The Wavelength Dependence of the Propagation of Near-Inertial Internal Waves

Jemima Rama aResearch School of Earth Sciences and ARC Centre of Excellence for Climate Extremes, Australian National University, Canberra, Australian Capital Territory, Australia

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Callum J. Shakespeare aResearch School of Earth Sciences and ARC Centre of Excellence for Climate Extremes, Australian National University, Canberra, Australian Capital Territory, Australia

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Andrew McC. Hogg aResearch School of Earth Sciences and ARC Centre of Excellence for Climate Extremes, Australian National University, Canberra, Australian Capital Territory, Australia

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Abstract

Wind-generated near-inertial internal waves (NIWs) are triggered in the mixed layer and propagate down into the ocean interior. Observational and numerical studies have shown the effects of background vorticity and high shear on propagating NIWs. However, the impacts of the background mean flow on NIWs as a function of the waves’ horizontal wavelength have yet to be fully investigated. Here, two distinct cases are analyzed, namely, the propagation of wind-generated, large-scale NIWs in negative vorticity and the behavior of small-scale NIWs in high shear. The propagation and energetics of the respective NIWs are investigated using a realistic eddy-resolving numerical simulation of the Kuroshio region. The large-scale NIWs display a rapid vertical propagation to depth in negative vorticity areas, while the small-scale NIWs are confined to shallower depths in high-shear regions. Furthermore, the dominant energy sources and sinks of near-inertial energy are estimated as the respective NIWs propagate into the ocean’s interior. The qualitative analysis of NIW energetics reveals that the wind triggers the generation of both the large-scale and small-scale NIWs, but the waves experience further amplification as they draw energy from the background mean flow upon propagation in negative vorticity and high-shear regions, respectively. In addition, the study demonstrates that small-scale NIWs can be induced independently by wind fluctuations and do not necessarily rely on straining nor refraction of large-scale NIWs by mesoscale motions.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jemima Rama, jemima.rama@anu.edu.au

Abstract

Wind-generated near-inertial internal waves (NIWs) are triggered in the mixed layer and propagate down into the ocean interior. Observational and numerical studies have shown the effects of background vorticity and high shear on propagating NIWs. However, the impacts of the background mean flow on NIWs as a function of the waves’ horizontal wavelength have yet to be fully investigated. Here, two distinct cases are analyzed, namely, the propagation of wind-generated, large-scale NIWs in negative vorticity and the behavior of small-scale NIWs in high shear. The propagation and energetics of the respective NIWs are investigated using a realistic eddy-resolving numerical simulation of the Kuroshio region. The large-scale NIWs display a rapid vertical propagation to depth in negative vorticity areas, while the small-scale NIWs are confined to shallower depths in high-shear regions. Furthermore, the dominant energy sources and sinks of near-inertial energy are estimated as the respective NIWs propagate into the ocean’s interior. The qualitative analysis of NIW energetics reveals that the wind triggers the generation of both the large-scale and small-scale NIWs, but the waves experience further amplification as they draw energy from the background mean flow upon propagation in negative vorticity and high-shear regions, respectively. In addition, the study demonstrates that small-scale NIWs can be induced independently by wind fluctuations and do not necessarily rely on straining nor refraction of large-scale NIWs by mesoscale motions.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jemima Rama, jemima.rama@anu.edu.au

1. Introduction

Near-inertial internal waves (NIWs) are the second-largest source of energy to the ocean’s internal wave field (Waterhouse et al. 2014) and are primarily generated through high-frequency winds imparting energy at near-inertial frequencies to the ocean’s upper mixed layer. A large fraction of the near-inertial energy (NIE) dissipates locally in the upper ocean; however, a substantial part of the NIE propagates, both laterally and vertically, to depth away from the generation site in the form of NIWs (Alford et al. 2016). Large-scale (low vertical mode) NIWs tend to rapidly propagate away from their generation site (Simmons and Alford 2012), carrying a significant fraction of the total wave energy to the far field (Silverthorne and Toole 2009). In contrast, smaller-scale (higher vertical mode) NIWs have a slower propagation and contain relatively less energy but have higher shear (Polzin and Lvov 2011), which makes them likely players in small-scale mixing local to their generation site. Nevertheless, the wind-generated NIWs’ capacity to cause mixing in the ocean interior is yet to be determined. The intensity and location of mixing, either in the upper mixed layer or at depth, is governed by how NIWs interact with their surrounding environment, and thus depends on stratification, mesoscale shear and vorticity, and other internal waves. Despite the crucial contribution of NIWs to the ocean energy budget, large uncertainties still remain on the overall partitioning of energy and shear between the large- and small-scale NIWs, following their generation from wind events.

