Magnetohydrodynamic Equations

near an X-type Magnetic Neutral Line

For the purposes of our investigation, we shall use the same initial-value problem set up by Shivamoggi [15]. The system is two-fluid, quasi-neutral plasma near a two-dimensional X-type magnetic neutral line. In this system, the magnetic field is given

The solution to the system given in [15] is

where *γ*(*t*) satisfies the following differential equation

with the initial conditions

where *γ _{0}* and

Since the equations themselves do not seem to permit exact solutions, asymptotics can be used to approximate solutions for short and long times. Short and long times are compared to an Alfvén time

while for long time, equation (4) gives

But do these approximations satisfy compatibility conditions like the plasma incompressibility condition and the exact invariant? The equation for incompressibility is given by

while the invariant outlined in [15] is

Note that equations (2) and (3) have the following relationship

Substituting equation (11) into (9) yields

So the incompressibility of the plasma is satisfied regardless of

Substituting equation (13) into the invariant (10) gives

where

This shows that a proper

So the large time approximation for

We must use the Taylor polynomials for

Substituting equation (7) and (18) into (10) yields

where

where

and

We can see the following pattern emerge

where

is subject to *n *≥ 1;

is the number of unique ways of accomplishing * j _{i}* +

Note that equation (22) is equivalent to zero only for a properly chosen

If *γ* is written in that form, then taking *n* derivatives of *γ *eliminates the first *n* terms of *γ*.

If we evaluate equation (24) at * t* = 0 then only the
constant term will remain

Recall from equation (19) that *γ* = *γ _{0}t* + (

In order to solve for the missing

We may generalize equation (27) for arbitrary order and prescribe that it is used only when

Equation (28) is valid because when

So by recursively applying the formula for

Now that we have derived an effective method
for generating higher degree approximations for *γ*,
we naturally want an equation that quantifies
the approximations' quality. Because *γ* is a polynomial,
*I _{m}* is composed of a linear combination of polynomials.
As such,

.

As in the treatment of *γ* in equations (24)
and (25), we can take *n* derivatives of *I _{m}* and evaluate the
result at

Because

Now that we have derived a way to refine the small
time approximations for* γ* to higher degrees and have
determined its respective accuracy, let us get a better
appreciation for its process. Figure 1 shows the absolute
error of successive refinements of *γ* with *λ *=* k* = 1.5. We
can see in Figure 1 that depending on which order of
approximation for *γ* is used there is a point at which the
approximation becomes too inaccurate to be useful. The amount of time where the error is almost zero increases
for higher degree approximations.

The range of time
from *t* = 0 to the point of inaccuracy is a restriction of *γ*
denoted by *D* = [0; *t* *].The point of inaccuracy is defined
to be the point where the error has a magnitude equal
to 0.001. Not only does the degree of *γ *affect the value
of* t **, but the choice of the constants *γ _{0}*, σ,

To determine the time at which the singularity occurs,
we introduce the transformation

which reduces (4) to

where for simplicity we define the constant

Equation (34) admits the implicit solution

where we have used the initial condition *u*(0) = 1 to
obtain the lower point of integration. Note that, when* γ* becomes singular, this corresponds to *u *→ 0. So, if
*t* =* t*^{+}is a singular point, we have that *u*(*t*^{+}) = 0. Placing
this assumption into (36), we find that

We are concerned with t ≥ 0, so we take the positive of
the two solutions, obtaining

This gives the first singularity in the region *t* ≥ 0.
Equation (38) is only accurate in the parameter regime
*k* > 1, *σ*C > 0, *γ*_{0} > 0. It is interesting to note that for *λ* = *k* = 1.5, *t*^{+} ≈ 0.3439
which appears to be the limit
of the polynomial approximations' accuracy as *n* → ∞ and thus lim_{n→ ∞} *t*^{*} (*n*) = *t*^{+}. This point is marked with a
black X in Figure 1.

Now that we have established that a singularity for a
certain parameter regime exists, we may instead use a
different model for the behavior of *γ* based on [14],
which studies the ideal MHD model where *σ* = 0. In
[14], the differential equation is

and its large time approximation is given by

where *t*_{0} is the time when the singularity occurs. Note
that if we use our transformation (21) on (4), we will
get an identical differential equation; therefore, we may
use the inverse of the transform and create the long time
approximation

However, this approximation has an error of
at *t =* 0 and an approximately quadratic error decay
until *t = t ^{+}*. We thus introduce a quadratic correction
term to equation (41) to reduce the error.

Although this approximation has

and therefore is very accurate for large values of

But what of our original large time asymptotic
approximation for *γ*, equation (8)? The linear behavior
is only observed in a certain parameter regime where
*γ*_{0} = -*σ*C. With this condition, the Hall term
successfully prevents the finite-time singularity, and the
linear approximation is accurate. This conincides with
our previous note that *t* *→∞ as *λ* → 0 and thus the
singularity never occurs.