Parametric Subharmonic Instability Research Articles

Parametric Subharmonic Instability of the M2 Internal Tides in the Tokara Strait

AbstractThe Tokara Strait is a mixing hotspot due to the coexistence of complex bottom topographies and strong composite flow including both the Kuroshio and tidal currents. Although previous studies have revealed several mechanisms from the view of Kuroshio‐Topography interaction, the role of tides in driving mixing is still not clear. Given that it is located at the M2 critical latitude (29°N), parametric subharmonic instability (PSI) is expected as an important process responsible for the mixing. Here, we study PSI of the M2 internal tides in the Tokara Strait based on a high‐resolution model. Our model results indicate that intense near‐inertial waves are generated via PSI, which exhibit a horizontally layered structure and have much larger vertical wavenumbers than the M2 internal tides. Energy is transferred from the M2 internal tides to the near‐inertial waves around the generation sites, and most of the near‐inertial energy is dissipated locally. The dissipation rates of near‐inertial waves are comparable to those of the M2 internal tides. Simulations with and without the Kuroshio Current revealed the suppression of PSI along the Kuroshio path, which could be attributed to two mechanisms. First, the Kuroshio Current modifies the local minimum internal wave frequency by its horizontal and vertical shear, making the condition for PSI not satisfied. Second, the Kuroshio Current advects the near‐inertial waves downstream in the Okinawa Trough, which inhibits the accumulation of near‐inertial energy there. However, in most of the areas outside the Kuroshio path, PSI majorly contributes to mixing in and around the Tokara Strait.

Near- and Superinertial Internal Wave Responses and the Associated Energy Transfer after the Passage of Tropical Cyclone Fitow at a Midlatitude Shelf Slope

Abstract Observations from a mooring station at the East China Sea (ECS) shelf slope revealed both near- and superinertial dynamic responses to tropical cyclone (TC) Fitow. Different from the typical near-inertial response, near-inertial internal waves (NIWs) after TC Fitow showed a complicated phase pattern due to its superposition with parametric subharmonic instability (PSI)-generated M1 subharmonic waves. The wind-injected near-inertial kinetic energy (NIKE) was largely restrained to the upper 250 m. Wave packet analysis revealed the co-occurrence of enhanced NIKE, circularly polarized near-inertial currents, veering NIW propagation direction, and shrinking NIW vertical wavenumber at the base of the Kuroshio (∼180 m). This indicated the trapping and stalling of the TC-generated NIWs. Intense high-frequency internal waves (HFIWs) appeared immediately after TC Fitow which had an average period of ∼24 min and lasted ∼8 h. HFIWs also existed before the arrival of TC Fitow with a regular semidiurnal cycle. However, the HFIW after TC did not follow the semidiurnal cycle and had much larger amplitudes and longer-lasting periods. Local generation of supercritical flow over a slope or evolution from propagating internal tide as modified by TC may have induced these HFIWs. Along with the occurrence of intense HFIWs after TC Fitow, intense energy transfers from NIWs to HFIWs were identified. Due to the limited vertical propagation of TC-induced NIWs, it was the PSI-generated M1 subharmonic wave rather than the wind-induced NIW that contributed most to the energy transfer. Significance Statement Winds blowing over the ocean generate not only surface waves but also another type, so-called internal waves, which can travel down into the ocean. These internal waves carry a large amount of wind-injected energy into the ocean interior and are thought to play an important role in sustaining ocean turbulent mixing and circulation. We provided evidence that the passage of a tropical cyclone (TC) can induce different types of internal wave responses: near-inertial and high-frequency internal waves. Interactions and energy transfer between different internal waves occurred. With the TC-induced near-inertial internal waves trapped in the upper ocean, the background M1 subharmonic waves contributed most to the interactions with high-frequency internal waves. This fact is crucial in understanding the propagation and dissipation of wind-injected energy in the ocean, which plays an important role in controlling the ocean interior mixing and large-scale circulation.

