The charge removal from the thundercloud results in an excess electric field from a charge with opposite sign until the charge distribution in the thundercloud adjusts to the new charge configuration [Pasko, 2006]. The electric field above the thundercloud is dominated by the electric (Coulomb) field of a unipolar monopole charge because the higher-order multipole fields of the charge distribution in the thundercloud, if any, fall off more quickly with height. This Coulomb field Ec results from the total charge deposition described in equation (3),
In this first-order, quasi-static description, the electric field varies spatially only with height (in one dimension), where r is the vertical distance from the charge. The electric field will decay in the conducting atmosphere with the local dielectric relaxation constant τr = ɛ0/σ, which is described with a scalar, inhomogeneous, linear differential equation of first-order for the resulting electric field E,
derived from Ampère's law by neglecting magnetic fields. The right hand side of equation (5) results from the driving external displacement current
by use of equations (3) and (4). The solution of this differential equation for the electric field is given by [Füllekrug, 2006]
Note that the electric field at the highly conducting boundaries of the model (the Earth's surface and in the lower ionosphere) need to vanish, which is achieved by placing appropriate mirror charges inside the Earth and into the ionosphere. In a more general approach, it is possible to include a time-dependent ionosphere, i.e., the moving capacitor model, which can lead to a slowing down of the relaxation and even a growth of the electric field below the moving boundary [Pasko et al., 1997]. The electrical (Joule) heating from the resulting electric fields is calculated from the time integral of the quasi-electrostatic power
where W is the energy density deposited by the electric fields in the atmosphere and σ is the atmospheric conductivity. This atmospheric conductivity depends on the height z and it may be approximated with σ(z) = σ0exp((z − z0)/s), where σ0 ≈ 6.4 × 10−10 S/m is a scaling conductivity at z0 = 50 km height and s = 3.2 km characterizes the conductivity gradient in the atmosphere. It is evident that the total electrical energy deposition in the stratosphere is dominated by the energy density resulting from the slow continuing current (Figure 3).