ANEXO C: CHAEA

Cyclone-anticyclone asymmetry has long been observed in atmospheric data [Venn, 1887], laboratory experiments [Perret et al., 2006] and numerical sim- ulations [Polvani et al., 1994a, Theiss, 2004, Stegner and Dritschel, 2000, and references therein]. Polvani et al. [1994a] found a skewness (as defined in equa-

Figure 3.7: Evolution of Π (top row), ˜h (second row), δ/(2Ω) (third row) and γ/(2Ω)2 _{(bottom row) fields for case 2, shown at times t = 0, 5, 10. Only every} 8th contour of the PV field is plotted. The contour interval is 4π for Π, 0.001 for ˜

Figure 3.8: Evolution of Π (top row), ˜h (second row), δ/(2Ω) (third row) and γ/(2Ω)2 _{(bottom row) fields for case 3, shown at times t = 0, 5, 10. Only every} 5th contour of the PV field is plotted. The contour interval is 5π/2 for Π, 0.05 for ˜h, 0.01 for δ/(2Ω) and 0.1 forγ/(2Ω)2_.

Figure 3.9: Evolution of Π (top row), ˜h (second row), δ/(2Ω) (third row) and γ/(2Ω)2 _{(bottom row) fields for case 4, shown at times t = 0, 5, 10. Only every} 8th contour of the PV field is plotted. The contour interval is 4π for Π, 0.02 for ˜

t=0 t=5 t=10

Figure 3.10: Evolution of Π for case 1, shown from the north pole (top row), from the equator (middle row) and from the south pole (bottom row). Red indicates positive Π, blue indicates negative Π.

t=0 t=5 t=10

Figure 3.11: Evolution of Π for case 2, shown from the north pole (top row), from the equator (middle row) and from the south pole (bottom row). Red indicates positive Π, blue indicates negative Π.

t=0 t=5 t=10

Figure 3.12: Evolution of Π for case 3, shown from the north pole (top row), from the equator (middle row) and from the south pole (bottom row). Red indicates positive Π, blue indicates negative Π.

t=0 t=5 t=10

Figure 3.13: Evolution of Π for case 4, shown from the north pole (top row), from the equator (middle row) and from the south pole (bottom row). Red indicates positive Π, blue indicates negative Π.

tion 3.4) in favour of anticyclonic vorticity with the asymmetry becoming more pronounced with increasing Froude number. They also note the appearance of some large-scale cyclonic structures, thus demonstrating that, contrary to the suggestion of Cushman-Roisin and Tang [1990], cyclones do not just fall apart. In fact, as demonstrated below, cyclones, while fewer, tend to be much stronger than anticyclones.

Following Polvani et al. [1994a], we measure the cyclone-anticyclone asym- metry by computing the skewness

S(q)_≡ hq 3_i

hq2_i3/2, (3.4)

where _h._i denotes the spatial average, of various fields q. Polvani et al. [1994a] examined S(ζ), which is inappropriate here since ζ > 0 does not correspond to cyclonic circulation in both hemispheres. Instead, we examine S(˜h) and S(γ), both of which indicate cyclonic circulation forS >0 and anticyclonic circulation for S < 0. The time mean values over 20 _≤ t _≤ 40 of S(˜h) and S(γ) are tabulated in table 3.2. The first feature to note is that case 3 is the only case for which the average skewness is positive for either field. Case 1, as expected from its location in the Fr-Ro parameter space, exhibits the behaviour expected of a near-geostrophically balanced flow, in that the skewness stays close to, albeit less than, zero for the length of the run (not shown). In cases 2 and 4 asymmetry appears early on and levels out after about 10 days.

Another way to visualise the asymmetry is to examine the probability density function (PDF) of these fields. The depth field is shown in figure 3.14 (γ is qualitatively similar). As first noted by Venn (1887), departures from the mean surface pressure in atmospheric data do not form a Gaussian distribution. Here, in shallow water flows, the depth field plays the role of surface pressure through hydrostatic balance. Again, case 1 exhibits the behaviour expected of a near-

S(˜h) S(γ) (1) -0.19 -0.40 (2) -0.40 -3.14 (3) 0.08 0.51 (4) -0.26 -1.93

Table 3.2: Time averaged skewness of ˜handγover the last half of each simulation.

geostrophically balanced flow, in that its PDF is the most symmetric although even in this case there is a bias towards high pressure regions. The PDFs for cases 2 and 4 are significantly more skewed but both also exhibit a significant tail of low pressure values indicating that, although high pressure regions dominate, low pressure regions tend to be much more intense. This is the key observation made by Venn (1887) regarding surface pressure. This pattern is reversed in case 3. Looking at the final depth field for all four cases (figure 3.15), we see that, despite the skewness in favour of cyclonic circulation, the dominant feature in case 3 is a large, intense anticyclone which accounts for the PDF distribution in figure 3.14. This case of low Ro and high Fr may be the least representative of global atmospheric motion.

We now examine the asymmetry over the Fr-Ro parameter space. In figure 3.16 we plot the skewness S(˜h) against Fr and Ro. For the majority of the simulations the skewness is negative, indicating dominance of anticyclonic cir- culation. However, at low Rossby number and, to a lesser extent, high Froude number, some simulations have a strong positive skewness. As we saw above, the skewness does not reveal the whole picture and it is possible that the flows are in fact dominated by a few extreme vortices of the sign opposite to that which the skewness suggests. To investigate this we plot the ratio _|˜hmax/˜hmin| against Fr and Ro (see figure 3.17). This reveals that in the majority (57%) of simulations

Case 1 h~ -0.025 0.025 0 0.07 Case 2 h~ -0.02 0.02 0 0.07 Case 3 h~ -0.6 0.6 0 0.05 Case 4 h~ -0.3 0.3 0 0.04

Figure 3.14: Probability density functions of depth ˜h for the 4 cases. Dotted lines show the distribution at t = 0,10,20,30,40 and the solid line shows the time-mean distribution.

Case 1 Case 2

Case 3 Case 4

Figure 3.15: Depth field at t = 40 for all four cases, plotted with 20 contour intervals spanning 0 to ˜hmax.

Although 57% is on the margin of being significant (based on a rough estimate of the likely error in the 148 cases), this suggests that Venn’s observations of real data are consistent with shallow water turbulence over much of the Fr-Ro parameter space.