708 ISSN 2070-0482, Mathematical Models and Computer Simulations, 2017, Vol. 9, No. 6, pp. 708–715. © Pleiades Publishing, Ltd., 2017. Original Russian Text © R.Yu. Lukianova, 2017, published in Matematicheskoe Modelirovanie, 2017, Vol. 29, No. 5, pp. 122–132. Electric Potential in the Earth’s Ionosphere: a Numerical Model R. Yu. Lukianova a, b a Geophysical Center, Russian Academy of Sciences, Moscow, Russia b Space Research Institute, Russian Academy of Sciences, Moscow, Russia e-mail: r.lukianova@gcras.ru Received July 12, 2016 Abstract—Modeling the global distribution of the electric potential in the Earth’s ionosphere is based on the solution of a 2D continuity equation in the ionospheric-magnetospheric current circuit. The potential distribution is described by the boundary value problem for an elliptic system of partial dif- ferential equations on the spherical shell approximating the ionosphere, which is divided into three subregions with nonlocal boundary conditions. Implementation of the boundary conditions, which reflect the continuity of the common current circuit and potential equalization at the boundaries of the polar caps, is leading to the mutual dependence of the potential distribution within the northern and southern caps and their influence on the potential distribution in the midlatitude region. The problem is solved by an iterative method with a regularizing operator which is inverted using the sep- aration of the variables and the fast Fourier transform with respect to the azimuthal variable and the sweep method with respect to the latitudinal one. Keywords: numerical modeling, electrodynamics of the Earth’s ionosphere, ionospheric plasma con- vection, partial differential equations, iterative methods DOI: 10.1134/S2070048217060114 1. INTRODUCTION As the altitude in the Earth’s atmosphere increases, the main role start to be played by the electrody- namic processes controlled by the plasma flows leaving the sun: the solar wind (SW) and the interplane- tary magnetic field (IMF). As a layer with high electrical conductivity, the ionosphere is formed at an alti- tude of 80–500 km due to the ionization of air molecules by the high-frequency part of the electromag- netic solar radiation. The ionosphere can be considered as the base of the region occupied by the magnetosphere. The magnetosphere and the ionosphere are interrelated electrodynamically through field-aligned currents (FACs) flowing along highly conducting geomagnetic field lines from the boundary layers of the magnetosphere to the high-latitude ionosphere, where these currents excite the electric fields [1]. During the IMF‘s interaction with the Earth’s magnetosphere, approximately 10 12 W of energy enters the near-Earth space, which generates the movements of the magnetospheric and ionospheric plasmas and excites electric fields and currents. Since the geomagnetic field has a dipolar shape, these processes are especially active at high latitudes of the Earth. In the collisionless plasma, the occurrence of an electric field that is transverse to the magnetic field manifests itself in the convective movement of plasma at the rate of electrical drift. Thus, the distribution of electric potential in the ionosphere can be represented as a system where the convection lines are identical to the isolines of the electric field’s potential. The mathematical modeling of the distribution of the ionospheric electric potential is based on solving a 2D continuity equation in the ionospheric-magnetospheric current circuit. However, earlier this prob- lem was solved analytically using quite inaccurate approximations of the parameters mostly on one hemi- sphere regardless of the electrodynamic coupling of the ionospheres in the opposite hemispheres [2, 3]. In the empirical models constructed based on the statistical analysis of satellite and radar measurements, the Dirichlet homogeneous boundary conditions were usually assigned at the equatorial boundary of the high-latitude region of a hemisphere in order to simplify the solution method [4–6]. However, such a for- mulation of the problem does not allow calculating the global distribution of potential accurately, since a boundary condition for the normal current component is required for considering the current spreading over the entire ionosphere shell [7, 8]. It is important to formulate the problems of ionospheric electrodynamics in a double-hemispheric approximation due to the following main causes: (а) the tilt of the Earth’s axis and the mismatch between