In attempting to answer questions I think it is usually best to apply the following principles:
a. Match the complexity of the answer to that of the question.
b. Limit the range of discussion to that strictly necessary.
In this case the original question "what is spiral stability" suggested a need for a relatively low level of complexity. Applying the above principles in composing such an answer I deliberately avoided discussion of dynamic stability.
BOOKWORM
QUOTE. "It doesn't quite work like that. The combination of roll, yaw and sideslip get intertwined to produce three mixed modes.
In fact one is a heavily damped mode almost entirely in yaw called "roll damping" if the aircraft is rolling and you put the ailerons neutral, it stops rolling (gradually). And that's about all there is to that one."
COMMENT. The term dynamic stability describes the subsequent motion of an aircraft following a disturbance. If it oscillates, but the oscillations gradually subside then it is dynamically stable. If it oscillates but the oscillations increase in amplitude it is dynamically unstable. If it oscillates such that the oscillations neither subside nor increase then it is neutrally dynamically stable.
Roll damping is the principle factor in providing dynamic stability in roll. Although you state that roll damping is "almost entirely in yaw", your example actually illustrates its effect in roll. (Strangely) although a similar effect is the main factor in providing directional dynamic stability, this isn't (often) called yaw damping.
QUOTE. "Another is the Dutch roll mode. This is typically stable but underdamped, so if you induce Dutch roll the aircraft tends to wallow in the mode for a few oscillations which then disappear."
You are correct (in normal circumstance) but this is not always the case. If for example you switch off the yaw damper of a modern swept wing jet cruising at high altitude, you would probably experience a far more unstable and potentially destructive Dutch roll. The yaw damper prevents Dutch roll by artificially increasing directional stability so that it is stronger than lateral stability.
QUOTE. "The spiral mode, which is usually unstable as the name implies (otherwise you'd call it wing rock wouldn't you.) The instability means that with no control input bank gradually increases into a spiral dive. I think that it is possible to make an aircraft stable in this mode in principle, but you pay a high price in terms of handling, and it generally causes more trouble than its worth.
COMMENT. I agree. The spiral dive is quite predictable and easy to correct manually so it isn't worth putting in too much effort to eliminate it. But the original question indicates that the AB330 is spirally stable. Whether this effect is being generated aerodynamically or artificially matters little in attempting to answer the original questions (Which were, what is spiral stability and why does the AB330 go wings level at high speed and only to within the 33 degree limits at high alpha). If we accept that Shockwave has quoted the manual accurately and that the manual itself is correct, then we must accept the (effective) spiral stability of the AB330 as a given fact. I must confess that I went beyond the original questions in examining how this aircraft might actually use its spiral stability to achieve the specified effects.
QUOTE. "The tendency to Dutch roll does not necessarily imply a positive stability in the spiral mode, or vice versa."
COMMENT. I do not agree. Dutch roll requires that an aircraft repeatedly rolls from one side to the other. By your own definition a spirally unstable aircraft will not return to wings level, but will yaw towards the low wing. This will increase the roll which will in turn increase the yaw. It is a fundamental requirement for Dutch roll that the (bank reducing) rolling tendency due to lateral stability is stronger than the (bank increasing) rolling tendency due to directional stability.
GENGHIS
Returning to my comments above regarding giving a simple answer to a simple question, any answers involving term such as "partial derivative of" is probably aiming a bit too high in this case.
QUOTE. " Dutch roll (aerodynamic at-least) is largely down to the ratio of the two (lateral and dynamic stability). If lat/dir is greater than 1.0, then spiral stability should be positive.
COMMENT. I agree. In fact this is exactly what I said in my previous post!
QUOTE. " Dutch roll excitement is down to the lateral and directional damping. The damping, and the actual stability values are not closely related."
COMMENT. I agree to the extent that an aircraft that is heavily damped in roll and yaw will not exhibit significant Dutch roll. The fact remains however that unless a sideslipping aircraft rolls away from the sideslip it cannot exhibit any Dutch roll.
QUOTE. "The ratio of DR is roughly the same as the ratio of stabilities. So if the wingtip describes a circle during DR, expect roughly neutral spiral stability. Alternatively, if the wingtip describes a flat oval with its axis parallel to the horizon, expect it to be spirally divergent, because this indicates that directional stability is greater than lateral. "
COMMENT. I agree. This again reinforces my assertion that Dutch roll is caused by lateral stability being greater than directional.
SO WHERE ARE WE NOW?
Well, I think we have defined spiral stability pretty well, so we can probably move on to the second question of "why does the AB330 go wings level in an over speed / over bank, but only back to within the 33 degree limits in a high alpha / over bank. I freely admit that the suggestion in my previous post is little more than conjecture. It is of course based upon the assumption that the difference was intentionally created, rather than simply a fact of the aerodynamics or electronics. The suggestion by Shockwave that it might be due to "lower speed, higher AOA so less stability" is I think incorrect. Aircraft tend to become less (dynamically) stable as speed increases, due to a reduction in roll damping.