Internal Waves are all around us... - Physical Oceanography ...

Internal Waves are all around us...
Petrenko et al 00
17-18 isotherms
Baroclinic velocity
Google Earth
Eich et al 04

Erratic internal waves at SIO Pier
data and wavelet analysis courtesy of E. Terrill, SIO
Barotropic (depth-independent)
Baroclinic (depth-dependent)
semi-diurnal
diurnal
Time in hours
Time in hours

Simple interfacial internal wave
h = −h 0 cos(kx − ωt)
2π/k = one wavelength
U 1 = ωh 0
cos(kx − ωt)
H 1 k
H 1
U 1
h
U 2 = − ωh 0
cos(kx − ωt)
H 2 k
H 2
U 2
after Gill,
Atmosphere-Ocean Dynamics

Linearize equations of motion
∂u
= −⃗u · ∇u + fv − 1 ∂p
∂t
ρ ∂x + ν∇2 u
∂v
= −⃗u · ∇v − fu − 1 ∂p
∂t
ρ ∂y + ν∇2 v
∂w
= −⃗u · ∇w − 1 ∂p
∂t
ρ ∂z + ν∇2 w − g
∂ρ
∂t
= −⃗u · ∇ρ + κ∇ 2 ρ
∇ · ⃗u = 0
Internal wave equations
Low-mode versus high-mode
Try a solution of the form
u(x, y, z, t) = ûe −i[kx+ly+mz−ωt]
Get polarization and dispersion
relationships
ω 2 = (k2 + l 2 ) ∗ N 2 + m 2 ∗ f 2
k 2 + l 2 + m 2
Wave propagation direction
U = Ψ(z)cos(kx − ωt)
(Glenn Flierl)

What generates internal waves?
1) Barotropic tide sloshing over topography
Internal Tide: An internal wave
with a tidal frequency, usually
once in 12.4 hours = M2
Often generated at the continental
shelf break, with waves
propagating both on and off
shore.
(J. Nash)

What generates internal waves?
Spring Observations
2) Wind makes near-inertial internal waves
116 118 120 122 124 126 128 130
! / N m !2
0.4
0.2
Wind Stress
0
15
8
T
z / m
35
7
T / ° C
55
6
V bc
15
S 2
z / m
z / m
35
55
15
30
45
0.2
0
!0.2
!3
!4
!5
!6
log 10
(S 2 / s !2 )
V bc
/ m s !1
116 118 120 122 124 126 128 130
yearday
(MacKinnon and Gregg, JPO, Dec 05)

13 Dec 2003 18:52 AR AR203-FL36-12.tex AR203-FL36-12.sgm LaTeX2e(2002/01/18) P1: IBD
Internal wave breaking mixes the ocean
302 WUNSCH FERRARI
Waves break by shear instability
or convective overturning.
Woods 68
Wunsch and Ferrari 04
Figure 5 Strawman energy budget for the global ocean circulation, with uncertainties of
at least factors of 2 and possibly as large as 10. Top row of boxes represent possible energy
sources. Shaded boxes are the principal energy reservoirs in the ocean, with crude energy
values given [in exajoules (EJ) 10 18 J, and yottajoules (YJ) 10 24 J]. Fluxes to and from the
reservoirs are in terrawatts (TWs). Tidal input (see Munk & Wunsch 1998) of 3.5 TW is
the only accurate number here. Total wind work is in the middle of the range estimated by
Lueck & Reid (1984); net wind work on the general circulation is from Wunsch (1998).
Heating/cooling/evaporation/precipitation
drive the global
values are all taken
meridional
from Huang & Wang (2003).
Value for surface waves and turbulence is for surface waves alone, as estimated by Lefevre
& Cotton (2001). The internal overturning wave energy estimatecirulation
is by Munk (1981); the internal tide
energy estimate is from Kantha & Tierney (1997); the Wunsch (1975) estimate is four times
larger. Oort et al. (1994) estimated the energy of the general circulation. Energy of the
mesoscale is from the Zang & Wunsch (2001) spectrum (X. Zang, personal communication,
2002). Ellipse indicates the conceivable importance of a loss of balance in the geostrophic
mesoscale, resulting in internal waves and mixing, but of unknown importance. Dashed-dot
lines indicate energy returned to the general circulation by mixing, and are first multiplied
by Ɣ. Open-ocean mixing by internal waves includes the upper ocean.
May provide enough total power to

