Click
Here

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, C07029, doi:10.1029/2007JC004562, 2008

for

Full
ArticleInfragravity waves in the deep ocean: Generation, propagation,and seismic hum excitation
Yusuke Uchiyama1 and James C. McWilliams1
Received 19 September 2007; revised 7 February 2008; accepted 26 March 2008; published 22 July 2008.

[1] Oceanic infragravity waves are investigated as a possible source of seismic freeoscillations, often referred to as the ‘‘hum’’ of the Earth, using a numerical model ofdepth-independent, nondispersive, long-wave dynamics with a forcing from nonlinear
interactions among the primary wind waves (including swell). Because of near-resonant
amplification, the structure of the primary-wave forcing field in shallow water, and anedge-trapping mechanism, infragravity waves are excited very effectively near the coasts.
Deep-water infragravity waves are significantly influenced both by offshore leakage andpropagation of the coastally generated free waves and by deep-water primary-wave
forcing. With the inclusion of ‘‘mesoscale’’ variability on top of the more slowly varying
primary waves generated in synoptic storms, the deep-water infragravity waves are foundto have an amplitude on the order of a millimeter in height, which is consistent withfield observations and considered to be sufficient to account for local hum excitation in the
middle of the basin.
Citation: Uchiyama, Y., and J. C. McWilliams (2008), Infragravity waves in the deep ocean: Generation, propagation, and seismichum excitation, J. Geophys. Res., 113, C07029, doi:10.1029/2007JC004562.

1. Introduction[2] Seismologists have detected Earth’s hum, i.e., lowfrequency seismic free oscillations in discrete frequencies
between 2 and 7 mHz that occur even on seismically quiet
days [Nawa et al., 1998; Suda et al., 1998]. The frequenciesof the free oscillation coincide with spheroidal fundamental
modes of Earth, distinct from the microseismic noise at
frequencies peaked near 0.2 Hz. The amplitude of hum istypically equivalent to about a magnitude 6.0 earthquake
every day [Tanimoto and Um, 1999; Ekström, 2001]. The
source of the hum must be near Earth’s surface because thefundamental modes are all Rayleigh waves. Since the hum
occurs continuously, it cannot be explained by intermittent
large earthquakes nor by summing the contributions ofsmall earthquakes [Tanimoto et al., 1998]. The likely cause
is pressure fluctuations in the air or water at their bottom
solid surface. Microseisms are known to be caused byoceanic surface wind waves (i.e., ‘‘primary’’ waves, inclusive of remotely generated swell) that generate pressure
fluctuations that do not wane with depth [Longuet-Higgins,1950]. The same mechanism is not applicable to thegeneration of the much lower frequency hum. Some
researchers have speculated that atmospheric disturbances
distributed uniformly over both land and sea surface are apossible source of hum [Kobayashi and Nishida, 1998;
Tanimoto and Um, 1999; Nishida et al., 2000; Ekström,2001]. Recently, Rhie and Romanowicz [2004] used two
1
Institute of Geophysics and Planetary Physics, University ofCalifornia, Los Angeles, California, USA.

Copyright 2008 by the American Geophysical Union.
0148-0227/08/2007JC004562$09.00

arrays of broadband seismometers in California and Japanto detect the propagation of Rayleigh waves during 2000.
The strongest incident Rayleigh waves originate in the
Northern Pacific during the boreal winter and in thesouthern ocean in the austral winter. These are sites ofespecially strong storms and associated wind waves. Thisimplicates the mechanism of nonlinear interaction among
wind waves to generate long, low-frequency, oceanic infragravity waves, rather than atmospheric pressure fluctuations[e.g., Webb et al., 1991; Webb, 1998; Tanimoto, 2005;
Webb, 2007]. Estimating the size of seafloor pressure fromexisting measurements, Tanimoto [2005] found sufficientenergy in the infragravity fluctuations to excite the observed
seismic signals. Rhie and Romanowicz [2006] extended
their earlier analysis to compare the data from the seismicarrays with a global wave analysis and showed that one
source of hum was located along the U.S. West Coast
during the winter of 2000. They proposed that some coastal
infragravity waves locally generate bottom pressure fluctuations, while others propagate long distances and generate
fluctuations throughout the basin. The resulting low-frequency seismic waves propagate over the globe and constitute the hum.
[3] The theory of infragravity waves shows they are
generated by nonlinear interactions among the primarywaves, have depth-independent horizontal-velocity andpressure profiles, and propagate as nondispersive shallowwater waves [e.g., Hasselmann, 1962; Longuet-Higgins andStewart, 1962; McWilliams et al., 2004]. As part of the sameinteractions, modulations of primary wave height also
induce a bound or quasi-static depression of the sea level(i.e., wave set-up). Observed alongshore wave number
spectra [Munk et al., 1964; Huntley et al., 1981; Oltman-

C070291 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

Shay and Guza, 1987], weak energy levels in the deep
ocean [Webb et al., 1991], and cross-shore amplitude
variations [Okihiro et al., 1992] all suggest that infragravity
waves are refractively trapped on the shelf as edge modes
with only weak radiation to the deep ocean. Various models
suggest that free waves are generated close to shore [e.g.,
Longuet-Higgins and Stewart, 1962; Gallagher, 1971;
Bowen and Guza, 1978; Foda and Mei, 1981; Symonds etal., 1982], but the processes controlling the spatial andtemporal variability of infragravity waves are still poorly
understood. Because of their greater amplitude and instrumental accessibility, observations of infragravity waves
have been made mostly on continental shelves [e.g., Munk,
1949; Tucker, 1950; Guza and Thornton, 1982; Okihiro etal., 1992; Herbers et al., 1994, 1995a, 1995b; Sheremet etal., 2002]. In this nearshore context infragravity waves are
involved in surf beats, edge waves, foreshore beach erosion
during storms, formation of cuspate topography, harbor
oscillations, etc. Fewer measurements have been made for
infragravity waves in the deep ocean. The main reason isthat the amplitude of deep-water infragravity wave isextremely small and thus hard to measure accurately. For
instance, Webb et al. [1991] reported maximum amplitudesof deep-water infragravity waves inferred from bottompressure sensors ranging from O(104) m in the Atlantic
to O(103) m in the Pacific, although the sampling wasquite sparse. They also found that deep-water infragravity
wave energy is correlated not with the local wind-wave
energy but with swell energy averaged over all coastlines
from which free infragravity waves could originate. Furthermore, Rhie and Romanowicz [2006] suggest the hum isgenerated in deep water partly by freely propagating infragravity waves generated near the shoreline.
[4] Webb [2007] proposed a weakly nonlinear spectral
theory on coupling between ocean waves and seismic
modes by extending the microseismic excitation theory by
Longuet-Higgins [1950] and Hasselmann [1963], relying on
weak wave triads for energizing very low-frequency difference waves on the continental shelves. He approximately
reproduces the seismic spectrum observed, and confirmedthat ocean waves, rather than atmospheric turbulence, aredriving the seismic modes of Earth. However, this mechanism is somewhat doubtful for producing very low frequency
long waves from interacting primary waves with nearly
equal frequencies, since that requires phase coherence ofthe primary waves over the longer time and distance of the
product long wave (n.b., this objection is not as cogent for
shorter infragravity waves like surf beats). Instead, we makeuse of the more robust multiscale theory [McWilliams et al.,2004] where the long-wave response is forced by thespectrum-integrated primary-wave properties.
[5] This paper is aimed at clarifying the dynamics ofinfragravity waves in the deep ocean that are due to local
and remote excitation by primary gravity waves whose
wave-averaged properties vary with ‘‘synoptic’’-scale
storms and with finer, ‘‘mesoscale’’ fluctuations. The latter
forcing comprises empirically demonstrated modulations of
the primary-wave-averaged forcing on an intermediate
mesoscale, longer than the primary waves but shorter thanthe meteorological synoptic scale. Particular issues are
distinguishing the responses either bound to the forcing
field or freely propagating, assessing the degree of wave

C07029confinement by topography near the coastline, and examining
the relative efficiencies of coastal and abyssal infragravity
wave generation. By the generation mechanism we study, thefrequencies of interest are defined by those of the hum itself.
Therefore we have not pursued surf beats, shear waves,nearshore infragravity waves, 100 s wave-envelope modulations relevant to surf-zone infragravity waves, nor any of
the other motions with frequencies only moderately lower
than those of wind waves and swells (typically higher than
40 mHz) observed in surf zones. Nevertheless, since theinfragravity waves have their maximum amplitudes at theshoreline, we incorporate subgrid-scale forcing modulations
due to refraction and breaking in shallow water based on asimplified WKB model with an empirical breaking parameterization. This approach enables us to investigate deep-ocean
infragravity long-waves with a relatively large grid spacing,O(103) m. This focus on lower-frequencies and longerscales phenomena allows us to accurately rely on thenondispersive shallow-water approximation. We can further
reply on a linear wave-response dynamics where each
spectrum component of the primary wavefield modulation
at any scales independently forces its own response becauseof the observed (and simulated) small long-wave amplitudeof O(103) m. Therefore we investigate this problem with a
newly developed numerical model based on a barotropic
version of ROMS [Regional Oceanic Modeling System;
Shchepetkin and McWilliams, 2005]. A linearized long-wave
equation with general topography and shoreline shape andwith primary-wave-averaged forcing derived in a multiscale,
asymptotic wave-current interaction theory by McWilliams
et al. [2004] is used to represent infragravity waves onregional and basin scales.

2. Infragravity-Wave Dynamics[6] Long (infragravity) waves are driven by nonlinear
interaction among primary waves, whose sea level amplitude hp varies on a fast wave-scale (x, t). hp is written in a
Fourier integral representation as
Z 1

hp ðx; t Þ ¼

1^hp ðk; X; t Þ exp½ifk  x  wt g dk;

ð1Þ

along with the linear dispersion relation,
w2 ðkÞ ¼ gk tanh½kh :ð2Þ

w is a radian frequency; k = jkj = 2p/L is the magnitude of
the horizontal wave number vector k; and h(X) is the restingwater depth. The caret, ^, denotes the complex amplitude ofeach Fourier component. As in the asymptotic analysis of
McWilliams et al. [2004] (MRL04), we assume the waves
have a slowly varying envelope dynamics in the horizontal
and time coordinates (X, t), where X = bx, t = b t, and b 1,
including a long-wave component that does not oscillate on
the scale of the primary wave coordinates (x, t). The forced,
dissipative long-wave equations adapted from MRL04 are

2 of 25@q
þ gr~
z ¼ D;@t@ z~
þ r  ðhqÞ ¼ F :@t

ð3ÞUCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029D =  Cd q/h is the bottom drag force with a linear friction
coefficient Cd, and F is the nonlinear infragravity-wave
forcing averaged over the primary waves:
"

y

y #
A2 wk
@
A2 k
F ¼ r
2k tanh kh@t 2 sinh 2kh
"





E k
¼ r
rc k

y



 #
@
E cg 1 y@t rgh cp 2

C07029is an alternative representation of G(k) using the dispersion
relation [Komen et al., 1994] (Figure 1, left). The mean
wave period Tm and mean wave direction qm are evaluatedwith G by
Z 1Z pð4ÞGðw; q; X; t Þdqdw
;Tm ð X ; t Þ ¼ 2p Z 01 Z ppwGðw; q; X; t Þdqdw

ð10Þ

p

0

ð5Þ
Z 1Z psin qGðw; q; X; t Þdqdw
qm ðX; t Þ ¼ tan1 Z 0 1 Z p
p


@z s
:
¼  r  TSt þ@t

1
E ¼ rgA2 :2

ð7Þ

[7] The alternative expressions for F are written for asingle Fourier component in hp, and the contributions can besuperimposed for multiple components.

