This paper studies the nonlinear transformation of the semidiurnal internal tidal waves in the northern Gulf of California, based on spectral analysis of temperature and current fluctuations from moored instruments, and analytical simulation. Observations showed that: (a) The spectrum presented a quasilinear structure with peaks at frequencies ωn= n ω0, where ω0 is the frequency of the tidal harmonic M2 and n = 1, 2… is the subharmonics number. (b) The amplitudes of the even subharmonics M4 and M8 were of the same order, as were those of the odd subharmonics M6 and M10, but the last two were larger. (c) The energy of the subharmonics decreased as ω−3 with increasing n. These features were simulated by an analytical model spectrum of nonlinear internal waves; it produced a line structure formed by the harmonics whose energy depends on the distance traveled by the wave from the area of generation. In the approximation of quadratic nonlinearity, the spectrum of nonlinear long internal waves in the zone of wave breaking is asymptotically ∼ωn−2,6. Allowance for cubic nonlinearity leads to a non-monotonic decay of subharmonics energy depending on their numbern, similar to the observed spectrum, which indicates that the internal semidiurnal tide in the northern Gulf of California is a cubically nonlinear wave.
 The Gulf of California (GC, Figure 1) is a long (∼1000 km) and narrow (∼120 km) embayment whose entrance is in the tropical-subtropical transition zone of the eastern north Pacific [Lavín and Marinone, 2003]. Bottom depth in the southern part exceeds 2000 m, while the mean depth of the northern part is ∼200 m. Its dynamics and thermodynamics are strongly seasonal, and driven mostly by the Pacific Ocean. It is also a tidal sea that co-oscillates with the Pacific and its length and mean depth make it almost resonant to semidiurnal frequencies, with M2 the dominant frequency.
 Separating the northern and southern parts of the GC is the midriff archipelago, which contains several channels and sills (Figure 1a) where tidal currents can attain 1.5 ms−1. These tidal currents release large amounts of turbulent kinetic energy, which has a tremendous impact on the physics and biology of the GC [Filonov and Lavín, 2003].
 The existence of sills in the presence of high tidal energy, together with the strong stratification that characterizes the GC, are appropriate conditions for the generation of internal tides. In the initial stages of generation of internal motions by tidal flow the thermocline is depressed in the lee of San Esteban Sill. SAR images of the GC show that groups of short-period internal waves (IW) are generated with semidiurnal periodicity at San Esteban sill during spring tides, and spread northwest along the axis of Tiburon Basin [Fu and Holt, 1984].
Filonov and Lavín  analyzed data from two moorings (one for summer and one for winter) equipped with current and temperature sensors, located in Tiburon Basin. They found that the energy spectra of internal oscillations in the tidal frequency bands have a quasilinear structure with intense peaks at the tidal harmonic M2 and its subharmonics: M4, M6 and others.
 Quasilinear spectra of kinetic energy in the tidal range of frequencies with monotonic decay of subharmonic energy with increasing subharmonic number have been reported in several papers [Hosegood and van Haren, 2006; Xie et al., 2008], but this appears to be the first report of non-monotonic energy damping of higher subharmonics.Filonov and Novotryasov  proposed a quadratic nonlinear mechanism, based on the transformation of the internal tide spectrum, for the generation of higher harmonics of the M2 internal tide.
 Here we discuss new results on the nonlinear transformation of long internal tidal waves in the northern Gulf of California, specifically on the influence of cubic nonlinearity in the transformation of their spectrum.
2. Nonlinear Internal Tidal Waves Observations
 The area of study is the narrow Tiburon Basin (Figure 1a), which is elongated to the NW of the midriff archipelago. We analyzed time series of temperature and current, collected by an instrumented mooring (located at the flag in Figure 1a) which operated in a bottom depth of 440 m, from July 22 to August 7, 1990 [Godínez et al., 1994]. Current and temperature were sampled every 2 minutes with ENDECO current meters (accuracy 0.75 cm s−1) at six depths: 75, 110, 150, 250, 350, and 428 m. The vertical-mean velocity was removed from the time series to obtain the baroclinic fluctuations.
 Vertical profiles of temperature and salinity were obtained in the vicinity of the mooring during surveys from July 17 to 27 and from August 1 to 18, 1990. Average profiles of temperature and salinity, and the Brünt-Väisäla frequency profile are shown inFigure 1b. The buoyancy frequency from surface to bottom exceeds the semidiurnal frequency (0.081 h−1), which permits the existence of semidiurnal internal waves. The vertical displacement of isotherms was largest at 350 m depth, where maximum amplitude η0 ∼ 20 m (Figure 1c), and the maximum amplitude of the tidal baroclinic flow (Figure 1d) was u0 ∼ 0.30 ms−1; these amplitudes are related by the polarization ratio η0 = (uin/c)H(dφ/dz).
