A Frequency Selection Criterion in Spatially Developing Flows

The possible existence of global modes or self-excited linear resonances in spatially developing systems is explored within the framework of the WKBJ approximation. It is shown that the existence and properties of the dominant global mode may be deduced from the variations of the local absolute frequency ω₀ with distance X. The main results are summarized in two theorems: (1) A system with no region of absolute instability does not sustain temporally growing global modes with an O(1) growth rate. (2) If the singularity X, closest to the real X-axis of the complex function ω₀(X) is a saddle point, the most unstable global mode has, to leading order in the WKBJ approximation, a complex frequency ω₀(X_s). Thus, it will be temporally growing only if ω₀(X_s) is positive.     

The Effect of Profile Symmetry on the Nonlinear Stability of Mixing Layers

The nonlinear stability of arbitrary mixing-layer profiles in an incompressible, homogeneous fluid is studied in the high-Reynolds-number limit where the critical layer is linear and viscous. The type of bifurcation from the marginal state is found to depend crucially on the symmetry properties of the basic-state profile. When the vorticity profile of the mean flow is perfectly symmetric, the bifurcation is stationary. When the symmetry of the profile is broken, the bifurcation is Hopf. The nonsymmetry of the mixing layer also introduces some changes in the critical layer and the matching of flow quantities across it.     

An application of one-sided Jacobi polynomials for spectral modeling of vector fields in polar coordinates

A spectral tau-method is proposed for solving vector field equations defined in polar coordinates. The method employs one-sided Jacobi polynomials as radial expansion functions and Fourier exponentials as azimuthal expansion functions. All the regularity requirements of the vector field at the origin and the physical boundary conditions at a circumferential boundary are exactly satisfied by adjusting the additional tau-coefficients of the radial expansion polynomials of the highest order. The proposed method is applied to linear and nonlinear-dispersive time evolution equations of hyperbolic-type describing internal Kelvin and Poincaré waves in a shallow, stratified lake on a rotating plane.     

An Elementary Method on a Minimization Problem

一、ＴｈｅＰｒｏｂｌｅｍ .Ｌｅｔｆ(ｘ) =ａｘ2 ｂｘ ｃ ａｘ2 ｐｘ ｑ,ｗｈｅｒｅａ >0 ,ｂ2 - 4ａｃ≤ 0ａｎｄｐ2 - 4ａｑ≤ 0 .ＯｕｒＰｒｏｂｌｅｍｉｓ“Ｗｈｅｔｈｅｒｄｏｅｓｍｉｎｘ∈Ｒｆ(ｘ)ｅｘｉｓ?Ｍｏｒｅｏｖｅｒ ,,ｉｆｍｉｎｘ∈Ｒｆ(ｘ)ｅｘｉｓｔｓ,ｔｈｅｎｍｉｎｘ∈Ｒｆ(ｘ)=?ａｎｄｉｎｔｈｉｓｃａｓｅｘ =?” .Ｎａｔｕｒａｌｌｙ ,ｗｅｋｎｏｗｔｈａｔｍｉｎｘ∈Ｒｆ(ｘ)ｅｘｉｓｔｓｆｒｏｍｔｈｅｋｎｏｗｌｅｄｇｅｏｆｍａｔｈｅｍａｔｉｃａｌａｎａｌｙｓｉｓ.Ａｌｓｏ,ｗｅｃａｎｇｉｖｅｔｈｅ…     

The Coupling of Gravity Waves and Convection: Amplitude Equations and Planform Selection

Convective motion in a layer of fluid heated from below is considered where the boundaries are stress free and the upper surface supports interfacial gravity waves. Inviscid, immiscible, constant density, ambient fluid is separated from the convecting layer below by a stable density jump. An important parameter in the problem is δ representing the ratio of the interfacial density jump to the density change across the convecting layer. Amplitude equations are derived describing convective motion in the plane and a planform selection analysis performed. It is demonstrated that the breaking of the translational and Galilean invariance of the problem allows a strong coupling between a large-scale interfacial mode and convection. The resulting phase dynamics is third order in time.     

On the development of packets of surface gravity waves moving over an uneven bottom

A nonlinear Schrödinger equation with varying coefficients describing the evolution of onedimensional packets of surface gravity waves moving over an uneven bottom is derived in the paper and the disintegration of both an envelope soliton and an envelope-hole soliton as they propagate onto a shelf is studied.

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Zusammenfassung Eine nichtlineare Schrödinger Differentialgleichung mit veränderlichen Koeffizienten wird aufgestellt, welche die Entwicklung eindimensionaler Pakete von Oberflächenwellen beschreibt, die sich über einen unebenen Boden bewegen. Das Problem der Spaltung an der Schwelle wird für eine umhüllende solitäre Welle und eine konkave umhüllende solitäre Welle untersucht.

     

Similarity Solutions of Some Two-Space-Dimensional Nonlinear Wave Evolution Equations

Similarity reductions of the two-space-dimensional versions of the Korteweg-deVries, modified Korteweg-deVries, Benjamin-Davis-Ono, and nonlinear Schrödinger equations are presented, and some solutions of the reduced equations are discussed. Exact dispersive solutions of the two-dimensional Korteweg-deVries equation are obtained, and the similarity solution of this equation is shown to be reducible to the second Painlevé transcendent.