Latitude / °
Latitude / °
50
40
30
20
10
0
-10
-20
-30
-40
50
40
30
20
10
0
-10
-20
-30
-40
50
40
30
20
10
0
-10
-20
-30
-40
50
40
30
20
10
0
-10
-20
-30
-40
JFM
AMJ
JAS
OND
Cant’ explicitly resolve internal waves in climate models.
3 steps to parameterize their role:
1) Wave generation
1997
Internal-Tide Generation
-50
0 50 100 150 200 250 300 350
Longitude / °
0.1 1 5 30
10 3 Π() / Wm -2
Egbert and Ray 01
Wind-generated near-inertial internal waves
Alford 01
3058
/ m
Π(50 m) / mW m -2
6
4
2
0
150
100
50
0
Π() / mW m -2
3
2
1
0
a
b
c
2) Wave
ARTICLE
propagation
IN PRESS
Internal-Tide Propagation
37.5 N
37.5 S
J F M A M J J A S O N D
Month
Fig. 13. Baroclinic energy flux ^J and conversion rate ^C from our two-layer simulation, normalized according to (14) and
one tidal cycle. Only a subset of the flux vectors are shown. The zonal mean of the normalization factor (Eq. (14) and Fi
on the right side of the plot. Each vector component has been spatially smoothed with a bidirectional Hanning window w
to 2 degrees and then sub-sampled in order to give a meaningful large-scale picture of the general pattern and magnitude
visual clarity it was necessary to make the color range representing the conversion rate be different from Fig. 12, to clip
than 20 kW m 1 ; and to delete vectors shorter than 1 kW m 1 :
Figure 6. (a,c) The monthly mean flux is
plotted for each year 1996-1999 (thin lines,
solid=1996,dashed=1997,dashdot=1998,dotted=1999),
and for the 4-year mean (heavy line). Black lines are from
37.5 ◦ N, and gray lines are from 37.5 ◦ S. (a) The flux
without mixed-layer-depth correction, Π(50 m). (b) The
The conversion is mostly positive (energy
transfer is from barotropic to baroclinic motions),
and in all of our regional estimates the net
conversion is positive. However, over small spatial
scales there are regions that are negative. For our
two-layer simulations, the total barotropic-to-
H.L. Simmons et al. / Deep-Sea Research II 51 (2004) 3043–3068
3) Wave breaking ...
We note that a number of the regio
significant generators of baroclinic w
been commented on previously in the
and a global picture with many similari
can be found in Egbert and Ray (2001
referred to as ER. The level of spatial d
7

Statistical models of wave breaking
“With three parameters I can fit an elephant”
-Lord Kelvin
Generated waves
generally have large
(vertical) scales and low
frequencies
Resolution of GCMs
somewhere in between
Breaking waves generally
small (vertical) scale.
Small-scale waves have
the most shear.
Black Box
E low-mode
Ambient Conditions:
-stratification
-latitude
-mesoscale features
‘Universal’ spectra
Gargett et al 81
(9)
Dissipation
rate ( ɛ )
(4)
(1)

