Evidence for Infragravity Wave-Tide Resonance in Deep Oceans

ARTICLE

Received 29 Apr 2010 | Accepted 6 Sep 2010 | Published 5 Oct 2010 DOI: 10.1038/ncomms1083 Evidence for infragravity wave-tide resonance in deep oceans

Hiroko Sugioka1 , Yoshio Fukao1 & Toshihiko Kanazawa2

Ocean tides are the oscillatory motions 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-Pacifi c to Pacifi c scale. Ocean infragravity (IG) waves are sea-surface gravity waves with periods of several minutes and wavelengths of several dozen kilometres. Here we report the fi rst evidence of the resonance between these two ubiquitous phenomena, mutually very different in period and wavelength, in deep oceans. The evidence comes from long-term, large-scale observations with arrays of broadband ocean-bottom seismometers located at depths of more than 4,000 m in the Pacifi c Ocean. This observational evidence is substantiated by a theoretical argument that IG waves and the tide can resonantly couple and that such coupling occurs over unexpectedly wide areas of the Pacifi c Ocean. Through this resonant coupling, some of ocean tidal energy is transferred in deep oceans to IG wave energy.

1 Japan Agency for Marine-Earth Science and Technology, Institute for Research on Earth Evolution, 2-15 Natsushima, Yokosuka, K anagawa 237-0061 , Japan . 2 Earthquake Research Institute, University of Tokyo , 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan . Correspondence and requests for materials should be addressed to H.S. (email: hik [email protected]) .

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nfragravity (IG) waves are surface waves that are observed in In order to detect any semidiurnal or diurnal periodicity in the IG deep 1 and coastal2 oceans. IG waves are strongest near the shore- wave activity coherent across the stations, we divided the 1-year record line3 , where they cause harbour oscillations and near-shore proc- obtained from a station into ~ 30-day (2.8 × 10 6 s)-long segments I 4 6 esses such as sediment transport . Disturbances due to IG waves with an overlap of ~ 10 days (1.0 × 10 s). Each of the 30-day-long make it diffi cult to interpret long-period seismograms at the ocean segments was bandpass fi ltered between 4 and 20 mHz to extract the bottom5,6 . Forcing of IG waves is believed to be a major source of IG signals. Th e negative IG signals were reversed in sign in order to Earth’ s background free oscillations (also known as Earth’ s hum) 7 – 14 . obtain an absolute-value time series, which was used as a measure of IG waves are mainly generated through nonlinear interactions the IG wave activity. Th e persistency of temporal periodicity among between high-frequency (period: 20 – 5 s) wind waves 15,16 . Th ere diff erent stations in the same sampling period was examined by are a few reports on the tidal modulation of IG waves in coastal normalizing the maximum amplitude of each 30-days long record 17 – 19 oceans (water depth: 10 – 30 m) and that on the 1,000-m-deep and by calculating a cross-spectrum Q ij ( f ) between stations i and shelf6 , where the IG wave energy is reduced at low tide. Th e sug- j ( ≠ i ). Th e temporal persistence of the periodicity was examined gested mechanisms for tidal modulation emphasize the infl uence of by constructing cross-spectrograms. Th e cross-spectrogram for 19,20 the tide on the generation or decay of IG waves in shallow seas . a station pair i and j is a plot of the cross-spectral power | Q ij (f )| No tidal modulation has been reported for IG waves in open seas against time. For ease of illustration, we defi ne the stacked cross- or deep oceans. Here we present the fi rst evidence of the resonance spectrogram for the i th station as a stack of the cross-spectrograms between IG waves and the tide in deep oceans, and show that such obtained for all pairs of the i th station and the other stations in the resonance can occur over wide areas of the Pacifi c Ocean. same seismic array. Figure 4 shows the stacked cross-spectrograms f so defi ned. Th e frequency K1 (period: 23.93 h) of the principal solar Results tide K 1 , frequency 2 fM2 (period: 12.42 h) of the principal lunar tide Modulation of IG waves by ocean tide . We analysed the continuous M2 , and its double frequency 2 fM2 are indicated by arrows along vertical component records obtained from two broadband ocean- the frequency axis of each stacked cross-spectrogram. Note that the f f f bottom seismometer (BBOBS; sensor: Guralp CMG-3T) arrays: intervals between K1 and M2 and between M2 and 2 fM2 are not one (1 February 2003 – 31 January 2004) with 7 stations (FP equal to each other. Th e spectral peaks at these three frequencies array), deployed in French Polynesia21 and the other (1 December can be recognized in the spectrograms for almost all time periods

