Space–Time Scales of Shear in the North Pacific
MATTHEW H. ALFORD, JENNIFER A. MACKINNON, AND ROBERT PINKEL
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California
JODY M. KLYMAK
University of Victoria, Victoria, British Columbia, Canada
(Manuscript received 27 April 2017, in final form 18 July 2017)
ABSTRACT
The spatial, temporal, and directional characteristics of shear are examined in the upper 1400m of the
North Pacific during late spring with an array of five profiling moorings deployed from 258 to 378N (1330 km)
and simultaneous shipboard transects past them. The array extended from a regime of moderate wind gen-
eration at the north to south of the critical latitude 28.88N,where parametric subharmonic instability (PSI) can
transfer energy from semidiurnal tides to near-inertial motions. Analyses are done in an isopycnal-following
frame to minimize contamination by Doppler shifting. Approximately 60% of RMS shear at vertical
scales .20m (and 80% for vertical scales.80 m) is contained in near-inertial motions. An inertial back-
rotation technique is used to index shipboard observations to a common time and to compute integral time
scales of the shear layers. Persistence times are O(7) days at most moorings but O(25) days at the critical
latitude. Simultaneous shipboard transects show that these shear layers can have lateral scales $100 km.
Layers tend to slope downward toward the equator north of the critical latitude and are more flat to its
south. Phase between shear and strain is used to infer lateral propagation direction. Upgoing waves are
everywhere laterally isotropic. Downgoing waves propagate predominantly equatorward north and south of
the critical latitude but are isotropic near it. Broadly, results are consistent with wind generation north of the
critical latitude and PSI near it—and suggest a more persistent and laterally coherent near-inertial wave field
than previously thought.
1. Introduction
Shear instability is thought to drive most turbulence in
the open-ocean thermocline, making characterization of
the sources, scales, and variability of ocean shear an
important goal of physical oceanography. Additionally,
shear strongly modulates the propagation of internal
gravity and Rossby waves through the Taylor–Goldstein
family of equations. Away from major current systems
and fronts, internal gravity waves give rise to most shear
in the ocean, motivating a huge body of literature in the
1970s and 1980s to characterize the internal wave field
with an empirical spectral description known as GM76
(Garrett and Munk 1972, 1975; Cairns and Williams
1976; Munk 1981) and to develop the theory (McComas
and Müller 1981; Henyey et al. 1986) behind the means
by which energy is transferred throughout the internal
wave spectrum toward smaller shear-containing vertical
scales. Taking a stochastic and isotropic view of the wave
field, this work resulted in some key scalings that now
give the basis for the so-called finescale parameteriza-
tions (Gregg 1989; Polzin et al. 1995; Gregg et al. 2003)
by which turbulence can be estimated from measure-
ments of the low-wavenumber aspects of the spectrum.
More recent work has focused on the peaks in the
internal wave frequency spectrum, primarily the in-
ternal tides and near-inertial waves (NIW), which reside
near the inertial frequency f. These motions are more
deterministic and directional than the rest of the internal
wave continuum and are thought to provide the energy
for the continuum via nonlinear wave–wave interactions
(Polzin and Lvov 2011). Though the GM76 spectrum
contains an inertial peak, its variability and different
wavenumber content than the rest of the spectrum areCorresponding author: Matthew H. Alford, [email protected]
Denotes content that is immediately available upon publica-
tion as open access.
OCTOBER 2017 ALFORD ET AL . 2455
DOI: 10.1175/JPO-D-17-0087.1
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not captured (Garrett 2001). Because of their bluer
wavenumber spectrum compared to internal tides
(D’Asaro and Perkins 1984; Garrett 2001; Alford 2010),
NIW contribute much more to shear and in fact domi-
nate shear at most times and locations (though this has
not been documented systematically). The question
then becomes what leads to near-inertial shear in the
ocean?
Here, we report rare simultaneous spatial transects
and time series from an array of profiling moorings from
the Internal Waves Across the Pacific Experiment
(IWAP; Alford et al. 2007), in which we investigate the
percentage of the total shear contained in NIW, the
fraction going up versus downward, their lateral scales,
persistence times, and propagation directions, with the
goal of better understanding their behavior over a wide
latitude range. The long-term goal is to better un-
derstand how and where they are generated, where they
propagate, and the ultimate fate and effects of the
energy they carry.
One of the most well-known generation mechanisms
of NIW is by wind forcing at the ocean surface. Wind-
forced NIW arise when traveling storms set up inertially
rotating motions in the mixed layer (D’Asaro 1985).
Spatial variability in the wind, mesoscale flows (Weller
1982), and the b effect (D’Asaro 1989) subsequently
creates sufficiently large lateral gradients in mixed layer
flows to pump slightly superinertial motions below the
mixed layer. The modal distribution of these motions
depends on the stratification during the storm, but much
of the energy is generally in low modes (D’Asaro and
Perkins 1984; Alford 2010), which can propagate large
distances toward the equator (Nagasawa et al. 2000;
Alford 2003a; Furuichi et al. 2005; Simmons and Alford
2012). [Motions can equivalently be described in terms
of rays (Kroll 1975; Zervakis and Levine 1995; Garrett
2001)]. Because these low modes travel quickly, this
energy is presumed to dissipate far from the storm that
generated it, though the exact fate is unknown. The
generation and evolution of these low modes was rea-
sonably well characterized in the groundbreakingOcean
Storms Experiment (D’Asaro et al. 1995).
The slowly moving high modes (lz ’ 50–300m) that
are left behind dominate the shear and are our focus
here. Their evolution was poorly described by linear
theory (D’Asaro 1995). They have since been observed
many times propagating downward following winter
storms (Hebert and Moum 1994; Alford and Gregg
2001; Alford et al. 2012). However, upward-
propagating, near-inertial waves have been observed
at a variety of depths (Alford 2010) and in general
downgoing, near-inertial energy exceeds upgoing en-
ergy only little (Waterhouse et al. 2016), as found here.
Since high-mode waves have a slow vertical group ve-
locity and therefore would take many months to reach
the deep sea and reflect back upward, these puzzling
upgoing motions likely have other sources.
