The Dynamic Evolution of Twisted Omega-loops in a 3D Convective Flow W.P. Abbett 1, Y. Fan 2, & G. H. Fisher 1 W.P. Abbett 1, Y. Fan 2, & G. H. Fisher 1 1. Space Sciences Laboratory, University of California, Berkeley CA HAO, National Center for Atmospheric Research, PO Box 3000, Boulder, CO The strong magnetic field observed in and around sunspots and active regions is widely believed to originate via a dynamo mechanism at or near the base of the solar convection zone. If so, then magnetic flux must be transported through the convection zone in such a way as to exhibit the familiar structures and characteristics of active regions observed at the solar surface (eg. Hale’s polarity law, Joy’s law). This can be achieved by assuming that magnetic flux rises buoyantly through the interior in the form of discrete tubes, and that these tubes remain cohesive during their ascent through the convection zone (assumptions implicit in eg. the “thin flux tube” formulation). In contrast to 2D dynamic models of flux tube evolution, recent 3D numerical modeling has shown that, at least in the absence of convective turbulence, magnetic flux tubes positioned near the base of a rotating model convection zone can rise through a stratified interior without significant fragmentation, and without requiring a large amount of initial fieldline twist (see eg. Abbett, Fisher & Fan 2001). However, with the exception of the work of Dorch et al and Fan, Abbett, & Fisher 2003, multi-dimensional models ignore the effects of turbulent flow by modeling the rise of flux tubes in a static, adiabatically stratified model convection zone. This approach implicitly assumes that the magnetic field strength of the tube is sufficient to overcome the dynamic influence of the convective flow patterns. Here we present 3D MHD simulations (in the anelastic approximation) that explore this assumption, and extend the recent results of Fan, Abbett, & Fisher 2003 to active region scale flux ropes and Omega-loops of varying helicity, and morphologies in both rotating and non-rotating convective background states. We then investigate how different treatments of viscosity affect our results. The anelastic approximation results from a scaled variable expansion of the 3D compressible MHD equations about a nearly adiabatically stratified plane-parallel reference atmosphere (which we take to be a polytrope). The time-derivative term in the continuity equation is of sufficiently high order in the expansion that it can be neglected, thereby filtering fast-moving acoustic waves from the calculation. This approximation is well-suited for simulations of sub-sonic processes in the high-beta plasma deep in the solar convection zone. The computational savings inherent in this formulation allows for a large exploration of parameter space. We generate a number of distinct, field-free convective states. In each case, convection is initiated by a small random entropy perturbation within the computational domain, and the simulations progress until the model convection zone is thermally relaxed. We relax states both in the presence and absence of rotation, and with two standard treatments of the viscosity of a Newtonian fluid: the first assumes a constant coefficient of kinematic viscosity as a function of depth, the second a constant dynamic viscosity. Flux ropes with varying amounts of initial twist are inserted near the base of the field-free computational domain, and the flux ropes buoyantly rise toward the surface. Introduction Method Results Figure 1 shows two of the initial non-rotating convective states. The images in the left-hand column are horizontal slices taken from the simulation where dynamic viscosity is held constant, and the images in the right-hand column are from the run where kinematic viscosity is assumed constant. The top row shows slices from each simulation near the top of the domain; the bottom row shows slices closer to the lower boundary. In each run, the coefficients are set to the lowest value possible without introducing excessive numerical noise into the calculation. The domain spans ~5 pressure scale heights, and there is a density difference of a factor of ~20 between the upper and lower boundaries. Thus, the requirement that dynamic viscosity be held constant results in the strong depth dependence of the viscosity evident in the left-hand column of Figure 1. But do the different treatments affect the general characteristics of the buoyant Omega-loops in the simulations? Figure 2 shows how an initially untwisted, horizontal magnetic flux rope with an axial field strength of B 0 =3B eq (where B eq denotes the equipartition field strength with respect to the kinetic energy of the strong downdrafts) placed at the base of the box has evolved into a loop- like structure in both of the background states shown in Figure 1. This choice of axial field strength is optimal for this experiment, since it corresponds to the critical value of Fan, Abbett & Fisher 2003: B c =(H p /a) 1/2 B eq (where H p refers to the pressure scale height, and a refers to the initial radius of the tube). As shown in