Research papers

Spatial variability of internal waves in an open bay with a narrowsteep shelf in the Pacific off NW Mexico

A. Filonov a, M.F. Lavín b,1, L.B. Ladah c,n, I. Tereshchenko a

a Physics Department, University of Guadalajara, Blvd. Marcelino García Barragán #1421, Esq. Calzada Olímpica, C.P. 44430 Guadalajara, Mexicob Departamento de Oceanografía Física, CICESE, Carretera Ensenada-Tijuana 3918, Zona Playitas, 22860 Ensenada, Baja California, Mexicoc Department of Biological Oceanography, CICESE, Carretera Ensenada-Tijuana 3918, Zona Playitas, 22860 Ensenada, Baja California, Mexico

a r t i c l e i n f o

Article history:Received 26 February 2013Received in revised form16 January 2014Accepted 22 January 2014Available online 11 February 2014

Keywords:Internal tidesSemidiurnal tidal forcingNarrow shelfDiurnal wind forcingNW Mexico

a b s t r a c t

Small scale spatial patterns (o10 km) in nearshore internal wave fields are rarely reported on, yet canhave a large impact on nearshore mixing and productivity. In this study, the spatial pattern of internalwave characteristics were explored in Todos Santos Bay, Baja California (Mexico), using time series oftemperature and currents from moored and towed thermistor chains and acoustic profiling currentmeters, as well as cross-shore transects with a towed undulating CTD system. Spectra of temperature andcurrents showed significant spatial variability within the bay, with the northern sector dominated by theinternal tidally-forced semidiurnal signal, and the southern sector dominated by wind-forced, sub-inertial, baroclinic, diurnal fluctuations, which decreased with distance from shore. Semidiurnal internaltidal waves were generated by the barotropic tide at various sites on the continental slope to the west ofthe bay. They traveled toward the NE and reached the observation site in the northern part of the bay,after bouncing once or twice off the surface and the bottom. Despite the narrowness of the shelf, thesemidiurnal internal tides at this site presented a first-mode structure, although not completely formedat times. On average, the semidiurnal internal waves had a �9 kmwavelength, traveled in the form of anarc, and propagated with a phase velocity of �20 cm/s. When they reached shallow waters near thecoast, they disintegrated rapidly into groups of short, nonlinear internal waves, with 15–20 mamplitudes, 5–20 min periods, and 50–200 m wavelengths. The spatial patterns found in this study aremost likely due to variability in distance from generation sites, complex bottom topography, and smallscale (o10 km) spatial variability in meteorological conditions such as winds.

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1. Introduction

The internal tide plays an important role in continental shelfdynamics. It can be generated remotely or locally, at sites on theshelf edge or continental slope, from which it propagates acrossthe shelf. Once on the shelf, the energy of the internal tide isdispersed in various ways, either through the creation of otherinternal waves, and/or through mixing. The generation of internaltides depends on bottom inclination, tan α¼ ðdh=dxÞ, and ondensity stratification, as given by the Brünt–Väisäla frequency N(z) (Baines, 1982; Graig, 1987; Holloway, 1987), combined with aforcing factor, such as winds or tides.

Theoretically, freely propagating internal waves are restrictedto frequencies, ω, where f oωomaxN(z), and f is the inertial

frequency (LeBlond and Mysak, 1978). Therefore, for semidiurnalwaves, the critical latitude in either hemisphere is about 751, andfor diurnal waves the critical latitude is about 301. However, thereare many cases of subinertial internal waves occurring beyond thecritical latitude (Cudaback and McPhee-Shaw, 2009; Beckenbachand Terrill, 2008; Wallace et al., 2008; Albrecht et al., 2006, vanHaren et al., 2002). Subinertial internal waves can occur if theypropagate along a bathymetric barrier (such as the coastline or asloping bottom), confined to within approximately one internalRossby radius of the barrier (Emery and Thomson, 1997), and alsocan occur if they are generated locally, such as by local winds(Wallace et al., 2008).

Internal tides, commonly regarded as mode one internal waves,may be considered in a continuously stratified fluid as composed ofthe sum of two inclined waves traveling in the same horizontaldirection but with opposed vertical directions. The resulting internaltide is an inclined internal wave where the angle at which the crestsand troughs are inclined, θðzÞ ¼ arctgfðω2� f 2Þ=ðNðzÞ2�ω2Þg1=2, is

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Continental Shelf Research

http://dx.doi.org/10.1016/j.csr.2014.01.0150278-4343 & 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: lladah@cicese.mx (L.B. Ladah).1 Deceased.

Continental Shelf Research 78 (2014) 1–15

determined by the buoyancy frequency NðzÞ, the inertial frequency f,and the internal wave frequency ω, which is 0.081 cycle/h forsemidiurnal internal waves. The most effective transmission of energyfrom the barotropic to the baroclinic tide occurs at a critical value ofthe bottom slope angle (α) where α=θ� 1, when the slope angle ofthe shelf approximates the beam slope angle of the internal wave. Ifα4θ (supercritical), the energy travels offshore, and if αoθ (sub-critical), energy is reflected towards the continental shelf (Baines,1982; Miropolsky, 2001).

Because the generation, propagation and disintegration ofinternal tides depend on bottom topography and stratification, asdescribed above, internal tides can show significant spatial varia-tion. The velocity and vertical displacement of internal tides on theshelf can vary greatly over short distances (Rayson et al., 2012).Hydrographic conditions, as well, can vary on spatial scales of 5–30 km, also related with shelf morphology, with greater spatialvariability on steep narrow shelves than on wider ones (DiMarcoet al., 2010). It has been shown that spatial complexity of internalwaves increases in areas with multiple generation sites, and wherecomplex topography is present (Alford et al., 2006).

Because of their strong modulation by bottom slope, propaga-tion and disintegration of internal waves also depend greatly onthe width of the continental shelf. On wide continental shelves, aninternal oscillating bore forms as a result of the balance betweennonlinearity and dispersion. Under these conditions, internal tidalwaves usually have a first mode structure. With time, the oscilla-tions turn into trains of solitary waves, and, as the water becomesshallower, these waves are destroyed (Liu, 1988; New and Pingree,1990; Konyaev and Sabinin, 1992).

On very narrow shelves, the internal tide is an inclined wave,which propagates upward and onshore. Despite its reflection from

the bottom and from the surface, it can remain inclined and becompletely destroyed over the course of one wavelength (Konyaevand Sabinin, 1992). Due to the wave's reflection from the inclinedbottom, the horizontal and vertical wave numbers increase whenthe wave approaches shallow waters; howmuch the wave numberincreases depends on the inclination of the bottom. The waveundergoes nonlinear transformations and overturns, forming sev-eral homogeneous temperature layers up to tens of meters thick(Filonov and Konyaev, 2003). The most intense disturbances areoften observed near the bottom, where the slope angle approachesthe critical value. There are very few data on internal tides undernarrow-shelf conditions (Holloway, 1985; Rosenfeld, 1990; Filonovand Konyaev, 2006; Filonov, 2011).

