Internal Waves are all around us...

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1 Internal Waves are all around us... Petrenko et al isotherms Baroclinic velocity Google Earth Eich et al 04

2 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

3 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

4 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 U = Ψ(z)cos(kx ωt) (Glenn Flierl)

5 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)

6 What generates internal waves? Spring Observations 2) Wind makes near-inertial internal waves / N m Wind Stress bc S z/m T/ C 8 Vbc / m s V 15 log10 (S / s ) T z/m z/m yearday (MacKinnon and Gregg, JPO, Dec 05)

Wunsch and Ferrari 04 May provide enough total power

7 Internal wave breaking mixes the ocean Waves break by shear instability or convective overturning. Wunsch and Ferrari 04 May provide enough total power to drive the global meridional overturning cirulation Woods 68

8 Cant explicitly resolve internal waves in climate models. 3 steps to parameterize their role: 1) Wave generation Internal-Tide Generation 2) Wave ARTICLE propagation IN PRESS 3058 Internal-Tide Propagation H.L. Simmons et al. / Deep-Sea Research II 51 (2004) Egbert and Ray 01 Wind-generated near-inertial internal waves OND Longitude / Π(<H>) / Wm -2 Alford 01 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 : 3) Wave breaking... 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- 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

9 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)

10 Models of down-scale energy transfer Triad theory of weakly nonlinear wave-wave interactions (e.g.mccomas and Muller) U t =... U U U = U 1 e i(k 1 x ω 1 t) + U 2 e i(k 2 x ω 2 t) + U 3 e i(k 3 x ω 3 t) +... Eikonal calculations of small-scale waves propagating through spectrum of larger waves (Henyey et al) Experimental verification Garrett-Munk empirical spectrum E(m, ω) m 2 ω 2 Observed dissipation rates agree with predictions, based on measurements of shear and strain. (Gregg 89, Polzin et al 95) 1 ¼ N; F shear ðmþ; F strain ðmþ Lðv; NÞ Steady flux to smaller vertical scales F Ê2 N 2 Theoretically predicted family of steady-state spectra...hamiltonian... y power law of 3 d action spectrum r A(m, k) m y k x energy equipartition NATRE Eq.(5) FASINEX AIWEX PATCHEX MODE IWEX x power law of 3 d action spectrum GM SWAPP Lvov et al 04 Gregg 03 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)

11 n water masses of uniform density than across g waters of different densities. But the magnition of mixing across density surfaces are also OND Longitude / Π(<H>) / Wm -2 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 these exceptions, though localized, might dominate mixing in key places. Best 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) 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 - low mode of dissipation in accordance with. In- estimate our observations, from observations dissipation rates and ixing - explicitly across density resolve surfaces in GCM near the (?) Equator of those at mid-latitudes for a similar backal waves. Reduced mixing close to the Equator aken into account in numerical simulations of for example, in climate change experiments. re generated in the ocean when its stratification is surface waves, which propagate horizontally confined to the air sea interface, internal waves lly as well as horizontally, making flow disturbe 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 Doppler shifting of evolving test waves by a background wave field. The Doppler of down-scale shifting produces energy transfer a net energy to IW fluxcontinuum toward smaller scales that is, higher wavenumbers and ultimate breaking. The 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 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

Seems to be particularly efficient/resonant when the subharmonic frequency is equal to the local

12 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). Numerical evidence for fast PSI near 29 [next] MacKinnon and Winters 05 Furuichi et al 05

13 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 Pacific Ocean 2-1 Inferred diffusivity / m s Latitude Latitude Latitude Tian et al 06: (suggestive blue lines added by me)

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 Velocity

14 Observational evidence for fast PSI near 29 Internal Waves Across the Pacific (IWAP) Alford, Klymak, MacKinnon, Munk, Pinkel Altimetry 2 kw/m PEZHAT 2 kw/m Mooring MP6 TOPEX Pass 249 Velocity at the critical latitude 2 kw/m MP5 TS2 Latitude 30 TS1 MP4 28 MP3 MP2 TS3 26 MP1 Average near-inertial kinetic energy vs. latitude 24 Fench Frigate Shoals Nihoa Island - 22 Kauai Channel Longitude }

TOPEX Pass 249 2 kw/m PEZHAT MP6 2 kw/m 36 Mooring 2 kw/m East-west sections of inertially back-rotated velocity Neap 34 MP5 32 TS2 30 MP4 TS1 MP3 28 MP2 TS3 Spring Latitude

2 Kauai Channel 192 194 196 198 Longitude 200 202 204-6 -4-2 0 Depth / 1000 m 1: (Left) Bathymetry (colors; axis at lower right), measurement locations (black = moorue =

Reference arrows are at upper d the critical latitude, 28.8o N, is indicated with a dotted line. Note the PEZHAT model ends at 32o N.

