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

2. Formalism and selection rules

2.1. Notations

We develop all our computations in the well-known shearing sheet approximation, which consists in rectifying a narrow annulus around the corotation radius of the spiral wave into a Cartesian slab. The results given by an exact computation taking into account the cylindrical geometry of the galaxy would differ by some metric coefficients, but would not differ physically from the shearing sheet predictions, so that the main conclusions would remain valid. We call x the radial coordinate, with its origin at the corotation of the spiral, and oriented outward. We call y the azimuthal coordinate, oriented in the direction of the rotation, and z the vertical one, so as to construct a right-oriented frame xyz.

The hydrodynamic quantities are the density , the speed components , and the gravitational potential . We assume that the disk is isothermal, with a uniform temperature, in order to avoid unnecessary complexity. Thus we can write: , where P is the pressure and a the isothermal sound speed.

The perturbed quantities are denoted with subscripts 1, 2 or S referring to the wave involved (one of the two warps or the spiral wave), and equilibrium quantities with a subscript 0.

The epicyclic frequency is denoted by , the rotation frequency by , Oort's first constant by , and we have the relation:

We call µ the vertical characteristic frequency of the disk. We have:

(see Hunter and Toomre, 1969). This frequency plays the same role for bending waves as does for spiral waves; it should not be confused with the frequency of the vertical motion of individual particles, which is usually much higher. Indeed individual particles move vertically in the potential well of the disk, giving them the high frequency . On the other hand, when one considers motions of the whole disk, the potential well moves together with the disk and exerts no restoring force. This leaves only a weaker restoring force, giving the vertical frequency µ as discussed e.g. in Hunter and Toomre (1969) or Masset and Tagger (1995).

For each of the three waves we consider, we denote by its frequency in the galactocentric frame, m its azimuthal wavenumber ( for a two-armed spiral, and for an "integral-sign" warp), its frequency in a local frame rotating with the matter. In the peculiar case of the shearing sheet, we note where r is the distance to the galactic center, and the azimuthal wavenumber.

We will have to perform part of our computations in the WKB approximation, i.e. assume that the radial wavevector varies relatively weakly over one radial wavelength; in practice, in the shearing sheet, this reduces to the classical "tightly wound" approximation, . We will also use , the modulus of the "horizontal" wavevector. In the tightly wound approximation one has .

We also introduce , and various integrated quantities:

the perturbed surface density,

the equilibrium surface density, and:

the mean vertical deviation of a column of matter from the midplane, under the influence of a warp.

We call H the characteristic thickness of the disk. The vertical density profile in the disk is taken to be consistent (see Masset and Tagger, 1995), i.e. it must fulfil simultaneously the Poisson equation and the hydrostatic equilibrium equation, with the additional condition that the radial derivatives of the equilibrium potential do not depend on z throughout the disk thickness: this condition results from the hypothesis of a disk which is geometrically thin, , although we do resolve vertically the perturbed quantities along the vertical direction.

We summarize most of our notations in Figs. 1 and 2 and in Table 1.

Fig. 1. This figure summarizes our main notations relative to perturbed velocity and disk thickness.

Fig. 2. This figure summarizes our notations relative to waves (spiral or warps). The column of matter has a section unity. At equilibrium (or more generally without spiral wave) it contains a mass , and when perturbed by a spiral wave it contains the mass .

Table 1. In this table we summarize the main frequencies of our system, and some other quantities related to warps or spirals.

2.2. Notion of coupling and selection rules

2.2.1. Mode coupling

As discussed in the introduction, we consider the coupling between a spiral wave and two warp waves. This coupling is necessarily a non-linear mechanism. In a linear analysis, each wave can be studied independently, by projecting perturbed quantities onto the frequency and the wavevector of each wave. In particular, combining the hydrodynamic (continuity and Euler) equations and the Poisson equation leads to the dispersion relation which reads, in the WKB limit:

for a spiral and:

for a warp (see Masset and Tagger 1995, Papaloizou and Lin 1995).

Here we expand the hydrodynamic equations to second order in the perturbed quantities. Then the waves do not evolve independently anymore and, provided that they satisfy selection rules which will be discussed below, they can exchange energy and angular momentum.

Thus the problem we address can be described as follows:

We consider a spiral wave, with a given flux (its energy density multiplied by its group velocity), traveling outward, from its corotation to the Holmberg radius of the galaxy. We will not discuss here how this spiral has been excited and we do not try to justify its amplitude, but rather take it as an observational fact.

As it travels radially the spiral interacts with warps which are always present, at all frequencies, at a noise level associated with supernovae explosions, remote tidal excitations, etc.

Our goal is the following: can we find conditions such that, although the warps are initially at this low noise level, they can be non-linearly coupled to the spiral so efficiently that they absorb a sizable fraction of its energy and momentum flux ? And can these conditions be met commonly enough to explain the frequent (one might even say general) occurrence of warps and their main observational properties ?

We will first discuss the conditions, known as "selection rules", for the warps to be coupled to the spiral.

