A continuous approach to the lightning discharge
Dr. Beyza Aslan
Department of Mathematics, University of Alabama at Birmingham
January 11, 2008
219 Shelby Center
3:00 PM (Refreshments at 2:30 in 219 Shelby Center)
In this talk, I will describe a continuous model for the lightning discharge. In earlier work W.W. Hager developed the following approach for modeling a lightning discharge: The continuous partial differential equation describing the electric potential was discretized, and integrated forward in time. Whenever the electric field reached the breakdown threshold in some region of the atmosphere, the associated conductivity parameter in the discrete equation was taken to infinity. An explicit formula for the limiting potential was obtained. To develop a continuous version of this model, Maxwell's equations in three dimensions are analyzed, and a formula for the limiting potential as conductivity tends to infinity in a three-dimensional subdomain (the lightning channel) of the modeled domain is obtained. The limit is expressed in terms of the eigenfunctions for a generalized eigenvalue problem for the Laplacian operator. The potential in the breakdown region can be expressed in terms of a harmonic function which is constant in the breakdown region. The forcing term in the equation which is associated with the movement of charged particles by the wind can be estimated using the balloon-borne electric field sensors.
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