The Mediterranean Sea is often considered as a “miniature ocean”, due to its morpho-bathymetric characteristics and its thermohaline cell, where cold and/or salty waters sink during winter at specific regions and, subsequently, spread to intermediate and deep, near-bottom layers (Wüst, 1961; Bergamasco and Malanotte-Rizzoli, 2010; Millot and Taupier-Letage, 2005; Somot et al., 2006; Tsimplis et al., 2006; Schroeder et al., 2016; Artale et al., 2018). Moreover, among the Mediterranean-type climates, i.e., those defined by temperate, wet winters and hot/warm, dry summers over the western edges of five continents, the Mediterranean is the only one in which the atmospheric variability produces an intense deep-water formation (Seager et al., 2019).

Several analyses of the Mediterranean overturning circulation showed that this marginal basin is well ventilated, in comparison to the world ocean (Wüst, 1961; Malanotte-Rizzoli and Hecht, 1988; Pinardi and Masetti, 2000; Theocharis et al., 2002; Artale et al., 2006; Malanotte-Rizzoli et al., 2014; Schneider et al., 2014). Negative hydrological balance and cold-dry winds, blowing during the winter season, make the Mediterranean Sea an extraordinary place of intense ocean water mass transformation (Pinardi and Masetti, 2000), and thus, an ocean laboratory to study the main processes involved in the global ocean circulation.

Surface water from the Atlantic Ocean enters the Mediterranean basin through the Strait of Gibraltar (Fig. 1a), forming a shallow overturning circulation at the upper layer. Flowing eastward, along the African coasts, this Atlantic water increases its salinity and density, then sinking in the Levantine basin in the eastern Mediterranean and forming the Levantine Intermediate Water (LIW). The LIW flows back into the opposite direction, mixing with the surrounding water mass found along its travel back and leaving the Mediterranean Sea through the Strait of Gibraltar, which works as a salt-valve (Artale et al., 2006). Deep water exchanges between the western and the eastern Mediterranean basins occur within the shallow Sicily Channel (Fig. 1a), which makes the deep-water circulation of the two sub-basins rather independent from each other (Sparnocchia et al., 1999; Napolitano et al., 2003; Béranger et al., 2004; Schroeder et al., 2006). In the eastern basin, the Eastern Mediterranean Deep Water (EMDW) is formed, alternately, in separate regions of the Eastern Mediterranean Sea (Fig. 1a), i.e., the southern Adriatic Sea and the Aegean Sea (Gačić et al., 2010). This cold and high salinity winter water, after deep convection, flows into the Ionian Sea (Wüst, 1961; Roether and Schlitzer, 1991; Schlitzer et al., 1991; Roether et al., 1996; Bensi et al., 2013a, b; Bellacicco et al., 2016), i.e., the deepest sub-basin of the Mediterranean Sea, which has a maximum depth close to the mean depth of the global ocean (Fig. 1a). Such a thermohaline circulation is, therefore, characterized by a superposition of intermediate and deep, zonal and meridional overturning cells (Bergamasco and Malanotte-Rizzoli, 2010).

In the Ionian Sea boundary currents show intense downwelling due to their interaction with the topographic constraints while the deepest part of the basin is characterized by extremely weak stratification (Waldman et al., 2018), which induces a deepening and barotropization of the flow (Holte and Straneo, 2017; Send and Testor, 2017). These deep waters are characterized by IW-turbulence and mixing (van Haren and Gostiaux, 2011). Moreover, the Ionian Sea shows a semi-diurnal lunar tidal (M2) and a dominant motion that is around the inertial frequency, mainly as freely propagating IWs (Gerkema and Zimmerman, 2008). For this reason, the Ionian Sea constitutes a unique example for studying the fundamental dynamics and processes of the abyssal layers in the Mediterranean basin, also in the light of turbulent mixing due to IW breaking (Rubino et al., 2012; Artale et al., 2018; Giambenedetti et al., 2024).

Internal waves constitute a fundamental component of the Mediterranean Sea dynamics and play a key role in vertical energy transfer and in the long-term redistribution of heat and tracers. The semi-enclosed geometry of the Mediterranean, its mid-latitude setting, and the presence of steep and irregular bathymetry strongly shape the internal wave field, distinguishing it from that of the open ocean (Millot and Taupier-Letage, 2005; Schroeder et al., 2016).

In particular, the Eastern Mediterranean is characterized by weak deep stratification, quasi-homogeneous abyssal layers, and pronounced topographic gradients, conditions that make this sub-basin highly sensitive to internal wave processes and their variability (van Haren and Gostiaux, 2011). Despite the very low buoyancy frequencies typical of its deep layers, long-term high-resolution temperature measurements revealed intense internal wave activity in the deep Ionian Sea: vertical displacements reached several hundreds of meters, with significant vertical velocities occurring at near-inertial frequencies (van Haren and Gostiaux, 2011). These observations further showed that energetic internal waves advected through abyssal waters can generate strong vertical shear and episodic overturning, indicating that internal wave–induced mixing remains effective even under extremely weak stratification. Weak stratification does not inhibit internal wave dynamics; rather, it modifies their geometry and frequency content, allowing motions with substantial vertical components to develop in the deep basin (van Haren and Millot, 2004, 2005).

Evidence of energetic internal wave variability has also been reported in other Mediterranean sub-basins, such as the Western Mediterranean, where studies in the Tyrrhenian Sea documented active internal wave fields associated with mesoscale circulation and boundary processes (van Haren and Millot, 2004; Buffett et al., 2017). Although these studies focus on different hydrographic and dynamical settings, they support the broader view that internal waves remain dynamically relevant throughout the Mediterranean Sea, including deep environments.

In addition to broadband internal wave activity, recent studies have highlighted the importance of internal tides as a persistent and organized contribution to the Eastern and central Mediterranean internal wave climate. Barotropic tides are relatively weak in the Mediterranean Sea and deep signals are embedded in internal-wave noise (Sannino et al., 2015; van Haren, 2024). However, their interaction with steep bathymetric features, particularly in straits and along major topographic gradients, can generate energetic baroclinic internal tides (Artale et al., 1989; Oddo et al., 2023). Observational and modeling studies in the central Mediterranean have shown that diurnal internal tides, often with Kelvin-like characteristics, are generated at the transition between shallow regions of the Sicily Channel and the deep Ionian basin, propagate away from their generation sites, and contribute significantly to the internal wave energy budget while inducing strong vertical shear (Oddo et al., 2023). Direct observations in the deep Ionian Sea further indicate that baroclinic internal tides penetrate into abyssal layers, modulating temperature and velocity fields even under weak stratification (Giambenedetti et al., 2023).

This body of evidence defines a Mediterranean internal wave climate that is both spatially heterogeneous and dynamically rich, shaped by the interplay between stratification, bathymetry, and tidal forcing. Within this framework, the Ionian Sea emerges as a key region where internal wave activity intersects with large-scale circulation variability. The hydrographic structure of Atlantic Water, Levantine Intermediate Water, and deeper water masses controls internal wave propagation paths and vertical structure (Budillon et al., 2010), while variability in this structure directly affects the efficiency and vertical reach of internal wave–induced mixing, particularly along continental slopes and over irregular bathymetry. On longer timescales, decadal changes in water mass properties further modulate the background conditions for internal wave dynamics and internal redistribution of momentum and tracers (Gačić et al., 2010).

