Toward A Heuristic Model To Quantify A Driven Model For Solar Flares (2)

The isogauss model contours in Pt. 1, specifically the end graphic:  

Sets up the basis for a rotary footpoint driven energy system. Again the neutral line is in red and the anti-polar (+, -) magnetic flux centers are D1 and D2 now in much closer proximity disclosing a mutual polarity intrusion.  Then the total power dissipated in solar elements or flux tubes can best be expressed:

P =  S E I_i  =   S (- V X B)_i  [a _i (F_i  ) / m₀]  = -S J [_i  [a _i (F_i  )] 

where the (i) subscript refers to the ith unit element of the magnetic configuration.for each (i),  the field line configuration quantified for the force free parameter (a) and the magnetic flux (F). The specifics of treatment are based on a particular geometry, in this case shown below in 3 dimensions.

As one or both footpoints rotate, the changing magnetic flux df/dt induces V_(ind)  in the loop, and this (induced)  emf acts to reduce the current density in the loop, in this case the Hall current density magnitude. This back reaction from the induced E-field is what keeps the system at marginal stability and J_H  »J_ms, the relevant current density.  If this were not so, Lenz’s law would be violated. In like manner, the relevant velocity for assessing the two-stream instability condition must apply to V_d as the associated drift velocity, not V _q, which is responsible for the back emf V_(ind)  though it generates the Hall E-field via a dynamo action.

One can describe the loop conductor cutting magnetic lines via motion perpendicular to them.  For AR2776, df/dt  »  3.3 x 10 ¹⁸ mx s^-1  thereby setting  the stage for a two-stream instability with an electron beam comprising a tail of a distribution with a beam velocity v _b  =  Ö(2E _o / m _e).

In a paper delivered at the 40^th meeting of the Solar Physics Division of the American Astronomical Society, I noted the shear angle magnitude Dj, was taken as a proxy for the increase in relative magnetic helicity (H _r )  :

D H _r  »  Dj  =  arctan (B _z  /B _f )

where the numerator and denominator denote the axial and poloidal field components respectively, each defined in terms of the relevant Bessel function (e.g. Lundqvist, 1951):

B _z      =   B_o J_o (aR)         and          B _f   =   B_o J₁ (aR)

where J_o (aR)  is a Bessel function of zero order, and  J₁ (aR) is a Bessel function of first kind, order unity.  In general we have (cf. Menzel, 1961, p. 204):

J_m (x) = (1/ 2^m m!) x^m [1 -  x ²/ 2² 1! (m + 1)  +  x⁴/ 2⁴2! (m + 1) (m + 2) -  ….

.(-1)^j x^2j / 2 ^2j j! (m + 1) (m + 2)……(m + j) +  …]

The form of the two functions (using Mathcad plots) is shown below, where J_i = J_o (aR)   and J_2i = J₁ (aR):

                                                                             

In the next graph shown below, we see the magnetic free energy cumulatively added during the interval of interest, as well as the spike showing the greatest flare energy release (this is for SID or geo-effective flares). Because the dotted line discloses cumulative energy in the region, it also shows loading of energy into the region – probably as a result of negative helicity injection via currents and shearing over the interval of maximum interest (Nov. 7-8).

 

The energy release and MFE accumulation profile for SID-producing flares over Nov. 5 – 8, 1980 for Mt. Wilson region 21862

By Nov. 7, for example, it could be inferred from the graph of the energy profile that almost 4.0 x 10 ²⁴ J had been ”loaded” into the active region- delta sunspot group complex. On Nov. 7, approximately 3.7 x 10 ²⁴ J had been released, leaving a residue of  » 3.0 x 10 ²³ J in the attendant field as MFE This increment would be available to add to the next (» 1 d.) accumulation of MFE, say over 7-8 Nov. By Nov. 8 (median Julian date), an additional » 1.2 x 10 ²⁴ J of MFE has been added. This would have been 4.9 x 10 ²⁴ J had the earlier flare energy not been released.

Is the process driven? We first note from the graphic below - with the actual portion of the Mt. Wilson magnetogram at left:

 That ¶B(x) effectively crosses four contours, or (4 x 250 G) = 1000 G. (Each increment of 0.25 along “Delta Y” or “Delta X” in the Mt. Wilson vector magnetogram denotes a change of 250 km while each contour difference is a separation of 250 G.)  The B-vector direction has been estimated as shown, and the components B(y) and B(x) can be worked out, along with the changes, e.g. ¶B(x)/ ¶y  and  ¶ B(y)/ ¶ x, say from one iso-contour to the next. 

The separation dy (or ¶y ) from the vertical axis amounts to » 3.2 unit(s) or 3.2 x 250 km » 800 km. Thus:

¶B(x)/ ¶y  =   (1000 G)/  800 km  =   1.25 G/ km

In a similar way, we find for ¶ B(y)/ ¶ x:

¶ B(y)/ ¶ x  » (1000 G) / (1.4 x 250 km) »

 1000 G/ 350 km »  2.8 G/ km

Then the current helicity density is:

H _Z (c)  = [¶B(x)/ ¶y  -  ¶ B(y)/ ¶ x] B _Z    

 » [1.25 G/ km  -  2.8 G/ km] 350 G 

H _Z (c)  »   - 542 G ² / km

The negative sign indicates that the force-free parameter a is also negative, and will have magnitude:

a  »  -[¶B(x)/ ¶y  -  ¶ B(y)/ ¶ x] /   B _Z       

» - [1.25 G/ km  -  2.8 G/ km] / 350 G 

a  »  - (1.55 x 10 ^-3 m ^–1) G /350 G   »  -4.4  x 10 ^-6  m ^–1

By “hemispheric helicity rule” the sign fixes the region in the Northern solar hemisphere.

We know: a  =  m_o J _z / B _z

Then:   B _z  = m_o J _z /  a

(Where: m_o   =  4p x 10^-7 H/m)

Solving for the magnitude of the magnetic field  z-component:

B _z  = (4p x 10^-7 H/m)( 0.012 A m^-2   ) / 4.4  x 10 ^-6  m ^–1

B _z  = 3.4 x 10 ^-3  T

Along with assumption of Coulomb gauge, we can write:

m  J _Z   =  - Ѳ A   =  l f (A,y) 

wherein we finally see the appearance of the shear parameter l f (A,y) which is directly proportional the vertical current density J_z  = J (A,y) and thence to the current helicity density H _Z (c)   we already deduced from the isogauss contour. Given the basic (heuristic) model includes both energy accumulation and dissipation it is concluded it conforms to the driven paradigm.

 In general, l f (A,y) can be interpreted as a generalized, quantitative index of the magnetic free energy, and vertical current density, stored relative to the potential or current-free force-free field. (For which Ñ X B = 0). In many cases, l f (A,y) will be a differential angle of shear (e.g. D f) , which measures the departure from the potential state.

 Whatever the physical form, when l f (A,y)  >   l _c  (l _c  a critical angle relative to the magnetic inversion line) then the field should be poised for explosive energy release. In other words, the quasi-static development and configuration is abruptly terminated so we get dissipation of the accumulated magnetic free energy.

(The magnitude of energy released as actual flares can  also always be obtained by integrating the area under the power-time curves of specific flares as obtained from the SMS-GOES soft x-ray (1 – 8 Angstrom) record. )  

See Also:

• Looking at Bessel Functions & Applications

And:

New Solar Research Confirms Why Delta Sunspots Are More Flare Worthy Than Other Magnetic Classes