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Essay

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 γ0 and k are externally determined parameters and γ0 > 0; C > 0; k > 1. The parameter k is taken to be greater than one so that the Lorentz force of the system is directed to maintain the prescribed initial stagnation-point flow. Using equation (4) with the initial conditions (5) yields

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 τA, which may be taken to be γ0-1. For short time, equation (4) gives

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 γ, since α is non-zero. Plugging equations (2) and (3) into (4) and completing the square yields

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

where

This shows that a proper γ solution that satisfies the ODEs will satisfy the invariant. For large t, plugging in equation (8) into (10) yields

So the large time approximation for γ trivially satisfies the invariant. For small time, plugging equations (2), (3) and (7) into (10) gives

We must use the Taylor polynomials for α2 and β2 because exponentials and polynomials cannot be directly compared. Using equation (7) for γ, the second degree Taylor polynomials for α2 and β2 are the following

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

where I2 denotes the invariant with degree two Taylor polynomials replacing α2 and β2. This shows that for small time, the invariant is only satisfied up to O(t); however, the invariant can be used to refine γ, making the following definition

where

and α2 and β2m, are the mth degree Taylor Polynomials for α2 and β2 respectively. Differentiating Im repeatedly with respect to time gives

We can see the following pattern emerge

where

is subject to n ≥ 1;

is the number of unique ways of accomplishing ji + ki = n + 2. If one does not like the sum, recall that

Note that equation (22) is equivalent to zero only for a properly chosen γ. Let us treat the equation as if γ is written in the following form:


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 = γ0t + (k2 - 1)t2 does not return a constant if directly plugged into Im. We can intuitively understand that c3 should be non-zero. We can effectively solve equation (22) for under the assumption that a proper m is used (meaning that α2m and β2 m are not functions using ),

In order to solve for the missing c3 term, we can use equation (25) and evaluate equation (26) using t = 0 and n = 2. (Note that higher order ζ are equal to the higher order derivatives of γ.) Using m = n results in α2m and β2m only using ζ (n) and since α2 = e and β2 = e-4ζ .

We may generalize equation (27) for arbitrary order and prescribe that it is used only when c1, c2 , . . . , cn are known (cn+1 , cn+2 , are all 0 at the time of evaluation of In(n)) and n ≥ 2, which results in

Equation (28) is valid because when cn+1 is zero, ζ(n+1) is zero, which in turn causes equation (22) to reduce to

So by recursively applying the formula for cn+1 while incrementing n by 1, we can solve for the unknown coefficients of γ and create a higher order numerical solution to the coupled non-linear ordinary differential equations for small time.

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, Im is composed of a linear combination of polynomials. As such, Im can be written in the following form:

.

As in the treatment of γ in equations (24) and (25), we can take n derivatives of Im and evaluate the result at t = 0 to isolate the constant.

Because Im is supposed to be a constant for a properly chosen γ, any non-zero bn for n ≥ 1 is considered to be an error. As seen in equation (19), plugging an O(t2) γ approximation into I2 led to an error of O(t2). Because equation (28) eliminates one order of error every time it recurses, plugging in γ to O(tn) into the invariant will return an error of O(tn); therefore, to receive the leading order error from plugging λ into the invariant, one can use the following formula:

En should be used under the same conditions that apply to the cn+1 equation. That is, only when c1 ,c2 , . . . , cn are known (cn+1 , cn+2, . . . are all 0 at the time of evaluation of ) and n ≥ 2. If we use En when cn+1 is its proper non-zero value, then En will return 0. En is used to visually and numerically determine the accuracy of small time γ approximations.

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 γ0, σ, C, and k, also affect t *. So, for the purpose of determining their affect on t * , we shall treat them as variables rather than constants and keep the order of γ fixed. To reduce the visual dimensionality of En from six to four, we shall use equation (15). Now

Because we cannot readily see 4-dimensional data in a single picture, we must make discrete slices of the 4-D surface to turn it into many 3-D surfaces. Slices along the time axis are used. However, this report is on paper, which is a 2-D surface, and 3-D images are hard to discern when projected onto paper. To reduce the dimension to two, we shall make a planar slice normal to the error axis. To extract those points from En (t, λ, k)|t=t* we can plot the intersection of the 3-D surface with the plane normal to the error axis and containing the point (t, λ, k, En) = (t *, 0, 0, 0.001). The point of inaccuracy is now a curve in the (λ, k) plane. Performing the aforementioned planar slice for various t * values and using a 10th order γ results in a contour map (see Figure 2). We can see in Figure 2 that as the values of λ and k increase, the value of t * decreases, and as λ → 0, t * → ∞ because En ≡ 0 when λ = 0. We may vary the method in which En is partitioned to create contours in other planes (see Figure 3 and Figure 4). We can also solve the differential equation using the Runge-Kutta numerical method. In doing so, we find a finite-time singularity for moderate values of Hall parameter δ, which is tied in our constant λ (see Figure 5).






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, σC > 0, γ0 > 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 limn 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 t0 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 O(t) error when plugged into the invariant, its absolute error when compared to the numerical solution to the differential equation is bounded by

and therefore is very accurate for large values of λ and k.

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 γ0 = -σC. 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.

Conclusion >>