When interacting with rough topography, internal waves and internal tides have been linked to the redistribution of heat and tracers in abyssal layers, contributing to the observed variability of heat content and influencing long-term basin-scale circulation (Artale et al., 2018). This mechanism becomes particularly relevant during periods of reduced deep-water formation, when internal mixing plays an increasingly important role in controlling abyssal ventilation. Internal tides, in particular, represent a continuous source of energy that sustains mixing in deep layers, complementing episodic convective events and reinforcing the coupling between the deep Ionian interior and boundary regions (Giambenedetti et al., 2023; Oddo et al., 2023). Overall, this internal wave climate provides the dynamical background against which internal wave–driven mixing processes operate and contribute to the thermohaline structure and long-term evolution of the Ionian Sea and the broader Eastern Mediterranean basins.

An intense observational activity was conducted in the Ionian Sea in the framework of the Cubic Kilometre Neutrino Telescope (KM3NeT) project, from April 2006 to May 2009 (Katz, 2006). During the KM3Net cruise (July 2007), hydrographic measurements were spread over a large area, including a section that almost synoptically spanned the central Ionian Sea, from Sicily to Greece. On the western side and up to the central abyssal plane, a marked stratification was found in the deep layers, with a warmer and saltier water on the bottom, below a fresher and slightly colder layer (located approximately at about 2750 m), probably due to deep water formation processes at different times (Sparnocchia et al., 2011). Deep current meter measurements collected in the westernmost part in the period 2007–2008 showed a quite energetic and impulsive abyssal thermohaline circulation pattern, strongly affected by the bathymetric constraints, with maximum velocities up to 15 cm s⁻¹ and cyclonic and anti-cyclonic mesoscale structures, with a period of 5-to-11 d, superimposed on the background geostrophic flow (Rubino et al., 2012; Meccia et al., 2015); these measurements confirmed that tides are “limited” even at depth (i.e., order of ∼ 1 cm s⁻¹).

Artale et al. (2018), by analyzing water column characteristics of the Ionian Sea, observed during the last three decades, focused on the hydrological processes occurring at the bottom layer. They found that the ocean circulation in this abyssal plain is strongly affected by an interplay between advection and diffusion. The progressive warming and salinification of this sub-basin produced warmer near-bottom waters, causing an anomalous heat storage of ∼ 1.6 W m⁻², i.e., a value three times larger than the equivalent climate trend occurring in the same period at global scale (Bindoff et al., 2007). The analysis of Artale et al. (2018) investigated the triggering of a diapycnal mixing due to rough bathymetry. To explore topographic-induced mixing, these authors estimated dissipation rates from the “CTD strain-based” parameterization (Kunze et al., 2006), showing the role of the sea bottom in enhancing isopycnal vertical strain. In particular, they highlighted that the turbulent kinetic energy generated by breaking internal waves is large within a few hundred meters of rough-bottom topography and decays to weaker values farther up in the water column; the bottom enhancement of turbulence reflects the generation of energetic waves, impinging over topography and breaking locally.

From the synergetic use of LADCP and CTD data, collected in the Ionian Sea during the KM3Net cruise in 2007, here we expand the analysis of Artale et al. (2018) to investigate shear variance in addition to strain variance, also exploring the role of topographic characteristics, such as bottom slope and roughness. In particular, our new analysis aims at providing a spatial distribution of shear-to-strain ratio rather than a temporal analysis carried out over a specific region (i.e., the deep area of the Ionian Sea) as in Artale et al. (2018). We explore different seafloor characteristics, in terms of bathymetric gradient and bottom roughness. LADCP profiles, synchronous with CTD casts, allowed estimation of regional values of shear-to-strain ratio Rw (Kunze et al., 2006). In the frame of the Garrett-Munk parametrization (Polzin and Lvov, 2011; See Data and methods), we explore the relation between Rw and topographic features of the Ionian Sea. This approach employs CTD profiles to determine the isopycnal vertical strain (<ξz2>insitu), and LADCP profiles for the vertical shear of horizontal velocities (<Vz2>insitu), assuming that the variance is due to the presence of IWs (see Data and methods). These variances are used to estimate the local energy's level of the IW field, to modulate a canonical value of the dissipation rate of turbulent kinetic energy ε0 (Garrett and Munk, 1975).

