Shear flow instabilities in stratified magnetohydrodynamics

Shear flows are ubiquitous in nature and man-made settings and they play a fundamental role in the mixing and transition to turbulence of fluid flow. Shear flow instabilities of electrically-conducting fluids play a critical role in the dynamics of astrophysical objects, such as planets, stars, accretion disks and galaxies. An important motivating example is the thin stably-stratified region in the Sun known as the tachocline, wherein the turbulent dynamics are believed to be driven by shear flow instabilities. In this thesis we investigate how stable stratification and magnetic fields affect instabilities of a Kolmogorov shear flow in different parameter regimes of the Boussinesq magnetohydrodynamic (MHD) framework, extending the two-dimensional (2D) model from Balmforth (2002) to include a uniform horizontal magnetic field aligned with the direction of flow streaming. Using linear stability analysis, we find that the Kolmogorov flow is unstable to three types of linear instabilities in stratified MHD: Sinuous, Varicose and Oscillatory, extending the observations from Algatheem (2023), Algatheem (2024) and Fraser (2022) for non-zero stratification. Sinuous instabilities persist into the ideal regime and are stabilised by both stratification and magnetic fields. We determine that the critical Reynolds number and critical horizontal wavenumber of these instabilities vary according to a power law as a function of stratification for non-zero magnetic field strength. Varicose instabilities require non-zero magnetic field strength to exist and we empirically observe a stratification threshold above which they do not exist. We find that interestingly, an increase in weak stable stratification can increase the linear growth rate of Oscillatory instabilities at significant magnetic field strength in both 2D and 3D geometry. We derive also an analytical growth rate for Oscillatory instabilities in 2D geometry and see that the destabilising effect of weak stable stratification on Kolmogorov flow persists as the parameter regime is extended towards the astrophysically relevant case. Using weakly nonlinear theory, we derive amplitude equations for the two cases of Pe,Rm∼O(1) and Pe,Rm»1, where Pe (the ratio between thermal convection and thermal diffusion) and Rm (the ratio between magnetic induction and magnetic diffusion) denote the Péclet and magnetic Reynolds numbers respectively. This extends the work presented in §4-§5 of Balmforth (2002) to include magnetic field effects. By numerically solving the amplitude equation at Pe,Rm∼O(1), we see that a weak magnetic field halts the inverse energy cascade, in a similar manner to how Balmforth (2002) showed that weak stratification does. At Pe,Rm»1, the weakly nonlinear model predicts boundary layers in the total temperature and magnetic potential fields at the inflection points of the Kolmogorov flow that expand in time. When linearised, both of these amplitude equations give analytical growth rates, which are comparable to Sinuous instabilities in the respective limits of Pe and Rm. We simulate each of the linear instability types to nonlinear saturation. By considering the energetics of the full nonlinear model, we investigate the role of different energy fluxes, sometimes finding different effects in the linear and nonlinear stages of evolution of the instability. We also study the temporal and spatial structure of the mean flow, field and temperature. We verify the emergence of vertically-expanding in time boundary layers (predicted by the weakly nonlinear theory) by numerically solving the fully nonlinear governing equations for high Pe and Rm. A notable finding for a range of Rm»1 is that the Maxwell stress takes over as the dominant energy contributor to the magnetic energy in the nonlinear phase, initiating a sustained nonlinear amplification of the magnetic field strength over the time interval of simulation.<p></p>

Author dc:creator

• Velizar Kirkow (21043709)

• MHD
• Magnetohydrodynamics
• Shear flow instabilities
• Stable stratification
• Boussinesq approximation
• Periodic boundary conditions
• Idealised model
• Solar Tachocline
• Linear stability
• Weakly nonlinear theory
• Asymptotic methods
• Nonlinear simulations

Statement dc:rights

• All rights reserved
• Open Access after 2027-07-06

Identifier

10779/exe.32907506.v1

OAI identifier oai:identifier

oai:figshare.com:article/32907506