(R1966) Semi Analytical Approach to Study Mathematical Model of Atmospheric Internal Waves Phenomenon

This research aims to study atmospheric internal waves which occur within the fluid rather than on the surface. The mathematical model of the shallow fluid hypothesis leads to a coupled nonlinear system of partial differential equations. In the shallow flow model, the primary assumption is that vertical size is smaller than horizontal size. This model can precisely replicate atmospheric internal waves because waves are dispersed over a vast horizontal area. A semi-analytical approach, namely modified differential transform, is applied successfully in this research. The proposed method obtains an approximate analytical solution in the form of convergent series without any linearization, perturbation, or calculation of unneeded terms, which is a significant advantage over other existing methods. To test the effectiveness and accuracy of the proposed method, obtained results are compared with Elzaki Adomain Decomposition Method, Modified Differential Transform Method, and Homotopy Analysis Method

The time-fractional mZK equation for gravity solitary waves and solutions using sech-tanh and radial basic function method

In recent years, we know that gravity solitary waves have gradually become the research spots and aroused extensive attention; on the other hand, the fractional calculus have been applied to the biology, optics and other fields, and it also has attracted more and more attention. In the paper, by employing multi-scale analysis and perturbation methods, we derive a new modified Zakharov–Kuznetsov (mZK) equation to describe the propagation features of gravity solitary waves. Furthermore, based on semi-inverse and Agrawal methods, the integer-order mZK equation is converted into the time-fractional mZK equation. In the past, fractional calculus was rarely used in ocean and atmosphere studies. Now, the study on nonlinear fluctuations of the gravity solitary waves is a hot area of research by using fractional calculus. It has potential value for deep understanding of the real ocean–atmosphere. Furthermore, by virtue of the sech-tanh method, the analytical solution of the time-fractional mZK equation is obtained. Next, using the above analytical solution, a numerical solution of the time-fractional mZK equation is given by using radial basis function method. Finally, the effect of time-fractional order on the wave propagation is explained. &nbsp

Frequency staircases in narrow-gap spherical Couette flow

This is the author accepted manuscript. The final version is available from Taylor & Francis via the DOI in this record.Recent studies of plane parallel flows have emphasised the importance of finite-amplitude self-sustaining processes for the existence of alternative non-trivial solutions. The idea behind these mechanisms is that the motion is composed of distinct structures that interact to self-sustain. These solutions are not unique and their totality form a skeleton about which the actual realised motion is attracted. Related features can be found in spherical Couette flow between two rotating spheres in the limit of narrow-gap width. At lowest order the onset of instability is manifested by Taylor vortices localised in the vicinity of the equator. By approximating the spheres by their tangent cylinders at the equator, a critical Taylor number based on the ensuing cylindrical Couette flow problem would appear to provide a lowest order approximation to the true critical Taylor number. At next order, the latitudinal modulation of their amplitude a satisfies the complex Ginzburg-Landau equation (CGLe) ∂a/∂t=(λ+ix)a+∂2a/∂x2−|a|2a, ∂a/∂t=(λ+ix)a+∂2a/∂x2-|a|2a, where −x-x is latitude scaled on the modulation length scale, t is time and λλ is proportional to the excess Taylor number. The amplitude a governed by our CGLe is linearly stable for all λλ but possesses non-decaying nonlinear solutions at finite λλ, directly analogous to plane Couette flow. Furthermore, whereas the important balance ∂a/∂t=ixa∂a/∂t=ixa suggests that the Taylor vortices ought to propagate as waves towards the equator with frequency proportional to latitude, the realised solutions are found to exist as pulses, each locked to a discrete frequency, of spatially modulated Taylor vortices. Collectively they form a pulse train. Thus the expected continuous spatial variation of the frequency is broken into steps (forming a staircase) on which motion is dominated by the local pulse. A wealth of solutions of our CGLe have been found and some may be stable. Nevertheless, when higher-order terms are reinstated, solutions are modulated on a yet longer length scale and must evolve. So, whereas there is an underlying pulse structure in the small but finite gap limit, motion is likely to be always weakly chaotic. Our CGLe and its solution provides a paradigm for many geophysical and astrophysical flows capturing in minimalistic form interaction of phase mixing ixaixa, diffusion ∂2a/∂x2∂2a/∂x2 and nonlinearity |a|2a|a|2a.This paper was inspired by AMS’ attendance (19–21 March 2014) of the LMS Society Conference “Nonlinear stability theory: from weakly nonlinear theory to the verge of turbulence” to celebrate the 85th birthday of Professor J.T. Stuart. AMS gained further perspectives from attending (24–27 March 2014) the KITP program “Wave-flow interaction in geophysics, climate, astrophysics, and plasmas” at UCSB, where this research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. We are grateful to the referees for their helpful comments

Balanced initialisation techniques for coupled ocean-atmosphere models

Interactive dynamical ocean and atmosphere models are commonly used for predictions on seasonal timescales, but initialisation of such systems is problematic. In this thesis, idealised coupled models of the El Ni~no Southern Oscillation phenomenon are used to explore potential new initialisation methods. The basic ENSO model is derived using the two-strip concept for tropical ocean dynamics, together with a simple empirical atmosphere. A hierarchy of models is built, beginning with a basic recharge oscillator type model and culminating in a general n-box model. Each model is treated as a dynamical system. An important step is the 10-box model, in which the seasonal cycle is introduced as an extension of the phase space by two dimensions, which paves the way for more complex and occasionally chaotic behaviour. For the simplest 2-box model, analytic approximate solutions are described and used to investigate the parameter dependence of regimes of behaviour. Model space is explored statistically and parametric instability is found for the 10-box and upward versions: while it is by no means a perfect simulation of the real world phenomena, some regimes are found which have features similar to those observed. Initialisation is performed on a system from the n-box model (with n = 94), using dimensional reduction via two separate methods: a linear singular value decomposition approach and a nonlinear slow manifold (approximate inertial manifold) type reduction. The influence of the initialisation methods on predictive skill is tested using a perfect model approach. Data from a model integration are treated as observation, which are perturbed randomly on large and small spatial scales, and used as initial states for both reduced and full model forecasts. Integration of the reduced models provides a continuous initialisation process, ensuring orbits remain close to the attractor for the duration of the forecasts. From sets of ensemble forecasts, statistical measures of skill are calculated. Results are found to depend on the dimensionality of the reduced models and the type of initial perturbations used, and model reduction is found to result in a slight improvement in skill from the full model in each case, as well as a signifi�cant increase in the maximum timestep

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described