Our analysis is based on CTD and LADCP profiles acquired in July 2007 during a research cruise aboard the R/V Urania, conducted within the framework of the KM3NeT project (https://www.km3net.org/ (last access: 22 July 2026); Borghini et al., 2025). CTD-LADCP stations were distributed over a wide area of the northwestern Ionian Sea (Fig. 1a), including strategic sites for the KM3NeT (Neutrino Telescope) infrastructure that had been identified by the particle physics community (Katz, 2006).

CTD data were acquired from the sea surface to the bottom by using a SBE911-plus calibrated before the cruise at the NATO center in La Spezia and were processed by the SBE Sea Soft program, following the standard procedure suggested by the manufacturer. LADCP profiles were collected at a subset of stations using a dual-head system installed on the rosette made of two RDI BB/WH 300 kHz instruments looking upward and downward respectively. The instruments were set to acquire 20 bins with size 10 m and the LADCP data were processed using the LDEO LADCP software Version IX.13 (Visbeck, 2002), using CTD and respective GPS fixes as auxiliary data.

The small spatial scales (order of centimeters) and intermittent temporal nature (from minutes to hours) involved in internal-wave-driven turbulent mixing, imply that direct measurements are difficult to achieve. Indeed, measurements resolving small-scale turbulence rely on ship-based microstructure profilers, not yet in wide use and, in any case, not on board during the 2007 KM3NeT research cruise. To fill this gap, finescale parameterizations are being used to estimate the turbulent kinetic energy dissipation rate and diapycnal diffusivity from more common instruments, such as CTDs, ADCPs, and Argo profiles (Whalen et al., 2020; Polzin et al., 2014). Finescale parameterizations aim to infer the centimeter-scale turbulent energy dissipation rate, operating on the intermediate [O(10–100) m] vertical wavelengths that are assumed to mediate energy transfer between large and small scales in the ocean (Polzin et al., 2014). Finescale parameterizations are formulated by reference to the Garrett-Munk 75 model (hereafter, GM) (Garrett and Munk, 1975; Cairns and Williams, 1976; Polzin and Lvov, 2011), which provides an empirical expression for the internal wave energy density in the spectral domain. Strain- and shear-based parameterizations allow to estimate the in-situ internal wave energy level, which serves to establish a deviation from the GM model, and to adjust its canonical value for the turbulent dissipation rate ε0 to a more realistic estimate. In particular, the turbulent dissipation rate can be expressed through the form of the finescale parameterization proposed by Kunze et al. (2006) as: 1ϵiw=ϵ0N2N02EEGM2FRwLf,N where E is the in situ internal wavefield variance and N the in-situ buoyancy frequency, while EGM,ε0=7×10-10Wkg-1 and N0=5.2×10-3 rad s⁻¹ are the GM internal wavefield variance, dissipation rate and Brünt-Väisälä frequency, respectively (Garrett and Munk, 1975; Munk, 1981). Depending on whether the parameterization is based on shear or strain, Ein-situ/EGM2 in Eq. (1) can be estimated from the shear variance as <Vz2>in-situ2/<Vz2>GM2, or from the strain variance as <ξz2>in-situ2/<ξz2>GM2.

