In a series of three papers, the dynamical interplay between environments and dark matter haloes is investigated, while focusing on the dynamical flows through the virtual virial sphere. It relies on both cosmological simulations, to constrain the environments, and an extension to the classical matrix method to derive the responses of the halo. A companion paper (Paper I) showed how perturbation theory allows us to propagate the statistical properties of the environment to an ensemble description of the dynamical response of the embedded halo. The current paper focuses on the statistical characterization of the environments surrounding haloes, using a set of large-scale simulations; the large statistic of environments presented here allows us to put quantitative and statistically significant constrains on the properties of flows accreted by haloes.

The description chosen in this paper relies on a `fluid' halocentric representation. The interactions between the halo and its environment are investigated in terms of a time-dependent external tidal field and a source term characterizing the infall. The former accounts for fly bys and interlopers. The latter stands for the distribution function of the matter accreted through the virial sphere. The method of separation of variables is used to decouple the temporal evolution of these two quantities from their angular and velocity dependence by means of projection on a 5D basis.

It is shown that how the flux densities of mass, momentum and energy can provide an alternative description to the 5D projection of the source. Such a description is well suited to regenerate synthetic time lines of accretion which are consistent with environments found in simulations as discussed in the Appendix. The method leading to the measurements of these quantities in simulations is presented in detail and applied to 15000 haloes, with masses between 5 × 10¹² and 10¹⁴M_solar evolving between z = 1 and 0. The influence of resolution, class of mass, and selection biases are investigated with higher resolution simulations. The emphasis is put on the one- and two-point statistics of the tidal field, and of the flux density of mass, while the full characterization of the other fields is postponed to Paper III.

The net accretion at the virial radius is found to decrease with time. This decline results from both an absolute decrease of infall and a growing contribution of outflows. Infall is found to be mainly radial and occurring at velocities ~0.75 times the virial velocity. Outflows are also detected through the virial sphere and occur at lower velocities ~0.6V_c on more circular orbits. The external tidal field is found to be strongly quadrupolar and mostly stationary, possibly reflecting the distribution of matter in the halo's near environment. The coherence time of the small-scale fluctuations of the potential hints a possible anisotropic distribution of accreted satellites. The flux density of mass on the virial sphere appears to be more clustered than the potential, while the shape of its angular power spectrum seems stationary. Most of these results are tabulated with simple fitting laws and are found to be consistent with published work, which rely on a description of accretion in terms of satellites.