Nonlinear growth of the three-dimensional undular instability of a horizontal magnetic layer and the formation of arching flux tubes

We use an anelastic MHD code to simulate the nonlinear evolution of the three-dimensional undular instability of a horizontal magnetic layer with a fixed field line direction, embedded in an adiabatically stratified atmosphere. We consider the limit of very high plasma beta, representing the condition at the base of the solar convection zone. We show that, in the limit of high plasma beta and nearly adiabatic stratification, the anelastic formulation gives an accurate description of the magnetic buoyancy instabilities. We specify the thermodynamic conditions of the magnetic layer such that it is stable against pure interchange modes (with zero wavenumber in the direction of the magnetic field) and is unstable only to three-dimensional undular modes (with nonzero wavenumbers in both horizontal directions parallel and perpendicular to the field). Our simulations show that distinct arching flux tubes form as a result of the growth of the three-dimensional undular instability. The apices of the arching tubes become increasingly buoyant because of the diverging mass flow from the apices to the troughs. The field strength at each loop apex decreases with height at a significantly smaller rate in comparison with that for the rise of a horizontal flux tube, because of the stretching of the loop field lines. Even though the initial magnetic field is untwisted, it is found that the upward moving tube cross sections of the arching tubes maintain their cohesion as they rise through the distance of about 1 density scale height included in the simulation domain. The difference in motion between the apices and the troughs causes bending and braiding of the longitudinal field lines, whose restoring tension force improves the cohesion of the rising flux tubes in comparison with previous two-dimensional simulations of the buoyant rise of horizontal flux tubes with no initial twist. In addition, the fact that both the buoyancy and the tension forces grow self-consistently from zero as the tubes arch is also a crucial factor for the cohesion of the rising tubes. The result of our simulations suggests that the minimum value for the ratio of poloidal field strength over toroidal field strength (i.e., twist) at the base of the solar convection zone, necessary to ensure a cohesive rise of magnetic flux through the solar convection zone, may be far less than that suggested by the two-dimensional calculations of the buoyant rise of infinitely long horizontal tubes.