localized and extremely strong [*poloidal*]{} magnetic fields.

Formulation and numerical method
================================

Since we employ the formulation developed by [@Tomimura_Eriguchi_2005], [@Yoshida_Eriguchi_2006], and [@Yoshida_Yoshida_Eriguchi_2006], here we summarize the main scheme briefly and explain newly introduced parts in detail.

Assumptions and basic equations
-------------------------------

We make the following assumptions for the magnetized stars.

1.  The system is in a stationary state, i.e. $ \P{}{t} = 0$  .

2.  When stars are rotating and have magnetic fields, the rotational axis and the magnetic axis coincide.

3.  The rotation is rigid.

4.  The configurations are axisymmetric about the magnetic or the rotational axis, i.e. $ \P{}{\varphi} = 0 $, where we use the spherical coordinates $(r, \theta, \varphi)$.

5.  The configurations are symmetric with respect to the equator.

6.  There are no meridional flows.

7.  The star is self-gravitating.

8.  The systems are treated in the framework of non-relativistic physics.

9.  The conductivity of the stellar matter is infinite, i.e. the ideal magnetohydrodynamics (MHD) approximation is employed.

10. No electric current is assumed in the vacuum region.

11. The barotropic equation of state is assumed : $$\begin{aligned}
       p = p(\rho) \ .\end{aligned}$$

Here $p$ and $\rho$ are the pressure and the mass density, respectively. 