[10] The Stokes transport and the wave set-up in (6) can
be calculated in spectral form [Kenyon, 1969, 1970; Battjes,
1972]:

h2p ðX; t Þ ¼

0

z s ðX; t Þ ¼ 

Gðw; q; X; t Þdqdw;

ð8Þ2pkGðk; X; t Þ
@w=@k

ð9Þ

with

ð12Þ

Z 1Z p0Gðw; q; X; t Þk ðwÞ
dqdw:
2sinh 2k ðwÞhðXÞ
p

ð13Þ

4. Mesoscale Forcing Modulation[11] We can further express the slow-wave number and frequency dependencies for wave-averaged properties such
as E, TSt, and z s through a Fourier transform into (K, W)
space (Figure 1, right). For example,
Z 1Z 11pGðw; q; X; t Þ ¼Gðw; q; X; t Þwkðw; qÞdqdw
p 2k ðwÞ tanh k ðwÞhðXÞ

T ðX; t Þ ¼

TSt ðX; t Þ ¼

Z 1Z p0

Z 1Z pSt

3. Spectral Representation of Wave-Averaged
Quantities[8] The primary wavefield hp is described by a twodimensional (2D) wave number spectra G(k; X, t) on the
wave-averaged scales. G is nonzero only for k values within
the primary wave range, and the wave properties such as
amplitude and spectrum shape change over the slowly
varying coordinates (X, t). In this study we will make useof the global 2D frequency-directional wave spectral reanalysis provided by ECMWF [ERA-40, cf. Bidlot et al.,
2002; Janssen et al., 1997], G(w, q), at each point in theiranalysis (a 1.5° geographical grid), where q is the wave
number vector direction. The ECMWF wave analysis is
made four times a day by the WAM cycle-4 global wave
model with data assimilation. G is discretized with 25
frequency bins and 12 directional bins.
[9] The local variance of hp is represented using thespectrum as

ð11Þ

p

0

~z is the long-wave, free-surface elevation, and q is theassociated, depth-independent horizontal velocity. ()y denotes a band-pass filtering operator, averaging over the
primary waves and excluding slower variations on the scaleof oceanic currents (not modeled in this paper). A is wave
amplitude; cg = @w/@k is group velocity; and cp = w/k is phasespeed. Both cg and cp are associated with propagation in the
direction of k. The quantities TSt and z s in F are the verticallyintegrated Stokes drift (i.e., Stokes transport) and the quasistatic sea level response, often called the set-up. The wave
energy per unit area is:cos qGðw; q; X; t Þdqdw

ð6Þ

^ St ðK; WÞeifKXWtg dKdW;
T

ð14Þ

1

and no dispersion relation is implied between W and K.
Since these are primary-wave-averaged quantities, thespectral variance vanishes in the primary wave frequency
band wp, but there is variance at lower frequencies
associated with storms (Ws) and even with climate
variability. The spatial and temporal resolution of the
ECMWF wave analysis, on what we call the synoptic scale,is rather coarse: Dt = 6 h and DX = 1.5°. However, theprimary wave properties vary on even finer scales that wecall the mesoscale, with Ws < Wm < wp. There is no
information about the mesoscale component in the ECMWF
analysis. We formally separate these slowly varying
components, e.g.,St
TSt ðX; t Þ ¼ TSts ðX; t Þ þ Tm ðX; t Þ;

ð15Þ

the subscripts s and m stand for the synoptic and themesoscale components.
[12] A crude representation for the slowly varying (X, t)behaviors is to collapse the full primary-wave spectrum in G
into a single primary-wave component (denoted by subscript p) with an equivalent primary wave sea level varianceA2 /2. Practically speaking, the primary wavefield quantities
that matter for forcing the long waves (i.e., TSt and z s, both

3 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 1. Schematic diagrams of different scale ranges in the frequency w or horizontal wave number kspectra for sea-surface elevation h (left) or for the wave-averaged Stokes transport TSt (right). The
primary-wave frequency spectrum G(w) corresponds to the right peak in the left plot, and the slowly
varying time and space dependencies in G(X, t) contribute in the right plot to the slowly varying wave
number-frequency W-K spectrum. Infragravity-wave signals in ~z and z s appear as the middle peak in theleft plot.
of which have very steep spectra at high frequency or wave
number for a typical surface elevation spectrum, as evident
in their definitions in (5)) have almost all their variancefrom the primary-wave spectrum peak component, not the
shorter, faster components in the primary wave spectrum, so
their is nothing important lost by representing the primary
wavefield by its peak component. It is reasonable to identify
wp with 2p/Tm and qp with qm from (10) and (11). The
slowly varying fields can be expressed with separate synoptic and mesoscale components associated with separate
amplitudes, As and Am,
AðX; t Þ ¼ As ðX; t Þ þ Am ðX; t Þ;
 As ðX; t Þ  ½1 þ dm ðX; t Þ :ð16Þ

d m is defined as the nondimensional fractional amplitude of
the mesoscale component. Since we require A > 0 andanalyze the synoptic wave variability with As > 0, jd mj < 1. A
further simplification is to assume that the primary wave
properties vary only on the synoptic scale, except for thewave amplitude that varies on both synoptic andmesoscales. With both these simplifications the synoptic
and mesoscale contributions to the Stokes transport are
expressed asTSts ð X ; tÞ 

A2s wpcos qp ; sin qp ;2 tanh kp h

2
St
TSt
m ðX; t Þ  2d m þ d m Ts :[14] Furthermore, a similar collapse of the mesoscalevariability spectrum can be made into a single Fouriercomponent, again with an amplitude variance equivalent
to the spectrum integral over the mesoscale band. The
observed small long-wave amplitude, O(103) m, allows
us to use a linear wave-response dynamics that means thateach spectrum component of the primary wave modulation(mesoscale and synoptic) independently forces its own
response. Thus the response to a broadband modulation
field is merely the linear superposition of the independent
component responses, and there is nothing to be gained byconstructing a broadband mesoscale forcing that cannot belearned by using representative mesoscale components anddemonstrating how the long wave response depends on theiramplitude and wave number. There is even the interpretive
advantage of isolating representative spectrum components
since that allows a clear demonstration of how their properties influence the response (note that influence of ‘‘multidirectional’’ mesoscale variations is briefly examined witha realistic, basin-scale configuration in section 11 andAppendix E).
[15] If the mesoscale component has a spatial wave
number Km = 2p/lm, where lm is a wavelength of mesoscale fluctuations, thendm ðX; t Þ ¼ d0 sin½Km  X  Wm t ;ð19Þ

dm ðX; t Þ ¼ Km cgr ;ð20ÞKm ðX; t Þ ¼ Km cos qp ; Km sin qp :

ð21Þ

ð17Þ

ð18Þ

[13] An analogous expression to (18) can be given for z s,
and together these comprise the necessary forcing fields forequation (3). For the basin-scale experiment in the later
sections, we estimate TSt and z s using the ECMWF-ERA40
spectral wave data.

0 < d 0 < 1 is the relative magnitude of the mesoscale
amplitude. Wm is the mesoscale oscillation frequency
determined by the theoretical expectation that mesoscale

4 of 25

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029C07029Figure 2. Wave number spectra of significant wave height estimated with the TOPEX altimeter in the
Pacific Ocean averaged over cycles 269 through 272 (January – February 2000). (Left) Along Path #182
(eastern Pacific) and (right) Path #212 (western Pacific). Also shown are ECMWF wave analyses linearlyinterpolated for the same time and place as the TOPEX data.

variations in primary wave energy propagate with the groupspeed of the dominant primary wave with k = kp,

cgr ðX; t Þ ¼

wp kp
@w
kp ¼
2kp
@k


1þ


2kp h
:
sinh 2kp h

ð22Þ

The mesoscale forcing propagation direction coincides withthe primary phase propagation direction qp. Note thatreflection of the meso- and synoptic-scale disturbances fromthe shoreline is not considered in the present forcing
formalism.

[16] In calculating the infragravity-wave response to themesoscale forcing, we will choose K1m somewhat arbitrarily
as shorter than the scales resolved in the ECMWF analysis
but larger than the grid scale of our model, viewing it as acontrol parameter. On the other hand, an estimate for thefractional amplitude d 0 is evaluated using a combination of
TOPEX/Poseidon satellite radar altimetry (http://topexwww.jpl.nasa.gov), the National Data Buoy Center (NDBC)
pitch-roll buoy data (http://www.ndbc.noaa.gov), and the
ECMWF global wave reanalysis (http://www.ecmwf.int/
research/era). Band-integrated synoptic and mesoscale sea
level variances are calculated from these sources andFigure 3. Frequency spectra of significant wave height, Hs, measured by NDBC wave buoys and thecorresponding ECMWF reanalysis at two locations in Pacific Ocean for the year 2000: (left) #46001 near
Alaska and (right) #51001 near Hawaii.
5 of 25

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029

Table 1. Primary-Wave Amplitude Variances, A2 , and Mean
 From the TOPEX and ECMWF Wave Dataa
Amplitude, A,
Cycle

269

270

271

272

A2 (m2)

ECMWF
TOPEX
mesoscale

0.39450.60010.00880.3064
0.49330.00790.33550.51660.00750.37540.61320.0085

 (m)
A

ECMWF
TOPEX
mesoscale

0.81201.00730.12600.74090.93830.12160.74580.92910.11920.79581.01260.12500.15520.16410.15990.1571d0
a 2
 are estimated by taking averages of wave number spectra along
A and A
all the satellite paths that span the Pacific Ocean integrated over each
TOPEX cycles, #269 – #272.

averaged over the available samples, yielding A22 and A2m(see also Appendix A). This leads to the estimator,
2

d0 ¼

Am

!1=2

2

ð23Þ

:

As

[17] TOPEX/Poseidon data provides the significant wave
height of the ocean surface directly underneath the satellitewith an accuracy of about ±0.1 – 0.2 m [Callahan et al.,1994]. The altimeter wave height has a finer spatial spatial
resolution about 6 km along the satellite paths than that of
the ECMWF’s reanalysis (i.e., 1.5°) while temporal resolution is coarse since the repeat period of the satellite orbit isabout 10 d. On the contrary, NDBC provides significant
wave height once every hour with an accuracy of ±0.2 m,
although the buoys are sparsely distributed in space.
[18] Figure 2 compares wave number spectra of spatial
distributions of significant wave height measured by the
TOPEX radar altimetry along the paths 182 and 212 (bothin Pacific Ocean), averaged over Cycles 269 to 272 in
January and February 2000, and their counterparts estimatedwith the ECMWF reanalysis. By applying the cut-off wave
number using the ECMWF grid spacing in the spectral
domain to extract the synoptic and mesoscale fluctuations,
we obtain reference values for d 0 from (23) as summarizedin Table 1. Similarly, Figure 3 shows frequency spectra ofsignificant wave height measured by the two wave buoys
deployed in the Pacific Ocean and from the ECMWF’s
reanalysis. Table 2 is their summary. The comparison
between the measurements and reanalysis gives us d0
ranging from 0.076 to 0.263, suggesting that d 0 can be afunction of time and space. By averaging these estimates, d 0

C07029may be reduced to a single mean value, d 0  0.16 that wewill use in the present study because observations of
mesoscale wave variability, in particular wave groups inthe deep ocean, are rare [n.b., synthetic aperture radars haverecently begun to be utilized for detecting variability ofwave groups; Nieto Borge et al., 2004].

