  Gaseous, disk-halo interfaces are shaped by physical processes that are
critical to disk galaxy evolution, including gas accretion and galactic
outflows. However, observations indicate that extraplanar diffuse ionized gas
(eDIG) layers have scale heights several times higher than their thermal scale
heights. This discrepancy poses a challenge to our current understanding of the
disk-halo interface. In this paper, we present a spatially-resolved case study
of the eDIG layers in a nearby pair of sub-$L_*$ disk galaxies NGC$\,$3511/3513
using long-slit spectroscopy. We decompose optical nebular lines from the warm
interstellar medium and disk-halo interfaces into narrow and broad velocity
components. We show that in NGC$\,$3511, the broad component has three
distinctive characteristics in comparison to the narrow component: (1) elevated
[NII]$\lambda 6583/H\alpha$ and [SII]$\lambda 6716/H\alpha$ line ratios, (2)
significantly higher velocity dispersions (a median
$\langle\sigma\rangle_{\text{Broad}} = 24\,$km/s compared to
$\langle\sigma\rangle_{\text{Narrow}} = 13\,$km/s), and (3) a rotational
velocity lag. Together, these characteristics support an origin in an
extraplanar, thick gaseous disk. In NGC$\,$3513, the broad component is
consistent with localized outflows making their way out of the galactic disk.
Our findings demonstrate that gas circulation at the disk-halo interface is
present in both galaxies. Additionally, we test a dynamic equilibrium model
with pressure support supplied by thermal and turbulent motions. Based on
measurements of the eDIG velocity dispersion in NGC$\,$3511, we demonstrate
that turbulent motions increase the scale height by at least a factor of a few
above the thermal scale height, with $h_{z} \gtrsim 0.2 - 0.4$ kpc at $R = 3 -
5$ kpc. This highlights the importance of turbulent motions to the vertical
structure of the gaseous, disk-halo interface.
