Wednesday, May 23, 2012

What I Learned While On Blog Hiatus (1)

As noted in my blog of May 3rd, I planned to undertake a 2 ½  week project in numerical simulations for a solar flare model. In that blog, I basically summarized a few of the avenues toward resolving the quantitative complexities using some form of finite difference equations. I began by targeting a differential equation given as:

DIV^2 A + d/dA {½ B(y)^2} = 0

where B(y) is the magnetic induction or magnetic component in the y direction. Historically, two problems have been defined: 1) B(y) = f(A) for whcih the functional form of the field is pre-prescribed (obviously simpler) and 2) d = d (x) or the displacement d(x) of the footpoints (say in an arch model) from the x-axis are imposed. One then follows the evolution of the field through a series of force-free equilibria.

In the ideal world, one would find that the boundaries of the model neatly blend into exterior domains and so we say there exists the possibility of continuous solutions.  Alas, such a model wasn't possible for the 2-dimensional arcade (for a two ribbon flare) that I'd originally planned.  In my particular model, the shear angle magnitude φ , was taken as a proxy for the increase in relative magnetic helicity H( r) such that:

H (r) ~   O( φ ) = arctan (B z /Bφ  )

where the numerator and denominator denote the axial and poloidal field components respectively, each defined in terms of the relevant Bessel function. But...when one consults the appropriate Bessel functions, J0(aR) and J1(aR) one finds they must be truncated. Hence they offer discontinous boundaries, with respect to external fields, if one also intends to include the shear velocity v(φ, W). Readers can find more on Bessel functions in this earlier blog:

http://brane-space.blogspot.com/2007/12/remarkable-world-of-special-functions.html

The boundary conditions for the arcade field, defined at the (truncated)  dimensionless radius aR = 2.4 are:  B z   =  0,   and   B f   =   Bo J1 (aR)/ r.  The choice of the boundary at aR = 2.4 is consistent with a truncated Lundqvist solution, and obviates the problem of unobserved (and unphysical) field reversals which are peculiar to Lundqvist solutions. However, it doesn't resolve the problem of meshing the shear angle with continuous velocity changes of the form:

v (φ,W) = a sin (φ ) exp [1 - 2[φ ]/ W

This inconsistency forced me into a "Hobson's choice" with the simulations: Do I pursue the discretization of PDEs yielding strict Bessel function solutions,  which at least afford some continuity, e.g. with exterior fields joining smoothly to those interior to aR = 2.4 (Dimensionally, the boundary is set equal to a radius R = 5 x 10 7  m)  or do I forego that in favor of pursuing discontinous solutions that have the capacity to introduce the critical parameter v (φ,W)?  More on this in the next instalment.