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Astron. Astrophys. 318, 747-767 (1997)

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1. Introduction

Spiral galaxies often show a warping of their external parts, observed in HI. The warps appear as an "S" or an integral sign shape for galaxies observed edge-on (Sancisi, 1976).

This phenomenon, known since 1957, has led to theoretical difficulties. Warps are bending waves, and we correctly understand their propagation (see Hunter, 1969a for a dispersion relation in an infinitely thin disk, and see Nelson 1976a and 1976b, Papaloizou and Lin 1995, Masset and Tagger 1995 for the dispersion relation taking into account finite thickness and compressional effects), but we do not know the mechanism responsible for their excitation. It cannot be systematically justified by tidal effects (Hunter and Toomre 1969), since we observe warping of very isolated spiral galaxies. It cannot be explained by a temporary excitation, since a warp propagates radially in the disk is not reflected at its edge, so that it should disappear over a few galactic years (Hunter, 1969b). Attempts have been made to connect the existence of warps and the properties of haloes (Sparke, 1984a, Sparke and Casertano 1988, Hofner and Sparke 1994). The mechanism proposed by Sparke and Casertano (1988) implies an ad hoc misalignment angle between the normal axis of the galactic plane and the "polar" axis of the halo. This mechanism has recently encountered self-consistency difficulties (Dubinski and Kuijken 1995). Furthermore, it relies on the dubious hypothesis of a rigid and unresponsive halo. Several explanations have also been proposed, such as the infall of primordial matter (Binney, 1992), the effect of the dynamical pressure of the intergalactic medium, etc.... For a review of all the proposed mechanisms, see Binney (1992).

Binney (1978, 1981) has considered the possibility of a resonant coupling between the vertical motion of a star and the variation of the galactocentric force, due to a halo or a bar. He concluded that a bar could be responsible for the observed warps and corrugations. Sparke (1984b) has also explored this possibility, and considered the growth of a warp from a bar or a triaxial halo. She found that a bar was unlikely to be responsible for a warp, but she emphasized that a triaxial halo could quite well reproduce observed warps.

We consider here another mechanism, the non-linear coupling between a spiral wave and two warp waves. Non-linear coupling between spirals and bars has already been found (Tagger et al. 1987 and Sygnet et al. 1988) to provide a convincing explanation for certain behaviors observed in numerical simulations (Sellwood 1985), or observed in Fourier Transforms of pictures of face-on galaxies. The relative amplitude of spiral or warp waves (the ratio of the perturbed potentials to the axisymmetric one) is about [FORMULA] to [FORMULA] (see Strom et al. 1976). Non-linear coupling involves terms of second order in the perturbed potential, while linear propagation is described by first order terms. Classically one would thus believe that non-linear coupling is weak at such small relative amplitudes. However in the above-mentioned works it was found that the presence of resonances could make the coupling much more efficient if the wave frequencies are such that their resonances (i.e. the corotation of one wave and a Lindblad resonance of another one) coincide. At this radius the non-linear terms become comparable with the linear ones, so that the waves can very efficiently exchange energy and angular momentum. Indeed in Sellwood's (1985) simulations, as discussed by Tagger et al. (1987) and Sygnet et al. (1988), an "inner" spiral or bar wave, as it reaches its corotation radius, transfers the energy and angular momentum extracted from the inner parts of the disk to an "outer" one whose ILR lies at the same radius, and which will transfer them further out, and ultimately deposit them at its OLR. In this process the energy and momentum are thus transferred much farther radially than they would have been by a single wave, limited in its radial extent by the peaked rotation profile.

We will show here that a similar mechanism, now involving one spiral and two warp waves is not only possible (by the "selection rules" associated with their parity and wavenumbers), but also very efficient if the same coincidence of resonances occurs. This allows the spiral wave, as it reaches its OLR (and from linear theory deposits the energy and momentum extracted from the inner regions of the disk) to transfer them to the warps which will carry them further out.

Unlike Tagger et al. (1987) and Sygnet et al. (1988), we will throughout this paper restrict our analysis to gaseous rather than stellar disks, described from hydrodynamics rather than from the Vlasov equation. The reason is that our interest here lies mainly in the excitation of the warps, which propagate essentially in the gas (indeed the outer warp is observed in HI, and the corrugation is most likely (Florido et al. 1991) due to the motion of the gaseous component of the galactic disk). On the other hand, the spiral wave propagates in the stellar as well as the gaseous disk. The difference is important only in the immediate vicinity of Lindblad resonances, where the spiral wave is absorbed; as a consequence, its group velocity vanishes at the resonances. Since the group velocity of the waves will appear as an important parameter, we will choose to keep the analytic coupling coefficient derived from the hydrodynamic analysis, but we will introduce, for the spiral density wave, the group velocity of a stellar spiral. From the physics involved this will appear as a reasonable approximation; furthermore it should only underestimate the coupling efficiency, since it does not include the resonant stellar motions near the resonance.

On the other hand, we will show that non-linear coupling is efficient only in a narrow annulus close to the OLR of the spiral, over a scale length similar to the one of Landau damping. We will thus conclude that the two processes are in direct competition, with the spiral transferring its energy and momentum, in part to the stars by Landau damping, and in part to the warps which will transfer them further outward, the exact repartition between these mechanisms presumably depending on detailed characteristics of the galactic disk.

The paper is organized as follows: in a first part we will introduce the notations, and the selection rules relative to the coupling. In a second part, we will derive the coupling coefficient from the hydrodynamic equations expanded to second order in the perturbed quantities, and we will try and simplify it. In a third part, we will analyze the efficiency of the coupling, together with the locations where it may occur. In the last sections we will compare our predictions to the observations, and we will propose some possible observational tests of our mechanism.

Some of the computations are tedious and lengthy. For the sake of clarity, they are developed in appendices, so as to retain in the main text only the principal results and the physical discussions.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998