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Internal-Wave-Driven Mixing: Global Geography and Budgets
ERIC KUNZE
NorthWest Research Associates, Redmond, Washington
(Manuscript received 13 June 2016, in final form 6 January 2017)
ABSTRACT
Internal-wave-driven dissipation rates « and diapycnal diffusivitiesK are inferred globally using a finescale
parameterization based on vertical strain applied to;30 000 hydrographic casts. Global dissipations are 2.060.6 TW, consistent with internal wave power sources of 2.1 6 0.7 TW from tides and wind. Vertically in-
tegrated dissipation rates vary by three to four orders of magnitude with elevated values over abrupt to-
pography in the western Indian and Pacific as well as midocean slow spreading ridges, consistent with internal
tide sources. But dependence on bottom forcing is much weaker than linear wave generation theory, pointing
to horizontal dispersion by internal waves and relatively little local dissipation when forcing is strong.
Stratified turbulent bottom boundary layer thickness variability is not consistent with OGCM parameteri-
zations of tidal mixing. Average diffusivities K 5 (0.3–0.4) 3 1024 m2 s21 depend only weakly on depth,
indicating that « 5 KN2/g scales as N2 such that the bulk of the dissipation is in the pycnocline and less than
0.08-TW dissipation below 2000-m depth. Average diffusivities K approach 1024 m2 s21 in the bottom
500 meters above bottom (mab) in height above bottom coordinates with a 2000-m e-folding scale. Average
dissipation rates « are 1029W kg21 within 500 mab then diminish to background deep values of 0.15 31029W kg21 by 1000 mab. No incontrovertible support is found for high dissipation rates in Antarctic Cir-
cumpolar Currents or parametric subharmonic instability being a significant pathway to elevated dissipation
rates for semidiurnal or diurnal internal tides equatorward of 288 and 148 latitudes, respectively, althoughelevated K is found about 308 latitude in the North and South Pacific.
1. Introduction
Quantifying and understanding ocean mixing remains
one of the most challenging problems in physical
oceanography because of its spatial and temporal het-
erogeneity. Much of the turbulent mixing is concen-
trated in localized hot spots so that average mixing can
only be accurately assessed from large amounts of data
with well-distributed geographical coverage (Kunze
et al. 2006; Whalen et al. 2012; Waterhouse et al. 2014).
A wide range of features on time scales of months to
millennia, from precipitation in the western equatorial
Pacific (Jochum 2009) to the strength of the deep me-
ridional overturning circulation, equatorial upwelling,
and the Southern Hemisphere westerlies (Friedrich
et al. 2011; Melet et al. 2016), are sensitive to how dia-
pycnal mixing is parameterized in global OGCMs,
linking diapycnalmixing not just to the ocean circulation
but also biogeochemical cycles, weather, and long-term
climate. Jochum (2009) found that applying the latitude
dependence to mixing (Gregg et al. 2003) improved the
skill of OGCMs in reproducing equatorial SST and
precipitation, the spiciness of Labrador Seawater, and
the Gulf Stream path.
In the bulk of the stratified ocean interior, internal
wave breaking is the dominant source of turbulentmixing
(Munk and Wunsch 1998). The primary sources of
deep-ocean internal waves are tide/topography generation
of internal tides at ;1.0TW (Egbert and Ray 2001;
Nycander 2005); wind-forced, near-inertial waves at
0.2–1.1 TW (Alford 2001; Plueddemann and Farrar 2006;
Furuichi et al. 2008; Rimac et al. 2013); and subinertial
flow/topography generation of internal lee waves at
0.2–0.7 TW (Scott et al. 2011; Nikurashin and Ferrari 2011;
Wright et al. 2014). Thus, total internal wave power input
is 2.1 6 0.7 TW with most of the uncertainty in (i) near-
inertial wave production by winds, associated with the
temporal resolution of global wind products at high lati-
tudes and mixed-layer depth assumptions, and (ii) lee-
wave dissipation based on microstructure measurementsCorresponding author: Eric Kunze, kunze@nwra.com
Denotes content that is immediately available upon publica-
tion as open access.
JUNE 2017 KUNZE 1325
DOI: 10.1175/JPO-D-16-0141.1
� 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS CopyrightPolicy (www.ametsoc.org/PUBSReuseLicenses).
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being up to an order of magnitude below predictions
(Waterman et al. 2014). This generated internal wave en-
ergy is redistributed vertically and horizontally by propa-
gation of low-mode internal waves (Ray and Mitchum
1997; Alford 2001; Zhao et al. 2016) and from large to
small scales by wave–wave and wave–mean flow in-
teractions. Away from direct forcing by wind and currents
at boundaries, turbulence production is controlled by the
rate at which energy cascades from large to small vertical
scales. Internal wave–wave interaction theory (McComas
and Müller 1981; Henyey et al. 1986; Henyey 1991) has
provided a parameterization for turbulence production
that can be expressed in terms of finescale internal wave
shear Vz and/or strain jz (Gregg 1989; Gregg and Kunze
1991; Polzin et al. 1995; Gregg et al. 2003). Comparison
with direct microstructure measurements validates the
shear-and-strain finescale parameterizations to within
factors of 2–3 (Polzin et al. 1995; Polzin et al. 2014;Whalen
et al. 2015), although care is needed in its implementation
to avoid overestimation, particularly in low-N environ-
ments where sensor noise becomes problematic andwhere
N exhibits large changes with depth.
The finescale parameterization has recently been
reviewed by Polzin et al. (2014). In contrast to direct
microstructure measurements, its inferences represent
turbulent dissipation and mixing with built-in averaging
over internal wave time and space scales. It has been
used to (i) infer that elevated dye mixing in Santa
Monica Basin (Ledwell and Watson 1991) was being
driven by an energetic internal wave field on the slopes
(Gregg and Kunze 1991); (ii) predict elevated turbulent
mixing above seamount summits (Kunze et al. 1992)
before microstructure confirmation (Lueck and Mudge
1997; Toole et al. 1997; Kunze andToole 1997); (iii) infer
weak mixing over smooth bottom topography and ele-
vated mixing over rough topography (D’Asaro and
Morison 1992; Wijesekera et al. 1993; Kunze and
Sanford 1996; Mauritzen et al. 2002; Walter et al. 2005;
Kunze et al. 2006; Stöber et al. 2008; MacKinnon et al.
2008), consistent with deep direct microstructure mea-
surements (Toole et al. 1994; Polzin et al. 1997);
(iv) identify sites whereAntarctic Circumpolar Currents
interact with bottom topography to generate elevated
turbulence northeast of the Kerguelan Plateau (Polzin
and Firing 1997; Kunze et al. 2006) and inDrake Passage
(Naveira Garabato et al. 2004; Wu et al. 2011; Damerell
et al. 2012), subsequently verified with direct micro-
structure measurements (St. Laurent et al. 2012; Sheen
et al. 2013; Waterman et al. 2013); (v) contribute to the
argument that parametric subharmonic instability (PSI)
may transfer energy from low-mode internal tides to
high-wavenumber near-inertial shear of half the fre-
quency (Hibiya et al. 2006) and thence to turbulence
immediately equatorward of 288 (MacKinnon and
Winters 2005; Carter and Gregg 2006; Simmons 2008;
MacKinnon et al. 2013a,b; Sun and Pinkel 2013);
(vi) determine there is little seasonal variability in upper-
ocean mixing except under fall–winter storm tracks
(308–408) (Whalen et al. 2015); and (vii) assess the role of
turbulent diapycnal mixing in the meridional over-
turning circulation and large-scale property budgets in
the Indian Ocean (Huussen et al. 2012).
