On the meaning and inapplicability of the Zeldovich relations of magnetohydrodynamics

Considering a plasma with an initially weak large scale field subject to nonhelical turbulent stirring, Zeldovich (1957), for two‐dimensions, followed by others for three dimensions, have presented formulae of the form 〈 b 2 〉 = f ( R M ) $ {\bar B}^2 $ . Such “Zeldovich relations” have sometimes been interpreted to provide steady‐state relations between the energy associated with the fluctuating magnetic field and that associated with a large scale or mean field multiplied by a function f that depends on spatial dimension and a magnetic Reynolds number R M . Here we dissect the origin of these relations and pinpoint pitfalls that show why they are inapplicable to realistic, dynamical MHD turbulence and that they disagree with many numerical simulations. For 2D, we show that when the total magnetic field is determined by a vector potential, the standard Zeldovich relation applies only transiently, characterizing a maximum possible value that the field energy can reach before necessarily decaying. In 3D, we show that the standard Zeldovich relations are derived by balancing subdominant terms. In contrast, balancing the dominant terms shows that the fluctuating field can grow to a value independent of R M and the initially imposed $ \bar B $ , as seen in numerical simulations. We also emphasize that these Zeldovich relations of nonhelical turbulence imply nothing about the amount mean field growth in a helical dynamo. In short, by re‐analyzing the origin of the Zeldovich relations, we highlight that they are inapplicable to realistic steady‐states of large R M MHD turbulence. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)