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Gyrokinetics in Laboratory and Astrophysical Plasmas

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19th July 2010 to 13th August 2010
Bill Dorland [Maryland], University of Maryland
Sergey Nazarenko [Warwick], University of Warwick
Alex Schekochihin [Oxford], University of Oxford


Programme Theme

Perhaps the greatest challenge in plasma science is to understand the multiscale interaction between small-scale fluctuations and large-scale plasma dynamics. This is crucial both in fundamental astrophysical and space physics research (e.g., turbulence in the solar wind) and in more practical terrestrial contexts (e.g., the performance of the international fusion reactor, ITER, will be limited by the transport caused by small-scale fluctuations). Multiscale plasma dynamics also represent a formidable and fascinating mathematical challenge, as new analytical and numerical methods have to be developed in order for major breakthroughs to become possible.

In the last 25 years, a new mathematical approach, gyrokinetics, has been developed to treat low-frequency fluctuations in plasmas. In this approach, the fast orbital "gyromotion" is averaged to produce kinetic equations for rings of charge. This is a mathematically rigorous description that is far more tractable than the full kinetic theory. Despite some practical successes in code-building and simulations, the mathematical properties and physical implications of gyrokinetics are insufficiently well understood. In space and astrophysics, the wide applicability and power of the gyrokinetic theory has yet to be fully recognised and exploited. To realise the benefits of this approach, it is essential that gyrokinetics be put on a firm mathematical and physical footing.

Gyrokinetics is a nonlinear theory in a 5D phase space. When the collisionality is low, the ring distribution develops a broad range of scales in this phase space. This gives rise to a kinetic turbulent system that is richer and more complex than fluid turbulence. Some of the fundamental questions are:

  • Can the energy transfer from large to small scales be thought of as a cascade in phase space?
  • At what scales is the fluctuation energy transferred into heat and what is the relative efficiency of the heating of electrons, ions, and of fast non-thermal species?
  • What is the average effect of the small-scale fluctuations on the large-scale transport?
  • Is nonlinear gyrokinetics a mathematically well-posed system of equations? Under what conditions do regular solutions exist?

In the process of addressing these questions, many nontrivial methodological challenges arise:

  • What is the numerically efficient representation of the ring distribution in a 5D phase space?
  • What are the advantages and disadvantages of Eulerian and Lagrangian gyrokinetic solvers, global and flux-tube approaches, grid and PIC codes?
  • How does one account, analytically and numerically, for the role of collisions in a plasma where collision frequency is small, but some level of collisionality must be maintained in order for the entropy production and the consequent heating of the plasma to be correctly captured?
  • How can the techniques of nonlinear dynamics be utilised to construct statistical descriptions of gyrokinetic fluctuations?
  • What is the most appropriate, analytically and numerically, mean-field formalism to describe turbulent transport and how can local ("flux-tube") and global numerical simulations ("transport codes") be combined to produce quantitative predictions for evolving mean profiles?

The programme, while addressing these questions, will aim to step beyond the traditional focus on gyrokinetics for fusion plasmas: to confront a broader group of applied mathematicians and numericists with the practical challenges posed by gyrokinetics; to explore applications to space and astrophysical plasmas; and thus to help forge a cross-disciplinary and versatile community of physicists, astrophysicists and mathematicians, diverse in expertise and background, aware of the spectrum of problems, methods and applications in each other's areas, and dedicated to tackling the grand challenge of understanding kinetic plasma turbulence and transport via a broad collaborative effort.

Unofficial webpage for the event is here.

Additional Sponsor

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Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons