Using Simulations to Test Methods for Measuring Photospheric Velocity Fields W. P. Abbett, B. T. Welsch, & G. H. Fisher W. P. Abbett, B. T. Welsch, & G. H. Fisher Space Sciences Laboratory, University of California, Berkeley CA Space Sciences Laboratory, University of California, Berkeley CA REFERENCES: Abbett, W.P., Fisher, G.H., Fan Y., & Bercik D.J., 2003, ApJ submitted. Abbett, W.P., Fisher, G.H., Fan Y., & Bercik D.J., 2003, ApJ submitted. Abbett, W.P., Fisher, G.H. & Fan, Y., 2000, ApJ, 540, 548. Abbett, W.P., Fisher, G.H. & Fan, Y., 2000, ApJ, 540, 548. Demoulin, P. & Berger, M.A., 2003, Sol. Phys., in press. Demoulin, P. & Berger, M.A., 2003, Sol. Phys., in press. Longcope, D.W., Klapper, I., Mikic, Z., & Abbett, W.P., 2002, SHINE workshop, Longcope, D.W., Klapper, I., Mikic, Z., & Abbett, W.P., 2002, SHINE workshop, Banff. Banff. AUTHOR Introduction Coronal Mass Ejections (CMEs) are among the primary drivers of space weather, are magnetically driven, and are thought to originate in the low solar corona. Central to our understanding of these dynamic, eruptive events is the strong topological coupling of the coronal field to the photospheric magnetic field -- - since, to a good approximation, coronal magnetic fields are “line-tied” to the photosphere and evolve in response to changes in the Sun’s photospheric field. Thus, observations of the magnetic field at the Sun’s photosphere provide crucial data to aid in the forecasting and interpretation of space weather events. In order to better understand --- and ultimately predict --- the onset and evolution of CMEs, we must incorporate measurements of the vector magnetic field at the photosphere into numerical models of the low corona. Currently, the most common approach is to extrapolate a force-free or potential field into the corona for each magnetogram in a series, and study how the extrapolations’ topological structure evolves in time. These methods, while relatively easy to implement, suffer from the inability to smoothly follow changes in the topology of the corona as it responds to the evolving photosphere --- thus, the utility of static extrapolations as forecasting tools is somewhat limited. One way to extend our ability to predict eruptive events is to use high resolution vector magnetograms to drive MHD models of the corona, which can continuously follow topological evolution. Such models will provide insight into the physical conditions of the solar atmosphere prior to and during an eruption, and will allow researchers to test current theories of CME initiation processes. However, these numerical models require information about the photospheric flow-field in addition to the three components of the magnetic field (e.g. ideal MHD models often require information about the electric field along cell edges in order to properly evolve the magnetic field) --- data generally unavailable for a given series of vector magnetograms. Even if it is possible to obtain observations of the photospheric velocity field for a given time-series of vector magnetograms, there is no guarantee that the prescribed flows will self- consistently satisfy e.g. the induction equation at the driving boundary of the coronal model. This is problematic, since inconsistent velocities can lead to incorrect topological evolution and unphysical Lorentz forces in the coronal model. Thus we are faced with a type of data-assimilation problem, namely: Given a time series of photospheric vector magnetic field measurements, can we obtain a flow field physically consistent with observed photospheric field evolution? Given a time series of photospheric vector magnetic field measurements, can we obtain a flow field physically consistent with observed photospheric field evolution? Ultimately, if large scale 3D numerical models of the solar corona are to be used successfully as a predictive tool, then it is essential to be able to properly incorporate vector magnetogram data and information about photospheric flows into the lower boundary of a dynamic model corona. We specify the temporal evolution of the photospheric magnetic field along a surface using data obtained from high resolution, high cadence vector magnetograms. Since we have no information about the magnetic structure below the photosphere, we require that any velocity field used to drive an ideal MHD model corona at least satisfy the vertical component of the ideal MHD induction equation at the photospheric boundary: Method FIGURE 2: A comparison of the two velocity determination techniques using simulated, synthetic magnetograms where the associated flow field is known. The first column shows the transverse flows for all three cases, and the second column shows the vertical flows (thin contours denote negative vertical velocities, thick lines denote positive velocities, and dashed lines represent the velocity inversion line). The grayscale image in each frame corresponds to the vertical component of the magnetic field (along a horizontal slice near the top of the simulation domain) taken from a sub-surface simulation of an buoyant, untwisted Omega loop that has risen through a non-turbulent, stratified model convection zone. The top row is the simulated velocity field, the middle row is the velocity field obtained using MEF, and the bottom row is the velocity field obtained using EF. FIGURE 3: Same as Figure 2, except that the synthetic magnetogram was generated using a simulation of a twisted flux tube that ascends through a turbulent model convection zone. The simulated flow pattern includes super-granular scale