Statistics of active versus passive advections in magnetohydrodynamic turbulence

Active turbulent advection is considered in the context of magnetohydrodynamics. In this case, an auxiliary passive field bears no apparent connection to the active field. The scaling properties of the two fields are different. In the framework of a shell model, we show that the two-point structure function of the passive field has a unique zero mode, characterizing the scaling of this field only. In other words, the existence of statistical invariants for the decaying passive field carries no information on the scaling properties of the active field.

Figure 1

The elements of ${\mathcal{P}}_{n,20}^{\left(2\right)}$ for the scalar field Eq. (13) advected by the magnetohydrodynamic field Eqs. (5, 6), where the parameter was chosen to be $\alpha =2$. The times displayed are $.4$, $.55$, $.7$, $.85$, and $1$, respectively (measured in the natural time units of the model). The horizontal axis corresponds to the shell numbers. The units on the vertical scale are arbitrary. The simulation was done using a total of $35$ shells, with the first shell wave number $k_0=1∕16$ and a shell spacing of $\lambda =2$. All the fields were dissipated on the small scales with a term $\nu k_n^2$, with $\nu ={10}^{-12}$. The advecting fields were forced on shells $5–7$ with white delta correlated noise of amplitudes $1∕\sqrt{2}$, $1∕2\sqrt{2}$, and $1∕4$, respectively. Moreover, the phase of the forcing on shell $7$ was taken to be equal to the sum of the phases of the forcings on shells $5$ and $6$.

Figure 2

The curves of Fig. 1 collapsed according to Eq. (18).