 The inertial frequency at 23.5°N (0.0410 cycle/h) is so close to the semidiurnal frequency (0.0417 cycle/h), that it is impossible to separate their frequencies in a spectrum from a 17-day time series (0.00245 cycle/h).Figure 2shows the spectrum of the internal tide kinetic energy at 350 m depth; a deep current record was selected because horizontal internal motions are enhanced with respect to mid-depth. The data were analyzed with standard spectral analysis techniques [Emery and Thomson, 1997]. The spectrum was obtained by (i) dividing the resulting series into two 8.5–days-long segments, (ii) the calculation and averaging of spectral densities by segments, and (iii) the smoothing of the averaged spectral densities by a five-point Tukey's filter, providing a reasonable reliability of the results of spectral analysis with 88 degrees of freedom (the low-frequency trend was removed). The spectrum has the following characteristics: (a) A quasilinear structure with peaks located at frequenciesωn = nω0 where ω0 is the M2 frequency and n = 1, 2… is the subharmonic number. (b) The amplitude of the even subharmonics M4 and M8, and of the odd subharmonics M6 and M10 are of the same order among pairs, but the latter contain more energy than the former. (c) The energy of the subharmonic peaks decreased as ω−3 with increasing n. The inclined line in Figure 2 shows spectral density recession with increasing frequency. For the tidal band of frequencies, the spectral slope is ∼ω−3.
3. Theoretical Description of the Nonlinear Internal Waves
 The internal tidal waves in the study area are generated over San Esteban sill and spread along the axis of Tiburon Basin [Filonov et al., 2010], which extends to the NW for more than 200 km from the sill and has a maximum depth of ∼400 m (Figure 1a). We can assume that due to their horizontal length scale (30–40 km), semi-diurnal internal waves should reflect from the inclined walls of the channel, and return o the axial direction.
 The observations are interpreted in the framework of the 1D nonlinear shallow – water theory of internal waves in a rectilinear channel of rectangular cross-section and slowly varying depthH. The basic component of this theory is the extended Riemann (eR) equation. It is assumed that H/λ ⋘ 1, ω/N* ⋘ 1, a0/H ⋘ 1, and λ/L ⋘ 1, where λ, ω and a0 are a representative wavelength, frequency and amplitude of the wave, respectively, N* is a representative buoyancy frequency and Lis the characteristic scale of depth variations. The analytical solution of the eR equation is used to model the spectral frequency transformation of tidal-band internal waves in a channel. The internal tide is modeled as an initially sinusoidal waveform. For arbitrary stratification, the eR equation is written as
where η(x, t) is the vertical displacement of the pycnocline; x is a horizontal coordinate; t is time; c, α, α1, are the wave speed, quadratic and cubic nonlinear coefficients, respectively and q2 = (c0M0)(cM)−1, M(x) = N2φdz. Values with subindex “0” are the values at any fixed point x0. It is convenient to put the origin x0 = 0 at the gulf entrance. The wave speed c is determined from the eigenvalue problem for the modal structure function φ(z) of the vertical displacement in the linear long-wave limit (Boussinesq approximation and rigid-lid approximation are used):
where N(z) is buoyancy frequency and H is water depth. The nonlinear coefficients α, and α1 are given by equations (4) and (5) of Grimshaw et al. .
Equation (1) is solvable as an initial Cauchy problem. In our experiment we deal with measurement of time series of temperature and current velocity at some fixed points η(t, xi), and we want to predict the spatial development of the wave as it propagates along the channel, given that it is known as a function of time at the channel entrance. The mathematical problem can be called a boundary Cauchy problem for the eR equation [e.g., Osborne, 1995].
Equation (3)is valid to second-order accuracy in the wave amplitude (a0/H ≪ 1) and for long (H/λ≪ 1), low-frequency (ω/N* ≪ 1) waves in a gulf with slowly varying depth H (λ/L ≪ 1). The case α1 = 0 is known as the equation of the simple wave or the Riemann equation. It is necessary to solve it for the “initial” condition
 To interpret the field measurements it is necessary to know the Fourier frequency spectrum of the internal wave field parameters. Consider the spectral density of the vertical displacement of the thermocline governed by equation (3). Suppose that the parameter of cubic nonlinearity α1 = 0. In this case the eR equation (3)is transformed into a Riemann equation with quadratic nonlinearity whose solution can be represented as the Bessel-Fubini relation [Naugolnykh and Ostrovsky, 1998]
where Bn(x) = 2a0Jn(xnXbr−1)/(xnXbr−1) is the spectral amplitude or the harmonic amplitude, Jnis the Bessel function of n-order,n is the harmonic number, and Xbr is the breaking distance.
 Consider the dependence of the 2a0-normalized amplitude of the harmonic -bn = Bn(d)/2a0 on its number n and the distance from the entrance to the channel d = xXbr−1, expressed in Xbr units. The dependence of harmonic energy bn2 on n for fixed values d = 0.5 and d = 1.0 is shown in Figure 3. From this, the dependence n ≡ bn(1) is well approximated by n−2,6 for x ∼ Xbr.
