1. Infragravity Waves Forced by Surface Wind Waves in the Central North Pacific Ocean Yusuke Uchiyama and James C. McWilliams (CESR, IGPP, UCLA) Ocean bottom pressure spectra (Webb, 1998) Pa 2 /Hz Hz tide, inertial oscillation etc.. IG long- waves gravity waves capillary waves Pacific Atlantic Arctic

2. What are infragravity (IG) long-waves: Non-linear interaction between short primary waves (modulation) + varying topography ~ O(10 -2 )-O(10 -3 )Hz Forced (bound) & free waves [Herbers et al, 1994; 1995] Surf beat (surf zone) [Munk, 1949; Huntley et al., 1981] Edge waves (trapped & leaky) [Bowen & Inmann, 1971] IG waves are generally known to have small amplitudes in deep ocean only << O(10 -2 ) m.  amplified significantly in nearshore regions Is the hypothesis proposed by seismologists true? If so, how large is amplitude of IG waves? Dynamics: bound vs. freely propagating IG waves? Continuous seismic oscillations ~ “Earth’s hum”~M6 [Webb, 1998; Rhie and Romanowicz, 2004; Tanimoto, 2005]

3. Asymptotic equations developed in MRL04 (McWilliams, Restrepo and Lane, 2004) wave-averaged effects on currents & long waves primary waves ~ 2 nd order in wave slope (=Ak) scale separation in time and horizontal space Eulerian reference frame  observations & models Vortex force (curl u  V st ) Bernoulli head ~ pressure correction (set-up/down) Evolution due to Stokes drift vs. Classical “radiation stress” formalism (c.f. Longuet-Higgins and Stewart, 1960, 1962, 1964)

4. Equations for long-wave dynamics derived in MRL04 wave-averaged term Momentum: Continuity: Wave-averaged term: Momentum: Continuity:

5. Evaluation of the primary wave-averaged forcing term Stokes transport: wave set-down: Using the ECMWF 2D wavenumber (frequency-directional) spectral data, G (, ) [m 2 s /rad], every 6 hours on a 1.5 o grid (w/ interpolation) Data source: ECWMF/UCAR (http://dss.ucar.edu/datasets/ds123.0/)

6. based on 2D-ROMS with the wave-averaged term containing advection, Coriolis, bottom drag terms ~1/8 o geographical grid (1568  1152 cells) t e =18 s Bathymetry h (km) of the Pacific Ocean open boundary with a modified Orlanski condition Numerical Configuration

7. IG wave solution at 0 AM UTC on 27 th Julian day, 2000 wave energy H s & k  lw TmTmTmTm  s & T st

8. wave energy H s & k  lw TmTmTmTm  s & T st IG wave solution at 0 AM UTC on 28 th Julian day, 2000

9. wave energy H s & k  lw TmTmTmTm  s & T st IG wave solution at 0 AM UTC on 29 th Julian day, 2000

10. wave energy H s & k  lw TmTmTmTm  s & T st IG wave solution at 0 AM UTC on 30 th Julian day, 2000

11. wave energy H s & k  lw TmTmTmTm  s & T st IG wave solution at 0 AM UTC on 31 st Julian day, 2000

12. January 31, 2000 Seismically quiet, but “hum” was apparent in the IG frequency band ~ M6 (Rhie & Romanowics, 2004) Forced IG waves are evident, but free IG waves are unclear and amplitude is small ~10 -4 m. T st and  s simulated  lw wave-averaged term, F

13. (1) (2) Time series of  lw at two locations on January 31, 2000

14. 6h 1.6h 48min (1) Off Alaska (230 o 02’ E & 44 o 59’E) (2) West of Hawaii (170 o 03’ E & 34 o 58’E)

15. January 31, 2000 Applying Fourier low- pass time filter to extract IG wave energy  RMS for whole freq.  RMS for higher (~IG) freq. Fourier low-pass filtered f 1.38h)

16. Ratio of  RMS :  RMS for IG freq.  RMS for whole freq. Forced IG long-waves are predominant over slower variations in deeper ocean larger in the northern part because of storms tends to be larger near ridges, canyons and island chains  fairly consistent with seismologists’ hypothesis

17. Summary A 2D barotropic ROMS is modified by incorporating long-wave dynamics based on MRL04 for generation of infragravity wave in basin scale, ECMWF/UCAR 2D wavenumber spectral data is utilized to evaluate the wave-averaged forcing term, Long waves in the North Pacific are evidently exited as forced (far) infragravity waves. Remaining questions are : - peak frequency is slightly lower than IG freq. band. - amplitudes of IG waves are small  inconsistent with bottom pressure spectra. Why? - do free IG waves exist? (nearshore-generated?)

18. Dominant Frequency of the Model-produced  IG wave dispersion relation Why is dominant frequency lower than IG freq.? Long waves at T=100s have wave lengths of L < 10km A finer grid may be needed ~regional simulations

19. Comparison of wave data: ECMWF vs. NDBC buoys #46001 Off Alaska #51028 Off Hawaii Validity of Spatial/Temporal Resolution of Wave Data

20. ECMWF vs. NDBC buoys (off Alaska) significant wave height mean wave period principal wave direction magnitude of Stokes transport Julian day in 2000

21. Comparison of |T st | PSD: ECMWF vs. NDBC buoys #46001 off Alaska#51028 off Hawaii  Higher frequency (thus high wavenumber) components are not well resolved with less energy in the ECWMF data.

22. Primary surface wave field (magnitude of Stokes transport) apply filter function Fourier transform inverse Fourier transform

23. Comparison between vortex force & radiation stress formalisms c.f. Lane, Restrepo and McWilliams (2006, JFM) U=U= Scale separation both in time and horizontal space Substitute into momentum and continuity equations analogous to Reynolds equation analogous to Bernoulli equation

24. Radiation stress and vortex force formalisms are identical Radiation stressVortex force Bernoulli head Horizontal vortex force Not transparent Wave dynamics is non-linear, but weak compared to turbulence Non-linearity enters only through the surface B.C. Lowest order ~ radiation stress merely captures set-up effect IG wave equation using radiation stress

25. Incorporation of wave-averaged term into 2D ROMS + advection, Coriolis, linear bottom drag terms

26. Modified Orlanski’s Radiation Scheme for Open Boundaries (for 2D barotropic ROMS) c: phase speed of each variable (, u, v)  : nudging coefficient [T -1 ]  : coefficient for pressure-gradient mass correction (n.d.)

27. Volume (or area) averaged PE, KE,  and wave energy potential energy kinetic energy surface elevation  wave energy