In this paper, a global assessment of deep-ocean,
internal-wave-driven turbulent dissipation rates « and
diapycnal diffusivities K will be inferred by applying a
parameterization based on finescale internal-wave
strain jz (Gregg and Kunze 1991; Wijesekera et al. 1993;
Polzin et al. 1995; Gregg et al. 2003) to ;30 000 CTD
profiles. Data are absent from the Arctic, Weddell, and
Ross Seas and are limited poleward of 608S in the
Southern Ocean but otherwise are well-distributed with
latitude and longitude in all the major ocean basins.
Strain-based inferenceof internal-wave-driven turbulence
dissipation rates « and diffusivitiesK has seen widespread
use (Mauritzen et al. 2002; Sloyan 2005; Lauderdale et al.
2008; Wu et al. 2011; Whalen et al. 2012; 2015; Damerell
et al. 2012). The analysis here expands on previous shear-
and-strain estimates based on ;3500 full-depth LADCP/
CTD profiles (Kunze et al. 2006) with an order of mag-
nitude more data to provide much more comprehensive
global geographical coverage, expanding the scope to the
South Atlantic and improving sampling in the North At-
lantic, western Pacific, and Southern Oceans. The data
and methods are detailed in section 2, global maps and
sections are discussed in section 3, averages and budgets
are in section 4, zonally averaged structure is in section 5,
an evaluation of OGCM tidal mixing parameterizations is
in section 6, and finally the conclusions and discussion are
found in section 7. In a companion paper (Kunze 2017,
manuscript submitted to J. Phys. Oceanogr.), the in-
ferred diapycnal diffusivities K and dissipation rates
« are used to compute the interior internal-wave-driven
diabatic meridional overturning circulation and com-
pare it with diapycnal transports driven by near-bottom
buoyancy-flux divergence.
2. Data and methods
CTD profiles from WOCE/CLIVAR hydrographic
sections (https://cchdo.ucsd.edu/) were employed in this
analysis. These include about 73 sections from the In-
dian, 422 from the Pacific [although over 200 of these are
from theHawaii Ocean Timeseries (HOT) site], and 146
from the Atlantic with roughly 6700, 13 000, and 10 500
usable casts, respectively; casts shorter than 300m or
with resolution coarser than 2m are excluded. No data
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are included from the Arctic, Weddell, or Ross Seas and
are sparse south of 608S in the SouthernOcean. Roughly
10% of the retained profiles span less than 25% of the
water column. The T, S, and p files were downloaded
and converted to a uniform format. Density variables sp,
su, s3, and gn and stratificationN2 were then computed.
Internal-wave-driven turbulent dissipation rates « and
diapycnal diffusivities K are inferred from a finescale pa-
rameterization based on internal wave–wave interaction
theory (McComasandMüller 1981;Henyey et al. 1986) that
was first tested using 10-m vertical shear (Gregg 1989), then
modified to estimate variances spectrally in vertical wave-
number space and incorporate internal wave strain as both
(i) an independentmeans of inferring internalwave spectral
levels (Gregg and Kunze 1991; Wijesekera et al. 1993) and
(ii) to account for deviations of the internal wave aspect
ratio or frequency content from the Garrett–Munk (GM)
model (Polzin et al. 1995). The strain-based form of the
parameterization used here is
K5K0
hj2zi2
GMhj2zi2h(R
v)j
�N
f
�(1)
(e.g., Kunze et al. 2006), where K0 5 0.05 3 1024m2 s21
for a mixing efficiency g 5 0.2, hjz2i is the strain variance
with strain estimated as jz 5 (N2 2 N2fit)/N
2fit following
Polzin et al. (1995), and N2fit represents a quadratic fit to
half-overlapping, 256-m profile segments; alternative fit-
ting procedures were attempted including to log(N2)
rather than N2 and different functional forms, but these
were found to be biased compared to the simpler fit-
ting scheme. The normalizing GM model strain variance
GMhjz2i is computed over the same wavenumber band as
the observed strain. The GM75 model vertical wave-
number kz spectrum for strain is given by
SGM
[jz](k
z)5
pE0b
2
k2zj*
(kz1 k
z*)2
(2)
(Cairns and Williams 1976; Gregg and Kunze 1991),
where the canonical nondimensional spectral energy
level E0 5 6.3 3 1025, stratification length scale b 51300m, peak mode number j* 5 3, and corresponding
vertical wavenumber kz* 5 (pj*/b)(N/N0). Integrated to
10-m vertical wavelengths, the GM75 strain variance in
(2) is 0.24. The dependence on shear/strain variance
ratio Rv 5 hVz2i/(hN2ihjz2i) is
h(Rv)5
1
6ffiffiffi2
p Rv(R
v1 1)ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Rv2 1
p (3)
(Fig. 1a), which provides a crude measure of the wave
field’s aspect ratio (or frequency) content for internal
waves dominated by lower frequencies (Henyey 1991;
Kunze et al. 1990; Polzin et al. 1995). The dependence on
the ratio of buoyancy to Coriolis frequencies N/f is
j
�N
f
�5
fArccosh(N/f )
f30Arccosh(N
0/f30), (4)
where f30 5 f(308) 5 7.3 3 1025 rad s21 and N0 5 5.2 31023 rad s21. The latitude dependence [(4)] vanishes at
the equator because f goes to zero more rapidly than
Arccosh(N/f) goes to infinity; the appendix argues that
internal waves at the equator will be equatorially trap-
ped meridional modes with minimum frequencies set by
off-equatorial Coriolis frequencies at turning latitudes,
but it is not known how this will impact the cascade. The
FIG. 1. (a) Correction h(Rv) [(3)] for the strain-only turbulence
parameterization [(1)] as a function of shear/strain variance ratio
Rv. (b) Probability distribution function of Rv based on the
LADCP/CTD analysis of Kunze et al. (2006) with vertical lines
denoting the GM Rv 5 3 as well as the measured mean (10.6) and
mode (6) in the data. Almost identical distributions are foundwhen
the data are split into nine buoyancy frequencyN and nine latitude
bins, indicating that this distribution is a robust feature of the ocean
internal wave field. Means (modes) decrease for larger strain var-
iances that will contribute more to mixing, dropping to 8 (4) for
hjz2i/GM exceeding 1.
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dissipation rate « 5 KhN2i/g (Osborn 1980), where
mixing efficiency g 5 0.2 is assumed (Oakey 1982;
Itsweire et al. 1986;Moum 1996; St. Laurent and Schmitt
1999) following standard practice in the ocean micro-
structure observational community.
