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

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Appendix A: formal derivation of the coupling coefficient

A.1. Derivation without the shear terms

We use the hydrodynamical equations (continuity and Euler) and the Poisson equation.We write the first two equations in tensor formalism, in order to avoid too lengthy expressions. The continuity equation is:

[EQUATION]

where the vector [FORMULA] ([FORMULA]) is a tensor which represents [FORMULA], and where we implicitly sum over repeated indices (Einstein's convention).

The Euler equation is:

[EQUATION]

In this expression the last term of the L.H.S. represents the Coriolis acceleration, and [FORMULA] the Levi-Civita symbol. We also introduce a convenient notation, [FORMULA]. The reader can note that, in the convective derivative [FORMULA], we have not written the shear terms. We neglect them in this first approach, and we will check their influence at the end of this section.

In the Euler equation the pressure term can be expanded to second order as:

[EQUATION]

Finally, the Poisson equation is:

[EQUATION]

We want to know the energy injection rate into warp 1 from the spiral and the warp 2. In the linear regime, we would have: [FORMULA]. Here, we will have a non-vanishing expression due to the presence of the spiral and warp 2.

Since we treat the warp 1, we expand the hydrodynamic equations to second order in the perturbed quantities and project them onto [FORMULA].

For the continuity equation (A1) this gives:

[EQUATION]

and for the Euler equation (A2):

[EQUATION]

[EQUATION]

Our purpose is to write the temporal derivative of an expression which can be interpreted as the energy density of the warp (energy density per unit surface). For this we multiply equation (A5) by [FORMULA] and equation (A4) by [FORMULA], we integrate over z, add them and their complex conjugates. This gives:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

In this tensorial notation we will freely use the rule of integration by part (i.e. transpose derivative operators with a change of sign) in the linear terms of this equation: indeed if the index i is z, this is really an integration by parts; on the other hand if [FORMULA] or y the derivative is just a multiplication by [FORMULA] (in our WKB analysis), so that e.g. :

[EQUATION]

Furthermore, we must remember that the integration is only over z, so that we do not need to worry about boundary terms the integration by parts would introduce in an integration over x, and which would represent energy and momentum flux at the radial boundaries of the integration range.

However, we cannot blindly apply this simple formal integration by parts on the non-linear terms. A careful look at the details of the integration by parts for each value of the index i is necessary. For the dimensions ([FORMULA] and [FORMULA]) the formal integration by parts would give correct results, even on the non-linear terms. But for [FORMULA], an integration by part would imply a selection rule of the type [FORMULA], and we have intentionally avoided such an hypothesis. Since we want to stay, for simplicity, in the framework of tensorial formalism, we have to perform transformations that are correct whatever the index i and we do not integrate by parts the non-linear terms in this first part of the derivation.

Eq. (A6) already has the form we want : the first term in its L.H.S. is the temporal derivative of the kinetic and internal energy of warp 1. We can note that the term linked to Coriolis acceleration has vanished, since the Coriolis force is always normal to the velocity and thus does not work.

The second term in the L.H.S. must be rewritten to be interpreted as the temporal derivative of the potential energy. We integrate it by parts and then use equation (A4) to transform the result. We can also note that the first two terms of the R.H.S vanish together (integrating one of them by parts). We obtain:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The second term in the L.H.S still does not appear as the time derivative of a potential energy. We have to use the Poisson equation, which we have not used yet.

If we write [FORMULA] for the potential created at point [FORMULA] by a unit mass located at point [FORMULA], then:

[EQUATION]

where [FORMULA] represents the convolution operator.

Thus, [FORMULA], [FORMULA], etc....

We write [FORMULA] the form:

[EQUATION]

where the index E indicates that the integral extends over the whole space. Using the fact that S is a real function with spherical symmetry, it is an easy matter to check that (by writing explicitly the convolution and exchanging the integrals):

[EQUATION]

Hence [FORMULA] is a variational form, and one can write the following equalities:

[EQUATION]

(whether the integral extends only over z or over the whole space, since the integrated quantity does not depend on x or y).