Past numerical studies have analyzed the behavior of NIWs in varying background geostrophic fields and have highlighted the importance of vorticity in NIWs’ propagation. Klein et al. (2004b) observed the spatial reorganization of the NIWs by the geostrophic flow, in a field of barotropic mesoscale eddies. The effects of vorticity, producing “inertial chimneys” (Lee and Niiler 1998) whereby the NIWs are vertically fluxed downward in anticyclonic eddies, were further highlighted by more realistic ocean models. Furuichi et al. (2008) used a 6-hourly wind forcing over 13-km horizontal grid resolution while Zhai et al. (2005) imposed a daily wind stress in a model with 5-km horizontal grid resolution. Both studies suggested that at least 70% of the near-inertial wind energy input was dissipated in the mixed layer, with most of the NIE penetration to the interior facilitated by the mesoscale eddies. The trapping and focusing of NIWs in anticyclonic eddies initiate an intensification of the waves’ energy density; however, elevated energy in the vortex core could also be attributed to NIWs’ interaction with the mean flow (Barkan et al. 2017). Eddy-permitting numerical models (Polzin and Lvov 2011; Taylor and Straub 2016) proposed that near-inertial Reynolds stresses could act as an energy sink for mesoscale geostrophic motions, while asymptotic models (Wagner and Young 2016; Rocha et al. 2018), using energy conservation laws, demonstrated that NIWs can directly extract energy from balanced quasigeostrophic flows. The interactions of the NIWs with the strong eddy fields thus facilitate the penetration of NIE into the ocean interior.

Other numerical studies have highlighted the effects of not only vorticity, but also shear, on propagating NIWs. Using an idealized two-dimensional numerical model, Whitt and Thomas (2013) suggested that NIWs trapped in high vorticity areas are further amplified in regions of high shear. The latter highlighted the existence of stronger wave–mean flow interactions in high-shear regions, such as near fronts. The transfer of energy from the mean flow into the NIWs is promoted in the presence of frontogenetic strain (Thomas 2012) or when the wind work aligns with a laterally sheared geostrophic current (Whitt and Thomas 2015). Energy transfers between the mean flow and NIWs have been analyzed in three-dimensional numerical simulations (Thomas and Daniel 2020), whereby the wave–mean flow energy exchanges extend beyond the mixed layer near regions of cyclonic and anticyclonic eddies (Barkan et al. 2017). A more realistic numerical study of the Kuroshio extension by Nagai et al. (2015) suggests that NIWs can also get reabsorbed by, or lose energy to, the mean flow. However, Nagai et al. (2015) analyzed spontaneously generated NIWs (Danioux et al. 2012; Shakespeare and Taylor 2014), which have distinct properties to wind-generated NIWs; for instance, spontaneously generated NIWs are characterized by a preferential orientation with respect to the mean flow (Shakespeare and Hogg 2018). NIWs generated through frontogenesis will inevitably be present in addition to wind-generated NIWs in any wind-forced model. Therefore, both processes of frontogenesis and amplification of externally generated NIWs can contribute to elevated levels of NIE in the vicinity of fronts. The current study does not dissociate the two processes. Nevertheless, the contribution of NIE from spontaneously generated waves is smaller than the input from the wind-generated internal waves (Shakespeare and Hogg 2018), which are the focus of this paper.

The results of theoretical and numerical studies into the behavior of wind-generated NIWs propagating in a geostrophic field are supported by observational studies. Estimates of parameterized turbulence from Argo floats (Whalen et al. 2018) demonstrated that the response of the NIWs’ field to wind energy input is stronger in more energetic mesoscale eddy fields, such as near the Gulf Stream and the Kuroshio. Satellite altimetric data and surface drifters (Elipot et al. 2010) validated the ability of the geostrophic flow to rearrange the distribution of surface NIWs while fine-structure measurements confirmed the trapping of NIE in negative vortices (Joyce et al. 2013; Cuypers et al. 2013; Lelong et al. 2020). Amplification of the trapped NIWs, gaining energy at the expense of the mean flow, has also been observed in the POLYMODE experiment (Polzin 2008; Polzin and Lvov 2011), although more observational evidence is warranted to verify the details of the modeled wave–mean flow exchanges in the presence of vorticity (Barkan et al. 2017) and high shear (Whitt and Thomas 2015). Last, the view of numerical modeling studies (Furuichi et al. 2008; Zhai et al. 2005), suggesting that a trivial amount of NIE can penetrate beyond the mixed layer, has been disputed by observations of wind-generated, small-scale NIWs (high vertical modes with vertical wavelength less than 700 m) propagating past 800-m depth in the North Pacific (Alford et al. 2012) and transiting past 1000 m following the passage of a storm in the Indian Ocean (Cuypers et al. 2013). Energetic NIW packets were also observed at depths of 3500 m, radiating both upward and downward, near the Mendocino Escarpment in the North Pacific (Alford 2010). Such upward-propagating NIWs in this instance could be either from bottom generation (Nikurashin and Ferrari 2010) or parametric subharmonic instability (Nagasawa et al. 2000), or even represent remnants of surface-generated, downward-propagating NIWs that have reflected off the seafloor. Alford (2010) estimated that those NIW packets would likely take O(100) days to propagate to the seafloor. The scenario of the NIWs reflecting off the ocean’s bottom cannot be completely ruled out since the dissipation rate for small-scale NIWs is on the order of a few months in regions where the Richardson number is stable, such as below the pycnocline (Alford and Gregg 2001).