For the GM model, the shear/strain variance ratio
GMRv 5 3. But the ocean’s average Rv appears to be
higher (Fig. 1b), signifying that the ocean is more in-
ertial than the GMmodel on average. For measuredRv
data (Kunze et al. 2006), the distribution has mean 10.6
and mode 6.1, independent of buoyancy frequency N
and latitude; 80% of the data had Rv below 16. The
distribution’s mean and mode diminish for the larger
strain variances that will dominate internal-wave-
driven mixing (not shown). For shear/strain ratios
greater than 3, h(Rv) is an increasing function of Rv;
Rv 5 7 will be used here, which produces dissipation
rates « and diffusivities K a factor of 3 larger than for
Rv 5 3 and a factor of 3 smaller than for Rv 5 10
(Fig. 1a), so that factor of 3 uncertainties are expected;
Kunze et al. (2006) reported maximum factor of 2 dif-
ferences between shear-and-strain and strain-only dif-
fusivities for Rv 5 7.
Finescale parameterization (1) only accounts for
weakly nonlinear internal-wave-driven turbulence. It will
fail in environments where a weakly nonlinear wave-
number cascade is not expected either because of (i) lack
of bandwidth such as on continental shelves (MacKinnon
andGregg 2003; Carter et al. 2005), (ii) short-circuiting of
the cascade because of near-critical bottom reflection
(Carter and Gregg 2002; Nash et al. 2004; Kunze et al.
2012), or (iii) direct boundary forcing of turbulence (e.g.,
Klymak et al. 2008, 2010). It does not account for mixing
due to hydraulically controlled flow (Ferron et al. 1998)
or breaking solitary waves (MacKinnon andGregg 2003).
While these regions occupy a small fraction of the ocean,
theymay be important. For example, density overturns of
O(100) m [e.g., in Luzon Strait (Alford et al. 2011), Sa-
moan Passage (Alford et al. 2013), and Romanche
Fracture Zone (Ferron et al. 1998)] imply local diffusiv-
ities 104–105 times the background and so they need only
occupy 0.01%–0.1% of the ocean to produce basin-
averaged diffusivities of 1024m2s21. The shear-and-strain
parameterization overestimates turbulent dissipation rates
on the flanks of the Florida Strait (Winkel et al. 2002) and
overlying regions where lee-wave generation is expected
(Waterman et al. 2014).
Strain variance for (1) is estimated spectrally from
strain jz 5 (N2 2 N2fit)/N
2fit for half-overlapping, 256-m
profile segments starting at the bottomup to the depth of
the highest N2 in the upper 150m of the water column
(to exclude the surface mixed layer). This yields roughly
500 000 usable estimates. Strain variances are computed
by integrating the strain spectra S[jz](kz) from the
lowest resolved vertical wavenumber (lz5 256m) to the
wavenumber where variance exceeds a threshold value
of 0.05, which, for a GM-level spectrum, corresponds to
lz 5 50m, in part to avoid contamination by ship heave
near 10-m wavelengths (Polzin et al. 2014). By stopping
the integration short of the rolloff wavenumber, it is
assumed that the strain spectrum is flat like the GM
model; redder (more negative spectral slope) spectra, as
typically found here, will result in overestimation of the
strain variance by at most a factor of 1.3, while bluer
(more positive spectral slope) spectra will result in un-
derestimation. The 256-m profile segment length is a
compromise between resolution in depth and strain
variance (diffusivity). Strain variance is computed below
the rolloffwavenumberkc, whichbehaves as (0.2pm)EGM/E
(Fritts 1984; Gargett 1990; D’Asaro and Lien 2000).
With diffusivities K(EGM) ; 1025m2 s21 and K ; E2,
256-m segment lengths can resolve strain variance ra-
tios hjz2i/GMhjz2i less than 10 and diffusivitiesK less than
O(1023) m2 s21.
Strain is assumed to be dominated by finescale in-
ternal waves. While contamination by finescale geo-
strophic motions (Pinkel 2014), thermohaline staircases
(Gregg 1989), and interleaving cannot be ruled out on
dynamical grounds, in earlier high-resolution profiler
and CTD analyses, the only contamination signals that
stood out were sharp pycnoclines at low latitudes (Polzin
et al. 1995;Mauritzen et al. 2002; Kunze et al. 2006), with
contamination bywater-mass intrusions and geostrophic
motions appearing to be confined to wavelengths less
than 10m (Polzin et al. 2003) and greater than 200m
(Kunze et al. 2006). Therefore, the contamination is
largely filtered out here by the chosen 50–256-m band of
integration. Pinkel (2014) reports subinertial strain
confined to near the base of the mixed layer. Double-
diffusive interleaving may contribute where there is
strong water-mass variability on isopycnals (S. Merrifield
2016, personal communication). Double diffusion
tends to produce thermohaline staircases in only a few
known locales of weak internal-wave-driven mixing
(Gregg and Sanford 1987; Kunze 2003), which are of
little interest here, or at lateral water-mass boundaries.
Thermohaline staircases escape the spectral filter at
1200–1800-m depth beneath the Mediterranean salt
tongue between 308 and 408N in the eastern North
Atlantic and are also expected east of Barbados, in the
Tyrrhenian Sea, and under the Red Sea outflow in the
Arabian Sea (Schmitt 2003).
To avoid contamination by sharp pycnoclines, the
shallowest two segments (corresponding to the upper
380m of the water column) are omitted from anal-
ysis because they often, and unpredictably, exhibit
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unrealistically high strain variances. This problem has
previously been recognized and dealt with in a similar
manner by Mauritzen et al. (2002), Kunze et al. (2006),
andWhalen et al. (2012).Whalen et al. (2015) compared
Argo float strain-based diffusivities K to average mi-
crostructure K profiles at six sites and found that 81%
agreed to within a factor of 2 and 96% agreed to within a
factor of 3 below 250-mdepth.Amore conservative 380-m
depth was chosen here because of contamination by
very deep mixed layers at high latitudes. Qualitative
evidence that this is sufficient can be seen in Figs. 4, 6,
and 9 (shown below), where the shallowest plotted dif-
fusivities and strain variances just below 380m are
similar to those at greater depths. Profile segments were
also excluded if their average buoyancy frequency hNifell below the noise threshold 33 1024 rad s21, as these
are dominated by digitization noise (Whalen et al. 2015).
With the expectation thatN. 2f is a minimal frequency
bandwidth to allow internal wave–wave interactions,
segments with hNi less than 2f were also excluded; these
largely overlap with the hNi noise threshold, repre-
senting 10% of the data, 30% within 380 meters above
bottom (mab), and 17% within 1000 mab in abyssal
basins. Very low stratification is found (i) throughout
much of the water column at high latitudes in the
Southern Ocean, particularly the eastern Atlantic sec-
tor; (ii) at middepth around southern Greenland; and
(iii) in abyssal basins in the Caribbean, Angola Basin,
and North Pacific. These will have small diapycnal
buoyancy fluxes hw0b0i 5 2KhN2i, where w0 and b0 areturbulent vertical velocity and buoyancy fluctuations,
because their stratification is weak, so their omission has
little impact. While internal waves can exist for N , f
such that topographic generation of internal waves is
still possible in very weakly stratified bottom boundary
layers, these will be unable to propagate into regions
where N exceeds f and so will be confined near the
bottom. How this weak stratification might impact in-
ternal tide and lee-wave generation, or near-bottom
turbulence, has not been investigated to the author’s
knowledge.
