Reflected wave solution of Alfvén wave turbulence

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 51; no. 40; pp. 405501 - 405511
• Main Authors: Bell, N K, Nazarenko, S V
• Format: Journal Article
• Language: English
• Published: IOP Publishing 05.10.2018
• Subjects: Alfvén wave turbulence; Chaotic Dynamics; Nonlinear Sciences; numerical simulation; self-similar solutions
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/aad833

It was found numerically in Galtier et al (2000 J. Plasma Phys. 63 447-88) that the evolution towards the Kolmogorov-Zakharov (KZ) wave action spectrum of weak magneto-hydrodynamic turbulence (here is the wave number component perpendicular to the external magnetic field) consists of two stages. Stage one is characterised by a wave action spectrum with a front propagating to the maximum wave number kmax in a finite time t*, arguably with as , and with a power law which is steeper than KZ, . This stage was recently studied in Bell et al (2017 J. Phys. A: Math. Theor. 50 435501) by identifying it with a self-similar solution of the second kind. In particular, a corrected value of and the structure of the spectral front were found. In the present paper, we study the second (post-t*) stage which is characterised by formation of the KZ spectrum as an inversely propagating wave, from kmax toward lower wave numbers. We identify this stage with a self-similar solution of the third kind, whose similarity coefficients are fixed by the previous (pre-t*) self-similar stage rather than the energy conservation (as in the first-kind self-similarity) or by a nonlinear eigenvalue problem (as in the second-kind self-similarity).