Possible impact of equatorially trapped waves on the tropical cyclone drift

Abstract

The effect of equatorially trapped waves on the movement of tropical cyclones (TC) is studied numerically based on a two-dimensional barotropic model in a beta-plane approximation. According to recent studies, equatorially trapped waves contribute to the genesis of TCs. It is thus natural to assume that these waves affect also the movement of the TC. The effect of three types of equatorially trapped waves, namely Kelvin, Rossby, and n = 0 eastward inertio-Gravity (EIG) waves, on the TC trajectory is investigated with a focus on the sensitivity on some key physical parameters such as the wavenumber and wavespeed. Using a simple barotropic model forced by a prescribed baroclinic flow, the barotropic response to equatorially trapped waves is simulated for a period of 50 days, under various wave parameter configurations. This response is then used as a background flow where TC’s can evolve and propagate. TC-like flows are injected into this wavefield background at arbitrary times during the simulation, and the TC trajectories are tracked and recorded for 48h after the injection time. The resulting TC trajectory patterns with respect to the injection times and wave parameters appear to be stochastic and the mean paths and the associated standard deviations are calculated and reported here. The statistics are different for different wave types. Kelvin waves make shorter length of TC trajectories and small divergence of direction. On the contrary, Rossby waves cause rather dramatic changes in the TC path and yield longer trajectories. Meanwhile, TCs in EIG waves maintain fairly the same direction and typically have longer trajectories though less dramatic. A robustness test using a random forcing instead has also been conducted.

Appendix: Mean TC path and standard deviations for various tropical wave forcings

To undertake a more comprehensive investigation of the effect of the wave-induced background of the TC trajectory, we consider each one of the cases in Table 1 and vary the TC injection time $T_0$ from $T_0=10$ days to $T_0=30$ days with a uniform 2 day increment, yielding a total of 11 samples for each case. In Figs. 12 through 20, we report the mean path and the cone of trajectories represented by the mean path plus and minus one standard deviation, when $T_0$ is varied as just described, for each one of the test cases in Table 1. In all these panels, the blue-dashed curve labeled “ref” represents the zero-background reference trajectory, the red curve represents the statistical mean path and the thin black curves are the standard deviations. A fixed reference scale is used on all panels of each wave type to ease the comparison.

For each wave type, we consider six wavenumbers, grouped into three categories: planetary scale ($k=1,2$), upper synoptic scales ($k=4,5$), and lower synoptic scales ($k=8,10$), and three wave speeds, also representing three categories: very slow corresponding to a gravity wave speed $c_e=1$ $\text {m }\hbox {s}^{-1}$, slow corresponding to a gravity wave speed $c_e=5$ $\text {m }\hbox {s}^{-1}$ and moderate corresponding to $c_e=15$ $\text {m }\hbox {s}^{-1}$. We note that in the case of the Kelvin wave, the actual phase speed is the same as the imposed gravity wave speed ($c=c_e$) while in the other two cases the phase speed depends on both the gravity wave speed and the wavenumber. See Table 1.

From the plots in Figs. 12 through 20, we can see that both the mean path and the associated variance depend on the above wave parameters. Although there is no clear pattern, we can see that in the case of a Kelvin wave background that, apart from a few exceptions, the mean path remains fairly close to the reference path, especially in terms of the direction of propagation and that the variance is relatively small. In all cases, the average total distance travelled (as measured by the arc length of the mean path) is consistently smaller than its reference counterpart. The few exceptions are associated with the cases of slow phase speeds and synoptic scale wavenumbers, corresponding to $c=1,5$ $\text {m }\hbox {s}^{-1}$ and $k=4,5,8,10$ in Figs. 13 and 14 and display a remarkably smaller mean path distance and much larger variance. In retrospect, such cases are hardly realistic as Kelvin waves are typically observed to propagate at roughly 15 $\text {m }\hbox {s}^{-1}$ and have synoptic scale wavelength corresponding to wavenumbers in the range $k=4$ to $k=10$ (Kiladis et al. 2009). The cases of planetary scale wavenumbers $k=1,2$ and slow phase speeds $c=1,5$ $\text {m }\hbox {s}^{-1}$ can also be deemed to be realistic in the sense that they may be somewhat representative of the Madden Julian Oscillation type wave-disturbance which has comparable phase speeds and wavenumbers and whose zonal structure bears some resemblance to a Kelvin wave disturbance (Kiladis et al. 2009). The case of a Madden-Julian oscillation ought to be investigated on its own because of its importance in tropical atmospheric dynamics (Zhang 2005).

