Continuing on from the previous blog, the key issue was how to incorporate shear (φ) and velocity (v (φ,W) parameters together with magnetic field conditions (e.g. B z = 0, B f = Bo J1 (aR)/ r) into the same numerical simulation format - and to do so efficiently without having to resort to different (and possibly conflicting) simulations for shear, velocity and the ( B z , Bf ) fields.
A fortuitious bit of serendipity was the appearance of a paper entitled 'Efficient Algortihms for Solving Partial Differential Equations with Discontinuous Solutions' by Chi-Wang Shu, which appeared in The Notices of the American Mathematical Society (p. 615) in their May, 2012 issue.Shu gives an example of a (hyperbolic) PDE which displays attributes of the ideal sort of equation I was seeking to discretize. This was:
u_t + au_x = 0
where u = u(x,t) is a solution which depends on the spatial location, x, and time t while a is a constant (which could be the speed of wave propagation for his particular PDE, or in my case, perhaps the field line pitch, or tension). Also the use of subscripts simplifies the writing of the partial derivatives, so that u_t, for example, is really @u / @t where @ denotes the partial derivative symbol.
In the flare model context I had been considering, spatial location is automatically incorporated via the shear since as we saw in the (May 3rd) blog: shear is manifested via d (x) or the displacement d(x) of the footpoints (say in an arcade model). Also, it is well known (cf. E.R. Priest, Fundamentals of Cosmic Physics, Vol. 7, p. 363, 1982) that the shear can take the form d(x) = mx, where the shear gradient (m) is a montonically decreasing function, and also one can pose it in straight linear form via:
d(x) = x sinh b
where 'sinh' denotes the hyperbolic sine and b = (½ c)^½ a (kφ) cosh b, where kφ is the dimensionless shear. Thus, it's possible to define u(x) = x sinh b, but the problem remains for obtaining u(x,t). What 'blended' function u(x,t) might one then use? Shu provides a clue by first considering an initial condtion for his equation which is time-independent, viz.
u(x,0) = g(x)
and thence that it's easy to verify a unique solution of the original PDE is:
u(x,t) = g(x - at)
i.e. basically a shift in the initial condition with speed a. (This has a direct analogy to assorted wave equations in terms of the phase, or phase angle). In effect, one doesn't require a separate time aspect to the function u(x,t) if he's simply willing to allow shifts to occur via initial conditions at certain rates, a. So how would this translate to the shear system in an arcade model (or 2 ribbon) flare?
!/ -------------> x
One way to get around this is sketched in the diagram above, for which a 3rd dimensional shear parameter, z = mx is projected upon the xy plane (recall the context in the link from the May 3rd blog, showing a large solar active region). 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. (As one would easily see by constructing a 3D flexible arcade model then gradually shearing it to faithfully conform with the projections on to the x-z then x-y planes). 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.
The final trick then is to incorporate the fields into the mix, but how to do so? One wants as a priority criterion to have any field associations, functions related at least to x, or d(x) for the shear displacement. The one proposal that seems to work is the one originally given by Priest (op. cit.) but it is limited to the normal component at the photosphere, or ( B n ) and then:
B n (x) = x / R o [ B f ( R o ) ]
R o = [x^2 + h^2]^½
But where to from here? In particular, how does one drag the shear velocity :
v (φ,W) = a sin (φ ) exp [1 - 2[φ ]/ W
into the simulation?