Lecture Notes in Physics Founding Editors: W. Beiglbock,¨ J. Ehlers, K. Hepp, H. Weidenmuller¨ Editorial Board R. Beig, Vienna, Austria W. Beiglbock,¨ Heidelberg, Germany W. Domcke, Garching, Germany B.-G. Englert, Singapore U. Frisch, Nice, France F. Guinea, Madrid, Spain P. Hanggi,¨ Augsburg, Germany W. Hillebrandt, Garching, Germany R. L. Jaffe, Cambridge, MA, USA W. Janke, Leipzig, Germany H. v. Lohneysen,¨ Karlsruhe, Germany M. Mangano, Geneva, Switzerland J.-M. Raimond, Paris, France M. Salmhofer, Heidelberg, Germany D. Sornette, Zurich, Switzerland S. Theisen, Potsdam, Germany D. Vollhardt, Augsburg, Germany W. Weise, Garching, Germany J. Zittartz, Koln,¨ Germany The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching – quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced grad- uate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas • to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Pro- ceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series content is indexed, ab- stracted and referenced by many abstracting and information services, bibliographic net- works, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the managing editor at Springer: Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg / Germany [email protected] Jan-Bert Flor´ (Ed.) Fronts, Waves and Vortices in Geophysical Flows ABC Jan-Bert Flor´ LEGI (Laboratoire des Ecoulements Geophysiques´ et Industriels) Universite´ de Grenoble B.P.53X, 38041 Grenoble Cedex 09 France Jan-Bert Flor´ (Ed.): Fronts, Waves and Vortices in Geophysical Flows, Lect. Notes Phys. 805 (Springer, Berlin Heidelberg 2010), DOI 10.1007/978-3-642-11587-5 Lecture Notes in Physics ISSN 0075-8450 e-ISSN 1616-6361 ISBN 978-3-642-11586-8 e-ISBN 978-3-642-11587-5 DOI 10.1007/978-3-642-11587-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010922993 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Integra Software Services Pvt. Ltd., Pondicherry Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Foreword Without coherent structures atmospheres and oceans would be chaotic and unpre- dictable on all scales of time. Most well-known structures in planetary atmospheres and the Earth oceans are jets or fronts and vortices that are interacting with each other on a range of scales. The transition from one state to another such as in unbalanced or adjustment flows involves the generation of waves, as well as the interaction of coherent structures with these waves. This book presents from a fluid mechanics perspective the dynamics of fronts, vortices, and their interaction with waves in geophysical flows. It provides a basic physical background for modeling coherent structures in a geophysical context and gives essential information on advanced topics such as spontaneous wave emission and wave-momentum transfer in geophysical flows. The book is targeted at graduate students, researchers, and engineers in geophysics and environmental fluid mechanics who are interested or working in these domains of research and is based on lectures given at the Alpine summer school entitled ‘Fronts, Waves and Vortices.’ Each chapter is self-consistent and gives an extensive list of relevant literature for further reading. Below the contents of the five chapters are briefly outlined. Chapter comprises basic theory on the dynamics of vortices in rotating and strati- fied fluids, illustrated with illuminating laboratory experiments. The different vortex structures and their properties, the effects of Ekman spin-down, and topography on vortex motion are considered. Also, the breakup of monopolar vortices into multiple vortices as well as vortex advection properties will be discussed in conjunction with laboratory visualizations. In Chap. 2, the understanding of the different vortex instabilities in rotating, stratified, and – in the limit – homogenous fluids are considered in conjunction with laboratory visualizations. These include the shear, centrifugal, elliptical, hyperbolic, and zigzag instabilities. For each instability the responsible physical mechanisms are considered. In Chap. 3, oceanic vortices as known from various in situ observations and measurements introduce the reader to applications as well as outstanding ques- tions and their relevance to geophysical flows. Modeling results on vortices high- light physical aspects of these geophysical structures. The dynamics of ocean deep sea vortex lenses and surface vortices are considered in relation to their genera- v vi Foreword tion mechanism. Further, vortex decay and propagation, interactions as well as the relevance of these processes to ocean processes are discussed. Different types of model equations and the related quasi-geostrophic and shallow water modeling are presented. In Chap. 4 geostrophic adjustment in geophysical flows and related problems are considered. In a hierarchy of shallow water models the problem of separation of fast and slow variables is addressed. It is shown how the separation appears at small Rossby numbers and how various instabilities and Lighthill radiation break the separation at increasing Rossby numbers. Topics such as trapped modes and symmetric instability, ‘catastrophic’ geostrophic adjustment, and frontogenesis are presented. In Chap. 5, nonlinear wave–vortex interactions are presented, with an empha- sis on the two-way interactions between coherent wave trains and large-scale vor- tices. Both dissipative and non-dissipative interactions are described from a unified perspective based on a conservation law for wave pseudo-momentum and vortex impulse. Examples include the generation of vortices by breaking waves on a beach and the refraction of dispersive internal waves by three-dimensional mean flows in the atmosphere. Grenoble, France Jan-Bert Flór Contents 1 Dynamics of Vortices in Rotating and Stratified Fluids .............. 1 G.J.F. van Heijst 1.1 VorticesinRotatingFluids.................................... 1 1.1.1 Basic Equations and Balances . ........................ 2 1.1.2 HowtoCreateVorticesintheLab ....................... 9 1.1.3 The Ekman Layer . ................................... 12 1.1.4 Vortex Instability. ................................... 14 1.1.5 EvolutionofStableBarotropicVortices................... 15 1.1.6 Topography Effects . ................................... 18 1.2 VorticesinStratifiedFluids.................................... 20 1.2.1 Basic Properties of Stratified Fluids . .................... 20 1.2.2 Generation of Vortices . ............................... 22 1.2.3 Decay of Vortices . ................................... 24 1.2.4 Instability and Interactions. ........................ 30 1.3 Concluding Remarks . ................................... 33 References . ..................................................... 33 2 Stability of Quasi Two-Dimensional Vortices ....................... 35 J.-M. Chomaz, S. Ortiz, F. Gallaire, and P. Billant 2.1 Instabilities of an Isolated Vortex . ........................ 36 2.1.1 The Shear Instability . ............................... 37 2.1.2 The Centrifugal Instability . ........................ 37 2.1.3 Competition Between Centrifugal and Shear Instability . 40 2.2 Influence of an Axial Velocity Component . .................... 41 2.3 Instabilities of a Strained Vortex ............................... 43 2.3.1 The Elliptic Instability . ............................... 44 2.3.2 The Hyperbolic Instability.............................. 46 2.4 The Zigzag Instability . ................................... 47 2.4.1 The Zigzag Instability in Strongly Stratified Flow Without Rotation . ........................... 47 2.4.2 The Zigzag Instability in Strongly Stratified FlowwithRotation..................................