  Small-scale fluctuating magnetic fields of order $n$G are observed in
supernova shocks and galaxy clusters, where its amplification is likely caused
by the Biermann battery mechanism. However, these fields cannot be amplified
further without the turbulent dynamo, which generates magnetic energy through
the stretch-twist-fold (STF) mechanism. Thus, we present here novel
three-dimensional magnetohydrodynamic (MHD) simulations of a laser-driven shock
propagating into a stratified, multiphase medium, to investigate the post-shock
turbulent magnetic field amplification via the turbulent dynamo. The
configuration used here is currently being tested in the shock tunnel at the
National Ignition Facility (NIF). In order to probe the statistical properties
of the post-shock turbulent region, we use $384 \times 512 \times 384$ tracers
to track its evolution through the Lagrangian framework, thus providing a
high-fidelity analysis of the shocked medium. Our simulations indicate that the
growth of the magnetic field, which accompanies the near-Saffman kinetic energy
decay ($E_{\textrm{kin}} \propto t^{-1.15})$ without turbulence driving,
exhibits slightly different characteristics as compared to periodic box
simulations. Seemingly no distinct phases exist in its evolution, because the
shock passage and time to observe the magnetic field amplification during the
turbulence decay are very short ($\sim\!0.3$ of a turbulent turnover time).
Yet, the growth rate is still consistent with those expected for compressive
(curl-free) turbulence driving in subsonic, compressible turbulence.
Phenomenological understanding of the dynamics of the magnetic and velocity
fields are also elucidated via Lagrangian frequency spectra, which are
consistent with the expected inertial range scalings in the Eulerian-Lagrangian
bridge.
