2 Numerical calculations

2.1 2-D SPH methods

We perform two-dimensional SPH calculations (Lucy 1977; Gingold & Monaghan 1977), and reproduce gas motions in a weak stellar bar potential. We adopt the SPH formulation by Benz (1990), and use the spline kernel (Monaghan & Lattanzio 1985) with the modification for its gradient (Thomas & Couchman 1992). The correction term for viscosity (Balsara 1995) is taken into account to avoid large entropy generation in pure shear flows. The SPH smoothing length h varies in space and time, keeping the number of particles within the radius 2h at an almost constant of 32 according to the method of Hernquist & Katz (1989). The leapfrog integrator is adopted to update positions and velocities. We use $3\times10^4$ particles to represent the gas disk.

2.2 Galaxy disk and weak bar potentials

We take the galaxy potential for a weak bar used in Sanders (1977) and Wada & Habe (1992); the oval potential for a barred galaxy $\Phi(R,\theta)$ is represented in axisymmetric and bisymmetric parts by

$$\Phi(R,\theta) = \Phi_{0}(R) + \Phi_{b}(R) \cos 2\theta,$$ (2)

where the first term is an axisymmetric potential for a disk,

$$\Phi_{0}(R) = - C \frac{1}{(R^2 + a^2)^{1/2}},$$ (3)

and the second term is a bisymmetric one for a weak bar,

$$\Phi_b = - \epsilon_0 C \frac{a R^2}{(R^2+a^2)^2}\cdot$$ (4)

The normalization coefficient $C = (27/4)^{1/2} a v_{\rm max}^2$ is calculated from the core radius a and maximum rotation velocity $v_{\rm max}$. $\epsilon _0$ is a parameter to control the strength of the bar. R and $\theta$ stand for the galactocentric radius and azimuthal angle from the bar respectively. We are interested in the notion that noncircular motions in a bar apparently indicate a central massive component. We thus do not consider any massive central component in our potential, such as a bulge (Athanassoula & Bureau 1999) or a massive black hole (Fukuda et al. 1998, 2000).

Our potential model has the benefit of being capable of analytically investigating gaseous orbits in the bar potential (Wada 1994), and has been well-studied in numerical simulations for the bar-driven gas fueling into galactic centers (Wada & Habe 1992, 1995), the gas kinematics in the Galaxy (Wada et al. 1994), the spatial distribution of mass-to-light ratio in a galaxy NGC 4321 (Wada et al. 1998), and the effects of a central black hole (Fukuda et al. 1998, 2000).

We fix $a=\sqrt{2}{\rm ~ kpc}$ and $v_{\rm max}=220{\rm ~ km ~ s^{-1}}$; the corresponding rotation curve is shown in Fig. 1. The gas reaches the maximum circular rotation velocity at $R=2{\rm ~ kpc}$ with a rotational period of $56\rm ~Myr$. Figure 2 shows the radial changes of frequencies, $\Omega$ and $\Omega\pm\kappa/2$, where $\Omega$ and $\kappa$ are circular and epicyclic frequencies respectively. We set the pattern speed of the bar $\Omega _{\rm b}$ at 0.4, 0.8, and 1.5 times the maximum of $\Omega-\kappa/2$, indicated by horizontal lines. Models with $\Omega_{\rm b}=0.4$ and $0.8\times (\Omega-\kappa/2)_{\rm max}$ have two inner Lindblad resonances (ILR), while those with $\Omega_{\rm b}=1.5\times (\Omega-\kappa/2)_{\rm max}$ have no ILR. $\epsilon _0$ is set to 0.05, 0.10, and 0.15. Our nine models are listed in Table 1.

Figure 1: Rotation curve from the axisymmetric potential, Eq. (3), with the core radius $a=\sqrt{2}{\rm ~ kpc}$ and maximum rotation velocity $v_{\rm max}=220{\rm ~ km ~ s^{-1}}$.

Figure 2: Radial changes of frequencies, $\Omega (R)$ and $\Omega (R) \pm \kappa (R)/2$. Four horizontal lines represent the pattern speeds of the bar, i.e. 0.4, 0.8, 1.0, and 1.5 $\times (\Omega -\kappa /2)_{\rm max}$.

  

Table 1: Model parameters.