The aim of this study was to provide a first description of thecharacteristics of internal waves in Todos Santos Bay (Fig. 1), BajaCalifornia (Mexico), from direct measurements using moored andtowed instruments. We expected to find strong spatial variabilityin the internal wave signal and an incomplete formation of themodal structure prior to disintegration, due to the complexoffshore topography and the narrow, steep nature of the slope.We identified several likely generation sites, described the pat-terns of propagation, and determined the forcing factors in thedifferent parts of the bay.

2. Methodology

2.1. Study area

Todos Santos Bay is located on the west coast of the BajaCalifornia peninsula, 100 km south of the USA–Mexico border

Fig. 1. Bathymetric map of Todos Santos Bay, Baja California, Mexico. The mooring (circles) and current meter (crosses) positions are shown. Dotted blue lines show thelocation of vertical transects of temperature and salinity taken with an undulating CTD (SBE-19plus). Continuous red lines indicate transects taken with a pair of towedchains of thermographs and an ADCP. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

A. Filonov et al. / Continental Shelf Research 78 (2014) 1–152

(Fig. 1). Its western region is limited by two offshore islands, with anarrow (8 km), deep (up to 400 m) passage between the southisland and the mainland. North and West of the bay, naturalborders are formed by a sharp continental slope, where bottomslope angles reach 4–61. To the North, Todos Santos Bay isbordered by Salsipuedes Bay, which has an abrupt continentalslope forming part of the North American trench, with depths to5 km.

Most of Todos Santos Bay has a depth between 25 and 30 m,and the shelf is very narrow (Fig. 1). About 5–10 km from shore, ata depth of 40–50 m, the bottom inclination changes. Furtheroffshore, the continental slope angle increases to 3–51. The 200-m isobath is located less than 3–5 km from the shelf edge.

The barotropic tide in Todos Santos Bay shows a mixedcharacter with prevalence of the semidiurnal harmonic. Its max-imum amplitude is 71 m. Barotropic tidal currents rotate clock-wise around the ellipse, which tend to be elongated along theisobaths and to be particularly strong in the northern and westernparts of the bay (see Fig. 8 of Sánchez et al., 2009).

A ROMS model of the circulation in Todos Santos Bay (Mateoset al., 2008), forced with the California Current System and bysynoptic winds, suggests that summer circulation within the bayoscillates between two patterns. For a few days it is anticyclonicover the entire bay producing a large eddy, which then evolvesand splits into two counter-rotating eddies with the anticyclonic(cyclonic) one limited to the northern (southern) side of the bay.The transition between these two situations takes two to threedays and each phase lasts three to four days.

2.2. Measurements

The data used here were collected during the FLOO-07 (FluxesLinking the Offshore with the Onshore, 2007) experiment, whichtook place from August 5 to September 18, 2007 (Ladah et al.,2012). The thermohaline structure was observed using a towedautonomous CTD. At full vessel speed, about 20 km/h, the CTDskimmed the surface. When the boat stopped for the cast, theprofiler fell freely at a speed of 2 m/s measuring temperature,conductivity and pressure with a 0.5-s sampling interval. Whenresuming full vessel speed, the CTD surfaced again (Filonov et al.,1996).

Towed CTD onshore–offshore transects (performed twice onsome days) were made on August 6, 7, 15 and 16, 2007 (Fig. 1,dotted blue lines, Table 1, Table 2). Transects never took more thantwo hours, thereby almost representing the same phase of theinternal tide. Transects resulted in a total of 147 vertical profiles oftemperature and salinity with a spacing of 0.5–2.5 km, and amaximum depth of 380 m.

The detailed structure of groups of short-period internal waveswas observed with transects taken from August 8 to 11 (Fig. 1, red

lines), using a vessel-mounted ADCP and a towed antenna con-sisting of two vertical chains of thermographs. The first towedchain consisted of 6 thermographs (SBE-19, SBE-37 and SBE-39),which were evenly distributed from 4 m below the surface to adepth of 22 m, with a sampling interval of 5 s. The second towedchain was 150 m behind, fixed on a special float, and containedthree SBE-39 thermographs. The depth of the lower end of eachchain, monitored by a pressure sensor, was 2270.5 m. The towspeed (about 8 km/h) allowed the recording of temperature every11 m horizontally along each transect.

Ninety-four HOBO (Onset Computer Corp., Mass., USA) tem-perature recorders were installed at different depths on twentymoorings. Most thermographs had a sampling rate of 5 min (onlymoorings L1 and L5 had 1 min sampling intervals) and recordedfrom August 6 to 18, 2007. A 300 kHz RDI ADCP and a 500 kHzSonTek ADP were deployed on the seafloor (depth about 30 m);the former in the south of the bay near mooring 8 and the latternear mooring L5 in the northern part. They had a 1 min samplingrate in 1 m vertical bins. They were deployed on August 6 andrecovered on August 18 and 24, 2007, respectively.

Sea-level and temperature data were collected every minute for195 days (May 26–November 11) in 2005, using an SBE-39temperature-depth recorder (accuracy 0.002 1C for temperatureand 0.1 m for depth) deployed at the L1 mooring (at a depth of10 m) offshore of San Miguel, and were also analyzed in this study.These time series will be referred to as the “L12005 series”.

An Aanderaa meteorological station was installed 10 m abovesea level at San Miguel; it measured wind speed and direction, airtemperature, atmospheric pressure and relative humidity at 5 minintervals. Hourly wind data and hourly tidal information wereavailable from a meteorological station and a tide gauge installedat El Sauzal port, located 2 km south of the San Miguel study area.

2.3. Data analysis methods

2.3.1. Time series analysisCurrent meter data were analyzed for internal wave compo-

nents after removing the barotropic component of the flow bysubtracting the vertical mean flow for each sample. To analyze thevertical and temporal structure of currents in the internal tidal

Table 1Information about the rapid oceanographic transects which were made in the baysof Todos Santos and Salsipuedes 6–16 August, 2007.

Transect Date Runtime, h Length, km Casts Maximum depth, m

1A 6.08 3.03 34 32 3701B 6.08 2.68 34 29 3702A 7.08 1.32 18 7 2502B 7.08 1.25 18 7 2503A 15.08 5.57 59 24 3803B 15.08 6.08 59 23 3804 16.08 2.97 32 14 3805 16.08 2.37 24 11 380

Total 25.27 278 147

Letters A and B denote transects from the shore and to shore, respectively.

Table 2Information about thermographs HOBO installed on moorings in Todos Santos bay.

