T1-P3 Evidence for Infragravity Wave-Tide Resonance in Deep Oceans

Evidence for Infragravity Wave-Tide Resonance in Deep Oceans Hiroko Sugioka, Yoshio Fukao (JAMSTEC), Toshihiko Kanazawa (ERI, Univ. of Tokyo) Corresponding to H.S. e-mail: [email protected]

Ocean tide refers to the oscillatory motion of seawater forced by the gravitational attraction of the Moon and Sun with periods of a half to a day and wavelengths of the semi-Pacific to Pacific scale. Ocean infragravity waves are sea-surface gravity waves with periods of several minutes and wavelengths of several dozen kilometres. We report the first evidence of the resonance between these two ubiquitous phenomena in deep oceans mutually very different in period and wavelength. The evidence comes from long-term, large-scale observations with arrays of broadband ocean-bottom seismometers located at depths of more than 4000 m in the Pacific Ocean. This observational evidence is substantiated by a theoretical argument that infragravity waves and the tide can resonantly couple and that such coupling occurs over unexpectedly wide areas of the Pacific Ocean. Through this resonant coupling, some of ocean tidal energy is transferred in deep oceans to infragravity wave energy.

Broadband ocean-bottom seismic observations Modulation of IG waves by ocean tide Figure 3 A theory of the resonance between IG waves and the ocean tide We analysed the continuous vertical-component records obtained from two broadband ocean-bottom Figure 3 shows the FP array records that are bandpass-filtered between 4 and 20 A theory was developed for the coupling of IG waves with the tide in a long-wave approximation. The wave field is assumed to be a seismometer (BBOBS; sensor: Guralp CMG-3T) arrays: one (1 Feb 2003 to 31 Jan 2004) with 7 stations (FP mHz; in this frequency range, the signal due to the IG waves is dominant. The IG FP2 random, homogeneous superposition of free IG waves of the form u(k)cos(wt-kx) , where k is the horizontal wavenumber vector array), deployed in French Polynesia (Suetsugu et al., 2005) and the other (1 Dec 2006 to 30 Nov 2007) wave trains show temporal variations on a semidiurnal to diurnal scale; however, (Hasselmann, 1963). This assumption is justified by observations made in the deep ocean. The angular frequency w is related to k as FP3 with 11 stations (PS array), deployed in the northern Philippine Sea (Shiobara et al., 2009). Figure 1 shows visual inspection does not reveal any unique periodicity in time coherent through the c=w/k=(gH)1/2 , where c is the frequency-independent propagation speed; g, the gravitational acceleration; and H, the water depth. the geographical distributions of the stations in these two arrays, which are mostly located at depths of array. In order to detect any semidiurnal or diurnal periodicity in the IG wave activity FP4 The tidal motion in the local coordinates can be expressed as Ucos(Wt-Kx) so that the co-phase line of the ocean tide with an angular 4000‒5000 m. Details of the observations are provided in the Methods section. Figure 2a shows two coherent across the stations, we divided the one-year record obtained from a frequency W moves in the direction given by the wavenumber vector K and with a speed defined by C=W/K. The fluid-dynamic equation FP5 examples of seafloor noise spectra, one obtained from the FP array and the other obtained from the PS station into approximately 30-day (2.8x106 s)-long segments with an overlap of of motion including the advection term has a solution of the form R. ksin(Wt-Kx)sin(wt-kx) showing the tide-modulated IG waves, array. It has been reported that the spectral peak appearing at around 10 mHz is a signature of the seafloor approximately 10 days (1.0x106 s). Each of the 30-day-long segments was FP6 where the amplification factor R is proportional to U/[(w+/-W)(c|k+/-K|)]. In a homogenous random wave field, there is always a disturbance caused by IG waves (Webb et al., 1991; Webb, 1998). This interpretation is supported by bandpass-filtered between 4 and 20 mHz to extract the IG signals. The FP7 wavenumber vector k such that (w+/-W)-c|k+/-K|=0 with which R undergoes resonant divergence, if c/C>=1/(1+0.5W/w). Figure 2b, which shows that the depth dependence of the peak frequency in the IG band can be attributed cross-spectrogram for a station pair i and j is a plot of the cross-spectral power |Qij to the hydrodynamic filtering of IG waves. (f)| against time. For ease of illustration, we define the stacked cross-spectrogram FP8 Comparison of the observations and theory th for the i station as a stack of the cross-spectrograms obtained for all pairs of the 18-Sep 19-Sep 20-Sep 21-Sep 22-Sep 23-Sep 24-Sep 25-Sep Figure 1 -10˚ Figure 5a th FP2 j station and the other stations in the same seismic array. Figure 4 shows the We calculated for the Pacific Ocean by using the global Figure 3: Example of the FP array records (18 to 25 Sep 2003) stacked cross-spectrograms so defined. The frequency fK1 (period: 23.93 h) of the bathymetry model of ETOPO2 (Smith & Sandwell, 1997) and FP3 bandpass-filtered between 4 and 20 mHz. The dominant signal in this band is the seafloor disturbance due to IG waves. The IG wave trains -20˚ principal solar tide K1, frequency fM2(period: 12.42 h) of the principal lunar tide M2, an ocean tidal model developed by Matsumoto et al.(2000). show temporal variations on a semi-diurnal to diurnal scale. The FP4 and its double frequency are indicated by arrows along the frequency axis of each theoretical tidal sea-level changes are shown below each record by We set the w value to the peak frequency in the IG wave band blue lines. FP8 FP5 stacked cross-spectrogram. The M2 signal is much stronger in the FP spectrograms of the seafloor noise at each observational site, as shown by FP7 -30˚ FP6 than in the PS spectrograms, whereas the intensity of the K1 signal is almost the the arrow in Figure 2a. Figures 5a and 5b show the maps of R same in the FP and PS spectrograms. The second harmonic of the M2 signal is 210˚ 220˚ 230˚ for the M2 and K1 tides, respectively. The area coloured in red considerably weaker than the first harmonic in both the FP and PS spectrograms. is essentially in resonant state, as defined by the above inequality , which may be approximated as c>=C. T09 Figure 4 30˚ T06 Figure 5 demonstrates how different the resonance in the T07 T12 T03 real ocean is from that in the uniform model ocean. In Figure T08 T13 T05 5, the white area is understood to be in a near-resonance T11 T02 state, where the precise value of is not calculable according 20˚ T01 to our first-order perturbation theory. We can see that 130˚ 140˚ 150˚ unexpectedly wide areas of the Pacific Ocean are in the Figure 5b Figure 1: Map of two broadband ocean-bottom -8000 -4000 0 4000 seismic arrays (left): French Polynesia (right, top) resonant or near-resonant state. A major part of the FP array Topography(m) and Northern Philippine Sea (right, bottom). is within the resonant area for the M2 tide, while a major part Figure 2a of the PS array is outside the resonant area; this would be the