The tendency for NIWs to propagate relatively undissipated, or to become unstable and locally break, depends largely on the background flow in which they propagate and on other internal waves they might encounter. With respect to the effects of the background vorticity, critical layers exist at the base of anticyclonic eddies, whereby propagating NIWs might become unstable and dissipate (Kunze 1985). Refraction by the mean flow vorticity, or inertial chimneys, modulates the lower bound of the internal wave band to an effective Coriolis frequency, feff = f + ζ/2, where ζ is the relative vorticity (Mooers 1975). NIWs formed within negative vorticity structures are hence subinertial and can only freely propagate in those anticyclonic structures (Zhai et al. 2005; Lee and Niiler 1998). However, as the background vorticity decreases with depth, feff approaches the local inertial frequency. The downward-propagating subinertial NIWs eventually reach a critical depth where they become evanescent; their vertical wavelength and group velocity shrink to zero, and as per the conservation of the wave action, the wave energy amplifies to compensate (Kunze and Sanford 1986). The sink for the accumulation of energy at the critical depth can be enhanced dissipation, energy loss to the mean flow or high-frequency waves (Kunze et al. 1995), or internal reflection (Whitt and Thomas 2013).

While the background horizontal shear modifies f to feff, the background vertical shear can also give rise to critical levels through the action of the Doppler shift. The Doppler shift can modulate the intrinsic frequency (ωi) of the waves to the local inertial frequency f (or in the presence of vorticity effects feff), at which point the waves hit a critical level or a separatrix as defined by Whitt and Thomas (2013). Critical levels represent an extreme effect of the Doppler shift, while more common outcomes of the Doppler shift are the amplification and decay of the NIWs’ energy. Based on the conservation of the wave action (Bretherton and Garrett 1968), the wave energy scales with the intrinsic frequency of the waves. Therefore, NIWs propagating with a decreasing mean flow will experience an increase in their intrinsic frequency and a corresponding amplification of their wave energy, whereas NIWs propagating against the same varying mean flow will experience a decrease in their intrinsic frequency and a corresponding decay of energy.

NIWs of different scales are predicted to have distinct dynamics and behavior when encountering varying fields of vorticity and baroclinicity. Large-scale (low mode) and small-scale (high mode) NIWs equally experience the effects of vorticity. However, small-scale NIWs are more prone to the effects of the Doppler shift (ωo=ωi+ku¯, where ωo is the observed Eulerian frequency), especially in regions of high shear. Since the Doppler shift scales with ku¯, where k is the horizontal wavenumber and u¯ the horizontal mean flow (Gerkema et al. 2013), the waves with a higher k (smaller wavelength) will experience a stronger Doppler shift than waves with a lower k. Doppler shifting associated with frontal vertical circulation has been shown to enhance radiation of wave energy for small-scale NIWs (Thomas 2019) but to our knowledge, the distinct dynamics of large- and small-scale NIWs, based on their wavelengths and the background flow in which they propagate, has not been well characterized in previous studies.

The aim of this study is to investigate how the vertical propagation of the NIWs is affected by the mesoscale geostrophic flow and to analyze the respective sources and sinks for those internal waves as they propagate into the ocean’s interior. We first outline the model’s configuration and set up the context for our experiment (section 2). We then describe the data processing which involves the extraction of the NIWs component through the Lagrangian filtering (section 3a), and the separation of the waves into the upward- and downward-propagating signal through Fourier masking (section 3b). Spectral methods are additionally used to separate the waves into large and small scales (section 3c). We then consider the waves’ kinetic energy equation to estimate the sources and sinks of NIE for the propagating NIWs (section 4). In section 5a, we highlight how the NIWs’ distribution is governed by the mesoscale field, namely, by the background relative vorticity and vertical shear provided by the geostrophic flow. Section 5b describes the large-scale NIWs’ propagation in negative vorticity and section 5c the small-scale NIWs’ propagation in high-shear regions. In sections 5b and 5c, we describe the evolution of the terms, in the wave’s kinetic energy equation, with depth and over time. We conclude with a discussion highlighting the energy sources and sinks for the large- and small-scale NIWs as they propagate into the interior (section 6).

2. Model configuration

The Massachusetts Institute of Technology Global Circulation Model (Marshall et al. 1997) is used to simulate a subsection of the western North Pacific (Fig. 1a). The selected region provides an ideal domain for the study of the propagation of near-inertial waves in a rich mesoscale field: the extension of the Kuroshio current creates a strong front with high shear while the prevailing turbulent eddy field provides a wide range of vertical vorticities.

Fig. 1.
Fig. 1.