Mooring Maximumdepth, m

Number ofdevices

Horizons, m Number ofsamples

1 48 5 12, 17, 22, 27, 32 30972 82 5 12, 17, 22, 27, 32 26413 40 3 12, 17, 22 26174 20 3 10, 13, 16 25905 25 3 8, 10, 13, 16 30886 25 5 10, 13, 16, 19, 21 30888 28 8 9, 11, 13, 15, 17, 19, 21, 25 28909 25 5 9, 11, 13, 15, 17 3124

10 15 4 8, 10, 12, 14 310311 17 4 8, 10, 12, 14 292012 20 5 9, 11, 13, 15, 17 291113 23 3 13, 15, 17 288714 8 2 9, 10 309415 8 1 9 292616 8 2 9, 10 312417 26 3 9, 13, 16 313318 25 3 8, 10, 17 311519 14 3 8, 10, 13 3097L1 16 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

11, 12, 13, 14, 1518,636

L5 30 11 0, 2, 5, 8, 11, 14, 17, 20,23,26, 29

18,621

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waves, the ADP and ADCP time series were smoothed with aHamming filter of 1-h width.

Rotational spectral analysis of the current meter data was usedto calculate the direction of semidiurnal internal waves, using theformulae given in Appendix A. For the direction of wave propaga-tion (up to71801), the orientation aligns with the principal axis ofthe internal tidal ellipse. The ellipse stability function, E, was alsocalculated, with the formula presented also in Appendix A. Theellipse rotation was considered stable if the value of E at thefrequency of the internal tide was higher than the correspondingconfidence interval (Gonella, 1972; Emery and Thomson, 1997).

To estimate the parameters of semidiurnal internal wavesspatial spectral analysis was used (Appendix B) for clusters ofmoorings with thermograph chains and an acoustic profiler. The“South Cluster” consisted of moorings 8, 9, 12 and 17, whichformed a symmetrical spatial antenna with sides 2.5–3 km longwith the ADCP near mooring 8 (Fig. 1). The “North Cluster”consisted of moorings 5, L1 and L5, with the ADP near mooringL5 (Fig. 1).

Linear and autoregressive spectrum estimates of semidiurnalinternal waves were calculated from the temperature time seriesat 9 m depth for the North Cluster and at 8 m depth for the SouthCluster, where the thermocline was located. Initially the spectralmatrices (see Appendix B), gm;jðωÞ, were calculated for thesedepths from an 8-day time series using a time shift of 6 h betweenthem (necessary for spectral matrix inversion hm;jðωÞ ¼ gm;jðωÞ�1.See Konyaev, 1990, p. 142). Matrices were then averaged and usedto calculate the spatial spectra.

We calculated the theoretical parameters of the linear semi-diurnal internal waves at the two clusters using the mean profilesof the buoyancy frequency and numerically solving the waveequation: Wzzþk2hððNðzÞ2�ω2Þ=ðω2� f 2ÞÞW ¼ 0, with the bound-ary conditions W (�H)¼0, W (0)¼0 (LeBlond and Mysak, 1978;Miropolsky, 2001). Here, ω is the internal wave frequency, f is theCoriolis parameter, kh is the horizontal wave number, W is theeigenfunction (normal mode) and N is the buoyancy frequency.

To evaluate the possible bias of local inertial frequency in thestudy area, we estimated circulation and vorticity at 0, 5, 10, 15 and20 m depth. The calculations were made using the August outputsof the ROMS numerical model of Mateos et al. (2008).

2.3.2. Stratification and beam analysisBackground stratification was calculated from repeated mea-

surements of temperature and salinity available from the undulat-ing CTD and towed sensor antennas. We did not use the typicalaveraged profiles of buoyancy oNðzÞ4 , as this leads to verticalspreading of the thermocline, resulting in a much thicker thermo-cline than exists in reality. This is due to the influence of internalwaves, where the individual density profiles may be very differentfrom the background profile. Therefore, to obtain undistortedprofiles of buoyancy frequency, we used a procedure proposedby Bondur et al. (2009) using sets of separate vertical soundingsmade during different phases of the internal tide. For the i-thdensity profile of the conditional density array stðzÞ, the verticalgradient ∂stiðzÞ=∂z was calculated. The gradients were then aver-aged for each density value for all profiles o∂stðzÞ=∂z4 and theaveraged gradients were used to calculate the background buoy-ancy frequency profile Nf ðzÞ.

This method of estimating the background stratification can bedistorted due to the horizontal heterogeneity of the density field.It can lead to inflated values of the buoyancy frequency in theupper mixed layer, where it is constant with depth, and canchange from sounding to sounding. During our measurements,the thermocline was near the surface, so that these distortionswere not very large, although the background profiles and the

average buoyancy frequency showed different curvatures in thepycnocline.

To estimate the generating properties of the shelf and con-tinental slope for the study area, calculations of the bottominclination angle α, and angle of beam slope, θ, were made usingthe bathymetry matrix and the background buoyancy frequencyprofile. Because our measurements of stratification were made todepths of 380 m, we used historical stratification data taken inAugust for layers down to 1200 m.

2.3.3. Potential energy of the internal tideUsing data from field measurements, we assessed the internal

wave's total potential energy in August 2007, in the northeasternpart of the bay, using the relation (Phillips, 1977; Fu and Holt,1984): PE¼ 1=2ρ0LX

R 0H N2ðzÞζ2ðzÞdz, where ρ0 is the average water

density; L is the width of the wave front of the semidiurnalinternal waves on the line from San Miguel towards the northerntip of Todos Santos Islands (see Fig. 1); X¼1 m is the widthperpendicular to the wave front; N2ðzÞ is the vertical distributionof buoyancy frequency averaged during the observations; ζðz; tÞ isthe displacement field near mooring L5. To simplify the calcula-tions, we assumed that the wave energy did not change along thefront, the width of which was taken to be 10 km, and used theaverage depth of this line, which was 30 m.

Because temperature profiles were smoother than the salinityprofiles, they were used to determine the isopycnal verticaldisplacements. The conversions from measured temperature pro-files at the mooring to isopycnal vertical displacements (which aremainly determined by internal waves) require a number ofintermediate operations (Filonov and Konyaev, 2003; Filonov,2011). The original time series of temperature fluctuations mea-sured at 11 levels on mooring L5 were interpolated onto a single1 m depth grid. We then calculated hourly profiles, TjðzÞ, andthe average profile for the entire period of observation,T ðzÞ ¼ ð1=JÞ∑jT jðzÞ; where j¼1:J is the profile index (number).The mean profile T ðzÞ describes the unperturbed position ofisopycnal water layers. The inverse functions zj(T) and zðT Þ existif the hourly profiles Tj(z) and the mean profile are monotonicfunctions, so the vertical displacements of water layers from theunperturbed position are defined as Δhj(T)¼zðT Þ�zj(T). Next,temperature was replaced with depth by means of the meanprofile zðT Þ to give the relationship between displacement anddepthΔhj(z). Hence, the vertical displacements of water layers canbe defined as ζj(z)¼z�Δhj(z).

