Statistical laws of a one-dimensional model of turbulent flows subject to an external random force

• Published in: Nonlinearity Vol. 36; no. 8; pp. 4283 - 4302
• Main Authors: Tsuji, Yuta, Sakajo, Takashi
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.08.2023
• Subjects: 35R60; 37L55; 5C30; 76F20; 76M35; energy cascade; hydrodynamic equation; inertial range; random partial differential equations; structure function; the generalized polynomial chaos; turbulence
• DOI: 10.1088/1361-6544/acdfa3
• ISSN: 0951-7715

This paper provides mathematical and numerical analysis of a one-dimensional model of turbulent flow generating the anomalous cascade of the inviscid conserved quantity. The model is based on the generalized Constantin–Lax–Majda–DeGregorio (gCLMG) equation with viscous dissipation under a large-scale forcing. Suppose that the forcing function and the initial data are random variables defined on a certain probability space. Then, the equation is regarded as a random partial differential equation. We prove the global existence of a unique solution to the gCLMG equation, from which a stochastic process is defined. In addition, by approximating the solutions numerically by Galerkin approximation of random variables with generalized polynomial chaos, we confirm the existence of a steady distribution. We find that the steady distribution reproduces qualitatively the same cascades of the energy and the enstrophy spectra as those of a turbulent flow generated by randomly moving pulse (Matsumoto and Sakajo 2016 Phys. Rev. E 93 053101). We also investigate the structure functions, showing intermittency.