5. Nearshore Forcing[19] The primary waves vary significantly in shallow
water near the shoreline because of refraction, shoaling,
and breaking. This is relevant to the wave-averaged forcing
F in (6). However, the width of this nearshore zone is often
small compared to the regional and basin scales that are our
focus, so special care must be taken with the representationof its effects in a more coarsely resolved model. In addition,nearshore infragravity waves are most energetic at theshoreline and are important for exciting the hum on continental shelves, as documented empirically by Rhie andRomanowicz [2006], and rationalized theoretically by Webb[2007]. Instead of ignoring the surf zone effects totally as
Webb [2007] did, we have incorporated the minimum
effects of wave breaking and associated modification of
the nearshore forcing into our model as described below.
[20] We take the view that the offshore wavefield isspecified as described in sections 3 –4, and kp and h2p are
modified near the coast using ray theory, an empiricalbreaking parameterization, and grid-scale averaging. In
combination with the dispersion relation (2), ray theory[e.g., Lighthill, 1978; Mei, 1989] implies@kþ rw ¼ 0;@t@Abþ r cg A ¼  : ð24Þw@tr  k ¼ 0;

A = E/w is the wave action, and b is dissipation rate ofwave energy E due to breaking. In applying these equations
near the coast, we assume that the primary wavefield is
locally steady and represented by its equivalent monochromatic wave; alongshore derivatives of the local topography
and wave number vector are negligible compared to crossshore derivatives; and shoreline reflection is unimportant.
Then the ray equations reduce to

rw ¼

dw
¼ 0;
dx?

dkk
¼ 0;
dx?

d  ðx? Þ 
c E ¼ b ;
dx? g

ð25Þ

 Evaluated by Integrating Frequency Spectra of the NDBC
Table 2. Primary-Wave Amplitude Variances, A2 , and Mean Amplitude, A,
and ECMWF Wave Height Variations for the Eight Locations in Pacific Ocean Measured for a Year in 2000
buoy #

46001

46002

46006

46035

46059

46066

51001

51028

Longitude, °W
Latitude, °N

104.17
56.30

86.36
42.58

93.48
40.80

133.58
57.05

86.00
37.98

110.9852.70

118.21
23.43

109.870.02

A2 (m2)ECMWF
NDBC
mesoscale

0.6967
1.11550.01240.57270.82370.00640.70190.91100.00660.75161.08310.05790.40810.75430.01151.06201.16120.0062

0.19520.27560.01270.04180.05290.0029

 (m)
A

ECMWF
NDBC
mesoscale1.1804
1.49370.15731.07031.28350.11291.18481.34980.11501.22601.47180.34020.90351.22820.15181.4574
1.52400.11140.62480.74250.15940.28900.32530.07610.13320.10550.09710.27750.16800.07640.25510.2634d0

6 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 4. Analytical amplification factor jMrj from (29) as a function of depth h and forcing phasespeed Uf. (Left) phase speed of the synoptic storm, Cs, and (right) group speed of primary wave (i.e.,phase speed of the mesoscale forcing) cgr computed from the primary wave period Tp. Both synoptic andmesoscale forcings are capable of inducing resonant amplification in shallow water (h < O(102) m), but
the deep-water response is always off-resonant.
where x? denotes the cross-shore spatial coordinate and kk is
the alongshore wave number component. Equation (25)
implies that w and kk are constant approaching the shoreline.
We can use the dispersion relation and the known
topographic shoaling profile h(x?) to determine thehorizontal wave number magnitude k(x?), hence theperpendicular wave number component k?(x?) and the wave
number angle relative to the shoreline. Shoaling h(x?)
generally implies increasing k(x?) and turning of the ray
toward the shoreline. When b = 0 (i.e., outside the surf zone),(7) and (25) imply that
A2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
kk2 þ k?2
cg k?

 P1

6. Resonant Amplification[22] In this section we evaluate an analytical solution to
the forced long-wave equation (3) in a simplified setup, viz.,one-dimensional (1D) problems with flat bottom topography
and no bottom friction. The governing equations reduce to~z tt  c2 ~z XX ¼ Ztt ;ð26Þ

is constant along a ray and independent of x? approaching theshoreline, and with (22) this yields a formula for A(x?). As hdecreases, A generally increases since cg decreases. Within
the surf zone where wave breaking occurs and b 6¼ 0, we
assume that the wave amplitude is limited to be a fraction of
the local depth,1Aðx? Þ ¼ ghðx? Þ:2where its two expressions match and because F has asingular amplitude as h ! 0. The net result is reasonably
satisfactory in limiting the sensitivity to the grid size and theshallowest depth near the shoreline hmin.

F dt:

ð28Þ

pﬃﬃﬃﬃﬃ
c = gh is the phase speed of long waves, and X denotes thespatial coordinate. We consider a moving disturbance Z =Z(X, t) in the primary wavefield with a specified
propagation speed Uf. The particular solution to (28) is
designated by ~z = Mr Z, where Mr is an amplification
factor in the response; hence,1
Mr ¼ 1  Uf2 =c2:

ð27ÞThis simple parameterization is generally supported according to laboratory experiments [cf. Battjes, 1972] with g = 0.73
[Battjes and Stive, 1985]. By this rule, as h(x?) decreases,
A(x?) must also decrease, which is the opposite of the
shoaling effect in (26). So our shallow-water prescription for
A(x?) is to take the minimum value between the two
expressions (26) and (27).
[21] In discretization of the forcing term (6), we have found it
necessary to apply a local averaging and smoothing operator inevaluating F (Appendix B). This is both because the composite
rule for A(x?) has a discontinuous cross-shore derivativeZ t
Ztt ¼ ð29ÞThus the resonance condition to (29) is c = ± Uf, consistent
with that by Longuet-Higgins and Stewart [1962] based on
the radiation stress formalism for long waves generation.
In deep water a characteristic depth is about 4000 m so thatc  200 m/s, whereas a typical Uf due to synoptic storm isabout 10 m/s. The group speed of a deep-water primarywaves (i.e., the phase speed of mesoscale forcing), cgr, isc,also approximately 10 m/s. Either case gives rise to Uf
so the long-wave response ~z is always off-resonant andquasi-static in deep water.
[23] Equation (28) also provides a free-wave solution if
appropriate boundary conditions are given [e.g., Proudman,

7 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 5. Amplification factor M in the same format as Figure 4 for (a – b) low-frequency band(f < 0.139 mHz) and (c– d) high-frequency band (f  0.139 mHz) evaluated from (32) with 1D solutions
with a flat bottom. The cross marks denote discrete forcing conditions for the calculations. Overall thepattern of M resembles jMrj in Figure 4.

1953]. Suppose that there is a shoreline at X = 0 such thatbarotropic velocity is q(0, t) = 0, and Z moves toward theshoreline. Then we obtain the general solution of (28) as asum of the particular solution (above) and a general solutionof the homogeneous form of (28):Uf~
z ¼ Mr Z t  X =Uf  Z ðt  X =cÞ :c

ð30Þ[24] This equation illustrates that in shallow water, where
the response to the traveling disturbance Z comes close toresonant condition, there are two types of response: 1)
bound (slaved) waves that travel with the forcing fieldandpﬃﬃﬃﬃﬃ2) free waves that propagate at a phase speed of c =gh. Figure 4 shows jMrj as a function of depth h, a phasespeed of synoptic storm Cs, and a primary wave period

Tp, which is relevant to the phase speed of mesoscale
forcing, cgr. Hence the forcing phase speed Uf in (29) is
given by Cs or cgr(h, Tp). Long waves are excitedintensively in shallow water either by synoptic storms
or mesoscale forcing, if h < O(102) m.
[25] Next, a series of one-dimensional (1D), nondissipative, forced long-wave experiments with a flat bottom is
conducted to reproduce resonant response to given synthetic
forcing fields. The forcing imposed here is a combination ofa synoptic-scale storm traveling at a phase speed of Cs andmesoscale forcing amplitude modulation as described in
section 4 with lm = 20 km and d 0 = 0.16. A total of 10
depths ranging from 20 to 4000 m are chosen to represent
the flat topography to cover from shallow to deep waterconditions. The length of the domain is 256 km with a grid
spacing of 1 km. The initial condition is a resting state. A
periodic condition is applied to ~z and q at the two open

8 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 6. Amplification factor M for the high-frequency band (f > 0.1 mHz) calculated from (32) withthe 2D, nonbreaking cases for constant-slope cross-shore topographies: (a) without and (b) with ashoreline by applying a Flather-type radiation condition and a flux-blocking condition on the shallowest
side of the domain. The bottom slopes change with the minimum depth of the domain hmin while theoffshore maximum depth is fixed at hmax = 3000 m. The resonant parameter jMrj with Tp = 6 s (i.e.,cgr  5 m/s in deep water) is indicated by the solid curves to demonstrate that jMrj is close to M.
Including the shoreline increases the response by a factor of 2 due to shoreline reflections.

boundaries. The primary wave amplitude As resulting froma synoptic storm traveling in the X direction is defined as
As ð X ; t Þ ¼ A0 þ Af cosð Ks X  Ws t Þ;

ð31Þ

where A0 and Af are the mean and fluctuating amplitudes(= 3.25 and 1.75 m, respectively). The storm wavelengthis ls = 2p/Ks = 256 km, and Cs = Ws/Ks is its phasespeed equivalent to the group speed of the primary waves
as represented by (22) with a primary wave period, Tp,
and the linear dispersion relation (2). We conduct 1D
simulations with ten Tp ranging from 2.5 to 40 s with a constantCs (= 10 m/s), and ten Cs values ranging from 1 to 30 m/s with aconstant Tp (= 10 s) for each of the ten flat topographies; hence
a total of 200 configurations are examined.
[26] An amplification factor M is calculated from thenumerical solutions to compare with the theoretical estimateMr:
M ¼ Z t~z R:M :S::

mesoscale responses. Figure 5a is seemingly similar to
Figure 4a, and Figure 5d resembles Figure 4, demonstratingthat long-wave responses to the given forcing at an arbitrary
depth are well approximated by the simple expression jMrj
in (29). For example, M tends to have the maximum not at
the shallowest depth but slightly deeperpﬃﬃﬃﬃﬃ (Figures 4a and 5a),since jMrj and M peak where gh = Cs (e.g., a peak
appears at h  40 m if Cs = 20 m/s). On the contrary, M
monotonically increases as depth decreases, being thelargest at the shallowest part (Figure 5b). The overall patternof M in Figure 5b,c is essentially independent of Tp and Cs.
[28] These results clearly indicate that the low-frequency
and high-frequency responses are amplified according to Cs
and cgr(h, Tp), respectively: the mesoscale fluctuations in
the forcing are responsible only for high-frequency mesoscale responses, while synoptic-scale variability in theforcing induces low-frequency responses. The forcing phasespeed Uf due to synoptic storms (Cs) and mesoscale fluctuations (cgr) are both important parameters that control
resonant amplification of the infragravity waves.