3. Geography
Maps of depth-integrated dissipation rateЫ5 r0
Ыdz1
from 380-m depth to the bottom (Fig. 2) show elevated
values associated with abrupt topography and slow
spreading ridges, particularly (i) in the western In-
dian over the Southwest Indian Ridge (308–408S,408–608E), the Seychelles and Mascarene Ridge east of
Madagascar (28–208S, 508–608E), in the western Arabian
Sea over the Owen Fracture Zone (108N, 608E) and
Carlsberg Ridge, and in the wake of the Kerguelan
Plateau (508–608S, 708–808E; Polzin and Firing 1997);
(ii) in the western Pacific over abrupt ridge/trench to-
pography associated with subduction; (iii) in the central
Pacific associated with island archipelagos such as the
Hawaiian Island chain (208–308N, 1508–1808W) and
Tuamotu Archipelago (208–308S, 1308–1608W); and
(iv) in theAtlantic over continental slopes such as in theBay
of Biscay (458–508N, 08–108W), the Flemish Cap (408–508N,358–408W), and Mid-Atlantic Ridge. Low values are
found over smooth abyssal basins such as (i) the
central Arabian Sea (108–158N, 558E) and Bay of Bengal
(108–208N, 908E) and south Indian Basin south-
southeast of Sri Lanka (08–208S, 808E) in the Indian
Ocean, (ii) in the subpolar North Pacific and eastern
Pacific Ocean, (iii) in the Angola Basin in the eastern
South Atlantic (08–208S, 08–208E), and (iv) in the
Southern Ocean south of ;588S, where N is low
throughout the water column. Overall, high vertically
integrated dissipation rates are consistent with predicted
sites of high internal tide generation (e.g., Egbert and
Ray 2001; Simmons et al. 2004; Nycander 2005). Low
values in the Southern Ocean do not support lee waves
being a major dissipative conduit for the Antarctic
Circumpolar Currents (Nikurashin and Ferrari 2011;
Scott et al. 2011), consistent with microstructure measure-
ments at Kerguelen Plateau and Drake Passage finding
dissipation rates up to an order of magnitude below lee-
wave generation predictions (Waterman et al. 2014).
The WOCE/CLIVAR hydrography is not well suited
for examining the upper ocean’s response to storm
forcing because of its temporal intermittency. Argo
profiling floats provide better temporal sampling of the
seasonal cycle of upper-ocean mixing, which appears to
be confined to 308–408 latitude (Whalen et al. 2012).
Binning vertically integrated dissipation ratesЫ by
longitude shows that elevated values in the western In-
dian and western to central Pacific are related to to-
pography (Fig. 3), not tidal flows, which is more uniform.
This interpretation augments that of Hibiya et al. (1999),
who predicted western intensification of turbulence in
the North Pacific because of the hot spot of near-inertial
wave generation near 408N and west of the date line
(Alford 2001). On average, most of the dissipation oc-
curs in the pycnocline rather than near the bottom
(Fig. 3a), in contrast to OCGM tidal mixing parame-
terizations (Jayne and St. Laurent 2001; Decloedt and
Luther 2010).
1 A factor of g21 5 5 coding error in Kunze et al. (2006) that
underestimated integrated dissipation ratesЫ in their Figs. 5–12
has been corrected here. Their conclusion that there is insufficient
dissipation to account for internal wave power sources is invalid.
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FIG. 2. Maps of the vertically integrated dissipation rateЫ5
Ðr« dz in mWm22 for the full
water column excluding the upper 380m. Low values are found in the Southern Ocean, the
Indian’s Bay of Bengal, and South Atlantic’s Angola Basin. High values are associated with
abrupt topography in the western Indian, western and central Pacific, and over midocean
ridges. Because of the N2 scaling of «, dissipation rates integrated over the pycnocline
(,2000m) show similar patterns and magnitudes.
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Repeat sections of inferred diapycnal diffusivity K
illustrate that elevated variability related to topography
is reproducible. While elevated diffusivities K above
weak rough topography are sometimes confined to
within 500mab over stronger and more extensive to-
pography, it often extends throughout the entire water
column (Fig. 4), consistent with Fig. 3a and in contrast
with the fixed decay scale of 500 m implemented in
OGCM tidal mixing parameterizations (Simmons et al.
2004; Saenko andMerryfield 2005; Jayne 2009; Friedrich
et al. 2011) based on microstructure measurements on
theMid-Atlantic Ridge bounding the eastern side of the
Brazil Basin (St. Laurent et al. 2001). This is more
clearly illustrated in the joint probability density
function of diffusivity above 2000-m depthKpycno versus
below 2000-m depth Kdeep (Fig. 5), which reveals a
correlation between diffusivities in these two depth
ranges with Kpycno ; Kdeep/2. Since « 5 KN2/g and N2
exhibit more variability than K, most of the dissipation
will be in the high stratification of the pycnocline.
Figure 4 also illustrates some of the variety of bottom
geometries that can contribute to elevated strain
variance.
Equatorial crossings consistently show elevated strain
variance within628 of the equator (Fig. 6). There is littlecorresponding signal in diapycnal diffusivityK because a
reduced cascade rate as f / 0 in (4) allows more vari-
ance to accumulate at lower wavenumbers for the same
FIG. 3. Vertically integrated dissipation ratesЫ (a) reveal that most of the dissipation
occurs in the pycnocline (red). Peaks in the western Indian and western Pacific (left two gray
bars) appear to bemost correlated with 30 km3 30 km topographic roughness h2 (b), whereas
rms tidal velocities (c) are more uniform. Elevated dissipation rates and topography over
2858–3008E (608–758W) are associated with both Drake Passage and the Caribbean. TheseЫ
exclude the upper 380m.
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dissipation rate (Gregg et al. 2003). This contrasts with
the stripes of elevated diffusivity K or integrated dissi-
pation rateЫ reported flanking the equator by Whalen
et al. (2012); the absence of this signal here appears to be
due to the hydrography casts missing frequent but in-
termittent mixing bursts associated with negative La
Niña and neutral ENSO conditions (C. B. Whalen 2016,
personal communication).
FIG. 4. Repeat sections of inferred turbulent diapycnal diffusivityK show the influence of rough topography including (a05) crossing of
the Mid-Atlantic Ridge near 258N, (p14) crossing of the Aleutian Island Ridge, (p06) crossing of the Colville Ridge and Louisville
Seamount Chain near 338S in the subtropical South Pacific, (i05) crossing the ridges in the western Indian near 348S, and (p06) crossing
between eastern Australia and north of New Zealand near 308S. While elevated diffusivities K over rough topography are sometimes
confined near the bottom, they often extend through the entire water column, which will produce particularly strong dissipation rates in
the pycnocline where N is high.
