Energy Transport by Nonlinear Internal Waves
J. N. MOUM
College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon
J. M. KLYMAK
University of Victoria, Victoria, British Columbia, Canada
J. D. NASH, A. PERLIN, AND W. D. SMYTH
College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon
(Manuscript received 3 May 2006, in final form 13 November 2006)
ABSTRACT
Winter stratification on Oregon’s continental shelf often produces a near-bottom layer of dense fluid that
acts as an internal waveguide upon which nonlinear internal waves propagate. Shipboard profiling and
bottom lander observations capture disturbances that exhibit properties of internal solitary waves, bores,
and gravity currents. Wavelike pulses are highly turbulent (instantaneous bed stresses are 1 N m
2),
resuspending bottom sediments into the water column and raising them 30 m above the seafloor. The
wave cross-shelf transport of fluid often counters the time-averaged Ekman transport in the bottom bound-
ary layer. In the nonlinear internal waves that were observed, the kinetic energy is roughly equal to the
available potential energy and is O(0.1) megajoules per meter of coastline. The energy transported by these
waves includes a nonlinear advection term uE  that is negligible in linear internal waves. Unlike linear
internal waves, the pressure–velocity energy flux up includes important contributions from nonhydrostatic
effects and surface displacement. It is found that, statistically, uE 
2up. Vertical profiles through these
waves of elevation indicate that up(z) is more important in transporting energy near the seafloor while
uE(z) dominates farther from the bottom. With the wave speed c estimated from weakly nonlinear wave
theory, it is verified experimentally that the total energy transported by the waves is up  uE 
cE .
The high but intermittent energy flux by the waves is, in an averaged sense, O(100) watts per meter of
coastline. This is similar to independent estimates of the shoreward energy flux in the semidiurnal internal
tide at the shelf break.
1. Introduction
The circulation over Oregon’s continental shelf is
principally wind driven (Huyer et al. 1978). The large-
scale cross-shelf circulation is thought to be determined
by Ekman dynamics (Perlin et al. 2005a). Predomi-
nantly northerly winds in spring/summer drive near-
surface fluid offshore, generating a cross-shelf density
gradient that sustains a southward coastal jet, which in
turn drives near-bottom fluid onshore. In fall/winter,
winds are predominantly southerly, resulting in a north-
ward coastal jet and effectively reversing the cross-shelf
circulation pattern. As a consequence, there is a clear
distinction in the cross-shelf structure of the stratifi-
cation between upwelling and downwelling seasons
(Fig. 1).
In spring and summer, stratification is concentrated
near the surface. This provides a waveguide in which
nonlinear internal waves of depression are commonly
observed to propagate (Moum et al. 2003). In the left
panel of Fig. 1 two examples of these are evident. At 20
km offshore a large-wavelength wave is seen propagat-
ing onshore at about 0.3 m s
1; the leading edge of this
wave appears to be borelike. A train of smaller-scale
waves inshore at 1.5 km is aliased in our profiling ob-
Corresponding author address: J. N. Moum, College of Oceanic
and Atmospheric Sciences, Oregon State University, COAS Ad-
min. Bldg. 104, Corvallis, OR 97331-5503.
E-mail: moum@coas.oregonstate.edu
1968 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 37
DOI: 10.1175/JPO3094.1
© 2007 American Meteorological Society
JPO3094

servations. In both cases, the influence of the waves is
clear throughout the water column.
In winter, the water column is typically weakly strati-
fied, from the combination of wind-driven mixing, on-
shore transport of light near-surface fluid in the upper
Ekman layer, and offshore transport of dense near-
bottom fluid in the bottom Ekman layer. In the right
panel of Fig. 1, the lightest fluid is trapped inshore of 6
km. At times, this combination of motions leaves the
middle portion of the shelf completely unstratified.
However, at other times, a thin stratified layer exists
near the seafloor (Fig. 1, right panel) upon which non-
linear waves of elevation propagate onshore (Klymak
and Moum 2003). This particular example (Fig. 1)
shows a train of waves propagating onshore at 12 km.
Net fluid transport by these waves counters the off-
shore bottom Ekman transport.
When and where the internal tide steepens, it is fre-
quently found that a significant portion of the tidal en-
ergy goes to generation of a nonlinear internal wave
field [see Helfrich and Melville (2006) for a recent re-
view]. Since stratification is more typically strong near
the surface and this is where most observations have
taken place, the result that we typically observe is a
surface-trapped wave field (Sandstrom and Elliott
1984; Holloway 1987; MacKinnon and Gregg 2003).
When stratification is concentrated near the bottom,
tidal and mesoscale energy can also generate a bottom-
trapped nonlinear internal wave field (Klymak and
Moum 2003; Hosegood and van Haren 2003) that is
similar to the form of nonlinear internal waves ob-
served in the atmosphere (Smith et al. 1982).
Quantification of the energy transport in nonlinear
internal waves has been limited by the resolution of the
available observations (Pinkel 2000; Klymak and
Moum 2003; Lien et al. 2005; Klymak et al. 2006) or has
been based on a linear formulation of the energy flux
(Chang et al. 2006). An evaluation of nonlinear contri-
butions to the wave energy transport has only recently
been attempted from numerical simulations (Venaya-
gamoorthy and Fringer 2005; Lamb 2007) and from
observations obtained in nonlinear internal waves
of depression (Scotti et al. 2006). To define the en-
ergy transport mechanisms in the waves requires ve-
locity measurements with such detail that they are dif-
ficult to obtain from a moving ship. Our recent obser-
vations from a bottom lander fixed to the seafloor pro-
vide sufficient detail to fully evaluate the wave energy
transport terms in nonlinear internal waves of eleva-
tion.
The prevalence of bottom-trapped nonlinear internal
waves over the Oregon shelf in the winter downwelling
season is evident from a set of observations made in
January 2003. These include two sets of moored obser-
vations as well as shipboard profiling measurements
both across the shelf and, for 36 h, at the location of the
FIG. 1. Image plots of squared buoyancy frequency (red high; yellow low) from profiler measurements of density
made across the Oregon shelf at 45°00N in upwelling season (May 2001) and in downwelling season (January
2003). Contours of density are plotted at intervals of 0.5 kg m
3. These cross-shore distributions were constructed
from data obtained from 148 profiles in May 2001 and 202 profiles in January 2003. Each transect required
approximately 8 h to execute. (left) Nonlinear internal waves of depression at 2 km, and (right) a large internal bore
at 20 km. Nonlinear internal waves of elevation appear in the right-hand panel at 12 km.
JULY 2007 M O U M E T A L . 1969
Fig 1 live 4/C