ð32Þ

F dtjR:M :S:

7. Deep-Water GenerationR~z R.M.S. and t F dtjR.M.S. are standard deviations of theR
response ~z and time-integrated forcing t F dt computed ateach grid point for 14 h after the model reaches steady state(about 12 h after the initiation of the model). <> denotes aspatial-averaging operator.
[27] Figure 5 shows 1D solutions expressed by M. M forthe low-frequency band exposes synoptic-scale responses,
and the high-frequency band approximately represents the[29] Next we extend the 1D flat-bottom experiments to
2D cases on variable topographies in nonbreaking, deepwater conditions for the primary waves. The 2D topographies
employed here are planar beaches with constant slopes in the
cross-shore (X) direction but uniform in the alongshore (Y)direction, expressed by the following equation:

9 of 25

hð X ; Y Þ ¼ hmax  ðhmax  hmin ÞX =XL ;ð33Þ

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 7. Free-surface response ~z R.M.S. as a function of offshore distance for the 2D, deep-waterz R.M.S. is decomposed into (a) lowexperiments with nonbreaking forcing, for various hmin as used in (33). ~frequency variations (f < 0.1 mHz), (b) high-frequency (f >0.1 mHz) infragravity free propagation, and(c) high-frequency bound response. The first four cases in the legend in (c) are without a shoreline, while
the others have a shoreline. The shoreline modifies only the infragravity free component.

where hmax = 3000 m is the maximum depth at the seaward
boundary X = 0; hmin is minimum depth at the shoreward
boundary X = XL; and XL is the lateral length of the domain.
The domain is set to be 256  256 km, discretized with auniform grid spacing of 1 km. The linear drag coefficient Cd
is set to 1.0  103 m/s. We impose hmin = 100, 500, 1000,
2000, and 3000 m (flat bottom). A periodic condition isapplied in the alongshore Y direction, while a Flather-typeradiation condition is adapted at the seaward boundary, X = 0.
Two types of the shoreward boundary condition are
considered at X = XL: the Flather-type radiation condition
so that the domain has no solid boundary, and a flux-blocking
condition to make the shoreward boundary be a more
complete shoreline. The alongshore periodic forcing is
imposed by a traveling synoptic storm at a phase speed,
Cs = Ws/Ks = 10 m/s, and the wavelength of the travelingstorm is ls = 2p/Ks = 256 km. A mesoscale amplitude
variation is further considered on top of the synoptic-scale
forcing, with lm = 20 km and d0 = 0.16.
[30] The amplitude of the synoptic storm, As, at theoffshore-most position is defined asAs ð0; Y ; t Þ ¼ A0 þ Af cosðKs Y  Ws t Þ:

ð34Þ

A0 and Af are a mean amplitude (= 3.25 m) and a fluctuating
amplitude (= 1.75 m). On the contrary, primary wave period
and direction are set as constants, i.e., Tp = 6 s and qp = 30°.
Changes in wave number and wave amplitude in the crossshore direction are computed based on the WKB theory
described in Section 5 and Appendix B. The minimum
depth hmin = 100 m and primary wave period Tp = 6 s leadto kh > 11.17; hence, all the experiments are in deep-waterconditions. None of the cases here involves primary-wave
breaking (section 8).
[31] Figure 6 displays the amplification factor, M,
defined by (32), for the high-frequency band (i.e., frequency
f > 0.1 mHz) as a function of depth. The analyticallyderived resonance parameter, jMrj, given by (29) with Uf =cgr (h, Tp) is also plotted by the solid curves. 2  jMrj isshown in Figure 6b to approximately account for reflected
waves from the shoreline [Sheremet et al., 2002]. Since theprimary waves are in deep water (kh  1), the R.M.S. forcing
is independent of the water depth. M (thus the response ~z)appears to increase remarkably as depth decreases eitherwith or without the shoreline. The analytically derived
jMrj is broadly consistent with the M estimated from thenumerical solutions; so the amplification mechanism discussed in section 6 is also relevant to 2D wave excitation
on sloping topographies.
[32] We further apply a 2D frequency-alongshore wave
number (W-KY) FFT band-pass filter in order to separate thehigh-frequency component of the simulated responses into afree propagation and a forced response that are bound to thehigh-frequency, mesoscale forcing. A combination of the
cut-off frequency and wave number is defined by a phasespeed of alongshore infragravity waves, cY = W/KY = 10.0 m/s.
This filter works well to decompose the high-frequency signals
into free propagation and bound response if h >10m since thepﬃﬃﬃﬃﬃ
phase speed of the free propagation should be c = gh >10 m/s,
although the bound component travels at c = cg, i.e.,group speed of the primary wave, which is about 5 m/s
for the given primary wave period (i.e., Tp = 6 s). A
combination of a high-pass frequency filter and 2D WKY filter is used to extract the three response components in the 2D regional-scale simulations.
[33] The decomposed response components are plottedagainst the offshore distance in Figure 7: (1) low-frequency
synoptic components, (2) high-frequency free propagation(f > 0.1 mHz), and (3) high-frequency bound components.
Inclusion of the shore produces evident differences in~z R.M.S. for only the high-frequency free component, notthe other two components. Although the bound response isequivalent to the free response in shallow water, it decreases
sharply as depth increases and the free components make agreater contribution to ~z R.M.S. in deep water than the boundcomponent. Thus free wave propagation from the shore can

10 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 8. Long-wave response ~z R.M.S. decomposed for (a) low-frequency band ( f < 0.1 mHz), (b) highfrequency (infragravity) free-propagation component, and (c) high-frequency bound component for thealongshore-uniform, idealized regional topography with several values for Hfmax (upper) and Hfmin (lower).
spread over the ocean to amplify the infragravity waves indeep water.

8. Nearshore Influences[34] The nearshore region plays three important roles in
the long-wave dynamics: (1) alteration of primary-waveaveraged forcing field particularly in the surf zone; (2) edgetrapping of free waves and thereby reduction of theiroffshore propagation; and (3) resonant amplification when
the forcing propagation speed comes close to the phasespeed of long waves. We examine the first influence byisolating the forcing in two regions, i.e., in shallow and deep
water. To do this two forcing depth scales are introduced,
Hfmin and Hfmax. Respectively, these are the minimum andmaximum depths where the forcing is applied. The forcing
field consists of a periodic alongshore-traveling synoptic
storm with a mesoscale modulation (as in section 7). The
topography here is an idealized U.S. West Coast topography(Appendix C). The parameterization of primary wave
breaking is described in section 5 and Appendix B.
[35] We assume alongshore-homogeneity in the topography by choosing Ac = 0, la = 1, hmin = 1 m, and hmax =2600 m in (C1) in Appendix C. The depth scales Hfmax andHfmax are chosen to vary from 20 to 1600 m. A caseunconstrained either by Hfmax or Hfmin is additionallycalculated out as the reference case. The long-wave
response in Figure 8 shows that the low-frequency andhigh-frequency free-wave responses are almost identicalregardless of Hfmax, but they depend quite sensitively on
Hfmin. Imposing a finite Hfmax is significant only in thehigh-frequency bound component, although the free component is always larger than the bound component. Thisimplies that the primary-wave forcing in the nearshore
region is remarkably important in generation and propagation of long waves throughout the regional oceans. Includingz R.M.S. in deep water,
Hfmax significantly alters the bound ~which is comparable in magnitude to the free ~z R.M.S. The
results with various Hfmin are shown in the three lower panels
in Figure 8. The wave responses for both the low-frequency
band and high-frequency free components are diminished as
Hfmin decreases to omit the shallow-water contribution fromthe forcing field. Removing the shallow-water forcing within only several grid points from the shoreline (e.g., for thez R.M.S. by an order ofcase with Hfmin = 20 m) attenuates ~magnitude compared to the case with no Hfmin constraint.
However, the wave amplitudes for the low-frequency band
converge in the deepest region regardless of Hfmin. Thissuggests that the offshore forcing is still responsible for mostof the deep-water wave amplitude through the high-frequency

11 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 9. ~z R.M.S. decomposed for (a) low-frequency band (f < 0.1 mHz), (b) high-frequency freecomponent, and (c) high-frequency bound component for the alongshore-uniform, idealized regional
topography: (upper) with mesoscale wavelength lm = 10, 20, and 40 km, and with no mesoscale forcing(lm = 1); (lower) with fractional mesoscale amplitude d0 = 0.05, 0.10, 0.15, 0.20, 0.25, and 0.30.
Including mesoscale modulation enhances the high-frequency infragravity response.

bound waves. The greater part of the high-frequency freewaves here are excited in shallow water and propagate
offshore; however, even if the nearshore forcing vanishes,
free waves are excited locally in deep water (as also shownin section 7). The free waves generated in deep waterpropagate shoreward and amplify by the mechanism discussed in section 6.

9. Effects of Mesoscale Forcing Modulation[36] Inclusion of mesoscale modulation on the synopticscale disturbances can cause distinctive enhancement of thewave-averaged forcing F in mass conservation (3). There
are three parameters to determine mesoscale forcing, viz., afractional amplitude d 0, a phase speed, cgr, and a wave
number, Km = 2 p/lm, of the mesoscale forcing modulation.
Among them, cgr is spontaneously given by the synoptic
wavefield as expressed in (22). The other two parameters
should be determined through observation or separate
estimations of mesoscale forcing development, although insitu data on mesoscale waves in deep water is rare. In thissection we investigate effects of these two mesoscaleparameters on the long-wave response with the present
formalism. Regional experiments are conducted on the same
topography and synoptic-scale forcing as those used in thepreceding section 8, i.e., on the alongshore-uniform idealized U.S. West Coast topography (Appendix C) with thetraveling synoptic storm at Cs = 10 m/s in the alongshore
direction with mesoscale modulation through lm and d 0.
The breaking rule for the primary waves is also introduced
as in section 5 and Appendix B. Two types of experiments
are performed: (1) various lm ranging from 10 to 40 km
with a constant d 0 = 0.16, and (2) various d 0 ranging from0.05 to 0.30 with a constant lm = 20 km. A case without amesoscale component is also calculated for reference.
z R.M.S. with various lm[37] Figure 9 shows the simulated ~(upper) and various d 0 (lower) as a function of depth for (1)low-frequency band, (2) high-frequency free componentand (3) high-frequency bound component. Since we imply
lm < ls, inclusion of mesoscale does not influence the lowfrequency forcing by the definition in section 4. Hence lm
and d 0 do not alter the simulated low-frequency ~z R.M.S.,
whereas mesoscale forcing plays a role in altering the highfrequency responses. As indicated in the upper panels, the12 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 10. ~z RM.S. for (a) low-frequency band (f < 0.1 mHz), (b) high-frequency bound component, and(c) high-frequency freely propagating component for alongshore-varying topography with various crossshore topographic amplitudes (Ac = 0, 5, 10, 20, 40, 60 and 80 km) and the standard forcing defined
before (39). The alongshore topographic wavelength la is 128 km.

larger Km (i.e., the smaller lm) is, the more energetic are thehigh-frequency free and bound infragravity waves. The
mesoscale forcing,

@z sm
;
F m ¼  rTSt
þ
m
@t

ð35Þ

polynomials [e.g., Mei, 1989]. Ursell [1952] extended thistheory to discover the complete edge wave spectrum without
the assumption kh! 0 by introducing a wave velocity
potential. The resultant discrete edge wave dispersion
relation is
w2n ¼ gk sin ½ð2n þ 1Þb ;is approximately proportional to Km, since the terms
proportional to r(2dm + d 2m) and @t(2d m + d2m) are dominantin (35). Similarly, the high-frequency ~z R.M.S. depends on d 0
as shown in the lower panels in Figure 9. In addition, from(20) with Wm = Kmcgr, a larger Km induces higher-frequency
fluctuations in both of forcing and thus response, and ofcourse a larger d0 also strengthens the high-frequency wave
response.

