The stable atmospheric boundary layer (SABL) is broadly classified into two subtypes: the nocturnal SABL, driven primarily by surface cooling, and the long-lived SABL, in which ambient stratification coexists with surface cooling and persists well beyond a single diurnal cycle. Despite its prevalence in polar regions and over open oceans, the long-lived SABL has received comparatively little attention, and a systematic direct numerical simulation (DNS) study distinguishing it from the nocturnal SABL has not previously been reported. Here we present DNS of long-lived SABLs at $Re_D=900$, contrasting weak and strong ambient stratification cases against a nocturnal SABL. A striking feature of the long-lived SABLs is the emergence of a multi-layered thermal structure -- including an intermediate layer of reduced static stability capped by a buoyancy inversion -- driven not by enhanced turbulent mixing, but by a stratification-induced reorganisation of the buoyancy field. Budget analyses identify a turbulent potential energy-centred transport mechanism absent in nocturnal SABLs, lending support to total-energy-based closure approaches. Assessment of an extended Monin--Obukhov similarity theory incorporating a composite length scale $L_*$ shows strong collapse of dimensionless gradients of velocity $ϕ_m$ and potential temperature $ϕ_h$, and a new similarity function for $ϕ_h$ is proposed. Combining DNS with linear stability analysis, we construct a regime map delineating linearly stable, weakly stable, and very stable regimes within a two-parameter dimensionless space, demonstrating the inherently multi-parameter nature of long-lived SABLs and the limitations of single-parameter subgrid-scale parameterisations.

33 pages, 18 figures