For the Rossby wave case in Figs. 15, 16, 17, the dependence is somewhat reversed compared to the Kelvin wave case. For the small wavenumber cases, $k=1,2$ in Fig. 15, there is a large variance in TC trajectories while the mean path itself seems to depend strongly on the phase speed. For $k=1$ and $c=-0.332$ $\text {m }\hbox {s}^{-1}$ ($c_e=1$ $\text {m }\hbox {s}^{-1}$), the TC path distribution is severely pulled to the west of the reference trajectory. With $k=1$ and $c=-1.637$ $\text {m }\hbox {s}^{-1}$ ($c_e=5$ $\text {m }\hbox {s}^{-1}$), we see a somehow similar pattern though the reference path now appears to run along the eastern boundary of the standard deviation cone of trajectories but with $k=1$ and $c=-4.743$ $\text {m }\hbox {s}^{-1}$ ($c_e=15$ $\text {m }\hbox {s}^{-1}$), the trajectory distribution shifts completely to the east side of the reference path. In all three cases, the mean distance traveled (the mean path arc length) appears to be significantly larger than the reference. For the cases corresponding to $k=2$ on the right panels, the effect is less dramatic. While we see a somewhat similar effect as in the case $k=1$ when $c=-0.329$ $\text {m }\hbox {s}^{-1}$ ($c_e=1$ $\text {m }\hbox {s}^{-1}$), the mean path direction is more or less the same as the reference for the two other cases. The variance seems to increase with increasing wavenumber. With the exception of the case $k=8$ and $c= -1.120$ $\text {m }\hbox {s}^{-1}$ on the bottom left of Fig. 17, for all the synoptic scale wavenumber cases, $k=4,5,8,10$, the effect of the Rossby-wave induced background on the TC trajectory is minimal. Because Rossby waves are mainly planetary scale in nature and appear to move at speeds around − 4 $\text {m }\hbox {s}^{-1}$ and slower (Kiladis et al. 2009), only the cases in Fig. 15 maybe deemed realistic, as such we can conjecture that in reality Rossby waves may have a stronger impact on TC trajectories compared to Kelvin waves but in both cases, the effect may depend strongly on the wave parameters such as the phase, the phase speed, and the wavenumber.

Somewhat similar to the Kelvin wave case, only Figs. 19 and 20, corresponding on the synoptic scales ($k=4,5,8,10$), appear to show a significant impact of the MRG wave-induced background on the TC path. The planetary scale cases in Fig. 18 show very little difference between the TC trajectories corresponding to an MRG wave and the reference case, as suggested by the plots of the mean path and associated small variances. Also, as in the Kelvin wave case, the trajectory direction remains stable under the influence of the MRG-induced background while the average distance travelled is however slightly larger than the reference. This is in contrast to the Kelvin wave case which shows a systematically smaller mean distance. From observational point of view, MRG waves move at roughly 20 $\text {m }\hbox {s}^{-1}$ and mostly peak at the synoptic scale. It is thus fair to stress that the cases at the bottom in Figs. 19 and 20 are more representative of the real world and may conjecture that MRG waves have in general a small impact on the TC trajectory although some dependance on wave parameters such as phase speed, wavenumber and wave phase are to be expected.

Fig. 12

Mean path and its deviations of TC trajectories in Kelvin wave with (from left to right, top to bottom) $k=1, c=1$ m/s, $k=2, c=1$ m/s, $k=1, c=5$ m/s, $k=2, c=5$ m/s, $k=1, c=15$ m/s, $k=2, c=15$ m/s. In each cases The statistics were taking for TC varying injection times $T_0=10,12,14,\ldots ,30$ days

Fig. 13

Same as Fig. 12 but for $k=$4, 5

Fig. 14

Same as Fig. 12 but for $k=$8, 10

Fig. 15

Same as Fig. 12 but for the Rossby wave

Fig. 16

Same as Fig. 15 but for $k=4$, 5

Fig. 17

Same as Fig. 15 but for $k=$ $k=8$, 10

Fig. 18

Same as Fig. 12 but for the MRG wave

Fig. 19

Same as Fig. 18 but for $k=4$, 5

Fig. 20

Same as Fig. 18 but for $k=8$, 10