Spatially Resolved Properties of Extraplanar Diffuse Ionized Gas in NGC$\,$3511 and NGC$\,$3513

Gaseous, disk-halo interfaces are shaped by processes that are critical to galaxy evolution, including gas accretion and outflows. Extraplanar diffuse ionized gas (eDIG) layers are characterized by scale heights that largely exceed those predicted by their temperature, suggesting the presence of turbulent energy injection from star formation feedback. However, the origin of this large scale height remains uncertain. To explore the connection between eDIG and star-forming disks, we present a spatially-resolved case study of a nearby pair of sub-$L_*$, intermediately inclined disk galaxies NGC$\,$3511/3513. We decompose optical nebular lines observed using long-slit spectroscopy into narrow and broad velocity components. In NGC$\,$3511, the broad component has three distinctive characteristics in comparison to the narrow component: (1) significantly higher velocity dispersions (a median $\langle\sigma\rangle_{\text{Broad}} = 24$ \kms compared to $\langle\sigma\rangle_{\text{Narrow}} = 13$ \kms), (2) elevated [NII]$\lambda$6583/H$\alpha$ and [SII]$\lambda$6716/H$\alpha$ line ratios, and (3) a rotational velocity lag. These characteristics support the origin of the broad component in an extraplanar, gaseous disk. In NGC$\,$3513, the broad component reveals disk-halo circulation via localized outflows at radius $\lesssim 1$ kpc. For NGC$\,$3511, we test a vertical hydrostatic equilibrium model with pressure support supplied by thermal and turbulent motions. Under this assumption, the eDIG velocity dispersion corresponds to a scale height $h_{z} \gtrsim 0.2 - 0.4$ kpc at $R = 3 - 5$ kpc, a factor of a few above the thermal scale height ($h_{z} \lesssim 0.1$ kpc). This highlights the importance of turbulent motions to the vertical structure of the gaseous, disk-halo interface.