  To understand the mechanism behind high-$z$ Ly${\alpha}$ nebulae, we simulate
the scattering of Ly${\alpha}$ in a $\rm H\,I$ halo about a central
Ly${\alpha}$ source. For the first time, we consider both smooth and clumpy
distributions of halo gas, as well as a range of outflow speeds, total $\rm
H\,I$ column densities, $\rm H\,I$ spatial concentrations, and central source
galaxies (e.g., with Ly${\alpha}$ line widths corresponding to those typical of
AGN or star-forming galaxies). We compute the spatial-frequency diffusion and
the polarization of the Ly${\alpha}$ photons scattered by atomic hydrogen. Our
scattering-only model reproduces the typical size of Ly${\alpha}$ nebulae
($\sim 100\,$kpc) at total column densities $N_{\rm HI} \geq 10^{20} \rm
cm^{-2}$ and predicts a range of positive, flat, and negative polarization
radial gradients. We also find two general classes of Ly${\alpha}$ nebula
morphologies: with and without bright cores. Cores are seen when $N_{\rm HI}$
is low, i.e., when the central source is directly visible, and are associated
with a polarization jump, a steep increase in the polarization radial profile
just outside the halo center. Of all the parameters tested in our smooth or
clumpy medium model, $N_{\rm HI}$ dominates the trends. The radial behaviors of
the Ly${\alpha}$ surface brightness, spectral line shape, and polarization in
the clumpy model with covering factor $f_c \gtrsim 5$ approach those of the
smooth model at the same $N_{\rm HI}$. A clumpy medium with high $N_{\rm HI}$
and low $f_c \lesssim 2$ generates Ly${\alpha}$ features via scattering that
the smooth model cannot: a bright core, symmetric line profile, and
polarization jump.