The factor L(f,N) in Eq. (1) represents the latitude effect, calculated as L(f,N)=fcosh-1(N/f)/f0cosh-1(N0/f0) with f the Coriolis parameter and f0 the Coriolis's frequency at 30°. The term Rw represents the internal wavefield's aspect ratio and the bulk frequency content, and it is defined as the ratio of the buoyancy-normalized shear variance to the strain variance: 2Rw=(<Vz2>/N2)/<ξz2>), whose effect on dissipation rates εiw is modeled by the term FRw: 3aFRw=h1Rw=3/41+1/Rw2/Rw-1(for shear-based parameterization)3bFRw=h2Rw=RwRw+1/62Rw-1)(for strain-based parameterization) Equations (3a)–(3b) point out that the error associated with incorrectly assuming Rw is not symmetric (Chinn et al. 2016). For Rw=3 (i.e., the GM value), the correction functions are equal to 1 and do not affect εiw. When Rw>3, shear variance dominates, low-frequency internal waves contribute more strongly, and h1 decreases (thus reducing εiw), while h2 increases (thus increasing εiw). When Rw<3, strain variance dominates, high frequency internal waves contribute more to the ratio: h1 increases (thus increasing εiw), h2 decreases (thus reducing εiw). This makes the shear-based parameterizations more effective when Rw>3, and conversely the strain-based parameterizations are more effective when Rw<3 (Chinn et al. 2016).

The determination of Rw requires shear and strain variance measurements through LADCP and CTD observations, respectively. The strain ξz is defined as the vertical derivative of the isopycnal displacement and is generally calculated as ξz=(N2-Nfit2)/N‾2, where N2 is the Brünt-Väisälä frequency, N‾2 a mean value over the vertical segment of water column, and Nfit2 a polynomial quadratic fit: N2-Nfit2 represents the IW's density perturbation, normalized by N‾2 to obtain the vertical derivative of the isopycnal displacements.

Instead of determining ξz, we follow the approach of Ferron et al. (2014) and we determine ξ from the density fluctuations as ξ=(g/ρ0)(ρHP/N‾2), with g/ρ0 the standard gravity on the reference density, and ρHP density filtered by a high-pass Butterworth filter with a cutoff wavelength of 160 m to remove the density fluctuations associated with the background stratification, whose signal would contaminate the integration of the strain spectra (Kunze et al., 2006). This advantageously avoids fitting a polynomial to N2 (e.g., quadratic fit in Kunze et al., 2006), that could be inconsistent with the shape of the density profile in case of strong vertical gradients. This method is a variant from Kunze et al. (2006), and follows Ferron et al. (2014). Strain variance is then calculated as the integral of the isopycnal displacement spectra between kmin⁡ and kmax⁡: 4<ξz2>=∫kmin⁡kmax⁡k2S(ξ)dk. Similarly, the shear variance <Vz2> is obtained from the velocity spectra as 5<Vz2>=∫kmin⁡kmax⁡k2S(u,v)dk. To integrate the spectra, kmin⁡ and kmax⁡ must be carefully determined, as shear and strain rely on different instrument limitations and can be sensitive to distinct phenomenological contributions. In general, kmin⁡ is fixed by the choice of the data segment length kmin⁡=2π/L rad m⁻¹, where L is generally of 200-to-500 m, i.e., a compromise between the larger vertical wavelengths to be considered (the spectral resolution to be employed) and the desired vertical resolution. For strain, in particular, the integration is performed from kmin⁡=2π/O(150 m) (Kunze et al., 2006; Pollmann et al., 2017; Pollmann, 2020), to exclude contributions from larger-scale background stratification, and continues up to kmax⁡=2π/10 m (the GM upper limit).

For shear, a velocity noise is expected to dominate at high wavenumber, especially in weakly stratified layers lacking turbulent microstructures or suspended material (“water-column reflectors”) (Kunze et al., 2006), which tend to reflect the acoustic signal emitted by Doppler profilers like the LADCP. Their absence reduces the quality of the acoustic return, leading to higher measurement uncertainty and potentially spurious velocity variance, especially at small vertical scales. Thus, wavenumbers larger than kmax⁡=2π/O(100–50 m) represent the upper limit range of usable vertical wavenumbers for integration.

An additional saturation criterion is applied to limit the integration of spectra (Gargett, 1990; Kunze et al., 2006). Specifically, the integration is stopped when reaching the saturation value of 0.66N2 for the shear variance and 0.22 for strain. This threshold represents the onset of internal wave breaking or turbulence, where the linear internal wave regime breaks down. The corresponding vertical wavenumber kc at which this value is reached is retained as the upper limit for the integration. This approach constrains the finescale parameterization to physically realistic shear and strain levels, avoiding overestimation of internal wave energy due to instrument noise or unresolved scales, particularly in regions of enhanced energy.