organisms R E V I E W. LMost E T T Eof R Sthis mixing occurs
26 MARCH
where
2004
internal waves
wav
velocity fluctuations in the Pacific and Atlantic oceans between
break, overturning the density stratification of the ocean and
statistically steady, or varies slowly in time, the rate at which Wri
Below
Models
Atlantic thermocline,
of down-scale
(26 N, 29 W):
energy
m 2:75 !
transfer
0:6 428 N and 28 S to extend that result to equatorial regions. We find
(for
breaking dissipates Experimental energy verification
approximately equals the rate Dop at
creating l wave 1 < patches ! < 6 cpd) of[21].
turbulence. Predictions of the rate at which a strong latitude dependence of dissipation in accordance with
the energy
ed to These deep ocean observations (Fig. 1) exhibit a higher
the predictions is transferred 3 . In ourfrom observations, large to dissipation small scales. rates This andequiv
The
internal waves dissipate 2,3 were confirmed earlier at midlatitudes
4,5 . Here we present observations of temperature and
scale
umber degree of variability Wave-wave than one might Interactions
anticipate for a
allows accompanying the dissipation mixing rate across1density to besurfaces expressed near the in Equator terms of
Triad theory of weakly nonlinear wave-wave interactions
func in
ta sets universal spectrum. Moreover, the deviations from the
waveare parameters.
less Garrett-Munk than 10%
To
of
check
those empirical at
their
mid-latitudes spectrum numerical
for a similar
simulations,
background
of internal waves. Reduced mixing close to the Equator
asso H
velocity tailed (e.g.McComas GM h spectral power and Muller) laws form a pattern: they seem to
Theory: fluctuations Energy transfer infrom the Pacific large toand small-scale Atlantic internal oceans waves between
st list roughly fall upon a curve with negative slope in the x; y
Wright willand have to Flatte be taken 3 formulated into account in annumerical wave-
strong ∂U
plane. We show in the next section that the predictions of
E(m, ω) ∝ m −2 analytic
ω −2 simulations model based of o
428 N and 28 S to extend that result to equatorial regions. We find
ocean dynamics—for example, in climate change experiments.
alimit,
Doppler shifting of evolving test waves by a background wave
wave latitude
T
turbulence = ... dependence −theory U · ∇U are consistent of dissipation with this pattern. in accordance with
The Doppler Internal waves shifting are generated produces in the aocean netwhen energy its stratification flux toward is
ons
be theap-predictions of the field.—In this section we assume that the internal wave
scales—that
∂tA wave turbulence 3 . In our formulation observations, for the internal dissipation wave rates and disturbed. Observed Unlikedissipation surface waves, rates which agree propagate with predictions,
horizontally
because they
is,
are
higher
confined
wavenumbers—and
to the air–sea interface,
ultimate
internal waves
breaking term
accompanying
field
U
can
=
be
mixing Uviewed 1 e i(k 1·x−ω
across 1
as a field t) density +
ofU weakly 2 e i(k surfaces
2·x−ω 2
interacting
t) + near U 3 ethe i(k 3·x−ω
Equator 3 t) + ... based on measurements of shear and strain. (Gregg F
functional str
propagate dependence vertically as well of 1 as(which horizontally, is expressed making flowindisturb-
ances89, at the Polzin surface et oral bottom 95)
Stra
units of
waves, thus falling into the class of systems describable
W
are less than 10% of those at mid-latitudes for a similar backgroundenergy
and Munk
felt in the interior. Garrett and Munk 6
(2)
associated with this flux is the product of two terms:
by wave turbulence. Wave turbulence is a universal
dens
fficient
Eikonal
of internal transfer
statistical calculations
waves. through
theory for the of small-scale
Reduced resonant
description ofwaves mixing triads:
an ensemble propagating
close to the Equator
7 modelled internal waves in terms of a wavenumber–
of
men
is its
frequency spectrum; they assumed that the wave field is constant in
will have through weakly to be spectrum interacting taken particles of into larger account or waves. in This (Henyey numerical theoryet has
obse
al) simulations of
1 ¼ 1 308 N; F shear ðmÞ; F strain ðmÞ £ Lðv; NÞ
0 is a
k
contributed to our 1 = k
understanding 2 + k 3 ω
of spectral 1 = ω
energy 2 + ω 3
depe
ocean dynamics—for example, in climate change experiments.
transfer in complex systems [22], and has been used for
The first term, 1 at the 308 GM model reference latitude, de
waves, it Internal I: Continuous describing waves Steady surface spectrum are flux generated water to smaller waves of waves insince the vertical pioneering ocean when scales worksits stratification is
ormed
on stratification (via N) and on internal wave characteristi
disturbed. by Hasselmann Unlike surface [23], Benney waves, and which Newell [24], propagate and horizontally
wavey
the they are confined to the air–sea interface, internal waves
eragingZakharov over randomly [25,26]. distributed terms of spectra of vertical shear, F shear (m), and of vertical
because
The dynamics F of ∝oceanic Ê2 wave
internal N 2 phases and directions leads to slow, ineffint
energy easily transfers, vertically describede.g. in asisopycnal months well as(i.e., horizontally, to density) transfercoordinates,
energy making out of flow a mode-one disturb-internal tide
waves can be most
F
propagate m,
strain (m), where m is the vertical wavenumber in cycles per m
Strain fluctuations are variations of the vertical separations be
ances atwhich the surface
allow foror a simple
bottom
andfelt intuitive the
Hamiltonian
interior.
description
[12]. 82, Muller To describe et al 86] the wave field, we introduce
density surfaces, that is, fluctuations of ›z/›z. Previous me
lbers and Pomphrey Garrett and Munk 6
and Munk Theoretically predicted ‘family’ of steady-state
two 7 variables: modelled a velocity internal potential waves r; inand terms an isopycnal
spectrum; straining ...hamiltonian....
4 verified the shear and N dependence in 1 308 , and subse
of a wavenumber– ments
frequency spectra
it II: Distinct individual
r; they . The
waves assumed horizontalthat velocity theiswave given by field is constant in
the isopycnal gradient r of the velocity potential,
observations 5 and numerical simulations 8 confirmed the
m:
dependence. The second term contains the latitude (v) depen
nsider a coherent internal A(m, k) tide. ∝ mExplore −y k −x basic physics with idealized numerical
2.5
ertical
NATRE
2
Lðv; NÞ ¼
f cosh21 ðN=f Þ
periments, in both nearfield and farfield of wave generation sites...
f 308 cosh 21 ðN 0 Gregg =f 308 Þ03
major
ur best
uency
ODE),
0 0 W):
servaa
ther-
].
EX),
cline,
riment
a ther-
y−power law of 3−d action spectrum
1.5
1
0.5
0
−0.5
−1
energy equipartition
Eq.(5)
FASINEX
AIWEX
PATCHEX
MODE
IWEX
−1.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x−power law of 3−d action spectrum
GM
SWAPP
Lvov et al 04
ONR – p.4
Figure 1 Reduction of dissipation rates produced by breaking internal waves near the
Equator. The predicted latitude effect L is unity at 308 because the standard internal wave
spectrum, Garrett and Munk 6 , and the calculations based on it are referenced to that
latitude. Ratios of average 1 observed to 1 308 are plotted as short horizontal bars, and upper
and lower 95% confidence limits are shown by extents of the vertical lines. Multiple values
at the same latitude v come from different depths or locations. The shaded curve gives the
predicted L(v,N) for the range of N encountered. Because the N dependence is weak, the
height of the shaded band is small. Upper and lower solid lines are twice and one-half the
prediction, and approximately bound the scatter in the data. In spite of the scatter, the
observations confirm the predicted sharp cut-off of dissipation at low latitude.
Numerical simulations of internal-wave interactions.
Confirm GM spectrum as steady state and predicted
rates of down-scale energy flow. ( Hibiya et al,
Winters and D’Asaro, etc)
Figur
strati
latitud
the la
the E
water

