1.4. Transient growth and asymptotic stability

William Julian had, like Toomre, Kalnajs and Shu, been an undergraduate student at MIT. After receiving his bachelor's degree in mathematics in 1961, he continued on as a graduate student and soon took a course on galactic astronomy from Woltjer when he visited MIT. That roused Julian's interest in galaxy dynamics, and the time, personified by Lin and Toomre, magnified it. When the latter completed his axisymmetric-stability study of flat stellar galaxies (Toomre 1964a), he determined to encompass the asymmetric task, and this motif guided him and Julian into their work on "Non-axisymmetric responses of differentially rotating disks of stars" (Julian and Toomre 1966; hereinafter JT), which started in the spring of 1964. The news soon about parallel studies at the English Cambridge gave them still more incentive to struggle along, upon which Toomre promptly and in detail informed Lynden-Bell about the steps the MIT duet had done and planned to do. ¹⁷

"Since this May, a graduate student named W. Julian and I have been involved in much the same sort of an analysis as you describe in your Part II, but for the somewhat more complicated case of a thin sheet of stars with not insignificant random motions in the plane of the disk. [...] My interest in your problem dates back to the sheared non-axisymmetric disturbances for the case of negligible pressure, which were among the things I reported in the recent ApJ. Even at the time I did those, I realized that any inclusion of pressure forces to remove the shortest instabilities would leave a typical situation that was at first stable, when the disturbance was still tightly wrapped in the `unnatural' sense, then unstable for a while, and finally stable again. (In fact, if one were to choose the unwrapped wavelength long enough, and the pressure quite small, I felt one would even find two periods of temporary instability! Have you tried this admittedly unrealistic case on your computer?) ¹⁸ However, I felt then that the situation did not merit a detailed calculation, since it could not be terribly relevant to the spiral problem to discuss such disturbances to a supposedly uniform disk of gas in view of the observational evidence about the gross unevenness of the existing gas distributions in galaxies. [...]

Certainly, you arrive at a most worthwhile result in observing that under circumstances in which the axisymmetric instabilities (locally at least) would be avoided, there is still the distinct possibility of temporary non-axisymmetric instabilities, and that this could not help but provide a bias in any situation with a somewhat random excitation in favor of waves with the `natural' wrapped-up orientation. [...] Where I would at present reserve my judgment is in your conclusion that your result is directly pertinent to the spiral problem. Julian and I had our own burst of enthusiasm on this when we obtained our very similar results, but lately it has become a little more difficult for us to envisage the exact connections. But surely it cannot be altogether irrelevant!" (Toomre 1964b) ¹⁹

Using kinetic methods, Julian and Toomre described non-axisymmetric responses in a thin Cartesian model of a small stellar-disk region of a non-barred galaxy (JT model). In so doing, they actually managed to conquer a considerably more difficult technical problem via the collisionless Boltzmann equation than the one that GLB had needed to solve for their idealized gas. ²⁰ Help from the Volterra-type integral equation to which the authors had converted the problem enabled them to track the evolution of an impulsively applied disturbance and to see the shared waves damping. They damped as well at a finite-time imposition of disturbances, while asymptotically, as t . ²¹ "This means, JT concluded, that a collisionless star disk, if it strictly obeyed our model equations, could not even sustain self-consistent non-axisymmetric waves set up by previous gravitational disturbances, let alone admit modes that grow indefinitely" (JT, p.819). This plainly conflicted with the Lin-Shu self-sustained and tightly wrapped wave scenery, while still giving it formally, as the axisymmetric limit, a saving chance in an indefinitely slow damping. ²²

These results enabled Julian and Toomre to speak of the stability in the strict sense. Because the heat and shear parameters played no quantitative role (as well as k_y, they were only demanded not to be infinitesimally small), JT stated that the technically correct criterion for their model disk had to be the axisymmetric one, Q > 1. Yet "this curiously simple conclusion" is but an asymptotic result, they stressed: "as such, it does not preclude the amplification of disturbances during a conceivable intermediate time period" (JT, p.819). And JT did compute "a remarkable transient growth of these wavelets while swinging" (p.821), very similar to that revealed by GLB in gas models.

