On the possibility of helicity oscillations in the saturation of the Tayler instability

Abstract

Recent numerical results of current-driven instabilities at low magnetic Prandtl number and high Hartmann number support the possibility of a saturation state characterized by helicity oscillations. We investigate the underlying mechanism by analyzing this possibility using a higher order Landau-Ginzburg effective Lagrangian for the weakly nonlinear amplitude dynamics, where the magnetic and velocity perturbations are linearly dependent. We find that, if the mirror symmetry between left-and right-handed modes is spontaneously broken, it is impossible to achieve an oscillating helical state. We argue that the result is likely to hold also for adding higher order terms and in the presence of an explicit symmetry breaking. We conclude that an oscillating saturating state for the Tayler instability is unlikely to depend on the interaction of chiral modes. KEYWORDS instabilities, magnetic fields, magnetohydrodynamics (MHD) INTRODUCTIONThe Tayler instability is the instability of a toroidal field in a stably stratified medium due to the presence of an electric current along the axis of symmetry (Tayler 1973). In its simplest realization, in cylindrical geometry with axial symmetry, it can be shown that a purely toroidal field B (r), where r is the cylindrical radius, is stable against axisymmetric perturbations if it satisfies the condition d(B ∕r)∕dr < 0, and to non-axisymmetric perturbations if d(r B 2 )∕dr ≤ 0. In recent times, this problem has attracted a considerable amount of analytical and numerical investigations (Bodo et al. 2013;Bonanno & Urpin 2011;Ibáñez-Mejía & Braithwaite 2015;Jouve et al. 2015) as well as experimental verification (Seilmayer et al. 2012), mainly for its relevance for our understanding of magnetism in stellar radiative regions (Braithwaite & Spruit 2015;Kitchatinov & Rüdiger 2008) but also for its implications in the tachocline stability problem and the solar dynamo (Bonanno 2013;Kitchatinov & Rüdiger 2007).In particular, it was noted in Bonanno & Urpin (2008) that the unstable modes are helical and, in the presence of a vertical component B z , also invariant under the symmetry transformation P LR z ∶ (m, ) → (−m, − ), with = B z ∕B and m the azimuthal wavenumber. This has led to the speculation that this instability can produce a magnetic -effect which, in turn, could produce a dynamo action (Rüdiger et al. 2012b; Spruit 2002). The effect of a finite Prandtl number has been investigated in cylindrical geometry (Rüdiger et al. 2012a), while the role played by a finite electrical resistivity in liquid conductor has been discussed in Weber et al. (2013). A recent work in this direction discussed the numerical evidence that the saturation state of the Tayler instability at low magnetic Prantdl number and high Hartmann number is characterized by helicity oscillations: a result of important consequences if confirmed .The aim of this paper is to extend the approach to the nonlinear saturation phase of the Tayler instability discussed in Bonanno et al. (2012) in order to account for additiona...

The Tayler instability at low magnetic Prandtl numbers: chiral symmetry breaking and synchronizable helicity oscillations

“…The observed helicity oscillation seems to be a generic feature of the saturated TI for large enough Ha at low Pm (note that Bonanno and Guarnieri [14] could not find helicity oscillations in a dissipation-free model). We ask now for its possible implications for Tayler-Spruit type stellar dynamos.…”

A Model of a Tidally Synchronized Solar Dynamo

“…Beyond flux-tube approximation, the existence of any TI-related α-effect is still a matter of debate. Various authors (Chatterjee et al, 2011;Gellert, Rüdiger, and Hollerbach, 2011;Bonanno et al, 2012Bonanno et al, , 2017 had evidenced spontaneous symmetry breaking between left-and right-handed TI modes, leading indeed to a finite value of α, but mainly for comparably large values of the magnetic Prandtl number [P m], i.e., the ratio between viscosity and magnetic diffusivity. Things are different, though, for the case of low P m, as it is typical for the solar tachocline region where P m is believed to lie in the range 10 −3 ...10 −2 .…”

On the Synchronizability of Tayler–Spruit and Babcock–Leighton Type Dynamos

“…The emergence of such a TI-related α effect is far from trivial, though. For comparable large values of the magnetic Prandtl number [P m], Chatterjee et al (2011);Gellert, Rüdiger, and Hollerbach (2011); Bonanno et al (2012Bonanno et al ( , 2017 found evidence for spontaneous symmetry breaking between left-and righthanded TI modes, leading indeed to a finite value of α. Things are different, however, for low P m (as typical for the solar tachocline) for which we observed a tendency of the TI to produce oscillations of the helicity and the α-effect related to it (Weber et al, 2013(Weber et al, , 2015.…”