n water masses of uniform density than across
g waters of different densities. But the magnition
of mixing across density surfaces are also
Earth’s climate as well as the concentrations of
t of this mixing occurs where internal waves
ng the density stratification of the ocean and
of turbulence. Predictions of the rate at which
dissipate 2,3 were confirmed earlier at midwe
present observations of temperature and
ons in the Pacific
Sources
and Atlantic oceans between
extend that result to equatorial regions. We find
dependence of dissipation in accordance with
. In our observations, dissipation rates and
- low mode
- estimate from observations
50
40
6
30
1997
a
20 JFM
10
0
4
ixing-10
- explicitly
across density
-20 resolve
surfaces
in GCM
near the
(?)
Equator
-30
-40
2
50
40
30
0
20
b
10
AMJ
150
aken into 0 account in numerical simulations 100 of
-10
-20
50
-30
for example, 0
-40
in climate change experiments.
Latitude / °
50
40
3
30
20
10
JAS
2
surface 0 waves, which propagate horizontally
-10
-20
1
-30
-40
Latitude / °
50
40
30
20
10
OND
0
-10
-20
-30
-40
-50
0 50 100 150 200 250 300 350
Longitude / °
0.1 1 5 30
10 3 Π() / Wm -2
Figure 5. Seasonally averaged maps of energy flux from
the wind into near-inertial mixed-layer motions for the year
1997. Variability in mixed-layer depth is included by seasonally
averaging the Levitus climatology.
and the model gives zero currents. A few (< 1%) seasonallyaveraged
values are < 0 (loss of energy to the atmosphere),
but all of these have magnitudes less than 0.1 mWm −2 , and
so are also plotted in blue. The rough land edges reflect the
2.5 ◦ grid spacing.
◦
/ m
Π(50 m) / mW m -2
Π() / mW m -2
c
Doppler shifting of evolving 7 test waves by a background wave field.
37.5 N
scales—that is, higher 37.5 S wavenumbers—and ultimate breaking. The
Figure 6. (a,c) The monthly mean flux is
plotted for each year 1996-1999 (thin lines,
solid=1996,dashed=1997,dashdot=1998,dotted=1999),
and for the 4-year mean (heavy line). Black lines are from
37.5 ◦ N, and gray lines are from 37.5 ◦ S. (a) The flux
without mixed-layer-depth correction, Π(50 m). (b) The
monthly mean, zonally averaged Levitus and Boyer [1994]
mixed layer depth, < H >, at 37.5 ◦ N (black) and 37.5 S
(gray). (c) Mixed-layer-depth corrected flux, Π(H).
4.2. Seasonal Cycle
Wave propagation
through resolved mesoscale
(4)
(1)
But the problem is that there are some exceptions where f 308 cosh wave 21 ðNenergy 0 =f 308 Þ doesn’t
cascade in this nice way, and
the importance
these
of the inertial
exceptions,
component of the windthough fields,
localized, might dominate
rather than their overall magnitude, in generating near-inertial
mixing in key places. Best motions. to illustrate through examples...
(9)
Wave breaking
(small scales)
E t ( ⃗ k, ⃗x, t) = S o (k) − ∇statistically x · (C
steady, or
g E)
varies−slowly ∇in time,
k · F
the
(k)
rate at which
− Swave
i (k)
of those at mid-latitudes for a similar backal
waves. Reduced mixing close to the Equator
re generated in the ocean when its stratification is
confined to the air–sea interface, internal waves
0
lly as well as horizontally, making flow disturb-
J F M A M J J A S O N D
Month
e or bottom felt in the interior. Garrett and Munk 6
lled internal waves in terms of a wavenumber–
m; they assumed that the wave field is constant in
significantly more energetic.
Numerical simulations of energy transfer by interactions within
the internal wave field show a net flux Spectral towards flux small spatial scales
that increases the shear variance (›u/›z) (down-scale)
2 until it overcomes the
stratification and the waves break 2,3 . When the wave field is
breaking dissipates energy approximately equals the rate at which
the energy is transferred from large to small scales. This equivalence
allows the dissipation rate 1 to be expressed in terms of internal
wave parameters. To check their numerical simulations, Henyey,
Wright and Flatte 3 formulated an analytic model based on the
}
Parameterize unresolved dissipation rate as rate
of down-scale energy transfer to IW continuum
The Doppler shifting produces a net energy flux toward smaller
functional dependence of 1 (which is expressed in units of W kg 21 )
associated with this flux is the product of two terms:
1 ¼ 1 308 N; F shear ðmÞ; F strain ðmÞ £ Lðv; NÞ ð1Þ
The first term, 1 at the 308 GM model reference latitude, depends
on stratification (via N) and on internal wave characteristics (in
terms of spectra of vertical shear, F shear (m), and of vertical strain,
F strain (m), where m is the vertical wavenumber in cycles per metre).
Strain fluctuations are variations of the vertical separations between
density surfaces, that is, fluctuations of ›z/›z. Previous measurements
4 verified the shear and N dependence in 1 308 , and subsequent
observations 5 and numerical simulations 8 confirmed the strain
dependence. The second term contains the latitude (v) dependence:
Lðv; NÞ ¼
f cosh21 ðN=f Þ
ð2Þ

