Time Scalings of Entropy Transfer Process in Free Convection

Abstract

To study one elementary process of entropy cascade in hard turbulence, we propose a shell model which governsthe temperature gradient rotated through the angle +\frac{π}{2}, χ ≡ (∂_yT, -∂_xT), and the velocity gradient, i.e., strain rates.This model is based on the correspondence of the dynamics ofχin a two-dimensional free convection system to that of the vorticity in a three-dimensional Navier-Stokes system.We obtain the following results: 1) A steady solution which satisfies Bolgiano-Obukhov (BO) scaling in the viscous case exists.2) After a small disturbance of large scale is added to the steady solution, time series of the modes in the inertial range become similar to each other under the proper scaling, and each mode has its own characteristic time predicted by BO scaling.3) When the same disturbance is added to the null state, the modes also evolve similarly in the inertial range.The characteristic times of the modes are, however, explained by a dimensional analysis based on total entropy which is different from BO scaling.