Figure 3, tubes with initial axial field strengths below this critical value have their evolution dominated by convective flows, while tubes where B 0 significantly exceeds B c evolve as if the convective turbulence were absent. Though the detailed morphology of the tubes shown in Figure 2 differ, the general characteristics of loop evolution remain the same (for example, the dependence of loop fragmentation on initial twist and loop geometry). With B 0 ~ B c, we find that Omega-loops form quite naturally as a result of low entropy downdrafts perturbing a buoyant flux rope (see Figure 2), or flux layer (see Figure 4) positioned near the base of the model convection zone. In the presence of rotation, these loops are acted on by the Coriolis force in such a way that on average, those positioned at high latitude emerge with a large tilt angle, and those at a low latitude emerge with a correspondingly smaller tilt angle (see Figure 5). Though these simulations seem to suggest that active regions can result from a convective perturbation of field stored near the base of the convection zone, we note that Hale’s law (and Joy’s law) can only be reproduced in the simulations if a strong axial field parallel to the E-W direction is a priori imposed in the lower layers. Further, it remains to be seen whether the strong, narrow downdrafts that perturb the flux near the base of the simulation box will continue to penetrate to these deep layers if the Rayleigh number is substantially increased. Figure 1: Entropy of two of the thermally relaxed field-free convective states along horizontal slices toward the top of the computational domain (top row) and near the base of the box (bottom row). Darker colors indicate cooler plasma. Figure 2: |B| along a vertical slice through the center of the domain for two untwisted horizontal magnetic flux tubes initially positioned near the base of two different model convection zones: one where the dynamic viscosity is assumed constant (top) and one where the kinematic viscosity is assumed constant (bottom). The “dips” in the flux ropes coincide with areas of particularly strong downdrafts. Figure 3: Apex cross-section of |B| (left column) as a function of time, and a corresponding slice along the loop (right column) for tubes with no initial twist, but three different values of initial axial field strength: B 0 =B eq (top), B 0 =3B eq (middle), and B 0 =5B eq (bottom). Note that B c ~3B eq. These starting states differ from that of Figure each tube is given an initial entropy perturbation such that the ends are neutrally buoyant. This gives us the ability to investigate the effects of field strength and convection on loops of different geometries. Note that for B 0 =5B eq (bottom frames) the evolution is exactly that of simulations performed in the absence of convective turbulence (see Abbett, Fisher & Fan 2000,2001). Figure 4: Bottom frame: Omega-loops formed from convective turbulence acting on a thin, horizontal sheet of flux inserted near the base of a thermally relaxed, rotating model convection zone positioned at 15 degrees latitude with a magnetic Rossby number of unity. Top frame: corresponding volume rendering of the entropy. Cells size is that of super-granulation. Figure 5: Artificial magnetogram (image of the vertical component of the magnetic field near the upper boundary) for one of the flux ropes formed as a result of a convective perturbation of a flux sheet positioned near the base of a rotating model convection zone positioned at 15 degrees latitude. Note that a line drawn between the leading and trailing polarities of the bipole is inclined from the horizontal E-W direction. This theoretical active region spans roughly 70Mm. Of primary importance in the simulations is the choice of initial (axial) field strength of a flux tube (B 0 ) positioned near the bottom of the simulation domain. If B 0 is less than the critical value of Fan, Abbett, & Fisher 2003 (B c ) by more than a factor of two, then other parameters --- such as the amount of initial twist, other initial perturbations of the flux system at the base of the box, the value of the magnetic Rossby number (a measure of the effects of rotation and the Coriolis force) --- have little to no impact on the subsequent evolution of magnetic structures. In these cases, convective turbulence dominates, tubes lose their cohesion, and after several turnover times, a typical pattern of magneto-convection is evident (eg. magnetic flux in the upper layers becomes concentrated in the intergranular lanes and in low entropy downdrafts). However, if B 0 is equal to or greater than B c, then the other parameters (twist, loop geometry, latitude) play the dominant role in the evolution of the magnetic field, and flux ropes can survive their ascent through the turbulent model convection zone. REFERENCES: Abbett W.P., Fisher G.H. & Fan Y., 2000, ApJ 540:548. Abbett W.P., Fisher G.H. & Fan Y., 2001, ApJ 546:1194. Dorch, S.B.F., Gudiksen, B.V., Abbett, W.P., & Nordlund, A., 2001, A&A 380: , A&A 380:734. Fan Y., Abbett W.P., & Fisher G.H., Jan ApJ (in press) AUTHOR Conclusions