When last we left off, we had seen the putative basis for a linearly evolving non-dimensional shear d(x) which could be mapped onto the x-y plane overlaid on a solar active region. See e.g. the diagram in:
As we change the (dimensionless) shear through progressive numerical substitutions, the shear gradient m increases so that the dimensionless shear kφ increases to a maximum of 1, then decreases. Likewise, in doing this, one finds the line y = mx altering with m to yield evolving arcade footpoint axes (in relation to the potential or y = 0 configuration) of the arcade when in the potential or current free condition.
We are left with incorporating three functions:
1) d(x) = x sinh b
2) B n (x) = x / R o [ B f ( R o ) ]
3) v(φ,W) = a sin (φ ) exp [1 - 2[φ ]/ W
into one partial differential equation, e.g.
u_t + au_x = 0
which is amenable to an efficiently generated numerical simulation algorithm.
Note first that (2) can be recast as:
x = B N (x) [R o [ Bf ( R o ) ]
And we can choose as a first value for a: a = dx(kφ,W) /dt
where v (φ, W) in (2) is now dx(kφ,W) /dt
The final functional solution which we obtain for possible discretization after obtaining its PDE and after substituting for x in x sinh b, is:
u(x,t) = [ B n (x) [R o [ Bf ( R o ) ] sinh b + dx(kφ,W)/dt
And we note that the dimensionless shear kφ is actually a time -dependent function of the gradient, e.g. m(t).
Alas, the complexity of the (differentiated) equation became evident as I went through several initial runs merely testing for errors and stability for two methods, the finite difference and spectral method (which uses Fourier series)). What did I learn ? Well, that I bit off a bit more than I could chew in a 2 1/2 week project. But hey, at least I made a start!