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Influence of small scale $E_M$ and $H_M$ on the growth of large scale magnetic field

We investigated the influence of small scale magnetic energy ($E_M$) and magnetic helicity ($H_M$) on the growth rate ($γ$) of large scale magnetic field ($\overline{\bf B}$). $H_M$ that plays a key role in MHD dynamo is a topological concept describing the structural properties of magnetic fields. So, it is not possible to differentiate the intrinsic properties of $H_M$ from the influence of $E_M$, and vice versa. However, to understand MHD dynamo the features of helical and nonhelical magnetic field should be made clear. For this, we made a detour: we gave each simulation set its own initial condition ($IC$, same $E_M$(0) and specific $H_M$(0) at $k_f=5$), and then drove the system with positive helical kinetic energy($k_f=5$). According to the simulation results, $E_M$(0), whether or not helical, increases the growth rate of $\overline{\bf B}$. The positive $H_M$(0) boosts the increased growth rate, but the negative $H_M$(0) decreases it. To explain these results two coupled equations of $H_M$ and $E_M$ were derived and solved using a simple approximate method. The equations imply that helical magnetic field generates the whole (helical and nonhelical) magnetic field but quenches itself. Nonhelical magnetic field also generates the whole magnetic field but quenches itself. The initially given $E_M$(0) modifies the electromotive force ($\langle {\bf v}{\bf \times} {\bf b}\rangle$, $EMF$) and generates new terms. The effects of these terms depend on the magnetic diffusivity $η$, position of initial conditions $k_f$, and time. But the influence disappears as time passes ($\sim e^{-ηk_f^2 t}$), so the saturated magnetic fields are independent of the initial conditions.