Two free parameters, i.e. bar pattern speed $\Omega _{\rm b}$ and bar strength $\epsilon _0$, and radii of inner Lindblad resonances (ILR), corotation resonance (CR) and outer Lindblad resonance (OLR), maximum density ratio of the bar over the disk $(\rho\rm _b/\rho_0)_{\rm max}$, and rotational time-scale of the bar $T_{\rm bar} [\equiv 2 \pi / \Omega_{\rm b}]$ at $R=2{\rm ~ kpc}$ are tabulated.

2.3 Initial conditions

The gas is initially distributed in a uniform-density disk with an $8{\rm ~ kpc}$ radius, following pure circular-rotation that balances the centrifugal force. The gas temperature is assumed to be a constant $10^4{\rm ~ K}$, corresponding to the sound speed of about $10{\rm ~ km ~ s^{-1}}$, throughout evolution. The total gas mass is assumed to be 5% of the total stellar mass within the radius of $8{\rm ~ kpc}$. The results are not significantly affected by the total gas mass, because thermal pressure is much smaller than the rotational energy, and we do not calculate self-gravity of the gas. We advance the calculations up to about $500\rm ~Myr$.

  
2.4 Gas dynamical evolution

Figure 3: Gas dynamical evolution in model E: ( $\Omega _{\rm b}$, $\epsilon _0$) = (0.8, 0.10). The stellar bar runs horizontally, and the gas rotates counterclockwise. After $t\sim250~{\rm Myr}$ the system reaches a state of quasi-equilibrium.

Gas dynamics in a barred potential have been well studied in numerical simulations (Wada & Habe 1992; Heller & Shlosman 1994; Piner et al. 1995; Fukuda et al. 1998; Athanassoula & Bureau 1999). Our models evolve consistently with these simulations. Figure 3, model E, shows a typical evolution. Three phases of the evolution can be seen in this model: (a) linear perturbation phase, $t \sim 0{-}50\rm ~Myr$, (b) transient phase, $t \sim 50{-}250\rm ~Myr$, and (c) quasi-steady phase, $t>250\rm ~Myr$.

The characteristic structure appearing during the evolution depends strongly on the positions of resonances, i.e. the pattern speed of the bar $\Omega _{\rm b}$. In phase (a), leading and trailing spiral arms are formed around the inner ( $R=1.1{\rm ~ kpc}$) and outer ( $3.1{\rm ~ kpc}$) ILRs respectively at $t=36\rm ~Myr$. These resonant-driven spirals are expected in a linear theory (Wada 1994). While the outer trailing arms remain with increasing density contrasts, the inner leading arms evolve into an oval ring, or a gaseous bar (t= 71-107$\rm ~Myr$), i.e. phase (b). The oval ring first leads the stellar bar ( $71\rm ~Myr$), rotating opposite to the gas rotation ( $107\rm ~Myr$), and being aligned with the stellar bar ( $250\rm ~Myr$), and thereafter, the system develops toward a quasi-steady phase, i.e. phase (c). The ellipticity of the nuclear ring grows as high as $e \sim 0.8$. The ripple seen in the outer arms at $t=394\rm ~Myr$ would originate in the Kelvin-Helmholtz instability (Piner et al. 1995). Gas dynamics and structure in the inner region of the disk are not affected by this instability.

Figure 4: Final snapshots for all nine models. Different pattern speeds $\Omega _{\rm b}$ and bar strengths $\epsilon _0$ are arranged vertically and horizontally, respectively. The stellar bar runs horizontally, and the gas rotates counterclockwise.

Figure 5: Final velocity fields for all nine models. The arrangement is the same as in Fig. 3. Arrows are drawn for 1500 out of total 30 000 gas particles.

Figures 4 and 5 display the final snapshots and velocity fields for nine models. Different pattern speeds  $\Omega _{\rm b}$ and bar strength  $\epsilon _0$ are arranged vertically and horizontally, respectively. It is evident that the final structure depends strongly on  $\Omega _{\rm b}$, while  $\epsilon _0$ changes only the density contrasts. Model A, B, and C have no ILRs, thus no spiral arms or ring in their inner regions are formed. The outer spiral arms are formed outside the radius of the corotation resonance (CR) due to the outer Lindblad resonance (OLR). Model D and F resemble model E. Model G, H and I also have arms and rings similar to those in model E, but at different radii, corresponding to the location of the ILRs. Figure 5 clearly show that, in models D, E, F and G, most gaseous orbits are x₁-like, while in model H and I, the large separation and low density between the two ILRs suffice to leave the gases on x₂-orbits, which form a stable oval ring, nearly perpendicular to the stellar bar.