A mechanism of enhanced subsurface near-inertial kinetic energy in the East China Sea associated with successive typhoons

Near-inertial waves (NIWs) play an important role in diapycnal processes and energy dissipation. A mooring observation deployed on the continental shelf in the East China Sea captured anomalously intensified subsurface near-inertial kinetic energy (NIKE) during the passage of Typhoon Danas (2013). An early study has investigated the role of Parametric Subharmonic Instability (PSI) induced by internal tides in the intensification of the subsurface intensified near-inertial velocity. However, results based on regional numerical simulations reveal that strong subsurface near-inertial velocity persists even in the absence of tidal effects, implying the existence of additional sources of NIWs. Our analyses showed that after excluding the effect of PSI, approximately 30% of the remaining subsurface NIKE can be attributed to another Typhoon Fitow (2013), which occurred a week prior to Typhoon Danas. Constrained by the Kuroshio current and the continental shelf, the NIKE generated by Typhoon Fitow propagates northward and reaches the mooring location, leading to the intensified subsurface NIW signal. Our simulation, together with the observations, suggests complicated NIW dynamics in continental shelf regions, involving interactions between successive typhoons, topography and background current, and differing from the open ocean. These interactions will further influence vertical mixing on the continental shelf along the pathway of NIW.

Observations of Parametric Subharmonic Instability of Diurnal Internal Tides in the Northwest Pacific

Abstract Based on yearlong observations from three moorings at 12°, 14°, and 16°N in the northwest Pacific, this study presents observational evidence for the occurrence and behavior of parametric subharmonic instability (PSI) of diurnal internal tides (ITs) both in the upper and abyssal ocean around the critical latitudes (O1 IT: 13.44°N; K1 IT: 14.52°N), which is relatively less explored in comparison with PSI of M2 ITs. At 14°N, near-inertial waves (NIWs) feature a “checkerboard” pattern with comparable upward- and downward-propagating components, while the diurnal ITs mainly feature a low-mode structure. The near-inertial kinetic energy at 14°N, correlated fairly well with the diurnal KE, is the largest among three moorings. The bicoherence analysis, and a causality analysis method newly introduced here, both show statistically significant phase locking between PSI triads at 14°N, while no significant signals emerge at 12° and 16°N. The estimated PSI energy transfer rate shows a net energy transfer from diurnal ITs to NIWs with an annual-mean value of 1.5 × 10−10 W kg−1. The highly sheared NIWs generated by PSI result in a 2–6 times larger probability of shear instability events at 14°N than 12° and 16°N. Through swinging the local effective inertial frequency close to either O1 or K1 subharmonic frequencies, the passages of anticyclonic and cyclonic eddies both result in elevated NIWs and shear instability events by enhancing PSI efficiency. Particularly, different from the general understanding that cyclonic eddies usually expel NIWs, enhanced NIWs and instability are observed within cyclonic eddies whose relative vorticity can modify PSI efficiency. Significance Statement Parametric subharmonic instability (PSI) effectively transfers energy from low-mode internal tides (ITs) to high-mode near-inertial waves (NIWs), triggering elevated mixing around critical latitudes. This study provides observational evidence for the occurrence of PSI of diurnal ITs in the northwest Pacific and its role in enhancing shear instability. Generally, anticyclonic eddies act to trap NIWs while cyclonic eddies tend to expel NIWs. Here we document elevated NIWs and shear instability within both anticyclonic and cyclonic eddies, which shift the local effective inertial frequency close to either O1 or K1 subharmonic frequencies, thereby enhancing PSI efficiency. Processes associated with PSI and the modulation of PSI efficiency by mesoscale eddies have significant implications for improving mixing parameterizations in ocean circulation and climate models.

Observations of near-inertial internal wave amplification and enhanced mixing after surface reflection

