Magnetohydrodynamic turbulence follows Kolmogorov scaling

The problem of scaling in isotropic magnetohydrodynamic (MHD) turbulence has remained unresolved, with competing predictions of $k^{-5/3}$ (Kolmogorov) and $k^{-3/2}$ (Iroshnikov-Kraichnan) scalings. In this talk I will present results of high-resolution numerical simulations on $8192^2$ and $1024^3$ grids. The computed energy spectra of the total energy, cross helicity, and Elsasser variables are closer to $k^{-5/3}$ than $k^{-3/2}$. More importantly, our detailed analyses of structure functions, intermittency exponents, and energy fluxes demonstrate robust support for Kolmogorov scaling. For example, the fluxes of the Elsasser variables are different for imbalanced MHD, and the third-order structure functions for proportional to $l$. The magnetic energy shows $k^{-5/3}$ spectrum, but the kinetic energy exhibits $k^{-3/2}$ spectrum; the latter spectrum is due to the energy transfers from the magnetic field to the velocity field. We also discuss critical balance and Boldyrev et al.'s dyanmic alignment in relation to anistropic MHD turbulence. Our findings firmly establish Kolmogorov scaling in MHD turbulence, significantly improving the theoretical foundations for modeling astrophysical turbulence and dynamo processes.