−70 FP5 at depth of 4393 m −70 T05 at depth of 4866 m 2nd reason for the M2 signal being stronger in the FP array than in K1 M2 −80 −80 M2 −90 −90 the PS array (Figure 4). On the other hand, a major part of −100 −100 −110 −110 either array is within the resonant area for the K1 tide, −120 −120 Figure 2: (a) Seafloor noise spectra at station FP5 for the /Hz) in d B /Hz) in d B 4 −130 4 −130 period 14 Jan 2003 and 14 Jan 2004 and those at T05 explaining that the K1 signal intensities in the FP and PS arrays /s /s 2 2

m −140 m −140 for the period 11 Nov 2006 to 11 Nov 2007. Each of the are not very different (Figure 4). −150 −150 power spectra is calculated using the standard procedure PSD( PSD( −160 −160 followed in seismology (Peterson, 1993), for a −170 −170 record-section length of 4096 s and sampling rate of 100 −180 −180 Hz. The 10 spectral lines with the lowest noise amplitudes −190 −190 0.01 0.1 1 10 0.01 0.1 1 10 in the frequency range 0.02‒0.05 Hz are blackened to Discussion Frequency (Hz) Frequency (Hz) show the noise spectra typical to quiet days. The spectral peak around 0.01 Hz (indicated by the arrow) represents We have shown that free IG waves in the deep ocean are the disturbance due to infragravity (IG) waves. The two red curves indicate the New Low Noise Model (NLNM) and amplified by the resonance with ocean tides and that such 0.03 Figure 2b New High Noise Model (NHNM) of Peterson (1993). (b) 2nd Plot of the peak frequency F (in Hz) in the infragravity K1 M2 resonance can occur over wide areas of the Pacific Ocean. M2 0.02 band against seafloor depth H (in m). Data are obtained This implies that ocean tides dissipate a certain amount of from all the BBOBS stations shown in Figure 4. The plots are approximated by the relation F=(2H)1/2. This relation energy through resonant coupling with IG waves. Our findings 0.01 is understood as kH~1.4, which can be expected if the Frequency (Hz) hydrodynamic filtering of IG waves (Webb et al., 1991) is suggest the existence of a possible driving force for IG waves responsible for the peak. Figure 4: Stacked cross-spectrogram for each of the stations of the (a) FP and (b) PS arrays. The stacked cross-spectrogram for a station is a stack of the in deep oceans and a possible dissipation mechanism of Figure 5: Maps of amplification factor R for the infragravity wave coupled with the ocean tide 0.00 cross-spectrograms for all the pairs of that station and the other stations in the same seismic array. Each record is normalized by its maximum amplitude. The arrows along 1000 2000 3000 4000 5000 6000 is a measure of how large the coupled-wave amplitude is relative to the amplitude of the the frequency axis indicate the principal solar tide K1, the principal lunar tide M2, and the second harmonic of M2, respectively, in the order of increasing frequency. ocean tidal energy in deep oceans (Munk & Macdonald, 1960; Depth (m) undisturbed wave. The area that is practically in a resonant state is coloured in red. The whitish Munk & Wunsch, 1997). area may be understood to be in a near-resonant state. The tidal co-phase lines are contoured. White dots indicate the amphidromic points. Map (a) for the M2 tide and (b) for the K1 tide.