LAUSR

We examine the critical viscous mode of the Taylor–Couette strato-rotational instability, concentrating on cases where the buoyancy frequency $N$ and the inner cylinder rotation rate $\unicode[STIX]{x1D6FA}_{in}$ are comparable, giving a detailed account for $N=\unicode[STIX]{x1D6FA}_{in}$. The ratio of the outer to the inner cylinder rotation rates $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}_{out}/\unicode[STIX]{x1D6FA}_{in}$ and the ratio of the inner to the outer cylinder radius $\unicode[STIX]{x1D702}=r_{in}/r_{out}$ satisfy $0<\unicode[STIX]{x1D707}<1$ and $0<\unicode[STIX]{x1D702}<1$. We find considerable variation in the structure of the mode, and the critical Reynolds number $Re_{c}$ at which the flow becomes unstable. For $N=\unicode[STIX]{x1D6FA}_{in}$, we classify different regions of the $\unicode[STIX]{x1D702}\unicode[STIX]{x1D707}$-plane by the critical viscous mode of each region. We find that there is a triple point in the $\unicode[STIX]{x1D702}\unicode[STIX]{x1D707}$-plane where three different viscous modes all onset at the same Reynolds number. We also find a discontinuous change in $Re_{c}$ along a curve in the $\unicode[STIX]{x1D702}\unicode[STIX]{x1D707}$-plane, on one side of which exist closed unstable domains where the flow can restabilise when the Reynolds number is increased. A new form of viscous instability occurring for wide gaps has been detected. We show for the first time that there is a region of the parameter space for which the critical viscous mode at the onset of instability corresponds to the inviscid radiative instability of Le Dizès & Riedinger (J. Fluid Mech., vol. 660, 2010, pp. 147–161). Focusing on small-to-moderate wavenumbers, we demonstrate that the viscous and inviscid systems are not always correlated. We explore which viscous modes relate to inviscid modes and which do not. For asymptotically large vertical wavenumbers, we have extended the inviscid analysis of Park & Billant (J. Fluid Mech., vol. 725, 2013, pp. 262–280) to cover the cases where $N$ and $\unicode[STIX]{x1D6FA}_{in}$ are comparable.