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

3. The coupling coefficient

Once we have the set of selection rules, we can derive an expression of the coupling coefficient between the spiral and the two warps. In a linear analysis, each wave propagates independently and is thus subject to a conservation law of the form:

where stands for the partial derivative with respect to any quantity u, E is the energy density of the wave, its group velocity. This relation simply means that globally the energy of the wave is conserved and advected at the group velocity (Mark, 1974). The r and factors in the spatial derivative come from the cylindrical geometry of the problem. In the framework of the shearing sheet we neglect them and we may write:

The vanishing right-hand side results from the fact that the wave does not exchange energy with other waves or with the particles, and thus is neither amplified nor absorbed. Mode coupling introduces in the right-hand side a new term describing the energy exchange between the waves. Since E is quadratic in the perturbed amplitudes, and mode coupling corresponds to going one order further in an expansion in the perturbed amplitudes, this new source term will be of third order. Its derivation is lengthy and technical, and we give it in separate appendices. We summarize it in the next sections.

3.1. First step: exact expression of the coupling

The first step consists in deriving an exact expression for this coupling coefficient. To do this we integrate the hydrodynamic equations over z after transforming their linearized parts to write them as variational forms. After a few transformations, we find that the total time derivative of a variational form, which involves the perturbed quantities associated with each wave (and which we will interpret as its energy density), is equal to a sum of terms involving the three waves, which we will interpret as the coupling part, i.e. the energy exchanged between the waves. This derivation is done in Appendix A for the case of one of the warp waves. Let us just mention that this first step has the following features:

• The only approximations we make are that the disk is isothermal with a uniform temperature, and that the horizontal velocities in each wave, U and V, are in quadrature, a reasonable assumption which is asymptotically true in the limit of the WKB regime.
• We fully resolve the thickness of the disk. Thus, the coupling coefficient involves integrals over z of products of perturbed quantities related to each wave.
• For the sake of compactness, the derivation is made in tensor formalism. This allows us to have a restricted number a terms (in fact eight terms in the coupling coefficient), although at this step of the derivation we do not make any hypothesis on the perturbed motions associated with a spiral or a warp.

Finally, at the end of this primary step, we obtain for the time evolution of the energy density of warp 1:

where means . The L.H.S. appears as the total derivative of the energy density of warp 1, as expected, and the R.H.S. represents the coupling term, since each of the integrals involve the perturbed quantities of the other waves (spiral and warp 2).

3.2. Second step: analytic result in the WKB approximation

Eq. (1) is too complex to be used directly. We show in Appendix B how it can be simplified by expanding the implicit sums, and explicitly writing the perturbed quantities associated with the warps or the spiral, and then by using in the evaluation of this term the eigenvectors (i.e. the various components of the perturbation) derived from the linear analysis. We make some assumptions:

• We assume that the spiral involves no vertical motions, i.e. , and that horizontal motions do not depend on z, i.e. . These two assumptions lead to an eigenvector which is the one of the infinitely thin disk approximation, although we vertically resolve the disk.
• Symmetrically, we assume that the vertical speed in a warp is independent of z, i.e. : , so that the perturbed density is obtained by a simple vertical translation of the equilibrium density profile. This is consistent only if there is no horizontal motions. On the other hand, the compressibility of the gas disk does introduce horizontal motions in the warp wave (Masset and Tagger, 1995); thus, in order not to loose a possible coupling through these horizontal motions, we retain in the expression of the coupling coefficient the horizontal velocities due to the warps. This may be important near the Lindblad resonances of the warp, where horizontal motions can become dominant.

At the end of this second step, we get a new expression for (where is the energy density of warp 1) where we have performed the integrals over z, and which can be written:

where is a long expression which does not need to be reproduced here, and which depends on x. Eq. (2) shows that warp 1 is coupled to the product of the amplitude of both warps ( and ) and to the amplitude of the spiral (). In particular, we see that if the warp does not exist at all in the beginning (i.e. it has a vanishing amplitude), it will never grow. Thus it has to pre-exist at some noise level in order to be allowed to couple with another warp and the spiral. We see here an important difference with the harmonic generation by a single wave, since in that case the harmonic will be generated even it it does not pre-exist. Here we are rather concerned with the production of warps as "sub-harmonics" of the spiral. They cannot be spontaneously created, as this would correspond to a breaking of azimuthal symmetry.

Mathematically, we see that the L.H.S. of the expression above is second order with respect to the perturbations amplitudes, and that the R.H.S. is third order. In fact, the L.H.S. comes from the linear part of hydrodynamic equations, and has been transformed into a second order expression (a variational form), and the R.H.S., which comes from the non-linear part of the equations, has simultaneously been converted into a third order expression.

In the same manner we can derive, by swapping indices 1 and 2, an expression for the time evolution of the energy density of warp 2:

In order to close our set of equations, we also have to follow the behavior of the spiral. In the absence of any coupling to other waves, for reasons of global conservation of energy, the time evolution equation of the spiral is given by:

We choose to neglect coupling to other waves here since they either belong to the dynamics of the spiral itself (e.g. generation of harmonics), and are irrelevant here since we take the spiral as an observational fact, or involve other warps: but, as mentioned above, we will find below that one pair of warps is preferentially driven by the spiral.

3.3. Final step: the simplification

The third and final step is to give an estimate of the coupling coefficients . This is done in Appendix C. Simplifying this coefficient implies much discussion on the physics involved, e.g. the behavior of resonant terms near the Lindblad resonances, or the order of magnitude of ratios of characteristic frequencies for a realistic galactic disk, etc. These discussions are fully developed in Appendix C, and lead us to the result that the coupling coefficient, anywhere in the disk, is always of the order of , i.e. :

where represents the frequency of vertical oscillations of a test particle in the rest potential of the disk (it must not be confused with µ, which is the frequency of global oscillations of the galactic plane, achieved when the whole disk - stars and gas- is coherently moved up and down). We see that this coefficient does not contain any dependency on 1 and 2, so that .

An important difference occurs here with the case of coupling between spirals or bars, analyzed by Tagger et al. (1987) and Sygnet et al. (1988). In that case the coupling coefficients, obtained from kinetic theory, were found to involve two resonant denominators, corresponding to the resonances of two of the waves; this made the coupling very efficient if the two denominators vanished at the same radius. We do not find such denominators here, because the warp vertical velocity (represented by Z in equation 3) does not diverge at the Lindblad resonance, while the horizontal velocities associated with the spiral do. On the other hand we will recover here a similar property of strong coupling close to the resonances, because the group velocity of the waves becomes small (it goes to zero in the asymptotic limit), so that the waves can non-linearly interact for a long time. This will be discussed in more details below.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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