Physical mechanisms of the linear stabilization of convection by rotation

Despite the well-known limitations of linear stability theory in describing nonlinear and turbulent flows, it has been found to accurately capture the transitions between certain nonlinear flow behavior. Specifically, the transition in heat flux scaling in rotating convective flows can be well predicted by applying a linear stability analysis to simple profiles of a convective boundary layer. This fact motivates the present study of the linear mechanisms involved in the stability properties of simple convective setups subject to rotation. We look at an idealized two-layer setup and gradually add complexity by including rotation, a bounded domain, and viscosity. The two-layer setup has the advantage of allowing for the use of wave interaction theory, traditionally applied to understand stratified and homogeneous shear flow instabilities, in order to quantify the various physical mechanisms leading to the growth of convective instabilities. We quantitatively show that the physical mechanisms involved in the stabilization of convection by rotation take two different forms acting within the stratified interfacial region, and in the homogeneous mixed layers. The latter of these we associate with the tendency of a rotating flow to develop Taylor columns (TCs). This TC mechanism can lead to both a stabilization or destabilization of the instability and varies depending on the parameters of the problem. A simple criterion is found for classifying the influence of these physical mechanisms.

Figure 1

Sketch of the relevant elements in the two-layer convection setup that is used.

Figure 2

Schematic of the growth of a convectively unstable interface in a nonrotating frame. The red color is chosen to highlight the relationship between interfacial vorticity and vertical velocity at the interface; they are $\pi /2$ radians out of phase such that upwards (downwards) velocities are present in crests (troughs) of the interface displacement, inducing a growth of the interface.

Figure 3

(a) Plot of the dispersion relation for inviscid, unbounded, rotating convection in the two-layer setup. (b) The effects of including a boundary at a dimensionless distance of $H_{*}\equiv H/L_R$ from the interface. The different colored solid curves in (b) represent growth rates at differing values of $H_{*}$. The dashed curves in both panels are the rotating and nonrotating scalings.

Figure 4

Illustration of the mechanisms acting in rotating convection that lead to increased vertical velocities at the interface, as well as the misalignment of the interfacial vorticity with the ${\vec{k}}_{\perp }$ direction of baroclinic vorticity production. For brevity, only the interface and upper mixed layer are shown. Colors are used to group the vorticity sources with their associated velocities.

Figure 5

Growth rates in bounded rotating convection using an alternate nondimensionalization with $H$ as the length scale. The different colored curves represent growth rates at differing values of $\mathrm{\Pi }\equiv \left|\mathrm{\Delta }B\right|/\left(2f^{2}H\right)=(0.100,0.316,1.00,3.16,10.0)$. The gray region corresponds to conditions where the Taylor-column (TC) effect leads to a stabilization of convection, with destabilized convection at larger $\tilde{k}H$ outside.

Figure 6

Plot of the dispersion relation for viscous, nonrotating convection in the two-layer setup. Dimensional parameters are chosen to match the experiments of Davies-Wykes and Dalziel [37], namely, $\left|\mathrm{\Delta }B\right|=0.0137\mathrm{m}{\mathrm{s}}^{-2}$, $\nu =1.0\times {10}^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. The two asymptotic scalings corresponding to inviscid buoyant growth [i.e., as in (18)], and the viscous damping are shown as the indicated dashed lines.

Figure 7

Plot of the dispersion relation for viscous, rotating convection in the two-layer setup. (a) and (b) correspond to rotation rates of $f=0.3,10{\mathrm{s}}^{-1}$, respectively. All other dimensional parameters are chosen as in Fig. 6. The three asymptotic scalings corresponding to inviscid buoyant growth [i.e., as in (18)], the viscous roll-off, and rotation-dominated convection are shown as the indicated dashed lines.

Figure 8

Sketch of the heuristic interpretation of the interfacial mechanism of stabilization by rotation. Labels (a), (b), and (c) refer to various aspects of the sketch that are referenced in the text, and colors indicate the alterations that are present when rotation is included.