The energy method for calculating quasi-stationary atmospheric electric fields

A new mathematical model is proposed to describe the quasi-stationary atmospheric electric fields with an approximate consideration of the ionosphere conductivity. Under the assumption of verticality of the geomagnetic field, a two-dimensional model of the ionosphere conducting layer is used that is customary for the large-scale fields. Within the framework of this model, the ionosphere can be described by a special boundary condition in a boundary value problem for the atmospheric electric field. A linear boundary value problem is stated with a symmetric positive definite elliptic operator. The minimum principle is substantiated for the quadratic functional of energy. The existence and uniqueness of a generalized solution are proved. Under study is also the imprecision of the approximate description of the ionosphere conductor.