Analysis of Helicity Variation Via Collision of 2 Solar Loops In Relative Proximity (Pt. 1)

 Magnetic helicity properties and their variability  has only relatively recently come under the purview of solar physics.  The likely reason is that much of the content has emerged from topology, a mathematics discipline outside the purview of most solar physicists. Also, the fact that magnetic helicity is usually defined via the magnetic vector potential  A  which is hardly a readily observed quantity and must usually be extracted from vector magnetograms or via modelling using in situ magnetic field measurements. Nonetheless, it is important in providing a key physical marker without knowing the resistivity in a given active region.

In this post  and the next we will be considering magnetic helicity based on the collision of two proximate loops, of differing relative helicity as depicted below (Fig. 1), in three dimensions and in two in the (x-y ) plane of the solar photosphere. In addition, the quantity twist  T will be used as a proxy for A.

 

                         Colliding coronal loop scenario for which helicity H(r,r’) may be exchanged.

To simplify the treatment it is assumed that T is the only portion of the relative helicity changing for the loops. Then the respective changes for each loop are defined in Fig. 2 below for a presumed relative footpoint motion:

                                           Configuration for the relative loop footpoint motion

 And this may be written:

d H(a1-a2)/ dt =  d H(r’ ) [T_a1a2] / dt    =  d [ (L1 B _q(a))/  (a  B _z (a))]

d H(b1-b2)/ dt =  d H(r ) [T_b1b2] / dt    =  d [ (L2 B _q(b))/  (b  B _z (b))]

and: L1 >  L2

where B _z  is the longitudinal component of the photospheric magnetic field and B _q  is the poloidal component. And reckoning in the radius a for L1 and the radius b for L2, each of which is assumed to remained constant. Thus, the respective aspect ratios are: L1/a and L2/b.

When a collision occurs between the loops, we expect:

d H(a1-a2)/ dt  +  d H(b1-b2)/ dt  = 

d [ (L1 B _q(a))/  (a  B _z (a))]  + d [ (L2 B _q(b))/  (b  B _z (b))]

and the increase in relative helicity in terms of the linear velocities (projected on the axes: +x, -x) is:

D H(r’) [T_a1a2]  = v(x _a1a2)  =   d/dt [L1 cos w t]  =  - w L1 sin w t

D H(r) [T_b1b2]  = v(-x _b1b2)  =  d/dt [-L2 cos w t]  =  w L2 sin w t’

Therefore:

D H(r’) [T_a1a2]   + D H(r) [T_b1b2]   = - w L1 sin w t + w L2 sin w t’

If the relative rotary motions of each loop L1 and L2 are synchronous, then:

w t = w t’  = 2 p

One obtains (where r is the size ratio of the flux tubes: L1/a and L2/b) :

D H(r’) [T_a1a2]   + D H(r) [T_b1b2] »   2 r sin (2 w t - w t’ )/[ p (a + b)/2)²]   » 0

In other words there is essentially no net change in helicity. This is more likely to be true for a line-tied case, and the zero net change means an instability is unlikely.

However, for:  w t =  3 p/2,    w t’  = p/4

D H(r’) [T_a1a2]   + D H(r) [T_b1b2]   = - w L1 sin (3 p/2) + w’ L2 sin (p/4)

D H(r’) [T_a1a2]   + D H(r) [T_b1b2]   =  w L1  + w’ L2 [Ö2/ 2]

and a flare via helicity exchange will be likely once the twist  T for at least one loop: 

T > 2.5 p

 

Priest (1983) examined the effect of line-tying on the ideal kink (m = 1) mode and found that stability predominated up until the twist T > 3.3p^[1]. At that point the loop “becomes unstable to a range of  k” for a particular perturbation. Clearly, line-tying provides a greater safety factor in terms of preserving reliability – making it less likely the twist will exceed the critical threshold – or, if it does,. it will take a longer time.

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[1] E.R. Priest, Plasma Physics, Vol. 25, No. 2, p. 161, 1983.