In our case, strain and shear are derived from CTD and LADCP measurements from the bottommost 640 m of each vertical profile. Signals are then linearly detrended and Fourier transformed with a Hamming tapering window. Strain and shear variance estimates are obtained by integrating the 640 m segments, overlapping every 10 m from the bottom to the surface, to account for non-homogeneity in the statistical distribution of the water column properties and to have a statistically robust representation of the fine-scale variability. Final variances are obtained by averaging these overlapping estimates within finite-sized vertical bins (here 80 m), which are independent from the 640 m spectral window length and chosen as a compromise between statistical stability and vertical resolution.

The shear integration upper limit is identified from the LADCP velocity uncertainty. The LDEO LADCP processing (velocity-inversion method; Visbeck, 2002) provides, for each station, a depth-dependent single-bin velocity error profile, i.e. the estimated uncertainty associated with each vertical velocity bin, which is used to constrain the usable shear wavenumber range. The average V‾ERR=0.075 m s⁻¹ is used to estimate a velocity error by segment as VNOISE=V‾ERR/640m/10m. Its associated variance VNOISE2 is assumed to arise from white velocity noise, uncorrelated in space and therefore uniformly distributed across vertical wavenumbers. In the spectral domain, this corresponds to a flat velocity noise spectrum which, once differentiated, produces a shear noise spectrum used to identify the noise-dominated upper wavenumber limit.

The resulting spectra are then averaged by stratification bins, where we highlight their overall shape and the noise-free bandwidths we retain for integration, from 640 to 107 m (i.e., from 0.01 to 0.06 rad m⁻¹). This choice is conservative, as our lowest stratification levels are close to the NERR=5×10-4 s⁻¹ (discussed in Kunze et al., 2006), and consistent with the integration set up of Ferron et al. (2014) that stopped the integration at 107 m in the most limitative case. Additionally, this treatment combined both data from downward and upward profiles, in relatively small bin size (10 m), limiting the expected variance loss at short scales due to the various steps of data processing (Polzin et al., 2002; Thurnherr, 2012). For strain, integration is led from 128 to 10 m (i.e., from 0.05 to 0.63 rad m⁻¹; the GM upper limit).

Since shear and strain are integrated on their respective bandwidths, instead of using Eq. (2) directly, the shear-to-strain ratio is calculated as: 6Rw∗=Rw∗GM<Vz2>/<Vz2>GM/<ξz2>/<ξz2>GM, where Rw∗GM=3, and both shear and strain factors are divided by their corresponding GM values evaluated over the same spectral bandwidths used for each variable. Because Rw∗ in Eq. (6) is derived from variance integrated over finite wavenumber bands, it does not correspond to a single-frequency quantity, but represents a broadband statistical descriptor of the internal-wave shear–strain partition (see Supplement).

Traditional finescale parameterizations of turbulent kinetic energy dissipation rely on the assumption that the internal wave field follows a GM-like structure. In this framework, a fixed value of the shear-to-strain ratio, typically Rw≈3, is used to represent the relative contributions of vertical shear and isopycnal strain to the finescale energy budget. This fixed value implicitly assumes a broadband, statistically stationary internal wave field, with no significant spatial or temporal variability in spectral composition. However, these models sometimes fail to capture the true dynamics, for instance, near mixing hotspots where the internal wave spectra can be significantly distorted, showing bias toward higher or lower frequencies (Chinn et al., 2016; Polzin et al., 2014; Ferron et al., 2014).

