  We investigate, via numerical simulations, the phase ordering kinetics of a
two- dimensional soft-spin O(2) Ginzburg-Landau model when a reversible mode
cou- pling is included via the conserved conjugate momentum of the spin order
parameter (the model E). Coarsening of the system, when quenched from a dis-
ordered state to zero temperature, is observed to be enhanced by the existence
of the mode coupling terms. The growth of the characteristic length scale L(t)
exhibits an effective super-diffusive growth exponent that can be interpreted
as a positive logarithmic-like correction to a diffusive growth, i.e., L(t) ~
(t ln t)^{1/2}. In order to understand this behavior, we introduced a simple
phenomenological model of coarsening based on the annihilation dynamics of a
vortex-antivortex pair, incorporating the effect of vortex inertia and
logarithmically divergent mobility of the vortex. With a suitable choice of the
parameters, numerical solutions of the simple model can fit the full simulation
results very adequately. The effective growth exponent in the early time stage
is larger due to the effect of the vortex inertia, which crosses over into late
time stage characterized by positive logarithmic correction to a diffusive
growth. We also investigated the non-equilibrium autocorrelation function from
which the so called {\lambda} exponent can be extracted. We get {\lambda} =
1.99(2) which is distinctively larger than the value of {\lambda} = 1.17 for
the purely dissipative model-A dynamics of non-conserved O(2) models.