2006– 30 November 2007) with 11 stations (PS array), deployed and across all the stations. Th e M2 signal is much stronger in the FP in the northern Philippine Sea22 . Figure 1 shows the geographical spectrograms than in the PS spectrograms, whereas the intensity of distributions of the stations in these two arrays, which are mostly the K1 signal is almost the same in the FP and PS spectrograms. Th e located at depths of 4,000 – 5,000 m. Details of the observations are second harmonic of the M2 signal is considerably weaker than the provided in the Methods section. Figure 2a shows two examples of fi rst harmonic in both the FP and PS spectrograms. seafl oor noise spectra, one obtained from the FP array and the other obtained from the PS array. It has been reported that the spectral A theory of the resonance between IG waves and the ocean tide . peak appearing at ~ 10 mHz is a signature of the seafl oor disturbance A theory was developed for the coupling of IG waves with the tide caused by IG waves 1,5 . Th is interpretation is supported by Figure 2b, in a long-wave approximation. Details of this theory are provided in which shows that the depth dependence of the peak frequency in the the Methods section. Th e wave fi eld is assumed to be a random, homo- − IG band can be attributed to the hydrodynamic fi ltering of IG waves. geneous superposition of free IG waves of the form uko()cos( w tkx ) , = Figure 3 shows the FP array records that are bandpass-fi ltered where k is the horizontal wavenumber vector and kk|| (ref. 23). between 4 and 20 mHz; in this frequency range, the signal due to the Th is assumption is justifi ed by observations made in the deep IG waves is dominant. Th e IG wave trains show temporal variations ocean1 . Th e angular frequency ω is related to k as ckgH==w / , on a semidiurnal to diurnal scale; however, visual inspection does where c is the frequency-independent propagation speed; g , the not reveal any unique periodicity in time coherent through the array. gravitational acceleration; and H , the water depth. Th e tidal motion

abc –10°

FP2

T09 30° T06 FP3 –20° T07 T12 T03 T08 FP4 T13 T05 T11 FP8 FP5 T02 ° ° FP7 20 T01 –30 FP6

130° 140° 150° 210° 220° 230°

–8,000 –4,000 0 4,000 Topography (m)

Figure 1 | Station distribution map. Geographical distributions of the stations in the French Polynesia (FP) array21 (close-up in b ) and Philippine Sea (PS) array22 (close-up in c ). These stations are mostly located at depths of 4,000 – 5,000 m. We analysed the continuous vertical records at 7 FP-array stations (red circles) available between 1 February 2003 and 31 January 2004 and those at 11 PS-array stations (orange circles) available between 1 December 2006 and 30 November 2007. For the PS array, only the results obtained for the seven western stations are shown in Figure 4 . The stations of the PS array not considered in the analysis are shown by orange dots. The BBOBS stations not belonging to the FP and PS arrays are shown by white circles in a .

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a –70 FP5 at depth of 4393 m–70 T05 at depth of 4866 m –80 –80 FP2 –90 –90 –100 –100 –110 –110 FP3 –120 –120 /Hz) in dB /Hz) in dB 4 4 –130 –130 /S /S 2 2 –140 –140 FP4 –150 –150 –160 –160 PSD (m PSD (m –170 –170 –180 –180 FP5 –190 –190 0.010.11110 0.01 0.1 10 Frequency (Hz) Frequency (Hz) FP6

b 0.03 FP7

0.02 FP8

0.01 18-Sep 19-Sep 20-Sep 21-Sep 22-Sep 23-Sep 24-Sep 25-Sep Frequency (Hz)