convective cells, and the magnetic field strength of the simulated active region is roughly in equipartition with the kinetic energy of the strongest downflows. Thus, magnetic field is advected away from the center of the flux rope, resulting in the relatively complex morphology. As in Figure 2, the top row represents the flow fields of the simulation, the middle row represents the velocity field generated by MEF, and the bottom row represents the velocity field generated using EF. So how do each of these methods fare? Figure 1 shows the flow field that results from the EF method applied to a sequence of reduced IVM vector magnetograms of AR8210, the CME producing active region of May 1, The red vector field represents horizontal motions determined directly from LCT, and the blue vector field represents the transverse component of the EF velocity field. The contours show the vertical component of the velocity obtained by EF. These initial results are promising --- the areas where EF predicts strong, positive vertical flows correspond to regions where flux is emerging, and the transverse flows obtained via EF have the magnitude and direction expected from the observed evolution of the magnetic structures. However, we wish to move beyond a qualitative assessment of the success or failure of these techniques. To do so, we use the sub-surface simulations of Abbett et al. 2000, 2003 (3D MHD simulations of the evolution of active region scale flux ropes embedded in a model convection zone) to generate synthetic magnetograms that we use to test each method of determining photospheric velocities. Figure 2 compares the velocity fields determined by EF and MEF with flows directly obtained from a horizontal slice near the upper boundary of a sub-surface simulation of an untwisted flux rope emerging through a non-turbulent, stratified model convection zone. In this case, EF reproduces the characteristic transverse velocities reasonably well (in the magnetized region), but fails to capture the vertical flow pattern; Conversely, MEF fails to adequately describe the transverse flows, but is nominally better at reproducing the vertical flows. However, a similar comparison performed using a simulation where a relatively weak flux rope emerges through a turbulent convection zone yields very little agreement, as shown in Figure 3. It is perhaps not surprising that both EF and MEF fail to accurately reproduce the simulated flows of the MHD simulations --- after all, the velocity field obtained from a solution of only the vertical component of the induction equation is not guaranteed to yield the vector field that satisfies the entire MHD system of equations. Thus, further development and testing of velocity inversion techniques is needed before having confidence that the flows so prescribed are those that will produce the self-consistent boundaries necessary for numerical models of the solar corona. FIGURE 1: AR8210 baby. Clearly, if only the magnetic field is known, this equation is under-determined --- to derive the velocity field, additional information is required. Recently, Longcope et al developed a method (dubbed MEF for “Minimum Energy Fitting”) whereby all three components of the flow field are obtained by simultaneously satisfying a finite-difference approximation of the above equation and minimizing the spatially integrated square of the velocity field (the minimization provides the additional constraint necessary to solve the equation). Another way to determine a velocity field consistent with the above equation is to use Local Correlation Tracking (LCT), a widely used technique that cross-correlates successive images to find the displacement of observed features, to determine an empirical flow field --- keeping in mind that horizontal motions obtained in this manner implicitly include the effects of flux emergence (see Demoulin & Berger 2003). Then, in the above equation, we may write the expression in parenthesis as a sum of a gradient of a scalar function and the curl of another, use the Demoulin & Berger 2003 hypothesis to equate this with u LCT B z (obtained empirically), and solve for the two scalar functions. All that remains is to use the fact that flow along field lines doesn’t affect the evolution of the magnetic field at the photosphere, and we can obtain a velocity field that is both consistent with the vertical component of the induction equation, and with the velocities obtained via LCT (u LCT ). We refer to this technique as EF for “Empirical Fitting”. Of course these solutions are by no means unique, and MHD models of the corona have stencils which generally require additional sub-surface information to properly evaluate the derivatives or fluxes necessary to advance a particular algorithm. Nonetheless, these methods provide a means of determining flows which are at least minimally physically self-consistent --- a necessary first step in the effort to incorporate reliable, verifiable, physically-based data assimilation techniques into the photospheric layers of large scale, global dynamic models of the solar corona. Results FIGURE 1: Shown is the vertical magnetic field (grayscale) of one of the IVM vector magnetograms of the May 1, 1998 CME producing active region AR8210 (19:40). The horizontal flow field obtained via LCT is represented by the red arrows. Also shown are the horizontal velocities (blue arrows) and vertical velocities (blue contours) derived using the EF technique. Solid contours indicate outward-directed vertical flows, while dotted contours indicate inward-directed flows.