the high-resolution vertical structure of internal tides and near...

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Deep-Sea Research I 54 (2007) 533–556

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The high-resolution vertical structure of internal tides andnear-inertial waves measured with an ADCP over the

continental slope in the Bay of Biscay

Hendrik M. van Aken�, Hans van Haren, Leo R.M. Maas

Royal Netherlands Institute for Sea Research, P.O.Box 59, 1790AB Den Burg, the Netherlands

Received 15 November 2004; received in revised form 23 March 2006; accepted 21 July 2006

Available online 1 February 2007

Abstract

ADCP measurements of the velocity structure in the permanent thermocline at two locations over the continental slope

in the Bay of Biscay are presented. The vertical variation of the contribution of the inertia-gravity waveband to the kinetic

energy, vertical motion, and current shear are analysed. The semi-diurnal tides together with near-inertial waves appear to

provide over 70% of the high-frequency kinetic energy (41/3 cpd). Over the vertical range of the ADCP observations the

phase of the harmonic M2 tide changes up to 1551, while the kinetic energy varies in the vertical by a factor of 3.8, showing

the importance of the contribution of internal waves to the observed tidal motion. Both semi-diurnal internal tidal waves

and near-inertial waves have a vertically restricted distribution of the variance of the horizontal and vertical velocity, as in

internal wave beams. The short-term 14-day averaged amplitude and phase lag of the M2 tide shows large temporal

changes, with a characteristic 40–45 day time scale. These changes are probably related to variations in generation sites and

propagation paths of the internal tide, because of changes in the temperature and salinity stratification due to the presence

of meso-scale eddies. The relatively large shear in the inertia-gravity wave band, mainly at near-inertial frequencies,

supports low-gradient Richardson numbers that are well below 1 for nearly half of the time. This implies that the large

shear may support turbulent mixing for a large part of the time.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Internal waves; Internal tides; Near-inertial waves; Bay of Biscay; Continental slope

1. Introduction

Internal inertia-gravity waves are waves wherepressure changes due to density variations in theearth’s gravity field as well as the Coriolis forcecontribute to the acceleration of water particles.

front matter r 2007 Elsevier Ltd. All rights reserved

r.2007.01.003

ng author. Tel.: +31222 369 416;

9 674.

ss: [email protected] (H.M. van Aken).

According to traditional theory, these waves canexist in the angular frequency (o) bandf(f)oooN(z) (Krauss, 1966). Here f(f) is the localinertial or Coriolis frequency, f(f) ¼ 2O sin(f),twice the local vertical component of the earthrotation vector O at latitude f, and N(z) is thedepth-dependent Brunt-Vaisala or buoyancy fre-quency, which depends on the adiabatic densitygradient: N2(z) ¼ (�g d ln(r)/dz)|ad with g the grav-itational acceleration and r the time averaged water

.

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ARTICLE IN PRESSH.M. van Aken et al. / Deep-Sea Research I 54 (2007) 533–556534

density. Such waves contribute significantly to thekinetic energy in the abyssal Bay of Biscay (Pingreeand New, 1991, 1995; van Haren et al., 2002). Thesewaves, especially at tidal and near-inertial frequen-cies, are assumed to be a main agent for thetransport of mechanical energy into the deepabyssal ocean worldwide. There the wave energycan be converted to turbulent kinetic energyavailable for diapycnal mixing that drives the globaldeep upwelling (Munk and Wunsch, 1998). Over theslopes of the Mid-Atlantic Ridge in the Brazil Basin,high levels of turbulence were observed, attributedto the breaking of internal waves that weregenerated by tidal currents flowing over theirregular steep topography of this ridge (Polzinet al., 1997; Ledwell et al., 2000).

Internal waves can be described in different ways.For a horizontally homogeneous ocean with a flatbottom, Vertical modes (vertically standing waves)are used to describe the internal waves with thelargest vertical scales (Fjeldstad, 1963). If thevertical wavelength of these waves reduces to scalessmaller than the scale of vertical variation of thedensity stratification, nearly plane internal wavescan be assumed to be slowly modulated so that thevertically changing stratification influences only thelocal vertical wavelength, phase and group velocity,and does not lead to energy scattering and trapping:the WKB approximation (Leblond and Mysak,1978). When internal waves of certain frequency arelocally generated over steep topography, e.g. asinternal tides, the waves can no longer be approxi-mated as nearly plane waves or vertical modes.Then the propagation of wave energy can beanalysed by the method of ray tracing. There themechanical energy of the internal waves is assumedto be spatially concentrated in narrow beams thatmay reflect at the sea surface or at the bottom of theocean (e.g. Thomas and Stevenson, 1972; Baines,1982; Gerkema, 2001). In a wave beam the conceptof wavelength, developed for plane waves, fails,since the beam is a superposition of a large numberof plane wave modes (Lighthill, 1978). However,one can still define a characteristic length scale, thedistance over which extrapolation of the observedphase shift would reach a value of 3601. In the caseof plane waves this length scale equals the wave-length.

Garrett and Munk (1972, 1975) devised a semi-empirical universal spectral model (GM spectrum)for the statistical description of the internal wavefield in the abyssal ocean. For this model the

existence of a vertical modal structure of theinternal waves was not vital. A basic assumptionfor the model was horizontal statistical isotropy, forwhich strong non-linear resonant interaction to re-distribute energy among wave numbers and fre-quencies would be expected. The model assumes novertical or horizontal flux of mechanical energy anda perfectly reflecting smooth horizontal oceanbottom. The resulting model spectrum was pre-sented as an equivalent continuum spectrum, with-out any spectral lines at tidal frequencies.

Shortly after the Garrett–Munk spectrum wasintroduced, Hayes and Halpern (1976) presentedevidence that near the continental slope off Oregonthe bulk of the kinetic energy of internal waves wasnot smoothly distributed over the frequencies, as inthe GM spectrum, but was found in the spectralbands near the semi-diurnal and inertial frequencies.The kinetic energy level in these bands was notstationary but appeared to vary over time by anorder of magnitude. From temperature observa-tions in the deep Bay of Biscay, Pingree and New(1991) found evidence that the energy of semi-diurnal internal tides propagated from their genera-tion area near the continental break into the openocean. There the internal wave beam reflected at thebottom. Gerkema (2001) has theoretically shownthat variation in the seasonal stratification may leadto changes in the propagation and structure ofinternal wave beams in the Bay of Biscay. It can beexpected that stratification changes due to thepresence of meso-scale eddies will lead to similarvariations in the internal wave structure.

In this paper, we focus on the vertical andtemporal structure of semi-diurnal internal tidesand near-inertial waves in the main thermoclineover the continental slope in the Bay of Biscay.Observations and modelling efforts have shown thatinternal tidal waves are generated at the continentalbreak and propagate mainly downward as an 800mthick wave beam (Pingree and New, 1991; Lam etal., 2004; Gerkema et al., 2004). The available dataset consists of two moored ADCP records from1995–96 and 1997–98 of the horizontal and verticalvelocity components (Table 1). The length of eachrecord is about 1 year, which allows accurateestimates of the internal wave statistics in theinertial and semi-diurnal frequency bands. Addi-tionally, data from a current meter mounted nearthe bottom at BB 17 (Fig. 1) were used. The highvertical resolution of the ADCP observations,compared with classic current meters, allows the

ARTICLE IN PRESS

10 W 9 W 8W 7 W 6W 5W 4W

Longitude

45 N

46 N

47 N

48 N

49 N

La

titu

de

Fig. 1. Bathymetry (m) of the northern Bay of Biscay with the

positions of the ADCP moorings. The insets show details of the

topography with depth contours every 200m. The 2000m isobath

is highlighted as a thick line.

Table 1

Information on the TripleB ADCP Moorings

Mooring Longitude Latitude Water

depth (m)

Bin range

(m)

Bin size

(m)

Sampling

period (min)

Ensemble

(pings)

Start date Number of

days

BB 1 51220W 461430N 1615 1075–723 16 15 80 20 July 1995 336

BB 17 81170W 471380N 1498 970–602 8 30 190 15 August

1997

370

H.M. van Aken et al. / Deep-Sea Research I 54 (2007) 533–556 535

analysis of the vertical change of amplitude andphase of internal waves and the inference of wavebeams. Attention is given to the permanent verticalstructure of the M2 internal tides as well as to thevariable part of the internal tides. The spectralbehaviour of the current shear, inherent in theinternal and inertial waves, is analysed too, sincesuch wave shear may drive the turbulence andturbulent mixing in the thermocline.

