Observations of internal wave coherence in the deep ocean

Deep-SeaRmearch,1974,Vol.21, pp. 597 to 610.Per~_men Pre~. Printedin GreatBritain.

Observations of internal wave coherence in the deep ocean* G.S~L~

(Received 18 September 1973; in revisedform 20 February 1974; accepted20 March 1974) Abstract--An experiment is described which was aimed at testing assumptions and predictions of the internal wave model suggested by GARRETrand MUNK(1972). Two moorings were set at a depth of 2660 m with a horizontal separation of 920 m only. The results of current and temperature measurements on these moorings indicate that the field of motion is probably horizontally isotropic in the inertic~gravitational wave band. The limiting frequency for horizontal coherence is three times the frequency predicted by the theoretical model. The phase of the vertical coherence is stable over a wide frequency range and the coherence decreases towards higher frequencies. This may be due to coherent motion contaminated by uncorrelated noise at high frequencies. The results are basically in agreement with the theoretical model when taking a number of modes below 10.

b Cm, Q I F f Subscript i •ffi 1, 2, 3 h No

N~ uj t

xt(i ffi 1, 2, 3) Ax Ax, Ct' O,

oh 2n

NOTATION Depth where N drops to N0e-1 Co- and quadrature spectrum for component m at location n Energy density spectrum Circular inertial frequency Cartesian components, x, downwards Mode number bandwidth at f Circular Brunt-Vaisala frequency Nfor x,---- 0 Velocity component Time Cartesian coordinates Horizontal separation Vertical separation Radian three-dimensional wavenumber of internal waves Radian horizontal wavenumber of internal waves Radian wavenumber component of internal waves for/-coordinate Circular frequency

1"

cot .f

Circular frequency where the coherence drops to ½ Period.

INTRODUCTION IT IS USUALLY assumed t h a t the frequency b a n d of oceanic variability between the inertial a n d the local Brunt-V~is~ili frequency is governed by inertio-gravitational internal wave dynamics. I n a n e x a m i n a t i o n o f m o o r e d c u r r e n t meter records by FOFONOFF (1969) a n d a study o f float m e a s u r e m e n t s by VOORHIS (1968), it was s h o w n that certain functions of spectral properties are consistent with results from linear i n t e r n a l wave theory. T h e m e a s u r e m e n t o f the energy-spectrum F(co, a') as a f u n c t i o n o f b o t h frequency co a n d w a v e n u m b e r a ' over a wide range is needed for a more com*Contribution No. 3140 from the Woods Hole Oceanographic Institution. tInstitut ffir Meereskunde an der Universitit Kiel, 23 Kiel, Diisternbrooker Weg 20, Germany. 597

598

G. S~DLEa

plete study of internal waves. This is not obtainable at present by moored instruments because of the limited number of current meters that can be employed in a moored array. Earlier measurements from single moored instruments supplied information about the auto-spectrum E(co) as a function of frequency alone which indicate that there is usually a peak at inertial and/or tidal frequencies and a slope proportional to c0-p at higher frequencies, withp close to 2 (FoFoNOFF, 1969 ; WEBSTER, 1969). The energy level at any frequency within this bound in the deep ocean decreases proportional to N (WEBSTER, 1969). Information is available for the auto-spectrum F(a) as a function of horizontal wavenumber a alone only for the mixed layer and the thermocline. When instruments like thermistor chains are towed at speeds of several kmh -1, the towing speed is much larger than the phase speed of the high mode number internal waves, and a 'frozen' field of internal waves can be observed. Available data were mainly obtained from shallow thermistor chain tows in deep water (see CH~,NOCK, 1965; LAFOND and LEE, 1969). Recently KATZ (1973) was able to determine the horizontal wave number spectrum by towing a temperature-conductivity--depth sensor package in deep water. His data indicate a power law with a-a, where q is somewhat below 2 forwavenumbers below 1 cycle km -1 and somewhat above 2 for higher wave numbers. Information related to the wavenumber bandwidth can be obtained from moored arrays by studying the coherence and the phase difference for instrument pairs. GARR~TT and MUr~K (1972) have summarized observational results on the vertical and horizontal coherence of current and temperature data. Figures 1 and 2 give 100

1000

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where the coherence drops to ~ as a function of vertical separation Axa,

according to earlier observations. The straight line indicates the rule given by WEBSTER(1972).