10. Trapped Modes and Offshore Leakage[38] Edge waves can be trapped near the shoreline on a
sloping planar topography [e.g., LeBlond and Mysak, 1978;
Mei, 1989]. The regional-scale experiments reveals thatshallow coastal topographies potentially excites both boundand free infragravity waves owing to a combination of alarger forcing amplitude and resonant amplification (section 6).
Ultimately all the infragravity-wave energy may be confined around the shoreline if nearshore bathymetry monotonically deepens into the interior. Suppose the nearshore
bathymetry consists of a uniform slope, viz., tan b = const.,
and a straight shoreline with an offshore boundary condition of ~z ! 0 as x ! 1. In such an idealized configuration, Eckart [1951] solved the confluent hypergeometricequations derived from the long-wave equation without
forcing and found the edge wave dispersion relation
corresponding to the following discrete values:
w2m ¼ gk ð2n þ 1Þ tan b;

ð36Þ

ð37Þ

with (2n + 1)b < p/2 and n = 0, 1, 2, . . .. Equations (36) and(37) indicate that higher modes edge waves tends topropagate further offshore than lower modes. If ~z has afinite amplitude at x ! 1, then we call this a leaky
solution that can exist for
w2 > gk:

ð38Þ[39] In reality obliquely incident waves and their reflections from the shoreline are refracted on a given sloping
topography to form edge waves. However, beyond a certain
incident angle, long waves reflected at the shoreline leakoffshore and rarely return to the nearshore region. So theleaky mode, as well as higher edge modes, can be responsible for increasing the wave amplitude in deep water.
[40] Edge waves are always found in the present solutions
with the sloping topography and shoreline. Given the waveaveraged forcing, the most likely cause that induces offshore propagation of excited infragravity wave is alongshore variability of topography. To assure this we perform
computations with Ac = 5, 10, 20, 40, 60, and 80 km (seeFigure C1 in Appendix C). We consider two types offorcing. The first one is the same as those imposed in thepreceding sections: the alongshore-traveling storm at Cs =10 m/s and Ts = 6 s with mesoscale amplitude modulation
using lm = 20 km and d 0 = 0.16, calling this the standard
forcing for convenience. Another type of forcing is alsoessentially the same as the standard forcing, but the primarywave period Tp on the synoptic storm varies as

where n = 0,1,2,. . . denotes discretized eigenmode numbers.
The associated eigenfunctions are proportional to Laquerre
13 of 25

Tp ð X ; Y ; t Þ ¼ Tm þ Tf cosðKs Y  Ws t Þ:

ð39Þ

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 11. Same as Figure 10 except for using broadband forcing defined after (39) and decomposing~z R.M.S. for the hum band ( f > 1.25 mHz) in Figures (b) and (c).

Tm is a mean period (= 8 s) and Tf is a fluctuating period(= 2 s), so that Tp varies from 6 to 10 s as the synoptic storm
evolves. Since dominant frequency of mesoscale forcing is
Wm = Km cgr and cgr is altered with Tp, the resultant forcing
spectrum becomes much broader than that with the basic
forcing. We call this broadband forcing.
[41] Figure 10 shows the decomposed wave responses toz R.M.S.
the standard forcing with various Ac. As Ac increases ~decreases for the low-frequency band, and the high-frequency
free component decreases in shallow water but increases indeep water. Consequently, the on-offshore differences in~z R.M.S. diminish and make the ~z R.M.S. (h) curves shallower as
the alongshore topographic variability increases (i.e., Ac
increases). The high-frequency bound component alsochanges as the forcing field is altered with Ac, but with amuch weaker dependence on Ac than that for the highfrequency free component. Thus the alongshore topographic
variability tends to release the offshore wave energy as freewaves from the nearshore into deep water, while the boundwaves remain concentrated in shallow water. This tendency
is also generally true for the cases with the broadband
forcing (Figure 11). In particular, on-offshore differencesin ~z R.M.S. in the hum-band free component decrease as Ac
increases.
[42] Figure 12 shows the frequency-wave number (reciprocal alongshore wavelength) spectra of the simulated
alongshore velocity at 20 m deep for Ac = 0 (viz., straightcoastline), 5 and 20 km with the broadband forcing. The
broken curves illustrate the discrete edge-wave dispersion
relation derived by Ursell [1952] as represented in (37) onan assumption that the nearshore bottom slope is approximated by b = 0.004, which is about the mean slope between
the shoreline to onshore edge of the shelf break. The solid
curves in the low-wave number range are the corresponding
leaky cut-off curves defined by (38). The spectra indicatethat the infragravity energy tends to concentrate on thediscrete, lower mode edge-wave dispersion curves. As Ac
increases, the response spectral energy is transferred fromthe low modes to the higher modes and even to the leaky
range. In the straight coastline case (Figure 12 (a)), spectral
density is mostly concentrated on the modes 0 and 1

dispersion curves in the region where reciprocal wavelengthis negative, coinciding with the incident wave direction.
6 0),
Once alongshore topographic variation is present (i.e., Ac ¼
spectral energy spreads in the positive reciprocal wavelength. The spectral energy is scattered to high-mode,
high-frequency, and low-wave number regions as A c
increases, hence the leaky modes become more significant.
These results clearly demonstrate that edge waves are
excited most effectively in the straight-coastline case,and the offshore propagation of long waves is less dominant. Alongshore topographic variation acts to transfer the
response from the discrete lower edge modes to the higher
edge and leaky modes as free waves, reducing ~z R.M.S. in
the nearshore and enhancing it offshore.

11. Basin-Scale Infragravity Waves[43] Rhie and Romanowicz [2004] report strong amplitude for Earth’s hum in the North Pacific in a boreal winter.
They focus on two distinct hum events between the 31st and34th Julian days in 2000. The inferred generation site for thehum shifts from the North Pacific to the Southern Ocean
between boreal and austral winter. Rhie and Romanowicz[2006] further examine these two hum events and detect a
lag time of several hours between the seismic signals at their
two arrays. Guided by a primary wind-wave analysis with a
global wave model, they infer that part of pressure fluctuation in infragravity waves generated by synoptic storms
converts to seismic waves locally and propagates long
distances as Rayleigh waves to produce the observed hum
events at remote sites. The other part is reflected back out
into the open ocean as oceanic infragravity wave, propagates across the Pacific basin, and couples to the seafloor
near the coast to excite hum remotely. Infragravity waves inthe deep ocean and the resulting seismic hum are attributed
to a combination of local and remote generation [Webb etal., 1991]; thus, basin-wide simulation is essential. In
addition, in the regional simulations in sections 7 – 10,
infragravity waves are most effectively excited near the
shore, but they are mostly trapped in shallow water as lowmode edge waves. Alongshore topographic variability can

14 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 12. Frequency-wave number (alongshore reciprocal wavelength) spectra of alongshore velocity
at h = 20 m for three cases with cross-shore amplitude of the alongshore topography (a) Ac = 0 (straightcoastline), (b) 5 km, and (c) 20 km with alongshore topographic wavelength la = 128 km. The discrete
theoretical edge-wave dispersion curves by Ursell [1952] are also plotted by assuming that the nearshore
topography is planar with a uniform bottom slope of 0.004. As Ac increases the spectral energy in the lowfrequency edge modes spreads over the high-frequency region into both edge modes and leaky modes.
increase the offshore propagation as high-mode edge waves
and leaky waves.
[44] To synthesize the influences of these competing
processes, realistic basin-scale simulations are necessary
to investigate oceanic infragravity waves in relation to
global-scale hum generation. We use the forced-dissipative
long-wave model in the Pacific Ocean with realistic primarywave-averaged forcing using ECMWF’s ERA-40 wave data
[Bidlot et al., 2002; Janssen et al., 1997], modulated bymesoscale variability (section 4). The horizontal grid spacing is chosen to be 1/8 °, and the time step is 18 s. For thesynoptic scale forcing, statistical wave quantities such as Hs,
Tm, qm, TsStand z ss are computed from the 2D spectral wave
data on the ECMWF’s analysis grid at a 1.5° resolution andthen interpolated on to the model grid with a cubic tension
spline interpolation. The nearshore wave number field is
modified for kh < p as described in section 5 and Appendix B.
This modification helps reduce an underestimation ofshallow-water infragravity-wave generation in the regional-scale configuration, and it plays a similar role in the
Pacific with its even larger grid spacing of 1/8°. On thebasis of our experience with different grid spacings, we
believe that the residual underestimation bias is not very
important for the simulated deep-water response in the
Pacific, although almost certainly the simulated nearshore
infragravity-wave amplitudes are too small. F is computedby centered finite differences scheme with second-orderaccuracy both in space and time. Mesoscale components
are evaluated with the parameter values lm = 100 km andd 0 = 0.16. Since the spatial domain is wider and the lengthof integration is longer in the basin-scale experiments thanthose in the regional cases, the effective wave number andfrequency due to mesoscale forcing may undesirably
become fast-varying functions. To minimize this problem,a piecewise local-origin treatment is implemented to evaluate the phase function of the mesoscale fluctuations as
specified in Appendix D.

11.1. Infragravity Waves on 31 January 2000
[45] Basin-scale experiments are made for the first 40 d in2000 on the 1/8 ° Pacific grid. We first make the calculation
as described immediately above, and call this the baseline
case. Figure 13 shows sequential snapshots of the longwave forcing and response fields every 6 h on the 31st day
in 2000 for the baseline case. Long waves are observed tospread over the basin and propagate everywhere with arelatively large wavelength compared to the mesoscalez iswavelength, lm = 100 km. In addition, the simulated ~observed to be generally large near the coasts, in the highlatitude regions (e.g., near the Aleutian Arc and along the
U.S. West Coast), and in semienclosed bights such as the
Yellow Sea, Sea of Japan, South China Sea, and Bering Sea.
[46] The temporal evolution of the long waves is displayedin Figure 14 at several latitudes. Progressive free waves are
ubiquitous in all the plots, yet quasi-static responses bound to
the forcing appears to be minor. Both easterly and westerly
propagations are nearly equally
excited with a phase speed
pﬃﬃﬃﬃﬃ
approximately equivalent to gh  200 m/s. Long-wave
energy is intensified near the coasts probably due to resonant
amplification and topographic trapping as examined in
sections 6 and 10. The dominant response period in thebasin is around 3 h, roughly corresponding to a dominant
forcing period of Tm = lm/cgr  3 h. Evolving patterns oflong waves are remarkably altered in embayments (i.e.,the Yellow Sea in Figure 14b and the Sea of Okhotsk in
Figure 14c) from those in the Pacific basin. There is noclear correlation with the generally westerly evolution of
the forcing field, indicating a nonresonant response.
[47] Figure 15 shows the R.M.S. wave amplitude averaged
over the North Pacific as a function of depth and decomposed
into four components: low-frequency (f 0.1 mHz); highfrequency free wave (f > 0.1 mHz and c = W/K > 15 m/s);
high-frequency bound wave (f > 0.1 mHz and c 15 m/s);
and hum band (f > 1.11 mHz). The high-frequency bound15 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 13. Six-hourly snapshots in the Pacific Ocean of (left) the forcing (the color is the significant
wave height Hs in m; the vector direction is mean wave direction qm, and the vector length is mean wave
z in mm on 31 January 2000. Freely propagating
period Tm in s), and (right) the wave response field ~waves are ubiquitous over the basin, and they do not simply correlate with the forcing.
component decreases more rapidly with depth than the othercomponents; the high-frequency free component has anamplitude of about 2.4  103 m in shallow water anddecreases slowly with depth / h1/4 to around 1.0  103m in deep water. We attribute the weak depth dependenceof ~z R.M.S. to the complexity of the topography that, which
can increase offshore propagation of coastally trapped
free waves (section 10), and to the more realisticallycomplex synoptic-scale forcing (plus its associated meso-

scale component) in the Pacific that magnifies the deepwater free-wave response. Moreover, the simulated highfrequency amplitude in deep ocean is observed to beapproximately consistent with the in situ infragravitywave amplitude reported by Webb et al. [1991] in the
Pacific Ocean. The high-frequency response is almost
equally composed of free and bound waves in shallow
water, while free waves dominate over bound waves indeep water. In the baseline case with mesoscale param-

16 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure 14. Temporal evolution of ~z in mm along (a) 6.7873°N, (b) 30.06°N and (c) 48.9431°N on
31 January 2000.
eters lm = 100 km and d 0 = 0.16, ~z R.M.S. for the humband is about an order of magnitude smaller than ~z R.M.S.
for the whole high-frequency band.
11.2. Mesoscale and Deep-Water Forcing[48] The baseline case is now compared to the three typesof experiments, i.e., the cases with no offshore forcing, no
mesoscale forcing variation, and with higher mesoscalewave numbers (i.e., lm = 20, 30, 40, and 60 km). Figure
16 shows the high-frequency component ( f > 0.1 mHz) for
four of these seven cases. Excluding deep-water forcing
reduces ~z R.M.S. to about 30% of the baseline case, especiallyin deep water. Excluding mesoscale forcing modulation
reduces ~z R.M.S. to merely about 10% of the baseline case.
z R.M.S. up to almost 10-fold,
Conversely, increasing Km boosts ~consistent with section 9 where F m and thus the infragravity response are proportional to Km in (35). While~z R.M.S. tends to be intensified near the coasts (e.g., off the
U.S. West Coast) for the four cases, it is clear that deepwater forcing is necessary for the long waves to be strong
in deep water (e.g., south of the Aleutian Arc where thesynoptic-scale forcing is generally intense; Figure 13).
z R.M.S. (h) relations are displayed in[49] Decomposed ~Figure 17 for all these seven cases. Nearshore responses forthe baseline and no offshore forcing cases are almost thesame, although offshore amplitude for the baseline case isabout 4 times larger than with no offshore forcing. The
cases with higher Km have a larger amplitude for all
frequency bands, especially for the hum band. As Km ishigher (lm shorter), the long-wave amplitude is larger. The
deep-water response is amplified by both nearshore andmesoscale forcing, but it also has a significant component
forced in deep water by synoptic storm waves. This latter
result is partly in contrast with the regional-scale cases in
section 8 with simpler topographies.
[50] The baseline case is further compared to a case withmultidirectional mesoscale variations as described inAppendix E, in order to examine importance of directional spreading in mesoscale forcing. It is demonstrated
that the multidirectional mesoscale variations do notsignificantly alter overall amplitudes of the infragravity
wave. Therefore the unidirectional, monochromatic representation used throughout the present study is essentiallycapable of estimating infragravity wave amplitudes without
a full spectral consideration of mesoscale modulations.

