Tubulence Numerics
How are the RMS of The Turbulence solver equations defined, and how big shall they be in order to stay in range of good convergence ?

The residuals? They are all in the documentation. Also guides for acceptable residual targets. Start reading.

i mean the residuals of the turbulence equations

All discussed in the documentation.
But often the turbulence residuals are not so important. Momentum and mass are critical but sometimes you can get away with turbulence being not so tight. But the only way to be sure is to run a sensitivity check on your case and find out for yourself. 
So after testing the turbulence numerics options i got this result:
With Turbulence Numerics turned on to High Resolution i got different results as with Turbulence Numerics turned on to First Order. In the countercurrent waterair Flow in my Channel everything is smoth at the same level of quite high velocities whereas at the first order (Turbulence Numeric discretization) mode got a kind of churn bubbly with drops flow?? 
First order discretisation will smear the turbulence field out, whereas the second order approach will enable it to capture finer turbulence features. But from your post I am not completely sure what you are saying.

With first order turbulence numeric i got such a hydraulic jump and the free surface is too much wavy and a kind of churn flow is arising at the crest of the waves whereas with high order turbulence numeric nothing occurs at the same inlet conditions.

A picture would help.

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