{"title":"Parallel Shear Flow: the Effects of Stratification","authors":"","doi":"10.1017/9781108640084.005","DOIUrl":"https://doi.org/10.1017/9781108640084.005","url":null,"abstract":"The Earth’s oceans and atmosphere tend to be inhomogeneous, with density differences that can strongly affect the motion. The primary reason for the inhomogeneity is heat radiation. At the equator, both ocean and atmosphere gain heat from the sun. Near the poles, the ocean loses heat to the atmosphere, while the atmosphere radiates heat into space at all latitudes. Ocean density is also governed by salinity. All of these processes create inhomogeneities which, under the action of gravity, tend to rearrange themselves into horizontal layers. We therefore say that these fluids are “density-stratified,” or just “stratified.” Based on our experience","PeriodicalId":276093,"journal":{"name":"Instability in Geophysical Flows","volume":"28 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":"122765719","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":"Parallel Shear Flow: the Effects of Viscosity","authors":"","doi":"10.1017/9781108640084.006","DOIUrl":"https://doi.org/10.1017/9781108640084.006","url":null,"abstract":"","PeriodicalId":276093,"journal":{"name":"Instability in Geophysical Flows","volume":"6 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":"129145097","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":"Homework Exercises","authors":"L. Bowen","doi":"10.1017/9781108640084.015","DOIUrl":"https://doi.org/10.1017/9781108640084.015","url":null,"abstract":"Please attempt at least one problem per set. The sets are intended to correspond to lectures. 1 Set #1 1. Prove the rank 2 free group 〈a, b〉 is non-amenable. 2. The Ornstein Isomorphism Theorem states that if two Bernoulli shifts over the integers have the same base entropy then they are isomorphic. That is, if H(μ) = H(ν) where μ ∈ Prob(A), ν ∈ Prob(B) then the Bernoulli shifts Zy(AZ, μZ) and Zy(BZ, νZ) are measurably conjugate. Prove that the same statement is true for the free group F2 := 〈a, b〉. Hint: identify Z with a subgroup of F2. You can build the isomorphism (AF2 , μF2)→ (BF2 , νF2) from an isomorphism for (AZ, μZ)→ (BZ, νZ) coset-by-coset. 2 Set #2 1. A point x ∈ AZ is periodic if there is some integer n such that xi = xi+n for all i. If x is periodic with period n then let μx := 1 n ∑n−1 i=0 δT ix be the uniform measure on its ∗supported in part by NSF grant DMS-1500389, NSF CAREER Award DMS-0954606","PeriodicalId":276093,"journal":{"name":"Instability in Geophysical Flows","volume":"6 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":"121598298","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":"Beyond Normal Modes","authors":"","doi":"10.1017/9781108640084.012","DOIUrl":"https://doi.org/10.1017/9781108640084.012","url":null,"abstract":"","PeriodicalId":276093,"journal":{"name":"Instability in Geophysical Flows","volume":"145 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":"133233346","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":"Instability in a Rotating Environment","authors":"","doi":"10.1017/9781108640084.009","DOIUrl":"https://doi.org/10.1017/9781108640084.009","url":null,"abstract":"","PeriodicalId":276093,"journal":{"name":"Instability in Geophysical Flows","volume":"1 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":"130777557","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":"Refining the Numerical Methods","authors":"","doi":"10.1017/9781108640084.014","DOIUrl":"https://doi.org/10.1017/9781108640084.014","url":null,"abstract":"","PeriodicalId":276093,"journal":{"name":"Instability in Geophysical Flows","volume":"12 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":"126787622","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":"Convective Instability in Complex Fluids","authors":"Ippei Naoi","doi":"10.1017/9781108640084.010","DOIUrl":"https://doi.org/10.1017/9781108640084.010","url":null,"abstract":"So far we have assumed that the equation of state is trivial: buoyancy is proportional to a single fluid property (e.g., temperature) that varies only due to advection and diffusion (section 1.5). Real geophysical fluids may be much more complicated than this. For example, buoyancy may be controlled by two or more scalar constituents with different chemical properties, or by phase changes (e.g., freezing). The equation of state may include nonlinearities; for example, a mixture of two constituents may be lighter or denser than either constituent individually. The scalars that control buoyancy may have complex properties such as being alive. All of these properties affect the stability of the fluid. In this chapter we consider a few simple examples in which buoyancy changes lead to convection.","PeriodicalId":276093,"journal":{"name":"Instability in Geophysical Flows","volume":"3 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":"125647780","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}
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