Formation of columnar baroclinic vortices in thermally stratified nonlinear spin-up

We investigate the mechanisms that affect the formation of columnar vortices for spin-up in a cylinder where the temperatures at the horizontal walls are prescribed. Numerical results from the three-dimensional Navier–Stokes equations show that a short-lived instability, suppressed by the combined effect of rotation and stratification, generates small temperature variations in the azimuthal direction. Temperature-gradient anomalies produce vorticity, and these vortices stir the fluid at the interface of the central vortex core thus reinforcing the temperature gradients. For sufficiently strong temperature gradients, the central vortex core breaks up into several columnar vortices. It is found, in particular, that small aspect ratios (height over radius of the cylindrical fluid layer) $\Gamma = 1, 2$ tend to inhibit the instability, while larger ones, $\Gamma = 3. 3$, have the opposite effect. The main source of instability is the baroclinic vorticity production and not the presence of a solid sidewall since, counter-intuitively, the flow is more unstable for a free-slip boundary than for a no-slip one. Finally the effect of the temperature boundary conditions (isothermal versus adiabatic) on the horizontal boundaries has been investigated. The adiabatic boundaries help to preserve for longer times the sloping density interfaces that feed, with their potential energy, the baroclinic vorticity production; this results in more unstable flows.