  Gravitational waves provide a unique and powerful opportunity to constrain
the dynamics in the interior of proto-neutron stars during core collapse
supernovae. Convective motions play an important role in generating neutron
stars magnetic fields, which could explain magnetar formation in the presence
of fast rotation. We compute the gravitational wave emission from proto-neutron
star convection and its associated dynamo, by post-processing three-dimensional
MHD simulations of a model restricted to the convective zone in the anelastic
approximation. We consider two different proto-neutron star structures
representative of early times (with a convective layer) and late times (when
the star is almost entirely convective). In the slow rotation regime, the
gravitational wave emission follows a broad spectrum peaking at about three
times the turnover frequency. In this regime, the inclusion of magnetic fields
slightly decreases the amplitude without changing the spectrum significantly
compared to a non-magnetised simulation. Fast rotation changes both the
amplitude and spectrum dramatically. The amplitude is increased by a factor of
up to a few thousands. The spectrum is characterized by several peaks
associated to inertial modes, whose frequency scales with the rotation
frequency. Using simple physical arguments, we derive scalings that reproduce
quantitatively several aspects of these numerical results. We also observe an
excess of low-frequency gravitational waves, which appears at the transition to
a strong field dynamo characterized by a strong axisymmetric toroidal magnetic
field. This signature of dynamo action could be used to constrain the dynamo
efficiency in a proto-neutron star with future gravitational wave detections.