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4. Averages and budgets
The global-integrated dissipation rate r0ÐÐ Ð
«dV,
computed as r0Ð h«(z)i dz times the ocean area,
where , . is the average of all the profiles, is 1.5 60.4TW (4.3 6 1.0mWm22 per unit area) below 380-m
depth. Averages per unit area are largest in the North
Pacific (5.3mWm22) and smallest in the South Atlantic
(2.3mWm22). The above value is missing the contri-
bution above 380-m depth, which is potentially signifi-
cant in light of theN2 scaling of «, so this fraction is now
estimated.Assuming a conservative, that is, low average
pycnocline diffusivity K 5 0.1 3 1024m2 s21 (Gregg
1989; Waterhouse et al. 2014) and « 5 KN2/g above
380m implies an additional 0.5 6 0.2 TW in the upper
380m, likely an underestimate because shipboard sam-
pling is biased toward fair weather so will miss some of
the wind-forced contribution in the 308–408 latitude band(Whalen et al. 2015). The total inferred dissipation of
2.06 0.6 TW is then consistent with the sum of internal
wave power inputs of 1.0–1.2TW from the tide (Egbert
and Ray 2001; Nycander 2005), 0.2–1.1TW from wind
(Alford 2001; Plueddemann and Farrar 2006; Furuichi
et al. 2008; Rimac et al. 2013), and 0.2–0.7 TW from lee-
wave generation (Scott et al. 2011; Nikurashin and
Ferrari 2011; Melet et al. 2014; Wright et al. 2014) and
thus is sufficient to close the internal wave energy bud-
get within the present large uncertainties for both
sources and sinks of internal waves.
Average profiles are similar in all ocean basins so
only global averages are shown (Fig. 7). Average GM-
normalized strain variances hjz2i/GM ; 2, almost in-
dependent of depth z, but increase to;3 at the bottom in
height above bottom coordinates h. Dissipation rates
« exhibit the most variability with respect to depth z and
buoyancy B ’ 2(ggn/r0), where gn is neutral density,
while GM-normalized strain variance and diapycnal dif-
fusivity K vary the most with respect to height above
bottom h. Dissipation rates « decrease with depth z.
They have a minimum of 0.2 3 1029Wkg21 between
1000 and 3000 mab and increase to a maximum of
1029Wkg21 within 500-mab of the bottom. But much of
this increase is contributed by the pycnocline. Aver-
aging only waters below 2000-m depth, the average dis-
sipation rate is 0.33 1029Wkg21 at the bottom and 0.131029Wkg21 above 1000 mab in height above bottom
coordinates.
Average diffusivities K are (0.3–0.4) 3 1024m2 s21
with little dependence on depth z but increase toward
1024m2 s21 at the bottom in height above bottom co-
ordinates h with an e-folding scale of ;2000m. This dif-
ference arises because the bottom is not always at the same
depth. Diffusivities K increase from 0.3 3 1024m2 s21 for
buoyancy B . 20.27ms22 to K5 0.7 3 1024m2 s21 at
lower buoyancy (higher density), showing weaker depen-
dence than the Lumpkin and Speer (2003) inverse esti-
mates, though lying within the latter’s uncertainties except
in 20.267 . B . 20.269ms22 (gn 5 28.0–28.2). While
average diffusivity profiles here are lower than the 1024m2s21
reported below 1000-m depth from 17 microstructure sites
in Waterhouse et al. (2014), comparison of finescale pa-
rameterization [(1)] inferences in the vicinity of these 17
sites were consistent with microstructure averages. Most of
the sites considered in Waterhouse et al. have predicted
sources larger than depth-integrated dissipation rates, im-
plying that the sites were mostly located in net internal
wave sources rather than net sinks. This suggests that (i) the
strain parameterization is reasonable on average and (ii)
microstructure sampling has been biased toward regions of
stronger forcing, which is consistent withWaterhouse et al.
reporting that internal wave sources exceeded sinks atmost
microstructure sites, which have undersampled regions of
low tidal power (their Fig. 5b).
The strain-based average diffusivities K are a factor
of 2–3 higher than shear-and-strain-based values in
Kunze et al. (2006) above 3000-m depth but comparable
below 3000-m depth and comparable to their strain-based
estimates. They are higher for heights above bottom h
below 1000 mab; Kunze et al. reported average dissipation
rates « increasingmonotonicallywith height above bottom.
Cumulative dissipation rates, substituting for the
upper 380m as above, are concentrated in the upper
FIG. 5. Probability distribution function of diapycnal diffusivity
K in the pycnocline (,2000-m depth) and in the deep (.2000-m
depth), illustrating that elevated pycnocline diffusivities are cor-
related with deeper diffusivities but weaker by a factor of 2 (lower
thin dotted line).
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pycnocline (Fig. 8), with 80% of the dissipation below
380-m depth contributed above 1000m. Only 0.08TW
dissipates below 2000-m depth, suggesting very little
local mixing in the abyss. Roughly 20% (30%) of the
dissipation is found below h, 500 mab (1000 mab), that
is, 0.4 TW (0.6 TW). Again, these differences reflect that
the bottom is not always at the same depth.
5. Average zonal and meridional structure
Zonally averaged depth–latitude sections reveal small
differences between the three major oceans (Fig. 9). All
the oceans show a pool of elevated dissipation rates
« shallower than 2000-m depth for latitudes equatorward
of 508–608 (Fig. 9c), coinciding with the higher stratifica-
tion in the main pycnocline (Fig. 9b). This is not reflected
in the diffusivity K, which is vertically more uniform
(Fig. 9d), consistent with average dissipation rates
« scaling as N2 (Gregg and Sanford 1988). Weaker dif-
fusivities straddling the equator reflect the N/f scaling in
(4) but may be biased low (Whalen et al. 2012; Thurnherr
et al. 2015) because the rich equatorial wave field is
outside the scope of the internal gravity wave–wave in-
teraction theory behind the finescale parameterization.
Indian and Atlantic diffusivities K exceed 1024m2 s21 at
all depths for latitudes poleward of 408–508, associatedwith weaker high-latitude stratification, while the Pacific
diffusivities are more uniformly weak at subpolar
FIG. 6. Repeat sections illustrating equatorial crossings including (i04) in the Indian Ocean
along 808E and (p18) south of Baja California in the eastern Pacific. Strain variance is elevated
near the equator because the cascade proceeds more slowly, as represented by the j(N/f ) term
in (1). TheN/f scaling [(4)] compensates for the excess strain near the equator formore uniform
diffusivitiesK that are forced to zero at the equator as f/ 0. In the Pacific section, there is also
elevated strain variance associated with rough topography at 58S and 108N.
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latitudes and its stratification is stronger. An alternative
explanation is that this latitude band is associated with
internal lee-wave generation by Antarctic Circumpolar
Currents interacting with topography, where the finescale
parameterization overestimates turbulence dissipation
rates « by as much as an order of magnitude (Waterman
et al. 2014). In the Indian and Atlantic, diffusivities seem
to weaken slightly south of 608S, while they becomemore
elevated north of 608N in the Atlantic.
