Steady state existence of passive vector fields under the Kraichnan model

TL;DR

This paper investigates the conditions under which steady states exist for passive vector fields in the Kraichnan model, encompassing MHD, LPM, and LNS equations, across various dimensions and initial conditions.

Contribution

It extends previous results by identifying the parameter ranges for steady state existence in the linearized Navier-Stokes model within the Kraichnan framework.

Findings

Steady state existence conditions for MHD and LPM models confirmed.

New bounds for the Kraichnan roughness parameter in LNS model derived.

Results applicable to both isotropic and anisotropic initial conditions.

Abstract

The steady state existence problem for Kraichnan advected passive vector models is considered for isotropic and anisotropic initial values in arbitrary dimension. The model includes the magnetohydrodynamic (MHD) equations, linear pressure model (LPM) and linearized Navier-Stokes (LNS) equations. In addition to reproducing the previously known results for the MHD and linear pressure model, we obtain the values of the Kraichnan model roughness parameter $\xi$ for which the LNS steady state exists.