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The Hosking integral (Hosking & Schekochihin 2021, PRX 11, 041005) has recently been recognised as a key invariant constraining the decay of magnetic fields that are statistically homogeneous, isotropic and non-helical. In this talk, I provide a theoretical background on the decay of primordial MHD turbulence, and I construct von-Karman-Howarth-Monin relations for the Hosking correlator (i.e., the two-point correlation function of the magnetic-helicity density) for the first time, and provide exact scaling relations for its behaviour in both incompressible and compressible magnetohydrodynamic (MHD) turbulence. I will show that the condition of rapid decorrelation of velocity and magnetic fields are indeed satisfied, so that the Hosking integral becomes a conserved quantity of ideal, non-helical incompressible magnetohydrodynamic (MHD) turbulence. Furthermore, I provide explicit forms of the relevant fourth and fifth-order two-point correlation functions. Moreover, I outline conditions for its conservation in compressible MHD turbulence. |
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