Showing papers on "Method of matched asymptotic expansions published in 1985"

Do Rossby-wave critical layers absorb, reflect, or over-reflect?

Abstract: The Stewartson-Warn-Warn (SWW) solution for the time evolution of an inviscid, nonlinear Rossby-wave critical layer, which predicts that the critical layer will alternate between absorbing and over-reflecting states as time goes on, is shown to be hydrodynamically unstable. The instability is a two-dimensional shear instability, owing its existence to a local reversal of the cross-stream absolute vorticity gradient within the long, thin Kelvin cat’s eyes of the SWW streamline pattern. The unstable condition first develops while the critical layer is still an absorber, well before the first over-reflecting stage is reached. The exponentially growing modes have a two-scale cross-stream structure like that of the basic SWW solution. They are found analytically using the method of matched asymptotic expansions, enabling the problem to be reduced to a transcendental equation for the complex eigenvalue. Growth rates are of the order of the inner vorticity scale Sq, i.e. the initial absolute vorticity gradient dq,/dy times the critical-layer width scale. This is much faster than the time evolution of the SWW solution itself, albeit much slower than the shear rate du,/dy of the basic flow. Nonlinear saturation of the growing instability is expected to take place in a central region of width comparable to the width of the SWW cat’s-eye pattern, probably leading to chaotic motion there, with very large ‘eddy-viscosity ’ values. Those values correspond to critical-layer Reynolds numbers A-’ Q 1, suggesting that for most initial conditions the time evolution of the critical layer will depart drastically from that predicted by the SWW solution. A companion paper (Haynes 1985) establishes that the instability can, indeed, grow to large enough amplitudes for this to happen. The simplest way in which the instability could affect the time evolution of the critical layer would be to prevent or reduce the oscillations between over-reflecting and absorbing states which, according to the SWW solution, follow the first onset of perfect reflection. The possibility that absorption (or over-reflection) might be prolonged indefinitely is ruled out, in many cases of interest (even if the ‘eddy viscosity’ is large), by the existence of a rigorous, general upper bound on the magnitude of the time-integrated absorptivity a(t). The bound is uniformly valid for all time t. The absorptivity a(t) is defined aa the integral over all past t of the jump in the wave-induced Reynolds stress across the critical layer. In typical cases the bound implies that, no matter how large t may become, I a(t) I cannot greatly exceed the rate of absorption predicted by linear theory multiplied by the timescale on which linear theory breaks down, say the time for the cat’s-eye flow to twist up the absolute vorticity contours by about half a turn. An alternative statement is that I a(t) I cannot greatly exceed the initial absolute vorticity gradient dq,/dy times the cube of the

An introduction to Maslov's asymptotic method

Abstract: Summary. Familiar concepts such as asymptotic ray theory and geometrical spreading are now recognized as an asymptotic form of a more general asymptotic solution to the non-separable wave equation. In seismology, the name Maslov asymptotic theory has been attached to this solution. In its simplest form, it may be thought of as a justification of disc-ray theory and it can be reduced to the WKBJ seismogram. It is a uniformly valid asymptotic solution, though. The method involves properties of the wavefronts and ray paths of the wave equation which have been established for over a century. The integral operators which build on these properties have been investigated only comparatively recently. These operators are introduced very simply by appealing to the asymptotic Fourier transform of Ziolkowski & Deschamps. This leads quite naturally to the result that phase functions in different domains of the spatial Fourier transform are related by a Legendre transformation. The amplitude transformation can also be inferred by this method. Liouville's theorem (the incompressibility of a phase space of position and slowness) ensures that it is always possible to obtain a uniformly asymptotic solution. This theorem can be derived by methods familiar to seismologists and which do not rely on the traditional formalism of classical mechanics. It can also be derived from the sympletic property of the equations of geometrical spreading and canonical transformations in general. The symplectic property plays a central role in the theory of high-frequency beams in inhomogeneous media.

Singular Perturbation Analysis of Discrete Control Systems

Abstract: Singular perturbation analysis of difference equations in classical form.- Modelling and analysis of singularly perturbed difference equations in state space form.- Three-time-scale difference equations with application to open-loop optimal control problem.- Singularly perturbed nonlinear difference equations and closed-loop discrete optimal control problem.

