  We experimentally and numerically investigate the effect of wind forcing on
the spectral dynamics of Akhmediev breathers, a wave-type known to model the
modulation instability. We develop the wind model to the same order in
steepness as the higher order modifcation of the nonlinear Schroedinger
equation, also referred to as the Dysthe equation. This results in an
asymmetric wind term in the higher order, in addition to the leading order wind
forcing term. The derived model is in good agreement with laboratory
experiments within the range of the facility's length. We show that the leading
order forcing term amplifies all frequencies equally and therefore induces only
a broadening of the spectrum while the asymmetric higher order term in the
model enhances higher frequencies more than lower ones. Thus, the latter term
induces a permanent upshift of the spectral mean. On the other hand, in
contrast to the direct effect of wind forcing, wind can indirectly lead to
frequency downshifts, due to dissipative effects such as wave breaking, or
through amplification of the intrinsic spectral asymmetry of the Dysthe
equation. Furthermore, the definitions of the up- and downshift in terms of
peak- and mean frequencies, that are critical to relate our work to previous
results, are highlighted and discussed.