0.00 Figure 3 | Seismograms fi ltered through the IG-band. Example of the FP 1,000 2,000 3,000 4,000 5,000 6,000 array records (18– 25 September 2003) bandpass fi ltered between 4 and Depth (m) 20 mHz. The dominant signal in this band is the seafl oor disturbance due Figure 2 | Infragravity wave signature on the seafl oor noise spectra. to IG waves. The IG wave trains show temporal variations on a semidiurnal ( a ) Seafl oor noise spectra at station FP5 for the period 14 January 2003 to diurnal scale. The theoretical tidal sea level changes are shown below 25 and 14 January 2004 and those at T05 for the period 11 November 2006 – 11 each record . November 2007. Each of the power spectra is calculated using the standard procedure followed in seismology29 , for a record-section length of 4,096 s and sampling rate of 100 Hz. The 10 spectral lines with the lowest noise uniform Earth. Th is implies that for a 4,000-m-deep uniform ocean, ϑտ amplitudes in the frequency range 0.02 – 0.05 Hz are blackened to show the resonance is expected only at high latitudes ( 65 ° ). Figure 5 dem- noise spectra typical to quiet days. The spectral peak ~ 0.01 Hz (indicated onstrates how diff erent the resonance in the real ocean is from that by the arrow) represents the disturbance due to infragravity (IG) waves. in the uniform model ocean. In Figure 5, the white area is under- The two red curves indicate the New Low Noise Model (NLNM) and New stood to be in a near-resonance state, where the precise value of R High Noise Model (NHNM) of Peterson29 . (b ) Plot of the peak frequency is not calculable according to our fi rst-order perturbation theory. We can see that unexpectedly wide areas of the Pacifi c Ocean are f 0 (in Hz) in the infragravity band against seafl oor depth H (in m). Data are obtained from all the BBOBS stations shown in Figure 1. The plots are in the resonant or near-resonant state. A major part of the FP array − 1 / 2 is within the resonant area for the M tide, whereas a major part of approximated by the relation f 0 = ( 2H ) . This relation is understood as 2 ≈ 1 the PS array is outside the resonant area; this would be the reason k 0 H 1.4, which can be expected if the hydrodynamic fi ltering of IG waves is responsible for the peak. for the M2 signal being stronger in the FP array than in the PS array ( Fig. 4 ). On the other hand, a major part of either array is within the

resonant area for the K1 tide, explaining that the K1 signal intensities in the FP and PS arrays are not very diff erent (Fig. 4). UtKxcos(Ω− ) in the local coordinates can be expressed as 0 so that the co-phase line of the ocean tide with an angular frequency Discussion Ω moves in the direction given by the wavenumber vector K and Although the above-mentioned observations clearly indicate that with a speed defi ned by C ≡ Ω / K . Th e fl uid-dynamic equation of IG waves in the deep ocean recorded by the BBOBSs are persist- motion including the advection term has a solution of the form ently modulated by tidal disturbances, there is a possibility of these Ru⋅−⋅−⋅()sin( kΩ t K x )sin(w t kx ) 0 showing the tide-modulated waves being apparently modulated by the tide-related sensitivity IG waves, where the amplifi cation factor R is proportional to change of the instrument. Such a sensitivity change may be caused UckK/[()w ±−Ω ||] ± 0 . In a homogenous random wave fi eld, there by tilting of the sensor axis due either to the tidal currents or to ()w ±−Ω ck|| ±= K 0 is always a wavenumber vector k such that , the solid Earth tide. Either possibility, however, is unlikely, as will with which R undergoes resonant divergence, if be explained below. Figure 6a,b shows comparisons of the theoretical tides with the tidal-band records of the FP array corrected for c 1 the instrumental response. In Figure 6a, the comparison is made ≥ . 1 Ω for the gravity component (corresponding approximately to the ver- C 1+ 2 w tical displacement) of the theoretical solid Earth tide 25 . In Figure 6b, the comparison is made for the sea level change (corresponding Comparison of the observations and theory . W e c a l c u l a t e d R approximately to the horizontal velocity) of the theoretical ocean for the Pacifi c Ocean by using the global bathymetry model of tide 25 . Th e value attached to each pair of the observed and theo- ETOPO2 24 and an ocean tidal model developed by Matsumoto retical traces is a lag-time τ to obtain the best waveform match. In et al.25 We set the ω value to the peak frequency in the IG wave Figure 6a, the observed and theoretical traces are in almost in phase band of the seafl oor noise at each observational site, as shown by the ( τ always < 0.4 h), indicating that the observed traces are almost π / 2 arrow in Figure 2a . Figure 5a,b show the maps of R for the M2 and K1 out of phase with the tilt component of the solid tide. In Figure 6b, tides, respectively. Th e area coloured in red is essentially in resonant the observed and theoretical traces are not in phase, where τ state, as defi ned by inequality (1), which may be approximated as increases systematically (from − 3.3 to + 1.2 h) as we move south- c ≥ C. For a 4,000-m ocean depth, c ≈ 200 m s − 1 , whereas the tidal co- westward through the array (from FP2 to FP8, Fig. 1 ). Clearly, the phase line at latitude ϑ is expected to move westward with a speed vertical records in the tidal band are dominated by the gravity signal C ≈ 450 × cos ϑ m s − 1 , if a laterally uniform ocean covers a laterally due to the solid Earth tide. Th ere is no indication for the vertical

