4. The coupling efficiency
We have to solve, or at least analyze, the behavior of a system of coupled differential equations which is non-linear (since the right-hand-sides are products of the unknowns). Such a system can have subtle and varied solutions, such as limit cycles or strange attractors (chaotic behavior). This goes far beyond the scope of this paper and we will restrict ourselves to looking for stationary solutions, which might prove to be unstable but still will inform us on the efficiency of the coupling mechanism. Thus we suppress the partial t derivatives in the total derivative , making solution much simpler.
4.1. Search for a stationary solution
We can write the system of three coupled equations in a slightly different manner, by introducing the energy fluxes () of the waves, and by relating density energies and amplitudes through constants derived in Appendix D:
(see for the derivation of ); the energy densities are proportional to the square of the amplitudes, while of the order of magnitude of the sound speed, as expected for an acoustic wave, and and are of the order of the "spring constant" ).
With these new notations we get:
In order to be able to estimate the behavior of the solutions of system (I), we must analyze the geometry and localization of the coupling.
4.2. Localization of the coupling
To this point we have considered the coupling between three waves: the spiral wave, and two low amplitude warp waves which satisfy the selection rules. But any pair of warps obeying the frequency selection rule will also obey the other ones, leaving us potentially with a continuous set of warp pairs coupled to the spiral. The question is now to determine whether one or several couples of warps can be preferentially amplified by the spiral wave, and if so, which one(s) and where. In order to clarify the situation, we present the problem graphically in Fig. 3. In this figure, each wave is presented by a curve (a segment in the shearing sheet approximation, but this does not lead to a loss of generality). For each curve, x is given by the implicit relation , where we have taken and . It is an easy matter to check that the slope of the curves describing the warps, at a given x, must be twice the slope of the curve describing the spiral. For the reason already explained in Sect. 2.2.2, if two warps satisfy the -selection rule at some x, they satisfy it at any x. Since x has its origin at the corotation of the spiral, the curve of the spiral starts from (0,0). The warps cannot propagate in the vertical forbidden band , since their dispersion relation is (in the infinitely thin disk limit) (taking into account the finite thickness of the disk and possible compressional effects maintains the existence of such a forbidden band, see e.g. Masset and Tagger 1995). For a given warp, the forbidden frequencies can be converted into a forbidden band in x as shown in Fig. for warp 2.
We have depicted a narrow forbidden band since µ is expected to be small compared to if the rotation curve is nearly flat. In the ideal case where the rotation curve is really flat, giving .
According to the expression of system (I), we see that the coupling can be strong where one or several group velocities vanish. Physically, this corresponds to the fact that the waves take a long time to propagate away from the region where coupling occurs, leaving them ample time to efficiently exchange energy and momentum.
As already mentioned, we shall now use the group velocity of a stellar spiral for the spiral wave, although we keep our purely gaseous coupling coefficient. By doing so, we avoid the tedious handling of a mixed kinetic-hydrodynamic formalism without a great loss of precision. The group velocity of the spiral vanishes near the Lindblad resonances (here at the Outer Lindblad Resonance, OLR) and the group velocity of the warp vanishes at the edge of the forbidden band, where an incident warp must be reflected 1.
For these reasons we have four qualitatively distinct cases to consider:
Far from the OLR of the spiral:
At the OLR of the spiral:
4.2.1. Far from the OLR
Let us analyze the first case. The graph is presented in Fig. 4. We choose a value of x such that neither warp is close to its forbidden region. We define by . Summing the two first lines of system (I), and after some transformations, we obtain:
Our purpose is to determine whether the coupling will be sufficient in these conditions to extract the warps from the noise level. As long as the warps remain at this low amplitude, their influence on the flux of the spiral is negligible. We then assume that is constant. This naturally leads us to define an e -folding length for the warps. If this length happens to be short (in a sense which will be defined below), the warps can be strongly amplified from the noise level. Here, the e -folding length is:
Taking into account the expression of given in Appendix D, and noting that the group velocity of warp i can be written as , we can rewrite this as:
where we have taken the typical value (see the diagram) .
Now let us follow these two warps in their motion outward. After traveling over a distance , their quadratic total flux has been amplified by a factor . Will amplification at this rate be sufficient for the warps to extract a significant fraction of the flux of the spiral ? For this to occur, the spiral and the beat-wave of the warps should be able to maintain a well-defined relative phase as they travel radially, lest the coupling term oscillates and gives alternatively a positive and negative energy flux from the spiral to the warps. We meet here a condition which is the WKB equivalent of the selection rule on the radial wavenumbers: since the WKB wavenumbers of the waves do not a priori obey this selection rule, they will decorrelate over a few radial wavelengths, , where is the radial wavenumber of any of the waves involved. Thus if is small the warps can be strongly amplified before they decorrelate from the spiral, whereas if is of the order of 1 or larger the coupling will not significantly affect them. Here, we have:
Then the waves are in an adequate phase condition over one e -folding length at best, and we conclude that the warps cannot be extracted from the noise level in this first case.
The second case, illustrated in Fig. 5, still corresponds to coupling far from the OLR of the spiral, but with one warp near the forbidden band. It is an easy matter to check from the linear dispersion relation that the group velocity of a warp near its forbidden band is given by:
where is the distance to the forbidden band. This formula has been obtained in the WKB limit, but we do not expect very different results in the general case. This leads us to replace in the expression of a term of the type: by an integral of the type:
We have adopted the length scale H because it corresponds to the range over which equation (5) is valid, close to the forbidden band.