(Right) Raw (blue) and highpass-filtered (red) meridional averaged from 0-1000 m depth on a southward transit from MP6 to MP1 at yearday nce the transit took 4.

15 TOPEX Pass kw/m PEZHAT MP6 2 kw/m 36 Mooring 2 kw/m East-west sections of inertially back-rotated velocity Neap 34 MP5 32 TS2 30 MP4 TS1 MP3 28 MP2 TS3 Spring Latitude Problem II: coherent waves TOPEX <V> Filtered <V> Altimetry MP <V> / cms Fench Frigate Shoals SSHA / cm Nihoa Island Kauai Channel Longitude Depth / 1000 m 1: (Left) Bathymetry (colors; axis at lower right), measurement locations (black = moorue = shipboard time series; white = ship track), and internal-tide energy fluxes (yellow HAT numerical model; black=altimetry; red = moorings). Reference arrows are at upper d the critical latitude, 28.8o N, is indicated with a dotted line. Note the PEZHAT model ends at 32o N. Moored flux variability is indicated with gray curves, which enclose the ely 50% of observed values. (Right) Raw (blue) and highpass-filtered (red) meridional averaged from m depth on a southward transit from MP6 to MP1 at yearday nce the transit took 4.5 days to complete, the ship passed through northbound modes generated at Hawaii s neap tide near 33o N, and at the following spring tide at 26o N at left). SSHA from TOPEX/POSEIDON track 249 (plotted at left, black) is overplotted hip reference frame (black; see text). The axis limits for V " and SSHA (shown below) al for a mode-1 free wave. 4 North-south section of back-rotated velocity

16 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 Velocity Temperature Dissipation rate Klymak et al 07

17 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. Velocity Temperature Klymak et al 07

18 Directly breaking internal tide on Oregon slope Similarly huge overturns, but in a place not predicted by global internal tide generation maps. Nash et al 07

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

20 Extreme soliton breaking FIG. 14. Example acoustical snapshot of a propagating ISW within which is embedded a sequence of rollups identical in nature to Kelvin Moum et al 03 MacKinnon and Gregg % of total average daily mixing occurs in these few minutes Solitons around the world Apel et al 06

gligible shear. Starting with this simple motion field, the goal is to predict its spectral nature in s-l and Eulerian frames. Is the spectrum a myth?