The overreflection process of near-inertial internal waves (NIWs) has been theoretically predicted for several decades; however, to the best of our knowledge, this phenomenon has never been comprehensively investigated in real ocean scenarios. Based on the buoy observations collected several days after the passage of Typhoon Lekima in the Yellow Sea, a NIW surface overreflection event is clearly captured. The observed NIWs undergo nearly total reflection meridionally but are amplified zonally after reflection by approximately 20% in amplitude and 56% in vertically integrated horizontal kinetic energy. Ray tracing analysis indicates that the NIW was generated in the wake of Typhoon Lekima in the area north of the Shandong Peninsula and may propagated to the buoy station as coastal-trapped internal Kelvin waves. A simulation using a slab mixed layer model suggests that local wind work was insufficient to generate the amplified NIWs. The temporal evolution of near-inertial energy also implies that the intensified near-inertial waves cannot be attributed to the spontaneous generation resulting from unbalanced flows or the parametric subharmonic instability of M2 internal tides during the reflection period. We found a high temporal correlation between the zonal NIW enhancement and the duration of a meridional lens-type shear flow after reflection, which is consistent with the Stern’s overreflection theory (Stern, 1977) that perpendicular background shear flow can feed energy to the incident NIWs. This indicates that the enhanced NIW may be stimulated by the near-surface reflection and the rotation effect plays a crucial role in the NIWs overreflection process in the real ocean. Furthermore, enhanced instability are found between the ocean surface and the upper thermocline after reflection. This study provides observational evidence that the background field could inject energy into the near-inertial band through NIW overreflection process, and may shed some light on understanding upper ocean mixing caused by NIW reflection.

Energy cascade in the Garrett–Munk spectrum of internal gravity waves

We study the spectral energy transfer due to wave–triad interactions in the Garrett–Munk spectrum of internal gravity waves based on a numerical evaluation of the collision integral in the wave kinetic equation. Our numerical evaluation builds on the reduction of the collision integral on the resonant manifold for a horizontally isotropic spectrum. We evaluate directly the downscale energy flux available for ocean mixing, whose value is in close agreement with the finescale parameterization. We further decompose the energy transfer into contributions from different mechanisms, including local interactions and three types of non-local interactions, namely parametric subharmonic instability, elastic scattering (ES) and induced diffusion (ID). Through analysis on the role of each mechanism, we resolve two long-standing paradoxes regarding the mechanism for forward cascade in frequency and zero ID flux for the GM76 spectrum. In addition, our analysis estimates the contribution of each mechanism to the energy transfer in each spectral direction, and reveals new understanding of the importance of local interactions and ES in the energy transfer.

Parametric subharmonic instability of inertial shear at ocean fronts

The two-dimensional stability of vertically sheared inertial oscillations at ocean fronts is explored through a linear stability analysis and nonlinear simulations. Baroclinic effects reduce the minimum frequency of inertia-gravity waves to an extent determined by the balanced Richardson number ${{Ri}}$ of the front. Below a critical value of ${{Ri}}$ , which depends on the strength of the inertial shear, the inertial oscillations become unstable to parametric subharmonic instability (PSI) resulting in growing perturbations that oscillate at half the inertial frequency $f$ . Since the critical value is always greater than 1, PSI can occur at fronts stable to symmetric instability. Although modest in weak inertial shear, growth rates exceeding $f/2$ can be achieved for inertial shear greater than or equal to the thermal wind shear. Our formulation allows for non-hydrostatic perturbations and can be applied to initially unstratified geostrophic adjustment problems. We find that PSI will almost totally damp the transient oscillations that arise during geostrophic adjustment. The perturbations gain energy at the expense of the inertial oscillations through ageostrophic shear production. The perturbations then themselves become unstable to secondary Kelvin–Helmholtz instabilities creating a pathway by which the inertial oscillations can be dissipated rapidly. In contrast to symmetric and baroclinic instabilities that draw on a front's kinetic or potential energy, PSI acts to increase the energy stored in the balanced front as the convergence and divergence of the eddy-momentum fluxes set up a secondary circulation in the sense to stand up the front.

Stability of internal gravity wave modes: from triad resonance to broadband instability

A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter $\mu$ , compares to the basic-state amplitude parameter, $\epsilon \ll 1$ . For $\mu ={O}(1)$ , the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that $\mu \ll 1$ but $\alpha =\mu /\epsilon \gg 1$ , this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as $\mu$ is further decreased holding $\epsilon$ fixed, higher-frequency perturbations than these two subharmonics come into play, and when $\alpha ={O}(1)$ Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when $\alpha ={O}(1)$ PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.

