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.