where \(\omega\) is frequency, and \(k\) and \(l\) are the zonal and meridional wavenumbers, respectively. \(\beta\) is the gradient of the Coriolis parameter at specific latitudes, and we take \(\beta\) at 10.5°N to be 2.25 × 10-11 m-1s-1. Note that, in Eq. (1), the Rossby radius is assumed to be much larger than the wavelength. The comparison was performed under the assumption that the TIW and TIW-induced BTRW have the same frequency and zonal wavenumber (Farrar, 2011).
The frequencies and zonal wavenumbers of TIW were estimated to be 1.8 × 10-6 s-1 < \(\omega\)< 2.9 × 10-6 s-1 (periods of 25−40 days) and −6.4 × 10-6 m-1< \(k\) < −2.9 × 10-6m-1 (zonal wavelengths of 9°–20° of longitude) based on the two-dimensional PSDs of Farrar SSH (see Figure 2a). The meridional wavenumbers of BTRW were calculated by substituting the frequencies and zonal wavenumbers of TIW into Eq. (1). The theoretically possible range of frequencies and wavenumbers of TIW-induced BTRW appears in the wavenumber space to be the purple region in Figure 4. On the other hand, zonal and meridional wavenumbers are also calculated using the first-mode CEOF phases as follows. The zonal wavenumbers of Ubt and Vbt by using phases at two points (10.5°N, 126.3°W and 10.5°N, 136.3°W) are estimated to be −4.18 × 10-6 m-1 and −4.31 × 10-6 m-1 and the meridional wavenumbers of Ubt and Vbt by using phases at the two points (15.5°N, 131.3°W and 5.5°N, 131.3°W) are estimated to be −5.06 × 10-6 m-1 and −4.03 × 10-6 m-1. Note that the co-phase line is shown as the black lines in the CEOF phase maps (Figures 3e and 3f). The estimated wavenumbers based on Ubt and Vbt, are marked by blue and pink small circles in the wavenumber space (Figure 4). It is quite encouraging that they fall within the possible range of frequency and wavenumber estimated earlier, supporting the fact that the first CEOF mode is quite compatible with the TIW-induced BTRW. The direction of group velocity corresponding to the estimated possible wave frequencies and wavenumbers is northward (Figure 4). This result suggests that our in-situ near-bottom current measurements enable fluctuation due to the northward propagation of energy of TIW-induced BTRW estimated by CEOF analysis of GLORYS12V1 results.
The non-significant coherence between our in-situ near-bottom current measurements and Farrar SSH along 10°N indicates that the barotropic signals in SSH near 10°N is not correlated with the in-situ near-bottom current. This zonally non-correlated band could be observational evidence to support the speculation of Farrar (2011) that the barotropic signal in SSH is distorted by the superposition of TIWs and TIW-induced BTRWs.