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a French Polynesia FP2 FP3 FP4 FP5 FP6 FP7 FP8 Log freq. (Hz) Log freq. (Hz) Log freq. (Hz) Log freq. (Hz) Log freq. (Hz) Log freq. (Hz) Log freq. (Hz) –5.2 –4.8–4.4.–4.0 –5.2 –4.8 –4.4.–4.0 –5.2 –4.8 –4.4.–4.0 –5.2 –4.8 –4.4. –4.0 –5.2 –4.8 –4.4. –4.0 –5.2 –4.8–4.4. –4.0 –5.2 –4.8 –4.4. –4.0 7,200 6,000 4,800 3,600

Time (h) 2,400 1,200 0

K1 M2 2nd M2 b Philippine Sea T01 T02 T03 T05 T08 T11 T12 Log freq. (Hz) Log freq. (Hz) Log freq. (Hz) Log freq. (Hz) Log freq. (Hz) Log freq. (Hz) Log freq. (Hz) –5.2 –4.8 –4.4. –4.0 –5.2 –4.8 –4.4. –4.0 –5.2 –4.8 –4.4. –4.0 –5.2 –4.8 –4.4. –4.0 –5.2 –4.8 –4.4. –4.0 –5.2 –4.8 –4.4. –4.0 –5.2 –4.8 –4.4. –4.0 7,200 6,000 4,800 3,600 Time (h) 2,400 1,200 0

K1 M2 2nd M2 01 Cross-spectral power in logarithmic scale

Figure 4 | Stacked cross-spectrograms. 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 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 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. records being aff ected by tilting of the instrument due either to the by carrying out long-term, dense-array observations of the pres- seafl oor tilt or to the tidal current at the BBOBS site. Th e observed sure changes at the sea bottom. tidal modulation of IG waves should be an actual process occurring We have shown that free IG waves in the deep ocean are ampli- in the ocean, rather than an apparent phenomenon caused by the fi ed by the resonance with ocean tides and that such resonance can instrumental tilting. occur over wide areas of the Pacifi c Ocean. Th is implies that ocean Th is is further confi rmed in Figure 6c , which compares the phase tides dissipate a certain amount of energy through resonant cou- obs solid fij ()fM of the observed cross-spectrum Qfij()M with fij ()fM pling with IG waves. Our fi ndings suggest the existence of a possible ocean2 2 solid 2 and fij ()fM in the same sampling period. Here, fij and driving force for IG waves in deep oceans and a possible dissipation ocean 2 mechanism of ocean tidal energy in deep oceans 26,27 . fij are the cross-spectral phases of the tilt component of the theoretical solid Earth tide and the sea surface elevation of the theoretical ocean tide 25 , respectively. Th e cross-correlation function of Methods the observed time series takes only positive values, whereas the A theory of the resonance between IG waves and the ocean tide . Herein, we cross-correlation functions of the theoretical tides take both posi- present a theory of the resonant coupling between IG waves and the ocean tide. In a long-wave approximation, the two-dimensional equation of motion and the tive and negative values. To account for this diff erence, a phase in equation for conservation of mass are written as the range 0 – 2π is reduced to that in the range 0– π . In Figure 6c , solid ocean the values of f ()f and f ()f are plotted against ∂u ij M2 ij M2 +⋅∇=−∇−∇uu gxf, obs obs ∂t (1) fij ()fM for the FP array for the cases where ||Qfij ()M i s v e r y 2 obs 2 ∂x large (more than 0.95 times the maximum of | Qfij ()| over the +∇[()Hu +x ] =0, ∂t frequency range shown in Fig. 4). It is apparent that fobs()f i s ij M2 ocean solid = ξ in better agreement with fij ()fM than with fij ()fM . Th e where uuv(,) is the horizontal velocity vector; , the sea surface elevation; g , the 2 2 φ 28 agreement of fobs()f w i t h focean()f and the absence of gravitational acceleration; , the tidal potential; and H , the water depth . When ij M2 ij M2 H = const and ζ Ӷ H , ocean tide signals on the records in the tidal band ( Fig. 6c ) indi- ∂2u ∂ ∂ (2) cate that tidal modulation occurs by coupling between IG waves + ()uugHu⋅∇ − ∇ () ∇⋅ =− ∇f. and the ocean tide in the deep ocean, and not by tide-induced ∂t2 ∂t ∂t To obtain the fi rst-order solution, we ignore the nonlinear advection term in (2): tilting of the sensor axis. If this is the case, the propagation of IG 1 ∂2u ∂ waves may not be perfectly isotropic , but preferentially parallel to −∇∇⋅=−gH() u ∇f. (3) the direction of the co-phase lines. Th is hypothesis can be tested ∂t2 ∂t

