Magnetic field correlations in kinematic two-dimensional magnetohydrodynamic turbulence

The scaling properties of the second order magnetic structure function D_2^{(B)}(r) and the corresponding magnetic correlation function C_2^{(B)}(r) are derived for two-dimensional magnetohydrodynamic turbulence in the kinematic regime where the ratio of kinetic energy to magnetic energy is much larger than one. In this regime the magnetic flux function $\psi$ can be treated as a passive scalar advected in a two-dimensional turbulent flow. Its structure function $D_2^{(\psi)}(r)$ and the one for the magnetic field D_2^{(B)}(r) are connected by an exact relation. We calculate D_2^{(\psi)}(r) and thus D_2^{(B)}(r) within geometric measure theory over a wide range of scales r and magnetic Prandtl numbers Pr_m. The magnetic field correlations follow a r^{-4/3} - scaling law and show an anticorrelation at the beginning of the Batchelor regime indicative of the formation of strongly filamented current sheets. Differences to the full dynamic regime, where the ratio of kinetic to magnetic energies is smaller than in the kinematic case, are discussed.