Model for the cyclonic bias of convective vortices in a rotating system

In convective fluids, small vortices develop between neighboring convective cells. Familiar in the atmosphere in the form of dust devils and water spouts, these convective vortices have been seen in simulations of oceanic convection where the vortices exhibit an unexplained bias towards cyclonic rotation despite having a large Rossby number. Here we use large eddy simulations (LES) of an idealized oceanic convective mixed layer and vary independently the Coriolis acceleration and the surface buoyancy flux to investigate the development of the cyclonic bias of convective vortices. While large convective vortices are biased for sufficiently large values of the Coriolis parameter, small convective vortices do not exhibit a clear bias. Using Lagrangian particles, we find that the large convective vortices develop through the merger of many small convective vortices. We propose a statistical theory to predict the cyclonic bias of large convective vortices composed of many small unbiased convective vortices and test the theory using LES results. We apply the theory to typical convective conditions and find that convective vortices in upper ocean mixed layers are expected to exhibit a bias, while convective vortices in the terrestrial and Martian atmospheres are expected to be largely unbiased.

Figure 1

Horizontal cross sections at $t=16.5$ h of the vertical velocity at $z=-1\mathrm{m}$ (a), vertical velocity at $z=-10$ m (b), pressure at $z=-1$ m (c), and a vertical cross section of pressure isosurfaces for ${\delta }_p<3\times {\sigma }_p$ Pa (d) in a simulation with $B_0=-4.24\times {10}^{-8}{\mathrm{m}}^2/{\mathrm{s}}^3$ and $f={10}^{-4}{\mathrm{s}}^{-1}$. Circles highlight locations of the largest four convective vortices [(a)–(c)] and the dashed line indicates the bottom of the mixed layer (d).

Figure 2

(a) Horizontal cross sections at time $t=16.5$ h of the vertical vorticity averaged over the top 5 m in a simulation with $B_0=-4.24\times {10}^{-8}{\mathrm{m}}^2/{\mathrm{s}}^3$ and $f={10}^{-4}{\mathrm{s}}^{-1}$. (b) Probability density function of vertical vorticity at $z=0$ for points identified as large (solid) and small (dashed) convective vortices for $B_0=-4.24\times {10}^{-8}{\mathrm{m}}^2/{\mathrm{s}}^3$ and $f={10}^{-4}{\mathrm{s}}^{-1}$ (red) and $f={10}^{-6}{\mathrm{s}}^{-1}$ (blue).

Figure 3

Horizontal slices of pressure at $z=-1$ m with surface particle position (magenta) superimposed for $t=5$ h [(a) and (c)] and $t=5.5$ h [(b) and (d)]. Panels (c) and (d) show a zoomed in section of (a) and (b) containing the large convective vortex.

Figure 4

Horizontal slice at $z=-1$ m of vertical vorticity when $t=0$ h (a) and $t=0.8$ h (b) and of vertical velocity when $t=0.8$ h (c) with boundaries of basins of attraction superimposed (black dashed line) for $\xi =0.025{\mathrm{m}}^2/\mathrm{s}$ and $f=2\times {10}^{-5}{\mathrm{s}}^{-1}$.

Figure 5

Horizontal slices of vertical vorticity at $z=-1$ m with the position of four surface particles (A, B, C, and D) superimposed in a simulation with $\xi =0.025{\mathrm{m}}^2/\mathrm{s}$ and $f=2\times {10}^{-5}{\mathrm{s}}^{-1}$ at $t=10$ min, (a), $t=20$ min, (b), $t=30$ min, (c) and $t=40$ min (d). The sequence shows the interaction and merging of four small vortices of opposing signs into one larger cyclonic vortex. In (c), particles C and D are at the same location and in (d) particles A, B, C, and D are all colocated.

Figure 6

(a) Median (solid) and 25th and 75th percentile (upper and lower dashed) particle vorticity when $\xi =0.01{\mathrm{m}}^2/\mathrm{s}$ (black) and $\xi =0.025{\mathrm{m}}^2/\mathrm{s}$ (red) for $f=2\times {10}^{-5}{\mathrm{s}}^{-1}$. (b) The predicted proportion of cyclones versus $f$ for different values of $\xi$ (lines) and observed proportion of cyclones in simulations (points).

Figure 7

Predicted probability contours (colored lines) and ${\mathrm{Ro}}^{*}=1$ line (black dashed) for constant $H$. We include ${\mathrm{Ro}}^{*}$ and the observed bias, $b$, for simulations (points).

Figure 8

Predicted probability contours (colored lines) and ${\mathrm{Ro}}^{*}=1$ line (black dashed) for constant $f$. We highlight the approximate parameter spaces for convective vortex regimes.

Figure 9

Probability density function of vertical vorticity at $z=0$ for points identified as convective vortices [(a) and (c)] and all points in the domain [(b) and (d)] as the domain size [(a) and (b)] and resolution [(c) and (d)] are varied.

Figure 10

Three-dimensional visualization of small (a) and large (b) convective vortices in a simulation with $B_0=-4.24\times {10}^{-8}{\mathrm{m}}^2/{\mathrm{s}}^3$ and $f={10}^{-4}{\mathrm{s}}^{-1}$. (a) The contour of perturbation pressure at $\delta p={\sigma }_p$ up to a depth $z=-7$ m colored by vertical vorticity with a horizontal pressure slice overlain. (b) The contour of perturbation pressure at ${\delta }_p=5\times {\sigma }_p$ colored by vertical vorticity for the large vortex highlighted in the white box in (a).

Figure 11

PDF of initial mean basin vorticity for $\xi =0.025{\mathrm{m}}^2/\mathrm{s}$ and Gaussian with $\mu =0,\sigma =2.4\times {10}^{-5}{\mathrm{s}}^{-1}$ (black dashed line).

Figure 12

Scaled probability density function of vertical vorticity in large-scale upwelling regions for simulations with $f={10}^{-4}{\mathrm{s}}^{-1}$ with a Gaussian approximation to the scaled curve (dashed line) at $z=-1$ m.