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a Without taking into account the detailed form of the tidal potential, we assume 60° that the tidal response can be locally expressed in terms of the horizontal velocity as follows: K uU=⋅−⋅cos(Ω tKx ), (8) 0 K 30° so that the co-phase line of the tide moves in the direction defi ned by the wavenumber = vector KKK(,)xy , with a speed defi ned by C ≡Ω/.K (9) ° 0 Th e height variation associated with (8) is hh=⋅−⋅cos(Ω tKx ), (10) 0 with –30° h U 00= . (11) H C Th e fi rst-order solution u is then written as 1 ° –60 =−⋅+⋅−⋅k K Ω uk1 ∫∫uˆ() cos(wtkxkk )ddxy U0 cos(tKx ). (12) 120° 150° 180° 210° 240° 270° 300° k K In a more general case, we assume that the fl uid velocity can be expanded in a =++⋅ 23 b perturbation series uu12 u , where the fi rst-order term is given by (12) . 60° To determine u2 , we must make allowance for the nonlinear term. Th e solution can be obtained by solving the following equation: ∂2 ∂ u2 −∇∇⋅=− ⋅∇ (13) 2 gH() u211 (uu ). 30° ∂t ∂t When (12) is introduced on the right-hand side of (13) and only the interaction terms between free waves and the tide are retained, (13) reduces to ∂2u 2 −∇∇⋅gH() u ° ∂ 2 2 0 t =−1 kK +ΩΩ + ⋅ − ⋅+ −⋅ ∫∫Uk0uˆ() (Kk )(w )cos[( tKKx)(w t kx )]dd kxy k (14) 2 kK 1 kK −−−∫∫Ukuˆ() (Kk )(Ω w )cos[(()()].Ω⋅tKx − ⋅ −w tkxkk − ⋅ dd –30° 0 xy 2 kK Th e particular solution of (14) is given by =+−−kK ⎡ ΩΩ⎤ uU2 0 ∫∫uˆ() k ⎣FKkFK ( , ;ww , ) ( , ; , k )⎦ ° kK –60 × −⋅Ω − ⋅ (15) cos(wttkx)cos( tKxkk )]ddxy 120° 150° 180° 210° 240° 270° 300° −−kK ⎡ Ω Ω −−⎤ Uk0 ∫∫uˆ()⎣FKk ( , ;w , ) FFK(,;w , k )⎦ kK –2–1 0 1 ×−⋅−⋅sin(wtkx )sin(Ω tKxkk )]dd , Log (R) xy where ++Ω Figure 5 | Maps of ampliﬁ cation factor R for infragravity waves– ocean Ω = ()()Kk w (16a) FKk(,;,)w 22 2, tides coupling. R is a measure of how large the coupled-wave amplitude ()Ω +−w cKk|| + is relative to the amplitude of the undisturbed wave. The area that is and practically in a resonant state is coloured in red. The whitish area may be FK(,;ΩΩ−−=−−ww , kF ) ( , Kk ;,). (16b) understood to be in a near-resonant state. The tidal co-phase lines are Condition for the resonance between IG waves and the ocean tide: contoured. White dots indicate the amphidromic points. Map (a ) for the