Unsteady lifting-line theory as a singular-perturbation problem

Abstract: Unsteady lifting-line theory is developed for a wing of large aspect ratio oscillating at low frequency in inviscid incompressible flow. The wing is assumed to have a rigid chord but a flexible span. Use of the method of matched asymptotic expansions reduces the problem from a singular integral equation to quadrature. The pressure field and airloads, for a prescribed wing shape and motion, are obtained in closed form as expansions in inverse aspect ratio. A rigorous definition of unsteady induced downwash is also obtained. Numerical calculations are presented for an elliptic wing in pitch and heave; compared with numerical lifting-surface theory, computation time is reduced significantly. The present work also identifies and resolves errors in the unsteady lifting line theory of James (1975), and points out a limitation in that of Van Holten (1975, 1976, 1977).

Nonlinear Cusped Caustics for Dispersive Waves

Abstract: A multiple-scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple-valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrodinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann-Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple-phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple-valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearcey integral.

Asymptotic behavior of solutions of Volterra integro-differential equations

Abstract: The asymptotic behavior of solutions of Volterra integrodifferential equations of the form x'(t) = A(t)x(t) + J K(t, s)x(s) ds + F(t) is discussed in which A is not necessarily a stable matrix. An equivalent equation which involves an arbitrary function is derived and a proper choice of this function would pave a way for the new coefficient matrix B (corresponding A) to be stable.

Countercurrent convection in a double-diffusive boundary layer

Abstract: Countercurrent flow may be induced by opposing buoyancy forces associated with compositional gradients and thermal gradients within a fluid. The occurrence and structure of such flows is investigated by solving the double-diffusive boundary-layer equations for steady laminar convection along a vertical wall of finite height. Non-similar solutions are derived using the method of matched asymptotic expansions, under the restriction that the Lewis and Prandtl numbers are both large. Two sets of asymptotic solutions are constructed, assuming dominance of one or the other of the buoyancy forces. The two sets overlap in the central region of the parameter space; each set matches up with neighbouring unidirectional similarity solutions at the respective borderlines of incipient counterflow.Interaction between the buoyancy mechanisms is controlled by their relative strength R and their relative diffusivity Le. Flow in the outer thermal boundary layer deviates from single-diffusive thermal convection, depending upon the magnitude of the parameter RLe. Flow in the inner compositional boundary layer deviates from single-diffusive compositional convection, depending upon the magnitude of .

On The Role of a Non-Newtonian Fluid in Short Bearing Theory

Abstract: The importance of rheological properties of lubricants has arisen from the realization that non-Newtonian fluid effects are manifested over a broad range of lubrication applications. In this paper a theoretical investigation of short journal bearings performance characteristics for non-Newtonian power-law lubricants is given. A modified form of the Reynolds’ equation for hydrodynamic lubrication is studied in the asymptotic limit of small slenderness ratio (i.e., bearing length to diameter, L/D = λ→0). Fluid film pressure distributions in short bearings of arbitrary azimuthal length are studied using matched asymptotic expansions in the slenderness ratio. The merit of the short bearing approach used in solving a modified Reynolds’ equation by the method of matched asymptotic expansions is emphasized. Fluid film pressure distributions are determined without recourse to numerical solutions to a modified Reynolds’ equation. Power-law rheological exponents less than and equal to one are considered; power-law fluids exhibit reduced load capacities relative to the Newtonian fluid. The cavitation boundary shape is determined from Reynolds’ free surface condition; and the boundary shape is shown to be independent of the bearing eccentricity ratio.

An Introduction to the Technique of Reconstitution

Abstract: In many physical problems it is recognised that the solution is dominated by a particular structure. It is usually possible to derive a differential equation which approximately describes the spatial and/or temporal evolution of this dominant structure. Such an evolution equation is valid only for a limited range of the parameters. The technique of reconstitution provides a rationale and a mechanistic method for correcting such evolution equations by including terms representing higher-order physical interactions and thus to significantly extend the parameter range in which the equation is valid. To obtain some feel for how this technique works it is applied to a simple pair of coupled nonlinear differential equations. The results show clearly how the solutions to the approximate evolution equations, of varying accuracy, relate to the exact solution of the full problem.