3. Results

3.1. Circulation

There were differences in background currents from the dailymean flow at different depths (not shown), measured by the ADCPand ADP, in the southern and northern parts of the bay. Inagreement with Mateos et al. (2008), amplification and attenua-tion of flows with a three-day period were evident. At the SouthCluster (ADCP), currents were �12 cm/s and mostly westward inthe top 10 m during the first three days of observation. On theother days, strong currents covered the entire water column. Atthe North Cluster (ADP near San Miguel), in the 10–20 m layer,flows had a northwest direction at 8–12 cm/s. Speed decreased inthe top and bottom layers. These observations did not meet thecirculation model described by Mateos et al. (2008) (theirFigs. 2 and 3).

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3.2. Stratification

Salinity in Todos Santos Bay changed very little with depth;therefore the vertical profiles of water density and buoyancyfrequency were principally controlled by temperature.

The maximum background buoyancy frequency was 15 cycles/h, at a depth of 11 m, and the thickness of the pycnocline wasabout 50 m (Fig. 2a). As mentioned in Methods, the verticaldistribution of N(z) obtained by the average density profile (dotted

line, insert, Fig. 2b) underestimated the maximum of the back-ground buoyancy frequency.

3.3. Wind, barotropic tides and baroclinic tides

Time series of tides and winds during the experiment as well asthe daily average wind velocity for the 15 days of observation areshown in Fig. 3. A consistent breeze is apparent, with a maximum

Fig. 2. Background buoyancy frequency profiles in Todos Santos Bay, according to measurements taken in August 2007. (a) The set of st ðzÞ profiles in a layer down to 385 mand (inset) down to 50 m. (b) The buoyancy frequency profile Nf ðzÞ down to 385 m; insert shows the two mean profiles oNðzÞ4 and Nf ðzÞ in the upper 50 m.

Fig. 3. (a) Hourly fluctuations of sea level and (b) wind velocity at El Sauzal during the experiment. (c) Average hourly wind velocity for the 15 days of observation.

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wind speed of 3–4 m/s to the NE (onshore towards the coast)between 10:00–17:00 h, dropping to zero at night.

To compare parameters of the barotropic and baroclinic tides, aspectral analysis was performed on the sea-level and temperaturefluctuations (representing baroclinic fluctuations) of the L12005time series. Sea level amplitude (Fig. 4a) showed four harmonics:the two semidiurnals M2 and S2, and the two diurnals K1 and O1.The amplitude of the semidiurnal peak in the temperature timeseries (Fig. 4b) was four times greater than the diurnal peak. Thespectrum of hourly temperature fluctuations (Fig. 4c) showed

distinct peaks at semidiurnal, diurnal and �3 day periods. How-ever, the spectral analysis of meteorological data (temperature,pressure, wind) from the weather station at San Miguel did notshow a peak of spectral density at 3 days (not shown).

Theory indicates that freely propagating internal waves canonly exist at frequencies f rωrNmax (or f ZωZNmax, but only atgreat depths) (Konyaev and Sabinin, 1992; Miropolsky, 2001).Because Todos Santos Bay is located north of 301 (f¼1.06 cyclesper day at latitude 31.881N), diurnal period internal waves aresubinertial ðω24 ho f Þ (Fig. 4b). However, similar to Fig. 4c, all

Fig. 4. Spectra (periodogram) of the amplitudes of: (a) barotropic tide and (b) baroclinic tide (according to a thermograph 10 m below the surface; the L12005 series).(c) Smoothed frequency spectra of hourly temperature fluctuations at the thermograph. The inclined lines represent the spectral density attenuation law as a function offrequency. The vertical line shows the 80% confidence interval.

Fig. 5. Spectra of hourly temperature fluctuations at 10 m depth at (a) moorings 1 and 14, and (b) at moorings 8, 11, 12 and 15. The inclined lines in panel (a) represent thespectral density attenuation law as a function of frequency. The vertical line shows the 95% confidence interval. Arabic numerals indicate the mooring numberscorresponding to the spectra.

A. Filonov et al. / Continental Shelf Research 78 (2014) 1–156

spectra of temperature time series presented a peak with a diurnalperiod; we propose that this was due to the breeze, and had norelation to the diurnal baroclinic tide.

The spectra of smoothed temperature fluctuations for thenorthern and southern parts of the bay were qualitatively different(Fig. 5). For northern mooring number 1, a semidiurnal tidal peakwas well defined and it was an order of magnitude greater than

the diurnal peak (Fig. 5a). The spectrum showed sub-harmonicsfor periods of 8 and 6 hours. The decrease of the spectrum withincreasing frequency was close to the ω�3 law. In contrast, thespectrum of data from mooring 14, which was located in shallowwater in the southeastern part of the bay (20 km from mooring 1),was dominated by the diurnal peak (Fig. 5a). This spectrum had aslope of ω�2. On average, the spectral densities over the entire

Fig. 6. (a) Mooring location (bold dots with numbers). (b) Spatial distribution of the ratio between the mean-square amplitude of the semidiurnal tidal and diurnal tidalvariations in temperature ðβ¼ a12:4=a24Þ in northern and (c) the southern parts of Todos Santos Bay. (d) Spatial distribution of vorticity at 10 m depth in August.

Fig. 7. Bathymetry (isolines, in m) and spatial distribution of the ratio of the bottom slope angle and the characteristic beam inclination angle ðα=θÞ for semidiurnal internalwaves in and around Todos Santos Bay (color).

A. Filonov et al. / Continental Shelf Research 78 (2014) 1–15 7

range of analyzed frequencies from mooring 14 were five timesgreater than in the spectrum from mooring 1.

In the southern part of the bay, especially close to shore, thediurnal spectral density was higher than the semidiurnal one(Fig. 5b). This is illustrated by the distribution of the ratioβ¼ a12:4 =a24 , where a12:4 is the mean-square amplitude of thetemperature variation at the semidiurnal tidal period, and a24 is

the corresponding amplitude at the diurnal period. In relation to β,the bay was divided into the northern and southern sectors(Fig. 6a–c). For thermistors in the northern sector (Fig. 6b), theamplitude of the semidiurnal variation was much greater than thediurnal variation; in the coastal area adjacent to San Miguel, β wasin the range of 3.3–3.7, with its maximum value of 7.1 just to thenorth, near mooring 5. In the southeastern part of the bay (Fig. 6c),

Fig. 8. Computed trajectories of characteristic beams of internal semidiurnal tides on the Todos Santos Bay shelf. Arrows show the group velocity direction. (a) Along themain axis of the bay, (b) perpendicular to the shelf in the northern part of the bay. The inserts show, with circles, the main areas of internal wave generation and, witharrows, the principal direction of energy propagation.

