Toward A Heuristic Model To Quantify A Driven Process For Solar Flares (1)

An ongoing interest in the solar flare process has incepted a rivalry between two paradigms: unloading and driven. The former has received most prominence in the Magnetosphere-Ionosphere (M-I) coupling models explored by Kan et al (1983, 1994) that invoke such auroral mechanisms as V-, S-potentials, double layers and field aligned potential drops. Such models focus almost entirely on energy generation, while the paradigm we call "driven" incorporates both energy accumulation and dissipation.

Rather than employing space physics auroral-substorm mechanisms, it is shown in this post and the next that solar physics processes and structures already exist that comport with the storage of free magnetic energy under solar conditions and portend their subsequent release. Hence, these storage processes are driven in nature, and can be mathematically modeled.  A conceptual basis for such a theoretical model will be presented which I believe comports fully with the criteria for a driven process. 

To illustrate how magnetic helicity change works in tandem with magnetic shear of localized fields to trigger flares, I modeled the large solar active region –sunspot group designated as Mt. Wilson 21862 which displayed complex polarities and severe polarity intrusion over Nov. 5- 8, 1980, according to the magnetograms..

Modeling entailed computation of rotation rates of 2 elementary flux centers (denoted by D1 and D2) each weighted by magnetic flux, F_m (D1, D2).  At all stages the governing equation is (cf. DeMoulin et al, 2002, DeMoulin and Berger, 2006):

d H(R)/ dt =  {[T]    +   [W]} d H(R)  / dt   

where [T] denotes the ‘twist’ component of relative helicity change, and [W] denotes the writhe component. A sketch of the first two days of magnetic topology as it affected the sunspot group is shown below:

Following the method of DeMoulin et al (2002) we situate the magnetic dipole D1-D2 within a Cartesian frame and define the size of the region of interest as:

S =  ïr +    r-ï

where r(+) =   ò _Bn>0   |B_n| r dS/F              r(-)=    ò _Bn<0   |B_n| r dS/(-F)             

Two key parameters are S (above) and W the size of the shearing region. Also critical is the ratio (W/S), since for small W/S the twist and writhe components of helicity are nearly the same, so H(r) is a monotonic function of shear distance. We can obtain this ratio as a good approximation from the sketches, namely that for Nov. 5, 1980 with the neutral line (NL) shown. From this the size S is reckoned as 7.5 deg and W » 1 deg, so that W/S = 0.133.

For the shear motion we follow the advice of DeMoulin et al, in using:

v(x) =  a sin (y) exp (1 – 2 |y| /W)

 but with y specifically adapted to the conditions prevailing in the active region over 5- 8 November, 1980. Thus:  y =  f(t_i)  + W 

If we normalize the value of W at Nov. 5 to 1, it becomes possible to see the radical change in v(x) as W decreases over the time interval of interest.

We show this for different values of W below:

Changing horizontal velocity profile as W (shear width) narrows. The horizontal scale is for y as a fraction f of S (fS) in mM, and the vertical in ms^-1.

Note that for W = 0.2, near y = 0.2, the shear velocity v(x) »  0, which indicates the dimension of |y|  is close to exceeding the width, W.  In order to model the isogauss contours for the field, we incorporate a rotation rate (dq/ dt) into the mapping function such that:  r =  J0(x) cos q (dq/ dt)  +  f(t_i)

 where the Bessel function J0(x) is associated with the axial magnetic field – which must increase dramatically as magnetic shear increases. For the Cartesian coordinate changes in the magnetic changing topology:  x = Dr cos q (dq/ dt)  -  f(t_i)   and:

y =  (p/8) sin q (dq/ dt)  +  c

where ‘c’ denotes a correction factor to preserve the conditions noted earlier, such that

W/S » 0.2, and |y|  » W. 

 The associated isogauss contours are shown below:

                                                                  

Top frame: Model magnetic contours on Nov. 5 (left) and Nov. 6 (right), with pole positions shown. The neutral line is in red and shifts in a counter-clockwise sense denoting the injection of negative relative helicity in the region. The overall effect is to bring D1 and D2 closer together and alter the magnetic field gradients.

Lower frame:  Model magnetic contours on Nov. 7 (left) and Nov. 8 (right) in Mt. Wilson group 21862, with pole positions (D1, D2) shown. Note how full mutual polarity intrusion has now occurred by the 8^th

In part 2 I will show how the energy dissipation arises, starting with the features of the last (Nov. 8) isogauss contour.