Statistical State Dynamics of Couette MHD Turbulence

Abstract

The roll streak structure (RSS) is ubiquitous in shear flow turbulence and is fundamental to the dynamics of the self-sustaining process (SSP) maintaining the turbulent state. The formation and maintenance of the RSS in wall-bounded shear flow suggest the presence of an underlying instability that has recently been identified using statistical state dynamics (SSD). Due to the parallelism between the Navier-Stokes equation and the induction equation, it is reasonable to inquire whether the RSS in wall-bounded shear flow has a counterpart in the MHD equations formulated as an SSD. In this work we show that this is the case and that an analytic solution for the composite velocitymagnetic field RSS in the MHD SSD also arises from an instability, that this instability equilibrates to either a fixed point or to a turbulent state, that these turbulent statistical equilibria may be self sustaining, and that both the fixed point and the turbulent states may correspond to large scale coherent dynamos.

Figures (12)

• Figure 1: Panel(a): stability diagram for turbulence excitation parameters $\epsilon_{\boldsymbol{u'},\boldsymbol{u'}}$ and $\epsilon_{\boldsymbol{B'},\boldsymbol{B'}}$. Panel (b): streamwise velocity contours and spanwise/cross-stream velocity vectors of most unstable eigenmode. Panel (c): streamwise magnetic field contours and spanwise/cross-stream magnetic field vectors of most unstable eigenmode. Eigenfunction structures have been normalized so that the maximum value of $\delta \overline{u}_x$ is unity. The location of the unstable mode at $\epsilon_{\boldsymbol{u'},\boldsymbol{u'}}=4$ and $\epsilon_{\boldsymbol{B'},\boldsymbol{B'}}=5$ is indicated with a star in panel (a).
• Figure 2: Equilibrium regime diagram for turbulence excitation parameters $\epsilon_{\boldsymbol{u',u'}}$, $\epsilon_{\boldsymbol{B',B'}}$ for $Re=Re_m=400$
• Figure 3: Fixed point equilibrium RSS supporting finite amplitude components in both the mean velocity field $\overline{\boldsymbol{u}}$ and the mean magnetic field $\overline{\boldsymbol{B}}$. On the left: streak velocity $\overline{u}_{xs}=\overline{u}_x-[\overline{u}_x]_z$ (contours) and roll velocity $(\overline{u}_{y},\overline{u}_z)$ (vectors). On the right: streak magnetic field $\overline{B}_{xs}=\overline{B}_x-[\overline{B}_x]_z$ (contours) and roll magnetic field $(\overline{B}_y,\overline{B}_z)$ (vectors). Parameters: $\epsilon_{\boldsymbol{u'u'}}=2.35$ and $\epsilon_{\boldsymbol{B'B'}}=2.35$, $Re=Re_m=400$
• Figure 4: Snapshots of turbulent equilibria supporting finite amplitude components in both the mean velocity field $\overline{\boldsymbol{u}}$ and the mean magnetic field $\overline{\boldsymbol{B}}$. For each snapshot, on the left: streak velocity $\overline{u}_{xs}=\overline{u}_x-[\overline{u}_x]_z$ (contours) and roll velocity $(\overline{u}_{y},\overline{u}_z)$ (vectors). On the right: streak magnetic field $\overline{B}_{xs}=\overline{B}_x-[\overline{B}_x]_z$ (contours) and roll magnetic field $(\overline{B}_y,\overline{B}_z)$ (vectors). Parameters: $\epsilon_{\boldsymbol{u'u'}}=3.9$ and $\epsilon_{\boldsymbol{B'B'}}=3.9$, $Re=Re_m=400$
• Figure 5: Snapshots of self sustaining turbulent equilibria sustaining finite amplitude components in only the mean velocity $\overline{\boldsymbol{u}}$. For each snapshot, on the left: streak velocity $\overline{u}_{xs}=\overline{u}_x-[\overline{u}_x]_z$ (contours) and roll velocity $(\overline{u}_{y},\overline{u}_z)$ (vectors). On the right: streak magnetic field $\overline{B}_{xs}=\overline{B}_x-[\overline{B}_x]_z$ (contours) and roll magnetic field $(\overline{B}_y,\overline{B}_z)$ (vectors). Parameters: $Re=Re_m=400$, $\epsilon_{\boldsymbol{u'u'}}=\epsilon_{\boldsymbol{B'B'}}=0$