tokens: solves the MHD process of an isolated, twisted magnetic flux tube, initially placed in the convection zone, buoyantly rising into the atmosphere. The full set of basic resistive, adiabatic MHD equations in conservative form are solved by a finite difference scheme with the spatial derivatives by the fourth-order central differences and the temporal derivatives by the four-step Runge–Kutta scheme. Periodic boundary conditions are applied for the x -direction and symmetric boundaries for both the y - and z -directions. The initial background atmosphere of the simulation is gravitationally stratified, which consists of an adiabatically temperature-gradient convection zone (, where km is the length unit of the FES model), the cool isothermal photosphere/chromosphere (), and the hot isothermal corona (). The flux tube is initially placed at with the form of and , where r is the radial distance from the tube’s axis, the radius, kG the axial field strength, and the twist intensity (the negative sign indicates a left-handed twist). The middle of the tube is made buoyant by a small density deficiency and the tube started to emerge due to its own buoyancy. As the time-dependent bottom boundary data for the data-driven model, we extracted the time sequence of 2D data at fixed heights from the FES model. To test the behavior of the data-driven model with different inputs, data sets at three different slices from the FES model are used, including (referred to Z00 , representing the photosphere), ( Z10 , i.e., the chromospheric layer), and ( Z20 , i.e., the coronal base). The slices were sampled at every (where s is the time unit of the FES model) from 0 to 500. The slices spanned over , resolved by the uniform grid spacing of 200 × 200, thus the resolution is . The data sets include three components of magnetic and velocity field, i.e., , and v z . As we have mentioned in Section 1 , the Lorentz force in the boundary data is an important factor affecting the output of the data-driven model. Thus we quantitatively assess the Lorentz force as well as torque in the data of different layers. The sum of Lorentz force (and torque ) in a volume can be expressed by surface integral (Aly 1989 ; Sakurai 1989 ), and by neglecting contribution from the side and top boundaries, they can be written as integrals of the 2D data sets:andFollowing Duan et al. ( 2020 ), to assess the data with respect to the force-free condition, the forces can be normalized by the integrated magnetic pressure force F p that is given byand the normalized forces are ratios defined as , , and , respectively. Similarly, the normalized torques are defined as , , and , where T p is magnitude of the net torque induced by only the magnetic pressure force, whereFor a field being close to force-free, it must have ratios much less than unity, i.e., and . Metcalf et al. ( 1995 ) suggested that the magnetic field can be considered as force-free if the normalized forces are all less than or equal to
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