Observations of internal wave coherence in the deep ocean 50 5000

10 =,m I

-

1 i ' l I I,Ilml I M U N K et al. (i970)

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Frequency mt where the coherence drops to ½as a function of horizontal separation Ax, according to earlier observations.

similar up-dated summaries of the instrument separation as a function of the frequency cot where the coherence drops to ½ but further data have been added. For the vertical coherence, the limiting frequency cot generally decreases with increasing separation Ax8 and the current meter data fit a line with Axs proportional to co-x over a wide range (WeBsTER 1972). At frequencies very close to the minimum and the local Brunt-V[iis~il~t frequency this law does not seem to hold (SmDLER, 1971). The temperature data, however, are much more scattered. The horizontal coherence indicates a similar trend of decreasing cot for increasing separation, but no data were published for coherences in the deep ocean well off the bottom and offmajor topographic disturbances with c0t significantly above the inertial frequency. THE G A R R E T T - M U N K

MODEL

GARRETT and MUNK (1972) [henceforth G M (1972)] attempted to find a model for internal waves which would be consistent with the properties of current and temperature structure obtained from observations. The basic idea (MuNK and PHILLIPS, 1968) is that the internal wave field can be approximately described as a stationary and horizontally homogeneous ensemble of elementary wave trains with random phases. The energy integrals depend on the wave functions and on the processes of generation, interaction and dissipation of waves. To obtain the wave functions in a rather simple way, G M (1972) used an exponential form of the Brunt-V~tis~I~t frequency profile in the deep ocean neglecting the mixed layer at the surface:

600

G. SIEDLER

(1)

= N o oxp

P. Miiller (personal communication) considered the effect of using the WKBJ-approximation instead, and DESALraIES(1973) developed approximations for the turning depth where the local vertical wave number vanishes. They arrived at the conclusion that the basic results of the GM model are fairly accurate except near the turning point. As the measurements described later in this paper were carried out well away from this depth, the original GM model will be used throughout the following discussion. GM (1972) discuss the case of horizontal isotropy of the wave field. They also assume that the scale of the energy integral F(ct, co) as a function of the horizontal wave number et is proportional to the a-bandwidth ~(co), but that its shape is invariable. The spectrum as a function of o) alone is chosen to represent a peak at the inertial f r e q u e n c y f a n d a power law at higher frequencies. The energy is proportional to co-p at high frequencies and to a-q at high wave numbers. The constants p and q are related with a third constant r with the wave number bandwidth proportional to ¢o' at high frequencies: r - p -

1

(2)

q--1 It follows from the model that f o r f ,~ co < N the moored vertical coherence drops to 0-5 at the frequency 0)j:

(?)'-~ Axa-

1-9 No b

jf ~ N(z)"

(3)

The moored horizontal coherence drops to 0.5 for

((i)~ r-1co~ Ax 7]

2~

--

2.8 Nob

(4)

Yr~

Here Jl is the equivalent number of modes at the inertial frequency. THE EXPERIMENT An experiment was carried out at sea to compare the horizontal coherence scales predicted by the G M (1972) model with observed coherences. An instrumented array with 2 moorings was set close to Site D (~ = 39°08'N, 7~ = 69°59'W) at approximately 2660 m depth with 3 current/temperature meters and 2 additional temperature meters at each mooring. In order to obtain a coherence above 0.5 in the middle of the internal wave band betweenf/2n = 0-053 cph and N/2rc = 1 cph, a separation of less than 1 km is required according to the G M (1972) model if an appropriate density model for Site D is used. The depth level of the instruments had to be well away from the surface and the bottom in a depth range with steadily changing Brunt-V~tis~il/i frequency. Those requirements posed two specific technical problems. First, the separation of the moorings was only one-third of the water depth, and it is difficult to place