Fig. 9. Normalized wave number spectra of semidiurnal internal waves measured at two points of Todos Santos Bay. Shapes of (a) the South Cluster (moorings near theADCP) and (c) the North Cluster (moorings near the ADP) are shown in the top panels. The circle points with numbers indicate the thermograph mooring numbers. Thecorresponding spatial spectra are shown in the bottom panels (b and d). The solid lines correspond to linear estimation, and the dotted line the estimation by the maximumlikelihood method. The horizontal wave vector (not shown) is directed from the origin of the coordinate system. Isolines of the spectral density are drawn at 20% intervals ofits maximum value.

A. Filonov et al. / Continental Shelf Research 78 (2014) 1–158

the magnitude of the diurnal peak grew substantially as shorewas approached (see also Fig. 5b). In the shallowest sites inthe southern sector, β was only 0.1–0.2, i.e., the amplitude ofthe diurnal variations was more than 5–10 times greater than theamplitude of the semidiurnal ones.

3.4. Internal semidiurnal tidal wave generation

The distribution of the parameter α/θ in and around the baywas calculated using the bathymetry and the calculated θ (Fig. 7).Considering that internal tides which travel onshore can begenerated in the range 1:2rα=θr0:8, Fig. 7 indicates thatconditions were suitable for the generation of internal semidiurnalwaves not only along the shelf break, but also at numerous sites(platforms, banks) on the continental slope between 300 and1000 m depth.

In Fig. 7 we numbered six possible generation sites to the west(and outside) of the bay, fromwhich the semidiurnal internal tidescould reach the North Cluster observation site off San Miguel.There is also a clear generation site in Salsipuedes Bay, but most ofthe internal tidal energy would be radiated to the east, with verylittle directed towards Todos Santos Bay.

The trajectory of the internal semidiurnal tidal beams wascalculated along two transects starting at potential generationsites, using the formula xðzÞ ¼ x07

R z0 ctgðθðzÞÞdz (where x0 deter-

mines the beam location at the x-axis). We assumed that thebeams are perpendicular to the isobaths, but it should be remem-bered that oblique incident waves can be subject to refraction. Theresults (Fig. 8a–b) showed that the beams bounced only once ortwice at the surface and bottom, due to the narrowness of theshelf. The wavelength remained unchanged when the beam wasreflected from the ocean surface, but decreased considerably whenit was reflected from the bottom, which is inclined in the oppositedirection with respect to the beam. Each reflection of the wavefrom the bottom resulted in a reduction of its wavelength.It is this mechanism that creates the observed decrease in wave-length when the wave propagates towards the shore.

3.5. Characteristics and direction of propagation of internal tidalwaves

The two spatial spectral estimates (linear and adaptive) gavesimilar values of wavelengths and directions for a given cluster.However, for semidiurnal waves they differed spatially betweenthe two sectors of the bay. For the South Cluster (Fig. 9a–b), thesemidiurnal internal waves had an average wavelength of 9.1 km,phase velocity of 0.20 m/s and a direction of propagation of 1561.For the North Cluster (Fig. 9c–d) (in front of San Miguel) the firstmethod gave the following values: a wavelength of 8.3 km, adirection of 1051, and a phase velocity 0.19 m/s. The secondmethod resulted in a wavelength of 8.2 km, a direction of 881,and a phase velocity of 0.18 m/s.

The average direction of propagation of the semidiurnal inter-nal tidal waves at the North Cluster (96.51 clockwise from North)suggests that they were generated west of the bay, on thecontinental slope and at the edge of the shelf. Their direction inthe South Cluster (1561) suggests that once inside the bay theyrefracted gently (because of their great length) and movedsoutheastward.

The spatial spectra of the diurnal variation (not shown) for thetwo clusters of buoys gave similar results, with the followingaverage of the two estimates (linear and adaptive spectrumestimate): the wavelength was 22 km and the average directionwas 1461, i.e. from NW to SE along the axis of the bay.

The SAR satellite image of Todos Santos Bay (Fig. 10) taken onAugust 16, 2006, is in agreement with the interpretation thatinternal tidal waves were generated west of the bay and thenpropagated to the northeast across the shelf, to the area of SanMiguel and then into the bay along its main axis (that is, to thesoutheast). Spreading along the inside shelf, the tidal wavesquickly accumulated non-linearity and transmitted their energyto high-frequency waves, which are visible at the surface ascrescent-shaped slick bands. The groups of short-period internal

Fig. 10. Satellite SAR image (catalog ID: 1010010005251A11, Acquisition date:August, 16, 2006) of Todos Santos Bay. The crosses denote the positions of theADCP and ADP current meters. Note the group of short-period internal waves(solitary waves) approaching the southern shore of the bay.

Fig. 11. Rotary current spectra of clockwise (1) and counterlockwise (2) compo-nents of semidiurnal internal waves at all levels of the San Miguel ADP measure-ment. (3) Direction (α) of the wave's propagation (orientation of major axis ofelliptical orbits) and (4) the stability E of the ellipse orientation. Vertical dotted lineshows the 95% confidence interval for nonzero coherence.

A. Filonov et al. / Continental Shelf Research 78 (2014) 1–15 9

waves (possibly solitary waves) mark the leading edge of semi-diurnal internal tides.

The slick patterns show interference between some wavepackets, suggesting that the internal tides propagated from inde-pendent generation sites. In the leading group approaching thesouthern shore of the bay, there are eight short waves, withwavelengths decreasing from 0.5 to 0.3 km. The length of thewave group was about 2 km. The distance between the leadingfronts of two consecutive groups was about 4 km.

Using the rotational spectrum method (Emery and Thomson,1997) for the ADP data (North Cluster), the stability of wavepropagation direction (E) (Fig. 11, trace 4) was found to be higherthan the 95% confidence interval only in the 10 m surface layer andnear the bottom. The average direction of wave propagation in theupper layer was approximately 118741 (Fig. 11, trace 3), which is�221 larger than with the spatial spectrum method applied to thethermistor antennae. This difference is acceptable considering thatthese are statistical estimates based on different data sets. At thebottom, the direction of wave propagation had similar values(115721). In the 10–25 m layer, stability was lower than theconfidence level; hence the direction of wave propagation couldnot be estimated there.

3.6. Normal modes for semidiurnal internal tides

The elliptical orbits of the semidiurnal internal waves from theNorth Cluster ADP data were elongated in the direction of wavepropagation (Fig. 12). The orbits were greatest at the surface andbottom, and were significantly smaller in the center of the watercolumn, as expected for waves of the first baroclinic mode. Theratio of minor to major axis (f/M2¼0.55 in the ADP area) was alsoas expected for the first mode (Konyaev and Sabinin, 1992).

