  We use a large N-body simulation to study the characteristic scales in the
density gradient profiles in and around halos with masses ranging from
$10^{12}$ to $10^{15} h^{-1}{\rm M_\odot}$. We investigate the profiles
separately along the major (T_1) and minor (T_3) axes of the local tidal tensor
and how the characteristic scales depend on halo mass, formation time, and
environment. We find two kinds of prominent characteristic features in the
gradient profiles, a deep `valley' and a prominent `peak'. We use the Gaussian
Process Regression to fit the gradient profiles and identify the local extrema
to determine the scales associate with these features. Around the valley, we
identify three types of distinct local minima, corresponding to caustics of
particles orbiting around halos. The appearance and depth of the three caustics
depend significantly on the direction defined by the local tidal field,
formation time and environment of halos. The first caustic is located at a
radius r>0.8R_{200}, corresponding to the splashback feature, and is dominated
by particles at their first apocenter after infall. The second and third
caustics, around 0.6R_{200} and 0.4R_{200} respectively, can be determined
reliably only for old halos. The first caustic is always the most prominent
feature along T_3, but may not be the case along T_1 or in azimuthally-averaged
profiles, suggesting that caution must be taken when using averaged profiles to
investigate the splashback radius. We find that the splashback feature is
approximately isotropic when proper separations are made between the first and
the other caustics. We also identify a peak feature located at $\sim$
2.5R_{200} in the density gradient profile. This feature is the most prominent
along T_1 and is produced by mass accumulations from the structure outside
halos. We also discuss the origins of these features and their observational
implications.
