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 JZ (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:
to obtain:
By the
divergence-free condition, (Ñ·B) = 0, so the equation becomes:
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:
And since DIV 2(B) + (a)2 B = 0
we get:
For which the
(axially symmetric) Bessel function solution is:
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)
g µ A/ T + B2(1/T 2)
The fuller form of
the Dirac representation then becomes:
¶ 2 J /¶ t 2 + g J + (ac)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 ac. This is now the non-linear parameter,
compatible with impulsive alterations of the B-field components.
J(s) = I/m { exp (- s t0) / [s 2 + 2 gs
+ (ao)2 ]}
whence:
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
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].
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.
¶ / ¶ t { ∫v B2/2m dV} = 1/m ∫ v div[(v X B) X B] dV
- ∫v {h an | Jms |2 }dV
where the first term on the right side embodies footpoint motion, and the second, joule dissipation, but with Jms 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:
lm = 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 (Bz) and critical changes in the associated current density at marginal stability:
(Jzms)
such that: d(Ñ(+ Bz)
) Þ d Jzms Þ dJz ( dt)/ dt
where Jz is the vertical current density associated with putative footpoints magnetic induction (+ Bz ) and rate of change in |Bz| 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 (ℓB) 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.
[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.
- Looking at Bessel Functions & Applications
- And:
- Revisiting Basic Laplace Transforms In Solving Differential Equations
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