21 gligible shear. Starting with this simple motion field, the goal is to predict its spectral nature in s-l and Eulerian frames. Is the spectrum a myth? (the personalist approach) Real-world observations depart from the N-frame ideal through advective Intrinsic shear inertialto and vortical (f=0) There areonly fourat cases consider: frequencies, advected by horizontal currents Small vs large phase distortion Garrett-Munk seul = s0 exp(i k*x + k*(v t) - t) Normalized Arctic data Doppler shifting makes for broadened band at as higher wavenumbers. ects,frequency which appear phase distortions of the signal of interest: ie 1) ( k * V dt > / < ") Doppler model Stochastic vs deterministic distortion velocity, V Perhaps the internal-wave continuum is in the eye of the beholder, at least the The models introduced here relate the probability density function and / or the high-wavenumber part where all the shear lives. ectrum of the advecting velocity field, V, to the spectrum of the observed quantity. The Figure 22 a. a. A A wavenumber-frequency wavenumber-frequency spectrum spectrumof ofarctic ArcticOcean Oceanshear shearestimated estimatedfrom fromthe the Figure spectrum of Arctic Ocean shear estimated from the wavenumber-frequency proach follows a path suggested by Papoulis, The focus is on the temporal SHEBA Doppler Doppler sonar. sonar. The The white white rectangle rectangleindicates indicatesthe thesmoothing smoothingused usedtoto toachieve achieve SHEBA rectangle indicates the smoothing used achieve 150 degrees of of freedom. freedom. The The left left quadrants quadrants of of the thespectrum spectrumcorrespond correspond anti-cyclonic Pinkeltototo07 degrees quadrants of the spectrum correspond anti-cyclonic anti-cyclonic rotation in in time. time. The The lower lower left left and and upper upper right rightquadrants quadrantsrepresent representupward upwardphase phase rotation upper right quadrants represent upward phase propagation, which which for for the the internal internal wave wave component componentof ofthe theshear shearimplies impliesdownward downward propagation, wave component of the shear implies downward energy propagation. propagation. energy b. The The associated associated normalized normalized spectrum spectrumn( N(,z,,")=E( ")=E(z, ")/ ")/E( E(,z,,") ") d". d". First First b. normalized spectrum N( E( First zz ")=E(zz,, ")/ z z ")d". (white) and and second second (black) (black) spectral spectral moments momentsare areindicated. indicated.color Colorcontours contoursmimic mimicaaa (white) moments are indicated. Color contours mimic hourglass pattern. pattern. hourglass ocovariance function, R(#), which, for a line spectral process (in the N-frame), takes form

Such models appear to work some places, but many of these are boring places.

22 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.

24 Breaking internal waves N (Stratification) Even if you can t directly see turbulence, can parameterize turbulent mixing in terms of observed (low-resolution) internal-wave shear. S ( ) Internal Wave Shear (9) Black Box ε ( Turbulent ) Dissipation (4) (1) ling N Slf 1 MG (W kg ) N S

1 (above/below stairs) 1 (in staircase) Double-diffusive staircase PATCHEX 0.

1 (in thermocline) 1 (in undercurrent) Equatorial processes Atlantic deep 27.96 < n < 28.1 3 ± 1.5 4±2 Indian deep 4±1 Pacific deep 9±4 Atlantic bottom n > 28.

1 (thermocline) 1 (in NADW) First open-ocean tracer release <1./31%4=4)/3,;>215?)(%+%23/./3+@23/.,/?-.%3&%,&22+%315)63+%/3 BBTRE 0.

tex LaTeX2e(2002/01/18) 1 (near the ridge) =4)/3152'&515)()/()23.01A21(/4B,-2.)A/(+29*#CD;E2F)(/&)%,,-/(,)(%315)8/4%9%4G'1?2,1./1%1'+),/3+.23&%1'+),5/F),2?),/?-.%3& /F/%./G.

)G)1A))315)<@2(),7 budget K())3./3+/3+15)>(%1%,56,.),;H5)215)(15()).%3),-(2F%+)?)(%+%23/. Lumpkin & Speer 2006 density n Water class 42F)(/&)G'115)()%,3242?-()5)3,%F)@23/.

Ganachaud & Wunsch (2000) 9±2 12 ± 7 4±1 3 4 Hogg et al. (1982) 30 ± 10 500 Amirante Trench 10.6 ± 2.7 Romanche Fracture Zone 3 ± 1.5 4±2 1.5 4 100 $%&'()*+,-.

%15/3+D/3+A)..QRRST%315)63+%/3=4)/3 42.2(J42+)+G015)QU50+(2&(/-5%4.%3),; $%&'()*+,-./01203/4565%7846978&:%5;<)(%=%7869+65;><)5(%? 497@)4%85;)16?%A%??797(B?7=)=+>#C;>=(7&(6@;%?

(1998) (2002) found the remarkably high value of (30 ± 10) 10 4 m2 /s in the Scotia Sea. These and other estimates are listed in Table 1.