12. Influence on Earth’s Hum[51] Although infragravity waves have been recognized
as a likely cause for seismic hum, it has not been fully
understood how large the infragravity amplitude must be toaccount for observed hum magnitudes. Tanimoto [2005]

17 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

Figure 15. ~z R.M.S. as a function of depth, averaged over
the North Pacific Ocean between 0°N and 56°N and over all
24 h on 31 January 2000. ~z R.M.S. for the high-frequency( f > 0.1 mHz) bound component decreases more rapidly
with increasing depth than the other components. The
high-frequency free-wave component is around 103 m indeep water, and its depth dependence is approximately h1/4.

proposes a linear transfer function from stochastic oceanic
bottom pressure to seismic vertical acceleration due tonormal-mode excitation based on the elastic models of
Tanimoto and Um [1999] and Fukao et al. [2002]. This
proposal fits the characteristic shape and level of the lowfrequency background noise spectrum, and it indicates that

C07029the typical excitation area could be rather small (i.e., about
100  100 km2) at any given time. It may be controversial[Rhie and Romanowicz, 2006; Webb, 2007] whether thedifference in wavelength between infragravity and elastic
waves can be represented by the linear transfer function
model; nevertheless, Tanimoto [2005] indicates that a bottom pressure spectral density, < P2b >, of 103 to 104 Pa2Hz1
is required for hum generation by infragravity waves.
[52] To check whether our simulations match this requirement, the bottom pressure spectral density, <P2b ( f )>, isevaluated from ~z at four selected locations in the North
Pacific (Figure 18) for both the baseline and high Km (lm =20 km) cases. At low frequencies the long-wave response ismuch larger than the level required in the hum band. In thebaseline case, <P2b ( f )> for the hum band is mostly too
small compared to the required amplitude, by about an orderof magnitude except in the Bering Sea where ~z R.M.S. isrelatively large (Figure 16a). However, with the larger Km
value (Figure 16), the peak frequency shifts to around
0.5 mHz, and <P2b ( f )> becomes much more energetic in
the hum band, ranging from 103 to 105 Pa2Hz1 in the
Bering Sea and off Southern California down to 102 to
104 Pa2Hz1 west of Hawaii and southeast of Japan. So, with
an appropriate mesoscale wave number Km and fractional
amplitude d0, the simulated amplitude of the hum-band
infragravity waves in the Pacific Ocean is large enough toexcite Earth’s hum locally both in nearshore shallow wateror in the middle of the basin. Of course, our monochromatic
model of mesoscale waves is an oversimplification of thebroad intermediate, mesoscale band (section 4), and bothmore field measurements and a more elaborate representa-

Figure 16. Long-wave response z~R.M.S. for the high-frequency component (f > 0.1 mHz) on 31 January
2000 for (a) the baseline case; (b) no offshore forcing (F = 0 where h > 200 m); (c) no mesoscale forcing(d0 = 0); and (d) a higher mesoscale wave number (lm = 20 km). Note the different color scales used ineach plot. Excluding deep-water and mesoscale forcing significantly reduces ~z R.M.S., but increasing Km
amplifies it.
18 of 25

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029C07029Figure 17. Long-wave response ~z R.M.S. as a function of depth, averaged over the entire Pacific domainfor (a) all frequencies, (b) high-frequencies ( f > 0.1 mHz), and (c) hum band ( f > 1.11 mHz) for the fourcases in Figure 16. The baseline case has a larger wave amplitude in both shallow and deep water than thecases with no offshore forcing or no mesoscale forcing. Increasing of Km (i.e., decreasing lm from 100 km
for the baseline case) remarkably magnifies the response amplitude, especially for the hum band.
tion of the forcing are needed to refine the assessment ofhum generation by mesoscale infragravity waves.

13. Conclusions[53] This paper analyzes the generation and propagation ofoceanic infragravity waves as the cause of Earth’s continuous
background seismic free oscillations, i.e., hum. The dynamics
of long gravity waves is modeled by a linearized, forceddissipative, shallow-water equation and implemented in a 2D
version of ROMS. Long-wave forcing is due to quadratic
nonlinear interaction among primary wind- and swell-waves
on the basis of a multiscale asymptotic wave-current interac-tion theory. The slowly evolving primary-wave-averaged
forcing field is represented by a combination of synoptic-scalevariability and mesoscale amplitude modulation. Changes in
the primary wave number and wave breaking near the
shoreline are based on ray theory and an empirical breaking
parameterization.
[54] Resonant amplification is first examined with ananalytical solution and 1D, deep-water numerical experiments. The theoretical analysis gives a resonant amplification parameter, Mr in (29), indicating that an amplified
response is expected only in shallow water, while the deepwater response is always off-resonant. From (29) a phasespeed of the forcing is found to control amplification of long

Figure 18. Bottom pressure spectral density, <Pb2>, on 31 January 2000 at four locations in the North
Pacific Ocean: south of the Bering Sea (179.20°E, 49.02°N, h = 5140 m); off southern California(122.41°W, 36.37°N, h = 681 m); west of Hawaiian Ridge (152.60°W, 21.72°N, h = 5218 m); andsoutheast of Japan (137.15°E, 32.28°N, h = 4071 m), for (a) the baseline case and (b) the mesoscale wave
number case with lm = 20 km. A spectral density greater than about 104 Pa2Hz1 can excite normalmode seismic oscillations relevant to the observed hum according to the criterion of Tanimoto [2005].
19 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

waves. A traveling synoptic storm is responsible for lowfrequency forcing fluctuations and a bound response, while
the speed of mesoscale forcing (i.e., the group speed of the
primary waves) more effectively forces high-frequency,
freely propagating infragravity waves. Including a solid
shoreline amplifies the high-frequency infragravity freewaves due to reflection from the shore.
[55] Regional-scale experiments are performed including
the shoreline and the depth-induced wave breaking overidealized U.S. West Coast topographies. A baseline case ischosen with an alongshore-uniform topography and a relatively simple forcing by a periodic synoptic storm travelingin the alongshore direction at a constant phase speed, Cs =10 m/s with a mesoscale amplitude modulation with afractional amplitude of d0 = 0.16 and wavelength of lm =20 km. This forcing excites infragravity waves mostly inshallow water. The infragravity-wave amplitude, ~z R.M.S.
strongly diminishes as the depth increases, approximately
inversely proportional to the depth. Contributions in ~z R.M.S.
from free infragravity waves and high-frequency boundwaves are nearly unchanged as the depth changes, and freewaves are dominant over bound waves everywhere. To
investigate the relative contributions of shallow-water anddeep-water forcing, we make regional experiments varying
the spatial distribution of the primary-wave forcing. Excluding deep-water forcing alters ~z R.M.S. mainly in the highfrequency bound component in deep water. However,excluding shallow-water forcing significantly diminishes
the amplitude at all frequencies for the long-wave response.
[56] The effects of mesoscale forcing variability are
represented by two empirical parameters, lm and d 0 in(18) and (19). For simplicity our mesoscale forcing model
for lm and d0 is based on a single mesoscale component
and an average fractional amplitude. As lm decreases ord 0 increases, the mesoscale band forcing is amplified.
1 as observed, mesoscale modulation does not
With d0affect the low-frequency bound waves, but it is significant for
generating high-frequency infragravity waves. Because of (20),lm is also significant in determining the frequency of themesoscale forcing and the high-frequency response amplitude.
[57] Nearshore amplification by edge-trapping and offshore propagation are investigated in the regional-scaleconfiguration. Frequency-alongshore wave number spectra
show that freely propagating infragravity waves are refractively trapped near the coast to form edge waves. Horizontal
topographic variability is modeled with the cross-shore
amplitude of alongshore topographic variability, Ac. As Ac
increases offshore propagation of free infragravity waves
increases. This offshore propagation occurs as high-modeedge waves and leaky modes. As topographic complexity
increases, infragravity energy tends to attenuate in shallow
water and magnify in deep water.
[58] Basin-scale infragravity-wave simulations are madefor the Pacific Ocean in January and February 2000, tocompare to observations of Earth’s hum. The simulations
demonstrate that free propagation is predominant everywhere in the basin, and the quasi-static response bound to
the primary-wave forcing field is secondary. The infragravity-wave response is greater in shallow water and diminishes as the depth increases. Nevertheless, as compared tothe idealized regional results, deep-water generation isobserved to be more significant in the Pacific because of

C07029the complex coastlines that increase offshore propagation(section 10) and the realistic deep-water forcing that can bestronger than the simplified forcing imposed on the regional-scale experiments. With lm = 100 km and d0 = 0.16, theamplitude of infragravity ~z, integrated over all frequencies,
ranges from O(103) to O(102) m depending on the depth.
If lm is decreased to 20 km, the spectral energy of theinfragravity waves shifts to a higher frequency band, andthe hum band ( f > 1 mHz) response amplitude increases allover the basin to a level of O(102) m near the shore andaround O(103) m in deep water. The deep-water amplitude
is comparable to the in situ amplitude reported by Webb et
al. [1991], and it is considered to be large enough to excite
the hum locally [Tanimoto, 2005]. Therefore mesoscale
forcing variability in primary wind- and swell-waves probably is responsible for generation of oceanic infragravity
waves and the associated seismic hum in both shallow
coastal seas and the deep ocean.

