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.
Don't read what I wrote, read what I meant!
Thank you, yes, "almost entirely in roll"
You are correct [that Dutch roll mode is usually stable] (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.
It certainly improves stability but I don't think it can flip the sign. FAR 23.181 requires:
"Any combined lateral-directional oscillations ("Dutch roll") occurring between the stalling speed and the maximum allowable speed appropriate to the configuration of the airplane must be damped to 1/10 amplitude in 7 cycles"
FAR 23.672 requires the aircraft to be stable after a failure of "the stability augmentation system...trim, stability, and stall characteristics are not impaired below a level needed to permit continued safe flight and landing".
I'd be surprised if there's anything certified that's actually
unstable in Dutch roll even if the yaw damper fails. But it might be stomach churning and concentrate the mind somewhat!
The crux of my disagreement with what you previously wrote was in:
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.
You may be reading more into "tendency to Dutch roll" than I intended. All I meant was that the mode is lightly damped, not unstable. Surely it's unthinkable that the Airbus without artificial stability input has an unstable Dutch roll mode.
McCormick's
Aerodynamics, Aeronautics and Flight Mechanics has a nice section on lateral stability, including an example of a Cherokee after an impulsive rudder displacement. For 10 seconds it wallows around in Dutch roll, but over longer timescales the unstable spiral mode takes over.
But putting aside the interpretation, I don't see anything in the maths that prohibits a stable Dutch roll and stable spiral mode. Though I salute your quest for simplicity, I think you're in danger of oversimplifying. These are coupled oscillations.
As Genghis said (my brackets []),
Mathematically, lateral stability is the partial derivative of rolling moment with respect to sideslip [Lb]. Directional stability is similarly the partial derivative of yawing moment with sideslip. [Nb]
I think that there are a couple of other derivatives you have to bring in to think about the spiral stability. They are the rolling moment with yaw rate [Lr] and the yawing moment with yaw rate [Nr]. According to McCormick, the constant term in the characteristic equation is proportional to:
Nb*Lr - Lb*Nr
(that feels a bit more symmetrical than Nb -Lb or Nb/Lb, which feel like apples and oranges)
If that constant term changes sign, so does
one of the roots. That will doubtless be the spiral mode. But I don't see that Dutch Roll stability/instability (i.e. the sign of the root) is affected directly.
[To take a wild guess at John Farley's puzzle, did you start off at higher speed in the stable case which reduces Lr?]