Zonally averaged vertically integrated dissipation
ratesЫ (Fig. 10a) appear to correlate with topographic
roughness h2 (Fig. 10b), while tidal flows are more
FIG. 7. Global average profiles of, from left to right, the number of data points n, buoyancy frequencyN, GM-normalized strain variance
hjz2i/GM, diapycnal diffusivity K, and dissipation rate « as functions of (top) depth z, (middle) height above bottom h, and (bottom)
buoyancy B5 0:32 (ggn)/r0 with neutral density gn indicated along the rightmost axis. Values are not plotted for n, 300 and are plotted
gray for n, 3000. Dotted curves in height above bottom coordinates exclude data shallower than 2000m. Normalized strain variance and
diapycnal diffusivityK are nearly independent of depth (top row) at 1.5–2 and (0.3–0.4)3 1024 m2 s21, respectively, butK decreases from
1024 m2 s21 at the bottom to 0.153 1024 m2 s21 above 5000mab in height above bottom coordinates h (middle row)with a 2000-m e-folding
scale, and the dissipation rate « is elevated below 1000mab because of both increasingK as h/ 0 and elevatedN below 1000 mab; much of
the elevated N and « BBL is contributed by shallow water less than 2000m deep. Strain-inferred diffusivities exhibit less variability with
buoyancyB (bottom row) than the Lumpkin and Speer (2003) inverse estimates though agreeing within their uncertainties except over B520.267 to 20.269m s22 (gn 5 28.0–28.2).
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uniform (Fig. 10c). In the Southern Hemisphere, a peak
inЫ poleward of 308S is not consistent with the pre-
dictions of parametric subharmonic instability enhanc-
ing the cascade of low-mode internal tide energy
equatorward of 288 and 148 (MacKinnon and Winters
2005; Hibiya et al. 2006; Alford et al. 2007; MacKinnon
et al. 2013a,b). A signature of elevatedЫ equatorward
of 308N in the Northern Hemisphere may be tied to ei-
ther parametric subharmonic instability or elevated to-
pographic roughness over 108–228N (Fig. 10b) latitude in
both the Pacific and Atlantic.
Meridionally averaged depth–longitude sections show
widespread elevated diffusivities poleward of 508 in the
Atlantic and Indian (Fig. 11) and a slight tendency to-
ward higher values in the upper ocean near western
boundaries. In the 508–708S latitude bin, the longitude
dependence resembles that of wind forcing (Kilbourne
2015) with elevated values of O(1024) m2 s21 in the In-
dian and Atlantic sectors of the Southern Ocean but
weak mixing O(1025) m2 s21 in the eastern Pacific sec-
tor. However, this pattern is also seen in higher strati-
fication N2 in the eastern Pacific sector compared to the
Indian andAtlantic (Fig. 11), and, as alreadymentioned,
this is the latitude band where the finescale parameter-
ization overestimates turbulent dissipation rates in
Antarctic Circumpolar Currents (Waterman et al. 2014).
6. Tidal mixing parameterizations
The last decade has seen the development (Jayne and
St. Laurent 2001; Polzin 2004; Decloedt and Luther
2010) and implementation (St. Laurent et al. 2002;
Simmons et al. 2004; Saenko andMerryfield 2005; Jayne
2009; Friedrich et al. 2011; Melet et al. 2013, 2014) of
subgrid-scale parameterizations for local tidally driven
mixing in OGCMs. In general, the dissipation rate « can
be expressed as
«5qE(x, y)F(z) (5)
(Jayne and St. Laurent 2001), whereE(x, y) is the laterally
variable bottom forcing, F(z) is the vertical structure, and
q is the fraction lost to turbulent dissipation locally in the
overlying water column. OGCM implementations of (5)
have assumed constant q5 0.3 and a constant decay scale
of 500m in F(z). As already shown (Figs. 4–5), turbulent
bottom boundary layer thicknesses are extremely vari-
able, often extending through the entire water column.
This supports a more dynamically variable turbulent
bottom boundary layer thickness, such as Polzin (2004) as
implemented in Melet et al. (2013), or Olbers and Eden
(2013). Most of the dissipation occurs in the pycnocline
(Figs. 3, 7, 8, 9, 10). This suggests that bottom-generated
internal tides freely propagate up through the water col-
umn, largely dissipating in the upper ocean where higher
stratification amplifies the nonlinear cascade.
Likewise, it is known that q is much lower over steep
isolated topography (Althaus et al. 2003; Klymak et al.
2006) than over the abyssal hills’ topography character-
izing slow midocean spreading ridges (St. Laurent and
Garrett 2002). Here, we compare vertically integrated
dissipation ratesЫwith topographic forcing predictions
to reiterate that q is not constant but appears to decrease
with increasing forcing. In Fig. 12, the vertically in-
tegrated dissipation ratesЫ (Fig. 2) are binned with (i)
linear internal tide power input (Bell 1975)
E(x, y)5NU2hhx5 kNU2h2 , (6)
(ii) linear internal lee-wave generation theory (Bell 1975)
E(x, y)5N2Uh2 , (7)
and (iii) topographic roughness (height variance) h2,
whereN is the bottom buoyancy frequency;U is the rms
barotropic tidal velocity from TPXO.3 (Egbert and Ray
FIG. 8. Global cumulative dissipation ratesЫ as a function of
(top) depth z and (bottom) height above bottom h illustrate that
50% (80%) of the dissipation occurs above 500-m (700m) depth.
Dissipation rates are not accumulated above 400-m depth. Only
20% of the dissipation is found below h 5 500 mab.
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2001); h2 is the topographic roughness (height variance)
on length scales less than that of the mode-one internal
tide, which here is taken to be the variance in 30km 330km domains in the Smith and Sandwell (1997) global
bathymetric database (10km 3 10km domains yielded
similar dependences); and k is a characteristic horizontal
wavenumber that is taken as a free parameter to best
match rms global surface tide elevation (Simmons et al.
2004). These linear theories are applicable for weak to-
pography (topographic height hmuch less than the water
depthH and topographic slope smuch less than the wave
ray slope k/m), which is valid at the semidiurnal frequency
for 97% of the bottom based on Smith and Sandwell
(1997) bottom slopes (though only 75% above 1500-m
water depth). At 1.0–1.2TW (Egbert and Ray 2001;
Nycander 2005), internal tide generation [(6)] is thought
to dominate (Waterhouse et al. 2014) over the less certain
wind-generated inertial waves power input of 0.2–1.1TW
(Alford 2001; Plueddemann and Farrar 2006; Furuichi
et al. 2008; Rimac et al. 2013) and internal lee-wave gen-
eration [(7)] of 0.2–0.7TW (Scott et al. 2011; Nikurashin
and Ferrari 2011; Wright et al. 2014; Waterman et al.
2014), but neither these other sources nor remote tidal
dissipation are separable in our estimates.
Observed dependence forЫ on topographic forcing
(Fig. 12) is much weaker than one to one and similar to
the h1/2 dependence reported by Kunze et al. (2006).
Since saturation has been discounted above, we in-
terpret this as signifying that bottom-forced internal
waves are not all locally dissipated (q , 1), and hori-
zontal radiation redistributes a significant fraction of
the forcing before dissipation and mixing; that is, most
internal-wave-driven mixing is remote from sources.
This is consistent with Waterhouse et al. (2014), who
reported that most of the 17 microstructure measure-
ment sites they considered had excess forcing compared
to dissipation. The weak dependence ofЫ on forcing in
Fig. 12 points to q decreasing with increasing forcing, but
there is order-of-magnitude scatter, suggesting unre-
solved physics.
