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 thr ee-dimensional undular instability of a horizontal magnetic layer with a fi xed field line direction, embedded in an adiabatically stratified atmospher e. We consider the limit of very high plasma beta, representing the conditi on at the base of the solar convection zone. We show that, in the limit of high plasma beta and nearly adiabatic stratification, the anelastic formula tion 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 direc tion of the magnetic field) and is unstable only to three-dimensional undul ar modes (with nonzero wavenumbers in both horizontal directions parallel a nd 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 be cause 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, becau se of the stretching of the loop field lines. Even though the initial magne tic field is untwisted, it is found that the upward moving tube cross secti ons of the arching tubes maintain their cohesion as they rise through the d istance 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 w ith 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 sim ulations suggests that the minimum value for the ratio of poloidal field st rength over toroidal field strength (i.e., twist) at the base of the solar convection zone, necessary to ensure a cohesive rise of magnetic flux throu gh the solar convection zone, may be far less than that suggested by the tw o-dimensional calculations of the buoyant rise of infinitely long horizonta l tubes.