3.1 Discrete Modes

Imagine one has found the normal modes, of which there should be 2N, where N is the number of rings employed. Most of these modes will involve adjacent rings having radically different phases. Such modes are not physically meaningful from the point of view of the original stellar disk. The only modes of potential physical interest are those with characteristic wavelengths much larger than the inter-ring separation. Now imagine how these modes evolve as one increases N. More and more modes will have wavelengths long compared to the ever shrinking inter-ring separation. The question is, do the frequencies of this gathering host of modes crowd ever more densely on the real line, to form a continuum of frequencies, or do the frequencies of certain modes stand out in glorious isolation?

Suppose that the latter is the case - that for any sufficiently large value of N, at least one mode has a frequency clearly separated from that of every other. Then in the limit N -> this modes corresponds to a motion of the disk that can endure indefinitely. Lynden-Bell (1965) first conjectured that warps are such excited normal modes of disks.

Sparke (1984a) and Sparke & Casertano (1988) have investigated the normal modes of disks by breaking them up into N rings. They find that as N -> , disks with realistically fuzzy outer edges have at most one discrete mode. If the disk is not embedded in any kind of halo potential, this isolated mode is trivial: it represents the possibility of tipping the entire disk with respect to the coordinate system. Obviously if an isolated disk starts out in a plane tipped at some angle, it will remain in that plane as there is nothing to align it with the chosen coordinate system. Mathematically this possibility is represented by a zero-frequency normal mode.

If the disk is embedded in the flattened potential of a halo, which we take to be aligned with the coordinate system, this mode becomes more interesting. Now the halo potential tries to twist the disk back into alignment with the coordinate system, and, being a gyro, the disk precesses steadily in response. But being a floppy disk, it precesses in a warped configuration. Sparke & Casertano (1988) have successfully fitted the warp of NGC 4013 to the shape of such a ``modified tilt mode''.

The shapes of modified tilt modes depend on the radial variation of the halo's flattening. If the center of the halo is significantly flattened, the inner part of the disk is anxious to precess more quickly than the disk's perimeter. Harmonious agreement on a common mode frequency is established by the inner disk applying a torque to the outer disk in the same sense that the halo torques the disk, thus speeding the precession of the outer disk and slowing the precession of the inner disk. To ensure this system of torques, the line of nodes is straight and the inclination of the rings of the disks to the halo's symmetry axis increases outwards. Sparke & Casertano call this a ``Type I'' or ``fast mode''.

Conversely, if the inner halo is very spherical and the outer halo rather flattened, harmonious agreement on a common precession rate is established by the outer disk hastening the precession of the inner disk by torquing the inner disk in the same sense as does the halo. Hence in such a ``Type II'' or ``slow mode'' the line of nodes is straight and the inclination decreases outwards.

Fast modes curl rather sharply away from the plane of the inner disk about a scale length from the disk's edge. The curvature of slow modes is less localized.

The essential distinction between fast and slow modes is the relative precession rates in the halo potential of isolated inner and outer rings. This may be varied either by varying the halo's ellipticity 1 - q as a function of radius, or by varying the halo's radial density profile. Thus if the halo's core radius is large enough, the disk will display a slow mode even when is independent of R.

A disk embedded in a flattened halo does not necessarily support a discrete modified tilt mode. Indeed if the inner halo is too flattened, or the disk too flimsy or extensive, the self-gravity of the disk is unable to reconcile the divergent precession rates of inner and outer rings. For typical parameters only disks less than ~ 7.5 Rd in extent have a discrete modified tilt mode.

As with spiral structure, valuable insight into bending modes can be obtained by studying running waves - a mode is considered to be a superposition of outward and inward running waves. Each mode's frequency is determined by the condition that as the corresponding disturbance runs out to the edge, where it is reflected, in to the centre, where it is either transmitted or reflected, and finally back out to its starting point, its total phase increment should be a multiple of 2. To impose this quantization condition one needs the dispersion relation for running waves. This is obtained by exploiting tight-winding and WKB approximations, the former to make the relation between perturbed density and potential local, and the latter to follow the motion of waves through a medium of varying refractive index.

Hunter & Toomre (1969) used this running-wave approach to interpret their normal mode results. They discovered that the only discrete normal modes of isolated disks with realistically fuzzy outer edges are the trivial tilt mode and interpreted this result in terms of the ability of the disk's outer edge to reflect outward-running bending waves. They showed that unless the surface density (R) vanishes so abruptly at the edge that the integral dR / (R) is well defined (as it is not for an exponential disk), outward-running waves are not reflected from the edge. Instead these waves proceed to ever larger radii, increasing in amplitude as they propagate into an increasingly rarefied medium. Eventually they go non-linear and their energy serves to heat the outer disk vertically.