2. The data

As part of the Dutch ‘‘TripleB’’ research pro-gramme between 1995 and 1998, a moored ADCPwas deployed at two locations in the Bay of Biscayfor about 1 year in each deployment. The two sites,both near the 1500m isobath on the continentalslope (Fig. 1), were separated by 240 km. The

upward looking four-beam 76.8 kHz narrow-bandADCP, manufactured by RDI, was mounted on themooring so that the observational range of theinstrument (about 400m) covered the permanentthermocline. In the observational depth range theacoustic backscatter generally was relatively highover the year, because of the presence of a deepscattering layer with the highest amounts ofscatterers in the shallower levels of the observa-tional range. The instrument was fitted with anadditional external battery pack, in order toincrease the number of pings in the recordedensembles. Horizontal and vertical velocity compo-nents and an error velocity were obtained from theADCP. The latter was used as a measure for theaccuracy of the velocity measurements. For moor-ing BB 1 ensemble averages of 80 pings wererecorded every 15min, from which the velocitycomponents were derived in 25 bins, each 16m inrange. The data from ADCP bin 1 of mooring BB 1showed many spikes because of the occasionalpresence of a pick-up buoy in one of the ADCPbeams, while bin 25 at the end of the observationalrange showed many outliers. These bins were notused for further analysis because of their limitedquality. The 15min ensemble average of the errorvelocity of bins 2–24 had a standard deviation of0.9 cm s�1. This agrees with the white noise level,derived from the spectra of the vertical velocity,presented below. The error level hardly increasedfrom bins 2 to 24. For mooring BB 17 the number ofpings in each ensemble was increased to 190,recorded every 30min in 50 bins of 8-m length.For this mooring bins 1 and 2 were of limitedquality because of interference with the pick-upbuoy, while the last bin, bin 50, had many outliers.These three bins were also discarded for furtheranalysis. The standard deviation of the ensembleaverage of the error velocity in the remaining binswas only 0.3 cm s�1 and hardly varied with depth.While the position of mooring BB 1 was reasonablyunobstructed by topography on the continental


slope, mooring BB 17 was situated along a curvingpromontory of the Meriadzek Plateau. In both casesirregular topography, with canyons cutting into thecontinental slope, was nearby. A summary of themain ADCP settings is given in Table 1.

Nearby CTD stations, occupied before thedeployment and after the recovery of the ADCPmoorings, were used to determine the mean stabilitystructure for both moorings. The 400-m ADCPbeams were located in the main thermocline, whichwas found at pressure levels between 500 and1100 dbar. Maximum mean values of the Brunt-Vaisala frequency, N, in the main thermocline(38 and 43 cpd) were observed near 800 dbar(Fig. 2). Above the main thermocline a thermostadwas present with a Brunt-Vaisala frequency ofabout 25 cpd in the 200- to 400-dbar interval. Theshallow N maximum near a pressure of 50 dbar wasnot reached by our observations. Note for compar-ison that the local Coriolis frequency equals about1.5 cpd. In these circumstances semi-diurnal tides,

0 50 100 150 200

Brunt-Väisälä frequency (cpd)

1500

1000

500

0

Pre

ssure

(dbar)

BB 1

BB17

ADCP

observational

range

Fig. 2. Profiles of the Brunt-Vaisala frequency for moorings BB 1

(thick line) and BB 17 (thin line). These profiles are based on the

average of 8 (BB 1) or 7 (BB 17) nearby CTD stations occupied

before deployment or after recovery of the moorings. In the

calculation of this frequency, the adiabatic density gradient was

estimated from 50dbar vertical differences. The observational

ranges of the ADCP are indicated with the two-headed arrows.

with frequency oE2 cpd and foooN, can exist asfree internal waves.

Interference of a vertically propagating internaltidal wave and a barotropic tide may generate avertical structure of the net tidal amplitude andphase lag that hides the true structure of the internaltide. For the interpretation of our observations asrepresentative for baroclinic tides, an estimate of themagnitude of the barotropic tide is required. Withonly data from a 400m observational range and alocal ocean depth of about 1500m, a strict separa-tion between barotropic and baroclinic waves is notpossible. Measurements of barotropic tides over thecontinental slope in the Bay of Biscay are scarce.Along the continental slope the strength of thebarotropic tide varies considerably in strength(Pingree et al., 1983). Perenne and Pichon (1999)have determined the barotropic tidal amplitudeswith a bottom-mounted ADCP near the 300misobath at two positions, within 65 and 40 km of ourmoorings BB 1 and 17, respectively. When weassume that the barotropic mass flux connected withthese tidal currents is constant over the continentalslope, the major and minor semi-axes of the M2

tidal ellipse are �6 and 5 cm s�1 and 4 and 2 cm s�1

for the vicinity of moorings BB 1 and 17,respectively. Measurements with a towed ADCPfor �24 h along three cross-slope sections, locatedbetween the moorings of Perenne and Pichon(1999), reported by Lam et al. (2004) confirmedthe existence of internal wave beams at semi-diurnalfrequencies. Below 500m depth the baroclinicvelocity amplitudes were stronger than those ofthe barotropic tide. Those measurements also haveconfirmed that indeed the tidal mass transport isconstant over the continental slope within 30%. Theobserved along-slope gradient in the barotropicsemi-diurnal tidal amplitudes by Lam et al. (2004)confirms the modelling results of Pingree et al.(1983). This implies that at the BB 1 position the M2

barotropic tide is probably weaker than derivedfrom the results of Perenne and Pichon (1999). Themean cross-slope and along-slope tidal amplitudesat BB 1, estimated from extrapolation of the resultsof Lam et al. (2004), suggest a barotropic semi-diurnal tidal flow with cross-slope and along-slopeamplitudes of 6 and 1 cm s�1, respectively. Probablythe M2 tidal components at BB 1 are even smallersince the observations by Lam et al. (2004) have aspring tide bias. The available phase information ofthe barotropic M2 tide near our moorings is notaccurate enough to allow subtraction of the

ARTICLE IN PRESSH.M. van Aken et al. / Deep-Sea Research I 54 (2007) 533–556 537

barotropic tide from the ADCP observations.Therefore, the original data were analysed, being amixture of barotropic and baroclinic motions.However, most likely the magnitudes of thevariance of orthogonal current components of thebarotropic M2 tide estimated from the observationspresented here amount to about 20 cm2 s�2 and10 cm2 s�2 or less for moorings BB 1 and 17,respectively.

3. Results

3.1. Horizontal velocity spectra

In order to illustrate the characteristic contribu-tion of tidal and other frequencies to the observedhorizontal motion in the permanent thermocline, wehave determined vertically averaged power spectrafor the horizontal and vertical motion observedwith the ADCPs. Spectra were determined forsuccessive 70-day periods with a 35-day overlap

0 5 10 15 20

Frequency (cpd)

Pow

er

spectr

um

(m

/s)2

/cpd

1

1x10-1

1x10-2

1x10-3

1x10-4

1x10-5

a b

Fig. 3. Mean power spectra of the horizontal velocity, obtained by summ

The spectral estimates per bin, with 10 degrees of freedom, were aver

mooring BB 17 (b). The indicated 95% confidence interval is a worst ca

the complete spectra, the main figure the part of the spectra with prono

Vaisala frequency N (mean plus or minus twice the standard deviation)

That frequency range was not resolved for mooring BB 17 because of the

(D2), inertial (f) or a higher harmonic for these frequency bands. The g

continuum.

and averaged subsequently. End-point matching foreach 70-day period was used as a data windowingtechnique in order to prevent smearing of tidal andinertial spectral peaks due to side lobe leakage(Frankignoul, 1974). The resulting spectra, withabout 10 degrees of freedom, were then averagedover all available ADCP bins. The total number ofdegrees of freedom in these averaged spectra,approximated following Shcherbina et al. (2003,Appendix) and Bendat and Piersol (1986),amounted to at least 120, reflecting an accuratespectral estimate.

