Ocean motions at frequencies of the internal wave band are generally associated with freely propagating waves that are supported by stable vertical stratification in density. Previous analyses of yearlong current observations from the Bay of Biscay showed that a finestructure of semidiurnal tidal and near-inertial higher harmonics fills the spectrum. Here, a simple model is presented of forced nondispersive motions with forward energy cascade. The model fits the spectral shape of higher harmonics well within statistical significance and shows that such interactions imply maximum wave steepness in a balance between forcing and turbulent mixing. The single fitting parameter takes a value of approximately one, at which the barotropic tidal flow speed equals the internal wave phase speed. We infer that the barotropic tide sets a non-linear limit to baroclinic current scales without generating non-linear higher harmonics directly.

Following in situ observations (e.g.,

In theory, freely propagating internal gravity waves can exist in the IW frequency (σ) band between |f(ϕ)| < σ < N(z), N >> f. The frequency range is limited at the high end by the depth (-z) dependent buoyancy frequency N(z) = (-g(dlnρ/dz+g/c_{s}^{2}))^{1/2}, where g is the acceleration of gravity, and c_{s} the speed of sound describing compressibility effects. At the low end, it is limited by the latitudinal (ϕ) dependent inertial frequency f(ϕ) = 2Ωsinϕ, which is twice the local vertical component of the Earth’s rotation vector Ω (e.g.,

In this paper a simple heuristic model is proposed describing non-linear interactions that generate such higher harmonics peaks in the IW-band and fit the spectral shape of yearlong current meter observations from the deep Bay of Biscay. The model follows theoretical suggestions (

It is common that simplified models are based on strong assumptions. The simplified ‘cartoon’ model, proposed below, describes the cascade of tidal and inertial-tidal higher harmonics. It not only assumes that advection dominates other forces in the equation of motion, but also that advection of the ‘barotropic’ surface tide as well as of inertial motions does not play a role due to large scales compared to the small scale of the internal-tide. This scale-separation immediately clarifies why inertial-inertial interactions, leading to motions at frequencies 2f, 3f etc., and, consequently, advection of the ‘baroclinic’ internal tide by these inertial higher harmonics, that would e.g. lead to M_{2}+2f, are weak. M_{2} denotes the semidiurnal lunar tidal frequency.

Uniformly-stratified fluids may display both free, resonant as well as forced, non-resonant higher harmonics (

Currents were evaluated from two moorings deployed in the Bay of Biscay NE-Atlantic Ocean during 11 months, above the continental slope at 46°39’ N, 05°29’ W (water depth H = 2450 m) and above the abyssal plain at 45°48’ N, 06°50’ W (H = 4810 m), see

Current meter mooring sites in the Bay of Biscay with black contours of topography every 1000 m water depth. The coloured dots correspond with the spectra in Figure 2.

From a few CTD density profiles, obtained near the moorings, stratification was estimated N(z) = (1 ± 0.5)(20 + 0.0034z) cpd, –4480 < z < –2740 m (frequency was calculated in cycles per day, 1 cpd = 2π/86400 s^{–1}). The significant one-standard deviation of variations follows from computations over small 10-m vertical scales that will play a significant role in the modelling later on. The large-scale depth dependence changed abruptly above 2740 m. At 1500 m, N ≈ 28 cpd.

Tidal harmonic analysis (_{2} the barotropic signal is about twice the value of the baroclinic signal.

Observed kinetic energy spectra P_{KE}(σ) revealed larger energy at shallower depth (where N is larger), except at f (_{2}, respectively) and higher harmonics (indicated as M_{4}, M_{6},…and M_{2}+f, M_{4}+f…). These energy peaks exceeded the spectral continuum that sloped with frequency like P_{KE} ~ σ^{–1}, for f < σ < 7 to 10 cpd. A σ^{–1} spectral slope, ‘pink noise’ (^{–1} spectrum is broadly seen as bearing evidence of self-organized criticality (_{10} at the deeper site and up to M_{16} at the shallower site. The higher harmonics were found in a sequence of decreasing amplitudes. Recalling that the observations were made at single points in space, it is unlikely that they represent locally-forced, freely-propagating higher harmonic internal waves (e.g.,

