Modeling lightning as a 2-dimensional fractal in an electric field
Sam Castle
11 May 2012
Many natural phenomena, including tree branches, snowflakes, coastlines, and lightning, can be modeled using fractal patterns. A fractal is a mathematical set that uses a recursive formula to produce an often self-similar shape with a nonintegral dimension. This experiment uses a common algorithm in lightning research, known as the dielectric breakdown model, to model a lightning strike as a fractal. The user has the option to add another initial source of lightning (a region representing initial negative charge), and the user can add conductors to the ground, simulating lightning rods, trees, or buildings.
The Dielectric Breakdown Model:
The dielectric breakdown model (DBM) establishes a grid with regions of varying electric potential to simulate the electrical charge buildup during a lightning storm. Before a strike of lightning, the bottom of a cloud typically possesses a strong negative charge and the ground has a net relatively positive charge. To simulate the source of the lightning in the cloud, we place a small region of negative charge, electric potential ϕ = 0, near the top of the grid. Along the bottom of the grid, we set the boundary condition ϕ = 1 to simulate the positive charge on the ground. All other grid points are initially given a midway electric potential equal to 0.5. Then, we allow the physical system to react to the presence of the charged areas by iterating the Laplace equation ten times (our arbitrary choice) over the grid:
∇2ϕ= 0
Computationally,
we solve the
Laplace equation for a point on the grid by calculating the average
potential
of the four neighboring grid points:
ϕi,j = ¼ * (ϕi-1,j + ϕi+1,j + ϕi,j-1 + ϕi,j+1)
After solving this equation ten times, we examine all the grid points that are adjacent to a negative charge (ϕ = 0). One of these points is randomly chosen, given a potential ϕ = 0, and added to the growing lightning bolt. The probability of a grid point being chosen is weighted according to its electric potential and is calculated by the following equation:
![[probability equation for DBM]](LaplaceLightning_ProbabilityEquation.PNG)
Computing dimension:
The most common way to analyze fractals is to calculate their dimension. To compute the dimension of our fractal models, we have chosen to use the radius of gyration method. The radius of gyration is a root mean square calculation. First, we find the center of mass of the fractal by computing the average distance between each particle and the original seed particle. Then, find the sum of the distance squared between each particle on the fractal and the center of mass. The radius of gyration, Rg, is equal to the square root of this value. For a fractal cluster, Rg is proportional to the number of particles, N, raised to the exponent β. The exponent β is equal to the reciprocal of the fractal dimension. So, we plot ln(Rg) vs ln(N) for many different instances, and the slope of the linear fit for this plot is equal to the fractal dimension.
The Dielectric Breakdown Lightning Model was created by Sam Castle using the Easy Java Simulations (EJS) version 4.3.7 authoring and modeling tool.
You can examine and modify a compiled EJS model if you run the model (double click on the model's jar file), right-click within a plot, and select "Open Ejs Model" from the pop-up menu. You must, of course, have EJS installed on your computer. Information about Ejs is available at: <http://www.um.es/fem/Ejs/> and in the OSP comPADRE collection <http://www.compadre.org/OSP/>.