In this dissertation, we do not implement the lightning discharge described in
Section 2.2.3. Instead we focus on integrating the equations up to the discharge
initiation.
We take the divergence of Equation (3.1) to get
0E
eV*- + V-aE + V J = 0. (3.2)
Boundary conditions: In this model, we consider a bounded domain Q between
the surface of the earth and ionosphere. Let us consider boundary conditions on
9fO:
= 0 on the surface of the earth, since we treat the earth as a perfect
conductor.
= --1 on the ionosphere. 41 = 250,000 V (Adlerman and Williams [1]).
On the sides of the domain Q, assume that the potential 0 is the "fair field"
potential. That is, the current and the time derivative of the electric field are
both zero under fair-weather conditions. Hence, the potential satisfies
V aE = 0. (3.3)
Assume that both the potential and the conductivity a depend only on z
on the sides of the domain Q, and since E = V0, we have
a or =0, (3.4)
from Equation (3.3). In order to obtain the potential function on the sides of
the domain Q, the Equation (3.4) is solved with these boundary conditions:
the potential is zero on the surface of Earth and 41 on the ionosphere.
Therefore, we discover that
(z) = dz,
O /z) (Z)