摘要:

In the linear inviscid theory of shear flow stability, the eigenvalue problem for a neutral or weakly amplified mode revolves around possible discontinuities in the eigenfunction as the singular critical point is crossed. Extensions of the linear normal mode approach to include nonlinearity and/or wave packets lead to amplitude evolution equations where, again, critical point singularities are an issue because the coefficients of the amplitude equations generally involve singular integrals. In the past, viscosity, nonlinearity, or time dependence has been introduced in a critical layer centered upon the singular point to resolve these integrals. The form of the amplitude evolution equation is greatly influenced by which choice is made. In this paper, a new approach is proposed in which wave packet effects are dominant in the critical layer and it is argued that in many applications this is the appropriate choice. The theory is applied to two‐dimensional wave propagation in homogeneous shear flows and also to stratified shear flows. Other generalizations are indicate

ISSN:0022-2526    

DOI:10.1002/sapm19949111

年代:1994

数据来源: WILEY

摘要:

A simple model equation for western boundary outflow in the Stommel model of the large scale ocean circulation is obtained by evaluating the potential vorticity equation at the western boundary. A series solution to this model equation demonstrates similar behavior to the boundary layer solution of the potential vorticity equation, in particular that “resonances” are present at a discrete series of parameter values which necessitate the addition of logarithms to the series; these resonances occur because the model equation has a logarithmic branch point at these val

ISSN:0022-2526    

DOI:10.1002/sapm199491117

年代:1994

数据来源: WILEY

摘要:

Explicit solutions are found for the stream function satisfying the Navier Stokes equations representing the steady two‐dimensional motion of a viscous incompressible liquid. The solutions contain two arbitrary analytic functions and in general are confined to certain regions of thex,yplan

ISSN:0022-2526    

DOI:10.1002/sapm199491127

年代:1994

数据来源: WILEY

摘要:

This paper introduces Geissinger multiplication on the vector space generated by indicator functions of closed convex sets. Minkowski's mixed volume for compact convex sets is naturally represented in terms of the volume of the Geissinger multiplication of their indicator functions. Some properties of mixed volumes and new results are obtained by this representation, including a polynomial identity.

ISSN:0022-2526    

DOI:10.1002/sapm199491139

年代:1994

数据来源: WILEY

摘要:

The internal layer motion associated with the Ginzburg‐Landau equationis analyzed, in the limit , for various boundary conditions onx= ±1. The nonlinearityQ(u) results either from a double‐well potential or a periodic potential, each having wells of equal depth. Using a systematic asymptotic method, some previous work in deriving equations of motion for the internal layers corresponding to metastable patterns is extended. The effect of the various types of boundary conditions and nonlinearities will be highlighted. A dynamical rescaling method is used to numerically integrate these equations of motion. Using formal asymptotic methods, certain canonical problems describing layer collapse events are formulated and solved numerically. A hybrid asymptotic‐numerical method, which incorporates these layer collapse events, is used to give a complete quantitative description of the coarsening process associated with the Ginzburg‐Landau equation. For the Neumann problem with a double‐well potential, the qualitative description of the coarsening process given in Carr and Pego [4] will be confirmed quantitatively. In other cases, such as for a periodic potential with Dirichlet boundary conditions, it will be shown that, through layer collapse events, a metastable pattern can tend to a stable equilibrium solution with an internal layer

ISSN:0022-2526    

DOI:10.1002/sapm199491151

年代:1994

数据来源: WILEY