One such source of NIW is from parametric sub-
harmonic instability (PSI) of the internal tide, which
generates two waves near half the frequency (Mülleret al. 1986). Importantly, one is traveling upward and the
other downward, possibly providing an explanation for
upgoing waves. For semidiurnal tides, this is possible at
and equatorward of the critical latitude, lc 5 28.88, andexpected to be more efficient at the critical latitude
where the inertial frequency is exactly half the M2 tidal
frequency (MacKinnon and Winters 2005; Simmons
2008; Carter and Gregg 2006; Alford et al. 2007). The
observations reported on here span lc and were col-
lected in part to observe this process. As documented in
Alford et al. (2007) andMacKinnon et al. (2013a), shear
increases sharply at and south of the critical latitude. At
the same location, upgoing and downgoing shear be-
come more equal, consistent with this hypothesis.
MacKinnon et al. (2013b) demonstrated a statistically
significant bispectrum between the internal tides and the
near-inertial waves, proving the mechanism was active
[however, see caveats by Chou et al. (2014)], though
calculated transfer rates were not as great as predicted
by model simulations by MacKinnon and Winters
(2005), owing to multiple vertical modes, temporal and
spatial incoherence, and other factors in the real ocean
(Young et al. 2008; Hazewinkel and Winters 2011).
Several other mechanisms known to generate near-
inertial waves are briefly mentioned here, which could
complicate this simple two-mechanism view [see Alford
et al. (2016) for a recent review]. Equatorward- and
downward-propagating near-inertial waves have been
observed in the region emanating from the Subtropical
Front near 308N (Alford et al. 2013), evidently gener-
ated through wave–mean current interactions at the
front in like manner to that observed and modeled
by Nagai et al. (2015). Geostrophic flow over bottom
topography gives rise to lee waves, which lead to
upward-propagating, near-inertial waves in 2D numer-
ical simulations (Nikurashin and Ferrari 2010). Finally,
wave–wave interactions among poleward-propagating
waves can funnel energy toward the near-inertial peak
(Fu 1981).
As a step toward the long-term aim of determining the
degree and spatial distribution of mixing by NIW, we
here report observations of NIW during the IWAP
experiment (Alford et al. 2007). The experiment was
designed to observe the propagation of internal tides
northward from Hawaii but also provided an opportu-
nity to study and document numerous NIW over a large
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depth range (80–1400m). Crucial aspects of the mea-
surements are 1) simultaneous shipboard transects and
moored profiler time series, allowing characterization of
spatial structure and temporal evolution; 2) ability to
interpolate shear onto isopycnals to minimize Doppler
shifting by internal wave displacements; and 3) simul-
taneous shear and strain measurements allowing calcu-
lation of individual wave properties including lateral
propagation directions following Winkel (1998).
In the following, we first outline the data and the ex-
periment’s oceanographic context, including description
of low-mode, near-inertial fluxes (section 2). Moored
shear and strain are presented in section 3, and band-
passing and Fourier filtering is used to determine the
near-inertial portion of total shear, and the ratio of up-
going to downgoing shear. Back rotation is used to
compute the time scales of modulation of the near-
inertial shear layers. Wavenumber frequency spectra
of shear are presented next in isopycnal-following
coordinates. In section 4, shipboard transects are pre-
sented showing the horizontal spatial structure of
near-inertial waves. Their long lateral coherence scales
are quantified with a feature-tracking algorithm. In
section 5, we study the characteristics of upgoing and
downgoing, high-mode NIW using a plane wave fitting
method and estimate their lateral propagation
directions from the phase between shear and strain
(Winkel 1998). We end with a discussion and
conclusions.
2. Data and oceanographic setting
a. Experiment
IWAP was designed to examine the northward pro-
gression of internal tide energy from its generation re-
gion near French Frigate Shoals, Hawaii, past the critical
latitude and beyond. Besides the Hawaiian Ridge, the
bathymetry in the area includes broad areas of rough
bottom topography, notably the Musicians Seamounts
(Fig. 1), but is relatively smooth north of 328N. The
experiment took place in two cruises spanning about
62 days from April to June 2006. The general approach
was to obtain simultaneous spatial and temporal in-
formation by steaming along a main line with the
Hydrographic Doppler Sonar System (HDSS; Pinkel
2012), a specialized, high-power Doppler sonar, while
six moorings spaced along the line [McLane moored
profiler (MP);MP1–MP6] profiled to ascertain temporal
information. MP5 malfunctioned and is not included in
this analysis. Analyses of internal tides during IWAP are
reported in Alford et al. (2007), Zhao et al. (2010), and
MacKinnon et al. (2013a,b); this paper focuses on the
near-inertial signals.
b. Moored array
Six moorings of similar design were deployed span-
ning latitudes 258–378N, which are described in detail in
Zhao et al. (2010). The primary measurement on each
mooring was a McLane moored profiler, which climbed
up and down through the water column along the
mooring wire between 85 and 1400m, completing one
up or down profile each 1.5 h. This sampling pattern
leads to a variable temporal resolution ranging from 3h
at 85- and 1400-m depth to 1.5 h at middepths. This is
easily sufficient to resolve near-inertial motions, which
have a period ranging from 20 to 28h over the latitude
range of the moorings, but still aliases the highest in-
ternal wave frequencies, as will be seen in the spectra.
Each MP carried a Falmouth Scientific Instruments
(FSI) CTD with vertical resolution of 2m and an FSI
acoustic current meter with vertical resolution of about
10m and precision of about 0.01m s21 (Doherty et al.
1999; Silverthorne and Toole 2009; Alford 2010). Ad-
ditional instrumentation on the moorings included a
300-KHz Teledyne RD Instruments (TRDI) ADCP at
FIG. 1. Location of IWAP moorings (black diamonds) and ship
tracks (red dashed line). Isobaths are contoured every 1000m from
0 to 4000 m. Background color is the time-mean vertical energy
flux (1023Wm22) from the wind to near-inertial motions during
the experiment, estimated from the Pollard andMillard (1970) slab
model (see text). The critical latitude lc and the diurnal turning
latitude are indicated. Yellow arrows are low-mode, near-inertial
energy flux (appendix A). Fluxes south of the D1 turning latitude
(MP1, MP2, and MP3) are likely diurnal internal tides rather than
near-inertial waves.
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50m looking upward toward the surface and an
Aanderaa RCM8 current meter and a Sea-Bird temper-
ature logger at 3000m. MP3 also had a TRDI 75-KHz
ADCP looking downward from 50m.
Data were corrected for mooring knockdown (,20m
at all sites, owing to generally weak currents and high
wire tension) and interpolated onto a uniform 2-m, 1.5-h
grid (Fig. 2). All records are at least 40 days long with the
exception ofMP4, which stopped profiling early; all told,
the profilers collected 3184 profiles. Full-depth data are
used to compute low-mode energy fluxes, as described in
appendix A and presented briefly in section 2d; the
remainder of the paper focuses on the upper 1400m
resolved by the ADCPs and MPs.
c. Hydrographic Doppler sonar system
Spatial transects of velocity and shear (›u/›z) in the
upper 1000m were measured along the mooring line
(Fig. 1, dashed) with the Revelle’s HDSS (Pinkel 2012).
