Fig. 1: Schematic representation of the simulation framework for solar wind-Earth atmosphere interactions, incorporating terrestrial atmospheric profiles and the geodynamo model used in this study, along with the coordinate frames employed for analysis.

a Illustrated is the MHD computational domain used to model the interaction of the solar wind with the terrestrial atmosphere. Both the Earth and the Moon lie within the simulation box, while the solar wind impinges on the left boundary (y-z plane) with a velocity normal to the plane. The incoming wind is magnetized, with its field lines aligned along the y-axis. We used the Geocentric-Lunar Orbital Plane coordinate frame, which is tilted by the same angle as the Moon’s orbital plane, to denote the changing phase angle of the Moon in its orbit and to compute the solar and Earth wind number fluxes over a complete lunation. b Static isothermal atmosphere at the start of a run (Case–I) for a present-day magnetized Earth (refer to Equation 5) and an Archean unmagnetized Earth (see Equation 6). The interior of the planet is also assumed to be isothermal. The red and blue shaded regions represent the fixed total mass in the two subcases (~6.0 × 10¹⁸ kg), while the turquoise region illustrates the range of hydrodynamic escape boundary heights used in the binary mixing models. The kink in the red line denotes the \({\upbeta }_{{\rm{p}}}\) = 1 surface, outside of which the magnetic field governs the plasma outflow. c Left panel: Cross-section of the planetary magnetic field \(({\overrightarrow{B}}_{\small{\!\!GF}})\) in the noon-midnight meridian plane used in the simulations at the beginning of the runs (t = 0). The color contour shows the log-normalized strength of the B-field (B_⊕ = 3.12 × 10⁴ nT), and the arrows denote the direction of field lines. We use a cutoff boundary of 20 R_⊕ beyond which the influence of the intrinsic field is essentially insignificant. The horizontal axis aligns with the ecliptic plane (marked by the solid black line), while the polar axis (indicated by the dashed red line) points from dusk to dawn. Right panel: Zoomed-in slice of the field structure in the interior of the planet. The solid thick circle represents the planet’s surface.