While Figs. 4 and 12 point to problems with choosing
constant local dissipation fractions q and decay scales as
in existing tidal mixing parameterizations, it would be
premature to suggest better scalings. Order of magni-
tude scatter is evident in the raw scatterplots (Fig. 12)
that may be related to topographic details lost in the
coarse Smith and Sandwell topographic roughness h2
and rms tidal flows U used here. Some of the largest
internal tide sources are associated with abrupt topog-
raphy (Fig. 2) such as the Luzon Strait, Hawaiian Ridge,
Tuamotu Archipelago, Aleutian Island chain, and so on
(Ray and Mitchum 1997; Lee et al. 2006; Simmons et al.
FIG. 9. Depth–latitude sections of (a) number of data points n, (b) average buoyancy frequencyN, (c) dissipation rate «, and (d) diapycnal
diffusivityK for the (left) Indian, (center) Pacific, and (right) Atlantic. Dissipation rates « (c) are elevated in the pycnocline (latitudes, 508–608) mirroring the stratificationN in (b). DiffusivitiesK in (d) areO(0.13 1024) m2 s21 in much of the oceans but are elevated in the Indian
andAtlantic sectors of the SouthernOcean (latitudes below 408S) and in the northern NorthAtlantic (latitudes above 408N) and spottily near
the bottom. There are also hints of a band of elevated K just equatorward of the semidiurnal PSI critical latitude of 288 (Alford et al. 2007;
MacKinnon et al. 2013a,b) in the North and South Pacific but not the Atlantic or Indian. Black contours are density surfaces.
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2004; Zhao et al. 2016), for which the weak-topography
approximation does not apply. No effort was made to
isolate topographic scales relevant to internal tide and lee-
wave generation or to include fortnightly tidal modulation
or subinertial flows. We have no way of robustly isolating
tidal, lee-wave, and wind-forced dissipation sources. Nor
do we need internal wave sources and sinks to be corre-
lated in space/time because internal waves propagate (Ray
and Mitchum 1997; Alford 2001, Zhao et al. 2016), car-
rying energy away to dissipate elsewhere and at other
times. For example, the canonical q5 0.3 implies 70% of
the energy is not dissipated locally. We cannot distinguish
locally and remotely forced dissipations, both of which
Oka and Niwa (2013) find are necessary to explain the
Pacific thermohaline circulation.
Future research is planned to try to better tease apart
conditions needed for the turbulent bottom boundary
layer to extend through thewholewater column and howq
depends on topographic forcing, but, because of the limi-
tations in the data, process-oriented numerical modeling
may be the best way to explore this parameter space.
7. Conclusions
Afinescale parameterization for internal-wave-driven
turbulent dissipation rates « and diapycnal diffusivities
K was applied to ;30 000 CTD casts from all the major
oceans (Fig. 2), though excluding the Arctic, Weddell,
and Ross Seas. The global integrated dissipation of
2.0 6 0.6 TW is consistent with the 2.1 6 0.7 TW tide,
FIG. 10. Zonal averages of (a) vertically integrated dissipation ratesЫ, (b) topographic
roughness h2, and (c) rms tidal currents U as a function of latitude. Integrated dissipation
rates are for the water column below 380-m depth (black) in the pycnocline between 380- and
2000-m depth (red and below 2000-m depth (blue).
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wind, and lee-wave sources for internal gravity waves, so
there may be no need to invoke missing or ‘‘dark’’ tur-
bulent mixing on continental slopes and canyons (Kunze
et al. 2006; Waterhouse et al. 2014), though we caution
that uncertainties are large enough that, for example,
there need be little or no contribution from internal
lee waves (Waterman et al. 2014). Of this dissipation,
80%–90% occurs above 1000-m depth and less than
0.08 TW below 2000m (Fig. 8), compared to the 0.3 TW
required to maintain deep stratification in a vertical
advective–diffusive balance that ignores horizontal ad-
vection (Munk 1966; Munk and Wunsch 1998; Wunsch
and Ferrari 2004). As a caveat, because the bulk of
the dissipation occurs in the upper pycnocline, deep
(.2000-m depth) and abyssal (.4000-m depth) mixing
are poorly constrained by the global bulk budget.
Nevertheless, it can be concluded that most mixing is
remote from deep generation sites. The 256-m half-
overlapping spectral bins limit vertical resolution and
so may not resolve thin stratified turbulent bottom
boundary layers. Vertically integrated dissipation rates
Ы vary by three to four orders of magnitude (Fig. 2)
with elevated values in the western Indian and Pacific
Oceans associated with abrupt topography (Fig. 3),
consistent with internal tide generation site predictions
(Egbert and Ray 2001; Simmons et al. 2004; Nycander
2005). These do not scale with the predictions of linear
theories for internal tide or lee-wave generation (Bell
1975; Fig. 12), suggesting that the locally dissipated
fraction q decreases with increasing forcing, and sig-
nificant horizontal redistribution of wave energy occurs
before dissipation. Because there is little local dissipation/
mixing and considerable redistribution by internal wave
propagation, prediction of where and when turbulent
dissipation will occur is not straightforward.
Spatial patterns are repeatable (Figs. 4, 6) and show
variable turbulent bottom boundary layer thicknesses
that often extend throughout the whole water column
over rough topography (Figs. 4–5) in contrast to the
fixed 500-m decay scale assumed in OGCM tidal mixing
parameterizations (Simmons et al. 2004; Saenko and
Merryfield 2005; Jayne 2009; Friedrich et al. 2011). This
FIG. 11. Depth–longitude sections of turbulent diapycnal diffusivities K and buoyancy frequencies N binned by latitude. Diffu-
sivities areO(1025) m2 s21 in most of the ocean but become higher in the North Atlantic (608W–08) in the upper ocean of the western
boundaries and in parts of the Southern Ocean (308E –1808, 708–108W), where the stratification is weak. Elevated diffusivities in
the Southern Ocean correspond to the longitude bands where N is weak and there is elevated inertial wind forcing (Kilbourne 2015)
but are also at latitudes of the Antarctic Circumpolar Current where the finescale parameterization overestimates turbulence
(Waterman et al. 2014). Elevated diffusivities in the shallowest layer at high latitudes are likely biased high by sharp pynoclines so are
not to be trusted.
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supports use of a more dynamically motivated parame-
terization such as Polzin (2004) as implemented inMelet
et al. (2013), or Olbers and Eden (2013). Further testing
is needed to better determine how local dissipative
fraction q and decay scale depend on topography, tidal
flows, and other environmental properties.
The global-averaged turbulent diapycnal diffusiv-
ity K is almost independent of depth z at (0.3–0.4) 31024m2 s21 but increases from 1025m2 s21 at 6000mab to
1024m2 s21 at the bottom (h5 0) in height above bottom
h coordinates with an average e-folding scale of;2000m
(Fig. 7), though this decay scale is not constant (Fig. 4).
