  The Lyman alpha forest power spectrum has been measured on large scales by
the BOSS survey in SDSS-III at $z\sim 2.3$, has been shown to agree well with
linear theory predictions, and has provided the first measurement of Baryon
Acoustic Oscillations at this redshift. However, the power at small scales,
affected by non-linearities, has not been well examined so far. We present
results from a variety of hydrodynamic simulations to predict the redshift
space non-linear power spectrum of the Lyman Alpha transmission for several
models, testing the dependence on resolution and box size. A new fitting
formula is introduced to facilitate the comparison of our simulation results
with observations and other simulations. The non-linear power spectrum has a
generic shape determined by a transition scale from linear to non-linear
anisotropy, and a Jeans scale below which the power drops rapidly. In addition,
we predict the two linear bias factors of the Lyman Alpha forest and provide a
better physical interpretation of their values and redshift evolution. The
dependence of these bias factors and the non-linear power on the amplitude and
slope of the primordial fluctuations power spectrum, the temperature-density
relation of the intergalactic medium, and the mean Lyman Alpha transmission, as
well as the redshift evolution, is investigated and discussed in detail. A
preliminary comparison to the observations shows that the predicted redshift
distortion parameter is in good agreement with the recent determination of
Blomqvist et al., but the density bias factor is lower than observed. We make
all our results publicly available in the form of tables of the non-linear
power spectrum that is directly obtained from all our simulations, and
parameters of our fitting formula.
