In the absence of the toroidal flux, two coupled quasi two‐dimensional elliptic equilibrium equations have been derived to describe self‐consistent three‐dimensional quasi‐static magnetospheric equilibria with isotropic pressure in an optimal (ψ,α,χ) flux coordinate system, where ψ is the magnetic flux function, χ is a generalized poloidal angle, α = ϕ‐δ(ψ,α,χ), ϕ is the toroidal angle, δ(ψ,α,χ) is periodic in ϕ, and the magnetic field is represented as B = ∇ψ × ∇α. A three‐dimensional magnetospheric equilibrium code, the MAG‐3D code, has been developed by employing an iterative metric method. The MAG‐3D code is the first self‐consistent three‐dimensional magnetospheric equilibrium code. The main difference between the three‐dimensional and the two‐dimensional axisymmetric solutions is that the field‐aligned current and the toroidal magnetic field are finite for the three‐dimensional case, but vanish for the two‐dimensional axisymmetric case. The toroidal magnetic field gives rise to a geodesic magnetic field curvature and thus a magnetic drift parallel to the pressure gradient direction, which gives rise to the field‐aligned current.
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