M 2 tide and (b ) for the K1 tide. If c 1 ≥ , (17) 1 Ω Under the assumption that the wave fi eld is random and homogeneous23 , the free- C 1 + 2 w wave solution can be given as there is a wavenumber vector k , such that k =−⋅ˆ uuk∫∫ () cos(wtkxkk )ddxy , (4) +−ΩΩ + −− − = k {}()wwck|| K{}() ck|| K 0, (18) where in a homogeneous random wave fi eld. With such a k value, either F (,;,)Ω Kkw o r −−Ω w ==w FKk(,;,) becomes infi nite in (15), and hence, the IG wave resonates with c gH , (5) the tide. Th e resonance area in the Pacifi c Ocean, as defi ned by (17), is shown for k the M 2 tide in Figure 5a and for the K1 mode in Figure 5b . and kk=||. Th e free-wave (IG wave) propagates in the direction defi ned by the No resonance occurs if wavenumber vector = with a frequency-independent speed c . We express c 1 kkk(,)xy < , the free-wave-induced sea surface elevation as 1 Ω (19) C 1 + 2 w xx=−⋅∫∫ ˆ()cos(ktkxkk w )dd . xy (6) where the sine– sine modulation term dominates the cosine– cosine term in (15). k Substituting this expression in the linearized second equation in (1), we obtain Th e amplitude of the IG wave becomes maximum when the wavenumber vectors and K are in the same direction, so that ˆ ukˆ() x()k (7) K k = . uRk≈−∫ ()uk ()sin(w t −Kx ⋅ )sin(Ω t − Kx ⋅ )d k , (20) c H 2 K K

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a c FP2 –0.21 h FP3 –0.37 h FP4 –0.07 h 3 FP5 –0.14 h FP6 2.5 +0.27 h FP7 –0.10 h FP8 2 +0.38 h 1-Aug 1-Sep b 1.5 FP2 –3.27 h FP3 –3.16 h 1

FP4 Observed phase difference (rad) –1.04 h FP5 –1.30 h 0.5 FP6 –1.37 h FP7 +0.82 h 0 FP8 0 0.5 1 1.5 2 2.5 3 +1.21 h Calculated phase difference (rad) 1-Aug 1-Sep

Figure 6 | Seismograms and comparison of observed and calculated cross-spectral phases. ( a , b ) Example of the FP array records (31 July– 31 August 2003) ﬁ ltered through the tidal band and corrected for the instrumental response. Comparison is made with (a ) the gravity component (or vertical displacement) of the theoretical solid Earth tide25 and (b ) the sea level change (or horizontal velocity) of the theoretical ocean tide25 . The value attached to each paired trace indicates the relative time shift that yields the best waveform match. (c ) Plots of theoretical cross-spectral phases for the tilt of the solid

Earth tide and for the sea level change of the ocean tide against the observed cross-spectral phase for the IG wave activity at the M2 tidal frequency. Plots are made for every station pair in the FP array if the observed cross-spectral power at the M2 tidal frequency is > 0.95 times the maximum cross-spectral power. A phase in the range 0 – 2 π is reduced to that in the range 0 – π by appropriately subtracting π . If the observed and theoretical cross-spectral phases coincide with each other, the plots should lie on the dotted straight line.

