  We study the spin mixing dynamics of ultracold spin-1 atoms in a weak
non-uniform magnetic field with field gradient $G$, which can flip the spin
from +1 to -1 so that the magnetization $m=\rho_{+}-\rho_{-}$ is not any more a
constant. The dynamics of $m_F=0$ Zeeman component $\rho_{0}$, as well as the
system magnetization $m$, are illustrated for both ferromagnetic and polar
interaction cases in the mean-field theory. We find that the dynamics of system
magnetization can be tuned between the Josephson-like oscillation similar to
the case of double well, and the interesting self-trapping regimes, i.e. the
spin mixing dynamics sustains a spontaneous magnetization. Meanwhile the
dynamics of $\rho_0$ may be sufficiently suppressed for initially imbalanced
number distribution in the case of polar interaction. A "beat-frequency"
oscillation of the magnetization emerges in the case of balanced initial
distribution for polar interaction, which vanishes for ferromagnetic
interaction.
