I'm reading papers regarding the 2D Navier-Stokes equations. But I don't have a physics background. Would anybody come up with some introductory references of 2D turbulence which contain the following (part of) issues?

  • energy power law
  • enstrophy cascade
  • palinstrophy
  • enerygy dissipation
  • enstrophy dissipation
  • how enstrophy is transferred through length scales

2 Answers 2


I would recommend the article 2D Turbulence from Boffetta and Ecke published in the Annual Review of Fluid Mechanics in 2012. It gives equations and statistical informations in spectral space with known scalings on energy spectra and third-order structure function. Then details about methods and examples on different approaches are mentioned and finally, results are shown using both numerical and experimental data. A clear summary and some future issues are addressed at the end. In the introduction, they mention others reviews which could help you evaluate the progression in the subject. I especially recommend :

  • Two-dimensional turbulence by Kraichnan and Montgomery in 1980. It displays analytical developments and numerical results. Very interesting but maybe not the best paper to begin with, may be hard to get the idea for someone without a good mathematical background.
  • Two-dimensional turbulence: a review of some recent experiments by Kellay and Goldburg in 2002. As it says, their review is based on experimental results with both decaying and forced turbulence as well as mixing of a passive scalar. They recall the basics of 2D turbulence in a small introduction.

A forth source could be the book of Lesieur, Turbulence in Fluids. The eighth chapter is dedicated to two-dimensional turbulence. You will find everything, statistical tools, enstrophy, inverse cascade, transfer, etc. He also mentioned how to study 2D turbulence with a spectral model using an EDQNM closure.

Those four references will certainly give you a complete view on the 2D turbulence fundamentals. Feel free to complete if I missed a major article.


It is nit an easy task to find a good reference which contains all of the points you would like to know about in one reference.

However, for the first approach to turbulence I do suggest the book of "David Wilcox" called "Turbulence modeling for CFD"https://www.amazon.com/Turbulence-Modeling-Third-David-Wilcox/dp/1928729088. In this book, some of the fundamental and important points on turbulence are discussed very nicely such as energy power law, law of the wall, Reynolds stress, and energy dissipation. For a further domain, you can check out the "Turbulent Flows" by Pope https://www.amazon.co.uk/Turbulent-Flows-Stephen-B-Pope/dp/0521598869 is an excellent book which covers alot. Do not waste your time with "An Introduction to Turbulent Flow" by John Mathieu.

Furthermore, you can get state-of-the-art of turbulence from some of the journals such as Flow, Turbulence and Combustion, Physics of fluids, and Journal of Turbulence.

Good luck learning about this very interesting side of Fluid Dynamics.

  • $\begingroup$ What kind of prerequisites are needed for these two books? Care to explain why Mathieu's book is a waste of time? $\endgroup$
    – Kyle Kanos
    Apr 24, 2019 at 15:34
  • $\begingroup$ The prerequisites are mainly calculus, fluid dynamics, partial differential equations, also good if you know thermodynamics. Well, it is not completely a waste of time, but there are other better references out there. $\endgroup$ Apr 24, 2019 at 15:39

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