(a) The mean surface speed over the full model domain. (b) Time average of wind work on total inertial motions during the main storm event (4–11 Apr 2006). The black box defines the subregion for all analyses. (c) Cumulative sum (divided by the total time) of the wind work on total inertial motions, horizontally averaged in the defined subregion.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

The model solves the hydrostatic primitive equations on a rotating sphere, with an implicit free- surface and discretized on a C grid. A Mercator grid, with horizontal grid size decreasing toward the northern latitudes, is employed; the largest horizontal grid spacing is 4.95 km at the southern boundary of our domain. The vertical grid consists of 200 vertical levels over 6-km depth, with 50 levels in the upper 100 m and vertical cells reaching a thickness of 100 m at the bottom, with a partial-step representation of the topography. The topography is constructed from the 1/12° ETOPO5 bathymetry data (Smith and Sandwell 1997). The parameters for background viscosity (Laplacian vertical eddy viscosity: 1 × 10−5 m2 s−1, biharmonic lateral viscosity: 4 × 109 m4 s−1) are chosen so as to ensure model stability, primarily near steep topography. Turbulence in the surface boundary layer is further parameterized by the KPP vertical mixing scheme (Large et al. 1994). The temperature, salinity and horizontal velocity fields are restored to 1/12° horizontal resolution Copernicus data (E.U. Copernicus Marine Service Information) on a 12-h time scale over the 100 km of the domain directly adjoining the model boundary. The damping of the velocity fields creates a buffer zone that prevents reflection of internal waves into the model’s interior.

The surface is forced with a daily net heat flux, computed from shortwave, longwave, latent, and sensible heat fluxes. An hourly wind stress is also imposed at the surface. Both the heat fluxes and the wind stress are constructed from ERA5 reanalysis datasets, which are on a horizontal grid of 30-km spacing (Hersbach et al. 2020). The net heat flux ensures that the seasonal evolution of the sea surface temperature and the mixed layer depth is resolved properly. The mixed layer thickness can affect the partitioning of energy into the respective NIWs’ wavelength (Gill 1984) and is hence a key player in properly resolving NIW generation by the wind and their subsequent propagation into the interior. In addition, an hourly forcing of the wind is sufficient to stimulate the inertial resonance in the upper mixed layer, with or without eddies (Klein et al. 2004a). There is no tidal forcing imposed in the model.

3. Data processing

The model is forced using data starting from 1 January 2006 and is allowed to equilibrate from January to March 2006, so as to avoid any contamination of the wave field by the equilibration processes. The model output for analysis is taken from April 2006 during which period a rapidly developing storm moved east of Japan and strengthened to a hurricane storm. The storm traversed the selected study region (Fig. 1b) between 4 and 8 April, providing high wind stress, with the spatially averaged wind work on inertial motions in the study region peaking to ∼4 mW m−2 (Fig. 1c). Fields of pressure (p) and velocity (u) are saved from all depth levels at hourly intervals during the 4 weeks of the analysis period. All data processing and analyses are restricted to the defined substudy region that encompasses high wind energy input (Fig. 1b). In the following subsections, we describe the three preprocessing steps (Fig. 2) performed prior to the data analysis.

Fig. 2.
Fig. 2.

Summary of the preprocessing steps for the filtering of the model’s output.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

a. Lagrangian filtering

The near-inertial wave energy flux is computed from the velocity pressure correlation 〈up′〉 (Nash et al. 2005), where prime means the perturbation associated with the wave field. To compute the wave energy flux, there is a need to separate the high-frequency wave field from the overall flow fields, which also comprise lower-frequency nonwave components due to eddies, jets, the western boundary current, and other motions. In this study, the wave component is extracted from the overall field using a semi-Lagrangian filtering technique (Shakespeare et al. 2021). The Lagrangian filtering advects water parcels passively with the horizontal velocity (u, υ) for 4 days (2 days forward and 2 days backward from the time of interest). The model’s output of velocity (u, υ, w) and pressure (p) are interpolated onto the parcels’ trajectory and are then high-pass filtered with a cutoff of f = 5.699 × 10−5 s−1, the inertial frequency at 23°N, the model’s southernmost latitude. Signals with frequencies lower than the cutoff are the nonwave component (mean, geostrophic flows), expressed by the overbar (α¯) and signals with higher frequencies represent the wave component (α′), where the total field (α) is α=α¯+α.