Turbulence Generated through Critical Reflection of Internal Waves off the Seafloor due to Nontraditional Effects

Abstract Recent work has shown that, when nontraditional (NT) effects associated with the horizontal component of the Coriolis parameter are taken into account, equatorial waves (EWs) experience critical reflection when they reflect off the seafloor at the latitude where their frequency is equal to the inertial frequency. As a result, the vertical shear associated with the wave is strongly enhanced locally and results in bottom-intensified mixing. Using an off-the-shelf parameterization for mixing, these studies have shown that this process could play an important role in driving diapycnal upwelling in the abyssal ocean, but the specific mechanisms generating the mixing have not been studied yet. In this work, we address this limitation by running two-dimensional, high-resolution, nonhydrostatic simulations of the critical reflection of internal waves modified by NT effects. These simulations can resolve the instabilities triggered when the wave reflects off the bottom, allowing us to characterize the energy cascade to smaller scales and to estimate the mixing it generates. We find that shear instabilities drive elevated turbulent diffusivities between 10−1 and −10−3 m2 s−1 over a critical layer of 100–300 m thick. The shear instabilities result directly from the enhancement of kinetic energy in the reflected wave that is confined against the seafloor during the critical reflection process. Simultaneously, higher harmonics are generated and flux energy upward in the water column. These higher harmonics are unstable to parametric subharmonic instability, which absorbs their energy and drive enhanced dissipation above the critical layer, to a height of O(1000) m off the bottom. We show how these results depend on key elements of the EWs and of the medium and discuss the implementation of a parameterization of these effects in global ocean models.

Parametric Subharmonic Instability of Diurnal Internal Tides in the Abyssal South China Sea

Abstract Internal waves close to the seafloor of abyssal oceans are the key energy suppliers driving near-bottom mixing and the upwelling branches of meridional overturning circulation, but their spatiotemporal variability and intrinsic mechanisms remain largely unclear. In this study, measurements from 10 long-term moorings were used to investigate the internal wave activities in the abyssal South China Sea, which is an important upwelling zone. Strong near-inertial internal waves (NIWs) with current velocity pulses exceeding 5 cm s−1 were observed to dominate the near-bottom internal wave field at approximately 14°N. These abyssal NIWs were phase-coupled with diurnal internal tides (D1), and both displayed common seasonal variations that were larger in winter and summer, providing evidence of diurnal parametric subharmonic instability (PSI) near its critical latitudes (CLs). Emitted from the bottom, near-inertial kinetic energy rapidly decreased by one order of magnitude from depths of ∼120 to ∼620 m above the bottom. Near rough topographies, the abyssal PSI was shifted poleward to approximately 14.8°N by negative relative vorticities of passing anticyclonic eddies or topographic Rossby waves. Compared with flat topography, PSI near rough topography was significantly promoted by topographic-localized strong D1 with high-mode structures, creating abyssal NIW bursts. Bottom-reaching shipboard conductivity–temperature–depth profiles revealed that the bottom mixed layers became much thicker when approaching CLs, suggesting that abyssal PSI potentially accelerates the ventilation and upwelling of bottom water. The observational results presented here illustrate notable spatiotemporal variations in abyssal NIWs regulated by PSI and call for consideration of PSI to better understand near-bottom mixing and upwelling.

Enhanced turbulent mixing in mesoscale eddies near the critical latitude of the M2 internal tides

Mesoscale eddies can elevate turbulent mixing, but the mechanisms have not been fully understood. Based on a field experiment in two mesoscale eddies near the critical latitude of the M2 internal tides in the northwestern Pacific Ocean, enhanced turbulent mixing was found both in and below the mixed layer. Microstructure observations indicate that enhanced turbulent mixing mainly appeared outside the eddy center in the mixed layer (approximately 117 m), which was speculated to be related to geostrophic current shear. By analyzing the CTD and Argo measurements through a fine-scale parameterization method, enhanced turbulent mixing was found to be concentrated at the southern flank of the anticyclonic eddy below 300 m, which showed little correlation with topographic roughness and local winds. Based on a ray-tracing model, the wind-generated near-inertial waves at higher latitudes were also excluded as the cause of the enhanced turbulent mixing. The enhanced turbulent mixing below 300 m was generally consistent with the features of subthermocline eddies and eddy-modulated parametric subharmonic instability (PSI) of the M2 internal tides. However, due to the limitation of observations, the eddy-modulated PSI mechanism was not fully validated.

