We propose the mean field dynamo model for the generation of strongest magnetic fields, $Bsim 10^{15},{rm G}$, in a neutron star (NS) accounting for the chiral magnetic effect (CME) driven by the shock in a supernova (SN) progenitor of that NS. The temperature jump at a narrow shock front, where an initial magnetic field existing in inflowing matter rises sharply, is the source of the CME that prevails significantly the erasure of the CME due to the spin-flip through Coulomb collisions in plasma. The growth of the magnetic field just behind the shock given by the instability term $ ablatimes (alpha {bf B})$ in induction equation, stops after a successful SN explosion that throws out the mantle of a protoneutron star. As a result, such an explosion interrupts the transfer of strongly magnetized plasma from the shock onto NS surface and leads to the saturation of the magnetic field. Assuming the rigid protostar rotation, we employ the mean field dynamo, which is similar to the $alpha^2$-dynamo known in the standard magnetohydrodynamics (MHD). The novelty of our model is that $alpha^2$-dynamo is based on concepts of particle physics, applied in MHD, rather than by a mirror asymmetry of convective vortices in the rotating convection.
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