Magnetic Suppression of Zonal Flows on a Beta Plane

Abstract

Zonal flows in rotating systems have been previously shown to be suppressed by the imposition of a background magnetic field aligned with the direction of rotation. Understanding the physics behind the suppression may be important in systems found in astrophysical fluid dynamics, such as stellar interiors. However, the mechanism of suppression has not yet been explained. In the idealized setting of a magnetized beta plane, we provide a theoretical explanation that shows how magnetic fluctuations directly counteract the growth of weak zonal flows. Two distinct calculations yield consistent conclusions. The first, which is simpler and more physically transparent, extends the Kelvin-Orr shearing wave to include magnetic fields and shows that weak, long-wavelength shear flow organizes magnetic fluctuations to absorb energy from the mean flow. The second calculation, based on the quasilinear, statistical CE2 framework, is valid for arbitrary wavelength zonal flow and predicts a self-consistent growth rate of the zonal flow. We find that a background magnetic field suppresses zonal flow if the bare Alfvén frequency is comparable to or larger than the bare Rossby frequency. However, suppression can occur for even smaller magnetic fields if the resistivity is sufficiently small enough to allow sizable magnetic fluctuations. Our calculations reproduce the η/B 2 0 = const. scaling that describes the boundary of zonation, as found in previous work, and we explicitly link this scaling to the amplitude of magnetic fluctuations.

Potential Vorticity Mixing in a Tangled Magnetic Field

“…If the magnetic field is large enough, then the zonostrophic instability switches off, as shown numerically on a β −plane Durston & Gilbert 2016) and on a spherical surface (Tobias et al 2011). Theoretically, this suppression of the zonostrophic instability has been described via a straightforward application of QL theory (Tobias et al 2011;Constantinou & Parker 2018), though as we will show here, this approach does not capture the relevant physics.…”

“…Neither tackles the strong stochasticity of the ambient tachocline field. Recent progress on this subject has exploited theoretical approaches based on quasilinear (QL) theory or wave turbulence theory (Constantinou & Parker 2018). These are unable to take into account for the stochasticity of the ambient field; i.e.…”

A Geometric Interpretation of Zonostrophic Instability

“…This statistical state dynamics (SSD) is tractable only with a closure assumption, as a straightforward calculation leads to an infinite hierarchy of equations for the moments (Hopf 1952). A large number of studies in the literature on diverse physical problems ranging from quasi-geostrophic (DelSole 2004;Ioannou 2008, 2009a;Marston 2010) and stratified turbulence (Fitzgerald and Farrell 2018a,b) to turbulence in astrophysical flows (Farrell and Ioannou 2009b;Tobias et al 2011;Parker and Krommes 2013;Constantinou and Parker 2018) and in pipe flows (Constantinou et al 2014b;Farrell et al 2017) have shown that a second-order closure of the SSD is accurate in capturing the characteristics and dynamics of the dominant largescale structures. Such closures of the SSD are either referred to as stochastic structural stability theory (S3T) (Farrell and Ioannou 2003) or second-order cumulant expansion (CE2) (Marston et al 2008).…”

Nonlinear saturation and oscillations of collisionless zonal flows

“…The linear stage of the MI is generally understood, but the dynamics of ZFs at the nonlinear stage is not sufficiently explored. Some progress in this area has been made by applying quasilinear (QL) models (section 2.4), such as the secondorder cumulant expansion theory (CE2) [9][10][11][12][13][14], or the stochastic structural stability theory [30][31][32]; however, those are not particularly intuitive. A more intuitive paradigm was proposed based on a simpler QL model known as the wave-kinetic equation (WKE) [6,7,[33][34][35][36][37][38][39].…”