Accumulating evidence from both observations and modeling (Ijichi and Hibiya, 2015, 2017; Takahashi et al., 2021; Dematteis et al., 2024) indicates that this assumption is often violated, especially in regions influenced by complex bathymetry, variable stratification, or energetic boundary currents. In these settings, the internal wave spectrum is distorted, leading to departures from the canonical balance between shear and strain. As a result, Rw can vary significantly in space and time, reflecting changes in the dominant frequency content of the wave field.

Ijichi and Hibiya (2017) used 3D eikonal equations to simulate how internal waves transfer energy across distorted spectra, including variations in the wave energy level, the local buoyancy frequency, and the inertial frequency. Their results confirm that energy transfer rates are consistent with the Henyey et al. (1986) model, which predicts dissipation from internal wave-wave interactions and highlights the importance of the spectral shear-to-strain ratio Rw. The study questions the accuracy of existing finescale parameterization models, particularly in environments with distorted spectra near boundaries or topographic features. To address these limitations, Ijichi and Hibiya (2015, 2017) propose a revised parameterization that explicitly considers both broadband and narrowband spectra. This leads to more accurate dissipation estimates by incorporating a dynamically varying Rw instead of assuming a constant value based on idealized spectral shapes. Their approach was further developed by Takahashi et al. (2021), who showed how vertical wavenumber spectra distortions (such as spectral humps) can bias traditional estimates, emphasizing the need to resolve the spatial variability of Rw.

From a physical point of view, Rw represents the ratio between the buoyancy-normalized vertical shear variance and the isopycnal strain variance. The spectral structure of the internal wave field determines whether energy is primarily stored in the shear (associated with horizontal motion) or in the strain (associated with vertical isopycnal displacement). The theoretical foundation for this lies in the internal wave dispersion relation: 7w2=N2kh2+f2kz2kh2+kz2, where w is the wave frequency, kh=kx,ky and kz the horizontal and vertical wavenumbers, respectively.

Equation (7) shows the intrinsic relation between horizontal (vertical) wavenumber and the shear-to strain ratio. When w≈f, it follows that kz≫kh and that the wavefield is dominated by horizontal motion, yielding high shear ∂u/∂z, i.e., the wavefield varies rapidly with depth and more slowly in the horizontal direction. This configuration results in particle motions that are predominantly horizontal but vary significantly over short vertical distances.

In observed internal wave spectra, low-frequency components (near the Coriolis frequency f) tend to concentrate energy in horizontal motions that change rapidly with depth, making shear-dominated fields a hallmark of low-frequency wave regimes (Kunze et al., 2006; Musgrave et al., 2022).

Conversely, when w≈N, kh≫kz and the particle motion becomes primarily vertical, enhancing isopycnal displacement and thus the strain ∂ξ/∂z (i.e., significant vertical displacements of isopycnals). It results that the wave propagates with a nearly vertical beam and the wave field varies slowly with depth. For this condition, although the vertical structure of the wave is broad (i.e., fewer oscillations along z), the amplitude of vertical displacements is large. As a result, the isopycnals experience stronger stretching and compression, leading to enhanced strain energy. This explains why internal waves at high frequencies, despite their low vertical wavenumber, contribute disproportionately to the strain variance. In observed internal wave spectra, higher-frequency components tend to be associated with enhanced isopycnal displacement relative to velocity shear, leading to strain-dominated regimes (Garrett and Munk, 1975; Kunze et al., 2006).

A propagation angle of an internal wave ray can be determined using the dispersion relation: 8tan⁡2α=N2-w2w2-f2, where tan⁡2α=kz2kx2+ky2. From Eq. (8), the angle of the wave trajectory (i.e., the beam angle α) relative to the horizontal can be determined. The angle α describes a cone of propagation that supports the wave: 9α=tan⁡-1N2-w2w2-f2. The wave energy propagates with the group velocity cg, perpendicular to the wave direction k, at an angle β=90°-α. For Rw it is possible to derive the following expression (see Kunze et al., 2006; Ijichi and Hibiya, 2015): 10Rw=Vz2ξz2N2=w2+f2w2-f2≃1+2f2kz2N2kh2. From Rw∗ in Eq. (6) we derive an effective mapped value w∗ and the associated effective beam slope β∗, using the classical internal-wave relation only as a geometric mapping. These quantities are not interpreted as the frequency or propagation angle of a single wave, but as broadband statistical diagnostics of the observed shear–strain variance partition. We then compare β∗ with the local bathymetric slope to organize the observations with respect to bottom geometry.