Energy Transfer From PSI‐Generated M1 Subharmonic Waves to High‐Frequency Internal Waves

AbstractInternal tides (ITs) represent one of the major energy sources available for small‐scale mixing in the ocean. Here, the energy pathway associated with ITs is examined based on measurements obtained at the East China Sea shelf slope. We reveal that after the occurrence of parametric subharmonic instability (PSI), energy can be further transferred from the PSI‐generated subharmonic waves to high‐frequency internal waves (HFIWs) through nonlinear interactions between them. This interprets the observed similar varying trends of magnitude between internal waves at near‐inertial and high‐frequency bands. Bispectrum and bicoherence estimates confirm that the M1 subharmonic wave interacts with a pair of HFIW which have larger vertical scales and a broad range of frequencies. The averaged net value of the energy transfer rate reaches 2 10−7 W kg−1, which has implications for a further significant role of this energy pathway on the spatial distribution of ocean mixing.

Parametric instabilities of a stratified shear layer

A Boussinesq fluid inside a stably thermally stratified square container whose walls are inclined ${45}^{\circ }$ with respect to gravity, with two opposite walls kept at constant temperatures and the other two insulated is nearly isothermal in the regions above and below the horizontal diagonal. The flow is concentrated in the wall boundary layers and a shear layer centred about the horizontal diagonal. The equilibrium is maintained by the balance between dissipation in the shear and boundary layers, the heat fluxes at the constant temperature walls, and the induced flow resulting from the no-flux condition at the inclined insulated walls. The dynamical response of the fluid to vertical oscillations of the container is studied over a range of forcing frequencies. For a small forcing amplitude and below a viscosity-dependent cutoff forcing frequency, this response exhibits a modal cellular structure localized about the shear layer. With increasing forcing amplitude, the response experiences instabilities, studied here numerically at a forcing frequency above the cutoff frequency, that are similar to those encountered in the Faraday wave problem, such as parametric subharmonic instability, triadic resonance and resonant collapse.

Near-Surface Reflection and Nonlinear Effects of Low-Mode Internal Tides on a Continental Slope

AbstractTwo sets of mooring data were collected at two sites (MA and MB) along a cross-slope section on the northeastern continental slope in the South China Sea (SCS). These data are used to investigate evolution and energy decay of low-mode semidiurnal (M2) internal tides on a subcritical slope with respect to M2. At the deep portion of the slope (~1250 m; MA), the M2 internal tides show upward energy propagation, while vertically standing M2 internal tides are often observed at shallow MB (~845 m). A two-dimensional linear internal tide model with realistic topography and stratification reproduces the observations, suggesting that low-mode M2 internal tides incident on subcritical slopes evolve into vertically propagating internal waves due to topographic scattering, propagate upward to the boundary, and reflect from the sea surface. The reflection point largely depends on the phase between the modal components of the incoming flux. In the near-surface reflection region, two kinds of nonlinear effects are observed to decay energy of the incoming internal tides. One is the resonant parametric subharmonic instability which transfers M2 internal tides to diurnal subharmonics M1 (=M2/2), but the instability is found to mainly depend on the incident waves. The other one is the nonresonant wave–wave interaction, producing two higher-harmonic M4 (=2M2) rays with opposite vertical propagation. A strong westward mean flow is observed in the interacting region, with amplitude comparable to that of the incident waves. This mean flow also appears to be generated by the nonlinear reflection of the M2 internal tides.