The spectra for the east and north componentswere summed to obtain information on the fre-quency distribution of the kinetic energy. Theresulting power spectra (Fig. 3) show a spectralcontinuum with, contrary to the smooth Garret–Munk spectrum, a series of spectral lines in thefrequency range up to 20 cpd for mooring BB 1 andup to 12 cpd for mooring BB 17. The dominantspectral lines were located at the M2 and S2

0 8 12

Frequency (cpd)

Pow

er

spectr

um

(m

/s)2

/cpd

1x10-1

1x10-2

1x10-3

1x10-4

1x10-5

1

4

ing the power spectra of the east and north velocity components.

aged over bins 2 to 24 of mooring BB 1 (a) and bins 3 to 49 of

se estimate according to Shcherbina et al. (2003). The insets show

unced peaks. In the inset of mooring BB 1 the range of the Brunt-

in the observational range is indicated with a two-headed arrow.

30min sampling. The spectral peaks are indicated as semi-diurnal

rey line illustrates the o�0.9 character of the background spectral


semi-diurnal (D2) tidal frequencies. The third high-est spectral peak in the internal wave band spectraof the horizontal velocity was observed near theCoriolis frequency, f, indicative for near-inertialwaves. At near multiples of the semi-diurnalfrequency a regular pattern of peaks was observed,primarily consisting of a linear combination ofhigher harmonics of the semi-diurnal tidal frequen-cies (D2n, n ¼ 1, 2, y). Additional spectral peakswere observed at frequencies that formed linearcombinations of the semi-diurnal tides and theirhigher harmonics and the Coriolis frequency(D2n7f). The spectrum for mooring BB 17 alsoshowed a peak at diurnal frequencies, centred on theK1 tidal frequency. The presence of spectral peaks athigher harmonics of the semidiurnal tides and atlinear combinations of these frequencies with theCoriolis frequency is not unique for the slope regionor thermocline in the Bay of Biscay. Van Haren etal. (2002) already reported the presence of suchpeaks in a spectrum derived from observations at adepth of 3819m, 1000m above the bottom in theBiscay Abyssal Plain. These are assumed to becaused by non-linear interaction, especially advec-

0 5 10 15 20

Frequency (cpd)

Po

we

r sp

ectr

um

(m

/s)2

/cp

d

1x10-4

1x10-5

1x10-6

a

Fig. 4. Vertically averaged power spectra of the vertical velocity. The in

BB 1 (a) and bins 3–49 of mooring BB 17 (b). The insets show the

pronounced peaks. In the inset of mooring BB 1 the range of the Bru

deviation) in the depth range of the ADCP range is indicated with a tw

tion of slower moving high-frequency waves by thedominant M2 internal tides.

Because of the sampling period of 15min formooring BB 1 we were able to determine the spectrafor that mooring up to frequencies well above theBrunt-Vaisala frequency in the main thermocline(29–44 cpd). The complete spectrum for this moor-ing (inset in Fig. 3(a)) shows an increase in thespectral slope at the lower boundary of thisfrequency range. For both moorings the back-ground spectral continuum had an empiricallydetermined o�0.90(70.01) character in the frequencyrange 2–24 cpd.

3.2. Spectra of vertical motion

The vertically averaged spectra of the verticalvelocity W, measured with the ADCP, also show anear-white background spectral continuum, inter-spersed with spectral lines (Fig. 4). The spectralcontinuum is nearly constant until it shows a slightincrease near the Brunt-Vaisala frequency. Athigher frequencies (at least for BB 1 where thesefrequencies were resolved) a definite decrease of the

0 5 10 15 20

Frequency (cpd)

Po

we

r sp

ectr

um

(m

/s)2

/cp

d

1x10-3

1x10-4

1x10-5

1x10-6b

dividual spectra per bin were averaged over bins 2–24 of mooring

complete spectra, the main figure the part of the spectra with

nt-Vaisala frequency N (mean plus or minus twice the standard

o-headed arrow.

ARTICLE IN PRESS

0 5 10 15 20 25 30 35 40 45

Frequency (cpd)

1x10-6

1x10-5

1x10-4

Pow

er

spectr

um

(m

/s)2

/cpd

BB 1vertical velocity power spectra

ADCP derived data

Temperature derived data

Range of N

Fig. 5. Power spectra for the vertical velocity derived from the

temperature sensor, mounted on the ADCP (thin black line), and

from the velocity measured in bin 2.40m above the temperature

sensor (thick grey line). The ratio of the half-width spectral

integral of the first 6 spectral peaks (T derived spectrum/ADCP

derived spectrum) amounts to 1.02 (70.16). The range of the

Brunt-Vaisala frequency, N, is indicated with a two-headed

arrow.


spectral values with increasing frequency occurred(see inset of Fig. 4(a)). At frequencies above 45 cpd,the spectral level approached the error level of theADCP measurements. When we integrate the levelof that high frequency spectral tail in Fig. 4(a)(�1.6 � 10�6m2 s�2 cpd�1) over the total frequencyrange (0–48 cpd), and then take the square root, wecome to an estimate of the noise level in the verticalvelocity of 0.9 cm s�1. This value equals the stan-dard deviation of the error velocity for mooringBB 1. The form of the spectral continuum agreesclosely with the vertical displacement spectrumabove the semi-diurnal frequencies, derived byCairns (1975) from temperature observations withan oscillating mid-water float, multiplied by thesquare of the frequency. The multiplication isrequired to transform a displacement spectrum toa velocity spectrum. Recently, Lien et al. (2005) alsopublished spectra of the vertical velocity in LuzonStrait that had a form similar to our Fig. 4. Thelevel of the smooth background spectral continuumappeared to decrease upwards over the observa-tional range by 45% and 30% for, respectively,BB 1 and 17.

Narrow spectral peaks were observed at the solardiurnal S1 tidal frequency in the spectra of thevertical velocity. This is probably due to the diurnalvertical migration of biological scatterers (Weisbergand Parish, 1974). As diurnal migrations are quitediscrete and offset from being perfectly sinusoidalthe presence of higher harmonics of this S1 signal isexpected. These harmonics were indeed observed asnarrow and small peaks at the S2 to S5 frequenciesin the w-spectrum for BB 1, and up to S4 for thew-spectrum for BB 17.

The dominant spectral peaks for the verticalmotion were found at semi-diurnal tidal frequencies(D2) and at higher harmonics of this frequency(D2n). The vertical velocity measured at mooring BB17 also showed relatively low but very narrow peaksat the D3 and D5 odd tidal harmonics, presumablyindicating tidal advection of the diurnally movingscatterers. Linear combinations of the Coriolisfrequency with tidal frequencies, e.g. D2+f, andD4+f, were also observed in the spectra of thevertical velocity while the inertial peak itself wasabsent. The absence of an f peak is to be expected,since near-inertial motion in the free water columnis predominantly horizontal. The magnitude of theD2 spectral peak at the semi-diurnal frequencies wasan order of magnitude larger for mooring BB 17compared with BB 1. In both cases this peak was

determined mainly by the semi-diurnal lunar M2

tide.Traditionally the vertical velocity, W, in an

internal wave field was determined from tempera-ture fluctuations, according to the relation (Krauss,1966):

W ¼ �@T

@t

�@T

@z(1)

where @T=@t is the time derivative of the observedtemperature and @T=@z the mean vertical tempera-ture gradient. We have compared the spectrum ofthe vertical velocity, determined from temperaturemeasurements on the body of the ADCP frommooring BB 1, with the spectrum of the verticalvelocity measured in the nearest ADCP bin 2, 40mabove the ADCP body (Fig. 5). Both spectra showsimilar spectral peaks of the same magnitude andthe decrease of the background continuum spec-trum in the Brunt-Vaisala frequency band. The


main difference between the two spectra is the lowerlevel of the background continuum of the tempera-ture-derived spectrum at low frequencies (o4 cpd)and at the highest frequencies (440 cpd). For alarge part this difference can be explained by theinfluence of the noise in the ADCP signal. In orderto determine whether the tidal peaks are wellrepresented by the temperature fluctuations accord-ing to (1) we have determined the integral over thehalf-width of the first 6 spectral peaks. The meanratio of the spectral integrals of the first 6 spectralpeaks (temperature-derived spectrum/ADCP de-rived spectrum, semi-diurnal to bi-hourly peaks)amounted to 1.02 (70.16 standard deviation). Thisagreement of the variance of the tidal peaks in theADCP-derived vertical velocity spectrum with thosein the temperature-derived spectrum indicates thatthe determination of the vertical tidal velocity withthe ADCP is reliable.

