  We present analytic expressions for the $s$-parametrized currents on the
sphere for both unitary and dissipative evolutions. We examine the spatial
distribution of the flow generated by these currents for quadratic
Hamiltonians. The results are applied for the study of the quantum dissipative
dynamics of the time-honored Kerr and Lipkin models, exploring the appearance
of the semiclassical limit in stable, unstable and tunnelling regimes.