The first-mode structure was also shown by the rotary currentspectra (clockwise and counterclockwise components) of semi-diurnal internal waves (Fig. 11, traces 1 and 2, respectively), withhigher spectral densities in the surface and bottom layers than inthe 10–25 m layers.

These results show that, despite the narrowness of the shelf,the modal structure of the internal semidiurnal tides did form inthe North Cluster (San Miguel) area, with a dominant first mode.

Using the averaged vertical profile of the buoyancy frequency atthe two cluster sites (Fig. 13a) we calculated the amplitude of thevertical displacement of the two principal modes of the semidiur-nal internal tide (Fig. 13b) as well as their wavelengths and phasevelocities, which are shown in Table 3. The similarity of the

wavelengths (8–9 km) and phase velocities (�0.2 m/s) of the firstmode with the values obtained from the spatial spectral analysisindicates that the semidiurnal internal waves in the bay were ofthe first mode. Apparently, the internal mode was formed in justone or two cycles of reflection of the characteristic rays from thebottom and the sea surface (Konyaev and Sabinin, 1992; Filonovand Lavín, 2003).

Because the thermocline in the bay was pressed close to thesurface, the maximum vertical displacement (W, Fig. 13b) was alsoshifted to the surface. Near the North Cluster (ADP position), themaximum amplitude of vertical displacement (Fig. 13b, continuousline) was at a depth of 7 m and in the South Cluster (ADCPposition, dotted line Fig. 13b) at a depth of 12 m.

3.7. Temporal structure of diurnal and semidiurnal long internalwaves prior to disintegration

Below we present an analysis of temperature and currentmeasurements which show that semidiurnal internal waves werewell developed in the northern sector of the bay, while the

Fig. 12. Semidiurnal internal wave hodographs according to the ADP data, onAugust 2007. The coordinate axes u, v are rotated 251 clockwise. The major axes ofthe ellipses are oriented in the direction of maximum variability of currents.

Fig. 13. (a) Mean profiles of buoyancy frequency (N, cycles/h) on August 15–16,2007, in the North Cluster near the ADP (solid line) and in the South Cluster nearthe ADCP location (dotted line); (b) Dimensionless vertical displacement W of first(1) and second (2) normal modes of linear internal semidiurnal waves in both sites:North Cluster ADP (solid line), South Cluster ADCP (dotted line).

Table 3Theoretical parameters the two modes of the semidiurnal internal tide for thepoints of the ADP and ADCP.

Point Depth, m First mode Second mode

λ, km Сf λ, km Сf

ADP 29 7.99 0.18 4.08 0.09ADCP 29 9.17 0.20 5.11 0.11

A. Filonov et al. / Continental Shelf Research 78 (2014) 1–1510

southern part was dominated by baroclinic diurnal fluctuations,presumably caused by the breeze.

3.7.1. The South Cluster (mooring 8 and ADCP)At the South Cluster, from August 9 to 19, the thermocline

gradually shifted to the surface and the vertical amplitude of theoscillations decreased by the end of the period (Fig. 14a). Forinstance, the 14 1C isotherm shoaled from 20 m to almost 10 mdepth. Temperature fluctuated almost 6 1C at 10 m depth, similarto the average vertical temperature range. Semidiurnal and dailyfluctuations were evident in the temperature and current data(Fig. 14).

The intensity of diurnal flow varied greatly, being particularlyintense on August 8–10, and showed vertical and temporalvariability (Fig. 14b), with maximum velocities at the surface andbottom. The region of phase shift (zero velocity) occurred in theupper layers, at 10–12 m depth.

The frequency spectrum of the South Cluster ADCP currentfluctuations (Fig. 16a) shows that currents were dominated bydiurnal variations, with maxima near the surface and bottom. Thediurnal spectral density had two peaks: a large one at the surface(depth �5 m) and a second, smaller one, near the bottom (�23 mdepth). The three-day current periodicity was expressed only inthe surface layer, with a maximum at �3 m depth. Semidiurnaloscillations and the 8-hour sub-harmonic were weak in thissouthern sector of the bay, expressed only in the upper layer,and had a maximum spectral density at 4 m depth. The spectraldensity of semidiurnal fluctuations was only half of the spectraldensity of the diurnal fluctuations.

3.7.2. The North Cluster (mooring L5 and ADP)At the North Cluster, semidiurnal temperature fluctuations

were well defined and dominant (Fig. 15a). Their maximumamplitudes were at depths of 7–9 m. Vertical fluctuations nearthe bottom and the surface were minimal.

The internal waves at mooring L5 (and in other moorings closerto shore) showed the form of an oscillating bore (Fig. 15c), withsteep isotherms at the leading edge and gently sloping isothermsat the rear of the bore, pressed to the surface. The front slopes

were populated by several short-period oscillations, similar to asequence of solitary waves, with periods of 5–20 min and heightsof 15–20 m (Fig. 15c). Surface slick bands near buoy 5 showedwavelengths from 50 to 200 m. The fluctuations of horizontalcurrents were along the line 115–2951, i.e. the waves propagatedalmost along the shore. Because of the large wavelength andshallow depth of the bay (the ratio of depth to wavelength wasapproximately 0.003), internal tidal waves refracted poorly. Incontrast, the groups of (solitary) short waves refracted strongly.According to our visual observations, they always traveled towardsthe shore. That is, they moved along the crest of the tidal waves,roughly to the east.

The horizontal currents along the semi-major variability axis(251 clockwise from the east direction) changed approximatelytwice a day, were in the opposite direction at the surface andbottom, and showed minimal (almost zero) velocities at a depth of12–14 m (Fig. 15b and d). Maximum velocity of orbital currents inthe semidiurnal internal waves was about 20 cm/s. However, themaximum velocity of the baroclinic semidiurnal currents at thebottom and the surface were not shifted exactly 1801, but ratherhad a phase shift of 1–2 h (Fig. 15d).

The frequency spectrum of the current fluctuations in theNorth Cluster ADP shows (Fig. 16b) that the semidiurnal variationswere dominant, with maximum spectral density at the surfacelayer and a slightly lower peak near the bottom, i.e. the firstbaroclinic mode of oscillation. The spectral density of the surfacepeak (5 m depth) was approximately three times higher than thatof the bottom peak (�25 m depth). The three-day current peri-odicity was again present in the surface layers (o10 m depth)with a maximum at 3 m depth. Diurnal oscillations were veryweak and the 8-hour sub-harmonic was completely absent.

3.8. Potential energy of the internal tide

The vertical oscillations of the water layers at mooring L5 werenot uniform in time or with depth. They occurred throughout theentire water column, but due to the influence of stratification, inthe upper part the amplitudes were 4–7 m, while in the lowerportion they were 9–11 m (see average amplitude in Fig. 17). Theeffect of stratification can be corrected with the WKB

Fig. 14. Data from the South Cluster. (a) Hourly temperature fluctuations at mooring 8. (b) Hourly current fluctuation at the ADCP (along axis rotated clockwise by 561 fromthe east direction) on August 2007. The vertical average flow has been removed.

