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MA6L0 Advanced Topics in Fluids

Not running in 2017/18.

Lecturer: Sergey Nazarenko

Term(s): Term 2

Status for Mathematics students: List C

Commitment: 30 lectures

Assessment: Three hour written examination

Prerequisites: MA371 Qualitative Theory of ODEs, MA3D1 Fluid Dynamics, MA3G1 Theory of PDEs, or similar modules from other departments or universities.

Leads To:

Topics will include several of the following themes:
• Linear and nonlinear waves in fluids and other continuous media, such as plasmas, MHD fluids, Bose-Einstein condensates, superfluid helium, nonlinear optics crystals. Waves in inhomogeneous or/and moving media, scale separation, WKB and ray tracing approach, Born approximation for wave scattering on inhomogeneities and vortices. Hamiltonian and Lagrangian formulations for nonlinear waves. Solitons. Waves in excitable media, eg. spiral waves in cardiac tissue.
• Classical turbulence theory. Richarson cascade and Kolmogorov spectrum. Single and dual cascade systems. Structure functions and intermittency. Scalings in stationary and in evolving turbulence. Near-wall turbulence. Pipe turbulence. Rapid distortion theory.
• Quantum turbulence. Polarised and unpolarised tangles of quantized vortex lines. Biot-Savart-Rios description. Vortex line reconnections. Kelvin waves on vortex lines. Classical-quantum crossover scales. Sound emission by moving vortices.
• Turbulence in Bose-Einstein condensates. Gross-Pitaevskii equation model. Dark solitons and quantized vortices. Inverse cascade and condensation phenomenon. Wave turbulence description. Bogoliubov transformation. Berezinskii_Kousterlitz-Thouless and Kibble-Zurek phase transitions.
• Astrophysical and plasma turbulence. Alfen waves and drift waves. Wave turbulence approach to weak MHD and drift turbulence. Strong turbulence and critical balance.
• Large-scale waves and vortices in atmosphere and oceans. Quasi-geostrophic model. Planetary Rossby waves. Anisotropic cascades. Generation of zonal jets. Transport barriers. Two-layer model. Interaction of barotropic and baroclinic modes.

•To provide a useful course for our 1st year PhD students, Master students, DTC students, 4th year MMATHs, Master of Advanced Study (MASt) interested in fluid dynamics related subjects, nonlinear waves, superfluids, plasmas, geophysical flows, Bose-Einstein condensates, turbulence in all of these settings.
•Have a module which is flexible enough to adjust to the needs of the current students and to the expertise of available lecturers by choosing a topic from a broad range of interrelated themes.
•Build on entry knowledge towards topics of current interest or research.

(By the end of the module the student should be able to....)

  • Appreciate universality of the fluid dynamics processes in diverse applications, from quantum fluids to astrophysical systems.
  • Understand the nonlinear phenomena in fluids within the considered application and in the general fluid flow. The nonlinear processes are omitted from most UG fluids courses.
  • Be able to use statistical techniques for fluid systems arising in turbulent flows, e.g. manipulating spectra, structure functions and probability density functions, averaging over ensemble, space, time or initial data, derive and use turbulent closures, e.g. the kinetic equations, derive Kolmogorov spectrum and its analogues.
  • Be able to recognise that similar techniques may be used to study fluids and other physical systems described by nonlinear PDE’s, e.g. non-harmonic crystals or electromagnetic waves. Be capable to use these techniques in future research projects.

Whitham, G.B., Linear and Nonlinear Waves, 2011, Wiley
Frisch, U. Turbulence: The Legacy of A. N. Kolmogorov, 1995, Cambridge University Press
Nazarenko, S., Fluid Dynamics via Examples and Solutions, 2015, CRC Press
Sulem, C., Sulem, P.L., The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, 1999, Springer
Biskamp, D., Nonlinear Magnetohydrodynamics, 1997, Cambridge University Press
Pitaevskii, L., Stringari, S., Bose-Einstein Condensation (International Series of Monographs on Physics), 2003, Oxford University Press
Nazarenko, S., Wave Turbulence, 2011, Springer
Sinha, S., Sridhar, S., Patterns in Excitable Media: Genesis, Dynamics, and Control, 2014, Taylor & Francis
Donnelly, R.J., Quantized Vortices in Helium II, 1991, Cambridge University Press
Pomeau, Y., Pismen, L.M., Patterns and Interfaces in Dissipative Dynamics, 2006, Springer Berlin Heidelberg
Kadomtsev, B.B., Collective Phenomena in Plasmas, 1982, Elsevier Science Limited
McWilliams, J.C., Fundamentals of Geophysical Fluid Dynamics, 2006, Cambridge University Press

Additional Resources