where days to ensure long-term seismic observations. We have practiced more than 50 long-term BBOBS experiments since 1999, including several array observations ⎡ ()()Kk++Ω w ()()−+Kk −+Ω w ⎤ Rk()= U ⎢ − ⎥ at the Pacifi c Ocean seafl oor. Th e locations of all the BBOBS stations are shown in 0 ⎣()()Ω +−w 22cK + k 2()(−+Ω w 22 −cK −+ k)2 ⎦ Figure 1 . (21) ≈ Uk0 c . Ω()1 − References C 1 . Webb , S . C . , Zhang , X . & Crawford , W . Infragravity waves in the deep ocean . J. Geophys. Res. 96 , 2723 – 2736 ( 1991 ). Th us, in the case of (19), the interaction of an IG wave packet with the tide results 2 . Tucker , M . J . Surf beat: sea waves of 1 – 5 min period . Pro c. R. Soc. Lond. A 202 , in a tide-modulated IG wave packet whose amplitude is R times that of the original 565 – 573 ( 1950 ). wave packet. Th is amplifi cation factor R may be rewritten as follows, by using (5) 3 . Elgar , S . , Herbers , T . H . C . , Okihiro , M . , Oltman-Shay , J . & Guza , R . T . and (11): Observations of infragravity waves . J. Geophys. Res. 97 , 15573 – 15577 ( 1992 ). 4 . Holman , R . A . & Bowen , A . J . Bars, bumps, and holes: models for the ho C w Ω generation of complex beach topography . J. Geophys. Res. 87 , 457 – 468 ( 1982 ). w ≈ H c (22) R() c . 5 . Webb , S . C . Broadband seismology and noise under the ocean. Rev. Geophys. 1 − 36 , 105 – 142 ( 1998 ). C 6 . Dolenc , D . , Romanowicz , B . , Stakes , D . , McGill , P . & Neuhauser , D . For the non-resonant area in the Pacifi c Ocean, as defi ned by (19), R is plotted Observations of infragravity waves at the Monterey ocean bottom broadband for the M2 tide in Figure 5a and for the K 1 mode in Figure 5b . Note that the plane- station (MOBB) . Geochem. Geophys. Geosyst. 6 , Q09002 ( 2005 ). wave approximations, as in (8) and (10), may not be valid near the amphidromic 7 . Rhie , J . & Romanowicz , B . Excitation of Earth’s continuous free oscillations by point around which the tidal system rotates 28 . Th e behaviour near the amphidromic atmosphere-ocean-seafl oor coupling. Nature 431 , 552 – 556 ( 2004 ). point is beyond the scope of our theory. 8 . Rhie , J . & Romanowicz , B . A study of the relation between ocean storms and the Earth’s hum . Geochem. Geophys. Geosyst. 7 , Q10004 ( 2006 ). Broadband seismic observation at sea bottoms . Th e BBOBS system has been 9 . T a n i m o t o , T . Th e oceanic excitation hypothesis for the continuous oscillations developed by the Earthquake Research Institute of the University of Tokyo since the of the Earth . Geophys. J. Int. 160 , 276 – 288 ( 2005 ). 1990s. A BBOBS unit is a self-pop-up type, designed to rise from the seafl oor aft er 1 0 . Kurrle , D . & Widmer-Schnidrig , R . Th e horizontal hum, of the Earth: a global receipt of an acoustic command. Th e BBOBS with a three-component CMG-3 T background of spheroidal and toroidal modes . Geophys. Res. Lett. 35 , L06304 broadband sensor (Guralp Systems) senses ground motions at periods from 0.02 to ( 2008 ). 360 s with a sampling rate of 100 Hz at 24-bit resolution. All of the seismic instru- 1 1 . W e b b , S . C . Th e Earth’s ‘ hum ’ is driven by ocean waves over the continental ment components, including the sensor, data logger, transponder and batteries, shelves . Na ture 445 , 754 – 756 ( 2007 ). are packed into a 65-cm diameter titanium alloy pressure housing, which allows 1 2 . B r o m i r s k i , P . D . & G e r s t o ft , P . Dominant source regions of the Earth’s ‘ hum ’ are for a maximum operating depth of 6,000 m. Th e system runs for as long as 500 coastal. Geop hys. Res. Lett. 36 , L13303 ( 2009 ).