Now:

[EQUATION]

since S is time independent. This is linked to the use of the Poisson equation, i.e. to the fact that the potential propagates instantaneously. Then one can write:

[EQUATION]

Thus we see that we can write this term as a potential energy associated to the warp, and this gives:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Hence we have expressed the temporal derivative of the energy density (kinetic plus internal plus gravitational) [FORMULA] of warp 1 as a sum of integrals which involve the amplitude of each warp and of the spiral. In particular, one may note that in the linear case the energy of the warp is conserved. One important remark concerning the linear case is that, if we neglect the coupling term, we should not have [FORMULA] but [FORMULA] since energy is transported at the group velocity [FORMULA]. Here the convective term has disappeared in the integration by parts of one of the terms in the RHS of equation (A6): for simplicity we have neglected the integrated terms, assuming periodicity in x. From a more complete derivation we would recover a linear term [FORMULA], without affecting the non-linear ones. We will thus re-write (A8) as:

[EQUATION]

A.2. Effect of the shear term

As emphasized above, we did not take into account the shear term when writing down the Euler equation. Writing this term can a priori modify the warp energy and the coupling term.

Formally, in tensorial notation, the shear can be written as:

[EQUATION]

where the matrix A is, in the frame xyz:

[EQUATION]

With these notations, the convective term of the Euler equation reads:

[EQUATION]

which can be rewritten as:

[EQUATION]

The first of these four terms, proportional to [FORMULA], is null (this is generally the case when the velocity is constant along a current line).

The last term is the one we had without shear. The influence of this term on the coupling coefficient has already been derived in the previous section.

The intermediate terms are linear, and thus have no incidence on the coupling term.

Hence, the temporal derivative of the energy density of the warp is modified by the addition of the two corresponding terms, obtained by multiplying by [FORMULA] and adding the complex conjugates. This gives:

[EQUATION]

where [FORMULA] stands for the real part. For the waves under consideration both of these terms are null:

  • The first one because, in the WKB approximation, the perturbed horizontal velocities U and V are in quadrature (epicyclic motion). Thus this term, given the expression of A, is proportional to [FORMULA], and therefore purely imaginary.
  • The second one because it is related to the azimuthal derivative of terms related to the warp energy density, and thus vanishes.

Hence the addition of shear does not modify the coupling equation.

Appendix B: analytical computation of the coupling term

B.1. Approximation of the eigenfunctions

In the framework of our expansion to second order in the perturbed quantities (assuming weak non-linearities), the coupling integrals derived in the preceding appendix involve the linear eigenfunctions associated with the spiral and the warp. These eigenfunctions consist in the amplitude and phase relation between the perturbed quantities (velocities, density, potential...), and their spatial variations. An exact, analytical or numerical, knowledge of these eigenfunctions is beyond the scope of this work, and in fact would add little to it since we are not interested here in deriving detailed numbers appropriate to a given galaxy model but rather in the physics of non-linear mode coupling. We will thus use approximate expressions, which allow an easier access to this physics. Furthermore, our results are given in terms of variational forms. These are well known to preserve the important invariants in the problem (the energy and action densities), and to be good estimates even with poor approximations to the eigenvectors.

Thus we remain in the WKB formalism, and make the following assumptions:

  • We assume that the motion in the spiral is purely horizontal and independent of z.
  • We assume that the vertical velocity in warps is independent of z, and that the perturbed density is:

    [EQUATION]

From these assumptions we can deduce the perturbed quantities relative to a warp. We focus hereafter on the warp 2, but the results would evidently apply also to warp 1.

One gets [FORMULA] et [FORMULA], the horizontal components of the perturbed velocity, from the relation:

[EQUATION]

Using the WKB hypothesis ([FORMULA]), one obtains:

[EQUATION]

[EQUATION]

where [FORMULA] is known from [FORMULA].

It is noteworthy that these quantities are only a rough approximation to the eigenfunction of a warp; in particular they do not fulfil the continuity equation. On the other hand our reason for taking them into account is that near the Lindblad resonances, where horizontal motions are large, we suspect that they might strongly contribute to the coupling terms. We will find later that this is not the case, so that forgetting these terms altogether would not change the result to leading order.