A. Filonov et al. / Continental Shelf Research 78 (2014) 1–15 11

approximation of initial data (Konyaev and Sabinin, 1992; Filonovand Lavín, 2003). However, we did not follow this procedurebecause of the shallow depth at the observation point and thesmooth profile of the buoyancy frequency.

To simplify the calculations, we assumed that the wave energydid not change along the front, on the line from San Miguel to thenorthern tip of Todos Santos Islands; the width of which wasassumed to be 10 km and used the average depth of this line of30 m. For the average density and buoyancy frequency profiles weused the profiles shown in Fig. 2a and b.

The potential energy in one tidal cycle in the water layer of30 m depth, 1 m of width perpendicular to the front (average of 25cycles of the semidiurnal tide) at mooring L5 had a value of2.94�105 J (vertically integrated energy at mooring L5 respec-tively was 29.4 J/m2). The vertical profiles of buoyancy frequencyand potential energy were in good agreement. The rate of energyconversion from the barotropic tide to internal waves (along theinternal wave front) was approximately 7.79�105 J/s (0.779 MW).

4. Discussion

In this study, spectral analyses of temperature and currentsdetected wind-forced diurnal-period internal waves in the south-ern part of Todos Santos Bay, and semidiurnal tidally-forcedinternal waves dominating the northern part of the bay. Thepresence in Todos Santos Bay of semidiurnal internal waves andhigh-frequency short waves have been reported before (Ladahet al., 2005, 2012), but this is the first time that diurnal internalwaves are reported for this region.

The dominant diurnal signal found in the southern sector of thebay was unexpected due to its subinertial character at the bay'slatitude. Although a number of recent articles discuss baroclinicdiurnal oscillations poleward of the critical latitude off the coast ofCalifornia (Lerczak et al., 2001; Pidgeon and Winant, 2005;Beckenbach and Terrill, 2008; Hendrickson and McMahan, 2009;Cudaback and McPhee-Shaw, 2009) and in southern Chile (Kaplanet al., 2003), it is hard to discern the source of those oscillations as

Fig. 15. Data from the North Cluster. (a) Hourly temperature fluctuations at mooring L5. (b) Hourly current fluctuations at the ADP (along axis, rotated clockwise by 251 fromthe east direction) on August 2007. The vertically-averaged flow has been removed. (c) A zoom segment of the time series of temperature, showing groups of short periodwaves on the forward front of a tidal internal wave. (d) Segment of time series showing first internal mode current fluctuations.

A. Filonov et al. / Continental Shelf Research 78 (2014) 1–1512

mean stratification and buoyancy frequency were often notreported. Most of these studies attributed the observed diurnalvariations to diurnal heating and/or to the breeze. One study inparticular (Beckenbach and Terrill, 2008) found that for a steeptopographic ridge at 32.61N the diurnal baroclinic mode wastidally forced, and that although the response resembled the firstmode, it was topographically distorted near the bottom. In thesouthern sector of Todos Santos Bay, we found that the spectraldensity of the diurnal peak decreased almost linearly withdistance from shore, suggesting that in our case the diurnaltemperature oscillations were caused by the breeze.

Currents and eddies can alter the spatial structure of internalwaves, depending on the incident angle relative to the flow (along it,across it, or at an angle). Currents, depending on their vorticity, canincrease or decrease the frequency of internal waves (e.g. Lerczaket al., 2001), resulting in a slight displacement of the diurnal andsemidiurnal peaks (Kunze and Sanford, 1986; Konyaev and Sabinin,

1992). In particular, the flow may alter the effective inertial frequencyf ef ¼ f ð1þΩv=f Þ1=2, which may be higher or lower than the localinertial frequency f, where Ωv ¼ ðdv=dx�du=dyÞ is vorticity. Themean absolute value of vorticity in the study area (rectangle inFig. 6d), calculated with the outputs of the numerical model ofMateos et al. (2008), is equal to 1.872�10�3 cycles/h (0.052� cycle/s).The inertial frequency at the latitude of the central part of the bay(31.83 1N) is 0.04395 cycles/h (a period of 22.75 h). The maximumvalue of vorticity would give an effective inertial frequency of0.0449 cycles/h (a period of 22.28 h). Hence, the calculated differencefrom the local inertial frequency was negligible and vorticity couldnot have caused a significant displacement of inertial frequencytoward the diurnal frequency. This supports the idea that the causeof the observed daily internal fluctuations was the breeze. In TodosSantos Bay, the effect may be amplified by the specific orography ofthe coastline, as well as by the presence of islands. Unfortunately, theavailable wind observations do not permit a detailed analysis of theforcing of diurnal internal waves by wind.

One of the main results of this study was that the semi-diurnalinternal waves were not equally represented in the different partsof the bay. The internal waves come into the bay from two mainsites of generation. One is located north of the bay, on SalsipuedesBay, but this source was weak. Waves generated there propagateddirectly to the southeast along the axis of the bay, reflecting fromthe surface and bottom (Figs. 7 and 8a), and quickly disintegratedand disappeared; the internal mode was not formed in the centeror in the southern part of the bay.

The most regular internal semidiurnal waves with large ampli-tudes were always observed in the North Cluster, near San Miguel.They traveled to that site along the shortest path from the varioussites of generation (see Figs. 7 and 8b), which were located atthe edge of the shelf and at several platforms (shelves) on thecontinental slope. Despite the continental shelf's narrowness,which allows only one or two reflections of the characteristicbeams before reaching the observation site, a modal structure wasformed, with dominance of the first mode. The values of rotarycurrent spectra (clockwise and anticlockwise components) of thesemidiurnal internal waves at mid-depths were much lower thanin the surface and bottom layers, also suggesting that the firstinternal mode was formed. The mode was probably constructedfrom waves coming from the various generation sites, but wecould not separate the sources due to the shortness of the timeseries and limitations of the mooring arrangement; therefore wehave obtained average estimates of wave propagation directionand wavelength. Subsequently, the semidiurnal internal wavespropagated along the shore into the bay (towards the southeast)and very quickly disintegrated into high-frequency, solitary-likewaves near shore, which dissipated entirely prior to the arrival ofthe next internal wave.

Generally, for a first mode structure, currents in the upper andlower layers should be shifted in phase by 1801 (Konyaev andSabinin, 1992; Miropolsky, 2001). Although a first mode structurewas often found in this study, there were occasions when themaximum velocity of the baroclinic semidiurnal currents at thebottom and the surface were not shifted by exactly 1801, but had aphase shift of 1–2 h.

