Jets and instabilities in forced magnetohydrodynamic flows

Abstract

Magnetic fields are present in the solar system and astrophysical bodies (e.g. the Sun’s field, the Earth’s field, and the fields of giant planets, stars and galaxies). Our research examines the effect of magnetic fields on these systems, extending the work of Meshalkin and Sinai (1961) and Manfroi and Young (2002). The results will be useful for understanding the effects of the magnetic field in more turbulent regimes, although this study is concerned with the instabilities associated with classical laminar flow. We aim to investigate the role played by the magnetic field in modifying the stability properties of planar-forced fluid flows. In the absence of magnetic fields, the flow found by a body force, and nonlinear interactions with Rossby waves result in the generation of strong zonal flows. However, we find that the presence of a weak magnetic field suppresses the zonal jet generation. Here we study the instabilities of the Kolmogorov flow. We consider u0 = (0, sin x) as a 2D incompressible flow. In the presence of a mean magnetic field, the dynamics are governed by the Navier–Stokes equations and the induction equation. We perform a classical linear analysis, in which growth rate, stability criteria, and MHD effects are derived. Instabilities are investigated associated with two magnetic field orientations, which can be x-directed (horizontal) or y-directed (vertical) in our two-dimensional system to give an MHD version of Kolmogorov flow. In a basic equilibrium state magnetic field lines are straight for the case of vertical field and sinusoidal for horizontal field with an additional component of the external force balancing the resulting Lorentz force. As the basic state is independent of the y-coordinate we use Fourier analysis to study waves of wavenumber k in the y-direction, using the methods of classical stability theory and numerical solution of eigenvalue problems.

Instability can occur at a single Fourier mode in the y-direction, which can be identified as hydrodynamic instability for sufficiently small wavenumber k, while sometimes occurs at different Fourier modes by introducing Floquet wavenumber l in the x-direction. The classic hydrodynamic instability is suppressed by increasing field strength B0. The linear stability problem can be truncated to determine the eigenvalues of finite matrices numerically, by applying perturbation theory to the limit k → 0 and l → 0. For the vertical field case, there are strong-field branch destabilised Alfv ́en waves that occur when the magnetic Prandtl number P < 1 is less than unity, as found in recent work conducted by Fraser, Cresswell and Garaud (2022). A significant influence of the magnetic field is that it can lead to enhanced in- stabilities in parameter regimes in the case of an imposed horizontal magnetic field. The basic state is formed by a steady solution of field and flow with the driving body force balancing both viscosity and the Lorentz force. Increasing the magnetic field from zero suppresses the purely hydrodynamic instability once again, however, stronger fields reveal a new branch of instabilities. Introducing a non-zero Floquet wavenumber l allows a new branch of instability. A further point of interest is that the most unstable modes can occur for horizontal magnetic field γ → π/2, whereas the system stabilizes when γ → 0, where γ is the angle between the field and y-axis. We also present some results using analytical approximations of eigenvalue perturbation theory in the limit of k, l → 0, for which the instability is large scale compared to the jets. Analytical approximations are provided to represent both weak field and strong field instabilities, as well as to determine the growth rates and thresholds that are in good agreement with the calculated values. Analytical approximations also provided the thresholds of instability with the combination of vertical and horizontal fields for a general magnetic field with variation in the angle γ.

Exploration of the nonlinear evolution of a Kolmogorov flow under MHD effects is an interesting subject that we address in the final chapters of the thesis. Fluid dynamics can be observed in a variety of ways through the nonlinear interactions between Kolmogorov flow and Alfv ́en waves in the case of MHD instabilities. The onset of instability is characterised by the exponential growth in perturbation energy of flow and field specifically kinetic energy, enstrophy and magnetic energy. Perturbation quantities such as energy grow as a result of initial instability. In addition to this, we then justify the choice of magnetic field strength and magnetic Prandtl number and discuss the instability’s behaviour during non-linear evolution. First, we focus on the nonlinear evolution parameterised by the inverse Reynolds number effects in hydrodynamic systems, before investigating the instability evolution of MHD effects. Nonlinear simulation based on two systems is undertaken using the Dedalus frame- work; the hydrodynamic simulation demonstrated the effect of viscosity on instability, checking instability linear growth rate from simulations against the theory. We have found that a small viscosity has a destabilizing effect while increasing viscosity leads to stabilization. Introducing the magnetic field in our simulations exhibits different characteristics of instability, the presence of a weak vertical magnetic field can give hydrodynamic behaviour. While an oscillatory behaviour is obtained for a strong vertical magnetic field and a small magnetic Prandtl number. We also obtained an oscillatory behaviour for a large Prandtl number in the case of the horizontal magnetic field. Another phenomenon can be observed with the horizontal magnetic field; the system exhibits a tearing mode instability as B0 increases. The Flouquet wavenumber l allows large scales in the x-direction and even with a weak horizontal magnetic field in some cases we observe an inverse cascade and the formation of jets governed by the nonlinear properties of the system.