Much work has been done on low-dimensional models of turbulence and fluid systems in the 16 years since the first edition of this book appeared. In preparing the second edition, we have not attempted a comprehensive review: indeed, we doubt that this is possible, or even desirable. Rather, we have added one chapter and several sections and subsections on some new developments that are most closely related to material in our first edition. We have also made minor corrections and clarifications throughout, and added comments in several places, as well as correcting a number of errors that readers have pointed out. Here, to orient the reader, we outline the major changes.

Clancy Rowley (the new member of our team) has contributed a chapter on balanced truncation, a technique from linear control theory that chooses bases that optimally align inputs and outputs. Over the past ten years this has led to the method of balanced proper orthogonal decomposition (BPOD), which is especially useful for systems equipped with sensors and actuators. Since low-dimensional models provide a computational means for studying control of turbulence, we feel that BPOD has considerable potential. This new chapter (5) now closes the first part of the book (readers familiar with the first edition must therefore remember to add 1 to correctly identify the following eight chapters). The only other entirely new sections are 7.5, a discussion of traveling modes in translation-invariant systems, 12.6, a review of work on coherent structures in internal combustion engines, and 12.7, which gathers a miscellany of recent results.