Problem I: Fast wave interactions
Parametric Subharmonic Instability (PSI): Decay of a initial wave (e.g. low-mode
internal tide), into two smaller-scale waves of half the frequency. Seems to be
particularly efficient/resonant when the subharmonic frequency is equal to the local
intertial frequency (28.9 N/S).
Display Image
Numerical evidence for fast PSI near 29
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02/27/2006 12:45 PM
MacKinnon and Winters 05
FIG. 1. Time variations of the energy spectrum at (a) 49°, (b) 28°, and (c) 18°N in the two-dimensional
wavenumber space after the injection of the lowest-vertical-wavenumber M 2 internal tide energy. Note that each
spectrum is scaled by the unforced, freely decaying, reference spectrum. The red triangle at the lower left in each
panel shows the spectral location of the lowest-vertical-wavenumber M 2 internal tide. Numerals on the solid lines
denote the wave period.
Furuichi et al 05

JANUARY 2006 T I A N E T A L . 41
Observational evidence for something special near 29
‘Dissipation rate’ of internal tide, as inferred from convergences in internal tide
energy fluxes measured with satellite altimetry.
Atlantic Ocean
Indian Ocean
JANUARY 2006 T I A N E T A L . 41
JANUARY 2006
Pacific Ocean
T I A N E T
2 -1
Inferred diffusivity / m s
FIG. 9. Latitudinal distribution of the mixing rate caused by M 2
internal tides in (a) the Pacific, (b) the Atlantic, and (c) the Indian
Oceans.
Latitude Latitude Latitude
large as 50 m, the relative error of the dissipation rate
would still be no larger than 20%.
The calculations are based on 10 years of TOPEX/
Poseidon altimeter tidal data. The nominal bias of tidal
data is 1.5 cm (Benada 1997). The wavenumber of baroclinic
M 2 internal tides is of the order of 0.01 km 1 , the
average depth of the thermocline is assumed to be 500
m, and the average water depth is about 4000 m. The
estimated bias of the energy flux in Fig. 6 is about 1000
W m 1 . We also estimate the error caused by the nomi-
FIG. 9. Latitudinal distribution of the mixing rate caused by M 2
internal tides in (a) the Pacific, (b) the Atlantic, and (c) the Indian
Oceans.
There are some volumes in which the fluxes are insufficient
to calculate the summation. There are also a
few volumes near large topography, for example, near
the east of Australia, in which the sum of fluxes is negative.
This means that Eq. (4) does not apply in such
areas and, as a consequence, not all energy fluxes in Fig.
6 are used to calculate the energy dissipation shown in
Fig. 7.
The error is as large as one-quarter of the mixing rate
caused by M 2 internal tides near 28.9°N/S and is even
Tian et al 06:
(suggestive blue lines added by me)
i
O

Observational evidence for fast PSI near 29
Internal Waves Across the Pacific (IWAP)
Alford, Klymak, MacKinnon, Munk, Pinkel
38
36
Altimetry
2 kW/m
PEZHAT
2 kW/m
Mooring
MP6
TOPEX
Pass 249
TOPEX
Filtered
Velocity at the ‘critical’ latitude
2 kW/m
34
32
MP5
Neap
TS2
Latitude
30
TS1
MP4
28
MP3
MP2
TS3
26
24
MP1
Spring
-16 -8 0 8
/ cms -1
Average near-inertial kinetic energy vs. latitude
22
Fench Frigate Shoals
Nihoa Island
Kauai Channel
-14.4 -7.2 0 7.2
SSHA / cm
192 194 196 198 200 202 204
Longitude
-6 -4 -2 0
Depth / 1000 m
ure 1: (Left) Bathymetry (colors; axis at lower right), measurement locations (black = moors;
blue = shipboard time series; white = ship track), and internal-tide energy fluxes (yellow
}

38
36
Altimetry
2 kW/m
PEZHAT
2 kW/m
Mooring
2 kW/m
MP6
TOPEX
Pass 249
Problem II: coherent waves
TOPEX
Filtered
34
32
MP5
Neap
East-west sections of inertially back-rotated velocity
TS2
Latitude
30
TS1
MP4
28
TS3
MP3
MP2
26
Spring
24
MP1
-16 -8 0 8
/ cms -1
22
Fench Frigate Shoals
Nihoa Island
Kauai Channel
-14.4 -7.2 0 7.2
SSHA / cm
192 194 196 198 200 202 204
Longitude
-6 -4 -2 0
Depth / 1000 m
: (Left) Bathymetry (colors; axis at lower right), measurement locations (black = moore
= shipboard time series; white = ship track), and internal-tide energy fluxes (yellow
AT numerical model; black=altimetry; red = moorings). Reference arrows are at upper
the critical latitude, 28.8 o N, is indicated with a dotted line. Note the PEZHAT model
nds at 32 o N. Moored flux variability is indicated with gray curves, which enclose the
ly 50% of observed values. (Right) Raw (blue) and highpass-filtered (red) meridional
averaged from 0-1000 m depth on a southward transit from MP6 to MP1 at yearday
ce the transit took 4.5 days to complete, the ship passed through northbound modegenerated
at Hawaii’s neap tide near 33 o N, and at the following spring tide at 26 o N
t left). SSHA from TOPEX/POSEIDON track 249 (plotted at left, black) is overplotted
ip reference frame (black; see text). The axis limits for 〈V 〉 and SSHA (shown below)
l for a mode-1 free wave.
4
North-south section of back-rotated velocity