Observations of internal wavecoherencein the deep ocean

601

moorings with a positioning accuracy of an order 100 m. The solution was found by setting the first mooring (No. 418) with the usual buoy-first technique and using the anchor-first method for the second mooring (No. 419) and towing it to the appropriate position found by transponder distance measurements relative to the first mooring. The second problem was in obtaining information about the horizontal instrument separation and its variations in time. This was solved by monitoring the relative mooring motion from aboard the drifting ship by a combination of two acoustic transponders. Figure 3 gives a schematic diagram of the moorings, indicating the acoustic paths used for the monitoring. The transponders were interrogated three times within a time interval of 2-4 rain. Different pairs of transmissions and reception frequencies were used for each measurement during this period. On transmission of a pulse with frequency f l the transponder at A replied with a signal with frequency f2 travelling to the ship and simultaneously to the transponder at B. There it triggered a

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Fig. 3. Mooredarray and schematicdiagramof transponderoperation.The arrowsindicatethe direction of acoustictransmissionwith frequencyf~.

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Fig. 4. Mooringseparationdata. Dots indicatedata for a distancesmallerthan 3.7 km between ships and closest transponder, circles indicate data for larger distances. All data are removed where the ship's drift exceeded2 km h-t.

pulse with frequency]'8 that also propagated to the ship. During the first measurement with an outgoing signal with f l and an incoming signal with f2 the time delay between these signals was proportional to the path length C-A. When transmitting again with fl, and receiving with fs, the delay time indicated the path length C-A-B-C. Correspondingly, a transmission with fs and a reception with fs gave the path length C-B. The transponder separation could then be calculated by subtracting the path lengths C-A and C-B from C-A-B-C. Difficulties were encountered because of the ship's drift during the time interval needed for the three distance measurements and because of poor transmission for large separations between the transponders and the ship. To remove or indicate possible errors introduced by these effects, all data were removed where the ship's drift exceeded 2 km h -1, and data were marked when the separation between ship and transponder exceeded 3.7 km. The resulting set of mooring separation data is shown in Fig. 4. It turned out that typical separation changes were a few meters only and that the mean value of the separation was 920 m. The first set of data in Fig. 4 was obtained immediately after launching the second mooring. As is indicated by the separation changes of order l0 m at the beginning of this data series the mooring apparently needed about 10 h to settle down to its final catenary. Although there were depth changes with inertial period during the whole mooring period as indicated by a pressure meter on one mooring, the changes in mooring separation were amazingly small, typically considerably less than 1% except for the settling-down period. The instrument array is shown in Fig. 5. All current/temperature meters were crystal-clock controlled tape-recording EG & G type 850 current meters with additional temperature circuits. All temperature recorders were strip-chart recording meters with

Observations of internal wave coherence in the deep ocean

No.4'18

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603

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TRANSPONDER PRESSURE

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CURR./TEMR TEMR

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CURR./TEMR

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ITRANSPONDER DEPTH LEVEL

533m

585m

4t94 ] CURR./TEMP. 449; ITEMR

4193I l CURR./TEMR

920m

CURR./TEMR TEMP.

4185 [~4t86

635m

41941 I CURR./TEMP. 44951 I TEMP. I

,,

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Fig. 5.

Instrument array.

