Speed and Evolution of Nonlinear Internal Waves Transiting the South China Sea
MATTHEW H. ALFORD,* REN-CHIEH LIEN,1 HARPER SIMMONS,# JODY KLYMAK,@ STEVE RAMP,&
YIING JANG YANG,** DAVID TANG,11 AND MING-HUEI CHANG1
* Applied Physics Laboratory, and School of Oceanography, University of Washington, Seattle, Washington
1 Applied Physics Laboratory, University of Washington, Seattle, Washington
# University of Alaska Fairbanks, Fairbanks, Alaska
@ School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada
& Monterey Bay Aquarium Research Institute, Moss Landing, California
** Department of Marine Science, Naval Academy, Kaohsiung, Taiwan
11 Institute of Oceanography, National Taiwan University, Taipei, Taiwan
(Manuscript received 16 October 2009, in final form 6 January 2010)
ABSTRACT
In the South China Sea (SCS), 14 nonlinear internal waves are detected as they transit a synchronous array
of 10 moorings spanning the waves’ generation site at Luzon Strait, through the deep basin, and onto the
upper continental slope 560 km to the west. Their arrival time, speed, width, energy, amplitude, and number
of trailing waves are monitored. Waves occur twice daily in a particular pattern where larger, narrower ‘‘A’’
waves alternate with wider, smaller ‘‘B’’ waves. Waves begin as broad internal tides close to Luzon Strait’s two
ridges, steepening to O(3–10 km) wide in the deep basin and O(200–300 m) on the upper slope. Nearly all
waves eventually develop wave trains, with larger–steeper waves developing them earlier and in greater
numbers. The B waves in the deep basin begin at a mean speed of ’5% greater than the linear mode-1 phase
speed for semidiurnal internal waves (computed using climatological and in situ stratification). The A waves
travel ’5%–10% faster than B waves until they reach the continental slope, presumably because of their
greater amplitude. On the upper continental slope, all waves speed up relative to linear values, but B waves
now travel 8%–12% faster than A waves, in spite of being smaller. Solutions of the Taylor–Goldstein
equation with observed currents demonstrate that the B waves’ faster speed is a result of modulation of the
background currents by an energetic diurnal internal tide on the upper slope. Attempts to ascertain the phase
of the barotropic tide at which the waves were generated yielded inconsistent results, possibly partly because
of contamination at the easternmost mooring by eastward signals generated at Luzon Strait’s western ridge.
These results present a coherent picture of the transbasin evolution of the waves but underscore the need to
better understand their generation, the nature of their nonlinearity, and propagation through a time-variable
background flow, which includes the internal tides.
1. Introduction
The strongest nonlinear internal waves (NLIW) in the
world’s oceans occur in the South China Sea (SCS; Fig. 1),
where their horizontal and vertical velocities can exceed
2 and 0.7 m s21, respectively (Klymak et al. 2006).
These flows and the associated downward displacements
of .200 m are great enough to hamper surface and
submarine navigation. In addition, their fluid speeds can
exceed their wave celerities, leading to trapped cores and
elevated dissipation (R.-C. Lien et al. 2010, unpublished
manuscript). The associated mixing (St. Laurent 2008)
and/or transport of nutrients and prey cause pilot whales
to forage preferentially in their wakes (Moore and Lien
2007).
In spite of their strength and obvious scientific and so-
cietal importance, much remains unknown regarding the
waves’ dynamics and generation. They are always directed
westward and usually arrive twice a day in a quasi-
predictable pattern, making it clear that they originate via
interactions with the barotropic tide and the two-ridge
system at Luzon Strait. Because strongly nonlinear waves
are not seen in satellite imagery east of about 120.58E
(Zhao et al. 2004), the waves do not appear to be gen-
erated as lee waves directly at the sill. Instead, the most
attractive theory for their formation is 1) generation of
Corresponding author address: Matthew H. Alford, 1013 NE
40th St., Seattle, WA 98105.
E-mail: malford@apl.washington.edu
1338 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 40
DOI: 10.1175/2010JPO4388.1
 2010 American Meteorological Society

a westbound internal tide via flow of the barotropic tide
over the ridges followed by 2) formation of narrow non-
linear waves via instability or steepening of the internal
tide (Lien et al. 2005). Still, in spite of ongoing numerical
modeling efforts, the geometry and degree of nonlinearity
of the internal tide generation is poorly known, as is the
mechanism by which they break down into NLIW. Factors
hampering progress include the large and complicated
two-ridge system, the potential for three-dimensional
effects such as interference between multiple sources
along the ridge, the strong lateral stratification gradient
and shear associated with the Kuroshio, and a relative
lack of data near the generation region.
In addition, rotation fundamentally changes the dy-
namics of nonlinear wave formation and evolution. Un-
like NLIW observed at lower latitude (Apel et al. 1985),
rotation cannot be neglected in the SCS. With regard to
formation, rotational dispersion can prevent or slow the
breakdown into NLIW of the internal tide (Gerkema
et al. 2006; Helfrich and Grimshaw 2008; Farmer et al.
2009). In addition, the internal tides themselves are dis-
persive, causing the semidiurnal and diurnal constituents
to travel at different speeds (Zhao and Alford 2006,
hereafter ZA06). Finally, the rotating analog of the
Kortevrieg–de Vries (KDV) equation (which describes
weakly nonlinear solitary waves in a nonrotating frame)
has not been solved analytically, but it appears to not
have solitary solutions (Leonov 1981).
Instead, several families of solutions have been de-
termined numerically (Grimshaw et al. 1998) but have
not been adequately explored or verified observationally.
In all of these solutions, the nonlinear waves and internal
tides appear to interact strongly. Hence, with rotation we
lack simple expressions from KdV theory such as that
for the phase speed,
ckdv 5 ce 1
1
3
ah, (1)
where h is the wave amplitude, ce is the linear irrota-
tional phase speed (appendix A), and a is a function of
the stratification.
In contrast to KdV waves, whose speed exceeds the
linear value by a prescribed amount via (1), rotational
nonlinear solutions can travel either faster or slower
than their linear counterparts (Helfrich and Grimshaw
2008). Accurate measurements of wave speed and com-
parison to linear values may therefore constrain ongoing
efforts to numerically model the waves.
FIG. 1. (top) Map of study region showing depth (colors), mooring locations (magenta), the
propagation track assumed in calculating wave speed (black line), the assumed generation site
(G), and the location of each mooring projected onto it assuming cylindrical spreading (white;
see text). Selected NLIW crests from 1998 to 2001 from synthetic aperture radar (SAR)
(courtesy of Z. Zhao 2007, personal communication) are shown in white. Asterisks and pluses
indicate the locations of CTD casts (appendix A). (bottom) Bathymetry along the track.
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