes the physics of a shallow layer of fluid subject to external pressure forcing. Then, with the help of Hirota’s direct method, the analytic solutions of the fKdV equation are studied and some exact vortex solutions are given as examples, from which one can see that the solitary waves and vortex multi-pole structures can be excited by external pressure forcing in atmosphere, such as pressure perturbation and waves. It is worthy to point out that cyclone and waves can be excited by different type of moving atmospheric pressure forcing source.

Exact Solutions to Time-Fractional Fifth Order KdV Equation by Trial Equation Method Based on Symmetry

We study a fifth order time-fractional KdV equation (FKdV) under meaning of the conformal fractional derivative. By trial equation method based on symmetry, we construct the abundant exact traveling wave solutions to the FKdV equation. These solutions show rich evolution patterns including solitons, rational singular solutions, periodic and double periodic solutions and so forth. In particular, under the concrete parameters, we give the representations of all these solutions.

On the critical free-surface flow over localised topography

Flow over bottom topography at critical Froude number is examined with a focus on steady, forced solitary wave solutions with algebraic decay in the far field, and their stability. Using the forced Korteweg–de Vries (fKdV) equation the weakly nonlinear steady solution space is examined in detail for the particular case of a Gaussian dip using a combination of asymptotic analysis and numerical computations. Non-uniqueness is established and a seemingly infinite set of steady solutions is uncovered. Non-uniqueness is also demonstrated for the fully nonlinear problem via boundary-integral calculations. It is shown analytically that critical flow solutions have algebraic decay in the far field both for the fKdV equation and for the fully nonlinear problem and, moreover, that the leading-order form of the decay is the same in both cases. The linear stability of the steady fKdV solutions is examined via eigenvalue computations and by a numerical study of the initial value fKdV problem. It is shown that there exists a linearly stable steady solution in which the deflection from the otherwise uniform surface level is everywhere negative.

ANALYTICAL APPROXIMATE SOLUTION FOR THE FORCED KORTEWEG-DE VRIES (FKDV) ON CRITICAL FLOW OVER A HOLE USING HOMOTOPY ANALYSIS METHOD

Free surface flows in a two-dimensional channel past over a hole is studied using shallow water forced Korteweg-de Vries (fKdV) equation. The forcing term of fKdV equation represents the hole shaped bottom topography. Froude number (Fr), which represents the ratio of flow speed to the wave speed, will also be used in solving fKdV equation. The fKdV equation is solved using Homotopy Analysis Method (HAM). HAM is an approximate analytical technique used to obtain series of solutions for the nonlinear problems where HAM has an auxiliary parameter coto adjust and control the convergence region of the series solution. Solitary wave solutions are obtained from the series of solutions of HAM and wave flows are observed at particular time. The HAM solution shows the hole shaped bottom topography plays an important role in determining the evolution of solitary waves.

Approximate Analytical Solution for the Forced Korteweg-de Vries Equation

The forced Korteweg-de Vries (fKdV) equations are solved using Homotopy Analysis Method (HAM). HAM is an approximate analytical technique which provides a novel way to obtain series solutions of such nonlinear problems. It has the auxiliary parameterℏ, where it is easy to adjust and control the convergence region of the series solution. Some examples of forcing terms are employed to analyse the behaviours of the HAM solutions for the different fKdV equations. Finally, this form of HAM solution is compared with the analytical soliton-type solution of fKdV equation as derived by Zhao and Guo. The results is found to be in good agreement with Zhao and Guo.

Subcritical, transcritical and supercritical flows over a step

Free-surface flow over a bottom topography with an asymptotic depth change (a ‘step’) is considered for different ranges of Froude numbers varying from subcritical, transcritical, to supercritical. For the subcritical case, a linear model indicates that a train of transient waves propagates upstream and eventually alters the conditions there. This leading-order upstream influence is shown to have profound effects on higher-order perturbation models as well as on the Froude number which has been conventionally defined in terms of the steady-state upstream depth. For the transcritical case, a forced Korteweg–de Vries (fKdV) equation is derived, and the numerical solution of this equation reveals a surprisingly conspicuous distinction between positive and negative forcings. It is shown that for a negative forcing, there exists a physically realistic nonlinear steady state and our preliminary results indicate that this steady state is very likely to be stable. Clearly in contrast to previous findings associated with other types of forcings, such a steady state in the transcritical regime has never been reported before. For transcritical flows with Froude number less than one, the upstream influence discovered for the subcritical case reappears.

Generation of upstream advancing solitons by moving disturbances

This study investigates the recently identified phenomenon whereby a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, periodically, a succession of solitary waves, advancing upstream of the disturbance in procession, while a train of weakly nonlinear and weakly dispersive waves develops downstream of a region of depressed water surface trailing just behind the disturbance. This phenomenon was numerically discovered by Wu & Wu (1982) based on the generalized Boussinesq model for describing two-dimensional long waves generated by moving surface pressure or topography. In a joint theoretical and experimental study, Lee (1985) found a broad agreement between the experiment and two theoretical models, the generalized Boussinesq and the forced Korteweg-de Vries (fKdV) equations, both containing forcing functions. The fKdV model is applied in the present study to explore the basic mechanism underlying the phenomenon.To facilitate the analysis of the stability of solutions of the initial-boundary-value problem of the fKdV equation, a family of forced steady solitary waves is found. Any such solution, if once established, will remain permanent in form in accordance with the uniqueness theorem shown here. One of the simplest of the stationary solutions, which is a one-parameter family and can be scaled into a universal similarity form, is chosen for stability calculations. As a test of the computer code, the initially established stationary solution is found to be numerically permanent in form with fractional uncertainties of less than 2% after the wave has traversed, under forcing, the distance of 600 water depths. The other numerical results show that when the wave is initially so disturbed as to have to rise from the rest state, which is taken as the initial value, the same phenomenon of the generation of upstream-advancing solitons is found to appear, with a definite time period of generation. The result for this similarity family shows that the period of generation, Ts, and the scaled amplitude α of the solitons so generated are related by the formula Ts = const α−3/2. This relation is further found to be in good agreement with the first-principle prediction derived here based on mass, momentum and energy considerations of the fKdV equation.