Problem III - internal tide ‘directly breaks’
Hawaiian Ocean Mixing Experiment (HOME)
Huge overturns as internal tide sloshes
up and down a steep slope
Levine and Boyd 06
Aucan et al 05
Dissipation rate
FIG. 1. R/P FLIP was moored in 1100 m of water over K
N, 158 ◦ 37.77 ′ W. Also shown are two moorings from the sa
larger near-bottom overturns have been observed.
Velocity
Temperature
Klymak et al 07

Problem III - internal tide ‘directly breaks’
Hawaiian Ocean Mixing Experiment (HOME)
Huge overturns as internal tide sloshes
up and down a steep slope
Levine and Boyd 06
Aucan et al 05
Dissipation rate
Gregg-Henyey parameterization works in
upper water column, but not for the
deeper, stronger mixing.
FIG. 1. R/P FLIP was moored in 1100 m of water over K
N, 158 ◦ 37.77 ′ W. Also shown are two moorings from the sa
larger near-bottom overturns have been observed.
Velocity
Temperature
Klymak et al 07

Directly breaking internal tide on Oregon slope
Similarly huge overturns, but in a
place not predicted by global
internal tide generation maps.
L01605
L01605 NASH ET AL.: MIXING HOTSPOTS L01605
profiler and shipboard CTD yields the turbulent dissipation
rate i = 0.64L 2 Ti N 3 i , where N i and L Ti represent the
Thorpe-resorted stratification and rms Thorpe displacement
for the ith overturning region [Dillon, 1982]. Each L Ti is
rejected if smaller than the expected (spurious) values for
depth and density imprecision [Alford et al., 2006]. Following
Gregg [1989] and Gregg and Kunze [1991], was
parameterized from XCP data and found consistent with
Thorpe estimates at stations where XCP and CTD measurements
coincided (stations 3.3, 4.3 and 5.3). For both
overturn and Gregg-Henyey , diffusivity was computed
as K r = 0.2N 2 [Osborn, 1980].
3. Observations
[11] High near-bottom turbulent energy dissipation rates
span the 20-km region of enhanced roughness (Figure 2,
43.21° N, 125.05–125.3° W). Within 500 mab, timeaverage
K r were 10 4 –10 3 m 2 /s above rough topography,
NASH ET AL.: MIXING HOTS
Nash et al 07
Figure 3. Timeseries at LADCP/CTD station L2.5 at the
2200-m isobath (first occupation): (a) observed (blue) and
with the r
observed
and Boyd
[13] Up
(and henc
velocity b
northward
(i.e., x = R
the large-s
and orien
during th
dissipation
near-botto
forms the
dissipation
However,
is the sam
vertical w
generation
[Baines, 1
[14] At
lence (
during bo
5 days; 3
eastern ed
green), w
this statio
the 2200-
within 15

Problem IV: Solitons (internal waves of unusual size)
nonlinear steepening
balanced by disperson
24 hours
Stanton and Ostrovsky
GRL 24(14) 1998
100 minutes