The perturbed potential of the warp is given by the expansion to lowest order in qH of the expression given by the Green functions (see Masset and Tagger 1995):

One obtains:

[EQUATION]

and then the other perturbed quantities associated with the warp can be expressed, as functions of [FORMULA] exclusively:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

In the same manner, with the hypotheses mentioned above, we obtain for the spiral:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Rewriting the coupling term obtained in the previous section, and expanding the sums over repeated indices, we get:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The final step consists in replacing the quantities involved in these integrals by the approximate perturbed quantities derived above.

B.2. Computation of the coupling integrals

In this appendix we compute the various terms in the RHS of the coupling equation (B1).

We will not write down all the computations, which are lengthy and tedious. Let us just notice that the spiral eigenfunctions do not depend on z, so that each integral over z involves in fact only two eigenfunctions of warps 1 and 2.

We will make intensive use of integrals of the type:

[EQUATION]

and

[EQUATION]

All the other integrals are either straightforward or can be deduced from [FORMULA] or [FORMULA].

For the evaluation of [FORMULA], we use the hydrostatic equilibrium in the unperturbed state:

[EQUATION]

Hence

[EQUATION]

Integrating by parts, one obtains:

[EQUATION]

Now

[EQUATION]

where [FORMULA], the radial laplacian of [FORMULA] is equal to [FORMULA] (Hunter and Toomre, 1969).

Using the Poisson equation, [FORMULA], one obtains:

[EQUATION]

where [FORMULA] is defined as:

[EQUATION]

Using an integration by parts, one can easily derive:

[EQUATION]

The derivation needs a frequent use of the following integral:

[EQUATION]

Expanding the square in the integral:

[EQUATION]

One can deduce:

[EQUATION]

In order to express the coupling term , we group the terms as follow:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

and we find for these terms the expressions:

[EQUATION]

We note that the coefficients in eq. (B2) are all imaginary, and that all the expressions above give a real factor times [FORMULA]. Thus if the relative phases of the three waves were such that this product is real the coupling term (once added to its complex conjugate) would exactly vanish, while it would be maximized by waves in quadrature. This variation with the phases is associated with the complex non-linear behaviors mentioned earlier, and we will not discuss it here (technically the phases can evolve non-linearly on the "slow" time scale of the mode growth and non-linear evolution, as compared with the "fast" time scale of the linear frequency). Hereafter we will for simplicity assume that the non-linear process has picked up waves from the background noise, or made them evolve, such that their phases maximize the coupling, i.e. [FORMULA] imaginary. Thus we will from now only consider real quantities in the coupling equation.

Factorizing [FORMULA], we obtain the following coupling equation:

[EQUATION]

[EQUATION]

[EQUATION]

where we have introduced:

[EQUATION]

We clearly see that we are concerned with inhomogeneous coupling since the coefficient depends on x, which implicitly comes from the [FORMULA]. In particular, the dimensionless coefficients [FORMULA] and [FORMULA] are of the order of unity, except near the Lindblad resonances of the corresponding warps where they can take very large values.

Appendix C: simplification of the coupling term

In this appendix we simplify the expressions obtained in appendix B to get an estimate of the coupling coefficient, i.e. of the efficiency of non-linear coupling. First we estimate the constants [FORMULA], [FORMULA], and [FORMULA] which appear in equation (B3). [FORMULA] can be written as [FORMULA] (see Appendix B). From the hydrostatic equilibrium of the unperturbed state we find:

[EQUATION]

This gives from the definition of [FORMULA]:

[EQUATION]

and thus:

[EQUATION]

Now we have to compute [FORMULA]. One can say that [FORMULA] is of the order of magnitude of [FORMULA], which is the density in the mid-plane of the disk. Now [FORMULA] is the frequency of vertical oscillations of a test particle in the rest potential of the galactic disk. We note this frequency [FORMULA], where the z index reminds that this frequency depends on the vertical excursion of the test particle. Hence:

[EQUATION]