Kinetic energy spectra from 11 months of current meter observations at 1000 m above the seafloor in H = 4810 m water depth (red) and H = 2450 m (blue). Spectra were moderately smoothed (v ≈ 30 df) and not offset vertically. The difference in energy levels between the spectra corresponded to the difference in N(z), which variation is indicated between the vertical bars in the top-right corner. This corresponds with the vertical distance between the sloping lines at fall-off rates σ^{–1} (solid and dashed corresponding to red and blue spectra, respectively). Constant slopes in log-log plot are indicated “–1,–2,–3” representing σ^{–1}, σ^{–2}, σ^{–3}, respectively. Spectra of model (2) are superposed for observed barotropic and baroclinic fundamental tidal amplitudes and fitting parameter γ. Three model examples are given, two for tidal-interactions (+) and one for inertial-tidal-interactions including frequency-corrections to coefficients (o). They fit well the observed energy levels for nearly the same γ (see text). In all cases, reference amplitude is the barotropic M_{2} current amplitude, indicated at f and M_{2} (leftmost o, +). Baroclinic M_{2}-variance are a quarter of peak M_{2}-values.

When the kinetic energy was large at f (deepest mooring, small N), energy at f-interaction frequencies (e.g. M_{2}+f) showed a spectral fall-off rate with frequency like P_{KE} ~ σ^{–3}, which was a typical fall-off rate, or even steeper, for higher tidal harmonics, see red plusses in _{n}+f, n = 2, 4,…, scaled like ~σ^{–2}. When smoothed strongly, the latter records showed overall spectral fall-off rate close to ~σ^{–2} for f < σ < N (^{–3}. This is significantly steeper than the canonical fall-off rate P_{KE} ~ σ^{p}, –2.5 < p < –1.5 for open-ocean internal waves (_{2}+2f, etc., motions do not exceed the continuum level (

In an inviscid uniformly-stratified fluid, a single-frequency free plane obliquely propagating internal wave (or a set of collinear propagating plane waves) is governed by linear equations as the non-linear advection terms exactly cancel in the equations for vorticity and buoyancy (

Sketch of internal tidal velocity and its direction of variation. _{g}.

However, in reality non-linearity is not expected to vanish. Like surface waves, internal waves may manifest themselves partially as displacement waves on layers of enhanced stratification. This is because the ocean rather consists of non-uniform, alternatingly weaker and stronger stratified layers (interfaces) (

Sofar, no general solution has been given to describe strongly non-linear internal wave motions using the full set of governing equations (

However, considering that internal waves occur intermittently (

Consider a deformed wave’s discrete kinetic energy spectrum P_{j}(σ_{j}) = ½Û_{j}Û_{j}*, the asterisk denoting complex conjugate, of IW scalar current components û_{j} = Û_{j}exp[_{j}ξ-σ_{j}t)] with frequency σ_{j}, wavenumber k_{j} and scalar amplitude Û_{j}. Here, ^{2} = –1 and j is a positive integer starting from the fundamental baroclinic component j = 1. The coordinate ξ is in the oblique energy propagation direction of the deformed wave, parallel to the phase lines. It is assumed to indicate the direction of the largest baroclinic current component û, associated with an internal tidal beam, but this beam is deformed by non-uniform stratification (_{1}. Compared to the horizontal barotropic current, the latter may have a relatively large component in the perpendicular direction. Similar along-beam variations have been considered in sophisticated models concerned with subharmonic internal wave generation (

Remark that the incident internal wave beam ~exp(-_{j}. Thus the ζ-dependence is suppressed here (it is assumed to be virtually uniform on the scale of the along-beam variations), and does not contribute non-linearly to the higher harmonics. This leaves the along-beam velocity, or the velocity of interface waves resulting from beam-transmission and –reflection, now having a ξ-dependence due to non-uniform stratification (_{j}.

In our model discrete spectrum only those parts are considered that are entirely governed by advection in the wave-energy propagation direction, ξ, and no advection perpendicular to this direction. The model expresses forced non-linear interactions between linear motions that result in bound non-freely-propagating motions. Rotation is neglected by assuming that interactions are fast compared with the inertial period, resulting in motions at frequencies σ >> f. A pressure gradient forcing is assumed being linear and only governing the fundamental (j = 1) tidal component. Diffusion is neglected. For clarity, we here exclude both advection by a barotropic surface tidal current û_{0} (Û_{0}, σ_{0} = σ_{1}, k_{0} ≈ 0), since we consider motions within a barotropic oscillating system, as well as advection of the barotropic component because its wavenumber k_{0} << k_{1}. We thus base the following cartoon model (1) on the shock-wave model of Platzman (

Only non-resonant higher harmonics σ_{j} = jσ and k_{j} = jk are found here as the compound wave’s frequency and wavenumber do not themselves satisfy the dispersion relation, required for resonant triads (

