1 Introduction

Dwarf novae are cataclysmic variables (CVs) which exhibit 2-6 mag recurrent outbursts (see Warner 1995 for an encyclopaedic review). It is generally accepted that these outbursts are due to a thermal-viscous instability that occurs in an accretion disc in which the viscosity is given by the $\alpha$-prescription (Shakura & Sunyaev 1973). The instability arises when hydrogen becomes partially ionized and the opacities vary steeply with temperature; this occurs when the disc material reaches temperatures of the order of 8000 K (see Lasota 2001 for a review of the model). A number of characteristics of many light curves (characteristic time scales, amplitudes, ...) are reproduced by the disc instability model (DIM) provided the viscosity parameter $\alpha$ is different in quiescence and in outburst. However several subclasses of dwarf novae show specific features that are not easy to explain with the standard version of the DIM.

SU UMa stars are defined as dwarf novae in which normal outbursts cycles are regularly interrupted by anomalously bright and long outbursts called superoutbursts. These outbursts are 0.7 mag brighter and last 5 to 10 times longer than the normal ones. The mean time $T_{\rm s}$ between two consecutive superoutbursts defines the period of a supercycle $T_{\rm s}$; in most SU UMa stars, $130 < T_{\rm s} <350$ days. There are however two extreme exceptions: ER UMa stars have anomalously short supercycles (between 19 and 45 days), while WZ Sge stars show only superoutbursts separated by very long quiescence periods that can reach 30 years. In addition, the light curve is modulated during superoutbursts at a period slightly longer than the orbital period. These so-called superhumps usually disappear shortly after the decline of the superoutburst. Whitehurst (1988) explained these superhumps by the precession of a distorted disc. The name "superoutburst'' is reserved for dwarf nova eruptions in which a superhump is present. This property differentiates superoutbursts from wide outbursts observed in many dwarf novae; van Paradijs (1983) suggested that superoutbursts are just wider outbursts in the bimodal outburst-length distribution. In addition, a 1985 outburst of U Gem had all the properties of a superoutburst (of WZ Sge stars) except for the superhump (see Kuulkers et al. 1999 and references therein). This could suggest that the superoutburst phenomenon is independent of the disc distortion. We discuss this possibility in Sect. 4.

As almost all SU UMa stars have periods below the period gap of CVs, they all have low secondary to primary mass ratio q, and the primary Roche lobe is large. The disc can therefore extend to relatively large radii, and this led Osaki (1989) to propose the thermal-tidal instability (TTI), in which superoutbursts are caused by the disc reaching the 3:1 resonance radius, as a model for SU UMa stars. SPH simulations by Whitehurst (1988) showed that the disc becomes eccentric and precesses when its radius reaches the 3:1 resonance radius, but it must be kept in mind that the hydrodynamical simulations of Stehle (1999) and Kornet & Rózyczka (2000) did not show such an effect.

Detailed observations of SU UMa stars point out several limitations of the TTI model. The mass transfer rate from the secondary is observed to be enhanced during outburst; this increase, possibly due to irradiation, is not taken into account, although it can drastically alter the light curves (Hameury et al. 2000).

The TTI model cannot easily reproduce the properties of ER UMa and WZ Sge subclasses. In order to obtain the very short ER UMa supercycles (19 to 45 days), one must assume that the tidal instability shuts off prematurely (Osaki 1995a) for very small mass ratios. The very long quiescence time of WZ Sge systems would on the other hand require a very small viscosity in quiescence (Smak 1993).

Occasionally, some systems show echo outbursts at the end of a superoutburst: several consecutive small outbursts with very short recurrence time are triggered before the disc returns to quiescence. During these echo-outbursts, superhumps are still present. These so-called late superhumps (LS) have also been observed in ER UMa stars. However, the TTI model simulations predict that the end of a superoutburst should coincide with the cessation of the tidal instability.

These different problems have lead several authors to investigate alternative possibilities. Hameury et al. (2000) have shown that if one takes into account the disc and secondary illumination in the standard DIM, one is able to reproduce light curves reminiscent of those of systems such as RZ LMi or EG Cnc. Recently Hellier (2001) suggested that some difficulties could be solved in the framework of a slightly modified TTI model; he proposed that the superoutburst must end before the end of eccentricity in low mass ratio systems, thus giving an explanation for LS.

We have included in our DIM model (Hameury et al. 1998; Buat-Ménard et al. 2001, hereafter Paper I) an enhanced tidal torque when the disc radius exceeds the 3:1 resonance radius. We do reproduce the earlier results of Osaki (1989), obtained using a simplified model. We show that in a number of cases, a cooling wave can end a superoutburst while the disc is still eccentric, thereby confirming the proposal of Hellier (2001). However, this by itself is not sufficient to reproduce the properties of ER UMa or WZ Sge systems, and additional effects such as illumination and mass transfer variations must be included.