Determining the Initial Conditions (pictures)

To my knowledge the properties of the postulated flux loop are not determined precisely, so I will design an approximation which appears reasonable. For simplicity, I will construct the $r$- and $z$-components separately, and then at time $t=0$ form the product

$$B(t=0) = \left(\sum_{i}{c_{i}\cos(k_i z)}\right) \left(\sum_{j}{d_j J_1(\lambda_j r)}\right)$$ (25)

which is a special case simplification of the above formula. Throughout this article, distances will be measured in light seconds, so distances and times are essentially the same units.

The flux loop appears to be very compact in both the $r$- and $z$-directions. Unfortunately compact functions are often the most difficult to approximate with Fourier-type series, but it is possible. In the $z$ direction I found that a Gaussian profile is most straightforward to construct, which is achieved by generating coefficients $c_i$ with a broad Gaussian distribution: $c_i = \exp(-i^2/(2*35^2))$, and with $k_i = 0.05 i$ lt-sec$^{-1}$.

In the radial dimension I found that the prescription, $d_j = J_0(i/7)-J_0(i/5)$ constructed a suitable radial profile for the toroidal field, where $\lambda_j = 0.05 j$ lt-sec$^{-1}$. This profile is very compact, being nearly constant and non-zero in the interval $r \in [3,4]$ lt-sec, but nearly zero everywhere else. Here I used 300 terms for both the $r$ and $z$ terms.

The following figure shows the geometry of the toroid with respect to the X-Y plane (the iso-surface is about 5% of peak value).