HDSS consists of a 140- and a 50-KHz Doppler sonar
reaching depths of approximately 250 and 1000m, re-
spectively. Velocity from the two sonars wasmerged and
gridded in 8-m, 5-min bins (1.2-km horizontal resolution
at a ship’s speed of 4m s21). Agreement between MP
FIG. 2. Depth–time maps of (left) zonal and (right) meridional velocity at all moorings. Isopycnals with a mean
spacing of 30m are superimposed in black.
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and HDSS data is excellent, as demonstrated by a
comparison during a close approach to MP3 (Fig. 3).
d. Oceanographic background
Mixed layer depth was shallow as typical of the region
for this period, about 20–25m toward the north and even
shallower to the south (Klymak et al. 2015), with a
seasonal thermocline below. Background currents
(Fig. 2) were generally weak (,0.2m s21), as expected
given the region’s generally weak eddy kinetic energy
(e.g., Rainville and Pinkel 2006b). Eastward flow nor-
mally seen around 308N in association with the sub-
tropical front is not clearly evident in the moored data,
apparently because the front was closer to 328N during
the time of our measurements (Klymak et al. 2015).
Wind forcing during the experiment (not shown) was
generally typical of late spring conditions, with generally
light and variable winds with the exception of a storm on
yearday 118. The work on inertial motions in the mixed
layer, an estimate of the power available for wind-
forced, near-inertial waves, is computed by forcing the
Pollard–Millard slab model (Pollard and Millard 1970)
with 6-h NCEP reanalysis winds (Kalnay et al. 1996)
following Alford (2003b). Time-mean wind work during
the experiment (Fig. 1) was weak compared with win-
tertime values and increased poleward approaching the
storm track.
To contextualize the high-mode motions that are this
paper’s focus, we compute low-mode, near-inertial en-
ergy flux at each mooring and present it here first while
giving details of the calculation in appendix A. Time-
mean profiles of near-inertial energy flux are shown
below in Fig. A1, and depth-integrated values are plot-
ted in Fig. 1. The low-mode (modes 1 and 2), near-
inertial energy flux is northeast at MP1–MP3 and south
at MP4 and MP6. Note that the diurnal and inertial
frequencies are too close at these latitudes to separate
with the bandpass filter. The northeast flux atMP1–MP3
is therefore likely dominated by the D1 internal tide
propagating northward from Hawaii.
MP4 and MP6 are north of the D1 turning latitude,
and the observed southward flux is likely near-inertial
energy propagating equatorward. A rough comparison
of the wind work to the observed low-mode flux at MP6
can be made with a simple box model following Alford
(2003a). Taking a box from 368 to 408N and assuming all
wind work goes into equatorward-propagating waves
results in an estimated depth-integrated flux of
270Wm21. The observed mean equatorward flux at
MP6 was 130Wm21, 48% of the flux estimated from
wind work. A similar calculation for MP4, using a box
extending from the mooring north to 35.38N (the
northernmost originating latitude for the near-inertial
band at MP4) gives a depth-integrated flux of 313Wm21.
The mean observed flux at MP4 was 180Wm21, 57% of
the wind estimate. These percentages are approximately
consistent with estimates from Alford (2003a) and from
the Ocean Storms Experiment (D’Asaro et al. 1995).
The wind-generated portion of the shear observed here
can be thought of as what is left behind after these en-
ergetic low modes have left the region.
3. Temporal and vertical scales
a. Depth–time maps of shear and strain
Shear is computed by first differencing the velocity
records at all moorings and smoothing over 10-, 20-, and
80-m vertical intervals. The 20m interval is used for
analyses unless otherwise noted. Total zonal shear
(Fig. 4, left) is dominated at all moorings by inertially
FIG. 3. Profiles of (left) velocity and (right) shear measured by
the MP at MP3 (black) and HDSS (gray) as the ship passed the
mooring at a distance of 1 km. The profiles were measured at 2015
UTC on 11 Jun (yearday 161.84). The gray region is examined later
in the spatial sections.
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oscillating rather than mean shear. Broadly, a tendency
for stronger shear at (MP3) and south of the critical
latitude (MP2) is seen, as stated in the introduction.
Shear is visibly greatest up shallow, a consequence of
WKB scaling (cf. Leaman and Sanford 1975). Super-
imposed on a continuum of motions, a variety of co-
herent upward- and downward-propagating features
(down and upward phase propagation with time, re-
spectively) is evident. Shear at MP3 is stronger and
qualitatively different, with large regions of checker-
board shear that do not show clear upward or downward
phase propagation. Alford et al. (2007) and MacKinnon
et al. (2013b) interpreted these as evidence of PSI
occurring near the critical latitude.
Though our focus is the shear field, the strain field will
be required for our estimates below of lateral propaga-
tion direction and so is presented in Fig. 5 in parallel
format with the raw strain on the left and the near-
inertial filtered quantity at right. Strain is computed in
like manner to shear as the vertical derivative of iso-
pycnal displacement with subsequent smoothing with a
20-m vertical boxcar window. Strain magnitude does not
decrease with depth as does shear, consistent with a N0
scaling expected from WKB compared to N2 or N3 for
FIG. 4. Depth–timemaps of (left) total and (right) near-inertial zonal shear at all moorings. Isopycnals with a mean
spacing of 30m are superimposed in black.
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shear (e.g., Alford and Pinkel 2000). Near-inertial strain
is a smaller percentage of the total than is shear, as ex-
pected from linear theory, which predicts a peak at f for
shear and a null for strain, and because of its greater
contribution from vortical motions (Pinkel 2014).
b. Persistence of near-inertial shear layers
A complex demodulation technique known as inertial
back rotation (D’Asaro et al. 1995) is used next to re-
move the sense of inertial rotation in time. Backrotated
shear has two advantages: first, it allows examination of
the persistence or equivalently the time scales of mod-
ulation of the envelopes of near-inertial packets; the
second is in interpretation of shipboard transects that
take large fractions of an inertial period to conduct.
Backrotated shear references measured shear to a
common time, giving a closer approximation of a spatial
snapshot.
