Dear Marcus,
thanks for the hint to your recent loop paper. I browsed through it
but I stumbled already on page three where you outline your philosophy:
You want to test, how close you can fit a potential field to a set of
(stereoscopically derived) field lines. The answer could have bee given
without numerical tests: as close as your parameter space allows.
Well, within a specific model, such as with unipolar charges or dipoles,
you can approximate a set of coronal loops only within a certain tolerance.
Once you reach a best fit, you cannot get closer, regardless how many
free parameters you add. For instance, if an observed loop is a "square",
every model consisting of multiple "semi-circles" cannot improve the fit
by adding more semi-circles. So, how close you can fit the data depends
very much on your chosen parameterization. In this study we tried with
buried magnetic charges, but it was not clear a priori if we get a closer
match to the STEREO loops than standard extrapolation models.
The
same would hold for a NLFField.
They also have constraints from the photospheric field that does not
allow for a better fit than what we measured with the misalignment angles.
We wanted to see if the misalignment is due to a wrong boundary condition
or due to the inadequacy of NLFFF models.
The field you find is probably not
unique, because you may superpose potential fields which vanish along
the field lines.
Of course, one best-fit solution is not unique, there are many solutions
with similar small misalignment.
Hence from a successful fit no conclusion can be drawn about the nature of the true field.
You are right, we cannot tell whether the true field is potential or not,
but we can narrow down the degree of non-potentiality and
have a more accurate magnetic field model that is useful for
many modeling tasks (like hydrodynamic modeling of an active region).
At the end of subsection 2.1: I do not know the paper by Gary (2010),
Gary's paper contains PTA (parameteric Transform Analysis) that
is a general magnetic field deformation concept that can be applied
to any model.
but Conlon and Gallagher (2010) is not NLFFF.
You are right, he uses only LFFF (I have to correct that, thanks!)
And finally: What do you mean by "Abelian"? Commonly this term is used for a
group property indicating the the group operation is commutative. What does "Abelian" mean for a magnetic vector field ?
Abelian in the sense of commutative and linear. I wanted to emphasize