Appendix A: Fractional Amplitude d0[59] To obtain Table 1, the synoptic-scale variability ofsignificant wave height Hs (thus amplitude As) is evaluatedfrom the ECMWF spectral data on their 1.5° grid with 6 hintervals, using (8) and (A1):qﬃﬃﬃﬃﬃ
1
As ðX; t Þ ¼ Hs ðX; t Þ ¼ 2 h2p :2

ðA1Þ[60] Then bilinear interpolation both in time and space isperformed to compute As at each data point corresponding
to the TOPEX satellite data,As ðX; t Þ ! As ðXT ; t T Þ;

ðA2Þ

where XT and t T are the along-satellite-path spatial
coordinate and time of the measurement. With As(XT, t T)
from the ECMWF data and its counterpart, i.e., AT(XT, t T)
directly obtained from the TOPEX data set, the wave
number spectra Ps(k) and PT(k) are estimated. From thedefinition,
A2s ¼

Z ks

A2T ¼

Ps ðk Þdk;Z kTPT ðk Þdk;

ðA3Þ

0

0

where ks and kT are the cut-off wave numbers,
ks ¼ 2p=Ls ;kT ¼ 2p=LT ;ðA4Þwith Ls and LT (Ls > LT, hence ks < kT) the spatial intervalsof the ECMWF and TOPEX wave data. The mesoscalevariance of wave amplitude, A2m , is then expressed as
A2m ¼

Z kT
PT ðk Þdk;

ðA5Þ

ks

since we define the cut-off wave number for the mesoscalevariations to be ks (section 4). Consequently, d0 can beevaluated by (23). A similar procedure has been applied for
the NDBC data (Table 2) with the cut-off frequency of the
wave amplitude spectra computed from the ECMWF data at

20 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure B1. Long-wave responses with various grid spacings with DX = 0.2, 0.5, 1, 2, and 4 km for(a) the whole band and (b) the hum band (f > 1.25 mHz) on an idealized regional domain withoutalongshore topographic variability. The mesoscale forcing is configured with lm = 20 km.

each buoy locations as the cut-off frequency for themesoscale amplitude variance.

singularity in wave amplitude A as h ! 0, a smoothing
operator for A is introduced.
2

31=m
m

Appendix B: Discrete Evaluation of Nearshore
Wave Parameterization[61] In regional and basin-scale configurations with alarge grid-spacing 103 to 104 m, the surf zone may be
represented by a single grid cell next to the shore, although
the primary wavefield could change substantially within thecell due to shoaling, refraction and breaking. So we can
include subgrid-scale variability into F . First, to avoid

1
6
7
2 gh

AðhÞ ¼ 6
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃm 7
45
1
1 þ 12 gh cg k? P 1 jkj

:

ðB1Þ

P 1 is constant along a wave ray in (26) at an arbitrary
offshore location where the primary wave amplitude, period
and direction and the local depth are all known. m is an
ad hoc blending parameter that controls smoothness aroundbreaking point. Here m = 8 is chosen to attain a sufficiently
smooth change in A in the vicinity of breaking point. A grid-

Figure C1. Left: depth profile of the idealized regional bathymetry in (C1) at y = 0 km with hmin = 1 m,
Ac = 60 km, la = 128 km, and xc = 140 km. The inset is a blow-up of the nearshore region. Right: a planview of the isobaths in m.
21 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Figure E1. Long-wave response ~z R.M.S. for the hum-band component (f > 1.11 mHz) on 31 January
2000 for (a) the baseline case (n = 1); and (b) n = 5 (±30 deg.). Inclusion of multidirectional mesoscale
components provide slightly more variability in ~z R.M.S. for each frequency band than the baseline case.
averaging operator is subsequently introduced for F, as well
as for (25), (B1), and the dispersion relation (2):@z sF ¼ r  TSt 
;
@t
!
Z ha
1
gAðhÞ2 kðhÞrdh
2 cðhÞ k ðhÞ
Dh hb
( Z

 )
ha
@
1
AðhÞ2 cg ðhÞ 1 dh ;@t Dh hb 2hcðhÞ 2!
! !
g
A2 kX !! A2 kX !!
tan bX
ck !ha ck !hb
2Dh
!
! !
g
A2 kY !! A2 kY !!
tan bYck !ha ck !hb
2Dh
"

#
N
@ 1 X
A2 ðhn Þ cg ðhn Þ 1:@t 2N n¼1 hncðhn Þ 2

ðB2Þ

[62] The overbar  denotes the grid-averaging operator;
ha and hb are the deepest and shallowest water depths inthe single grid cell of interest; Dh = ha-hb, tanbX andtanbY are the constant bottom slopes of the cell in the twohorizontal directions; (kX, kY) is the horizontal wave

number vector; and hn = hb + Nn Dh is a discretizedhnha. Wedepth in which n = 0, 1, 2,   , N and hb
set N to be 50 for a sufficient resolution of the changes in the
primary wavefield in the nearshore area. In practice, the gridaveraging is applied for the primary wavefield in the shallow
water where kh < p (h/L < 0.5) since the forcing term significantly varies only in this range. Otherwise, a simple centeredfinite difference scheme is adopted for the both terms in F using
the primary-wave quantities and the local depth at the cell center.
[63] With this approach we are able to investigate generation and propagation of deep-ocean infragravity longwaves with a relatively large grid spacing, which in practiceenables us to conduct large-region and basin-scale, phaseresolved, infragravity wave simulations (Figure B1). As apractical matter, one cannot simultaneously pursue a detailed resolution of the nearshore region and do broad-area
depiction of the hum-forcing long waves. We believe wehave done a sufficiently good job with the former in order tobe able to adequately address the latter.

Appendix C: Regional Configuration[64] We specify an idealized coastal topography by taking
am alongshore-average of the U.S. West Coast topography

Figure E2. Long-wave response ~z R.M.S. as a function of depth, averaged over the entire Pacific domainfor (a) all frequencies, (b) high-frequencies ( f > 0.1 mHz), and (c) hum band ( f > 1.11 mHz) for the twocases in Figure E1.
22 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

between about 40.6 and 48.2°N (off the Oregon Coast). The
resulting idealized topography is approximated by thefollowing equations:



Ac
2p
xs ð yÞ ¼y1  cos2lax  xs ð yÞ
xm ð yÞ ¼ max x; xc
xc  xs ð yÞhð x; yÞ ¼ a0 xm tanhðb1 xm Þ þ d1 f1  expða1 xm Þg( )xm  xp 2tanhðb2 xm Þ d2 exp a2
xphð x; yÞ ¼ max½hmin ; minfhð x; yÞ; hmax g :ðC1Þ

x (0) is distance cross-shore; y is the alongshore
coordinate; xs(y) is shoreline position from x = 0; xm is ashrunk cross-shore coordinate modified by xs and xc; xc isthe offshore distance beyond which the topography isuniform alongshore; Ac is a cross-shore amplitude ofsinusoidal shoreline topography; la is the alongshore
wavelength of the sinusoidal shoreline; d1 = 2850.0 m is
the maximum offshore depth; d2 = 970.0 m is the peak
depth at the shelf break; xp = 4.3  104 m is offshore
distance to the shelf peak; a0 = 0.9  103 is a nearshore
bottom slope; a1 = 1.7  105 and a2 = 1.2 arecrossshore spreading parameters; and b1 = 5.0  107 and b2= 1.3  104 are attenuation parameters to reduce thetopographic variability in a far field. The bathymetry is further
constrained by hmin and hmax, the minimum and maximum
water depths in the domain.
[65] An example of the idealized topography is illustrated
in Figure C1. The topography has a foreshore steep slope inthe vicinity of the shoreline defined at x = 0 to about a few
kilometers offshore as shown in the cross-sectional depth
profile (left panel), followed by a relatively mild continental
shelf extending for about several kilometers where the depth
ranges around 16 m. Afterward, a steeper shelf break emerges toincrease the depth down to 2600 m, and then a flat bathymetry
appears in the deep offshore area of x > xc = 140 km. Alongshore
variability is controlled by parameters Ac and la. In the rightpanel, Ac = 60 km and la = 128 km are assigned.
[66] In the regional-scale computations discussed in sections 8 –10, a periodic lateral boundary condition in the Ydirection and a Flather-type inflow open boundary conditionin the X-direction at the offshore boundary (X = 256 km) areapplied. Grid spacings in the both horizontal directions areconstant: DX = DY = 1 km. A time step of Dt = 3.2 s is set
by the computational-stability criterion.

Appendix D:
Variability

A Local Origin for Mesoscale[67] We specified a representation for the primary wave
amplitude that includes both the synoptic-scale variations
and a mesoscale component as expressed by equations (16)and (19) in section 4,
AðX; t Þ ¼ As ð1 þ d0 sin QÞ;

ðD1Þ

C07029where d 0 < 1 andQ ¼ Km cos qp X þ sin qp Y  Km cg ðX; t Þt
 km X þ ‘m Y  Wm t

ðD2Þis a phase function. The variables with subscript s vary on
the synoptic scale in space and time and are determinedfromthe ﬃ ECMWF wave analysis. The quantities Km =p
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
km2 þ ‘2m and d 0 are constants associated with themesoscale modulation. K1
m is assumed to be much smaller
than the synoptic scale. Thus the mesoscale wave numbers andfrequency (km, ‘m, and Wm) are also slowly varying functions.
[68] Our challenge is to evaluate (D1) and obtain a resultthat has oscillations only near the mesoscale wave numbers
and frequency that are slowly varying. A straightforward
evaluation of Q in (D2) does not meet this criterion. Thiscan be illustrated with a 1D example:^
F ðxÞ ¼ sin Q;^ ¼ k ðx Þx:QðD3ÞThe effective wave number for this function is^@Q@k¼k þx :
@x
@x

ðD4Þ

Even if @ xk is small (i.e., k is slowly varying), x can bearbitrarily large, and the effective wave number can be very
different from k, either much larger or smaller.
[69] A remedy for this bad behavior is to replace the sinQfunction with piecewise constant wave numbers and frequency. For (D1), the first step is to expand the oscillatory
function into separate coordinate function factors,
sin Q ¼ sin½ km X cos½‘m Y cos½wm Tþ sin½ km X sin½‘m Y sin½wm Tþ cos½ km X sin½‘m Y cos½wm T cos½ km X cos½‘m Y sin½wm T :ðD5Þ

If each of these 6 right-side functions can be appropriately
represented, then the total function will be.
[70] We can illustrate an appropriate representation using
the illustrative function (D3). First consider a piecewise
constant approximation where F is continuous between x
intervals but has a discontinuous derivative (i.e., is a C0function). First define intervals in x of size Dx:
xnþ1 ¼ xn þ Dx;

x0 ¼ 0;xnþ0:5 ¼ xn þ 0:5  Dx: ðD6Þ

Within each interval, xn x xn+1, k(x) is held constant at
a value of kn+0.5 = k(x n+0.5). This means that the second termin (D4) is missing and that the oscillations in F in (D3) will
have a wave number that is the local value of k. The
function itself is approximated by
F0:5 ðx Þ ¼ sin½k0:5 x ;x0
F1 ¼ F0:5 ðxÞ ¼ sin½k0:5 Dx :x

F1:5 ðxÞ ¼ F1 þ sin½k1:5 ðx  Dx Þ ;F2 ¼ F1:5 ðx2 Þ ¼ F1 þ sin½k1:5 Dx :...

23 of 25

x1 ;
ðD7Þ

x1

x

x2 ;

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

C07029Fnþ0:5 ðxÞ ¼ Fn þ sin½knþ0:5 ðx  nDxÞ ;xnxx nþ1 ; Fnþ1 ¼ Fnþ0:5 xnþ1 ¼ Fn þ sin½knþ0:5 Dx : ðD8Þ

Note that this can be made into a C1 function by doing
weighted averaging across the interval boundary: i.e., in x n mDx/2 x x n + mDx/2,no~n ðxÞ ¼ 1  ðmDx Þ1 ½x  xn þ mDx=2 Fn0:5 ðxÞFþ ðmDxÞ1 ½x  xn þ mDx=2 Fnþ0:5 ðxÞ:ðD9Þ

We choose m = 1 for computational efficiency, avoiding
overlap between the averaging intervals. In the basin-scale
experiments in section 11, dx = 500 when the mesoscale
wavelength lm ranges between 20 and 100 km.