Equation (9), along with the physical interpretation of Eq. (7), highlights that the wave geometry (i.e., the ratio kz2/kh2) dominates the shear–strain dynamics: for kz≫kh we have high shear and Rw values, and for kh≫kz high strain values and low Rw. In other words, high Rw values correspond to low-frequency, shear-dominated conditions, while low Rw indicates high-frequency, strain-dominated fields. The dispersion relation is used here as a mapping to define an effective propagation slope, rather than to infer a true wave frequency, following a geometric interpretation of the variance partition rather than a dynamical inference based on monochromatic wave theory. In the general case of a broadband internal wave field, this ratio (i.e., Eq. 10) represents a weighted integral over a distribution of wave components and cannot be reduced to a function of a unique frequency. To avoid confusion with the classical monochromatic internal-wave framework, we therefore introduced the notation Rw∗, w∗, and β∗ to denote the broadband statistical quantities derived from the shear–strain variance partition used in this study.

A growing body of evidence shows that turbulent mixing is significantly enhanced in regions of rough bottom topography and sloping boundaries (Polzin et al., 1997; Ledwell et al., 2000; Kunze et al., 2006; Ferrari et al., 2016; Mashayek et al., 2017; Naveira Garabato et al., 2025). Internal wave interactions with these features can lead to reflection, scattering, and energy transfer toward higher vertical wavenumbers, increasing the likelihood of wave breaking and dissipation (Nash et al., 2004; Legg and Adcroft, 2003; Musgrave et al., 2022). In addition to tidal processes, geostrophic flows impinging on small-scale topography generate broadband internal waves that can radiate and dissipate energy locally, contributing to abyssal mixing (Nikurashin and Ferrari, 2010a, 2010b). These mechanisms, collectively, explain much of the spatial variability in observed mixing rates and highlight the importance of accurately representing topographic interactions in ocean mixing parameterizations.

When an internal wave encounters a rough topography, its propagation is disturbed, promoting scattering, reflection, or the generation of higher-frequency waves. This tends to break down low-frequency, large-scale wave structures and shift spectral energy toward higher vertical wavenumbers.

Since strain is associated with large vertical displacements, characteristic of high-frequency waves, this shift leads to a systematic decrease of the shear-to-strain variance ratio Rw in rougher regions, as documented in Kunze et al. (2006), where departures from the canonical Garrett–Munk spectrum were observed in high-roughness environments. As a result, areas of high roughness typically exhibit lower Rw, indicating increased strain. In contrast, smoother topographic regions preserve the lower-frequency shear-dominated character, yielding higher Rw.

In our investigation of the impact of bathymetry on Rw∗ (as a proxy for Rw), we analyze the KM3NeT hydrological stations in relation to bathymetric features such as slope and topographic roughness, calculated using the GEBCO gridded data (https://www.gebco.net, last access: 22 July 2026) over the Ionian Sea area, at 15 arcsec intervals. In particular, slopes are estimated between the hydrological cast position and the points located apart in the eight cardinal directions around, then the average is retained. A distance of 10 km between the two points is chosen to avoid overestimation of the topographic slope, due to the presence of small bathymetric features. Topographic roughness is calculated from the variance of the bathymetry over the same neighboring area (i.e., 20-by-20 km²). Rw∗ values are then compared with the local bathymetric roughness, i.e., the logarithm of the variance of seafloor elevation (Fig. 1c, d).