The use of the Lagrangian filtering is particularly crucial when extracting the wave component of the pressure field (p). It can be shown that the pressure field, as opposed to the velocity field, on an Eulerian frequency spectrum does not exhibit a clear minimum between low frequencies (associated with the mean component) and the inertial frequency (associated with NIWs). The absence of such a minimum complicates the separation of the pressure signal into the wave and nonwave components (further details in appendix A). In addition, the present Lagrangian filtering unveils the subinertial waves that may have been buried in the nonwave component, and that would have otherwise been discarded if the Eulerian frame had been used for the high-pass filtering (Nagai et al. 2015). We recall that the NIWs’ Eulerian frequencies may become subinertial following the effects of straining in negative vorticity regions or the Doppler shift in strong sheared flows. The prevailing vorticity modifies the local inertial frequency such that the intrinsic frequency (ωi) is defined as (feff2m2+N 2k2)/m2. The Doppler shift subsequently modifies the observed Eulerian frequency (ωo) compared to the intrinsic frequency (ωi) such that ωo=ωi+ku¯. To account for both effects, an adaptive filter cutoff which varies with both latitude and local vorticity is warranted for the Lagrangian filtering to unveil all subinertial waves present in the model. The filter cutoff for the Lagrangian filtering in our study is fixed at f23N but the subinertial waves present in the study region (figure not shown) are correctly identified. For instance, the cutoff at f23N lies at the minimum (between nonwave and wave components) of the respective Lagrangian spectra (figure not shown) in both high-shear areas such as in the extension of the Kuroshio and in the large anticyclonic eddies at 37°N. The implementation of an adaptive filter cutoff was not possible at the time the study was conducted but is now available in the Lagrangian filtering package (Shakespeare et al. 2021). In view of the current study, the Lagrangian filtering allows the proper separation of the pressure field into its nonwave and wave component and uncovers any subinertial waves present in the defined study region. The Lagrangian filtering also minimizes the contamination of the waves by the submesoscale fields but does not completely eliminate the effects of the latter given the fixed cutoff frequency of the Lagrangian filtering.

b. Separation into up-going and down-going component

The Lagrangian filtered wave fields, for both u′ and p′, contain the upward-propagating and downward-propagating waves. To track the local sources of the NIWs and investigate their downward propagation following wind events, a separation into the upward and downward components is required. The up–down filtering relies on the vertical phase speed of the waves, assumed to be purely linear in this instance and defined as cp = ω/m, where ω is the frequency of the wave and m its vertical wavenumber. The internal wave vertical group velocity is further given by cg = ∂ω/∂m, and is of opposite sign to the vertical phase velocity cp (McWilliams et al. 2006). By definition, when ω and m are of the same sign, the vertical group velocity is negative; the waves are downward propagating. Conversely, when ω and m are of opposite sign, cg is positive, implying that the waves are propagating upward. In that view, the wave field is Fourier transformed onto the frequency domain (from time) and onto the vertical wavenumber domain (from vertical space) for each horizontal (x, y) grid cell. A masking is subsequently performed in the frequency–vertical wavenumber spectrum; opposite signs (pairs of ω < 0, m > 0 and ω > 0, m < 0) are masked out and the spectrum is inverse Fourier transformed to extract the downward-propagating wave component. A masking of the same signs (pairs of ω < 0, m < 0 and ω > 0, m > 0) and subsequent inverse filtering can yield the upward-propagating wave component. In this instance, we compute the up-going signal as a residual of the total wave field and the down-going wave field. The up–down separation hence results in a time series at each grid point in the model, but with the respective downward and upward-propagating wave signal separated out for fields of velocity (u′, υ′, w′) and pressure (p′).

We note that the Doppler shift may cause some bias in the up–down separation of the wave fields (appendix B), but this bias was found to be negligible in our results, except when estimating the vertical dissipation as described in section 4.

c. Separation into small-scale (35–80 km) and large-scale NIWs (80–500 km)

The final step in the data processing involves the separation of large and small wavelengths. A single storm event will generate many NIWs with varying frequencies and wavenumbers which may disperse as they propagate away from the source. To study the different behavior of the NIWs, as per their wavelengths, the upward and downward wave components are respectively separated into low horizontal wavenumber (low k) waves, subsequently referred to as large-scale waves or low modes, and high horizontal wavenumber (high k) waves, termed as small-scale waves or higher modes. The up–down wave signals (u′, υ′, w′, p′) are Fourier transformed from horizontal space onto a 2D horizontal wavenumber domain x ↦ k,y ↦ l at each time and on each vertical level of the model output. A tapered high-pass radial filter (Kaiser window with β = 60), corresponding to a wavelength cutoff of ≈80 km, is applied to the 2D horizontal wavenumber domain to mask out low K, where K is the modulus of the horizontal wavenumber, and the resulting output is inverse Fourier transformed to isolate small-scale waves. Large-scale waves are then computed as the difference between the unfiltered waves and small-scale waves. The filter cutoff (80 km) for the wavelength separation is selected to match the scale of the mesoscale eddies. Based on the selective trapping of the NIWs (Klein et al. 2008), the large waves are hereby defined so that the waves with a horizontal wavelength of ≈100 km would be trapped in the mesoscale anticyclonic structures present in the model, while the defined small-scale NIWs would propagate away unaffected by the effects of the mesoscale vortices. Finally, we note that the wavelength separation between the large-scale and small-scale NIWs is imperfect. There is no minimum on the NIW wavenumber spectra at the filter cutoff (corresponding to 80 km), therefore, some spectral leakage is expected following the wavelength separation. In addition, the values of velocities and pressure are taken as 0 in the topography when performing the 2D Fourier transform and its inverse. The filtered inverse signal hence introduces real values in the topography. However, given that our analyses focus on depths shallower than 2000 m, where there are relatively few and small topography outcrops, such artifacts are negligible.