Finally we consider [FORMULA]. [FORMULA] is of the order of 1, except in the vicinity of the Lindblad resonances, where it become large. This is essentially the same effect that was found in previous works (Tagger et al. , 1987; Sygnet et al. , 1988) to make non-linear coupling very efficient near the Lindblad resonances of the coupled waves (although in that case the coupling coefficient was obtained in a kinetic rather than fluid description, i.e. adapted to the stellar rather than gaseous component of the disk). On the other hand, our assumption of weak non-linearities certainly breaks down if [FORMULA] becomes too large, i.e. exactly at the Lindblad resonances. A convenient estimate is that this break-down occurs at one sound-crossing time [FORMULA] (or equivalently one epicyclic radius) from the resonances, giving an upper limit for [FORMULA]:

[EQUATION]

Now we can estimate the different terms of equation B3. The first term can be estimated as follows:

  • The term in inner brackets is of the order of 2. This estimate is of course expected to vary of a factor 2 or 3, but no more. In particular, it is irrelevant to speculate on a possible resonant effect due to a vanishing value of [FORMULA]: as we will find below, the spiral is tightly wound in the region of strong coupling.
  • It is an easy matter to see that [FORMULA] is always about 0.5, even in the vicinity of the Lindblad resonances, since [FORMULA].
  • Finally, [FORMULA] is of the order of [FORMULA], as seen above.

One can then deduce the order of magnitude of the first term [FORMULA]:

[EQUATION]

Let us evaluate now the second term [FORMULA]:

  • First, we compare [FORMULA] and [FORMULA]. We have:

    [EQUATION]

    Now [FORMULA] is larger than µ in the region of the galaxy where the rotation curve is nearly flat, hence [FORMULA].

  • On the other hand, since [FORMULA], we have [FORMULA].
  • Finally, we assume that [FORMULA] (this is a good approximation in the region of strong coupling; we will emphasize it again when we study the localization of the coupling).

We deduce that:

[EQUATION]

Let us note that [FORMULA], from eqs. (C1) and (C2), is of the order of:

[EQUATION]

where each factor is smaller or much smaller than unity. Hence the first term is always negligible compared to the second one.

Let us now find an estimate for [FORMULA]. We will derive it in the vicinity of a Lindblad resonance, maximizing [FORMULA]. We note that:

[EQUATION]

so that [FORMULA].

We estimate:

[EQUATION]

Hence [FORMULA] is smaller or comparable with [FORMULA] in the vicinity of a Lindblad resonance. Away from the resonances, the ratio [FORMULA] is still lower. Thus we deduce, since [FORMULA] is always negligible compared to [FORMULA], that the coupling term is always of the order of [FORMULA]:

[EQUATION]

Appendix D: expression of energies

The coupling equation written in the previous appendix involves both the energy and the amplitude of warp 1. They are actually linked by an expression we wish to derive.

The energy [FORMULA] reads (cf. eq. (1)):

[EQUATION]

After a straightforward calculation, in particular making use of the integrals of Appendix B, one obtains:

[EQUATION]

Far from the Lindblad resonances, it is easy to see that the first term of this expression of energy is negligible compared to the third, and a fortiori to the second. However, at the Lindblad resonances of the warp, the first term can become important and dominate the others. The physical interpretation is that near the Lindblad resonance the kinetic energy associated with horizontal motions, due to the compressibility of the gas, becomes dominant.

Hence, at the Lindblad resonance, on can have energy "hidden" in the horizontal motions associated with the warp, i.e. a large energy with a small vertical displacement. This is illustrated in Figs. 9 and 10.

[FIGURE] Fig. 9. This figure shows the velocity field of a warp away from the Lindblad resonance.

[FIGURE] Fig. 10. This figure shows the velocity field, which becomes nearly horizontal, of a warp near a Lindblad resonance.

On the other hand, for a spiral, the vertical motion never dominates even when compressibility becomes important. This can be directly seen from the expression of the energy of the spiral:

[EQUATION]

which does not show any resonant term. Thus the perturbed surface density [FORMULA] represents fairly well the energy of the spiral wave. There are no hidden motions (hidden in the sense that they don't have incidence on the observable) similar to the hidden horizontal motions in warps. Thus we have:

[EQUATION]

and

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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