Resonant Instability of mixed poloidal-toroidal magnetic configurations

Resonant Instability of mixed poloidal-toroidal magnetic configurations Giovanni Licciardello(1,3) ,Rainer Arlt(2) ,Alfio Bonanno(1,2) ,Domenico G. Meduri(1,2) 1INAF – Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123 Catania, Italy 2Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany 3Dipartimento di Fisica e Astronomia “Ettore Majorana”, Università di Catania, Via S. Sofia 64, 95123, Catania, Italy 1. Introduction Our understanding of stellar evolution remains incomplete. Observations like the slow, rigid rotation of radiative cores and strong internal magnetic fields (≈10 kG [1]) in red giants lack a clear theoretical explanation. MHD instabilities may play an important role by enhancing angular momentum transport or rearranging magnetic fields. Among the most studied MHD instabilities in an astrophysical context is the Tayler instability, which destabilizes purely toroidal fields even under strong stratification, but gets suppressed by poloidal components. Here, we present our findings on a resonant instability that destabilizes mixed configurations and remains active under stratification, even when Tayler instability is suppressed by poloidal fields. 2. Previous Studies The presence of a resonant instability affecting mixed configurations is well known in plasma physics. It was first identified by Suydam [2] through a local criterion involving magnetic shear and pressure gradients. It appears only for a narrow band of wavenumbers that satisfy k·B≈0, and has been observed in cylindrical geometry both in linear analyses and in direct numerical simulations [3]. Since realistic stellar fields are mixed, it is interesting to revisit this type of instability in spherical geometry and including stable stratification. Fig. 1 Snapshot of the density fluctuations arising in the direct numerical simulation [3], taken along the z= 0 plane. The nonaxisymmetric structure with a high m≈9is visible 3. Unstratified Case We performed a linear stability analysis in a spherical shell of radii rin and rout of a mixed poloidal–toroidal magnetic configuration, consisting of a radially increasing toroidal field Bϕ∝rsin θ and a uniform axial field Bz, with a ratio Bz/Bϕ= 0.1, treating the problem globally in the radial direction while adopting a local approximation in latitude and assuming: •Boussinesq approximation •No rotation Ω<< ωA •Negligible viscosity and magnetic diffusivity •Perturbations of the form f(r) exp [−i(lθ +imϕ) + σt] •Conductive temperature gradient T(r)∝1/r •Small but non-zero thermal conductivity r2 ink/ωA= 0.01 In this simplified setup, when gravity is neglected, we find instability, growing at the Alfvenic rate ωA, for all values of the azimuthal wavenumber mwhen the resonance condition k·B≈0is satisfied. 1 Fig. 2 (left) Adimensional growth rates Γ = σ/ωAcalculated at the equator as a function of the latitudinal wavenumber l. (right) Fastest growing eigenfunctions evaluated at a colatitude θ= 10◦. 4. Influence of Gravity When gravity is included, we find that the growth rates of modes with lower azimuthal wavenumbers mare progressively reduced as the strength of stratification increases, with growth rates decreasing as a function of the stratification parameter δ=N/ωA as Γ∝δ−2, while higher mmodes remain strongly unstable, growing at the Alfvenic rate even under strong stratification. 102103 / 10-4 10-3 10-2 10-1 100 ! Bz / B? = 0.1 m = 1 m = 5 m = 20 m = 50 m = 100 m = 200 Fig. 3 Adimensional growth rates Γ = σ/ωAat a colatitude θ= 10◦as a function of the ratio between the Brunt-Väisälä and Alfvén frequencies δ=N/ωA. 6. Acknowledgements Acknowledgements: GL acknowledges support from the research grant “Unveiling the magnetic side of the Stars” funded under the INAF national call for Fundamental Research 2023. DGM acknowledges support from the European Union - NextGenerationEU within the framework of the Italian National Recovery and Resilience Plan (NRRP), Mission 4: Education and Research, Component M4C2, Investment 1.2, through the “Bando Young Researcher 2024” Project Number SOE2024_0000097 “Elucidating the Transport of Angular momentum by MAgnetic fields in red GIAnts (ETAMAGIA)”. AB acknowledges support from the European Union – NextGenerationEU RRF M4C2 1.1 n: 2022HY2NSX “CHRONOS: adjusting the clock(s) to unveil the CHRONO-chemo-dynamical Structure of the Galaxy” and from the research grant “Unveiling the magnetic side of the Stars” funded under the INAF national call for Fundamental Research 2023. 5. Conclusions Our results suggest that mixed fields can undergo a resonant instability that, much like the Tayler instability, is able to operate under strong stable stratification, making it an interesting candidate as a source of turbulence in stably stratified stellar interiors. 6. References [1] Li, G., Deheuvels, S., Ballot, J., & Lignières, F. 2022, Nature, 610, 43 [2] Suydam, B. R., 1958, Proc. of the Second U. N. Internat. Conf. on the Peaceful Uses of Atomic Energy, United Nations, Geneva, 31, 157. [3] Bonanno, A. & Urpin, V. 2011, Phys. Rev. E, 84, 056310