If the measured shear field (written in complex nota-
tion Uz [ uz 1 iyz) has a time and space structure
given by
Uz(x, y, z, t)5 U
z(x, y, z)e2ift , (1)
then multiplication by eift gives the backrotated field
Uz(x, y, z). If near-inertial waves dominate the shear
FIG. 5. (left) Total and (right) near-inertial strain at IWAP moorings.
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field, then Uz(x, y, z) gives their slowly varying ampli-
tude. Shear at other frequencies than the back-rotation
frequency gives errors; for example, Uz(x, y, z) would
beat with a time scale Dv21, where Dv is the difference
between the true and assumed frequencies. Hence, long
persistence times in Uz(x, y, z) indicate both a domi-
nance of near-inertial shear as well as a prolonged
presence of a particular wave packet.
Applying the back-rotation operator at each mooring
(Fig. 6, left) removes the near-inertial carrier wave,
giving instead the envelope of the near-inertially rotat-
ing wave packets. The layers are advected vertically by
the isopycnal displacements (black) and are much flatter
when viewed in an isopycnal-following or semi-
Lagrangian reference frame (right), which is accom-
plished by first computing a set of reference isopycnals
with 1-m mean spacing and interpolating shear profiles
onto them at each time (Pinkel 1985). Phase information
is lost; for example, crests are no longer seen propa-
gating upward or downward in time, allowing inference
of vertical propagation direction, but wave duration is
seen more clearly.
A persistence time of O(5) days can be seen at most
moorings. MP3 again stands out as qualitatively differ-
ent than the other locations, with highly persistent fea-
tures lasting .20 days. This tendency is quantified by
calculating the autocovariance of isopycnal shear, for
which examples atMP3 andMP6 are shown in Fig. 7. An
integral time scale t can then be computed as the first
zero crossing, which gives t 5 25 days for MP3 and
5 days for MP6, supporting visual conclusions from the
maps. This calculation is repeated at each mooring and
at all depths. Profiles of t (Fig. 8) are O(3–6) days for
most moorings, increasing sharply to about 10–15 days
at MP3.
These analyses suggest that NIW dominate shear in
our data, since backrotated shear would not give long
persistence times otherwise. To quantify, the near-
inertial components of the shear field are isolated by
means of bandpass filtering. Because vertical Doppler
shifting of the shear layers by internal waves of other
frequencies (primarily internal tides) transfers vari-
ance from the inertial peak to sum and difference fre-
quencies (Pinkel et al. 1987; Alford 2001a; Pinkel
2008), it is important to do the filtering in the isopycnal-
following frame. At each mooring, a fourth-order
Butterworth filter with passband (1 6 0.3)f is run
over the time series for each isopycnal forward and
backward, and the result is plotted to right in Fig. 4.
The reduction of the frequency bandwidth of near-
inertial shear due to the filter is apparent. However,
also notable is a reduction in the vertical wavenumber
bandwidth; that is, the temporal filtering has also
removed the high-wavenumber signals, which would
not be expected if the spectrum were separable in
wavenumber and frequency as posited in the Garrett–
Munk (GM) spectrum (Garrett and Munk 1975; Cairns
and Williams 1976). The nonseparability of the ob-
served internal wave spectrum has already been
pointed out by Pinkel (1985) and will be examined in
more detail below.
c. Fractions of up, down, and near-inertial shear
The up- and downgoing waves can be examined sep-
arately at eachmooring by twomeans, which give nearly
identical results in our dataset. Following Leaman and
Sanford (1975), single profiles of complex shear Uz can
be separated into components rotating counterclock-
wise (CCW) and clockwise (CW) with depth, which are
up- and downgoing waves for a linear wave field. Al-
ternately, the depth–time series of Uz can be 2D Fourier
transformed and the four quadrants of the transform
interpreted as up/down phase propagation and CW/
CCW rotation in time. We employ the latter technique
as we will be examining the 2D wavenumber/frequency
spectra next. As an example, the zonal shear field at
MP1 (Fig. 9, top) is decomposed into the constituents
with upward (middle) and downward energy propaga-
tion (lower). While some regions are clearly a super-
position of up- and downgoing waves, at other times and
depths the separation filter successfully isolates packets
of waves that appear to be propagating downward and
upward. Their lateral propagation direction will be ex-
amined in section 5.
Conventional wisdom is that downgoing energy
exceeds upgoing energy in the upper ocean as ob-
served by Leaman and Sanford (1975) and Alford
et al. (2012), consistent with the dominance of wind
work at the surface as the forcing mechanism. This
assumption is examined here by repeating the same
separation as just done at MP1 on both the total and
near-inertially filtered fields for all moorings and
averaging in time (Fig. 10, left). Total shear (light
gray) and the upward and downward components of
near-inertial shear (darker grays) decrease with N
(black line) at all locations. The fraction of down-
going near-inertial shear is plotted in the middle
columns of 10. A slight excess of downgoing shear is
indeed seen at MP4 and MP6 as also found by Alford
et al. (2007) and MacKinnon et al. (2013b), but the
fractional excess is small compared with, for exam-
ple, Ocean Station Papa (Alford et al. 2012), which is
at much higher latitude and experienced much
stronger wind forcing. There, down energy exceeded
up energy by factors of 3–7 in the winter and 2–3 in
the summer.
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The total 20-m shear in near-inertial motions, given by
the border between dark and light gray, accounts for
approximately 60% of the total shear (Fig. 10, right).
The fraction is greater, closer to 75%, at MP3, possibly
suggestive of the generation of excess near-inertial shear
by PSI. Because shear extends to smaller vertical scales
than our measurements resolve, the ratio is somewhat
sensitive to the amount of vertical smoothing. As will be
clear in the spectra shown next, near-inertial motions do
not contain very small vertical scales in our data, such
that the ratio of near-inertial shear to the total decreases
to about 50% when shear is differenced over a 10-m
interval and increases to over 80% when an 80-m in-
terval is used. Stated another way, larger scales are
more dominated by near-inertial motions than the
smallest scales.
d. Spectra
Extending the wavenumber spectra presented in
MacKinnon et al. (2013a), we next document the verti-
cal wavenumber and frequency characteristics of the
shear measured at each mooring by means of wave-
number frequency spectra. Velocity was first WKB
scaled (Leaman and Sanford 1975) as
FIG. 6. (left) Depth–time maps of the real part of the backrotated shear uzr at all moorings. Isopycnals with a mean
spacing of 30m are superimposed in gray. (right) The uzr is plotted in an isopycnal-following frame.