3.3. Coherent horizontal tides

In many models on internal tides and theirgeneration the slight change in density stratificationdue to the seasonal cycle and meso-scale eddies hasimportant consequences at larger distances from thegeneration area. Wave propagation along wavebeams integrates the effects of stratificationchanges, so these may become dominant at largerdistances from their source (Gerkema, 2001). Herewe have assumed that our moorings over thecontinental slope are close to the generation areaof internal tides. Then a considerable part of theobserved internal wave motion is due to coherentbaroclinic tidal waves (implying that amplitude andphase lag were persistent on the time scale of themooring duration). To determine amplitude andphase of the internal tides we applied a harmonicanalysis for the horizontal velocity measured in theADCP bins. Each variable x(t) was approximatedby a function,

xðtÞ ¼X

i¼1;N

Ai cosðoit� fiÞ (2)

with the constants Ai, oi and fi being, respectivelythe amplitude, angular frequency and phase lag ofthe ith of N tidal components (Emery and Thom-son, 1997). In the analysis that we performed, themain semi-diurnal (M2, S2, and N2) and diurnal(K1, O1) as well as the M4 and M2S2 harmonicsfrom the D4 band were used. Amplitude and phaselag were determined by means of a multiple

regression least-squares fit. The phase lag wasdefined with reference to t ¼ 0 at 00:00 UTC onJanuary 1 of the deployment year. For comparisonwith other high-frequency phenomena, separatehigh-pass-filtered current records were constructedthat contained only variations at frequencies above1/3 cpd. These high-pass currents represent about80% of the total kinetic energy.

On average, the coherent harmonic tides con-tributed 54% of the observed high-frequency kineticenergy in both data sets. For both moorings, the M2

tidal component gave the most important contribu-tion, followed by the S2 and the N2 tidal compo-nents (order of magnitude of the amplitudes of 7, 3,and 1 cm s�1, respectively). The tidal componentsfrom the D1 and the D4 bands had velocityamplitudes well below 1 cm s�1.

At mooring BB 1 the variance of the horizontalvelocity due to the M2 component (Fig. 6(a)) wasmore or less constant from 720m to about 900m,halfway in the ADCP range (4271 cm2 s�2). Thisvalue is at least twice the variance contributionexpected from the barotropic M2 tide. From 900 to1075m this variance increased to about 150 cm2 s�2.The contribution of S2 to the velocity variance wasan order of magnitude smaller with a maximum at850m (12 cm2 s�2). The amplitudes of the N2 tidalcomponent were less than 2 cm s�1 at all depths,resulting in a variance of the order of 1–4 cm2 s�2,definitely smaller than the M2 and S2 contributions.The M2 tidal phase lag of both the east and northcomponent (Fig. 6(b)) increased monotonicallyupwards, indicative of a downward component ofthe energy transport (Phillips, 1977). The M2 phase-lag difference was 1061 between 720 and 1075m forthe east component and 1551 for the northcomponent.

A numerical simulation of the superposition of abarotropic tide and a vertically propagating planebaroclinic tide shows that in order to get amonotonic vertical phase change larger than 901,the baroclinic contribution has to be larger than thebarotropic contribution. The observed relativelylarge phase change, 106–1551, shows that thedominant part of the observed tidal motion at BB1 had a baroclinic character. One may assume thatthe low value of the current variance at �900m(�40 cm�2 s�2) occurs when the barotropic andbaroclinic currents are in counter-phase, while thehigh value at 1075m (�150 cm�2 s�2) occurs whenthe barotropic and baroclinic tides are in phase.Then the barotropic speed amplitude will amount to

ARTICLE IN PRESS

0

0

50 100 150 200 250

Variance(U)+Variance(V) (cm/s)2

1100

1050

1000

950

900

850

800

750

total

tides

M2

S2

1100

1050

1000

950

900

850

800

750

-180 -90 0

0

90 180

Phase lag (degrees)

M2 east

M2 north

5

10

15

20

25

0

5

10

15

20

25

BB 1 BB 1

50 100 150 200 250


950

900

850

800

750

700

650

600

950

900

850

800

750

700

650

600

0 90 180 270 360

Phase lag (degrees)

0

10

20

30

40

50

0

10

20

30

40

50

BB 17 BB 17

Depth

(m

)D

epth

(m

)

Bin

num

ber

Bin

num

ber

Fig. 6. Variance of the horizontal velocity (black circles), its total tidal contribution (open circles) and the contributions to the variance

from the M2 (thick crosses) and S2 (thin crosses) harmonic tidal components (panel (a) for mooring BB 1 and panel (c) for mooring

BB 17), and phase lag angles for the east (black symbols) and north (open symbols) components for the M2 harmonic tidal component

(panel (b) for mooring BB 1 and panel (c) for mooring BB 17).



�5.5 cm s�1 and the baroclinic amplitude to�12 cm s�1. This barotropic amplitude comparesreasonably with the estimate of the upper bound ofthe barotropic tide at BB 1, introduced earlier.However, the phase information in Fig. 6(b) doesnot agree with this interpretation. With a largeramplitude for the baroclinic tide, the M2 tide shouldbe in counter-phase at both levels, not 57–731 apart.Apparently a vertical change in the barocliniccurrent amplitude contributes to the observedvariance structure. The observed gradient in thephase lag above 900m, where the M2 velocityvariance was nearly constant, shows that also inthat low-variance part of the water column thecontribution of the baroclinic M2 tide to the tidalvariance is significant. The mean phase shift overthe observational range suggests a vertical wave-length (distance over which the phase changes 3601)of 800–1200m. Over 2/3 of the total kinetic energyin the M2 band was located in the lowest 200m ofthe observational range, an energy localization that,given the vertical phase change, cannot be attrib-uted to interference of a barotropic and a verticallypropagating plane baroclinic wave. According tomodel simulations by Gerkema et al. (2004) theinternal M2 tidal wave beam in the Bay of Biscaywill be observed at a depth of �1100m at the BB 1location, about 17 km from the 400m isobath.These model simulations also show that most ofthe kinetic energy in internal wave beams isconcentrated within a 1801 phase lag interval.Thomas and Stevenson (1972) and Lighthill(1978) have derived a total phase changeacross a non-viscous 2-dimensional wave beam ofonly 1801, increasing up to about 3601 for internalwaves in a viscous fluid. Lam et al. (2004) andGerkema et al. (2004) observed such baroclinicwave beams with a near �1801 vertical phase shiftacross the beam in the upper 800m of the ocean atnearby locations over the continental slope, similarto the model observations and the theoreticalstudies. Therefore, we interpret our observationsas evidence for the presence of a deep tidal internalwave beam near 1075m. The fact that 55% of thekinetic energy in this depth interval (71% at 1075m)is phase-locked to the harmonic M2 tide suggeststhat the source of the internal wave beam is quitenearby.

