Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2010-08-02
Nonlinear Sciences
Chaotic Dynamics
more details of the subtleties are provided in the version for JFM
Scientific paper
An electrostatic gyrokinetic model for the deviation from Maxwellian distribution is used to study the dual cascade feature in the magnetized plasma (kinetic) turbulence in a 2D slab geometry. Only a finite range of spacial Fourier modes are kept and the Gibbs statistics are calculated with one ($E$) plus a continuum ($G(v)$) of constants of motion. The covariance density with continuous velocity is obtained by doing functional inversion and it is found that kinetic effects greatly enrich the physics of the absolute equilibria; but, the qualitative feature in physical space is similar to other 2D fluid models and that the conventional dual cascade arguments may be carried over \textit{mutatis mutandis}. A finite extra contribution to the Fourier spectrum of $g^2$ emerges once a cutoff scale $\Delta v$ of velocity arrises from the numerical discretization/coarse graining or other physical mechanisms. This contribution may seriously deteriorate the equipartition of $G(v)$ over the wave vectors and may drastically change the relation between the spectrum of $G$ and that of $E$, which then may lead to some variations of the dual cascade in numerical simulations with under-resolved $v$ and in physical systems with a large collisional and/or dissipative scale of $v$. The treatment of the 2D gyrokinetic system can also be applied to another cases of similar feature with extra dimension besides space, and the calculation for $(3+2)$-D gyrokinetics is shown as an example.
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