  We study the local structure of the vorticity field and the $\Lambda$
polarization in Au+Au collisions in the energy range
$\sqrt{s_{\mathrm{NN}}}=7.7$--$200$ GeV and Pb+Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=2760$ GeV using A Multi-Phase Transport (AMPT) model.
We focus on the vorticity field arising from the non-uniform expansion of the
fireball, which gives the circular structure of the transverse vorticity
$\boldsymbol{\omega}_{\perp}=(\omega_{x},\omega_{y})$ around the $z$ direction
as well as the quadrupole pattern of the longitudinal vorticity $\omega_{z}$ in
the transverse plane. As a consequence, the three components of the
polarization vector $\mathbf{P}=(P_{x},P_{y},P_{z})$ for $\Lambda$ hyperons
show harmonic behaviors as $\mathrm{sgn}(Y)\sin\phi_{p}$,
$-\mathrm{sgn}(Y)\cos\phi_{p}$, and $-\sin(2\phi_{p})$, where $\phi_{p}$ and
$Y$ are the azimuthal angle and rapidity in momentum space. These patterns of
the local $\Lambda$ polarization are expected to be tested in future
experiments.