¹⁷ Lynden-Bell and Toomre already knew each other. They first met briefly in June 1962 at Woods Hole Oceanographic Institute. Toomre's cold axisymmetric modal calculations were being finished during his stay there, and he spoke of disk instabilities at a seminar with Lynden-Bell present. (The listener later recalled: "I fear that such are one's subjective impressions that my memory of your talk at Woods Hole is solely an irrelevance which I will not burden you with" (Lynden-Bell 1964b). "I think your sentence is Churchillian", then commented Toomre.) In June 1964 Mestel visited MIT, and he brought both Lin and Toomre preprints of two GLB papers from Lynden-Bell. "Figures 3-5 in their Paper (or really preprint) II resembled hugely what Bill Julian and I had managed both to discover and to plot all on our own just during the preceding 1-2 months - Toomre recalls. - We had at that point been doing our stellar dynamics only via truncated moment equations which were flawed in not including the strong (= vaguely Landau) damping toward short wavelengths that is very characteristic of the stellar rather than gaseous problem ... and it was for that slightly bogus reason that our results looked so similar." (Toomre) Back.

¹⁸ "We did not try any of the double growth period solutions (where oscillations take place in between growths) because unless the radial modes are unstable the double growth ones never get a decent acceleration." (Lynden-Bell 1964a) Back.

¹⁹ "I agree with almost all you say, Lynden-Bell responded, even to some extent the doubtfulness of whether the theory as outlined by us is really the mechanism." (Lynden-Bell 1964a) Back.

²⁰ "My main idea in spring 1964 had been to expand the perturbed phase density from the collisionless Boltzmann equation as a sum of products of Hermite polynomials in uand v multiplying the two-dimensional unperturbed Schwarzschild distribution. Closing them was not a big concern [... and] this was already some rather honest stellar dynamics. [...] But Bill and I were dismayed to learn during summer 1964 (or roughly a month or two after the GLB preprints had arrived) that such expansions looked as if they would need thousands (!) of terms to begin to capture reasonably accurately the later decay of vibrations due to what we realized eventually was just phase-mixing. It was this terrible inefficiency that prompted Bill to go looking extra hard at the alternative route of an integral equation. And it was definitely he who first realized that the ferocious kernel there could be integrated explicitly, an insight that suddenly made that route much more palatable than it had seemed at first" (Toomre).

"I guess Alar knew it also, but did not realize that the integrals could be worked out." (Julian) Back.

²¹ That non-axisymmetric waves damp is due to phase mixing of the perturbed star distribution function in the course of its averaging. It does not imply energy dissipation as long as the system obeys the isentropic collisionless Boltzmann equation. In the gradient-free JT model, the mixing effect comes from the shear that breaks down phase alignment of the stars on their epicycles, induced by previous disturbances. Waves of lengths _y > > 2r_e (r_e being the epicyclic radius) are almost uninfluenced, while those of _y 2 r_{e T} damp severely.

"Agris Kalnajs started hammering away on my dense skull from roughly spring or summer 1963 onward about the undamped axisymmetric vibrations even in the presence of ample (especially Q > 1) random epicyclic motions, - since they followed very naturally (as he well knew) from the kinds of plasma-like math. [...] I remained suspicious for a long time especially about his claims that there should be such undamped vibrations no matter how short one chose their wavelengths - I had somehow become over-convinced that strong phase mixing or loosely speaking `Landau damping' of any short waves in stellar sheets had to be the rule and could not be avoided! Of course on the latter point I was wrong, as even Julian and I had convinced ourselves [...], ...but intuition is a funny thing, and sometimes when wrong it takes a long time to get repaired." (Toomre) Back.

²² "Let me state again, a little more explicitly, why all that largely Russian `grumbling' about folks in this business having been genuinely `ignorant of plasma parallels' is beginning to get under my skin. In its linearized form, the collisionless Boltzmann equation is nothing more than a 1st-order quasi-linear PDE which almost any competent applied mathematician would (or should) recognize is solvable via very standard characteristic curves such as I did in my paper (Toomre 1964a). So I honestly still don't think that was any big deal, or something that C.C. or anyone else could not quickly rederive on their own. One most definitely did not need to go running to plasma physicists to see how they had handled something so `obvious'. [...But] those characteristics surfaced again in JT, and there in a situation with a shear flow which even the kind plasma physicists had probably not met! I also assert that the Volterra integral equation (21) from JT - with its kernel figured out as compactly and explicitly as it appears in eqn (23) thanks to my clever student Julian - is distinctly more remarkable than anything Lin & Shu 1966 managed to do on their own, esp. since that formalism not only contains `their' dispersion relation as a limiting case that we there only hinted at, but also because we unlike they went on to show right in JT that the shearing sheet `could not even sustain self-consistent non-axisymmetric waves', plainly contrary to what Lin & Shu 1966 would have implied for this same situation." (Toomre) Back.