2106 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y
VOLUME 33
period. (b), (d) Same for interval surrounding yearday 238.96.
Extreme soliton breaking
thermocline on yearday 238 and depth between 20 and
40 m during yearday 242 (Fig. 4). Other patches of high
diffusivity were associated with anomalously low stratification
but not with elevated dissipation, such as above
profiles of turbulent dissipation rate. The spacing of the
dissipation profiles illustrates the degree to which ou
profiler undersampled the event. The locations of two
reference isopycnals that bound the thermocline are
shown for each profile. Average dissipation within thi
density range remained around 10 6 W kg 1 in the third
through seventh profiles (until minute 30), then de
FIG. 14. Example acoustical snapshot of a propagating ISW within which is embedded a sequence of rollups identical in nature to Kelvin–
Helmholtz instabilities observed in the laboratory and in small-scale simulations. The vertical scale of the largest is more than 10 m, and
the horizontal scale Moum (in the direction et ofal wave 03 propagation) is about 50 m. Toward the trailing edge of the wave, the rollups become less
coherent but contribute a greater backscatter signal, suggesting breakdown to turbulence. At greater depth, denoted by arrows, are two more
layers of bright backscatter. These are presumably the same phenomenon, but smaller scale. Hence the echosounder resolution does not
permit a clear depiction of rollups.
given the density profile at the same location. First, the
ambient streamfunction in the wave’s reference frame
is obtained from the measured upstream velocity profile
u 0 (z) using
z
0
(z) [u (z) c ] dz. (3)
0
w
50-75 % of total average
daily mixing occurs in
these few minutes
At any location within the wave, we obtain the streamfunction
profile corresponding to the local density profile
1 (z) using
( ) ( ).
1 1 0 0 (4)
Last, we differentiate 5 to obtain the velocity profile and
transform into the earth’s reference frame:
1
u 1 c w . (5)
z
An appropriate value for was obtained by examining
density fluctuations in six profiles prior to and
including the profile selected for analysis. Thorpe reordering
was used to estimate density fluctuations in
turbulent overturns. The maximum fluctuation had absolute
value of 0.1 kg m 3 (Fig. 17). We adopted this
MacKinnon and Gregg 03
FIG. 10. The solid lines are contours of northward (onshelf) baroclinic velocity from 0.3 to 0.3 m s 1 in intervals of 0.1 m s 1 . Th
shaded areas are 4-m shear variance, ranging from 0 (white) to 3.5 10 3 s 2 (black) in increments of 5 10 4 s 2 . Profiles of dissipatio
rate are overlain, and correspond to the colorbar above. The slight slant of each profile represents the passage of time as the profiler descends
The black (upper) and magenta (lower) stars on each profile indicate the evolving locations of the 22.65 and 24 kg m 3 isopycnals, respectively
Solitons around the world
FIG. 15. (left) Observed profiles of horizontal velocity from 300-
kHz ADCP (2-m bin size) and (right) from Chameleon. These
coincide with the Chameleon profile noted by the black dot in Fig.
9 which corresponds to the time of maximum stratification at the
interface at 30-m depth, after which overturns and intense turbulence
first appear.
5
Differentiations and integrations were approximated using second-order,
finite differences on an uneven grid. Interpolations between
the various data grids were done using an equal mixture of
linear and cubic Hermite interpolants. (This controls the spurious
peaks that appear when cubic interpolation is used alone.)
Apel et al 06

ligible shear. Starting with this simple motion field, the goal is to predict its spectral
Is the spectrum a myth? (the personalist approach)
ature in s-L and Eulerian frames.
Real-world observations depart from the “N-frame” ideal through advective
Doppler shifting makes for broadened
frequency band at higher wavenumbers.
cts, which appear as phase distortions of the signal of interest: ie
Arctic data
38 38
38
s Eul = s 0 exp(i k*x + k*(V t) - ! t) 1)
Normalized
Intrinsic
There
shear
are
only
four
at
cases
inertial
to
and
consider:
vortical (f=0)
frequencies, advected by horizontal currents
Small vs large phase distortion (! | k * V | dt > / < ")
Stochastic vs deterministic distortion velocity, V
Garrett-Munk
The models introduced here relate the probability density function and / or the
ctrum of the advecting velocity field, V, to the spectrum of the observed quantity. The
roach follows a path suggested by Papoulis, 1984.
The focus is on the temporal
Figure 2 a. A wavenumber-frequency spectrum of Arctic Ocean shear estimated from the
SHEBA Doppler sonar. The white rectangle indicates the smoothing used to to to achieve 150
degrees of freedom. The left quadrants of the spectrum correspond Pinkel to to to 07
anti-cyclonic
rotation in time. The lower left and upper right quadrants represent upward phase
propagation, which for the internal wave component of the shear implies downward
energy propagation.
b. The associated normalized spectrum N(! z , ! z , z , ")=E(! z , ")/ ! E(! z , ") d". First
(white) and second (black) spectral moments are indicated. Color contours mimic a
“hourglass” pattern.
covariance function, R(#), which, for a line spectral process (in the N-frame), takes
form
Perhaps the internal-wave continuum is
in the eye of the beholder, at least the
high-wavenumber part where all the
shear lives.
Doppler model

Conclusions?
Statistical models of internal-wave breaking are based on rates of down-scale
energy transfer through steady, gaussian fields of incoherent waves. Such
models appear to work some places, but many of these are ‘boring’ places.
In other places, mixing is dominated by extreme events, mostly because ‘special
physics’ is taking place.
How much do extreme events matter?
Coastal ocean - probably quite a bit, in shallow water
most internal wave breaking is abnormal.
Open ocean - hard to say. Factor of 2 near hawaii,
averages out to not much basin-wide. But perhaps
extreme breaking events are undersampled.