We found that the slope of the internal wave spectra changedfrom “ω�3” to “ω�2”, which is typical and often occurs as thefrequency increases and the energy decreases for nonlinear inter-nal waves that disintegrate on the continental shelf (van Harenet al., 2002; van Haren, 2004). This behavior can be attributed tothe fact that internal waves, during their propagation on the shelf,redistribute their energy from larger to smaller fluctuations in anenergy cascade. An analytical model for the spectrum of nonlinearinternal waves and a detailed explanation for the different slopesof the spectrum are given by Filonov and Novotryasov (2005,2007).

Fig. 16. (a) and (b): Spectra Sðz;ωÞ of the current fluctuations whose graphs areshown in Figs. 14b and 15b respectively. The spectra for each depth were smoothedby 5 frequencies of the spectral density, which corresponds to an estimated 10degrees of freedom (Emery and Thomson, 1997).

Fig. 17. Hourly average amplitude of the vertical oscillations of the water columnon mooring L5.

A. Filonov et al. / Continental Shelf Research 78 (2014) 1–15 13

A change of amplitudes with depth was detected in our timeseries, which usually occurs as a result of interference betweentwo inclined waves propagating in opposite vertical directions, asin the vertical modes. In modes of vertical oscillations, nodes areformed at the surface and bottom, where amplitudes of fluctua-tions are reduced to zero. These waves were most likely generatedin different regions of the continental slope and arrived at theobservation point (our thermistor chains and current meters) afterbeing reflected from the surface and bottom one or more times(Konyaev and Sabinin, 1992; Miropolsky, 2001). This type ofinternal tide deformation has been observed in Australia(Holloway, 1985), and an example of an individual wave deforma-tion can be seen in the numerical model discussed by Vlasenkoand Hutter (2002).

The calculated potential energy of internal waves in TodosSantos Bay was two orders of magnitude lower than in otherstudies. For example, Fu and Holt (1984) estimated 500 MW forthe energy transfer from the barotropic tide to the internal tidegenerated near San Esteban Sill in the northern Gulf of California.In the Gulf, where barotropic tidal flows reach 1–2 m/s, energytransfer processes occur rapidly. However, in Todos Santos Bay,the energy transfer was an order of magnitude lower than thatcalculated for the Gulf of California. In Todos Santos Bay, theinternal tidal energy was ultimately dissipated inside the baythrough the generation of turbulence and vertical mixing.

5. Conclusions

According to the structure of the observed internal waves,Todos Santos Bay could be roughly divided into two parts: thenorthern part, where semi-diurnal internal waves predominate,and the southern part, dominated by baroclinic diurnal fluctua-tions caused by the breeze.

For the semi-diurnal baroclinic tide, generation occurs not onlyalong the shelf break, but also at numerous sites (platforms,banks) on the continental slope west of the bay, which arescattered between 300 and 1000 m depth (Fig. 7).

After generation, most of the energy of the semidiurnal internalwaves propagated towards the northeast, to the San Miguel region,where a modal structure with dominance of the first mode wasdetected, despite the narrowness of the shelf. The first mode wasprobably formed from signals arriving from the various generationsites, after one or two reflections of the characteristic beams fromthe surface and the bottom. A small part of the energy reached thecentral part of the bay, there undergoing non-linear transforma-tions. Nonlinearity was manifested, at first, in the distortion of thewave form, and secondly, in the formation of refracted groups ofsolitary-like short waves of large amplitude near the shore, whichrequires not only a sufficiently large degree of nonlinearity, butenough time for nonlinear wave evolution.

Although the latitude of Todos Santos Bay exceeds the criticallatitude (301N) for the existence of diurnal internal waves, allspectra of temperature and current fluctuations calculated frommeasurements on moorings and with the ADCP presented a peakwith a diurnal period, which we propose to be caused by thebreeze.

Acknowledgments

This paper is dedicated to the memory of Dr. Miguel F. Lavin,who passed away the day this paper was accepted. He was a greatman and an inspiring colleague, and leaves behind an impressivelegacy. May he rest in peace. This study was supported by theMexican Consejo Nacional de Ciencia y Tecnología (CONACYT

Project no. 105622, CONACYT project “LINK”, CONACYT project“FLOO”, by UCMEXUS-CONACYT and by TAMU-CONACYT). Wethank Dr. J.J. Leichter for providing the instrument that collectedthe L12005 data set. Dr. Efraín Mateos provided outputs of hisnumerical model of Todos Santos Bay, Dr. E.D. Barton helped withediting, Mr. Carlos Vargas-Aguilera participated in data collectionand in initial data processing, and Carlos Cabrera and ArturoOcampo assisted in the field.

Appendix A. Rotational spectral analysis of the current meterdata

For the direction of wave propagation (up to 71801), theorientation aligns with the principal axis of the internal tidalellipse, which is estimated with the formulaα¼ðarctgðð2PuvÞ=ðSuu�SvvÞÞ=2. The ellipse stability function, E, wascalculated with the formula jEj2 ¼ ððSuuþSvvÞ2�4ðSuuSvv�P2uvÞÞ=ðSuuþSvvÞ2�4Q2

uvÞ, which is similar to the coherence func-tion. The ellipse rotation is stable if the value of E at the frequencyof the internal tide is higher than the corresponding confidenceinterval (Gonella, 1972; Emery and Thomson, 1997). In theseformulae, the following values of spectral functions were used:Suu; Svv are the spectra and Puv; Quv are the co- and quadraturespectra of the u and v components of the current.

Appendix B. Spatial spectral analysis of the semidiurnalinternal waves

The spatial spectrum (wave number spectrum) of the semi-diurnal internal waves was calculated, using linear and adaptive(maximum likelihood) methods for the two clusters. To estimatethe spectrum by the linear method we used the equation (Barber,1963; Konyaev, 1990) Sðkx; ky;ωÞ ¼ a=p∑p

m; j ¼ 1gm;jðωÞexpð�i2πðkxΔxm;jþkyΔym;jÞÞ: Here a is the size of the area on whichthe samples of cross-spectra are arranged; j¼ 1; p is number ofthe cross-spectrum (number of the pair of the time series andspace shift); p is the number of pairs of time series; a=p is averagesize of the area per sample of the cross-spectrum; gm;jðωÞ is cross-spectra matrix; kx; ky are the horizontal components of theinternal wave number vector; ω¼0.081 cycle/h is the semidiurnalfrequency; and Δxm;j ¼ xm�xj; Δym;j ¼ ym�yj are the componentsof the distance between the moorings.

To estimate the wave number spectrum by the maximumlikelihood method, we used the equation: Sðkx; ky;ωÞ ¼ð∑p�1

m;j ¼ 0hm;jðωÞexpð� i2πðkxΔxm;jþkyΔym;jÞÞÞ�1, where hm;jðωÞ is a

whitening spectral function (inverse spectral matrix) andhm;jðωÞ ¼ gm;jðωÞ�1, from which we calculated a maximum like-lihood spectrum (Konyaev, 1990).

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