Speaker
V. Baran
Description
See the full Abstract at http://ocs.ciemat.es/EPS2018ABS/pdf/P1.4016.pdf
Weak drift wave turbulence and the statistics of random matrices
F. Spineanu, M. Vlad, V. Baran
National Institute of Laser, Plasma and Radiation Physics, Bucharest, Romania
A statistical analysis of the drift wave (weak) turbulence necessarily starts with the linear
eigenmodes. A weak nonlinearity can be seen as a vertex of an interaction where the elemen-
tary propagators correspond to the set of orthogonal eigenfunction of the linear operator and a
renormalization theory can be developed. The first nonlinearity is the interaction between the
non-adiabatic part of the density response and the electrostatic perturbation, which requires two
field calculation. We note, in the present work, the possibility of another technical approach
which introduces the nonlinearity through a perturbation of the complex roots of the functions
of the base.
For the drift wave in sheared magnetic field the eigenfunctions are Hermite polynomials.
With the order scaled by an artificial time parameter (which maintains the orthogonality) the
Hermite polynomials verify an equation of diffusion with a negative coefficient of diffusion. By
an inverse Hopf-Cole transformation one obtains the Burgers equation with the same negative
viscosity [Blaizot&Nowak, Phys Rev E 82, 051115 (2010)]. Evolving in the artificial time,
the solution of this equation exhibits a shock formation, which, due to the negative viscosity
is accompanied by oscillations. There is a connection between this solution and the average
resolvent of the hermitean matrix with Gaussian random entries.
Now we interpret the introduction of the weak nonlinearity of the turbulent drift waves as a
broken orthogonality of the modified set of functions, which depart from the linear drift wave
eigenmodes. Then the diffusion with a negative coefficient is modified by an averaged term
which acts as a source. However we adopt the approximative procedure to modify directly
the complex singularities that define the inverse Hopf-Cole transformed function. Since this is
connected with the resolvent of the hermitean random matrix set, the modification is reflected
in the density of the eigenvalues.
We discuss the possibility to use this technical approach in order to obtain renormalization of
the drift wave propagator in weak turbulence.