The three steps of Lagrangian filtering, up–down separation, and wavelength separation (Fig. 2) ultimately lead to the velocity and pressure fields being separated into up-going and down-going component for large- and small-scale NIWs, respectively. Figures 3a and 3b show a snapshot of the vertical velocity w′ at 200-m depth for the down-going large-scale NIWs and down-going small-scale NIWs, respectively.

Fig. 3.
Fig. 3.

Snapshot of the vertical velocity (w′) induced by (a) down-going large-scale (low k) NIWs and (b) down-going small-scale (high k) NIWs. Panels (a) and (b) are from the subregion outlined by the black box in Fig. 1b.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

4. Near-inertial waves’ kinetic energy

a. NIWs kinetic energy equation

We investigate the fate of the downward-propagating NIWs in our numerical model, by considering the wave kinetic energy (KE) equation. The available potential energy (APE) equation for the computation of the wave energy has been neglected in this study under the assumption that NIWs are characterized by nearly horizontal motions, with small vertical wavelengths (Alford et al. 2016). Considering that NIWs have frequencies ω close to f and given that their APE can be derived from APE=KE[(ω2f2)/(ω2+f2)], we assume that the NIWs’ APE is smaller than their KE. The governing equation for the near-inertial KE can be derived by taking the dot product of the horizontal momentum equations with the horizontal wave velocity, under the hydrostatic and Boussinesq approximation (Müller 1976; Polzin 2010), and then averaging through horizontal space and time as denoted by 〈 〉 such that
Ekt+(u¯Ek)=zνu¯hzuh(uhp)+zwp+wb uh(u)u¯hϵhϵυ.
As defined in section 3a, the nonwave fields are denoted by an overbar, and the wave components by a prime. Accordingly, u¯ is the velocity field for the nonwave component while u′ is the velocity field for the wave component. The term Ek is the wave KE, and p′ and b′ are the wave pressure and buoyancy fields, respectively. The term ν is the viscous stress coefficient, and ϵ represents the viscous dissipation of wave energy in the horizontal (ϵh) and vertical (ϵυ), respectively. We then integrate Eq. (1) vertically between z1 and z2, and choose z1 to be at the surface and z2 to be at 500-m depth. At the surface (z1 = 0 m), the viscous term [ν(u¯h/z)] is the wind stress (τ) whereas z2 is sufficiently deep for the viscous stress term to be neglected. In addition, at the surface, wp′ ∼ 0. Vertically integrating Eq. (1) within the limits of z1(0 m) and z2(−500 m) thus simplifies to
z2z1[Ekt+(u¯Ek)]dz=τuhz1z2z1(uhp)dz +wpz2+z2z1wbdz+z2z1[uh(u)u¯hϵυϵh]dz.

The change in the NIW KE, where Ek=(1/2)(uu+υ υ ), is governed by energy transfers into and out of the wave field through the sources and sinks of wave energy. We adopt a sign convention that is a positive value for contribution to wave energy and a negative value for energy sinks. The term τuh is the near-inertial energy input by the wind work at the sea surface, (uhp) is the convergence of the horizontal energy flux, and wp′ is the vertical energy flux. The term wb′ is the buoyancy flux responsible for the conversion of the wave APE to KE, and uh(u)u¯h is the mean-to-wave (MTW) conversion. The definitive energy sink is interior viscous dissipation of the waves, which is computed separately as the horizontal dissipation, ϵh=A4(h2uh)2, and vertical dissipation, ϵυ=(Aυ+AKPP)|uh/z|2.

b. Cross terms between nonwave and wave components

For the subsequent analyses, we diagnose each term in Eq. (2), to determine their relative contribution to the wave KE as the NIWs propagate into the interior. The cross terms, involving the product of any wave field (α′) and nonwave field (β¯) have been neglected from Eqs. (1) and (2). If we assume a temporal–spatial scale separation between the wave and nonwave fields (as is the case for the Lagrangian filtering), the spatiotemporal average of those cross terms is zero, β¯α=0, in a closed system (Shakespeare and Taylor 2014; Müller 1976). We can thus neglect the cross terms between the nonwave and wave fields when deriving the KE equation for the internal waves. Figure 4 shows the relative contribution of each term in Eq. (1) to the total wave KE. Each energy term is computed using the total NIW fields from the Lagrangian filtering, that is, the velocity and pressure fields prior to filtering in terms of the propagation direction and the wavelength. Each energy term is horizontally averaged over the study region shown in Fig. 1b and vertically integrated from the surface to 500-m depth [Eq. (2)]. The bar plots represent the time-integrated values over April 2006. The residual of the total NIW KE budget (Fig. 4) is small compared with the dominant sources (wind work) and sinks (vertical dissipation), albeit not zero. We recall that the horizontal averaging is not performed over the whole model’s domain. We are thus cutting through waves and discarding a relatively small amount of energy as we compute the horizontal spatial mean for each energy term. The respective energy terms are further time integrated over different time spans (over the first, first two, and first three weeks of April 2006, respectively) and horizontally averaged over smaller horizontal sections (removing 2°, 4°, and 6° bands, respectively, from the boundaries of the study region prior to averaging). Each set of horizontally averaged and time integrated energy term is then used to compute the standard deviation and form the uncertainty bars on Fig. 4. As expected, the resulting residual varies depending on how the internal waves’ phases have been averaged out, with instances of the residual approaching zero as the averaging region encapsulate more of the wave packets. Furthermore, the residual is considered to be negligible given that it is smaller than the dominant source (wind work) and dominant sink (dissipation) of the total NIWs. The relatively small magnitude of the residual also confirms that the contribution from the cross terms is minor and supports the assumption of αβ¯=0.