The difference between the z and h coordinate systems
arises because the bottom (h 5 0) is not always at the
same depth z. Diffusivities vary by two orders of
magnitude but cannot be estimated reliably for values
exceeding 10 3 1024 m2 s21 because of limitations in
the methodology. On average, the dissipation rate «
decreases with depth z and density (Fig. 7), though it
may display a weak increase with neutral densities
greater than 28.2. It is elevated to 1029Wkg21 in the
bottom 500 mab in height above bottom coordinates but
exhibits little variability over this bottom layer, the largest
gradient being between 700 and 1200 mab, and it is 0.1531029Wkg21 between 1000- and 3000-mab before in-
creasing slowly as h increases. This contrasts with the
commonly assumed exponential decay over 500 mab
above rough topography in OGCMs based on micro-
structure measurements in the Brazil Basin (St. Laurent
et al. 2001). This may reflect a difference between global
and regional behavior or that the finescale parameteri-
zation is underestimating near-bottom directly forced
turbulence. Consistent with the former interpretation,
an average profile that excludes the upper 2000m
(pycnocline) produces a turbulent bottom boundary
layer that more closely resembles the canonical decay
scale of;500 mab (dotted curves with h in Fig. 7). The
average dissipation rate is 0.2 3 1029Wkg21 for
FIG. 12. Full water column vertically integrated dissipation ratesЫ binned by (left) bottom roughness variance
,h2. over 30 km 3 30 km, (center) linear internal lee-wave generation [(7)], and (right) linear internal tide
generation [(6)]. The upper row displays the joint probability distributions and the lower row the means and
standard errors. All integrated dissipation ratesЫ exclude the upper 380m where strain estimates of « may be
contaminated by sharp changes in background stratification. Dotted curves in the lower row correspond to the
probability distributions of the lower-axis variable. Thick diagonal lines correspond to a linear dependence on the
horizontal axes (e.g., h2 in the left set of panels); thin lines indicate the corresponding square root and quartic root of
the same. Levels are not meaningful.
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buoyancy B , 20.268 (gn . 28.2), increasing to values
greater than 2 3 1029Wkg21 for B . 20.262 (gn ,27.0). Finescale inferences agree with average direct
microstructure measurements at the 17 sites high-
lighted in Waterhouse et al. (2014).
Zonal averages in all three oceans (Fig. 9) are similar,
with weak diffusivities along the equator despite elevated
strain variance (Fig. 6) because theN/f dependence in (4)
moderates the elevated strain; this contrasts with the off-
equatorial stripes of elevated dissipation rate reported in
the pycnocline by Whalen et al. (2012) based on more
extensive Argo profiling float sampling. Higher diffusiv-
ities are found at subpolar latitudes in the Indian and
Atlantic but not Pacific, reflecting both their stratification
and wind-forcing patterns. However, the finescale pa-
rameterization is also known to overestimate turbulence
in the Antarctic Circumpolar Current at these latitudes
(Waterman et al. 2014). The overall uniformity of K is
consistent with «;KN2, and most features in the zonally
averaged « can be related to variability in the stratifica-
tion rather than diffusivity. No compelling support for
tidal parametric subharmonic instability (PSI) enhancing
turbulence production equatorward of 148 and 288 lati-tudes was found (Figs. 9–10) since K is elevated near 308in the Pacific but not Atlantic or Indian. Likewise, in-
tegrated dissipation rates south of 408S are less than 0.03
TW (Figs. 2, 10), which does not support 0.1–0.3TW lee-
wave dissipation of Antarctic Circumpolar Currents
(Nikurashin and Ferrari 2011; Scott et al. 2011; Melet
et al. 2014; Wright et al. 2014) but is consistent with mi-
crostructure measurements finding dissipations as much as
an order of magnitude below theoretical predictions
(Waterman et al. 2014). Again, the finescale parameteri-
zation overestimates turbulent dissipation rates in Ant-
arctic Circumpolar Currents (Waterman et al. 2014).
The proxy dataset for global ocean mixing assembled
here has shown reasonable skill in reproducing direct
microstructure and preconceptions in a broad brush
view. While well suited for studying semisteady turbu-
lent sources such as internal tides, the datasetmay not be
suitable for studying wind-forced internal-wave-driven
mixing because of its limited temporal sampling; the
Argo profiling float dataset has proven more appropri-
ate for exploring these dependencies (Whalen et al.
2012) and finds little seasonal variability outside storm-
forced latitudes 308–408 (Whalen et al. 2015). The
finescale parameterization has also been found to
overestimate turbulence where internal lee waves are
expected to be the source (Waterman et al. 2014). How
well this dataset can reproduce the details of internal-
wave-driven mixing despite these known limitations
awaits further analysis and comparison, specifically
exploration of the decay scale and dissipative fraction q
of the stratified turbulent bottom boundary layer as-
sociated with internal tide generation in the context of
predictions based on internal wave–wave interaction
theory (Polzin 2004; Olbers and Eden 2013).
Acknowledgments. For Walter Munk, who started it
all, on his 100th birthday. The author acknowledges the
efforts of the hundreds of scientists and technicians who
collected, processed, and quality controlled the hydro-
graphic data used in this study. Barry Ma and Fiona Lo
assisted with data extraction. Discussions with Cimarron
FIG. A1. (left) Minimum frequency vmn (A3) and corresponding latitude (upper axis)
where vmn 5 f as functions of vertical wavenumber m (vertical coordinate). (right) Meridi-
onal length scale L 5 1/‘mn [(A2)] corresponding to the minimum frequency vmn [(A3)]
as a function of vertical wavenumber m. Low vertical wavenumbers are influenced by
latitudes;38, while finescale waves are much more closely confined within 20 km in their off-
equatorial influence.
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Wortham on fitting procedures were valuable. Com-
ments from two anonymous reviewers led to improvements
of the manuscript. This research was supported by
NSF Grant OCE-1153692. (Internal-wave-driven infer-
red turbulence dataset is available at ftp.nwra.com/
outgoing/kunze/iwturb.)
APPENDIX
Equatorial Internal Waves
Internal gravity waves on the equator are trapped
inside a meridional waveguide by b 5 ›f/›y such that
they feel off-equatorial rotation f 5 by by virtue of
having finite meridional scales. Therefore, the equato-
rial internal wave dispersion relationmight be written as
v5b
‘1
N2(k2 1 ‘2)‘
2bm2, (A1)
where ‘ is proportional to an effective meridional
wavenumber, k is the zonal wavenumber, and m is the
vertical wavenumber. Equation (A1) has a minimum
frequency at k 5 0 and ‘, satisfying
052b
‘21
3N2‘2
2bm20 ‘
mn5
�2
3
�1/4�bm
N
�1/2(A2)
with corresponding frequency
vmn
5
"�3
2
�1/41
1
2
�2
3
�3/4#�
bN
m
�1/2(A3)
(Fig. A1). If this scaling is correct, it may be more
appropriate to use vmn [(A3)] instead of f in the
N/f scaling [(4)] near the equator to allow finite tur-
bulence production, though Fig. A2 illustrates that
wavelengthsO(10)m are meridionally confined within
;0.48 of the equator and so will have very low vmn ;1026 rad s21 (;2-month periods). In contrast, low ver-
tical modes with O(1000)m vertical wavelengths are
confined within;28 latitude withvmn; 53 1026 rad s21
(;2-week periods).
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