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## Fractal Lightning: Stochastic Model

We want to generate a fractal model that can be parametrized by its fractal dimension. For this purpose, we follow Niemeyer et al. [1984] who proposed a two dimensional stochastic dielectric discharge model that naturally leads to fractal structures. In this model the fractal dimension D can be easily parametrized by a parameter . Femia et al. [1993] found experimentally that the propagating stochastic Lichtenberg pattern is approximately an equipotential. Then, the idea is to create a discrete discharge pattern that grows stepwise by adding an adjacent grid point to the discharge pattern generating a new bond. The new grid point, being part of the discharge structure, will have the same potential as the discharge pattern. Such local change will affect the global potential configuration, see Fig. 13.

Figure 13: Diagram of the discrete discharge model.

The potential for the points not on the discharge structure is calculated by iterating the discrete two dimensional Laplace's equation

until it converges. This method reproduces the global influence of a given discharge pattern as it expands. The discharge pattern evolves by adding an adjacent grid point. The main assumption here is that an adjacent grid point denoted by (l,m) has a probability of becoming part of the discharge pattern proportional to the power of the local electric field, which translates to

in terms of the local potential. Here we have assumed that the potential at the discharge is zero. The structure generated for , corresponding to a Lichtenberg pattern, is shown in Fig. 14.

Figure 14: Fractal discharge generated with .

Figure 15: The plot of vs for .

The color coding corresponds to the potential. Figure 15 shows a plot of vs. for the fractal discharge of Fig. 14, i.e. . Again the scaling behavior only occurs over a few decades, but it is very clear. The dimension of this structure is .

Figure 16: The dimension of the stochastic model as a function of with the estimated error bars.

Note that this model, and also the dimension of the discharge, is parametrized by . Intuitively we expect that when the discharge will have the same probability of propagating in any direction, therefore, the discharge will be a compact structure with a dimension D=2. If then the discharge will go in only one direction, hence D=1. Between these two limits, the dimension will be the function shown in Fig 16. As an example the corresponding structure generated for (Fig 17) has a dimension of .

Figure 17: Fractal discharge generated for .

To compute the radiated fields, we must describe the current along each of the segments of the fractal discharge. We start with a charge at the center of the discharge. The current is then discharged along each of the dendritic arms. At each branching point we chose to ensure conservation of current, but intuitively we know that a larger fraction of the current will propagate along the longest arm. Suppose that a current arrives at a branching point, and if is the longest distance along the branching arm, we intuitively expect that the current on the arm should be proportional to . Therefore, we satisfy charge (or current) conservation if the current along the branching arm is

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