As I had noted in an earlier blog (Sept. 21): some advances on the CME quantitative front have revolved around the rate of increase of the poloidal magnetic flux associated with a flux "rope", e.g.

dΦp(t )/dt

where the numerator denotes the increase in rope magnetic flux and the derivative indicates it's taken instantaneously as a function of time. In a simple flux rope model, think of taking a 6-7 cm section of thin rope and twisting it. As you twist it the rope 'humps up'. In real solar cases the magnetic counterpart 'humps up' through the solar surface. It is the increase in the poloidal (as opposed to vertical or longitudinal) component which allows this.

For a predictive basis, one requires the related function be adjusted for each CME to obtain a solution that best fits the observed time-height data. The function is usually given in terms of the electromotive force so that:

E(t ) ≡ −(1/c)dΦp(t )/dt

My primary aim is to tie this into the emf developed over time when solar double layers become unstable within coronal loops. It is already known that the evolving-increasing emf (V(b)) in a twisting solar active region can be computed from:

V(b) = ò (v X B) ds

Where v is the fluid velocity and B the magnetic intensity.

We can also obtain an independent estimate of V(b) from:

V(b) = L(dI/dt) + RI

where R is the ambient resistance of the active region, I is the current for the equivalent circuit and the inductance L is estimated from the characteristic timescale of electric current dissipation in the associated equivalent circuit. Meanwhile, in the presence of a collisionless, high temperature current sheet,

E » v

_{o}B_{o}/ c » 1- 10 V cm^{–1}To achieve 1 V/cm the typical distance affected would have to be 1.6 x 10

^{8}m. In fact, this represents about 0.17 of typical total observed length for coronal loops. In alternative terms, it represents roughly 10 separators ({E1/ Q

^{1/2}} of minimal length, say to trigger a solar flare where E1 is the cutoff or threshold energy for the region, for which the shortest separator is: ℓ1 = {E1/ Q

^{1/2}}

In more detailed form, the acquisition of free magnetic energy is given by:

¶ E

1/m ò

_{m}/ ¶ t =1/m ò

_{S}[(**v**X**B**) X**B**] dS - ò_{v}{ |**J**|^{2}/ s }dx dy dz
where the first term on the right side embodies footpoint motion, and the second joule (heat)dissipation. It is also known the flare distribution corresponds to a Poisson process of the form :

P(t) = exp (- l) l

^{t }/ t!More recently, Wheatland and Craig (Astrophysical J., Vol. 595, p. 458, 2003) have argued that the waiting time distribution (WTD) in

*individual active regions*is consistent with a Poisson process in time, which would conform to: P(t) = l exp (-lt) where l is the mean rate of flaring.

The work entails reconciling all these different facets as well as possible paradigms (double layer vs. twisted rope and increasing MFE).

Anyway, I'm sure there are plenty of blog posts still to be read (including from past years) and also problems to work! In the interest of spurring readers on to solutions, namely for the blogs on DE word problems, I provided a jump starter in the previous post pertaining to the solution of the ship problem (Applications of DEs, Pt. 3, Problem No. 1) .

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