2.2.2. Selection rules

The linearized set of equations governing the wave behavior (continuity, Euler and Poisson) is homogeneous and even in z, so that any field of perturbations to the equilibrium state of the disk can be considered as composed of two independent parts:

• Perturbations whose perturbed density is even in z, and thus, due to hydrodynamic equations, whose perturbed quantities are all even in z except W, which is odd. These perturbations are spiral waves, since they imply a perturbed density (a non-vanishing ), and a vanishing Z (so they do not raise the mid-plane of the disk).
• Perturbations whose perturbed density is odd in z, and thus whose perturbed quantities are all odd in z, except W, which is even. These perturbations are warps, since they imply a vanishing integrated perturbed density and a non-vanishing Z, which means that they involve a global motion of the mid-plane of the disk.

Furthermore, since the coefficients of the equations do not depend on time or azimuthal angle, Fourier analysis allows to separate solutions identified by their frequency and azimuthal wavenumber. On the other hand, since the coefficient do depend on the radius, Fourier analysis in r (or x in the shearing sheet) does not allow the definition of a radial wavenumber except in the WKB approximation. We will return to this below.

So let us consider a spiral wave for which we write:

and two warp waves with:

and

where represents any perturbed quantity. Hydrodynamic equations written to second order in perturbed quantities will contain terms involving the products , , , etc.

Let us consider for instance the term (rules for the other products are derived in a similar manner). Its behavior with time and azimuthal angle is:

More physically this product, which appears from such terms as or , can be interpreted as a beat wave. Fourier analysis in t and will give a contribution from this term, at frequency and wavenumber ; thus it can interact with the spiral wave if:

and

As discussed above, the equations for the spiral wave are derived from the part of the perturbed quantities which is even in z. Our third selection rule is thus that the product be even in z.

If we were in a radially homogeneous system, so that waves could also be separated by their radial wavenumber, a fourth selection rule would be:

Since this is not the case, we will write a coupling coefficient which depends on x. This coefficient would vanish by radial Fourier analysis if the waves had well-defined radial wavenumbers, unless these wavenumbers obeyed this fourth selection rule. We will rather find here that the coefficient varies rapidly with x and gives a coupling very localized in a narrow radial region. This is not a surprise since it also occurred in the interpretation by Tagger et al. (1987) and Sygnet et al. (1988), in terms of non-linear coupling between spiral waves, of the numerical results of Sellwood (1985). The localization of the coupling, as will be described later, is associated with the presence of the Lindblad resonances. We will find that, in practice, it results in an impulsive-like generation of the warp waves, in a sense that will be discussed in Sect. 4.2.2.

Let us mention here that we have undertaken Particle-Mesh numerical simulations in order to confirm this analysis. Preliminary results in 2D, with initial conditions similar to Sellwood (1985), do show the and spiral waves, at the exact frequencies and radial location predicted by Tagger et al. (1987) and Sygnet et al. (1988). The same absence of a selection rule for the radial wavenumber is observed. These results will be reported elsewhere, and in a second step the simulations will be applied in 3D to the physics described in this paper.

A radial selection rule would allow us to get a very simple expression for the efficiency of non-linear coupling - i.e. 0 for wave triplets that do not obey it, and 1 for triplets that do. Here we will have to rely on a more delicate integration of the coupling term over the radial extent where it acts. This will be done in Sect. 4.2. In particular in 4.2.2, we will show that in the vicinity of the OLR of the spiral, even though its WKB radial wavenumber is divergent, the main localization of the coupling comes from an (integrable) divergence of the coupling coefficient.

Let us summarize the selection rules:

• We first have the condition . This is obviously fulfiled by the most frequent warps (which have ) and spirals (with ). One should note here that, since we are computing with complex numbers but dealing with real quantities, each perturbation is associated with its complex conjugate, with wavenumber and frequency . Thus we find a contribution to the warp 1 by the coupling of the spiral with the complex conjugate of warp 2, i.e. from products of the form , etc.
• We then have the parity condition, which is obviously fulfiled by odd warps and an even spiral.
• Finally the frequency selection rule:

  

  gives us in principle an infinite choice of pairs of warp waves, since the latter can be presumed to form a continuous spectrum. One of our main tasks in the present work will be to determine which pair is preferentially coupled to the spiral. Our assumption of coupling between only three waves will be found valid when we find that actually one such pair is strongly favored, so that all the others can be neglected.

  The frequency condition can also be written as:

  

  showing that it is also true in the rotating frame at any radius, if it is true anywhere.

  A second remark concerning this selection rule is that can have an imaginary part. One is thus restricted to two possible choices : either working in a two-time-scales approximation, where the fast scale is that of the oscillations (), while the slow one is both that of linear growth or damping and of non-linear evolution; or simplify the problem by only looking for permanent regimes, keeping real. We will retain the second possibility, and in fact consider the non-linear evolution of the waves as a function of x as they travel radially, assuming that the inner part of the disk feeds the region we consider with spiral waves at a constant amplitude. This is sufficient for our main goal, which is to show that non-linear coupling is indeed possible and efficient to generate warps. On the other hand one should keep in mind that the permanent regime we will find may very well be unstable, since it is well known that mode coupling (in the classical case of a homogeneous system, much simpler than the one we consider) can lead to any type of complex time behavior, e.g. limit cycles or even strange attractors.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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