  The non equilibrium relaxational dynamics of the solid on solid model on a
disordered substrate and the Sine Gordon model with random phase shifts is
studied numerically. Close to the super-roughening temperature $T_g$ our
results for the autocorrelations, spatial correlations and response function as
well as for the fluctuation dissipation ratio (FDR) agree well with the
prediction of a recent one loop RG calculation, whereas deep in the glassy low
temperature phase substantial deviations occur. The change in the low
temperature behavior of these quantities compared with the RG predictions is
shown to be contained in a change of the functional temperature dependence of
the dynamical exponent $z(T)$, which relates the age $t$ of the system with a
length scale ${\cal L}(t)$: $z(T)$ changes from a linear $T$-dependence close
to $T_g$ to a 1/T-behavior far away from $T_g$. By identifying spatial domains
as connected patches of the exactly computable ground states of the system we
demonstrate that the growing length scale ${\cal L}(t)$ is the characteristic
size of thermally fluctuating clusters around ``typical'' long-lived
configurations.