6 NATURE COMMUNICATIONS | 1:84 | DOI: 10.1038/ncomms1083 | www.nature.com/naturecommunications © 2010 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1083 ARTICLE

13 . Nishida , K . , Ka w a k a t s u , H . , F u k a o , Y . & O b a r a , K . B a c k g r o u n d L o v e a n d a global model and a regional model around Japan. J. Oceanography 56 , 567 – 581 Rayleigh waves simultaneously generated at the Pacifi c Ocean fl o o r s . Geophys. (2000 ). Res. Lett. 35 , L16307 ( 2008 ). 2 6 . M u n k , W . H . & M a c d o n a l d , G . J . F . Th e Rotation of the Earth ( C a m b r i d g e 14 . Fukao , Y . , Nishida , K . & Ko b a y a s h i , N . S e a fl oor topography, ocean infragravity University Press , 1960 ) . waves and background Love and Rayleigh waves . J. Geophys. Res. 115 , B04302 27 . Munk , W . H . & Wunsch , C . Abyssal recipes II: energetics of tidal and wind ( 2009 ). mixing . Deep-Sea Res. 45 , 1977 – 2010 ( 1998 ). 15 . Longuet-Higgins , M . S . & Stewart , R . W . Radiation stress and mass transport 2 8 . A p e l , J . R . Principles of Ocean Physics ( Academic , 1987 ) . in gravity waves, with application to ‘ surf beats ’ . J. Fluid Mech. 13 , 481 – 504 29 . Peterson , J . Observations and modelling of seismic background noise . U SGS ( 1962 ). Open-File Report 93 – 322 ( 1993 ). 16 . Herbers , T . H . C . , Elgar , S . & Guza , R . T . Generation and propagation of infragravity waves . J. Geophys. Res. 100 , 24863 – 24872 ( 1995 ). 1 7 . G u z a , R . T . & Th ornton , E . B . Swash oscillations on a natural beach . J. Geophys. Acknowledgments Res. 87 , 483 – 491 ( 1982 ). We thank Hajime Shiobara, Earthquake Research Institute, University of Tokyo, and 18 . Okihiro , M . & Guza , R . T . Infragravity energy modulation by tides. J. Geophys. Daisuke Suetsugu, Japan Agency for Marine-Earth Technology (JAMSTEC), for Res. 100 , 16143 – 16148 ( 1995 ). organizing the BBOBS observation experiments. We are also thankful to the captains 1 9 . Th omson , J . , Elgar , S . , Raubenheimer , B . , Herbers , T . H . C . & Guza , R. T . T i d a l and crew of the research vessels and the Shinkai-6500 operation team of JAMSTEC for modulation of infragravity waves via nonlinear energy losses in the surfzone . their valuable support during the cruises. Th is study was supported by a Grant-in-Aid for Geophys. Res. Lett. 33 , L05601 ( 2006 ). Scientifi c Research from the Japan Society for the Promotion of Science (19740282). 20 . Dolenc , D . , Romanowicz , B . , McGill , P . & Wilcock , W . Observatio ns of infragravity waves at the ocean-bottom broadband seismic stations Endeavour (KEBB) and Explorer (KXBB). Geo chem. Geophys. Geosyst. 9 , Author contributions Q05007 (2008 ). H.S. carried out observations and analysed the obtained data; Y.F. proposed the theory; 2 1 . S u e t s u g u , D . et al. P r o b i n g s o u t h P a c i fi c mantle plumes with ocean bottom and T.K. developed the observation system. s e i s m o g r a p h s . EOS Trans. Am. Geophys. Union 86 , 429 – 435 ( 2005 ). 2 2 . S h i o b a r a , H . , B a b a , K . & Utada , H . Ocean bottom array probes stagnant slab beneath the Philippine Sea . EOS Trans. Am. Geophys. Union 90 , 70 – 71 ( 2009 ). Additional information 2 3 . H a s s e l m a n n , K . A statistical analysis of the generation of microseisms . Rev. Competing fi nancial interests: Th e authors declare no competing fi nancial interests. Geophys. 1 , 177 – 210 ( 1963 ). 2 4 . S m i t h , W . H . F . & S a n d w e l l , D . T . G l o b a l s e a fl oor topography from satellite Reprints and permission information is available online at http: //npg.nature.com/ altimetry and ship depth soundings . S cience 277 , 1956 – 1962 ( 1997 ). reprintsandpermissions/ 2 5 . M a t s u m o t o , K . , Takanezawa , T . & Ooe , M . Ocean tide models developed by How to cite this article: Sugioka, H. et al. Evidence for infragravity wave-tide resonance assimilating TOPEX/POSEIDON altimeter data into hydrodynamical model: in deep oceans. Nat. Commun. 1:84 doi: 10.1038 / ncomms1083 (2010).