mechanical docks. It was aimed to have exactly the same depth levels for horizontally corresponding instruments. By a comparison of the mean temperature, records by the meters at different levels, and by repeated STD-lowerings close to the moorings during the separation monitoring period it was possible to determine the actual depths of the moored instruments. It turned out that in spite of the different launching procedures and rope stretch histories the corresponding instruments were at the same depth level within approximately :k 5 m, and that the absolute levels were close to the numbers given in Fig. 5. Thus the vertical separation of the instruments in the array was at least one order of magnitude larger than the difference in depth of horizontally corresponding instruments. The moorings were set on I0 and 11 December 1971, and recovered on 20 December 1971. During 1-day periods immediately after launching the second mooring and three days before retrieving both moorings, the following measurements were carried out from aboard the ship: acoustic monitoring of mooring separation in half-hourly and later 5-min intervals, 4 hydrographic stations, 20 STD-stations with repeated lowerings over the depth range of the moored instruments on 7 of these stations, and 8 Expendable Bathythermograph lowerings. The results of the STD measurements are discussed elsewhere (SmDLEg, 1974). The vertical profile of the Brunt-Viis/iPt frequency as obtained from three of the hydrographic stations is given in Fig. 6. The BruntV~iisil~i frequency at the levels of the moored instruments appears to be close to 1 eph. The evaluation of the data records from the moored instruments showed that all

604

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FREQUENCY

Brunt-V~iis~il~i frequency distribution during ,the experiment.

instruments on mooring 418 had operated properly. On mooring 419 the current meter 4191 had no speed data, and 4194 had a completely bad record. This mooring had been towed over a period of almost 14 h because of problems in positioning, and the instruments had been shaken considerably during this time. All instruments except 4191 and 4194 had operated properly on this mooring. THE H O R I Z O N T A L

AND VERTICAL COHERENCE-

The data series obtained from the moored instruments were analyzed by using the Fast Fourier Transform. To reduce the effect of transient low frequency processes, a piccelength of 36 h was chosen for the analysis. As the dominant low-frequency motions in the internal wave band at Site D are of inertial (3 -- 19 h) and semidiurnal (3 = 12-4 h) frequency, one piece contained approximately 2 inertial and 3 tidal periods. All records were used for the time interval between 14 December, 12.00 and 19 December, 12.00, thus containing 4 pieces. Samples of current and modified temperature spectra can be found in SIEDLER(1974). Coherences for pairs including two different types of instruments were generally lower than those of corresponding pairs with instruments of the tape-recording crystalclock controlled type although the actual records looked very similar. Only records

Observations of internal wave coherence in the deep ocean

605

from the crystal-clock controlled instruments are therefore used in the following discussion. When assuming homogeneity of the internalwave field in the vertical over the short distance of 100 m, all coherence and phase estimates for a certain frequency band can be combined for pairs with equal horizontal or vertical separation. Furthermore, coherences for east--east and north-north component pairs were combined because of the assumed horizontal isotropy in the GM (1972) model. It was possible to obtain some indication as to whether this assumption may be valid by computing the 'ellipse stability' E as defined by GONELLA(1972): E(co) = (Cu,ul - - Cu@,) ~ + 4 C~1.,

(5)

P. Miiller (personal communication) was able to show that E can be expressed as a function of certain Fourier coefficients for a directional spectrum. Using clockwise '--) and antieloekwise ( + ) components, the total energy Fret can be split up in even (e) and odd (o) parts Ftoet and Fto°t :

F,o', (al, a,, ~0) = ½ [F,+o,(al, o~, co) + F,;~ (a. a,, co)]

(6)

Fto°t(ax, a2, co) = ½ [Ft+ (oh, a,, co) -- Ft~t (ax, a,, co)]. If the energy is mainly described by the even part, the ellipse stability can be written as:

E --

a~ 2 + b; ~

a~

'

(7)

where the constants on the right side of equation (7) are Fourier coefficients for developing Ft~t as a function of direction 9: Ft~t (a, ~, co) = -~-+ 1~ ( ~ cos n 9 + b~ sin n 9).

