  Fragmentation of spiral arms can drive the formation of giant clumps and
induce intense star formation in disc galaxies. Based on the spiral-arm
instability analysis of our Paper I, we present linear perturbation theory of
dynamical instability of self-gravitating spiral arms of magnetised gas,
focusing on the effect of toroidal magnetic fields. Spiral arms can be
destabilised by the toroidal fields which cancel Coriolis force, i.e.
magneto-Jeans instability. Our analysis can be applied to multi-component
systems that consist of gas and stars. To test our analysis, we perform ideal
magneto-hydrodynamics simulations of isolated disc galaxies and examine the
simulation results. We find that our analysis can characterise dynamical
instability leading arms to fragment and form clumps if magnetic fields are
nearly toroidal. We propose that dimensionless growth rate of the most unstable
perturbation, which is computed from our analysis, can be used to predict
fragmentation of spiral arms within an orbital time-scale. Our analysis is
applicable as long as magnetic fields are nearly toroidal. Using our analytic
model, we estimate a typical mass of clumps forming from spiral-arm
fragmentation to be consistent with observed giant clumps $\sim10^{7-8}~{\rm
M_\odot}$. Furthermore, we find that, although the magnetic destabilisation can
cause low-density spiral arms to fragment, the estimated mass of resultant
clumps is almost independent from strength of magnetic fields since marginal
instability occurs at long wavelengths which compensate the low densities of
magnetically destabilised arms.