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uwkb
(z)5 u
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN
o=N(z)
q, (2)
where No 5 0.0042 s21 is a reference buoyancy frequency
and N(z) is the time-mean stratification profile at each
mooring (Fig. 9), and then placed into the isopycnal-
following frame by interpolating onto isopycnals.
Complex isopycnal-frame velocity w[ uwkb 1 iywkb is
then demeaned and detrended in time and depth over
the regions plotted in Fig. 4, and the 2D Fourier trans-
form ~w is computed. The wavenumber frequency spec-
trum is then given as
Fy(v,m)5
j ~wfw*jDvDm
, (3)
where Dv [ T21 and Dm [ Z21 are the frequency and
wavenumber bandwidths, respectively; T and Z are the
length and vertical aperture of the records. The aperture
Z5 1100m for all moorings;T5 44 days for all moorings
except MP4, which is 22 days because of its early failure.
Positive and negative frequencies then correspond to
counterclockwise and clockwise rotation, respectively,
while positive and negative vertical wavenumbers m
correspond to upward and downward energy propaga-
tion, respectively. The shear spectrum Fshear(v, m) can
then be computed by multiplying at each frequency by
(2pm)2. Frequency spectra can then be computed by
integrating over the wavenumber, and wavenumber
spectra can be computed by integrating over frequency.
Shear wavenumber frequency spectra at all moorings
(Fig. 11, left) show prominent peaks at 2f. The lat-
itudinal shift of the f peak moving from MP6 to MP1 is
clear, as seen before in work such as van Haren (2005).
Tidal peaks are seen atM2 and K1 but are deemphasized
by examining shear, owing to their red vertical wave-
number spectrum. There is no energy at zero wave-
number at any frequency, owing to detrending and the
differentiation operator. Apparent in the Eulerian
spectra (not shown) at all moorings to different degrees
are prominent peaks at f6M2, which are because of the
vertical heaving of the near-inertial shear layers by tidal
motions (Williams 1985; Anderson 1993; Alford 2001a).
These are much less prominent when results are viewed
here in an isopycnal-following frame.
The spectra indicate that the near-inertial motions
have a dominant vertical scale, which is seen more clearly
at the right where each spectrum is plotted versus nor-
malized frequencyv/f in the vicinity of the negative, near-
inertial peak (black boxes at left in Fig. 11). Tidal peaks
appear at different values of normalized frequency de-
pending on the latitude; notably, the wavenumber-
dependent stripe at MP3 is likely the diurnal tides,
which cannot be separated from f there. All latitudes
show a dominant peak near f at vertical scales around
100–200m. Notable at all peaks is an absence of strongly
dominant downward propagation that has been seen in
summer by D’Asaro and Perkins (1984) and in winter at
Ocean Station Papa (Alford et al. 2012). Pinkel (2008)
argues that the hourglass shape of the spectra is owed to
lateral Doppler shifting of the near-inertial layers by
subinertial motions and waves of other frequencies. Even
in a semi-Lagrangian frame, horizontal advection pref-
erentially shifts the frequencies of the smaller-scale
waves. Using a fixed frequency band to define the near
inertial thus preferentially excludes the smaller-scale,
near-inertial constituents. With two-dimensional data,
hourglass-shaped wavenumber–frequency filters can be
FIG. 7. Autocorrelation of backrotated isopycnal shear at 750m at
MP3 (heavy gray) and MP6 (thin black).
FIG. 8. Depth profiles of integral time scale t of backrotated
isopycnal shear at all moorings, calculated as the first zero crossing
of the autocorrelation. Profiles are smoothed over 80m.
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used to isolate the near-inertial band in a Doppler tol-
erant manner (Pinkel 2014).
Integrals over all frequencies and wavenumbers give
the wavenumber and frequency spectra that are tradi-
tionally compared to the Garrett–Munk spectrum. The
spectra are clearly nonseparable, violating one of the
primary assumptions of the GM formulation, and it is
interesting to see how the different bands conspire to
give approximately the GM form, as originally pointed
out by Pinkel (1984). Frequency spectra of velocity
(Fig. 12, left) are plotted first as they are easily compa-
rable with common single-depth records. The CW part
of the spectrum quite closely follows GM76, with the
inertial and tidal peaks superimposed. The bandwidth of
the near-inertial peak is generally ’0.2f, as has been
observed in the past. Secondary peaks are seen slightly
higher than this range at both MP1 and MP6, which
could be interpreted as energy generated at higher lat-
itudes and propagating to the mooring site.
Of particular interest are the wavenumber spectra of
shear (Fig. 12, right), which are generally only slightly
above GM76 (dashed), with a broad hump super-
imposed near 100–200-m scales, as has been observed in
many other locations (e.g., Pinkel 1985; Alford and
Gregg 2001; Polzin and Lvov 2011). Examining the
wavenumber spectrum of only the near-inertial com-
ponents (thick lines), it is clear that the hump is due to
the near-inertial motions, which have a much narrower
bandwidth than the rest of the internal wave continuum.
Consequently, shear at 100–200-m scales is nearly all
near inertial, while shear at 10–20m scales is dominated
by higher frequencies. When viewed in an Eulerian
frame (not shown), the near-inertial peak drops off a
factor of 2 sooner in wavenumber, owing to the in-
creased spreading because of vertical Doppler shifting.
At MP3, and to a lesser extent MP2, narrower peaks
are also seen at a lower wavenumber in addition to the
100–200-m hump. Of these, the downgoing peak is at a
longer vertical scale. It is possible that these are the sig-
natures of PSI. In this interpretation, consider an inter-
acting wave triad mIT 1 mup 5 mdown, where mIT is the
vertical wavenumber of amode-1–3 internal tide, andmup
andmdown are an up- and downgoing, near-inertial wave.
Then, taking forMP3 the observedmup5 3.53 1023 cpm
andmdown5 2.53 1023 cpm givesmIT5 13 1023 cpmor
about 1000-m wavelength, a reasonable value.
4. Horizontal sections and lateral coherence scales
Having shown that shear in the region is dominantly
near inertial, we now examine its horizontal structure by
means of shipboard sections past themoorings. Profiles of
velocity and shear from MP3 and the HDSS as the ship
passed by themooring agree well (Fig. 3), confirming that
they are measuring the same features and allowing ex-
amination of their horizontal structure. The agreement is
due in part to the dominance of near-inertial waves in this
region and in part to their large horizontal scales.
Backrotated shear is computed from the shipboard
data using the inertial frequency at MP3; v 5 f(28.98).Since the operator is applied only over short intervals,
the precise frequency used makes little difference.