25 are we missing mixing still? Indirect estimates Inverse methods have been applied on global scales to estimate mixing rates Program/ Experiment Diffusivities O( 10-4) m2/s Notes C - S A LT / SFTRE 0.1 (above/below stairs) 1 (in staircase) Double-diffusive staircase PATCHEX 0.1 (thermocline) Thermocline processes Ganachaud and Wunsch 2000 density n Water class kv cm2 s-1 Tropic Heat 1/ Tropic Heat (in thermocline) 1 (in undercurrent) Equatorial processes Atlantic deep < n < ± 1.5 4±2 Indian deep 4±1 Pacific deep 9±4 Atlantic bottom n > ± 7 Indian bottom 9±2 Pacific bottom "# 5. Regional Results Dec :52 AR $%&'()*+%,-./0,,1/1%23.24/1%23,%315)63+%/378/4%9%4/3+:; NATRE 0.1 (thermocline) 1 (in NADW) First open-ocean tracer release <1./31%4=4)/3,;>215?)(%+%23/./3+@23/.,/?-.%3&%,&22+%315)63+%/3 BBTRE 0.1 (in thermocline) 1 (in NADW) >10 (in AABW) Basin-scale survey HOME AR203-FL36-12.sgm 0.1 (away from ridge) AR203-FL36-12.tex LaTeX2e(2002/01/18) 1 (near the ridge) =4)/3152'&515)()/()23.01A21(/4B,-2.)A/(+29*#CD;E2F)(/&)%,,-/(,)(%315)8/4%9%4G'1?2,1./1%1'+),/3+.23&%1'+),5/F),2?),/?-.%3& /F/%./G.);H5))I'/12(%/.8/4%9%4%,-/(1%4'./(.0A)..J,/?-.)+;6315) <1./31%47,/?-.%3&%,4239%3)+1215)32(15)(35)?%,-5)();H5())2F)(.0%3& Energy P1: IBD study "# <"*.%3),%315))/,1)(3-2./(&0()92(?/1(%/3&.)G)1A))315)<@2(),7 budget K())3./3+/3+15)>(%1%,56,.),;H5)215)(15()).%3),-(2F%+)?)(%+%23/. Lumpkin & Speer 2006 density n Water class 42F)(/&)G'115)()%,3242?-()5)3,%F)@23/.42F)(/&); Upper Deep Lower Deep 288 WUNSCH FERRARI Atlantic, bottom Indian, bottom Pacific, bottom Atlantic, deep Indian, deep Pacific, deep Scotia Sea Brazil Basin K " (10 4 m/s) Ganachaud & Wunsch (2000) 9±2 12 ± 7 4±1 3 4 Hogg et al. (1982) 30 ± Amirante Trench 10.6 ± 2.7 Romanche Fracture Zone 3 ± 1.5 4± $%&'()*+,-./01203/4565% &:%5;<)(%=% ;><)5(%? 497@)4%85;)16?%A%??797(B?7=)=+>#C;>=(7&(6@;%?9%8)4D $%&'()*/LM<NE8OEHN,1/1%23.24/1%23,/.23&A%15?)(%+%23/. G/150?)1(%4,.2-),HyPD?%15/3+D/3+A)..QRRST%315)63+%/3=4)/3 42.2(J42+)+G015)QU50+(2&(/-5%4.%3),; $%&'()*+,-./01203/4565% &:%5;<)(%=% ;><)5(%? 497@)4%85;)16?%A%??797(B?7=)=+>#C;>=(7&(6@;%?9%8)4D Heywood et al. (2002) Roemmich et al. (1996) Barton & Hill (1989) Saunders (1987) Ferron et al. (1998) (2002) found the remarkably high value of (30 ± 10) 10 4 m2 /s in the Scotia Sea. These and other estimates are listed in Table 1. Ganachaud & Wunsch (2000) described estimates of mixing inferred through $%&'()*?,-./01203/4565% &:%5;<)(=% ;><)5(%? 497@)4%85;)ED.59685%??797()=B?7=)=+>5;)4%F;>=(7&(6@;%?9%8)4D E75)5;65."*6G+68=?9%)69< @7A)6?;75;)(D < n < 28.1 n > 28.1 Reference 9±4 Samoan Passage Discovery Gap "#27.8 Bottom 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 to 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 27.0 < n < 27.8 kv cm2 s ± ± 0.5 3±1

Erratic internal waves at SIO Pier. data and wavelet analysis courtesy of E. Terrill, SIO

Erratic internal waves at SIO Pier data and wavelet analysis courtesy of E. Terrill, SIO Internal Surface Waves warm water Slow f ω N ~1day ~hours cold water Big tens of meters Internal Waves warm water