Mooring BB 17 showed a different verticalstructure of the velocity variance and tidal phaselag. The total velocity variance had a maximum of148 cm2 s�2 at 725m, while the contribution of the

harmonic M2 tide had a variance maximum of63 cm2 s�2 only 50m deeper (Fig. 6(c)). A currentmeter, mounted on the mooring at 1448m, 50mabove the bottom, had an even higher M2 varianceof 83 cm2 s�2. This maximum was over two times theminimum M2 velocity variance observed near600m. The S2-related variance had a broad max-imum of 20 cm2 s�2 situated between 820 and 850m.The contribution of the N2 component was every-where below 3 cm2 s�2. The phase structure of thetides at mooring BB 17 also differed from that atBB 1. On the large scale the phase lag of both theM2 and S2 velocity components increased down-wards, indicative for an upward energy transport(Fig. 6(d)). The M2 phase shift between 970 and650m was found to be 961 and 521 for the east andnorth components, respectively. The M2 phase lag,measured with the current meter at 1448m (2001and 1341 for the east and north components),suggests that between 968 (bin3) and 1448m thephase lag gradient remained roughly similar to thegradient in the observational range, on average 231/100m. In the upper parts of the observational rangethe phase lag profile became more complicated, forthe phase lag of the east component of the M2 tideincreased from 650 to 600m, while in that depthrange the phase lag of the M2 north component wasconstant. This suggests a possible reversal of thevertical energy transport of the M2 internal tideabove 650m. Vertical phase shifts of over 901 in theobservational range and of over 1801 between 650and 1448m suggest that also at this mooring adominant part of the observed coherent M2 tide wasbaroclinic. We have tried to reconstruct theobserved vertical change of phase and tidal ampli-tude from a superposition of a vertically propagat-ing plane internal wave and a barotropic tide.Satisfactory results were not possible unless weassumed a significant vertical change in the ampli-tude of the internal tide. Therefore, we will interpretthe localization of the M2 tidal energy near 775m asa baroclinic wave beam, combined with a smallerbarotropic tide.

The residual of the high-frequency variance(41/3 cpd), not explained by coherent tides, hardlyvaried vertically for either mooring, being5273 cm2 s�2 and 6273 cm2 s�2 for mooringsBB 1 and BB 17, respectively. This residual maybe attributed to non-coherent tidal contributionsinduced by temporal changes in stratification andcurrents, as well as to near-inertial motions andinternal waves at non-tidal frequencies.


3.4. Incoherent M2 tide

The statistics, shown in Fig. 6, present the long-term mean vertical structure of the horizontalvelocity field and of its tidal harmonics. On shortertime scales the velocity variance, the orientation ofthe tidal ellipse, and the vertical structure of thetidal phase and amplitude showed variability.Examples of the temporal and vertical variation ofthe velocity component in the direction of the majorand the minor axis of the tidal ellipse during threesuccessive days are shown in Fig. 7. The character-istic direction of the observed tidal ellipse for the3-day period was defined as the direction in whichthe total variance of the velocity component for allADCP bins was maximized. In these examples thesemi-diurnal tide is dominant for the variability ofthe velocity structure. Between 24 and 27 November1995 the vertical structure of the tide at BB 1(Fig. 7a,b) resembled that of the mean semi-diurnaltide with most kinetic energy in the lower half of theobservational range, although over these 3 days thecore of high current variance descended 100 to 150in depth. The phase of the current increasedupwards, similar to the phase lag of the M2

harmonic tide, about 1201 for the motion in thedirection of the major axis of the tidal ellipse, and1801 in the direction of the minor axis. This phaseshift agrees with a vertical wavelength of 800 to1200m. The major axis was oriented in the directionof 601 true north, about perpendicular to thecontinental slope, which runs from northwest tosoutheast.

From 18 to 21 January 1996 the major axis of thetidal ellipse was also directed more or less perpen-dicular to the local continental slope (651). Thevelocity in that direction showed a monotonic andsmooth vertical phase change, the motion at 1075mdepth leading the motion at 725m by nearly 6 h(Fig. 7(c)). This phase change of nearly 1801 can beinterpreted as characteristic for vertically propagat-ing internal waves with a wavelength of about800m. Contrary to the long-term mean M2 motion,which showed a downward increase in amplitude, atleast two high velocity cores were present.The most energetic of these cores was located near800m. The velocity component in the direction ofthe minor axis of the tidal ellipse showed two high-velocity cores (near �780 and �1050m), movingmore or less in counter-phase, with most of thephase change restricted to the 900 and 950minterval (Fig. 7(d)).

At BB 17, from 13 to 16 January 1998, thevelocity component in the direction of the majoraxis of the tidal ellipse (1751), also nearly perpendi-cular to the local continental slope, showed thatduring these 3 days most of the semi-diurnal tidalmotion was concentrated between 625 and 800m(Fig. 7(e)). The motion near 650m was on average1 h ahead of the motion at 900m. The semi-diurnalmotion perpendicular to the major axis of the tidalellipse was also concentrated mainly in the upperhalf of the observational interval, with a secondaryhigh velocity core near �875m (Fig. 7(f)). Thisvelocity component had, similar to the cross-ellipsemotion at BB 1 in January 1996, two high velocitycores nearly in counter-phase.

From 4 to 7 April 1998 the vertical velocitystructure in the direction of the major axis of thetidal ellipse had more resemblance with the harmo-nic M2 tide, with a high-velocity core halfway withinthe observational range (Fig. 7(g)). Similar to theharmonic M2 tide, the motion at 625m led themotion near 975m by �2.5 h. The structure of themotion in the direction of the minor axis (Fig. 7(h))was more irregular, with one or two high-velocitycores at varying depths. In this direction the motionat the top of the observational range was also aheadof the motion at deeper levels.

The main variation in the velocity structure forshort periods like those shown in Fig. 7 was in themagnitude and depth of the high velocity cores, withslight changes (10–201) in the orientation of the tidalellipse. That variability showed a strong fortnightlyspring-neap tide cycle, although significant varia-bility was also observed at longer time scales.Averaged over a year that variation resulted in thesmooth vertical change of amplitudes and phases ofthe harmonic tidal components shown in Fig. 6. Theobserved strong changes in amplitude of the semi-diurnal tides, and the limited vertical extent of thehigh-velocity cores, presented above, may be inter-preted as internal wave beams that pass through theobservational range.

The spring-neap tide cycle as well as changes instratification and long term currents due to meso-scale eddies and seasonal and interannual variabilitymay change the generation and propagation ofinternal tides, leading to local variations in ampli-tude and phase lag of the semi-diurnal internal tides,as is illustrated in Fig. 7. The latter parts of theinternal tides are often referred to as incoherentinternal tides (van Haren, 2004). To estimate thelong-term variation of the incoherent M2 tide and

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LA

SA

SA

LA

65°

155°

60°

150°

-35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35

Velocity Component (cm/s)

Fig. 7. Characteristic examples of the temporal-vertical distribution for 3 successive days of the horizontal velocity components in the

directions of the long or major axis (LA) and the short or minor axis (SA) of the tidal ellipse. Panels (a an b) are from mooring BB 1 in

November 1995, panels c and d from mooring BB 1 in January 1996, (e and f) from mooring BB 17 in January 1998, and (g and h) from

mooring BB 17 in April 1998. The isolines are 5 cm s�1 apart, while the thick line is the isoline for zero velocity.

H.M. van Aken et al. / Deep-Sea Research I 54 (2007) 533–556544

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pth

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Fig. 7. (Continued)


its contribution to the total velocity variance wehave determined the amplitude and phase of the M2

tidal component for successive 14-day periods, with

a 7-day overlap (Figs. 8 and 9). The resultingamplitudes and phase lags contain contributionsfrom the coherent as well as the incoherent tidal

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Dep

th (

m)

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390 420 450 480 510

1100

1000

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Dep

th (

m)

a

b

c

Fig. 8. Temporal change of the vertical structure of (a) the sum of the variances of the east and north velocity components (cm2 s�2), (b)

the phase lag (degrees) of the east and (c) the phase lag (degrees) of the north component of the M2 tide, measured in mooring BB 1.

January 1, 1995 equals day number 1. The data for these plots were obtained from a harmonic analysis over successive 14-day periods with

a 7-day overlap.


waves. The velocity variance at mooring BB 1 showsthat the kinetic energy of the M2 internal waveschanged by a factor of 2–5 with a characteristic timescale of 45 days (Fig. 8(a)). Occasionally, the sign ofthe vertical gradient of the velocity variance wasinverted, e.g., during a high-energy phenomenon atthe upper levels around day 360. The typical timescale of changes of the phase lag was also about 45days (Fig. 8b,c). Inversions in the sign of the verticalgradient of the phase lag hardly occurred, and if so,they were smaller than 101. This indicates that thedirection of the vertical energy transport by the

internal tide stayed downwards throughout themooring period.