 Next, suppose that the distribution of buoyancy frequency N(z) over depth in the gulf is such that the parameters of quadratic and cubic nonlinearity are of the same order and that H is constant. In this case, q2 = 1 and the solution of (3) has the form
whose Fourier transform is
 We calculated integral (7) numerically, and Figure 4 shows the dependence of the square of the spectral density on frequency, normalized by the maximum value at X = Xbr. The model spectrum has a quasilinear structure and has maximum peaks at frequencies ωn = nω0. In addition, the decay of the amplitude of the harmonics with increasing n is not monotonic; i.e., the amplitudes of the odd subharmonics (3 and 5) are larger than those of the neighboring even harmonics (2 and 4).
 Thus, the energy spectrum of long internal waves of finite amplitude at frequencies much smaller than the buoyancy frequency, based on the eR equation with quadratic and cubic nonlinearity, has the following characteristics: it is non-uniform in space, has a quasilinear structure and a different asymptotic behavior at various frequency ranges (Figure 4).
 This calculated spectrum adequately reflects the characteristics of the observed spectrum of horizontal kinetic energy (Figure 2) and vertical displacement of the thermocline in the GC [Filonov and Lavín, 2003, Figure 5] and may serve as a theoretical model of the energy spectrum for internal tidal waves in the northern Gulf of California and similar tidal embayments.
4. Discussion and Conclusions
 To the first order in amplitude, the horizontal velocity u and the vertical displacement η of internal waves are related by u ∼ cη(x, t)(dφ/dz)/H; that is, the power spectra of these parameters on a fixed depth have a similar form.
Filonov et al.  showed that semidiurnal internal waves propagating to the NW of San Esteban sill suffer highly nonlinear transformations. Their slope quickly reaches a critical value and they begin to break. The distance traveled by the waves from their point of generation to the area of breaking (the breaking distance) Xbr for a hypothetical channel of constant depth H can be estimated from Xbr = (c1/α)(λ1/2π)/a0, where c1, λ1 are the phase velocity and the length of the first mode M2 internal wave, respectively, a0 is its amplitude and α is a parameter of the quadratic nonlinearity, defined by equation (4) of Grimshaw et al. . These theoretical parameters and the eigenfunction for the first mode of internal waves were obtained by numerical solution of the boundary value problem (2a)–(2c), using the observed buoyancy frequency profile near the mooring. The magnitude of the phase velocity and wavelength were equal to c1 ≈ 1 m s−1 and λ1 ≈ 36 km, respectively, parameter α was 3 ⋅ 10−3 s−1, and α1 ∼ 1.3 ⋅ 10−4 m−1 s−1. Fluctuations in temperature at a depth of 350 m were δT ≈ 0.3 °C, and their average vertical gradient was 〈dT/dz〉 ≈ 0.03 °Cm−1. Hence, the amplitude of vertical displacement of the water layers near this depth was estimated as a = δT/〈dT/dz〉 = 10 m.
 As shown in Figure 2b of Filonov and Lavín , the eigenfunction had a maximum at 165 m depth, so that at 350 m it has only half that value. Considering that a0 = φ1(z*)/a, we find that the real amplitude of the first internal mode near the mooring was a0 = 0.5/10 ≈ 20 m and the calculated breaking distance was Xbr ≈ 100 km.
 Thus, our calculations show that the M2 internal wave runs from the area of generation to a distance of about 3λ1 and then starts to tip over because of the accumulated nonlinearity. As follows from (5), the nonlinear transformation causes a decrease in amplitude of the main tidal harmonic M2 with increasing distance from the area of generation and, in contrast, the amplitude of its subharmonics increases. As a result, the distribution of energy in the subharmonics becomes universal with the asymptotic behavior ∼(nω0)−2,6, where ω0 is the carrier frequency harmonic M2.
 Another feature of the spectrum of kinetic energy of current fluctuations (Figure 2) caused by nonlinear waves is that the peaks of the spectrum at M4 and M6 have the same order. However, the M8 peak is lower than that of M10. The calculations also showed that the quadratic and cubic terms in (1) are of the same order; therefore the internal semidiurnal tide in the northern GC is a cubically nonlinear wave.
 We have constructed a model spectrum of nonlinear internal tidal waves propagating in a channel of constant depth, based on the eR equation, and thereby shown that the transformation of these waves can be considered as a change of its spectral structure. Increased wavefront steepness corresponds to increased high-frequency subharmonics that are also included in the interaction if the amplitude of the internal tide is sufficiently large. As a result, the energy of the internal tidal waves is redistributed over the spectrum at higher frequencies.
 This work was supported by the Russian Basic Research Foundation, project 11-05-98564-r-vostok, by the Mexican National Council for Science and Technology (CONACYT project 105622) and by CICESE internal budget.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.