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The long-leg, full flutter kick generates continuous propulsion through the entire kick cycle, but uses short muscles with little mechanical advantage, so it's fatiguing.
The modified flutter, done from the knees with a flat body, uses a much more mechanically efficient technique. It minimizes downward water deflection. It is extremely useful when swimming into flow or current.
The frog kick directs water backwards. It uses efficient muscles and involves a rest period during each kick cycle. This also allows the diver, particularly the novice diver, to get a brief buoyancy check, because if they sink or rise during the glide phase, they realize they are not truly neutral. The frog kick is also the basis for the helicopter turn and back kick, both of which are useful for precise positioning for observing marine life, or taking photographs.
I do not understand how the modified flutter works ... it looks like the water is being moved downwards ... obviously it is not, but I would need to try it to understand.
You're right. The water is moving downward in a modified flutter. However, there is much less water moving per kick, it's not moving as forcefully, it's not moving as far, and - depending on how long your legs are - it's being moved downward from an additional two feet or so above anything you don't want to expose to downward moving water.
…Frog kick is not only more efficient from a propulsion standpoint... it's also "roll neutral" if you will...