(8)

nil

E will be 0 for isotropy and 1 for uni-directionality of the even part of the spectrum. If the odd spectrum can be neglected, E is an approximate measure of isotropy. The ellipse stabilities for the four current records are shown in Fig. 7. Following GON~LLA (1972), the 95 % confidence limit for zero ellipse stability was calculated and is indicated by the dashed line. The estimates are below this confidence limit in the whole inertiogravitational wave band and, furthermore, display no frequency dependence which repeats from one instrument to the next one. Therefore, the assumption of horizontal isotropy is probably correct in this case. The eoherences and phases for the currents are given in Fig. 8; the numbers of component pairs used are indicated. There is only one instrument pair, i.e. two component pairs, for the horizontal separation of 920 m. The horizontal coherence is high at low frequencies and drops below 0.5 at 0.35 eph and below the 95% confidence

606

G. SmOLER

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frequency (cph) Fig. 7. Ellipse stability obtained from the instruments indicated by the numbers.

level for zero coherence at 0.4 cph. When considering the bias these numbers would not differ sufficiently to change the conclusions of the later discussion. The corresponding horizontal temperature coherence (2 instrument pairs, i.e. 2 temperature pairs) in Fig. 9 behaves similarly, with a drop below 0-5 and below the 95% level at 0.3 cph and a peak between 0.5 and 0.65 cph. Thus, except for the peak close to 0.6 cph corresponding to the minimum Brunt-Viiisiilii frequency of the water column (SmDLER, 1971), the coherence shows a pattern as essentially predicted by the GM (1972) model, with a cut-off frequency in the middle of the internal wave band. Choosing N O = 3 cph and b = 700 m to represent the Brunt-Vfiisiilii frequency distribution in the depth range of the moored instruments, and following GM (1972) in taking r = 1 and Js = 20, we obtain from (4): co~ = 0.1 cph. 2n

(9)

It appears that the observed limiting frequency is three times the predicted frequency. A consistency of the model and the observational data can be obtained by either decreasing r o r j I. An exponent r < 1 seems unreasonable because it would mean increasing o t for increasing vertical separation, thus reversing the trend known from observations. One therefore arrives at the conclusion that the number of modes ought to be below I0 to described the high o)i for horizontal coherence within the framework of the G M (1972)model. The vertical current coherence for 50-m separation has a pattern similar to the horizontal coherence, with a drop below 0.5 at 0.15 cph and below the 95% level at 0.5 cph. The limiting frequency as calculated from the formula (WEnsa~R, 1972)

Observations of internal wave coherence in the deep ocean

607

MOORINGS 418-419 [hi

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Fig. 8.

2 PAIRS

Vertical and horizontal coherence of currents.

o~t _ 13 cph 2n Axs

(10)

obtained from earlier measurements at Site D is 0-26 cph, i.e. in reasonable agreement with these results. The temperature coherence over 50 m vertical separation drops below 0.5 at 0.3 cph, but is close to ½ over a fairly wide frequency range centered at 0.6 oph and a small range close to the local Brunt-Vitisgli frequency 1 cph. It is above the 95% confidence level in almost the whole internal wave band, and the phase is close to 0 ° from f up to the local N. The vertical current coherence over 100 m is below 0-5 everywhere in the internal wave band and close to the 95% level up to 0-45 cph, which also appears as the frequency above which the phase becomes unstable. Webster's formula predicts c0t/2x = 0.13 cph. The temperature coherence for this separation is low for inertial frequency and scatters around the 95 % level over the whole internal wave band. Except

608

G. Sm~LER MOORINGS 418-4~9 period [hi

~ ~ o 2 O ~o

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[ 0.8

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Fig. 9. Verticaland horizontal coherence of temperature.

for inertial motion, the phase, however, is fairly stable over the whole range. This difference in the frequency dependence of the vertical coherence and phase may be due to uncorrelated noise, possibly from fine structure, which diminishes the coherence but does not affect the phase (JovcE, in press). According to the GM (1972) model with r = 1 the internal wave motions should be vertically coherent at each frequency for separations below Ax3 given by (3), which is 65 m for j / = 20 and 200 m forjs = 7. G M (1972) suggest that the often observed drop in vertical coherence is in fact due to fine-structure contamination (GARRETTand MUNK, 1971). CONCLUSIONS

The assumption of horizontal isotropy of currents in the inertio-gravitational internal wave band is probably correct for the time and location of the described experiment. The data for horizontal coherence from this experiment are basically in agreement with the internal wave model suggested by G M (1972). The frequency limit

