applicability of hydrostatic approximation

[illilli]

As I could see from discussions with meteorologists,  in modern forecasting systems there is general tendency towards non-hydrostatic modeling.

Meanwhile on large scales (quasi)hydrostatic system is bearable.

The questions are:

1. For which minimal scales (and corresponding resolutions) hydrostatic modelling can give adequate results. F.e. can it be used for horizontal distances 10 or 5 or 1 km and with which resolutions.

2. As far as I could see, usually hydrostatic modeling assume stable atmosphere, with small scale convection, internal waves etc. being parametrized.
But there is also a number of papers, where hydrostatic model is used for large-scale convection. Despite the ill-posedeness of the problem with initial unstable profile, the solutions of such hydrostatic system can be constructed, and they can give some insight on some large scale problems.
So the question is: whether hydrostatic modeling with unstable vertical profile was ever used for weather( or climate) predictions, not just as some analytical exercises.

I've had some disagreement on the similar questions with the author of recently published paper. My position, which can also be found doubtful, can be found in here:
https://pubpeer.com/publications/446D764678B603CC6EF997C8C5EF00
https://assert.pub/papers/2001.08637
https://arxiv.org/abs/2001.08637

Thanks