Formation of eyes in large-scale cyclonic vortices

Abstract

We present numerical simulations of steady, laminar, axisymmetric convection of a Boussinesq fluid in a shallow, rotating, cylindrical domain. The flow is driven by an imposed vertical heat flux and shaped by the background rotation of the domain. The geometry is inspired by that of tropical cyclones and the global flow pattern consists of a shallow, swirling vortex combined with a poloidal flow in the r − z plane which is predominantly inward near the bottom boundary and outward along the upper surface. Our numerical experiments confirm that, as suggested by [1], an eye forms at the centre of the vortex which is reminiscent of that seen in a tropical cyclone and is characterised by a local reversal in the direction of the poloidal flow. We establish scaling laws for the flow and map out the conditions under which an eye will, or will not, form. We show that, to leading order, the velocity scales with V = (αgβ) 1/2 H, where g is gravity, α the expansion coefficient, β the background temperature gradient, and H is the depth of the domain. We also show that the two most important parameters controlling the flow are Re = V H/ν and Ro = V / (ΩH), where Ω is the background rotation rate and ν the viscosity. The Prandtl number and aspect ratio also play an important, if secondary, role. Finally, and most importantly, we establish the criteria required for eye formation. These consist of a lower bound on Re, upper and lower bounds on Ro, and an upper bound on Ekman number.

Dynamics of a trapped vortex in rotating convection

“…This is because oscillations are viscously damped. Increasing Ek further we found that no eye formed for Ek > 0.23, consistent with [19]. For Ek less that 0.1, we find that the critical Reynolds number for oscillations falls dramatically as viscous effects reduce.…”

“…They found that the flow was largely dependent on Reynolds number (and hence the temperature difference between the top and bottom of the vortex), with only a slight dependence on Prandtl number. The dependencies on aspect ratio, Ekman number (rotation rate) and Prandtl number (thermal diffusivity) were explored further in a follow up paper [19]. A detailed discussion of the dynamics of the tropical cyclone boundary layer, including how it may be swept into the eyewall, is given by Smith and Montgomery [20].…”

On the inertial wave activity during spin-down in a rapidly rotating penny shaped cylinder: a reduced model

“…Possible applications of this study include tropical cyclones in the atmosphere, which are characterised by an aspect ratio close to 10 (approximately 10 km in height and a few 100 km in radius) as well as a moderate turbulent Ekman number, owing to the fact they develop close to the equator (see also Oruba et al 2018).…”

Inertial wave activity during spin-down in a rapidly rotating penny shaped cylinder

“…2007; Fischer, Rogers & Reasor 2020), a feature that is often observed prior to rapid intensifications. In actual tropical cyclones one can estimate the Ekman number in the range (Oruba, Davidson & Dormy 2018), which is small enough to motivate the asymptotic study presented here. It should be stressed, however, that in actual tropical cyclones, nonlinear effects cannot be a priori neglected and will thus modify the wave dynamics.…”