{"title":"Instability and Turbulence","authors":"P. Shirshov","doi":"10.1017/9781108640084.013","DOIUrl":"https://doi.org/10.1017/9781108640084.013","url":null,"abstract":"","PeriodicalId":276093,"journal":{"name":"Instability in Geophysical Flows","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122768380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Instabilities of a Parallel Shear Flow","authors":"W. Smyth, J. Carpenter","doi":"10.1017/9781108640084.004","DOIUrl":"https://doi.org/10.1017/9781108640084.004","url":null,"abstract":"A parallel shear flow is a flow that moves in a single direction, and whose velocity changes in a perpendicular direction, e.g., u = U (z)ê(x). This class of flows includes wakes, jets, boundary layers and shear layers (e.g., Figures 3.2 and 3.3). Parallel shear flows are the simplest class of flows exhibiting the phenomenon of shear instability, in which a fluid is unstable because of spatial variations in its velocity. Here we will look at shear instability in its most basic form, free of complications due to viscosity, diffusion, buoyancy, or planetary rotation. We’ll need to solve an ordinary differential equation, the Rayleigh equation. This can be done analytically for some simple examples (section 3.3), but we’ll also make frequent use of the numerical methods introduced in Chapter 1. Not every parallel shear flow is unstable. We’ll prove a simple theorem that often allows us to identify stable cases without solving any equations. Even in an unstable flow, not all types of disturbances grow, so we’ll prove two additional results that allow us to distinguish those that do. Besides developing the needed mathematical theory, we will seek an intuitive understanding of the mechanism of shear instability. This latter goal is considerably more challenging than in the previous case of convection.","PeriodicalId":276093,"journal":{"name":"Instability in Geophysical Flows","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127785069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Nonparallel Flow: Instabilities of a Cylindrical Vortex","authors":"W. Smyth, J. Carpenter","doi":"10.1017/9781108640084.008","DOIUrl":"https://doi.org/10.1017/9781108640084.008","url":null,"abstract":"The cylindrical (or columnar) vortex is the simplest example of a non-parallel shear flow, and is a useful model for tornados and other geophysical vortices. Here we’ll examine two classes of vortex instabilities: (1) barotropic instabilities are closely analogous to the instabilities of a parallel shear flow, while (2) axisymmetric instabilities resemble convection, but with the centrifugal force playing the role of gravity. Consider a cylindrical coordinate system with radial, azimuthal, and axial coordinates r , θ , and z (Figure 7.1) and corresponding velocities u = dr/dt , v = r dθ/dt , and w = dz/dt . In geophysical applications, the axial direction usually coincides with the vertical. Assuming inviscid, homogeneous flow, the equations are","PeriodicalId":276093,"journal":{"name":"Instability in Geophysical Flows","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127227122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}