Infragravity Waves Forced by Surface Wind Waves in the Central North Pacific Ocean

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/H

z

Hz

tide, inertial oscillation etc..

IG long-waves

gravity waves

capillary waves

Pacific

Atlantic

Arctic

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]

Asymptotic equations developed in MRL04(McWilliams, Restrepo and Lane, 2004)

• wave-averaged effects on currents & long waves• primary waves ~ 2nd order in wave slope (=Ak)• scale separation in time and horizontal space• Eulerian reference frame observations & models

• Vortex force (curl u Vst)• 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)

Equations for long-wave dynamics derived in MRL04

wave-averaged term

Momentum:

Continuity:

Wave-averaged term:

Momentum:

Continuity:

Evaluation of the primary wave-averaged forcing term

Stokes transport:

wave set-down:

Using the ECMWF 2D wavenumber (frequency-directional) spectral data,G (, ) [m2 s /rad], every 6 hours on a 1.5o grid (w/ interpolation)

Data source: ECWMF/UCAR (http://dss.ucar.edu/datasets/ds123.0/)

• based on 2D-ROMS with the wave-averaged term• containing advection, Coriolis, bottom drag terms• ~1/8o geographical grid (1568 1152 cells)• te=18 s

Bathymetry h (km) of the Pacific Ocean

open boundary with a modified Orlanski condition

Numerical Configuration

IG wave solution at 0 AM UTC on 27th Julian day, 2000

wave

en

erg

yHHs s & k& k

lwlwTTmm

s s & T& Tstst

wave

en

erg

yHHs s & k& k

lwlwTTmm

s s & T& Tstst

IG wave solution at 0 AM UTC on 28th Julian day, 2000

wave

en

erg

yHHs s & k& k

lwlwTTmm

s s & T& Tstst

IG wave solution at 0 AM UTC on 29th Julian day, 2000

wave

en

erg

yHHs s & k& k

lwlwTTmm

s s & T& Tstst

IG wave solution at 0 AM UTC on 30th Julian day, 2000

wave

en

erg

yHHs s & k& k

lwlwTTmm

s s & T& Tstst

IG wave solution at 0 AM UTC on 31st Julian day, 2000

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 ~10-4m.

Tst and s

simulated lwwave-averaged term, F

(1)(2)

Time series of lw at two locations on January 31, 2000

6h

1.6h

48min

(1) Off Alaska (230o02’ E & 44o59’E)

(2) West of Hawaii (170o03’ E & 34o58’E)

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 filteredf<2 x 10-4 Hz (T>1.38h)

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

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?)

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

Comparison of wave data: ECMWF vs. NDBC buoys

#46001Off Alaska

#51028Off Hawaii

Validity of Spatial/Temporal Resolution of Wave Data

ECMWF vs. NDBC buoys (off Alaska)

significant wave height

mean wave period

principal wave direction

magnitude of Stokes transport

Julian day in 2000

Comparison of |Tst| 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.

Primary surface wave field (magnitude of Stokes transport)

apply filter functionFourier transform inverse Fourier transform

Comparison between vortex force & radiation stress formalisms

c.f. Lane, Restrepo and McWilliams (2006, JFM)

U=

• Scale separation both in time and horizontal space

• Substitute into momentum and continuity equations

analogous to Reynolds equation

analogous to Bernoulli equation

Radiation stress and vortex force formalisms are identical

Radiation stress Vortex force

Bernoulli head

Horizontal vortex forceNot 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

0

qHt x

SH

gt

1q

Incorporation of wave-averaged term into 2D ROMS

+ advection, Coriolis, linear bottom drag terms

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.)

Volume (or area) averaged PE, KE, and wave energy

potential energy

kinetic energy

surface elevation

wave energy