  (shortened) Planet forming discs are believed to be very weakly turbulent in
the regions outside of 1 AU. For this reason, it is now believed that
magnetized winds could be the dominant mechanism driving accretion in these
systems. However, there is today no self-consistent way to describe discs
subject to a magnetized wind, in a way similar to the $\alpha$ disc model. In
this article, I explore in a systematic way the parameter space of wind-driven
protoplanetary discs and present scaling laws which can be used in reduced
models \`a la alpha-disc. Methods: I compute a series of self-similar wind
solutions, assuming the disc is dominated by ambipolar and Ohmic diffusions.
These solution are obtained by looking for stationary solutions in the
finite-volume code PLUTO using a relaxation method and continuation. Results:
Self-similar solutions are obtained for values of plasma beta ranging from 10^2
to 10^8, for several Ohmic and ambipolar diffusion strengths. Mass accretions
rates of the order of 10^{-8} Msun/yr are obtained for poloidal field strength
beta=O(10^4) or equivalently 1 mG at 10 AU. The resulting magnetic lever arms
are typically lower than 2, possibly reaching 1.5 in weakest field cases.
Finally, I provide a complete set of scaling laws and semi-analytical wind
solutions, which can be used to fit and interpret observations. Conclusions:
Magnetized winds are unavoidable in protoplanetary discs as soon as they are
embedded in an ambient poloidal magnetic field. Very detailed disc microphysics
are not always needed to describe them, and simplified models such as
self-similar solutions manage to capture most of the physics seen in full 3D
simulations. The remaining difficulty to get a complete theory of wind-driven
accretion lies in the transport of the large scale field, which remains poorly
constrained and not well understood.
