Augmented rockets increase thrust (about twice) by mixing the exhaust of a rocket with ambient air in order to accelerate a greater mass of gas at a lower speed. Is the effectivness of this system proportional to the speed of the rocket or does the shroud around the exhaust have the capability to suck in enough air to significantly increase performance even when the engine is static and alight?
The only one I know about, NASA's early 2000s GTX proposal, used external air from liftoff to Mach 11. This Rocket-Based Combined Cycle vehicle operated in the following modes:
- Mach 0.0 to 25: air augmented rocket
- Mach 2.5 to 5.5: ramjet
- Mach 5.5 to 11.0: scramjet
- Mach 11.0 - orbit: pure rocket
I doubt the air inlet provided much augmentation while the vehicle was static, but that is a very short part of the ascent.
There are a lot of interesting papers about GTX on NTRS, here are some.
This is a late reply, but there's a more recent concept proposed in 2018: https://www.researchgate.net/publication/326263323_The_Turbo_Rocket_-_A_high_performance_air-breathing_rocket_propulsion_system_with_nuclear_and_chemical_configurations. This proposes both nuclear and chemical designs, the second one being relevant to this question.
Based on this graph, the augmentation is present right from the start, but it's not as much as at supersonic speeds.