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** = **cur**l 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** – **DIV ^{2} 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:

** DIV**^{2} = 1/
r [¶/ ¶r
( r ¶/ ¶r)]

And since **DIV** ^{2}(**B**)
+ (a)^{2 }**B** = 0

we get:

**B** + (a)^{2 }**B**
= 0

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

_{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 + B^{2}(1/T ^{2}) + a
sin (wt)

g µ A/ T +
B^{2}(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.

^{2}
J(s) + 2 gs J(s) + (a_{o})^{2} J(s)
= I
/ m
{exp (- s t_{0}) }

**s t _{0}**)
}is the Laplace transform of the Dirac delta function. The treatment proceeds
as follows:

J(s) = I/m { exp (- s t_{0}) / [s ^{2} + 2 gs
+ (a_{o})^{2} ]}

^{2} - (a_{o})^{2} }

_{o})^{2} > g ^{2} and then have
a ^{2 }=
(a_{o})^{2} - g ^{2}

whence:

_{0}) / (s – r1)
(s – r2) }

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 - t_{0})/ 2 ia +
exp (- g - ia )(t -
t_{0})/ -2 ia }

Simplifying:

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

^{2 }( wt) is depicted.

**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]}.

**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.

_{m} = B^{2}/2m ) given
by:

¶ / ¶ t { ∫_{v} B^{2}/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:

(J**z _{ms}**)
such that: d(Ñ(

__+__B

_{z}) ) Þ d J

**z**Þ dJ

_{ms}_{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.

*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.

[1] It
should be possible to show from this that J(t) = sin(2wt)/ 4mw [exp –g (t – t_{0}] 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

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