We first present four transects conducted over a pe-
riod of 8 days between 27.88 and 30.48N, passing by
mooring MP3. During this time period, velocity, shear,
and strain at MP3 (Figs. 13a,b,d) showed several near-
inertial waves including downgoing waves at 200 and
800m and a vertically standing wave near 500m, evident
by its checkerboard pattern (Alford et al. 2007). The
vertical heaving of the shear layers by the tidal dis-
placements discussed previously is more evident now
while zoomed in. Moored backrotated shear (Fig. 13c)
removes the sense of phase propagation with depth but
gives a sense of the time scale over which particular
near-inertial shear features persist as shown above
[O(10) days at this location and time] and also provides a
key anchor station for comparison with the shipboard
sections of backrotated shear (right).
Four sections were conducted past themooring, at times
indicated by the vertical dashed lines in the time series at
left. The mooring location is shown at right with a vertical
dashed line. In spite of the vertical advection of these
features by internal tides and other motions, backrotated
shear at a given location remains fairly constant for most
features between the successive occupations, suggesting
that they are the same near-inertial features sampled
multiple times—a conclusion bolstered by the long co-
herence times shown in the moored backrotated shear in
FIG. 9. Total, upward, and downward shear at MP1. Separation is
accomplished via 2D Fourier filtering (see text).
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FIG. 10. (left) Stacked profiles of time-mean RMS shear. Total shear is shown in light gray, with the downward and upward
near-inertial portions plotted in black and dark gray, respectively. Buoyancy frequency N, scaled by a factor of 5 to fit on the
same axes, is in black. (middle) Profiles of the time-mean ratio of downgoing near-inertial shear (NI_down) to total near-inertial
shear (NI_total). (right) Profiles of the time-mean fraction of near-inertial shear (NI) to the total.
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Fig. 13c. Agreement between moored and shipboard
backrotated shear at all flyby times is very good (as can be
seen in Fig. 3 and by comparing times and locations of the
dashed lines at left and right in Fig. 13), allowing in-
terpretation of the sections as the spatial structure of the
near-inertial motions seen at left.
Each of the layers of backrotated shear observed at the
mooring (Fig. 13d) is seen sloping upward to the right in
Figs. 13f–h, consistent with propagation downward and
toward the equator. The slope of the features is somewhat
less than that of a theoretical ray overplotted in Fig. 13h
from Garrett (2001). (Note that rays project identically
FIG. 11. Wavenumber frequency spectra of WKB-scaled rotary shear, computed in an isopycnal-following frame
from 120 to 1300m. Right is the zoom in on the box at left, plotted vs normalized frequency v/f. Boxes at lower left
in the panels at right show the spectral resolution in wavenumber and frequency.
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onto latitude independent of their lateral propagation
direction.) The downward sense of propagation is in
agreement with the observed upward phase propagation
at the mooring at 200 and 800m. Together, the moored
and shipboard sections at and north of the mooring are
broadly consistent with generation of high-mode, near-
inertial waves to the north, between 308 and 318N and
propagation downward and equatorward along wave
FIG. 12. (left) Rotary velocity frequency spectra and (right) wavenumber shear spectra computed by integrating the
wavenumber frequency spectra in wavenumber and frequency, respectively. Frequency spectra are of velocity rather
than shear for easy comparison with past results. Both quantities are calculated in an isopycnal-following frame. At
left, thick/thin lines correspond to CWand CCWmotions; at right, thin lines are for all frequencies; thick are for near-
inertial frequencies only. Gray/black lines in each correspond to up/down energy propagation, respectively. Small and
large bars give confidence limits for the total, and near-inertial waves are shown at far right.
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characteristics to the mooring site. The phase between
shear and strain (section 5 and appendix B) gives a
propagation direction toward the southwest, consistent
with these conclusions.
At and south of the critical latitude (dashed line), the
features are not sloping, which could possibly be evi-
dence that near-inertial energy at those locations is in
part due to PSI, which occurs at the inertial frequency
FIG. 13. (left) Moored data fromMP3 and (right) shipboard spatial transects past it. (a)–(d) Depth–time series of
velocity, zonal shear, backrotated zonal shear, and strain. Vertical dashed lines indicate times that ship passedMP3
on each transect at right. Isopycnals are plotted at 60-m intervals. (e)–(h) Backrotated zonal shearmeasured on four
consecutive shipboard transects, plotted vs latitude. The black dashed line in (h) is a ray beginning at 318N com-
puted using Eq. (16) from Garrett (2001) evaluated with N(z) at MP3. Gray lines are laterally connected maxima
identified with a feature-tracking algorithm of Shcherbina et al. (2009). Colored periods at the top of each ship
transect correspond to time periods at of mooring panels.
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and therefore gives zero vertical group velocity. This ten-
dency was also observed in our second set of shipboard
transects, which were past MP2 and MP3 on yearday 140
(Fig. 14). Three of the transects spanned both moorings,
while the fourth passed by MP3 only. The format of the
figure is slightly different than previous since the transects
passed both moorings, with the panels at the left repre-
senting shear and backrotated shear at the two moorings.
MP2 contains more noninertial shear than MP3 (Fig. 9),
and phase propagation at both moorings is complicated,
lacking regular packets of clear upward or downward phase
propagation. However, backrotated shear (Figs. 14b,d)
shows similar long time scales, as before. Backrotated shear
transects (right) again show features repeatedly observed in
all transects and long lateral coherence scales, with some
extending between moorings (e.g., 400–500m). The fea-
tures are everywhere fairly flat, consistent with the lacking
sense of vertical propagation seen in the moored records.
Given the extremely short lateral coherence scales of
O(3) km noted between moored fixed-depth current
meters by Garrett and Munk (1972), the long lateral
coherence scales of the shear features is striking, with
some features visibly extending at least 100 km. To
quantify, we employed a feature-tracking algorithm
developed for measuring the length of thermohaline
intrusions (Shcherbina et al. 2009) to identify loci of
laterally connected shear maxima and minima (gray
lines in Figs. 13 and 14). The PDF of feature length
(Fig. 15, thin black) peaks at horizontal scales of ’10–
15 km, but its integral (thick gray) shows that about 20%
of data are in features longer than 100 km.
We noted a visual tendency for layers in Figs. 13–14 to
slope downward toward the equator north of lc and to be
more flat south of it. To quantify, the slope of each of the
tracked layers is computed and its PDF plotted in Fig. 16
(gray). Shading represents one standard error in the
probability distribution, estimated by randomly sampling
many realizations of a Gaussian distribution with the
same sample size. The maximum likelihood is at zero,
indicating that most layers are flat. Near slopes of60.002,
negative slopes are more common, suggesting a tendency
for downward and equatorward propagation. Consider-
ing layers with mean location only north of (red) and
south of lc (blue), layers with slope magnitudes 0.001–
0.004 are significantly more likely to be sloping downward
toward the equator north of the latitude than south of it;
flat layers are significantly more likely south of lc than
north of it, confirming the visual conclusions drawn earlier.
5. Lateral propagation directions
We next employ a plane wave fitting method that al-
lows us to study the characteristics of individual waves
observed in the moored records. The simultaneous
measurement of shear and strain then allows us to esti-
mate lateral propagation directions for these waves
based on their relative phase following Winkel (1998)
andAlford andGregg (2001) and described in appendix B.
Plane wave fits were performed on inertially filtered
shear and strain in isopycnal coordinates using the
MATLAB optimization toolbox, which minimizes the
error between data and a plane wave model with variable
parameters. The model used is a combination of two
plane waves, one propagating upward and the other
propagating downward:
X5 a1Aucos(v
ut2m
uz1f
u)
1Adcos(v
dt2m
dz1f
d) , (4)
where a is a constant,Au,d are wave amplitudes,vu,d are the
wave frequencies, mu,d are the vertical wavenumbers, and
fu,d are the phases of downward and upward waves, re-
spectively. The fits are applied in overlapping windows with
vertical extent of 200m and a temporal extent of three in-
ertial periods. Fits done over a range of similar vertical and
temporal extents indicate that the results are not very sen-
sitive to these exact parameters. The phase difference be-
tween uz and yz is constrained to bep/2. The fit is applied to
shear and strain together, such that frequency and wave-
number are constrained to be the same for shear and strain.
Polar probability density functions are then computed
for the upgoing and downgoing waves, which had a
signal to noise ratio, defined as
hjX2jihju
z2Xj2i , (5)
greater than 1.6 for both shear and strain (20%–37% of
the total waves detected; Fig. 17). Downgoing waves at
moorings well away from the critical latitude (MP1,MP4,
andMP6) show a clear tendency to propagate toward the
equator, specifically, the southwest; consistent with wind
generation at slightly higher latitude and subsequent
propagation downward and equatorward past the moor-
ing. Propagation at MP2 and MP3 is isotropic, possibly
consistent with a lack of preferential lateral propagation
direction for the PSI-generatedwaves. Propagation of the
upgoing waves (blue) is isotropic at all moorings except
MP3, which shows an equatorward tendency. A variety of
reasons for isotropic propagation of the upgoing waves
can be imagined (discussion).
6. Conclusions
We have presented simultaneous depth–time obser-
vations of shear and strain in the upper ocean from
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profiling moorings and shear from shipboard
ADCP transects. The data are analyzed with the
goal of identifying the dominant source of shear
in the central North Pacific and quantifying its
spatial and temporal scales. The shear is found to
be predominantly near inertial, and aspects of
propagation and nonlinear interaction (PSI) are
explored using a spatial array of five profiling moorings
and simultaneous ship-base meridional transects. Our
conclusions are as follows:
FIG. 14. (left)Moored data fromMP2 andMP3 and (right) shipboard spatial transects past themoorings. The left
panels show time series of raw and backrotated zonal shear at MP2 and MP3. Isopycnals are plotted at 60-m
intervals. Vertical dashed lines indicate times that ship passed MP3 on each transect. The right panels show
backrotated zonal shear measured on four consecutive shipboard transects, plotted vs latitude. Colored periods at
the top of each ship transect correspond to time periods at the mooring panels.
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1) 60% of RMS shear at scales .20m is near-inertial,
increasing to about 70% at the critical latitude. For
scales .40m, the percentage increases to over 80%
at all moorings. Shear is propagating upward and
downward in approximately equal amounts, con-
trasting sharply with wintertime locations in the
storm track that show a strong downward excess.
Downward packets tend to be propagating equator-
ward except near the critical latitude where no
preferential direction is observed. Propagation is
isotropic at all latitudes for upgoing waves.
2) Shear measured at the critical latitude mooring MP3
stands out as greater, more dominated by near-
inertial shear, and more temporally persistent than
the other moorings, bolstering evidence presented in
Alford et al. (2007) and MacKinnon et al. (2013b) of
the importance of PSI at this latitude.
3) Shipboard transects of velocity and shear show the
horizontal spatial structure of near-inertial waves. Si-
multaneous moored measurements confirm that the
spatial features are near inertial. Much of the near-
inertial shear is coherent over large horizontal distances,
on the order of 80km.A change in character is seen near
the critical latitude for PSI of the M2 internal tide, close
to MP3. Between MP2 and MP3, features are more
horizontal. Features observed north of MP3 show a
tendency to slope more and are consistent with down-
ward, equatorward-propagating, near-inertial waves.
7. Discussion
This paper has explored the upper-ocean shear field
during weak, summertime wind forcing conditions,
through the unusual lens of simultaneous transects and
profiling moorings. The data show that even in summer,
when near-inertial energy is at a minimum (Alford and
Whitmont 2007), shear is dominated by near-inertial
waves. This allows back rotation to be used to examine
their lateral persistence, which is found to be tens of
kilometers, in striking contrast to past moored estimates
of very short horizontal coherence scales. It is clear from
Figs. 13 and 14 that the slope of the layers and their
heaving by displacements of other frequencies would
give very short coherence scales estimated from laterally
separated current meters at discrete depths (Garrett and
Munk 1972).
The long lateral coherence and downward slope to-
ward the equator of the shear layers are as expected for
energy traveling downward and toward the equator
along characteristics at the group velocity as depicted in
Garrett (2001). The observed persistence times and
lateral coherence scales are roughly consistent with this
view. Taking a meridional group velocity of 0.05m s21
for a wave of vertical wavelength 200m at 1.03f at
318N, a packet 50 km long would take 12 days to pass a
fixed point on the ray, the same order as observed.
The longer persistence times at the critical latitude are
suggestive of different physics, as concluded by
MacKinnon et al. (2013b). In contrast to the features at
higher latitude, which have been able to propagate
meridionally because their frequency has exceeded f,
PSI at lc would be expected to generate motions at
f, which would not propagate laterally or vertically.
As with any motions at f on a b plane (D’Asaro 1989),
the meridional variation of f would allow meridional
propagation as found by D’Asaro et al. (1995) for
FIG. 16. PDF of slopes of identified features. Thickness of lines is
one standard deviation of the spread inMonte Carlo simulations of
the PDF with the same sample size as the data. Blue and red lines
are subsampled north and south of the critical latitude,
respectively. The gray curve is all of the data.
FIG. 15. PDF (thin black) and cumulative distribution function
(CDF; thick gray dashed; axis at right) of lengths of shear features
identified in Figs. 13 and 14 using a feature-tracking algorithm.
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FIG. 17. Polar probability density functions of lateral propagation direction, computed from
the phase between shear and strain, for downgoing waves (red) and upgoing waves (blue) at all
moorings. Azimuthal axis is degrees true, such that 1808 represents a wave traveling southward.One standard error in calculation of the PDF for this sample size as determined from Monte
Carlo simulations is shown in the upper left panel in heavy black.
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wind-generated motions, which might impose the ob-
served tens-of-kilometers spatial scale on the PSI-
generated motions. For example, motions 100km north
and south of lc would have a frequency of ’(1 6 0.03)f,
sufficiently different from f to allow propagation and
decoherence. For comparison, Hazewinkel and Winters
(2011) found lateral coherence scales of 30km, consistent
with our results.
The observation of dominant equatorward propa-
gation for downgoing waves away from lc but not
near it supports the general picture drawn from the
equatorward/downward slopes to the north and flatter to
the south: midlatitude wind gives rise to a spectrum of
near-inertial motions that are constrained to propagate
down and toward the equator. At lc, PSI augments this
energy source, providing upgoing and downgoing waves
of approximately equal amounts that also propagate
equatorward.
The rates of both of these processes (and therefore
their relative importance) and their dependencies
are not yet well constrained. Globally, the energy
going into the mixed layer inertial motions has been
estimated at 0.3–1.3 TW but still has a large un-
certainty due to different wind sources (Alford
2001b, 2003b; Watanabe and Hibiya 2002; Jiang et al.
2005; Rimac et al. 2013) and uncertainty in the val-
idity of the slab model (Plueddemann and Farrar
2006). Additionally, the fraction of this energy that
propagates to depth is uncertain (Furuichi et al.
2008) but is known to depend sensitively on the me-
soscale field (Weller 1982; Lee and Niiler 1998;
Alford et al. 2012). Internal tide flux convergences
near the critical latitude computed by Simmons
(2008) are 5 3 1029W kg21, giving a global conver-
sion rate from internal tides to near-inertial waves of
about 0.2 TW, the same order of magnitude as the
wind work, suggesting that PSI is an important source
of NIW (as well as a significant sink for the internal
tide). The power input to the internal wave field
(primarily NIW) from wave–mean flow interactions
has been estimated at 0.36 TW by Nagai et al. (2015),
also the same order. For progress, these processes,
together with the nature of the upgoing near-inertial
waves, need to be better understood.
Acknowledgments. We are thankful for the expertise,
cheerful attitude, and hard work of R/V Revelle captain,
Tom Des Jardines, and crew as well as Eric Slater, Mike
Goldin, Mai Bui, Jon Pompa, Tyler Hughen, Eric Boget,
AndrewCookson,DaveWinkel, Oliver Sun, KimMartini,
Paola Eccheverri, Tom Peacock, and Carl Mattson. We
thank Andrey Shcherbina for providing the feature-
tracking code. This paper resulted in part due to the hard
work of Andy Pickering, who did much of the analysis for
this work for his master’s thesis but did not wish to be an
author on the publication. This work was supported by
NSF Grants OCE-0424717 and OCE-0425283.
APPENDIX A
Energy Flux
Near-inertial energy flux is computed from the band-
pass filtered velocity and displacement fields. The deep
velocity and temperature measurements allow fitting of
the data to the first two vertical modes for velocity
and displacement. Mode shapes are determined from the
full-depth World Ocean Database climatological strati-
fication (Boyer et al. 2013), which compares well with
measured stratification in the upper 100–1400m. Energy
flux is then computed using standard methods (Kunze
et al. 2002). Energy flux is defined as the covariance of
velocity and pressure perturbations:
F5 hu0p0i . (A1)
Perturbation pressure p0 is computed as
p0 5 psurf
1
ð0z
r0g dz , (A2)
where the r0 is the density perturbation.
The surface pressure psurf is not measured but can be
inferred from the baroclinicity condition that the depth-
averaged pressure perturbation must vanish:
1
H
ð 02H
p0(z, t) dz5 0: (A3)
The mode fits to pressure are used to determine the
correct offset to use since p0 cannot be integrated over
the entire water column (Rainville and Pinkel 2006a).
Total flux from (A1) (Figure A1, blue) then captures
the higher-wavenumber structures missed by the
modal fits (Figure A1, gray). Depth-integrated fluxes
(Figure 1, arrows) are similar from the two methods.
APPENDIX B
Determining the Propagation Direction ofNear-Inertial Waves from the Phase between
Shear and Strain
Starting from the standard internal wave polarization
relations for velocity and displacement, the polarization
relations for shear and strain are
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uz5ðN2 2s2Þðs2 2 f 2Þ
�2ik1
lf
s
�w , (B1)
yz5ðN2 2s2Þðs2 2 f 2Þ
�2kf
s2 il
�w, and (B2)
g5›h
›z5 imh5
2m
sw , (B3)
where the vertical velocityw5 exp[i(kx1 ly1mz2vt)],
s is frequency, and k, l, and m are zonal, meridional, and
vertical wavenumbers, respectively.
Tomatch the sign conventionused in thedata, expressions
for shear aremultiplied by21. The polarization relations for
shear and strain, matching the data conventions, are then
uz5
N2 2s2
ðs2 2 f 2Þ�ik2
lf
s
�w , (B4)
yz5
N2 2s2
ðs2 2 f 2Þ�kf
s1 il
�w, and (B5)
g5›h
›z5 imh5
2m
sw , (B6)
FIG. A1. Time-mean near-inertial energy flux profiles at IWAP moorings. Gray line is the sum of modes 1 and 2.
Blue is the total (not modal) energy flux; the correct p0 offset is determined from the first two modes.
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Computing the phase lag for all directions, we find
expressions for the propagation direction u:
uup52df
up, and (B7)
udown
5 1802 dfdown
(B8)
for upgoing and downgoing waves, respectively, where
u is propagation direction in degrees true (CW from
north), and df is the phase lag between zonal shear uzand strain (positive if strain leads shear).
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