Dirac Delta Function Basis For Modeling A Solar Flare Trigger (Part 2)

  Treatment of Dirac Delta function in the Solar Flare Context:

To apply the results of the previous (Part 1) post to the solar flare context we begin by positing an unknown current impulse function J_Z (t), which represents a “catastrophic” vertical current density pulse at time t ₀ , e.g. precipitating either the eruption of a solar filament (via the J x B force) or the sudden release of enough initial magnetic free energy to engender a “cascade” and an impulsive flare – which will have a profile similar to that in Fig. 3.

Following on from the previous instalment, we can write:

I =   ∫ _to  ^{to + t}  F(t) dt  = ∫ _to  ^{to + t}  a sin  (wt)  dt 

where the impulse function F(t) = a sin (wt)   is employed, but this can be expressed in multiple ways. The function I(t)  may then be expressed:

     I (t) =  ∫ _to  ^t  a sin  (wt)  dt  [d(t – t _o)]

where t in the impulse integral upper limit, may be taken arbitrarily small, but in any case leading to (t _o  +  t) = t.

Given these preliminaries, we can proceed with the development of the appropriate differential equations. We first derive the equation for the equilibrium or constant - a state: We begin by taking the curl of both sides of the force-free equation:

  curl curl B  = curl a (B)  or  Ñ x (Ñ x B) 

=  Ñ x (a (B) ) = a (Ñ x B)

In the above, substitute (Ñ x B) = a (B)  and use the vector identity:

 DIV DIV B – DIV² B =   Ñ(Ñ·B) - Ñ ²  B =  curl curl B  

to obtain:

  Ñ(Ñ·B) - Ñ ²  B   =  a ² (B) 

By the divergence-free condition, (Ñ·B) =  0, so the equation becomes:

    - Ñ ²  B   =  a ² (B)     or   Ñ ²  B   +  a ² (B)     = 0

A close examination reveals that the latter is just Helmholtz’s equation, but with sources absent. Expanding the equation (writing out the space derivatives of the grad operator Ñ) we obtain the partial differential equation in x, y and z coordinates:

     ¶ ² B /¶ x ² +   ¶ ² B /¶ y ²    + ¶ ² B /¶ z ²    +  a ² (B)     = 0 

The problem is that for most field applications, Cartesian coordinates will not be as much use as cylindrical ones. Thus, it makes sense to substitute the operator in cylindrical form:

  DIV²  =  1/ r  [¶/ ¶r  ( r  ¶/ ¶r)]

And since DIV ²(B) +  (a)² B = 0

we get:  

    1/ r  [¶/ ¶r  ( r  ¶/ ¶r)] B  +  (a)² B = 0

For which the (axially symmetric) Bessel function solution is:

   B_z (r)    =   B_o J _o (ar)

This pair will be found useful in the next section, when we address the instability and shearing of a magnetic arcade. For now, we need to use an equation for which the solutions will be more in line with a Dirac delta function. Since we wish to examine the impulse associated with the current density, we will use the differential equation:

    J _tt +  (a)² J =   ¶ ² J /¶ t ²  +  (a)² J  =   I(t) / m [( d (t – t ₀)}] 

where, as before,

I (t) =  ∫ _to  ^t  a sin  (wt)  dt 

is the extended impulse function   The above equation in J _tt denotes the Dirac representation, in terms of vertical current density, at the instant of the catastrophic, time-dependent impulse I(t).

As a further refinement to the equation, we include a damping constant, g.  This moderates the current impulse increase, which realistically does not become “infinite”. It would be analogous to adjusting the impulse function F(t) such that is has a “slow” and “fast” component, viz.:

F(t) =  F _s (t )   +  F _f(t)   =  A/ T +  B²(1/T ²) + a sin (wt)

 where the fast form is simply the function earlier defined I(t) and the slow form is a modulation of the total period from onset to peak. In effect, the damping constant:

g µ  A/ T +  B²(1/T ²)

The fuller form of the Dirac representation then becomes:

   ¶ ² J /¶ t ²  + g J +  (a_c)² J  =   ∫ _to  ^t   a cos ² (wt)  dt / m } d (t – t ₀)

Note in the above, that the linear force-free field parameter a, has been replaced by the catastrophic quantity a_c.  This is now the non-linear parameter, compatible with impulsive alterations of the B-field components.

 The solution may be obtained through the method of Laplace transforms. The Laplace transform version can be written, where we employ I as a general, non-specific impulse for the sake of simpler working:

  s ² J(s) + 2 gs J(s)  +  (a_o)² J(s) =  I  /  m    {exp (- s t₀) }

 where {exp (- s t₀) }is the Laplace transform of the Dirac delta function. The treatment proceeds as follows:

J(s) =  I/m { exp (- s t₀) / [s ²  + 2 gs   +  (a_o)²  ]}

 where the roots of the denominator are: r =  - g  ±  Ö{g ²  -  (a_o)²  }

 We let (a_o)²   > g ²   and then have  a ²  =  (a_o)²   -  g ²  

whence:

 J(s) =  I/m { exp (- s t₀) / (s – r1) (s – r2) }

 for which: 

 r1 = -  g + ia     and r2 = -  g -  ia    

So that:  (r1 – r2) =   2 ia     and  (r2 – r1) = -2 ia    

Then, on inverting using the Heaviside theorem:

J(t) =  I  /  m    {exp (-  g + ia )(t -  t₀)/ 2 ia      +   exp (-  g -  ia )(t -  t₀)/ -2 ia }

Simplifying:

J(t) =  I  /  m    {exp (- g )(t -  t₀) sin [w(t – t ₀)]   d (t – t ₀)

From which we find:

J(t) = 0    for t <  t ₀

J(t) =  I  /  m    {exp (- g )(t -  t₀) sin [w(t – t ₀)]     for t >   t ₀

 What about a more specific representation using an actual impulse function for I?  In Fig. 4 below, a functional impulse of the form f(t) =  10 cos ² ( wt) is depicted.

