  We investigate the dynamics of an Alfven surface (where the Alfven speed
equals the advection velocity) in the context of core collapse supernovae
during the phase of accretion on the proto-neutron star. Such a surface should
exist even for weak magnetic fields because the advection velocity decreases to
zero at the center of the collapsing core. In this decelerated flow, Alfven
waves created by the standing accretion shock instability (SASI) or convection
accumulate and amplify while approaching the Alfven surface. We study this
amplification using one dimensional MHD simulations with explicit physical
dissipation. In the linear regime, the amplification continues until the Alfven
wavelength becomes as small as the dissipative scale. A pressure feedback that
increases the pressure in the upstream flow is created via a non linear
coupling. We derive analytic formulae for the maximum amplification and the non
linear coupling and check them with numerical simulations to a very good
accuracy. We also characterize the non linear saturation of this amplification
when compression effects become important, leading to either a change of the
velocity gradient, or a steepening of the Alfven wave. Applying these results
to core collapse supernovae shows that the amplification can be fast enough to
affect the dynamics, if the magnetic field is strong enough for the Alfven
surface to lie in the region of strong velocity gradient just above the
neutrinosphere. This requires the presence of a strong magnetic field in the
progenitor star, which would correspond to the formation of a magnetar under
the assumption of magnetic flux conservation. An extrapolation of our analytic
formula (taking into account the nonlinear saturation) suggests that the Alfven
wave could reach an amplitude of B ~ 10^15 G, and that the pressure feedback
could significantly contribute to the pressure below the shock.
