Statistical State Dynamics of Large-Scale Structure Formation in Shallow Water Magnetohydrodynamic Turbulence

Abstract

Zonal jets (ZJ) are prominent coherent structures that spontaneously emerge from the background turbulent state in both stellar and planetary atmospheres. Although formation and maintenance of coherent jets from small scale hydrodynamic turbulence is well-documented, the mechanism underlying this phenomenon remains controversial. The dynamics of the Earth's polar jet and that of the quasi-biennial oscillation of the equatorial stratosphere have been analytically explained using the Statistical State Dynamics (SSD) framework applied to mid-latitude beta-plane and stratified turbulence of the equatorial equatorial,respectively (Farrell & Ioannou 2003). Extension of SSD to the shallow water equations of the equatorial beta-plane provided a corresponding theory for the dynamics of Jovian jets (Farrell & Ioannou 2009). However, the influence of Lorentz forces in the dynamics of a substantial subset of coherent structures observed in both planetary and stellar turbulence motivates the further extension of SSD analysis of coherent structure formation to magnetohydrodynamics (MHD) turbulence. In this work, we apply the SSD framework to shallow water MHD turbulence to study coherent structure dynamics in which both Reynolds and Maxwell stresses are involved. Perturbative and nonlinear equilibria SSD solutions reveal formation and statistical equilibration of zonal jet-toroidal field structure (ZJTFS) with both fixed point and time-dependent oscillation behavior with implications for understanding coherent structure formation in MHD turbulence including steady jets such as the solar super-rotation and time-dependent phenomena such as the 22 year old solar cycle.

Figures (19)

• Figure 1: Mean zonal velocity component, $\overline{u}_x$, of the fixed point SW hydrodynamic equilibrium at $\epsilon_{\boldsymbol{u}'\boldsymbol{u}'}=0.4, ~\epsilon_{\boldsymbol{B'}\boldsymbol{B}'}=0$, $\nu_u=0.09,~\nu_h=0.36,~\frac{\nu}{\eta}=0.25$, $r_B=r_u=r_h=0.1$, $f_0=0$, $\beta=1.8$, $g=11.1$.
• Figure 2: Mean magnetic field components of the unstable S3T eigenmode for the equilibrium velocity jet shown in figure \ref{['fig:zonaljets']}. Panel $(a):$ mean toroidal component, $\delta \overline{B}_x$. Panel $(b):$ mean poloidal component, $\delta \overline{B}_y$. Growth rate $Re(\sigma)=0.76$. Parameters: $\epsilon_{\boldsymbol{u}'\boldsymbol{u}'}=0.4,~ \epsilon_{\boldsymbol{B'}\boldsymbol{B}'}=0$$\nu_u=0.09,~\nu_h=0.36,~\frac{\nu}{\eta}=0.25$, $r_B=r_u=r_h=0.1$, $f_0=0$, $\beta=1.8$, $g=11.1$.
• Figure 3: Forcings of the mean poloidal field component, $\delta \overline{B}_y$, of the unstable eigenmode showing growth rate, $Re(\sigma) \cdot \delta \overline{B}_y$, dissipation, $\Pi_{\delta \overline{B}_yA}$, fluctuation-fluctuation advection, $\Pi_{\delta \overline{B}_yB}$, and fluctuation-fluctuation tilting, $\Pi_{\delta \overline{B}_yC}$. $Re(\sigma)=0.76$, $\frac{\nu}{\eta}=0.25$, $\epsilon_{\boldsymbol{u}'\boldsymbol{u'}}=0.4$, and $\epsilon_{\boldsymbol{B'}\boldsymbol{B}'}=0$.
• Figure 4: Stability diagram for the large scale dynamo instability of the fixed-point equilibrium velocity jet shown in figure \ref{['fig:zonaljets']} as a function of magnetic Prandtl number, $\frac{\nu}{\eta}$. Dashed black line is line of neutral stability. Parameters are: $k_x=6.25$$\epsilon_{\boldsymbol{u}'\boldsymbol{u}'}=0.4,~\epsilon_{\boldsymbol{B'}\boldsymbol{B}'}=0$, $\nu_u=0.09,~\nu_h=0.36$, $r_B=r_u=r_h=0.1$, $f_0=0$, $\beta=1.8$, $g=11.1$.
• Figure 5: Stability diagram for the large scale dynamo instability of the fixed-point equilibrium velocity jet shown in figure 1 as a function of zonal wavenumber, $k_x$. Dashed black line is the line of neutral stability. Parameters are: $\epsilon_{\boldsymbol{u}'\boldsymbol{u}'}=0.4,~\epsilon_{\boldsymbol{B'}\boldsymbol{B}'}=0$$\nu_u=0.09,~\nu_h=0.36,~\frac{\nu}{\eta}=0.25$, $r_B=r_u=r_h=0.1$, $f_0=0$, $\beta=1.8$, $g=11.1$.