Asymptotic Behavior of Elementary Solutions of One-Dimensional Generalized Diffusion Equations

Abstract: We give the asymptotic estimate for large $t$ of elementary solutions of one-dimensional generalized diffusion equations with regularly varying Green functions. As a corollary we obtain the precise asymptotic behavior of the semigroup $T_tf(x)$ for all $f \in L_1(dm)$ if the speed measure function $m(x)$ is regularly varying as $x \rightarrow \pm \infty$.

Atmospheric guidance law for planar skip trajectories

Abstract: The applicability of an approximate, closed-form, analytical solution to the equations of motion, as a basis for a deterministic guidance law for controlling the in-plane motion during a skip trajectory, is investigated. The derivation of the solution by the method of matched asymptotic expansions is discussed. Specific issues that arise in the application of the solution to skip trajectories are addressed. Based on the solution, an explicit formula for the approximate energy loss due to an atmospheric pass is derived. A guidance strategy is proposed that illustrates the use of the approximate solution. A numerical example shows encouraging performance.

Asymptotic expansions in the central limit theorem for compound and Markov processes

Abstract: For independent identically distributed bivariate random vectors (X 1, Y 1), (X 2, Y 2), ... and for large t the distribution of X 1 +...+ X N(t) is approximated by asymptotic expansions. Here N(t) is the counting process with lifetimes Y 1, Y 2,.... Similar expansions are derived for multivariate X 1. Furthermore, local asymptotic expansions are valid for the distribution of f(X 1)+ ...+ f(X N ) when N is large and nonrandom, and X i , i=1, 2,..., is a discrete strongly mixing Markov chain.

A Singularly Perturbed Free Boundary Problem Describing a Laser Sustained Plasma

Abstract: For a nonlinear diffusion equation with nonlocal terms which models a laser-sustained lasma, solutions are analyzed by the method of matched asymptotic expansions. Several stationary solutions are found and their stability is discussed as a function of the laser intensity. At a certain critical intensity, bifurcation of a stable stationary plasma into a pulsating plasma takes place. The pulsation process is of an unusual type, showing repeated formation of travelling plasma shells which die out later on.

On the asymptotic behaviour of solutions of a certain fourthorder differential equation

Abstract: In this paper sufficient conditions for the uniform global asymptotic stability of the zero solution of (1.1) are given.

Higher order effects in natural convection flow over a uniform flux horizontal surface

Abstract: Higher order boundary layer effects for natural convection flow over a horizontal plate prescribed with uniform heat flux is presented Using the method of matched asymptotic expansions the three terms in inner and outer expansions have been obtained It is shown that the contribution of the eigen functions to the three term inner expansions is identically zero

Asymptotic stability improvements of multiparameter nonlinear singularly perturbed systems

Abstract: In this note we study the asymptotic stability properties of multiparameter singularly perturbed systems. We show that by introducing a new fast variable, improved results are obtained for estimating the upper bound of the perturbation parameter and the region of attraction. An example is provided to demonstrate the improvements.

An Asymptotic Decomposition Method Applied to Multi-Turning Point Problems

Abstract: An asymptotic decomposition technique is developed. It is designed and used for 2 by 2 first order singularly perturbed linear differential systems. A new set of decoupled linear integral equations is introduced in the process of the asymptotic analysis. Its usefulness is demonstrated with multi-turning point problems. An adiabatic theorem in quantum mechanics is proved in a general case of degenerate energy levels.

Nonlinear hydrodynamic interaction in offshore loading systems

Abstract: The hydrodynamics of ships floating alongside one another at small separation distance is studied. Nonlinear and viscous effects are accounted for in the narrow gap between the ships, and the solution in this domain is matched to a linear outer solution using the method of matched asymptotic expansions. The matching equations are nonlinear ordinary differential equations that are solved numerically. Finally the forces on a ship carrying out forced harmonic oscillations are calculated. The nonlinear forces are compared with the linear added mass and damping forces. .