  We establish an asymptotic profile that sharply describes the behavior as
$t\to\infty$ for solutions to a non-solenoidal approximation of the
incompressible Navier-Stokes equations introduced by Temam. The solutions of
Temam's model are known to converge to the corresponding solutions of the
classical Navier-Stokes, e.g., in $L^3\_{\rm loc} (R^+ \times R^3)$, provided
$\epsilon\to0$, where $\epsilon>0$ is the physical parameter related to the
artificial compressibility term. However, we show that such model is no longer
a good approximation of Navier-Stokes for large times: indeed, its solutions
can decay much slower as $t\to+\infty$ than the corresponding solutions of
Navier-Stokes.