Equation (1) implies a forward cascade of energy from a source at a fundamental baroclinic internal (tidal) constituent (σ_{1}, k_{1}, Û_{1}) ≡ (σ, k, Û), determining the amplitudes of the successive harmonics,

recursively defining factors _{j} ≡ σ_{j}/k_{j} = c), as found here for all the higher harmonics of the baroclinic M_{2} tide, is remarkable. It not only implies non-dispersive wave steepening, but, since the internal wave and its higher harmonics are synchronized a spatio-temporal coherency. As we will see below, this synchronization is lost when waves of other frequencies and wave numbers advect the baroclinic tide. In that case the compound frequency and wave number do no longer grow at the same rate, implying varying c_{j} and loss of coherency. This translates in an absence of spectral peaks at frequencies as M_{2}+2f etc. The model (2) contrasts with the shock-wave model by Platzman (

From (2) a consistent model spectrum P_{m}(σ) = ∑_{j} P_{j}(σ_{j}) is obtained fitting the observed kinetic energy at discrete frequencies ∑_{j} P_{df}(σ_{j}) after tuning to γ = 0.48 ± 0.05 under the conditions |log(P_{df}(σ_{j})/P_{j}(σ_{j}))| < 10% for all j and σ < N (_{0} = (0.96 ± 0.11)c, using observed Û_{0}/Û = 2.0 ± 0.15 following harmonic analysis splitting the original signal into semidiurnal time-coherent signal and its remainder baroclinic signal. It suggests ∑_{j} P_{df}(σ_{j}) is the spectral representation of the spatio-temporal process of internal wave steepening and possibly breaking.

Thus, large scale barotropic Û_{0} is found setting a non-linear limit that determines baroclinic Û-length scales, whilst not generating non-linear constituents directly. This can be seen as for k_{0}→0 barotropic advection yields only forced, non-resonant, dispersive (σ_{j} = jσ, k_{j} = k) harmonics Û_{j} = (Û_{0}/c)^{j–1}Û/j!. Using the same γ, these constituents show a much faster energy drop with frequency than the observed σ^{–3} (for σ > M_{4}).

Advection of baroclinic M_{2}, M_{4}, …, by motions at other frequencies, such as at the inertial frequency f, leads to spectral amplitudes at M_{2}+f, M_{4}+f, ….. For areas having approximately the same amplitude at f as Û, the model fits the observations with a γ-value within the above error bounds (see red symbols in _{j} that include a frequency correction. As motions at f are supposed to be large-scale they will however not themselves be subject to advection. This implies that there is no spectral route to provide energy at M_{2}+2f, M_{2}+3f, …, or M_{4}+2f, …, etc. More seriously, as aluded to above, the combination frequencies M_{2}+f, M_{4}+f, …, loose their spatio-temporal synchronization. Their wave numbers do not increase at the same rate as the combination frequencies, implying a loss of coherence. This reasoning equally applies to motions at all neighbouring frequencies. In fact, _{2} band is transposed to the M_{2}+f/M_{4} band via the baroclinic tide following (2), and similarly to higher frequency bands. Hereby, the shape-shrink in frequency is attributable to log-log plotting and the self-similar variance-shrink to (2) that equally affects all frequencies in the primary band. The splitting of energy to neighbouring frequencies is in part due to interactions with the slowly varying stratification background (

Compared to background energy levels, large variance at tidal and inertial-tidal higher harmonics is observed at the deep site, more than 100 km from the continental slope. It may be questioned whether motions at these higher harmonics are reminiscent of non-linear interactions between and by deformation of internal waves in the ocean. Alternatively, these motions may be the result of effects of sloping underwater topography upon which internal waves break. The associated strong turbulence may reach far into the deep ocean. Observations are lacking of turbulence, but numerical internal tide modelling demonstrates the possibility of multiple interactions at such distances from topography (

This is because the observed similarity of semidiurnal particle displacement speed and phase speed, as in hydraulic jumps at the point of overturning, suggests a gradient Richardson number Ri ≈ 1. It also suggests a transition from weak wave-wave interaction to, strong, stratified turbulence (^{–1} as observed, for both inertial-tidal and tidal higher harmonics. Because γ was found independent of N, this length-scale may be fundamental for baroclinic non-linear transfer via advection. Like in shallow seas, the advection term seems to dominantly generate non-linear deep-ocean motions in the Bay of Biscay.

The simple model used here invites future clarification of a balance between internal wave forcing and diapycnal mixing in more sophisticated non-linear internal wave models. As existing numerical models are generally based on the Garrett and Munk (

The assistance of the crew of the R/V Pelagia was greatly enjoyed. The Bay of Biscay Boundary (TripeB; Hendrik van Aken) project was supported by grants from the Netherlands organization for the advancement of scientific research, NWO.

The authors have no competing interests to declare.