The character of the warm ionised medium.

$\textit{Herschel}$ 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.