Although the Sun has been somewhat quiescent, it now is beginning to find its "mojo" in terms of greater sunspot activity - which implies more solar flares. According to the latest stats from spaceweather.com the current proportion of "spotless days" for the year is 16% (41) compared to 71% (261 days) for 2009. Even as I write this a new sunspot designated '1106' is emerging.
All of this paves the way for inquiring how we might anticipate and predict the next giant flares such as shown in the attached H-alpha filtergram, taken of a large flare on September 17, 1980 at 04h 29m Universal Time. In the image a key reference line is the snaking (dotted) "neutral line" (NL)
To provide an analytical basis, one way is to reduce an activity region to something which can be quanitified in terms of power, magnetic energy storage and electrical currents. One modeled format is shown in the accompanying isogauss contour which resolves a number of "poles" (centers of high magnetic field) in one sunspot group.
In arriving at a credible driven (loading) energy process, the change in relative helicity must be taken into account, along with the impact of localized magnetic shear in altering the magnetic topology of a region. As we saw in a prior blog on solar modeling, the relative helicity can be expressed (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. In general both will be active and dynamic when an active region is reconfiguring toward the flare onset phase. A general approach is to follow the method of DeMoulin et al (Solar Phys., 2002)and situate a magnetic dipole D1-D2 within a Cartesian (x-y) frame and define the size of the region of interest as:
S = abs[r+ r-]
from which the magnetic gradient (grad B) can be obtained, viz.:
grad B = abs[+B_n - (-B_n)] / S
Where the numerator includes the magnitudes of the magnetic induction normal ('n') to the photosphere within the defined shear region extending over S. If I calculate grad B = 0.1 Gauss/km then I know a flare is 96% probable within 24 hrs. The larger the value the greater the probability that the flare will be intense.
A further quantity that one can use to check is the shear motion within a defined shearing region, given by:
v(x,y) = a sin (q) exp (1 – 2 [q] /W)
Where 'a' defines some normalizaton constant peculiar to the region (and generated from a mean of shearing scales by comparing regions in the same cycle), and [q] is the absolute value of shearing angle which will be defined from the neutral line orientation to the x-y axis, while 'W'is the size of the shearing region in appropriate units.
An analytical model, such as depicted in Fig. 2, can be useful in providing cross-checks. In the model shown, the line tied (at one end) dipole undergoes an approximate 28 degree proper motion over time interval (t2 - t1) during which the (-) polarity footpoint (or sunspot pole) is displaced from a1 to a2. This translates the location of f1 from roughly a1 = (-3k – 3ki) where each k = 10^ 4 km, to a2 (-k – 4ki).
Whence:
D = {(-3k – 3ki)^2 - (-k – 4ki)^2}^1/2
All of this paves the way for inquiring how we might anticipate and predict the next giant flares such as shown in the attached H-alpha filtergram, taken of a large flare on September 17, 1980 at 04h 29m Universal Time. In the image a key reference line is the snaking (dotted) "neutral line" (NL)
To provide an analytical basis, one way is to reduce an activity region to something which can be quanitified in terms of power, magnetic energy storage and electrical currents. One modeled format is shown in the accompanying isogauss contour which resolves a number of "poles" (centers of high magnetic field) in one sunspot group.
In arriving at a credible driven (loading) energy process, the change in relative helicity must be taken into account, along with the impact of localized magnetic shear in altering the magnetic topology of a region. As we saw in a prior blog on solar modeling, the relative helicity can be expressed (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. In general both will be active and dynamic when an active region is reconfiguring toward the flare onset phase. A general approach is to follow the method of DeMoulin et al (Solar Phys., 2002)and situate a magnetic dipole D1-D2 within a Cartesian (x-y) frame and define the size of the region of interest as:
S = abs[r+ r-]
from which the magnetic gradient (grad B) can be obtained, viz.:
grad B = abs[+B_n - (-B_n)] / S
Where the numerator includes the magnitudes of the magnetic induction normal ('n') to the photosphere within the defined shear region extending over S. If I calculate grad B = 0.1 Gauss/km then I know a flare is 96% probable within 24 hrs. The larger the value the greater the probability that the flare will be intense.
A further quantity that one can use to check is the shear motion within a defined shearing region, given by:
v(x,y) = a sin (q) exp (1 – 2 [q] /W)
Where 'a' defines some normalizaton constant peculiar to the region (and generated from a mean of shearing scales by comparing regions in the same cycle), and [q] is the absolute value of shearing angle which will be defined from the neutral line orientation to the x-y axis, while 'W'is the size of the shearing region in appropriate units.
