Conservation laws for tests of horizontal shear flow (left) and of white noise in the velocity field (right). For each case, the normalized variation in $J\overline{\rho }$ (left), Π_i (middle), and H_k (right) is presented as a function of time. For the horizontal shear flow test (see also Sect. 4.1.1), the modified mass conservation law ($J\overline{\rho }$) and the conservation law for total momenta (Π_i) and for kinetic helicity H_k are all satisfied to machine precision (note that for all of the panels, the y axis ticks are in units of 10⁻¹³). For the case with white noise in the velocity field, the conservation laws for both mass, $J\overline{\rho }$, and total momenta, Π_i, are fulfilled to machine precision as well, but not for the kinetic helicity, H_k (note that for the upper two panels, the y axis ticks are in units of 10⁻¹⁴). For comparison, a simulation with the same setup of white noise, but without collapse (i.e., U₀ = U_R0 = 0) was performed (“WN0”, dashed grey lines, see also the legend), which gives consistent results to WN. We thus anticipate that the deviation from conservation law for H_k is likely due to intrinsic characteristics of ATHENA++, which in certain situations is not strictly vorticity-conserving.