The zigzag path of buoyant magnetic tubes and the generation of vorticity along their periphery

We study the generation of vorticity in the magnetic boundary layer of buoy ant magnetic tubes and its consequences for the trajectory of magnetic stru ctures rising in the solar convection zone. When the Reynolds number is wel l above 1, the wake trailing the tube sheds vortex rolls, producing a von K arman vortex street, similar to the case of flows around rigid cylinders. T he shedding of a vortex roll causes an imbalance of vorticity in the tube. The ensuing vortex force excites a transverse oscillation of the flux tube as a whole so that it follows a zigzag upward path instead of rising along a straight vertical line. In this paper, the physics of vorticity generatio n in the boundary layer is discussed and scaling laws for the relevant term s are presented. We then solve the two-dimensional magnetohydrodynamic equa tions numerically, measure the vorticity production, and show the formation of a vortex street and the consequent sinusoidal path of the magnetic flux tube. For high values of the plasma beta, the trajectory of the tubes is f ound to be independent of beta but varying with the Reynolds number. The St rouhal number, which measures the frequency of vortex shedding, shows in ou r rising tubes only a weak dependence with the Reynolds numbers, a result a lso obtained in the rigid-tube laboratory experiments. In fact, the actual values measured in the latter are also close to those of our numerical calc ulations. As the Reynolds numbers are increased, the amplitude of the lift force grows and the trajectory becomes increasingly complicated. It is show n how a simple analytical equation (which includes buoyancy, drag, and vort ex forces) can satisfactorily reproduce the computed trajectories. The diff erent regimes of rise can be best understood in terms of a dimensionless pa rameter, chi, which measures the importance of the vortex force as compared with the buoyancy and drag forces. For chi2 much less than 1, the rise is drag dominated and the trajectory is mainly vertical with a small lateral o scillation superposed. When chi becomes larger than 1, there is a transitio n toward a drag-free regime and epicycles are added to the trajectory.