The variability of variance and phase lag of theM2 component at mooring BB 17 shows a similarvariability with a characteristic 40-day time scale(Fig. 9). Occasionally patches of high variance witha vertical scale of about 200m were observed in theobservational range (Fig. 9(a)), often accompaniedby abrupt changes in the phase lag (Fig. 9b,c).Similar to the mean phase lag change, the phase lagsof both the east and north components showed alarge-scale upward decrease, with only occasionally

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)

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Fig. 9. Similar to Fig. 8, for mooring BB 17. January 1, 1997 equals day number 1.


a local inversion (e.g. around day 340 for the eastcomponent). This reflects the fact that most of thetime the vertical component of the energy flux wasdirected upwards.

The characteristic 40–45-day time scale for thevariability of the M2 tide was probably associatedwith meso-scale eddy variability influencing thegeneration and propagation of internal tides in theBay of Biscay (van Haren, 2004). The meso-scaletemperature variability, measured with the ADCPs,can be considered to be characteristic for thebaroclinic eddies. The characteristic time scale ofthis variability (from the mean peak-to-peak dis-tance of the 7-day averaged signal) amounted to 47days, of the same order as the 40- to 45-day timescale for the tidal variability. For both moorings themean M2 velocity variance derived from successive

14-day periods was about 13 cm2 s�2 higherthan for the long term coherent M2 tide as shownin Fig. 6. The incoherent tides appear to contributean extra 10% to the total high-frequency kineticenergy, enhancing the total tidal contribution to64%.

3.5. Vertical component of the harmonic M2 tide

The spectra of the vertical motion (Fig. 4) showthat the semidiurnal M2 tide is the dominant tidalcontribution to the spectrum. A harmonic analysisof the vertical velocity was performed, whichconfirmed that the M2 tide was by far the mostimportant coherent tidal component, compared toother tidal frequencies. At mooring BB 1 the


coherent M2 tide explained about 3.3% of theobserved vertical motion (correlation ¼ 0.18) forthe lowest 160m, where the strongest verticalmotion was observed. At mooring BB 17, wherethe variance of W was distributed more evenly,about 13.5% of the variance of the vertical velocitywas explained by the coherent M2 tide(correlation ¼ 0.37). Incorporation of the incoher-ent tidal signal from the harmonic analysis ofsuccessive 14-day periods, as described for thehorizontal motion, increased these percentages to4.4% at BB 1 and 15.8% at BB 17. These low tidalM2 contributions to the vertical velocity, comparedto the M2 contribution to the horizontal velocityvariance (�65%), were probably connected with thestrong contributions to the variance of non-tidalinternal waves, as reflected in the near-whitecharacter over a large part of the continuumspectrum of W (Fig. 4).

At mooring BB 1 the M2 tide showed the largestcontributions at the lowest depths, similar to thehorizontal M2 tide (Fig. 10(a)). The phase lag of thevertical M2 tide increased upwards by 1901 over352m (Fig. 10(b)). The phase difference between1075 and 880m, where the variance nearly reachedits minimum, amounted to 911. Thus the verticaldistance of 195m agreed here with about a quarterof the vertical wavelength (3601 phase shift distance)of the internal tide, confirming the vertical wave-length estimate from the horizontal motion. In thisdepth interval the vertical motion was in phase withthe velocity in the direction of approximately 301true north, suggesting an opposite propagationdirection of the M2 internal wave beam towards2101. That is more or less perpendicular to thenearby continental slope, which can be consideredas the generation region for this internal wave beam.In the upper 100m of the observational range thepropagation direction of the internal tide wasrotated towards the west. The maximum verticaltidal excursion near 1075m, derived from theintegration over half a tidal period of a 14-dayharmonic M2 vertical tidal velocity in the lowestADCP bin, amounted to about 112m, over threetimes the mean vertical excursion due to theharmonic M2 tide, 34m. Such variations wererelated to the variations in the internal M2 tide,illustrated in Fig. 8.

The structure of the vertical tidal motion atmooring BB 17 differed strongly from that atmooring BB 1. The coherent vertical M2 tide hada definitely larger variance than found for mooring

BB 1 with less vertical change than the variance ofthe horizontal tidal velocity (Fig. 10(c)). The ratio ofthe mean variance of the vertical velocity to themean variance of the horizontal velocity was afactor of 10 larger for mooring BB 17 than formooring BB 1. Although the overall vertical shift inphase lag for the coherent vertical M2 tide had thesame sign as for the horizontal tidal velocity,decreasing upwards, it was in magnitude muchsmaller, the maximum phase lag difference amount-ing to 271 (Fig. 10(d)). This implies that at mooringBB 17 the vertical tidal motion was nearly in phaseover the whole observational range. Apparently theinternal tides, which were responsible for thevertical tidal motion in the thermocline, had arelatively large vertical scale near mooring BB 17, atleast of the order of the thickness of the thermo-cline. Heaving of the thermocline by the barotropictide over the continental slope may be a likelyinterpretation of these results, although mooring BB17 was relatively far (�20 km) from the 1000misobath. This heaving is part of the generation ofinternal tides (Lu et al., 2001). The vertical M2 tidalvelocity was in phase with the horizontal velocitycomponent in the north-westward direction at thedepth of the kinetic energy maximum, �750m. Thatdirection is more or less in the direction of the localslope (Fig. 1). However, for heaving of thethermocline by the barotropic tide over a slopeone can to first order expect the variance of thevertical velocity to increase with the square of thedepth, and to be in phase over the whole watercolumn. Most probably the deviation from aquadratic variance profile and the remaining ver-tical phase change are related to the co-existence ofthe baroclinic tide. The maximum vertical tidalexcursion, derived from the 14-day harmonic M2

vertical velocity varied from 129m near 970m, to94m above 750m, nearly double the mean verticalexcursion of 55m, derived from the long-termcoherent M2 tide.

3.6. Near-inertial motion

Near-inertial waves in the ocean are assumed tobe generated either locally as a response to localforcing, or remotely at lower latitudes as internalwaves with a frequency close to the inertialfrequency at the observation site (Fu, 1981). Thelocal generation of inertial waves may be due tochanges in the wind forcing at the sea surface(Ekman, 1905; Pollard and Millard, 1970) or due to


the geostrophic adjustment of currents in the oceaninterior (Rossby, 1938; Gill, 1982; van Aken et al.,2005).

0x100 2x10-2 4x10-2 6x10-2 8x10-2 1x10-1

Variance(W) (cm/s)2

0x100 2x10-2 4x10-2 6x10-2 8x10-2 1x10-1

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pth

(m

)

a b

BB 1

950

900

850

800

750

700

650

600

De

pth

(m

)

c d

BB 17

Fig. 10. Variance of the vertical velocity component measured with the

phase lag angle for BB 1 (b) and BB 17 (d) as function of depth.

The near-inertial motion cannot be determined bya harmonic method like that used for tides becauseof its strongly variable phase and amplitude due to a

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ADCP in mooring BB 1 (a) and mooring BB 17 (c), as well as the

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Variance of inertial motion (cm/s)2

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Variance of inertial motion (cm/s)2

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(m

)

a b

4 8 4 8

Fig. 11. Profiles of the summed variance of the east and north velocity component of the inertial band passed filtered signal for moorings

BB 01 (a) and BB 17 (b). The filter was so designed that only the angular frequencies from 0.9� f to 1.1� f went through.