Fig. 4.
Fig. 4.

Energy terms computed from the total NIWs, horizontally averaged over the study region (delimited in Fig. 1b) and integrated over the depth range 0–500 m. MTW is the mean-to-wave conversion, wb is the buoyancy flux term, and advh and advv are the horizontal and vertical advection, respectively. The bar plots demonstrate the cumulative sum (time integrated values divided by the total time) of the respective terms over the entire month of April 2006. The uncertainties are the standard deviations computed by averaging over different time span and over slightly smaller sections of the study region.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

c. Cross terms between up-going and down-going NIWs and large-scale and small-scale NIWs

Additional cross terms and biases are introduced when using the secondary filtered fields of NIWs for computing the energy terms of Eq. (2). For instance, filtering in terms of the propagating direction (section 3b) introduces cross terms between the up-going and down-going NIWs, while the wavelength separation (section 3c) introduces cross terms between the large-scale (low k) and small-scale (high k) NIWs. We recall that all cross-terms have been neglected from Eqs. (1) and (2) on the basis that the cross terms are close to zero when averaged over a closed system and with a clear separation method as in the Lagrangian filtering. Nevertheless, the assumption of the cross terms being negligible cannot be expected to hold in general, especially in cases where the scale separation (section 3c) is not perfect. In appendix C we verify whether the cross-terms are indeed negligible. The cross terms between the up-going and down-going waves for each energy term from Eq. (2) are determined to be relatively minor (cross terms form less than 35% of the |down-going| + |up-going| energy components) except for the vertical dissipation which is erroneously high in the mixed layer due to KPP component of the vertical viscosity acting on the filtered wave fields. On the other hand, the cross terms between the large-scale and small-scale NIWs for the respective energy terms of Eq. (2) are identified to be negligible except for the buoyancy flux. A high-pass filter in the Fourier space can introduce spurious features in the field being filtered, even when the employed filter is tapered. Unlike the separation of the nonwave and wave component in the Lagrangian frequency space where there is a natural dip between the two components, the wavenumber spectrum does not demonstrate any minimum at the wavenumber cutoff, which separates our defined low-k and high-k NIWs. The tapered radial filter (section 3c) introduces biases into fields that are separated into large-scale and small-scale NIWs. The bias from the wavelength separation is small for fields of velocity and pressure but is nonnegligible for buoyancy. We thus determine that the filtered wave fields (u′, υ′, w′, p′ from the down-going large- or small-scale NIWs) can be used to estimate the energy terms forming the KE equation, except for the vertical dissipation (ϵυ) and buoyancy flux (wb′).

Furthermore, the cross terms are only negligible when the spatiotemporal average is performed over a system without large boundary exchanges. Before proceeding we highlight that the nature of the spatial averaging in the subsequent analyses is not over the whole model’s domain (more details in section 5a) and consequently the cross terms might become important contributors or dampers of KE. Since the main analyses of this study is not over a closed system, we are not attempting to form a closed wave energy budget. We are rather seeking to estimate the sources and sinks of the wave KE. Furthermore, we are only considering energy terms that can be properly resolved following the separation of the NIWs in terms of propagation direction and wavelength, as described in this section.

5. Results

Figure 5 shows the vertical energy flux (wp′) at 200 m, for both downward-propagating and upward-propagating NIWs, averaged over a week starting from 7 April 2006. The time averaging period starts 3 days after the onset of the storm system over the study region (as defined in section 2 and Fig. 1b). Figures 5a and 5c show the mean vertical energy flux for the large-scale (low k) NIWs while Figs. 5b and 5d show the mean vertical energy flux for the small-scale (high k) NIWs. The mesoscale structure is overlaid on Fig. 5, where the contour lines outline the negative vorticity regions and the stippling represents high-shear areas at 200-m depth. The negative vorticity region is defined as Ro < −0.04, whereby the Rossby number Ro = ζ/f is used to characterize the relative vorticity of the background flow. On the other hand, the high-shear region is defined as log(Ri) < 2, where the Richardson number (Ri = N2/|∂u/∂z|2) characterizes the shear of the nonwave (mean) component in the domain. In addition, we observe the bias in the up–down separation, due to Doppler shift effects as discussed in section 3b: upward vertical energy fluxes contaminate the downward vertical energy flux inhigh-shear regions (Figs. 5a,b), and vice versa (Figs. 5c,d).

