  The long-time behavior of solutions of the three-dimensional Navier--Stokes
equations in a periodic domain is studied. The time-dependent body force
decays, as time $t$ tends to infinity, in a coherent manner. In fact, it is
assumed to have a general and complicated asymptotic expansion which involves
complex powers of $e^t$, $t$, $\ln t$, or other iterated logarithmic functions
of $t$. We prove that all Leray-Hopf weak solutions admit an asymptotic
expansion which is independent of the solutions and is uniquely determined by
the asymptotic expansion of the body force. The proof makes use of the
complexifications of the Gevrey-Sobolev spaces together with those of the
Stokes operator and the bilinear form of the Navier-Stokes equations.
