Statistics of active versus passive advections in magnetohydrodynamic turbulence

Abstract

Active turbulent advection is considered in the context of magneto-hydrodynamics. In this case, an auxiliary passive field bears no apparent connection to the active field. The scaling properties of the two fields are different. In the framework of a shell model, we show that the two-point structure function of the passive field has a unique zero mode, characterizing the scaling of this field only. In other words, the existence of statistical invariants for the decaying passive field carries no information on the scaling properties of the active field.PACS numbers: 47.27.-i,47.10.+g In the context of turbulent advection, the understanding of fluid turbulence has greatly improved in the recent years [1]. The anomalous scaling has been shown to be universal and connected to the existence of statistical integrals of motion [2]. In [3], it was shown that the statistically conserved structures of decaying passive turbulence dominate the statistics of forced turbulence, thus offering a rather general framework for understanding the universality of anomalous scaling in forced turbulence.Let φ be a decaying field transported by a stationary turbulent flow. The linearity of the advection implies the following relation for the correlation functions :where we used the compact notation r ≡ r 1 , . . . , r N to denote a collection of N position vectors. Equation (1) tells us there exists a linear operator P (N ) that propagates the nth order correlation function from time t 0 to time t. Without fresh input, that is in the absence of forcing, the correlation functions of φ decay due to dissipative effects. Nevertheless, as conjectured in [3], there exist special functions Z (N ) that are left eigenfunctions of eigenvalue 1 of the operator P (N ) ,such thatis preserved in time. I (N ) and Z (N ) are respectively called a statistical integral of motion and a statistically * thomas.gilbert@inln.cnrs.fr † dhruba@physics.iisc.ernet.in preserved structure of order N , also referred to as zero modes [15]. Now, consider the same passive advection problem with an external forcing, such that the system reaches a stationary state. Define, the correlation function of φ in that stationary state to be,where the symbol · f denotes averaging over the statistical stationary state. It was conjectured in Ref [3], that the anomalous part of F (N ) ( r) is dominated by the leading zero modes of the decaying problem, i. e. Z (N ) ∼ F (N ) . The conjecture was verified in the context of a shell model for passive scalar advection.In subsequent studies, it was discovered that the existence of statistical invariants of the motion for passive turbulence may help understand the statistics of active turbulence, a case where the advected quantity affects the dynamics of the advecting field. In [4] and [5,6], the case of thermal convection in the Boussinesq approximation was studied. There it was shown that the scaling of the active field is also dominated by the statistically preserved structures of auxiliary passive fields. It is yet unclear how gener...

Comparative experimental study of local mixing of active and passive scalars in turbulent thermal convection

“…For magnetohydrodynamics, Celani et al [10,12] showed that a passive scalar displays a direct cascade towards the small scales while the active magnetic potential builds up large-scale structures in an inverse cascade process. Gilbert and Mitra [14] further found that active and passive fields have different scaling properties in the framework of a shell model. Ching et al [13] argued that such different scaling behaviors are due to the fact that the active equations possess additional conservation laws and the zero modes of the passive problem is not the leading factor that dominates the structure functions of the active field.…”