Observations of internal wave coherence in the deep ocean

609

for horizontally coherent motion is found to be three times the value predicted from the G M model for 20 modes [equation (4)]. The data are consistent with the model when taking a smaller number of modes ( < 10) in this case. The vertical coherences in this experiment are similar to WEBSTER'S(1972) earlier results, with the coherence decreasing with increasing distance. The stable phase, however, may indicate a coherent motion up to rather high frequencies, contaminated by uncorrelated noise. This would be expected from the G M (1972) model [equation (3)] for the separations selected in this experiment when accepting a fine-structure contamination of coherence which is essentially independent of separation.

Acknowledgements--This work was supported by the Office of Naval Research, Contract No. NOOO14-66-CO241. The experiment at sea and the data evaluation were carried out with the aid of the Buoy Group at the Woods Hole Oceanographic Institution. I particularly benefited from discussions with K. HASSELMAr~,T. M. JOYCE,P. MtrLLEg, P. SAtmOEV,S and members of the Woods Hole Buoy Group. REFERENCES CSAmqOCK H. (1965) A preliminary study of the directional spectrum of short periodic internal waves. Proceedings of the 2nd U.S. Navy Symposium on Military Oceanography, pp. 175-178. D~AUBmS Y. J. F. (1973) Internal waves near the turning point. GeophysicalFluid Dynamics, 5(2), 143-154. FOrONOFF N. P. (1969) Spectral characteristics of internal waves in the ocean. Deep-Sea Research, Supplement to 16, 58-71. GA~ C. and W. MONK (1971) Internal wave spectra in the presence of fine-structure. Journal of Physical Oceanography, 1(3), 196-202. GARREa"r C. and W. MONK (1972) Space-time scales of internal waves. Geophysical Fluid Dynamics, 2, 225-264. GONF.LLAJ. (1972) A rotary-component method of analysing meteorological and oceanographic vector time series. Deep-Sea Research, 19, 833-846. GOULD J. (1971) Methods of measuring and the analysis of currents in coastal and oceanic waters. Ph.D. Thesis, University of Wales, 95 pp. and figs. HECHT A. and R. A. WRITE(1968) Temperature fluctuations in the upper layer of the ocean. Deep-Sea Research, 15, 339-353. JOYCE T. M. (in press) Fine structure contamination of moored temperature sensors: a numerical experiment. Journal of Physical Oceanography. KATZ E. J. (1973) Profile of an isopycnal surface in the main thermocline of the Sargasso Sea. Journal of Physical Oceanography, 3, 448--457. LAFOND E. C. and O. S. LEE (1969) Oceanographic activities of the Marine Environment Division, September 1968-September 1969. Report Naval Undersea Centre, San Diego, California, pp. 165, 414. MUNK W. H. and N. PHILLIPS(1968) Coherence and band structure of inertial motion in the sea. Review of Geophysics, 6, 447--472. MUNK W. H., F. E. SNODGRASSand M. WIMSUSH(1970) Tides off shore: transition from California coastal to deep sea waters. Geophysical Fluid Dynamics, 1, 161-235. PERKINSH. T. (1970) Inertial oscillations in the Mediterranean. Ph.D. Thesis, Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 155 pp. SmDLERG. (1971) Vertical coherence of short-periodic current variations. Deep-Sea Research, 18, 179-191. SIEDLERG. (1974) The fine structure contamination of vertical velocity spectra in the deep ocean. Deep-Sea Research, 21, 37-46. SCHOTTF. (1971) On horizontal coherence and internal wave propagation in the North Sea. Deep-Sea Research, 18, 291-307. UFFOROC. W. (1947) Internal waves measured at three stations. Transactions of the American Geophysical Union, 28, 87-95. VOORmSA. D. (1968) Measurements of vertical motion and the partition of energy in the New England slope water. Deep-Sea Research, 15, 599--608. WEnSTF.R F. (1968) Observation of inertial period motions in the deep sea. Review of Geophysics, 6, 473-490.

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