240 300 360 420 480 540 600

Day Number

0

25

50

75

100

125

150

175

Velo

city v

ariance (

cm

/s)2

1 October 1 April

Fig. 12. The temporal development of the velocity variance in the

f710% near-inertial wave band, average over the uppermost

55m of the observational range in mooring BB 17. Day 1 is

January 1 1997. The straight lines show the positions of the

beginning and end of the winter half year.


more or less random forcing. However, by anintegration of the spectrum over a relatively narrowfrequency band around the frequency f, we wereable to determine the variance of the velocity relatedto the near-inertial motion. This variance appearedto depend on depth and slightly increased upwards(Fig. 11). The magnitude of the summed variance ofthe near-inertial motion (at frequencies f710%) formooring BB 1 and 17 amounted t, respectively, 7and 10 cm2 s�2. The mean contribution of the near-inertial waveband to the total high frequency(41/3 cpd) kinetic energy was about 7%. Thetemporal variation of the variance in the near-inertial frequency band can also be determined fromthe band passed filtered data (Fig. 12). The variancetime series shows that the variance at near-inertialfrequencies was quite intermittent, with only 2events above 50 cm2 s�2 in a record of 11 months.Most inertial velocity variance, 8.3 cm2 s�2, wasfound in the winter half year (ONDJFM) againstonly 3.1 cm2 s�2 in the summer half year (AMJJAS),suggesting wind forcing as a generation process fornear-inertial motion. However, in most cases (56%)strong energy near-inertial wave events had their

strongest expression in the lower half of theobservational range. This suggests that processesother than local wind forcing contribute to energy inthe inertial wave band. Geostrophic adjustment ofthe deep boundary current near 1000m depth overthe continental slope was identified as a dominantsource of energy for strong inertial events nearGoban Spur in the northwestern Bay of Biscay (vanAken et al., 2005). Non-traditional Coriolis effects

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BB 17

day 349 to 357

day 363 to 371

day 473 to 481

a b

360°/260 m

360°/300 m

360°/310 m

360°/380 m

Fig. 13. Variance profiles of the velocity components of the approximated inertial waves during three high inertial energy eventsat

mooring BB 17 where the strongest near-inertial events were observed (a), and the corresponding phase lag profile of the east component

of the waves (b).


are also considered to be responsible for theenhancement and near-bottm trapping of nearinertial waves (Maas, 2001; Gerkema and Shrira,2005). As can be expected for near-inertial motions,the horizontal velocity vector followed a near-circular path and rotated clockwise in time.

The strongest near-inertial frequency events werefound in the data from mooring BB 17. For threehigh-energy inertial events, observed at mooring BB17, we have tried to resolve the vertical amplitudeand phase structure by approximating over 7-dayperiods the inertial band-passed (f710%) velocitysignal u(t), v(t) by a least squares fit with asinusoidal curve defined as

½u; v� ¼ ½U ;V � cosðft� fu;vÞ, (3)

where U(z) and V(z) are the vertically varyingamplitudes of the inertial waves and fu(z), fv(z) thecorresponding phase lags. The events chosen were:15–22 December 1997 (day 349–356), 29 December1997–5 January 1998 (day 363–370) and 18–25 April1998 (day 473–480). The velocity variance related tothe inertial oscillation, (U2+V2)/2 (Fig. 13(a)),showed for day 349–356 a maximum just belowthe top of the observational range near 600m and a

slightly enhanced variance region between 800 and950m. For the other two periods the variancemaxima were well embedded within the observa-tional range. The typical mean half width of thesevariance peaks was estimated to be 120m. In thesevariance peaks the mean phase lag differencebetween east and north components of the velocityis 90.71 (71.11 standard error), consistent with acircular clockwise rotation of the velocity vector. Inall four variance peaks the phase lag of the near-inertial oscillations increased upwards, indicative ofa downward energy transport. The rate of phaseincrease varied from 3601/380m to 3601/260m. Amean vertical wavelength (3601 phase shift dis-tance), estimated from these wave numbers,amounted to 380m. The approximate mean thick-ness of the variance peaks, 240m (twice the halfwidth), appeared to be definitely less than thiswavelength.

3.7. Vertical current shear

Internal and near-inertial waves are importantphenomena for the transport of mechanical energythrough the ocean. That energy ultimately can be

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a b

0 4 8 10

Frequency (cpd)

Po

we

r sp

ectr

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we

r sp

ectr

um

s-2

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1x10-6 1x10-6

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Fig. 14. Summed power spectra of the east and north components of the velocity shear over a 16m vertical interval, averaged over all

available bin pairs, for moorings BB 1 (a) and BB 17 (b). The insets show the complete spectra, the main figure the parts of the spectra with

pronounced peaks. The identified spectral peaks at tidal and inertial frequencies, their higher harmonics and linear combinations are

indicated.


made available for mixing in the interior of thestratified ocean by the shear production of turbulentkinetic energy. For the analysis of the phenomenaresponsible for the generation of turbulence, wehave derived the vertical current shear fromneighbouring 16-m ADCP bins and calculated theindividual shear spectra, summed these over the eastand north components, and averaged them for allthe available depth intervals (Fig. 14). For mooringBB 17 velocity values every 16m were obtained byaveraging over two neighbouring 8-m ADCP bins.Determining the shear over a vertical distance of16m relates the shear values to phenomena with avertical wavelength larger than twice this distance.The shear, related to smaller scale phenomena ispartly missed by this method, and partly aliased tolarger scales. For both moorings the variance of thevelocity difference per component over 16m waswell above the noise level, derived from the errorvelocity, up to a factor of 60 for mooring BB 17.

The shear spectra show a number of peaks thatcan be identified with specific combinations of tidaland inertial frequencies (Fig. 14). At frequenciesabove 20 cpd the spectra level off to nearly constantvalues. For mooring BB 17 this level is about twice

the high-frequency spectrum for mooring BB 1. Forboth moorings the level of the background spectralcontinuum appeared to decrease monotonicallyupwards over the observational range of nearly400m, with a factor 1.4 and 1.9 for BB 1 and 17,respectively.

The main spectral peaks for both moorings werefound at the frequencies of the semi-diurnal tides D2

(both M2 and S2 peaks) and at the inertial frequencyf. The shear variance in the inertial-semidiurnalfrequency band (1 to 2.5 cpd) at mooring BB 1 had aslight maximum near 900m, about 50% higher thannear 700 and 1100m. At mooring BB 17 the shearvariance in that frequency band decreased mono-tonically upwards with about 25% over theobservational range. Spectral peaks were also foundat the first higher harmonic of the semi-diurnalfrequencies D4. All other peaks involve sums anddifferences of tidal frequencies with f, i.e. D2�f

outside the internal wave band and D4�f, D2+f,D4+f and D6+f in the internal wave band. Thepeaks at frequencies involving f are larger in theshear spectrum than in the velocity spectrum. This isespecially clear in the spectrum for BB 17(Fig. 14(b)), where the f, D2�f and D2+f peaks


form three of the five highest spectral peaks. Theequal magnitudes of the D2�f and D2+f shearpeaks suggest support for the hypothesis of so-called ‘‘fine-structure contamination’’ as cause forthese peaks (Alford, 2001). Such contamination iscaused by vertical advection by the main internaltide of the near-inertial current shear, which has asmall vertical length scale. The prominence of thef-related shear suggests that the velocity structure atthe inertial frequency has a relatively small verticalscale compared with the tidal frequencies. One candefine such a characteristic vertical length L as

Lz ¼ 2pZ

B

Pvdf

�ZB

Pshdf

� �1=2

, (4)

where the power spectra of velocity, Pv and of shear,Psh, are integrated over an appropriate frequencyband B. For plane waves this definition agrees withthe vertical wavelength.

The estimates of Lz for the different frequencybands (Table 2) show that the characteristic lengthscale is largest for the semi-diurnal tides,O(L) ¼ 900m, followed by the quarter-diurnalharmonics, O(L) ¼ 430m. The near-inertial waveshave a smaller vertical length scale than the internaltides, O(Lf) ¼ 310m, while for the frequency bandsformed by combinations of the inertial and tidalfrequencies the smallest vertical length scales arefound, O(L) ¼ 180m. This result indicates thatwhile the internal tides are very dominant for thevariance of the horizontal velocity, the near-inertialwave band may be more important for the genera-tion of shear-induced turbulence.

