Determining flows from magnetic field evolution An outline of the approach we’ve adopted at UCB (Welsch, Fisher, Abbett, Regnier)

Determining a physically reasonable flow field is essential for data-driven MHD models. To what extent can flow velocities be inverted from time sequences of magnetogram data? We believe the ideal MHD induction equation should be obeyed…

The standard technique for determining horizontal motions is Local Correlation Tracking… There are many different implementations in use today. The original idea was described in November & Simon (1988) – try to define a velocity by moving sub-images of an image taken at an earlier time to best fit corresponding sub-images of an image taken later. No physics involved. Other versions in use at BBSO, LMSAL We developed our own implementation. For each pixel location, multiply the image by a gaussian centered at the given pixel location, and find the shift between the resulting 2 images that maximizes the cross-correlation function. The cross-correlation function is determined using standard FFT techniques. Our IDL LCT code is publicly available:

LCT tests show it works under a wide variety of conditions … Apply a velocity field to an image consisting of random hash – can LCT correctly recover the velocity?

Recovered velocity fields… Here, it did correctly find the applied horizontal velocity field… VxVx VyVy

Induction equation (z component) Define a new quantity U  such that then

Decomposing vertical and horizontal flow components: Once is known, are known if

Induction + LCT (ILCT) Welsch, Fisher, Abbett (UCB), Regnier (MSU) Induction Eqn. Constrains  Induction Eqn. Constrains  LCT Constrains 

Note that knowledge of transverse fields not needed to solve induction equation! Caveats: If knowledge of real vertical and horizontal velocities needed (as in an MHD code), then need B  to determine v z and v . The vertical component of induction equation doesn’t apply if the LOS and vertical direction differ substantially ILCT formalism needs to be extended away from disk center if it is to be used with scalar magnetograms

Warts, problems with LCT, ILCT With ILCT, velocity is poorly determined near neutral lines or in magnetically featureless areas LCT performs reasonably well in “moving paint” experiments, but does poorly in recovering flows from many MHD simulations. Our LCT algorithm is too slow. It needs to be made faster (is now written in IDL, should probably be re-written in Fortran or C. LCT, ILCT are unconstrained by flows parallel to the field. Magnetic evolution provides no constraint on parallel flows.