  Following Paper I we investigate the properties of atmospheres that form
around small protoplanets embedded in a protoplanetary disc by conducting
hydrodynamical simulations. These are now extended to three dimensions,
employing a spherical grid centred on the planet. Compression of gas is shown
to reduce rotational motions. Contrasting the 2D case, no clear boundary
demarcates bound atmospheric gas from disc material; instead, we find an open
system where gas enters the Bondi sphere at high latitudes and leaves through
the midplane regions, or, vice versa, when the disc gas rotates sub-Keplerian.
The simulations do not converge to a time-independent solution; instead, the
atmosphere is characterized by a time-varying velocity field. Of particular
interest is the timescale to replenish the atmosphere by nebular gas,
$t_\mathrm{replenish}$. It is shown that the replenishment rate,
$M_\mathrm{atm}/t_\mathrm{replenish}$, can be understood in terms of a modified
Bondi accretion rate,
$\sim$$R_\mathrm{Bondi}^2\rho_\mathrm{gas}v_\mathrm{Bondi}$, where
$v_\mathrm{Bondi}$ is set by the Keplerian shear or the magnitude of the
sub-Keplerian motion of the gas, whichever is larger. In the inner disk, the
atmosphere of embedded protoplanets replenishes on a timescale that is shorter
than the Kelvin-Helmholtz contraction (or cooling) timescale. As a result,
atmospheric gas can no longer contract and the growth of these atmospheres
terminates. Future work must confirm whether these findings continue to apply
when the (thermodynamical) idealizations employed in this study are relaxed.
But if shown to be broadly applicable, replenishment of atmospheric gas
provides a natural explanation for the preponderance of gas-rich but
rock-dominant planets like super-Earths and mini-Neptunes.