Appendix E:
Variations

Multidirectional Mesoscale[71] The mesoscale variations partially includes the familiar ‘‘group’’ modulations at typical scales of several
primary wavelengths. This second-order low frequency
modulation is intrinsically more broadbanded in direction
than the primary wave trains [e.g., Hasselmann, 1962].
Hence the unidirectional monochromatic approach takenin the present study might be a bit too crude, provided thatthe group modulations are predominant in the mesoscalevariations. We thus test the mesoscale forcing from thisperspective by introducing directional spreading throughlinear superposition of multiple mesoscale components
distributed around the spectral-peak primary wave direction,
qp. Therefore equations (19) and (21) are modified asdm ðX; t Þ ¼

X

dn sin½Kn  X  Wm t ;

ðE1Þ

n

Kn ðX; t Þ ¼ ½Km cos qn ; Km sin qn :ðE2Þwhere Kn, dn and qn are an n-th mesoscale wave number
vector, an n-th fractional amplitude, and a propagating
direction of the n-th mesoscale component (arbitrary
angular distribution function), respectively. d n is simply
determined to conserve of the total mesoscale wave-heightvariance,d2n ¼ d20 Pn ;X

Pn ¼ 1

ðE3Þ

nwhere P is an arbitrary probability density function.
[72] We show a result with five mesoscale components,
i.e., n = 5, Pn = (0.05, 0.15, 0.6, 0.15, 0.05), qnqp = (30°,
15°, 0°, 15°, 30°) and lm = 100 km so as to compare withthe baseline case (section 11). z R.M.S. for the hum band
(Figure E1) illustrates that spatial patterns and magnitude ofz R.M.S. are almost the same for the two cases whereas there
is slightly more spatial spreading with n = 5 than n = 1.
Spatially averaged ~z R.M.S. (Figure E2) also shows that themultidirectional mesoscale variability does not play a substantial role in modifying the overall amplitude of the long-

C07029wave response. We have conducted a few more tests withdifferent choices of Pn and qn, and obtained essentiallysimilar results. The spectrum-peak primary wave direction,
qp(X, t), is complicatedly distributed throughout the basinand changes with time (Figure 13) in response to thesynoptic-scale wavefield. In addition the resultant infragravity waves that mostly propagate as free waves readily
become broadbanded in direction through refraction on
the complex bathymetry and reflection from the shoreline.
Hence the simple, unidirectional, monochromatic mesoscale
amplitude expression (n = 1) instigates considerable directional variability in practice, which allows us to estimate
infragravity wave amplitudes without making a full spectral
consideration of the mesoscale modulations.
[73] Acknowledgments. This study was financially supported by the
Office of Naval Research with grant N00014-04-1-0166. The authors are
grateful to A. F. Shchepetkin (UCLA) for his support of the development of
the long-wave model based on ROMS. We also wish to express appreciation to P.A.E.M. Janssen and J.-R. Bidlot (ECMWF) and E. Huckle(UCLA) for help with processing the ERA40 global wave data.

References
Battjes, J. A. (1972), Radiation stresses in short-crested waves, J. Mar. Res.,30, 56 – 64.
Battjes, J. A., and M. J. F. Stive (1985), Calibration and verification of adissipation model for random breaking waves, J. Geophys. Res., 90(C5),9159 – 9167.
Bidlot, J.-R., D. J. Holmes, P. A. Wittmann, R. Lalbeharry, and H. S. Chen
(2002), Intercomparison of the performance of operational ocean wave
forecasting systems with buoy data, J. Phys. Oceanogr., 17, 287 – 310.
Bowen, A. J., and R. T. Guza (1978), Edge waves and surf beat, J. Geophys. Res., 83(C4), 1913 – 1920.
Callahan, P. S., C. S. Morris, and S. V. Hsiao (1994), Comparison of topex/
poseidon s0 and significant wave height distributions to geosat, J. Geophys. Res., 99(C12), 25,015 – 25,024.
Eckart, C. (1951), Surface waves on water of variable depth, in Wave Rep.,
100, 99 pp., Scripps Inst. of Oceanogr., La Jolla, Calif.
Ekström, G. (2001), Time domain analysis of Earth’s long-period background seismic radiation, J. Geophys. Res., 106(B11), 26,483 – 26,493.
Foda, M. A., and C. C. Mei (1981), Nonlinear excitation of long-trappedwaves by a group of short swells, J. Fluid Mech., 111, 319 – 345.
Fukao, Y., K. Nishida, N. Suda, K. Nawa, and N. Kobayashi (2002), A
theory of the earth’s background free oscillations, J. Geophys. Res.,
107(B9), 2206, doi:10.1029/2001JB000153.
Gallagher, B. (1971), Generation of surf beat by non-linear wave interactions, J. Fluid Mech., 49, 1 – 20.
Guza, R. T., and E. B. Thornton (1982), Swash oscillations on a natural
beach, J. Geophys. Res., 87(C1), 483 – 492.
Hasselmann, K. (1962), On the non-linear energy transfer in a gravity-wave
spectrum. part I: General theory, J. Fluid Mech., 12, 481 – 500.
Hasselmann, K. (1963), A statistical analysis of the generation of microseisms, Rev. Geophys., 1(2), 177 – 209.
Herbers, T. H. C., S. Elgar, and R. T. Guza (1994), Infragravity-frequency(0.005 – 0.05 hz) motions on the shelf. part I: Forced waves, J. Phys.
Oceanogr., 24, 917 – 927.
Herbers, T. H. C., S. Elgar, R. T. Guza, and W. C. O’Reilly (1995a),
Infragravity-frequency (0.005 – 0.05 hz) motions on the shelf. part II: Freewaves, J. Phys. Oceanogr., 25, 1063 – 1079.
Herbers, T. H. C., S. Elgar, and R. T. Guza (1995b), Generation and propagation of infragravity waves, J. Geophys. Res., 100(C12), 24,863 –
24,872.
Huntley, D. A., R. T. Guza, and E. B. Thornton (1981), Field observationsof surf beat: 1. Progressive edge waves, J. Geophys. Res., 86(C7), 6451 –6466.
Janssen, P. A. E. M., B. Hansen, and J.-R. Bidlot (1997), Verification of
the ECMWF wave forecasting system against buoys and altimeter data,
J. Phys. Oceanogr., 12, 763 – 784.
Kenyon, K. E. (1969), Stokes drift for random gravity waves, J. Geophys.
Res., 74(28), 6991 – 6994.
Kenyon, K. E. (1970), Stokes transport, J. Geophys. Res., 75(06), 1133 –1135.
Kobayashi, N., and K. Nishida (1998), Continuous excitation of planetary
free oscillations by atmospheric disturbances, Nature, 395, 357 – 360.

24 of 25

C07029

UCHIYAMA AND MCWILLIAMS: INFRAGRAVITY WAVES IN THE DEEP OCEAN

Komen, G. J., L. Cavaleri, M. Donelan, K. Hasselmann, S. Hasselmann,and P. A. E. M. Janssen (1994), Dynamics and Modelling of Ocean
Waves, Cambridge Univ. Press, Cambridge, UK.
LeBlond, P. H. and L. A. Mysak (1978), Waves in the Ocean, Elsevier,
Amsterdam, Netherlands.
Lighthill, J. (1978), Waves in Fluids, Cambridge Univ. Press, Cambridge,
UK.
Longuet-Higgins, S. (1950), A theory of the origin of microseisms, Philos.
Trans. R. Soc. London, Ser. A, 243, 1 – 35.
Longuet-Higgins, M. S., and R. W. Stewart (1962), Radiation stress andmass transport in gravity waves, with application to ‘‘surf beats’’, J. Fluid
Mech., 13, 481 – 504.
McWilliams, J. C., J. M. Restrepo, and E. M. Lane (2004), An asymptotic
theory for the interaction of waves and currents in coastal waters, J. Fluid
Mech., 511, 135 – 178.
Mei, C. C. (1989), The Applied Dynamics of Ocean Surface Waves, World
Scientific, Singapore.
Munk, W. H. (1949), Surf beat, Eos Trans. AGU, 30, 849 – 854.
Munk, W. H., F. E. Snodgrass, and F. Gilbert (1964), Long waves on thecontinental shelf: An experiment to separate trapped and leaky modes,
J. Fluid Mech., 20, 529 – 554.
Nawa, K., N. Suda, Y. Fukao, T. Sato, Y. Aoyama, and K. Shibuya (1998),
Incessant excitation of the earth’s free oscillations, Earth Planets Space,50, 3 – 8.
Nieto Borge, J. C., S. Lehner, A. Niedermeier, and J. Schulz-Stellenfleth(2004), Detection of ocean wave groupiness from spaceborne synthetic aperture radar, J. Geophys. Res., 109, C07005, doi:10.1029/2004JC002298.
Nishida, K., N. Kobayashi, and Y. Fukao (2000), Resonant oscillations
between the solid earth and the atmosphere, Science, 287, doi:10.1126/
science.287.5461.2244.
Okihiro, M., R. T. Guza, and R. J. Seymour (1992), Bound infragravity
waves, J. Geophys. Res., 97(C7), 11,453 – 11,469.
Oltman-Shay, J., and R. T. Guza (1987), Infragravity edge wave observations on two California beaches, J. Phys. Oceanogr., 17, 644 – 663.

C07029

Proudman, J. (1953), Dynamical Oceanography, Methuen, London, UK.
Rhie, J., and B. Romanowicz (2004), Excitation of Earth’s continuous freeoscillations by atmosphere-ocean-seafloor coupling, Nature, 431, 552 –556.
Rhie, J., and B. Romanowicz (2006), A study of the relation between ocean
storms and the Earth’s hum, Geochem. Geophys. Geosyst., 7, Q10004,
doi:10.1029/2006GC001274.
Shchepetkin, A. F., and J. C. McWilliams (2005), The Regional Oceanic
Modeling System: A split-explicit, free-surface, topography-followingcoordinate oceanic model, Ocean Modell., 9, 347 – 404.
Sheremet, A., R. T. Guza, S. Elgar, and T. H. C. Herbers (2002), Observations of nearshore infragravity waves: Seaward and shoreward propagating components, J. Geophys. Res., 107(C8), 3095, doi:10.1029/
2001JC000970.
Suda, N., K. Nawa, and Y. Fukao (1998), Earth’s background free oscillations, Science, 279, doi:10.1126/science.279.5359.2089.
Symonds, G., D. A. Huntley, and A. J. Bowen (1982), Two-dimensionalsurf beat: Long wave generation by time varying breakpoint, J. Geophys.
Res., 87(C1), 492 – 498.
Tanimoto, T. (2005), The oceanic excitation hypothesis for the continuous
oscillation of the Earth, Geophys. J. Int., 160, 276 – 288.
Tanimoto, T., and J. Um (1999), Cause of continuous oscillations of the
Earth, J. Geophys. Res., 104(B12), 28,723 – 28,739.
Tanimoto, T., J. Um, K. Nishida, and N. Kobayashi (1998), Earth’s continuous oscillations observed on seismically quiet days, Geophys. Res.
Lett., 25(10), 1553 – 1556.
Tucker, M. J. (1950), Surf beat: Sea waves of 1 to 5 min. period, Proc. R.
Soc. London, Ser. A, 202, 565 – 573.
Ursell, F. (1952), Edge waves on a sloping beach, Proc. R. Soc. London,
Ser. A, 214, 79 – 97.
Webb, S. C. (1998), Broadband seismology and noise under the ocean, Rev.
Geophys., 36, 105 – 142.
Webb, S. C. (2007), The Earth’s ‘‘hum’’ is driven by ocean waves over thecontinental shelves, Nature, 445, 754 – 756.
Webb, S. C., X. Zhang, and W. Crawford (1991), Infragravity waves in the
deep ocean, J. Geophys. Res., 96(C2), 2723 – 2736.


J. C. McWilliams and Y. Uchiyama, Institute of Geophysics andPlanetary Physics, University of California, Los Angeles, 3845 Slichter
Hall, Los Angeles, CA 90095-1567, USA. (uchiyama@atmos.ucla.edu)

25 of 25