Secular evolution of MHD wind-driven discs: analytical solutions in the
  expanded {\alpha}-framework
  The evolution of protoplanetary discs and the related process of planet
formation is regulated by angular momentum transport and mass-loss processes.
Over the past decade, the paradigm of viscosity has been challenged and MHD
disc winds appear as a compelling scenario to account for disc accretion. In
this work, we aim to construct the equivalent of the widely used analytical
description of viscous evolution for the MHD wind case. The transport of
angular momentum and mass induced by the wind is parameterized by an
$\alpha$-like parameter and by the magnetic lever arm parameter $\lambda$.
Extensions of the paradigmatic Lynden-Bell and Pringle similarity solutions to
the wind case are presented. We show that wind-driven accretion leads to a
steeper decrease in the disc mass and accretion rate than in viscous models due
to the absence of disc spreading. If the decline of the magnetic field strength
is slower than that of the gas surface density, the disc is dispersed after a
finite time. The evolution of the disc in the $\dot{M}_*-M_D$ plane is
sensitive to the wind and turbulence parameters. A disc population evolving
under the action of winds can exhibit a correlation between $\dot{M}_*$ and
$M_D$ depending on the initial conditions. The simplified framework proposed in
this work opens to a new avenue to test the effectiveness of wind-driven
accretion from the observed disc demographics and constitutes an important step
to include wind-driven accretion in planet population synthesis models.