Of course, this is not a unit impulse, but is given merely for illustration purposes.  But it still roughly approximates actual x-ray impulses collected e.g..

Subset of soft x-ray impulses from same date.

The integral will be:

∫ _to  ^t   10  cos ² (wt)  dt = 10 [t/2  + sin (2wt)/ 4(w)  ] ^{to + e}  _{to - e}

From which the transformed functions can be obtained in J(t) as given earlier^[1].

 This then is the representation, in the current (E-J  )paradigm for the non-equilibrium threshold for vertical current density in the evolution of a two-ribbon or “arcade” flare.  A sketch of a magnetic arcade model is shown below:

                       'Cartoon view' showing the progressive shear of a magnetic arcade

It is instructive to compare this with the vertical current density profile obtained by Parker (1996) for a twisted single loop^[2].  E.g. Parker writes (ibid.)

"The current is driven by the electric field, whose time rate of growth is described by Maxwell’s equation,

¶ E /¶ t  =  c (Ñ x B)  - 4 p  j

It is evident from inspection that E grows rapidly, forcing j into compliance with Ampere’s equation.  The energy to drive E comes from the magnetic field through the induction equation: ¶ B /¶ t  = - c (Ñ x E) "

Where j  is analogous to J in the current treatment. The most important observational step is to ascertain how well the equation fits an actual current density profile during the impulsive phase of a major flare. This may well have to await dramatic enhancement in observational resolution, both in respect of the initial unstable flare volume (and its location within a solar arcade) and improved vector magnetograph measurements in time resolution.

In regard to the latter, we also require vastly refined measurements of the transverse field component in order to assess current density profiles. These measurements should also be altitude dependent. In the meantime the energy density might be used as a proxy indicator of the vertical current density, especially during the initial impulsive phase.

 The concept of a Poisson-based “delay time” for build up of magnetic free energy, was first postulated by me in 1984 (see ref. #1 from Part 1),  for application to “SID” (sudden ionospheric disturbance-generating) flares, with the release attendant on a change in initial free magnetic energy (E _m = B²/2m ) given by:

¶ / ¶  t  { ∫_v  B²/2m  dV} = 1/m  ∫ _v div[(v  X B) X B] dV  

-   ∫_v  {h _an | J_ms |² }dV 

where the first term on the right side embodies footpoint motion, and the second, joule dissipation, but with J_ms the current density at marginal stability – since the marginal stability hypothesis is required for a driven process, and  h _an is the anomalous resistivity. In the same paper, it was shown how the flare distribution corresponds to a Poisson process of the form

P(t) = exp (-  l )    l ^t  / t!

where theoretically the Poisson mean rate of occurrence is:

l_m =   l Dt,

 with Dt = t,  assuming the time interval Dt = 1d. In reality, measuring constraints (say achieving uniformly equal time intervals between successive Mt. Wilson magnetograms), will usually ensure  Dt = 1d,  thereby introducing a selection effect variability, complicating computation of P(t).  It was also suggested, but not proven, that variability in l arises from  variability in vertical magnetic gradients (B_z) and critical changes in the associated current density at marginal stability:

(Jz_ms) such that: d(Ñ(+ B_z) ) Þ d Jz_ms Þ dJ_z  ( dt)/ dt

where J_z  is the vertical current density associated with putative footpoints magnetic induction (+ B_z ) and rate of change in |B_z| modulated by significant evolutionary changes (dt) in the lifetime of the magnetic field, especially critical if  dt < Dt

Since magnetic gradients and associated scale lengths (ℓ_B) also will change in time, there would be scope for accepting a Poisson process of form P(t) = f(dt, dℓ) which would embody an energy modulation with some inbuilt variance, with the latter having to be known to determine how much energy might be released and when. In other words, the differing scale factors inevitably introduced variabilities that were difficult to account for. The Poisson statistics therefore had to be able to take these differing modalities into account.

 More recently, Wheatland and Craig 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(t) exp (-lt)  where l(t) is the mean rate of flaring or “tick rate”.  It must be noted here that a priori l(t) »  l_m  Dt since the latter variability also takes into account variation in data indices, selection effects arising therefrom (already noted in a paper I had co-authored with Constance Sawyer, in Solar Physics, 1985,  98, 193.) In effect, we had forecast the later work of Wheatland and Craig by at least 18 years.  We had 'set the table' for articulation of the flare trigger but the exact formalism to describe it remains a work in progress, pending further advances in plasma physics.

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[1] It should be possible to show from this that J(t) = sin(2wt)/ 4mw  [exp –g (t – t₀] sin (wt), since the integration of t/2 within the defined limits and allowing e ® 0, removes that term.

[2] Parker, E.N. The Astrophysical Journal, Vol. 471, Nov. 1996, p. 491.

 

See Also:

• Looking at Bessel Functions & Applications
• And:
• Revisiting Basic Laplace Transforms In Solving Differential Equations