The ratio of these two quantities is:
and is typically of the order of 1 (the exponent does not allow it to change very much). Furthermore, the coefficient is now about , i.e. about . This is not sufficient to make the value of much smaller than 1.
Thus we can conclude that far from its OLR the spiral cannot transfer its energy and angular momentum to the warps.
4.2.2. Near the OLR
We turn now the last two cases, corresponding to coupling close to the Outer Lindblad Resonance of the spiral. What does close mean here ? We said that we must study separately the behavior of coupling near the OLR because the group velocity of the spiral in a stellar disk tends to zero as its approaches the resonance. In fact it does not really vanish (Mark, 1974), but we can in a first approximation consider that it does, with a decay length scale of the order of the disk thickness. Since this behavior is associated with the resonant absorption of the spiral wave at its OLR, by beating with the epicyclic motion of the stars, we expect that it should survive even in a realistic mixture of stars and gas. From the dispersion relation for a stellar spiral given by Toomre (1969), and using the asymptotic expression of Bessel functions, one can write the group velocity of the spiral near its OLR as:
So that now the group velocity is a power of the distance to OLR.
We can slightly transform our expression of , the e -folding length of the preceding section, as:
This expression is more appropriate in this new situation, since (at least when the fluxes of the warps remain small compared to the flux of the spiral) is constant (while is not) and now varies. As before, we have to see how many e -folding lengths are contained in a scale length of the order of , where we can use the above expression for the spiral group velocity. Then we have to replace in our previous estimate by , leading us to multiply the number of e -folding lengths over a scale H by:
Furthermore, we had taken in the previous section the mean value . Now since we are close to the OLR we have to take . Finally, with our typical figures, instead of having only one e -folding length for over a scale length H, we find eight such e -folding lengths, i.e. near the OLR the flux of the warps can be multiplied by . Of course this flux cannot become arbitrarily large, so that when it reaches the spiral flux we can consider that most of the spiral flux has been transferred to the warps. Even though our factor of could be reduced by many effects, we consider that mode coupling near the OLR is efficient enough to allow the warps to grow from the noise level until they have absorbed a sizable fraction of the flux of energy and angular momentum carried by the spiral from the inner regions of the galactic disk.
We now have to check in more details the effect of the localization of the coupling coefficient. Indeed from the system (I) we see that, if the amplitudes of the waves varied more rapidly than the coupling coefficient, their oscillations would reduce or even cancel over a given radial interval the efficiency of the coupling. This must be considered because the same effect which gives a vanishing group velocity for the spiral near its OLR also causes their radial wavenumber to diverge.
However, from the expression above for the group velocity of the spiral, we see that:
Now the phase of the spiral, in the WKB approximation, varies as
and the integral converges, so that in fact the phase tends to a finite limit (instead of varying rapidly as one might think from the divergence of ). Thus in the expressions on the RHS of system (I), the quantity which gives the dominant behaviour is indeed the (integrable) divergence of the coupling coefficient, in the vicinity of the OLR. This justifies that, in our estimates, we have integrated over the variations of this coefficient, keeping the other quantities approximately constant. It also justifies our statement, in Sect. 2.2, that the generation of the warps is impulsive-like.
This does not tell us yet which couple of warps is preferentially amplified at OLR, according to their frequencies. In fact the coupling coefficient still depends on these frequencies, through the term in . This factor, associated with the group velocities of the warps, further increases the efficiency for the most "external" pair: (by a factor compared to the most central pair of warps if we assume ). This difference is amplified geometrically at each e -folding length, and thus finally we can consider that the pair is preferentially emitted at the OLR. This result is illustrated in Fig. 6.
Now we have determined how and where the spiral is "converted" into warps. But we still have to find how the energy is distributed between the warps, and in which direction they propagate. In Fig. 6 we have illustrated the warp 2 with an outward-pointing arrow, since we know that it is necessarily emitted outward due to the presence of its forbidden band. But warp 1 can consist of two waves, propagating inward as well as outward, and we must now consider the detailed balance of the energy transferred to the three warps. This can be done using the conservation of angular momentum. The relation between the energy flux H and angular momentum flux J is, for the warps as well as the spiral (Collett and Lynden-Bell 1987):
since angular momentum and energy are advected at the same velocity . Angular momentum conservation gives:
where is the momentum flux lost by the spiral, while are the fluxes gained by the warps, and the et upper-scripts denote respectively the warps 1 emitted outward and inward. Writing , and using the the fact that the waves 1 have the same frequency, we get , giving from energy and momentum conservation the system of equations:
Defining and as the ratios and , we find .
Using the frequencies of the preferred pair of warps leads to:
A strictly flat rotation curve would give: and , while a nearly flat rotation curve with , would give: and .
Now it is an easy matter to check that the fluxes carried away by warps and are equal. The easiest way to do it is to write for the pair of warps 1 a system similar to the one used above. This system is degenerate, but introducing a slight offset between the frequencies and making it tend to zero we find that they share equally the flux . Physically, this corresponds to the fact that the source of the waves is "impulsional" (very localized), so that they are emitted equally without a preferred direction in space.
From the numerical values given above, we can deduce that each of the three waves (two emitted, one reflected) carries away one third of the flux extracted from the spiral, as a very good approximation.
4.3. Detailed physics at the OLR
The situation exposed above is idealized. Two remarks modify the simple computations, but should not qualitatively alter the main physical conclusions.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998