## 4. The coupling efficiencyWe 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 ## 4.1. Search for a stationary solutionWe 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 couplingTo 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,
We have depicted a narrow forbidden band since 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
For these reasons we have four qualitatively distinct cases to consider:
- We consider two warps, none of which close to its forbidden band;
- Or a pair of warps, one of which lies at the edge of its forbidden band.
- We consider two warps, none of which close to its forbidden band;
- Or a pair of warps, one of which lies at the edge of its forbidden band.
## 4.2.1. Far from the OLRLet us analyze the first case. The graph is presented in Fig. 4. We
choose a value of
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 Taking into account the expression of given
in Appendix D, and noting that the group velocity of warp 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 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 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 OLRWe turn now the last two cases, corresponding to coupling close to
the Outer Lindblad Resonance of the spiral. What does So that now the group velocity is a power of the distance to OLR. We can slightly transform our expression of
, the 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 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 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: so 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
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 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 OLRThe situation exposed above is idealized. Two remarks modify the simple computations, but should not qualitatively alter the main physical conclusions. - First, the group velocity of the spiral does not really vanish
at the OLR. This has been investigated by Mark (1974), who showed that
the group velocity of the spiral at the OLR was about a few kilometers
per second (typically 10 times lower than the value it has far from
the OLR). This leads us to replace the factor 4 in our computation by
about . Nevertheless, we still have a "large"
number of
*e*-folding lengths in the region "just before" the OLR. - The spiral, as it approaches its OLR, is linearly damped through
Landau effect. Mark (1974) gives an absorption length by Landau
damping which has typically the same order of magnitude as the
*e*-folding length we find for non-linear coupling. The two processes can thus compete, the spiral losing part of it energy and momentum flux to the stars (heating them and forming an outer ring), and to non-linearly generated warps, at comparable rates. It is thus very likely that the dominant process (non-linear coupling or Landau damping) will depend on the local characteristics of the galaxy under consideration.
© European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |