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, malford@ucsd.edu

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).

REFERENCES

Alford, M. H., 2001a: Fine-structure contamination: Observa-

tions and a model of a simple two-wave case. J. Phys.

Oceanogr., 31, 2645–2649, doi:10.1175/1520-0485(2001)031,2645:

FSCOAA.2.0.CO;2.

——, 2001b: Internal swell generation: The spatial distribution

of energy flux from the wind to mixed-layer near-inertial

motions. J. Phys. Oceanogr., 31, 2359–2368, doi:10.1175/

1520-0485(2001)031,2359:ISGTSD.2.0.CO;2.

——, 2003a: Energy available for ocean mixing redistributed

through long-range propagation of internal waves. Nature,

423, 159–163, doi:10.1038/nature01628.

——, 2003b: Improved global maps and 54-year history of wind-

work on ocean inertial motions.Geophys. Res. Lett., 30, 1424,

doi:10.1029/2002GL016614.

——, 2010: Sustained, full-water-column observations of internal

waves and mixing near Mendocino Escarpment. J. Phys.

Oceanogr., 40, 2643–2660, doi:10.1175/2010JPO4502.1.

——, and R. Pinkel, 2000: Observations of overturning in the

thermocline: The context of ocean mixing. J. Phys. Ocean-

ogr., 30, 805–832, doi:10.1175/1520-0485(2000)030,0805:

OOOITT.2.0.CO;2.

——, and M. C. Gregg, 2001: Near-inertial mixing: Modulation of

shear, strain and microstructure at low latitude. J. Geophys.

Res., 106, 16 947–16 968, doi:10.1029/2000JC000370.

——, and M. Whitmont, 2007: Seasonal and spatial variability of

near-inertial kinetic energy from historical moored velocity

records. J. Phys. Oceanogr., 37, 2022–2037, doi:10.1175/

JPO3106.1.

——, J. A. MacKinnon, Z. Zhao, R. Pinkel, J. Klymak, and

T. Peacock, 2007: Internal waves across the Pacific. Geophys.

Res. Lett., 34, L24601, doi:10.1029/2007GL031566.

——, M. F. Cronin, and J. M. Klymak, 2012: Annual cycle and

depth penetration of wind-generated near-inertial internal

waves at Ocean Station Papa in the northeast Pacific. J. Phys.

Oceanogr., 42, 889–909, doi:10.1175/JPO-D-11-092.1.

——,A.Y. Shcherbina, andM.C.Gregg, 2013:Observations of near-

inertial internal gravity waves radiating from a frontal jet.

J. Phys.Oceanogr., 43, 1225–1239, doi:10.1175/JPO-D-12-0146.1.

——, J. A.MacKinnon, H. L. Simmons, and J. D. Nash, 2016: Near-

inertial internal waves in the ocean. Annu. Rev. Mar. Sci., 8,

95–123, doi:10.1146/annurev-marine-010814-015746.

Anderson, S. P., 1993: Shear, strain and thermohaline vertical fine

structure in the upper ocean. Ph.D. thesis, University of

California, San Diego, 143 pp.

Boyer, T. P., and Coauthors, 2013: World Ocean Database 2013.

S. Levitus and A. Mishonov, Eds., NOAA Atlas NESDIS

72, 209 pp., http://doi.org/10.7289/V5NZ85MT.

Cairns, J. L., and G. O. Williams, 1976: Internal wave observations

from a midwater float, 2. J. Geophys. Res., 81, 1943–1950,

doi:10.1029/JC081i012p01943.

Carter, G. S., and M. C. Gregg, 2006: Persistent near-diurnal

internal waves observed above a site of M2 barotropic-

to-baroclinic conversion. J. Phys. Oceanogr., 36, 1136–1147,

doi:10.1175/JPO2884.1.

Chou, S. H., D. S. Luther, M. D. Guiles, G. S. Carter, and

T. Decloedt, 2014: An empirical investigation of nonlinear

energy transfer from the M2 internal tide to diurnal wave

motions in theKauai Channel,Hawaii.Geophys. Res. Lett., 41,

505–512, doi:10.1002/2013GL058320.

D’Asaro, E. A., 1985: The energy flux from the wind to near-

inertial motions in the mixed layer. J. Phys. Oceanogr.,

15, 1043–1059, doi:10.1175/1520-0485(1985)015,1043:

TEFFTW.2.0.CO;2.

——, 1989: The decay of wind-forced mixed layer inertial oscilla-

tions due to the b effect. J. Geophys. Res., 94, 2045–2056,

doi:10.1029/JC094iC02p02045.

——, 1995: Upper-ocean inertial currents forced by a strong

storm. Part III: Interaction of inertial currents and mesoscale

eddies. J. Phys. Oceanogr., 25, 2953–2958, doi:10.1175/

1520-0485(1995)025,2953:UOICFB.2.0.CO;2.

——, and H. Perkins, 1984: A near-inertial internal wave spectrum

for the Sargasso Sea in late summer. J. Phys. Oceanogr., 14,

489–505, doi:10.1175/1520-0485(1984)014,0489:ANIIWS.2.0.

CO;2.

——, C. E. Eriksen, M. D. Levine, P. Niiler, C. A. Paulson, and

P. V. Meurs, 1995: Upper-ocean inertial currents forced by a

strong storm. Part I: Data and comparisons with linear theory.

J. Phys. Oceanogr., 25, 2909–2936, doi:10.1175/1520-0485

(1995)025,2909:UOICFB.2.0.CO;2.

Doherty, K., D. Frye, S. Liberatore, and J. Toole, 1999: A moored

profiling instrument. J. Atmos. Oceanic Technol., 16, 1816–1829,

doi:10.1175/1520-0426(1999)016,1816:AMPI.2.0.CO;2.

Fu, L.-L., 1981: Observations and models of inertial waves in the

deep ocean. Rev. Geophys. Space Phys., 19, 141–170,

doi:10.1029/RG019i001p00141.

Furuichi, N., T. Hibiya, and Y. Niwa, 2005: Bispectral analysis of

energy transfer within the two-dimensional oceanic internal

wave field. J. Phys. Oceanogr., 35, 2104–2109, doi:10.1175/

JPO2816.1.

——, ——, and ——, 2008: Model-predicted distribution of wind-

induced internal wave energy in the world’s oceans.

J. Geophys. Res., 113, C09034, doi:10.1029/2008JC004768.

Garrett, C., 2001: What is the ‘‘near-inertial’’ band and why is it

different from the rest of the internal wave spectrum? J. Phys.

Oceanogr., 31, 962–971, doi:10.1175/1520-0485(2001)031,0962:

WITNIB.2.0.CO;2.

——, and W. Munk, 1972: Space-time scales of internal

waves. Geophys. Fluid Dyn., 3, 225–264, doi:10.1080/

03091927208236082.

——, and ——, 1975: Space-time scales of internal waves: A

progress report. J. Geophys. Res., 80, 291–297, doi:10.1029/

JC080i003p00291.

Gregg, M. C., 1989: Scaling turbulent dissipation in the thermo-

cline. J. Geophys. Res., 94, 9686–9698, doi:10.1029/

JC094iC07p09686.

——, T. B. Sanford, and D. P. Winkel, 2003: Reduced mixing from

the breaking of internal waves in equatorial waters. Nature,

422, 513–515, doi:10.1038/nature01507.Hazewinkel, J., and K. B. Winters, 2011: PSI of the internal tide

on a b plane: Flux divergence and near-inertial wave

2476 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 47

Unauthenticated | Downloaded 11/19/21 06:55 AM UTC

propagation. J. Phys. Oceanogr., 41, 1673–1682,

doi:10.1175/2011JPO4605.1.

Hebert, H., and J. Moum, 1994: Decay of a near-inertial wave. J. Phys.

Oceanogr., 24, 2334–2351, doi:10.1175/1520-0485(1994)024,2334:

DOANIW.2.0.CO;2.

Henyey, F. S., J. Wright, and S. M. Flatté, 1986: Energy and

action flow through the internal wave field: An eikonal

approach. J. Geophys. Res., 91, 8487–8495, doi:10.1029/

JC091iC07p08487.

Jiang, J., Y. Lu, and W. Perrie, 2005: Estimating the energy flux

from the wind to ocean inertial motions: The sensitivity to

surface wind fields. Geophys. Res. Lett., 32, L15610,

doi:10.1029/2005GL023289.

Kalnay, E. M., and Coauthors, 1996: The NCEP/NCAR 40-Year

Reanalysis Project. Bull. Amer. Meteor. Soc., 77, 437–471,

doi:10.1175/1520-0477(1996)077,0437:TNYRP.2.0.CO;2.

Klymak, J. M., W. Crawford, M. H. Alford, J. A. MacKinnon, and

R. Pinkel, 2015: Along-isopycnal variability of spice in the

North Pacific. J. Geophys. Res. Oceans, 120, 2287–2307,

doi:10.1002/2013JC009421.

Kroll, J., 1975: The propagation of wind-generated inertial oscilla-

tions from the surface to the deep ocean. J.Mar. Res., 33, 15–51.

Kunze, E., L. K. Rosenfeld, G. S. Carter, and M. C. Gregg, 2002:

Internal waves in Monterey Submarine Canyon. J. Phys.

Oceanogr., 32, 1890–1913, doi:10.1175/1520-0485(2002)032,1890:

IWIMSC.2.0.CO;2.

Leaman, K. D., and T. B. Sanford, 1975: Vertical energy propa-

gation of inertial waves: A vector spectral analysis of velocity

profiles. J. Geophys. Res., 80, 1975–1978, doi:10.1029/

JC080i015p01975.

Lee, D., and P. Niiler, 1998: The inertial chimney: The near-inertial

energy drainage from the ocean surface to the deep layer.

J. Geophys. Res., 103, 7579–7591, doi:10.1029/97JC03200.

MacKinnon, J. A., and K. B. Winters, 2005: Subtropical catastro-

phe: Significant loss of low-mode tidal energy at 28.98.Geophys. Res. Lett., 32, L15605, doi:10.1029/2005GL023376.

——, M. H. Alford, R. Pinkel, J. Klymak, and Z. Zhao, 2013a: The

latitudinal dependence of shear and mixing in the Pacific

transiting the critical latitude for PSI. J. Phys. Oceanogr., 43,

3–16, doi:10.1175/JPO-D-11-0107.1.

——, ——, O. Sun, R. Pinkel, Z. Zhao, and J. Klymak, 2013b:

Parametric subharmonic instability of the internal tide at

298N. J. Phys. Oceanogr., 43, 17–28, doi:10.1175/

JPO-D-11-0108.1.

McComas, C. H., and P. Müller, 1981: The dynamic balance of

internal waves. J. Phys. Oceanogr., 11, 970–986, doi:10.1175/

1520-0485(1981)011,0970:TDBOIW.2.0.CO;2.

Müller, P., G. Holloway, F. Henyey, and N. Pomphrey, 1986:

Nonlinear interactions among internal gravity waves. Rev.

Geophys., 24, 493–536, doi:10.1029/RG024i003p00493.

Munk, W. H., 1981: Internal waves and small-scale processes.

Evolution of Physical Oceanography, B. A. Warren and

C. Wunsch, Eds., MIT Press, 264–291.

Nagai, T., A. Tandon, E. Kunze, and A. Mahadevan, 2015: Spon-

taneous generation of near-inertial waves from the Kuroshio

Front. J. Phys. Oceanogr., 45, 2381–2406, doi:10.1175/

JPO-D-14-0086.1.

Nagasawa, M., Y. Niwa, and T. Hibiya, 2000: Spatial and temporal

distribution of the wind-induced internal wave energy avail-

able for deep water mixing in the North Pacific. J. Geophys.

Res., 105, 13 933–13 943, doi:10.1029/2000JC900019.

Nikurashin, M., and R. Ferrari, 2010: Radiation and dissipation of

internal waves generated by geostrophic motions impinging

on small-scale topography: Theory. J. Phys. Oceanogr., 40,

1055–1074, doi:10.1175/2009JPO4199.1.

Pinkel, R., 1984: Doppler sonar observations of internal waves: The

wavenumber-frequency spectrum. J.Phys.Oceanogr., 14, 1249–1271,

doi:10.1175/1520-0485(1984)014,1249:DSOOIW.2.0.CO;2.

——, 1985: A wavenumber-frequency spectrum of upper ocean

shear. J. Phys. Oceanogr., 15, 1453–1469, doi:10.1175/

1520-0485(1985)015,1453:AWFSOU.2.0.CO;2.

——, 2008: Advection, phase distortion, and the frequency spec-

trum of fine-scale fields in the sea. J. Phys. Oceanogr., 38,

291–313, doi:10.1175/2007JPO3559.1.

——, 2012: Velocity imprecision in finite-beamwidth shipboard

Doppler sonar: A first-generation correction algorithm.

J. Atmos. Oceanic Technol., 29, 1569–1580, doi:10.1175/

JTECH-D-12-00041.1.

——, 2014: Vortical and internal wave shear and strain. J. Phys.

Oceanogr., 44, 2070–2092, doi:10.1175/JPO-D-13-090.1.

——, A. Plueddemann, and R. Williams, 1987: Internal wave obser-

vations from FLIP in MILDEX. J. Phys. Oceanogr., 17, 1737–

1757, doi:10.1175/1520-0485(1987)017,1737:IWOFFI.2.0.CO;2.

Plueddemann, A., and J. Farrar, 2006: Observations and

models of the energy flux from the wind to mixed-layer

inertial currents. Deep-Sea Res. II, 53, 5–30, doi:10.1016/

j.dsr2.2005.10.017.

Pollard, R. T., and R. C. Millard, 1970: Comparison between

observed and simulated wind-generated inertial oscillations.

Deep-Sea Res., 17, 813–821, doi:10.1016/0011-7471(70)90043-4.

Polzin, K. L., and Y. V. Lvov, 2011: Toward regional character-

izations of the oceanic internal wavefield. Rev. Geophys., 49,

RG4003, doi:10.1029/2010RG000329.

——, J. M. Toole, and R. W. Schmitt, 1995: Finescale parameteri-

zations of turbulent dissipation. J. Phys.Oceanogr., 25, 306–328,

doi:10.1175/1520-0485(1995)025,0306:FPOTD.2.0.CO;2.

Rainville, L., and R. Pinkel, 2006a: Baroclinic energy flux at the

Hawaiian Ridge: Observations from the R/P FLIP. J. Phys.

Oceanogr., 36, 1104–1122, doi:10.1175/JPO2882.1.

——, and ——, 2006b: Propagation of low-mode internal waves

through the ocean. J. Phys. Oceanogr., 36, 1220–1236,

doi:10.1175/JPO2889.1.

Rimac, A., J.-S. von Storch, C. Eden, and H. Haak, 2013: The in-

fluence of high-resolution wind stress field on the power input

to near-inertial motions in the ocean. Geophys. Res. Lett., 40,

4882–4886, doi:10.1002/grl.50929.

Shcherbina, A. Y., M. C. Gregg, M. H. Alford, and R. Harcourt,

2009: Characterizing thermohaline intrusions in the North

Pacific subtropical frontal zone. J. Phys. Oceanogr., 39,

2735–2756, doi:10.1175/2009JPO4190.1.

Silverthorne, K. E., and J. M. Toole, 2009: Seasonal kinetic energy

variability of near-inertial motions. J. Phys. Oceanogr., 39,

1035–1049, doi:10.1175/2008JPO3920.1.

Simmons, H. L., 2008: Spectral modification and geographic re-

distribution of the semi-diurnal internal tide. Ocean Modell.,

21, 126–138, doi:10.1016/j.ocemod.2008.01.002.

——, and M. H. Alford, 2012: Simulating the long-range swell of

internal waves generated by ocean storms.Oceanography, 25,

30–41, doi:10.5670/oceanog.2012.39.

vanHaren,H., 2005: Tidal and near-inertial peak variations around

the diurnal critical latitude. Geophys. Res. Lett., 32, L23611,

doi:10.1029/2005GL024160.

Watanabe, M., and T. Hibiya, 2002: Global estimates of the

wind-induced energy flux to inertial motions in the surface

mixed layer. Geophys. Res. Lett., 29, 1239, doi:10.1029/

2001GL014422.

OCTOBER 2017 ALFORD ET AL . 2477

Unauthenticated | Downloaded 11/19/21 06:55 AM UTC

Waterhouse, A., E. Kunze, J. A. MacKinnon, H. Simmons,

R. Pinkel, and M. Nikurashin, 2016: Global patterns of

internal wave variability from observations of full-depth

rotary shear spectra. Eighth Int. Symp. on Stratified Flows,

San Diego, CA, University of California, San Diego, 1–7.

[Available online at https://cpaess.ucar.edu/sites/default/files/

meetings/2016/issf/papers/waterhouse_amy_article_0.pdf.]

Weller, R. A., 1982: The relation of near-inertial motions ob-

served in the mixed layer during the JASIN (1978) experi-

ment to the local wind stress and to the quasi-geostrophic

flow field. J. Phys. Oceanogr., 12, 1122–1136, doi:10.1175/

1520-0485(1982)012,1122:TRONIM.2.0.CO;2.

Williams, R. G., 1985: The internal tide off Southern California.

Ph.D. thesis, University of California, San Diego, 169 pp.

Winkel, D., 1998: Influences of mean shear in the Florida Current

on turbulent production by internal waves. Ph.D. thesis,

University of Washington, 138 pp.

Young,W.R., Y. K. Tsang, andN. J. Balmforth, 2008: Near-inertial

parametric subharmonic instability. J. FluidMech., 607, 25–49,

https://doi.org/10.1017/S0022112008001742.

Zervakis, V., and M. Levine, 1995: Near-inertial energy

propagation from the mixed layer: Theoretical consider-

ations. J. Phys. Oceanogr., 25, 2872–2889, doi:10.1175/

1520-0485(1995)025,2872:NIEPFT.2.0.CO;2.

Zhao, Z., M. H. Alford, J. A. MacKinnon, and R. Pinkel, 2010:

Long-range propagation of the semidiurnal internal tide from

the Hawaiian Ridge. J. Phys. Oceanogr., 40, 713–736,

doi:10.1175/2009JPO4207.1.

2478 JOURNAL OF PHYS ICAL OCEANOGRAPHY VOLUME 47

Unauthenticated | Downloaded 11/19/21 06:55 AM UTC