  $\textit{Herschel}$ observations of far infrared N$^+$ emission lines have
demonstrated that dense plasma, with $n_e\sim30\,{\rm cm^{-3}}$, is ubiquitous
in the inner Galactic plane. By combining the information from
$\textit{Herschel}$ with other tracers of ionised gas, we build a picture of
this dense plasma. We adopt a collisional ionisation model, so the analysis is
not tied to a specific energisation mechanism. We find that the dense plasma is
concentrated in a disk that is $130\;{\rm pc}$ thick, and makes a significant
contribution to radio pulsar dispersion measures in the inner Galactic plane.
The strength of the far infrared N$^+$ emission requires high temperatures in
the plasma, with $T \simeq 19{,}000\,{\rm K}$ indicated both by the ratio of
N$^+$ to C$^+$, and by the ratio of N$^+$ to microwave bremsstrahlung in the
inner Galactic plane. This parallels the situation at high Galactic latitudes,
where strong optical emission is observed from N$^+$ (and S$^+$), relative to
both H$\alpha$ and microwave bremsstrahlung, and suggests a common origin. If
so, the same gas provides a natural explanation for the extreme radio-wave
scattering phenomena that are sometimes observed in pulsars and quasars. We
therefore propose a new picture of the warm ionised medium as seen in emission,
in which the plasma is dense, hot, and localised in numerous structures of size
$\sim 10^2\,{\rm AU}$ that are clustered around stars.