An analytical model, such as depicted in Fig. 2, can be useful in providing cross-checks. In the model shown, the line tied (at one end) dipole undergoes an approximate 28 degree proper motion over time interval (t2 - t1) during which the (-) polarity footpoint (or sunspot pole) is displaced from a1 to a2. This translates the location of f1 from roughly a1 = (-3k – 3ki) where each k = 10^ 4 km, to a2 (-k – 4ki).
Whence:
D = {(-3k – 3ki)^2 - (-k – 4ki)^2}^1/2
= k {(-3 – 3i)^2 - (-1 – 4i)^2}^1/2
D = (4.064 + 1.23i)k = [4.064 + 1.23i]k = 4.246k (REM: Arg(z) and the complex function here!)
From this, an estimate of grad B can worked out, if the location and magnitude of the B_n(+), B_n(-) poles are known.
A much more complex flux map is depicted in the isogauss model of Fig. 3, with positive and negative polarity flux points denoted by P or N with a number ranking the magnetic intensity (e.g. ‘1’ is highest intensity). One extra negative polarity ‘pole’ lies on the line x = 0, and a separatrix ‘flow’ extends from the origin to just off the negative centers N1 and N3. The C1, C2, C3 and C4 denote differential isogauss contours.
The differences in coordinates (y – yi) and (x – xi) denote differences between a ‘center of mass’ coordinate system (x, y) based on active region total area, and a sub-region with coordinates (xi, yi) based on the position of the magnetic inversion or ‘neutral’ line (NL) – such that it forms the x(+) axis.
In this ansatz, we can regard the postive polarity magnetic centers: P1, P2, P3 as poles in the upper half plane such that:
INT C f(z) = 2 pi
for which f(z) is the particular function that specifies a given contour C and INT C f(z) is a "contour integral" which will yield "residues".
A further elaboration of this must await some exposure to analytic continuation and simple contour integrals, as well as obtaining residues (based on "the residue theorem") which I plan to introduce next month.
For now, we get an idea that the actual tactical approach to obtaining solar flare predictions can be hideously complex, depending on how much detail (as well as mathematical inputs, desription) we wish to incorporate. This is also why flare predictions often fall flat and about the only ones that hit the 90% or better mark are the negative ones, i.e. for "no flares".
Obviously, the theoretical infrastructure is only one aspect, and we also desperately need much higher resolution optics trained on solar active regions, especially as sunspot evolution rapidly configures toward the flare state. A Hubble dedicated to the Sun anyone?
From this, an estimate of grad B can worked out, if the location and magnitude of the B_n(+), B_n(-) poles are known.
A much more complex flux map is depicted in the isogauss model of Fig. 3, with positive and negative polarity flux points denoted by P or N with a number ranking the magnetic intensity (e.g. ‘1’ is highest intensity). One extra negative polarity ‘pole’ lies on the line x = 0, and a separatrix ‘flow’ extends from the origin to just off the negative centers N1 and N3. The C1, C2, C3 and C4 denote differential isogauss contours.
The differences in coordinates (y – yi) and (x – xi) denote differences between a ‘center of mass’ coordinate system (x, y) based on active region total area, and a sub-region with coordinates (xi, yi) based on the position of the magnetic inversion or ‘neutral’ line (NL) – such that it forms the x(+) axis.
In this ansatz, we can regard the postive polarity magnetic centers: P1, P2, P3 as poles in the upper half plane such that:
INT C f(z) = 2 pi
for which f(z) is the particular function that specifies a given contour C and INT C f(z) is a "contour integral" which will yield "residues".
A further elaboration of this must await some exposure to analytic continuation and simple contour integrals, as well as obtaining residues (based on "the residue theorem") which I plan to introduce next month.
For now, we get an idea that the actual tactical approach to obtaining solar flare predictions can be hideously complex, depending on how much detail (as well as mathematical inputs, desription) we wish to incorporate. This is also why flare predictions often fall flat and about the only ones that hit the 90% or better mark are the negative ones, i.e. for "no flares".
Obviously, the theoretical infrastructure is only one aspect, and we also desperately need much higher resolution optics trained on solar active regions, especially as sunspot evolution rapidly configures toward the flare state. A Hubble dedicated to the Sun anyone?
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