A criterion for the onset of turbulence thatderives its energy from the velocity shear is based

Table 2

Vertical wavelength Lz (3601 vertical phase shift), according to

(4), determined for different frequency bands

Mooring BB 1 BB 17

Frequency Lz Lz

Band (m) (m)

D2– f 194 194

f 276 339

D2 791 1011

D4�f 200

D2+f 200 220

D4 433 427

D4+f 170 163

D6+f 144 132

on the gradient Richardson number Rig, defined as:

Rig ¼N2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð@u=@zÞ2 þ ð@v=@zÞ2q . (5)

If Rig descends below a critical value (somewherebetween 0.25 and 1.0; Turner, 1973) turbulentmixing will maintain a vertical mass transport. Inwaves with circularly rotating shear vectors, like thenear-inertial waves, the high shear regime iscontinuously present as has been observed in thestratified North Sea (van Haren, 2000). We havedetermined the Richardson numbers from the shearmeasured with the ADCP over a vertical distanceDz ¼ 16m, averaged over 15min for mooring BB 1and over 30min for mooring BB 17. For mooringBB 1 Rig is less than 1 for 41% of the time, and lessthan 0.5 for 17% of the time, while for mooring BB17 these percentages amounts to 39% and 15%,respectively. For both moorings the percentage oftime that Rigo1 decreased smoothly upwards overthe observational range by about 10%. Althoughthe long-term average percentage of the time thatlow Richardson numbers occur varies smoothlywith depth, its distribution in space and time wasstrongly intermittent at shorter time scales, similarto the occurrence of strong near-inertial events. Andit is these events that generate high shear and lowRichardson numbers because of their relativelysmall vertical length scale. This frequency rangebetween f and D4+f provides 57% of the shearvariance for mooring BB 1 and 49% for mooringBB 17. Apparently the shear related to the near-inertial waves, and their combinations with tidalfrequencies, are a main factor for the large fractionof the time when the low Richardson number allowsturbulent mixing.

The near-inertial velocity variance shown inFig. 12 was positively correlated with the timefraction that Rig was below 0.5 in the same depthinterval (R ¼ 0.58). Many strong near-inertialevents could be linked to low Rig events. As anexample Fig. 15(a) shows the temporally coincidentoccurrence of a strong near-inertial event in theupper 55m of the observational range in December1997. The vertical structure of strong near-inertialevents and of low Rig events were repeatedly alsoquite similar. This is illustrated for the event inApril 1998 (Fig. 15(b)). This result confirms therelative importance of the near-inertial wave bandfor turbulent mixing in the ocean.

ARTICLE IN PRESS

0 20 40 60


Variance(U

)+V

ariance(V

) (c

m/s

)2

950

900

850

800

750

700

650

600

Depth

(m

)

0.1 0.15 0.2 0.25

Fraction Rig<0.5

348 352 356 360 364

Day Number (1997)

0

40

80

120

160

200

0.1

0.15

0.2

0.25

Fra

ction R

i g<

0.5

day 473 to 481upper 55 m

Fig. 15. Examples of the coincidence in time (a) and space (b) of strong near-inertial events (black dots) and events with a large time

fraction with low Richardson numbers (Rigo0.5, open symbols). The near-inertial event in (a) is the strongest event shown in Fig. 12. The

event in (b) is also shown in Fig. 13. The fraction of time with low Richardson number is a 7-day running mean value.


4. Discussion

Spectral analysis of the ADCP data showed thattidal motion was a dominant feature in thepermanent thermocline over the continental slopeof the Bay of Biscay. The harmonic analysis of theADCP data revealed that over 50% of the high-frequency (41/3 cpd) kinetic energy can be attrib-uted to coherent semi-diurnal tides with yearlongconstant phase lag and amplitude. Short-termvelocity records, however, showed a considerablevariation of the semi-diurnal motion, which wasattributed to incoherent tides with time-varyingamplitude and phase lag. When incoherent tidalcontributions were incorporated, about 2/3 of thehigh-frequency kinetic energy could be explained.The horizontal harmonic semi-diurnal tidal motionin the thermocline appeared to have a definitivevertical structure in amplitude and phase. Formooring BB 1 the highest M2 amplitude was foundnear 1075m depth, for mooring BB 17 near 750m.The phase lag of the tidal signal varied significantlyin the vertical, up to 1061/350m and 1551/350m for,respectively, the M2 east and north velocitycomponents of mooring BB 1. The observed verticalchange of phase lag and velocity variance indicated

that the largest part of the tidal motion had thecharacter of propagating internal waves. Therestricted vertical range of the variance maxima ofthe horizontal velocity suggested that the internaltidal motion was largely organized in wave beams oflimited vertical extent, about 400m, probablyemerging from a nearby generation area on thecontinental slope. This interpretation agrees withearlier observations and modelling results fornearby positions in the Bay of Biscay (Pingree andNew, 1991; Lam et al., 2004; Gerkema et al., 2004).

Because of the predominantly white noise char-acter of the spectral continuum of the verticalvelocity spectrum, the tidal peaks in this spectrumwere less dominant than the tidal peaks in thespectrum of the horizontal velocity. Nevertheless,the semi-diurnal and higher harmonics were clearlyrecognizable in the vertical velocity spectra. Har-monic analysis of the M2 contribution to the verticalmotion showed a vertical distribution of the velocityvariance and phase lag in general agreement withthe distribution of the velocity variance for thehorizontal motion. Including incoherent tides, theM2 band contributed on the order of 5 to 15% tothe total variance of the vertical velocity. Thisagreed with a maximum vertical tidal excursion in


the thermocline of about 110 to 130m. Unpublisheddeep (1200m) CTD yoyo time series from the Bayof Biscay over a tidal period confirm the existence ofsuch large internal tidal waves. Most of theremaining vertical velocity variance was connectedwith the spectral continuum. Probably only alimited part of this continuum was related toinstrumental noise, given the agreement of thetemperature derived spectrum with the ADCPderived spectrum (Fig. 5) and the agreement of thespectral continuum with the spectral form observedby Cairns (1975) and Lien et al. (2005).

Assuming that the M2 velocity signal at mooringBB 1 was purely baroclinic, the observed verticalchange of the phase shift was equivalent to a verticalwavelength of about 800m, about twice theestimated vertical scale of the tidal wave beam.This agreed in magnitude with the vertical wave-length derived from the harmonic vertical motion.The vertical gradient of the observed phase lagindicated that at mooring BB 1 the M2 tidal energywas transported downwards. From the correlationbetween horizontal and vertical motion it wasderived that the horizontal propagation directionof the internal tidal wave beam was 2101 true north,away from a probable generation region over thenearby upper continental slope.

For mooring BB 17 an upward energy transportwas derived from the vertical change of the phaselag of the horizontal M2 tidal motion. However, thestrong semi-diurnal vertical motion at differentdepths was nearly in phase. This was probablycaused by heaving of the thermocline water due totidal motion northwestward over the complicatedsloping topography near the Meriadzek Plateau.Apparently Mooring BB 17 was located in a regionwhere internal tides were generated or reflected.

The presence of near-inertial waves could berecognized from a spectral peak at the Coriolisfrequency. A seasonal cycle of near-inertial kineticenergy could be recognized, with most energy in thewinter period. This may be due to stronger windforcing of inertial motion in winter. However, thevertical distribution of strong near-inertial eventssuggested that a considerable part of the near-inertial energy entered the observational rangelaterally or from below, possibly forced by geos-trophic adjustment events. The analysis of indivi-dual near-inertial wave events showed acharacteristic vertical structure with relativelystrong current shear. During these events the wavesalso seemed to be organized in wave packets with a

vertical extent of about 240m. The rate of change ofthe phase suggested a mean vertical wavelength ofabout 380m. So the width of the near-inertial wavebeam appears to be smaller than the verticalwavelength, a result that compares well with ourfindings for the semi-diurnal internal tides.

The vertical velocity shear in the near-inertialfrequency band was more important than that in thetidal bands because of the small vertical extent ofthe inertial wave packets. Many strong near-inertialwave events appeared to be coincident in space andtime with low Rig events. The resulting annual meangradient Richardson number in the main thermo-cline turned out to be close to the critical value. Thissuggests that over the continental slope of the Bayof Biscay the circumstances are favourable for anearly permanent maintenance of turbulent mixing,driven by the energy supplied by the inertia-gravitywave band, especially at near-inertial frequencies. Inorder to improve the local parameterization ofmixing in ocean general circulation models, weshould improve the understanding of the roles anddistribution of both internal tides and near-inertialwaves and their generation, transmission andbreaking.

Acknowledgements

This research was funded by the board for Earthand Life Sciences (ALW) of the foundation forNetherlands Scientific Research (NWO) as part ofthe Bay of Biscay Boundary project. The threeanonymous reviewers, who made comments on theinitial manuscript, are acknowledged for theirefforts, which led to numerous improvements.

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