we used a leapfrog integrator with variable and individual time-steps. We used a Plummer softening length of $\epsilon = 30$ pc. This value is sufficiently small to resolve the three-dimensional structure of a disk galaxy [@Hernquist1987; @Bottema2003]. It is to be noted that the formation of the spiral arms is inhibited if the adopted softening length is considerably larger than the disk thickness ($\sim 300$ pc). The number of particles is 300 million.

  Component              Parameter                       Value
  ---------------------- ------------------------------- ---------------------------------
  Dark halo              Mass ($M_{\rm h}$)              $6.3\times 10^{11}$ M$_{\odot}$
  (Rigid)                Radius ($R_{\rm h}$)            $122$ kpc
                         Concentration ($C_{\rm NFW}$)   $5.0$
  Initial stellar disk   Mass ($M_{\rm sd}$)             $3.2\times 10^{10}$ M$_{\odot}$
  (Live)                 Scale length ($R_{\rm sd}$)     $4.3$ kpc
                         Scale height ($z_{\rm sd}$)     $0.3$ kpc

  : Model parameters for each mass component in $N$-body simulation

\[tbl:models\]

![image](fig1.eps){width=".9\textwidth"}

Global Evolution of Spirals {#sec:GlobalMorphology}
===========================

Figure \[fig:snapshotsMS\] shows the time evolution of the stellar disk. Hereafter, we use a rotational period measured at $R = 2 R_{\rm sd} (=8.6~\rm kpc)$, $T_{\rm rot}$, which corresponds to 325 Myr. The top panels show the face-on views of the stellar disk. The middle panels show the radial distributions of the spiral modes which are analyzed by performing a one-dimensional Fourier decomposition 