mber or frequency properties (P95).
3) The wavenumber of the test waves also
f waves is assumed to Breaking slowly evolve internal waves
simply related to observed large-scale mo
ratification and total wavefield energy
cording to WKB theory, horizontal waven
cal wavenumber of test waves depends
does not change in Na stratification field
al energy level. Specifically, it is choroude
spectra integrated out to a near-
(Stratification)
varies in the vertical. We therefore als
Even if you can’t directly see turbulence, characteristic horizontal wavenumber in
ber is order 1. Plugging in the anacan
parameterize turbulent mixing in unknown constant.
GM spectrum and the dispersion reree
factors terms onof the observed rhs of (low-resolution)
(6) can be With these( Internal assumptions,
S
)
the three terms in (
Wave
ε
Black
her in such a way internal-wave that at a particular shear.
Box
Shear (9)
(
Turbulent
ation rate scales as
dEˆ
E )
0 N
1488 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y
VOLUME 33
Dissipation
dm
(1)
m
(4)
ˆ
N 2
0 N0
2 1
f E
(W kg ), (7)
2
2
N 0
dU
Slf
S
erence buoyancy frequency and f is
dz
0
S 0
ency (G89).
kH
k H0.
s scaling with oceanic measurements
dlatitude locations, G89 uses the ratio The product of all three terms gives a dissi
at a fixed wavenumber (10 m) to the estimate that we will refer to as the MacKinn
Munk shear at that scale as a proxy (M–G) scaling
nergy level to get what we will sub-
FIG. 13. (a) Observed dissipation data averaged in bins of 4-m stratification (x axis) and 4-m shear variance (y axis). The
boundary of 4-m Richardson number 0.25 is shown for reference. (b), (c) Same as for the the Gregg–Henyey and MacKinnon–
Gregg parameterizations, respectively.
as the Gregg–Henyey (G–H) scaling,
observed dissipation rate. Nevertheless, the shear and
4 2
H model dissipation rate has an inverse relationship with
0
0 0
N Slf
1
MG
(W kg )
N S

are we missing mixing still?
Program/
Experiment
C - S A L T /
SFTRE
Diffusivities
O(×10 -4 ) m 2 /s
0.1 (above/below stairs)
1 (in staircase)
Notes
Double-diffusive
staircase
PATCHEX 0.1 (thermocline) Thermocline
processes
Tropic Heat 1/
Tropic Heat 2
NATRE
BBTRE
0.1 (in thermocline)
1 (in undercurrent)
0.1 (thermocline)
1 (in NADW)
0.1 (in thermocline)
1 (in NADW)
>10 (in AABW)
Equatorial
processes
First open-ocean
tracer release
Basin-scale
survey
ec 2003 18:52 AR AR203-FL36-12.tex HOME AR203-FL36-12.sgm 0.1 (away LaTeX2e(2002/01/18) from ridge) P1: Energy IBD budget
1 (near the ridge) study
288 WUNSCH FERRARI
TABLE 1 Spatial-average across-isopycnal diffusviities, as estimated by various
budget methods. In the first six rows, the bottom region lies between neutral surface 28.1
and the sea floor (see Figure 1), and from 27.96 to 28.07 for the deep layers. Generally,
these lie between about 3800 m and the sea floor, and between about 2000 m and 3800 m,
respectively, but with considerable spatial variations. The last six estimates are from
restricted basins or channels, but all values are spatial averages over the interior and
boundary layers
Ocean/Depth 〈K 〉 (10 −4 m/s) Reference
Atlantic, bottom 9 ± 4 Ganachaud & Wunsch (2000)
Indian, bottom 12 ± 7 ”
Pacific, bottom 9 ± 2 ”
Atlantic, deep 3 ± 1.5 ”
Indian, deep 4 ± 2 ”
Pacific, deep 4 ± 1 ”
Scotia Sea 30 ± 10 Heywood et al. (2002)
Brazil Basin 3–4 Hogg et al. (1982)
Samoan Passage 500 Roemmich et al. (1996)
Amirante Trench 10.6 ± 2.7 Barton & Hill (1989)
Discovery Gap 1.5–4 Saunders (1987)
Romanche Fracture Zone 100 Ferron et al. (1998)
! "#
5. Regional Results!
!
! $%&'()!*!+%,-./0,!,1/1%23!.24/1%23,!%3!15)!63+%/37!8/4%9%4!/3+!:;!
215!?)(%+%23/.!/3+!@23/.!,/?-.%3&!%,!&22+!%3!15)!63+%/3!
=4)/3!152'&5!15)()!/()!23.0!1A2!1(/4B,!-2.)A/(+!29!*#CD;!!E2F)(/&)!%,!
,-/(,)(!%3!15)!8/4%9%4!G'1!?2,1!./1%1'+),!/3+!.23&%1'+),!5/F)!,2?)!,/?-.%3&!
/F/%./G.);!!H5)!)I'/12(%/.!8/4%9%4!%,!-/(1%4'./(.0!A)..J,/?-.)+;!!63!15)!
! =(7&(6@;%?!9%8)4D!
!
$%&'()!*/L!!M=(7&(6@;%?!9%8)4D!
Indirect estimates
Inverse methods have been applied on
global scales to estimate mixing rates
Ganachaud and Wunsch 2000
Water class density ! k v cm 2 s -1
n
Atlantic deep
Indian deep
Pacific deep
Atlantic bottom
Indian bottom
Pacific bottom
!
27.96 < ! n < 28.1
“
“
! n >
28.1 “
“
Lumpkin & Speer 2006
3 ± 1.5
4 ± 2
4 ± 1
9 ± 4
12 ± 7
9 ± 2
Water class density ! k v cm 2 s -1
n
Upper Deep
Lower Deep
Bottom
27.0 < ! n < 27.8
27.8 < ! n < 28.1
! n >
28.1
0.3 ±
0.5
1 ± 0.5
3 ± 1
(2002) found the remarkably high value of (30 ± 10) × 10 −4 m 2 /s in the Scotia
Sea. These and other estimates are listed in Table 1.
Ganachaud & Wunsch (2000) described estimates of mixing inferred through
!