Why does the turn keeps getting tighter as the spiral dive develops? Use of elevator control makes sense as it would increase the horizontal lift vector inside the turn. But even without any elevator control the turn tends to get tighter eventually.
It's complicated. One factor driving the increase in bank angle is that the outboard wingtip is moving around a larger circle radius, and thus experiencing a higher airspeed, and thus creating more lift, than the inboard wingtip. Which contributes to a rolling-in torque.
However, there are also several factors that are acting to create a rolling-out torque, and for some of these factors, the steeper the bank angle and/or the higher the descent rate, the stronger they become.
Thus the bank angle will generally not tend to increase all the way to (or even approaching) 90 degrees, rather it will eventually reach some limit well short of that.
But, the wings may depart the aircraft before that point is reached. Especially if a pilot input suddenly increases the angle-of-attack of the wing. Or if the aircraft exceeds Vne.
And if the aircraft is in cloud and a disoriented pilot is actually contributing a rolling-in torque via an aileron input-- then a very steep bank angle, very high G-load, very high airspeed, and very high descent rate are virtually guaranteed.
PS the main reason that up elevator tends to increase the bank angle is not that a strong horizontal component of lift automatically equates to a roll torque, but rather that a small turn radius, and a lower airspeed than would exist if the pilot relaxed the pitch input, both tend to exacerbate the effect described in the first paragraph above.
It's a case of the planes "static stability" gone amok as the plane circles downward.
The lift of the banked wing causes a change in direction, which causes a change in the relative wind on the elevator/horizontal stabilizer. This has the same effect as pulling on the elevator in level flight: increasing the Angle of Attack of the wing ---> which increases lift ---> which further tightens the circle ---> which further changes the relative wind on the tail.
This is why it is imperative, when falling into a spiral, to:
- Cut power to avoid overspeed
- Reduce elevator back pressure
- Use ailerons to roll out of the spiral
- Recover from dive
- Add power back in
Unintentional spiral dives are rare in VFR, but are all too easy to get into when visibility is limited, and can be started by increasing back pressure on the yoke (in an effort to hold altitude) without realizing the plane is banked. Learning the attitude indicator and cross checking heading changes can be helpful in avoiding this situation.