Dynamics of a trapped vortex in rotating convection

We consider axisymmetric rotating convection in a cylindrical domain, focusing on the eye that can form at the center of a cyclonic vortex. Upon increasing the thermal forcing we observe that the system undergoes a Hopf bifurcation from a state with a steady eye to one in which the eye oscillates. For an aspect ratio, Ekman number, and Prandtl number of 0.1 we find that the critical Reynolds number at which this transition occurs is 398.

We examine the nature of the oscillations and propose that the behavior results from an inertial wave trapped in the eye, with the frequencies falling within the expected range for inertial waves, and the oscillations displaying clear similarities to a standing inertial wave in a cylinder. We also examine the effect of Ekman number on the oscillation, finding that there is an upper limit beyond which oscillations do not occur.

Figure 1

Sketch of the eye, eyewall, and lower boundary layer.

Figure 2

Schematic of the flow domain.

Figure 3

Plots of ${\omega }_{\varphi }/r$ overlaid with meridional streamlines $\mathrm{\Psi }$ for a variety of Reynolds numbers, showing the formation of a steady eye that grows as forcing is increased. In all cases $\mathrm{Ek}=\mathrm{Pr}=\mathrm{\Lambda }=0.1$. Solid streamlines represent the bulk, clockwise circulation, while dashed streamlines indicate anticlockwise motion.

Figure 4

Plots of ${\omega }_{\varphi }/r$ overlaid with contours of angular momentum $\mathrm{\Gamma }$ (black) and streamlines (yellow) for a range of Reynolds numbers with $\mathrm{Ek}=\mathrm{Pr}=\mathrm{\Lambda }=0.1$.

Figure 5

Normalized total angular velocity for the flow (top) and averaged in the $z$ direction (bottom), $\mathrm{Re}=300$.

Figure 6

Plots showing how streamlines and ${\omega }_{\varphi }/r$ vary over one complete oscillation cycle ($T$) for $\mathrm{Ek}=\mathrm{Pr}=\mathrm{\Lambda }=0.1, \mathrm{Re}=400.$

Figure 7

Time series for the case $\mathrm{Re}=480$. Time is scaled by $H/U$.

Figure 8

Fourier transforms of the $u_z$ time series for different levels of forcing (Re).

Figure 9

Plots showing how streamlines and ${\omega }_{\varphi }/r$ vary over one complete oscillation cycle ($T$) for $\mathrm{Ek}=\mathrm{Pr}=\mathrm{\Lambda }=0.1, \mathrm{Re}=650.$

Figure 10

Bifurcation diagram for the oscillating eye. The dashed line is proportional to ${(\text{Re}-{\mathrm{Re}}_c)}^{1/2}$.

Figure 11

Regime diagram for eyes showing Re plotted against Ek for the case $\mathrm{\Lambda }=\mathrm{Pr}=0.1$.

Figure 12

Variation of time and $z$ average of $\left(u_{\varphi }/r+\mathrm{\Omega }\right)/\mathrm{\Omega }$ in the eye, $\mathrm{Re}=480$.

Figure 13

${\mathrm{Ro}}_l$ in the eye (top) and bulk (bottom) for the case $\mathrm{Re}=350$.

Figure 14

$\varpi /\mathrm{\Omega }$ for the oscillations vs Re, along with selected values for a cylinder and frustum for comparison.

Figure 15

Figure 11 from Chen et al. [17] showing oscillations in a simulated Typhoon Hagupit. Color shows the time-height variation of water vapor convection while the black line shows the maximum wind speed in $\mathrm{m}{\mathrm{s}}^{-1}$ (scale on right). Reproduced with permission.

Figure 16

Plots showing how contours of $\mathrm{\Gamma }$ evolve in the eye over one cycle ($T$) for $\mathrm{Ek}=\mathrm{Pr}=\mathrm{\Lambda }=0.1, \mathrm{Re}=400$. Color = ${\omega }_{\varphi }/r$.

Figure 17

Plots showing how contours of $\mathrm{\Gamma }$ evolve in the eye over one cycle ($T$) for $\mathrm{Ek}=\mathrm{Pr}=\mathrm{\Lambda }=0.1, \mathrm{Re}=600$. Color = ${\omega }_{\varphi }/r$.

Figure 18

Plots showing how contours of $\mathrm{\Gamma }$ (top) and streamlines (bottom) evolve over one cycle (of period $T$) for a standing inertial wave in a cylinder.