Instabilities of finite-width internal wave beams: from Floquet analysis to PSI

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Near-inertial parametric subharmonic instability of internal wave beams in a background mean flow

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Disintegration of the K1 Internal Tide in the South China Sea due to Parametric Subharmonic Instability

AbstractThe disintegration of the equatorward-propagating K1 internal tide in the South China Sea (SCS) by parametric subharmonic instability (PSI) at its critical latitude of 14.52°N is investigated numerically. The multiple-source generation and long-range propagation of K1 internal tides are successfully reproduced. Using equilibrium analysis, the internal wave field near the critical latitude is found to experience two quasi-steady states, between which the subharmonic waves develop constantly. The simulated subharmonic waves agree well with classic PSI theoretical prediction. The PSI-induced near-inertial waves are of half the K1 frequency and dominantly high modes, the vertical scales ranging from 50 to 180 m in the upper ocean. From an energy perspective, PSI mainly occurs in the critical latitudinal zone from 13° to 15°N. In this zone, the incident internal tide loses ~14% energy in the mature state of PSI. PSI triggers a mixing elevation of O(10−5–10−4) m2 s−1 in the upper ocean at the critical latitude, which is several times larger than the background value. The contribution of PSI to the internal tide energy loss and associated enhanced mixing may differ regionally and is closely dependent on the intensity and duration of background internal tide. The results elucidate the far-field dissipation mechanism by PSI in connecting interior mixing with remotely generated K1 internal tides in the Luzon Strait.

Observed Internal Tides and Near‐Inertial Waves in the Northern South China Sea: Intensified f‐Band Energy Induced by Parametric Subharmonic Instability

AbstractThe variations in the internal tide (IT) and near‐inertial waves (NIWs) that occurred in the summer of 2006 over the continental shelf of the northern South China Sea were observed using three bottom‐mounted moorings equipped with acoustic Doppler current profilers. Due to the strong nonlinear interaction, large portions (~40–60%) of the semidiurnal IT energy at the continental shelf were found to be incoherent; this proportion gradually increased from the shelf break to the offshore region. More importantly, our observations showed that parametric subharmonic instability (PSI) can enhance the f‐band energy in the shallow water (~100 m) on the continental shelf. Using plane‐wave fitting and the bicoherence spectrum, we confirmed that the intensified NIWs and the observed D2‐f waves are nonlinear coupled waves derived from the semidiurnal internal tidal PSI. These two subharmonic waves were observed in the near field of a strong semidiurnal IT generation site at the shelf break. By tracking the M2‐ray path, we verified that the PSI occurred in the pycnocline in the path of the reflected wave beam that was characterized by strong nonlinear instabilities and it could be modulated by mesoscale eddies in this shallow shelf water.

Spontaneous superharmonic internal wave excitation by modal interactions in uniform and nonuniform stratifications

Internal waves in the ocean are well-recognized to play an important role in the global energy budget. Triadic resonance is one mechanism via which these internal waves transfer their energy to other spatial and temporal scales before dissipation, at locations blue both near and away from their generation sites. In this paper, we perform a combined theoretical and numerical study of triadic resonance in internal wave modes in a finite-depth ocean with an arbitrary stratification profile. Considering internal waves generated at spatially localized regions, the spatial evolution of the modal amplitudes within a resonant triad are derived based on the method of multiple scales. Two representative cases are considered: (i) modes 1 and 2 at a specific frequency ω0 in triadic resonance with the mode-1 superharmonic wave (frequency 2ω0) in a uniform stratification, and (ii) a self-interacting mode-3 at a specific frequency ω0 in triadic resonance with the mode-2 at frequency 2ω0 in an ocean-like nonuniform stratification. In case (ii), any initial energy in mode-3 at frequency ω0 gets permanently transferred to mode-2 at frequency 2ω0. Numerical simulations are performed to show the spontaneous excitation of superharmonic internal waves resulting from modal interactions in the aforementioned cases, and quantitatively validate the initial spatial evolution of the wave field predicted by the amplitude evolution equations. Furthermore, numerical simulations at off-resonant frequencies are used to identify the range of primary wave frequencies (around the resonant frequency) over which spontaneous superharmonic wave excitation occurs. Quantitative estimates of energy transfer rates within the resonant triads considered show that superharmonic wave excitation resulting from modal interactions should be an important consideration alongside other triadic resonances like parametric subharmonic instability (PSI). We conclude by giving estimates of the relative importance of superharmonic wave excitation in the ocean, and provide motivation for future studies.

Energy transfer in resonant and near-resonant internal wave triads for weakly non-uniform stratifications. Part 1. Unbounded domain

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