Fig. 5.
Fig. 5.

Average vertical energy flux at 200 m, computed from 7 to 14 Apr for (a) large-scale downward-propagating NIWs, (b) small-scale downward-propagating NIWs, (c) large-scale upward-propagating NIWs, and (d) small-scale upward-propagating NIWs. The bottom-left color bar of vertical energy flux is for the large-scale NIWs in (a) and (c) and the bottom-right color bar of vertical energy flux is for the small-scale NIWs in (b) and (d). As per the sign convention, the upward vertical flux is positive, and the downward vertical flux is negative. The contours outline regions of negative vorticity (Ro < −0.04), and the stippling define areas of high shear [log(Ri) < 2].

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

a. Selective trapping of near-inertial waves in the mesoscale field

Following the main storm event of 3–7 April 2006, we observe that the pattern of the mesoscale flows is imprinted on the downward vertical energy flux of both large (low k) and small (high k) NIWs. The downward vertical energy flux for small-scale NIWs is concentrated in regions of high shear, within fronts and filaments, and at the edge of eddy structures while the energy flux for large-scale NIWs is affected to a lesser extent by the prevailing mesoscale conditions at 200 m. The imprint of the mesoscale on large-scale NIWs is more readily observed at 500 m as shown in Fig. 6. The large-scale NIWs with wavelength matching the eddies’ length scale are trapped and rapidly fluxed down the inertial chimneys (in the large anticyclonic eddies along 37°N) following the onset of the storm system, while the remaining fraction of large waves propagate laterally away as is observed at 200 m in Fig. 5a. The selective trapping of the NIWs amidst the mesoscale background flow has been documented in earlier numerical studies (Balmforth and Young 1999; Klein et al. 2004b), whereby large-scale NIWs are concentrated in large anticyclonic eddies and smaller-scale NIWs in smaller negative vorticity structures. More recent studies have quantified the drainage of NIWs within inertial chimneys through the wave KE (Zhai et al. 2005) or the wave energy density (Asselin and Young 2020), while this study directly estimates the vertical energy flux from the NIWs. From the “up–down separation” (section 3b), the vertical energy flux can also be computed separately for the upward- and downward-propagating NIWs, hence facilitating comparison of the NIE propagation into the ocean’s interior with respect to the wind energy input at the surface. For instance, the downward vertical flux of large-scale NIWs at 500 m in the large anticyclonic eddy (at 37°N, Fig. 6) is 4 mW m−2, which is roughly 20% of the near inertial wind work (Fig. 1b). We note that our estimates for the downward flux represent a lower limit and that the cross terms (see appendix C) may also contribute to the downward flux.

Fig. 6.
Fig. 6.

Average vertical energy flux at 500 m, computed from 7 to 14 Apr for large-scale downward-propagating NIWs. The contours outline regions of negative vorticity (Ro < −0.04) and the stippling define areas of high shear [log(Ri) < 2] at 200 m.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

The average vertical energy flux for the upward-propagating large-scale NIWs (Fig. 5c) is approximately of the same magnitude (∼2 mW m−2) as its downward-propagating counterpart (Fig. 5a). There is no bottom source for the generation of upward-propagating internal waves since tidal forcing is not included in our model configuration and the coarse resolution of the bathymetry in the model inhibits any topographic generation of internal waves, in the form of lee waves (Nikurashin and Ferrari 2010). Thus, the only viable source for this upward energy flux is the reflection of the large-scale NIWs, which is expected given that their vertical structure spans the depth of the water column (Alford and Whitmont 2007; Simmons and Alford 2012). Conversely, the vertical energy flux for the upward-propagating small-scale NIWs (Fig. 5d) is weaker than its downward-propagating counterpart (Fig. 5b), indicating that reflection of small-scale NIWs is not as prevalent.

In view of the selective trapping of the NIWs, we attempt to form two separate wave KE equations (section 4) for the downward-propagating NIWs; one for the large-scale (low k) NIWs in negative vorticity regions (section 5b) and one for the small-scale NIWs (high k) in high-shear regions (section 5c). The negative vorticity region is defined, using the Rossby number, as Ro < −0.04 and the high-shear area is characterized, through the Richardson number, as log(Ri) < 2. Ro and Ri are computed from instantaneous fields at 200 m depth. Hourly values of Ro and Ri are then averaged over a day to obtain the mean background relative vorticity and shear, respectively. The negative vorticity area (Ro < −0.04) is used to extract and horizontally average the terms in the KE equation [Eq. (2)] for the large-scale NIWs. The same procedure is repeated for the small-scale NIWs but using the high-shear region [log(Ri) < 2] for masking and averaging. As mentioned in section 4, the horizontal averaging of the terms in Eq. (2) is not performed over the whole model’s domain, but over a localized mask which varies with time. Given that the spatial averaging is applied over a time-varying open boundary, the cross terms (either between the wave/nonwave, up-going/down-going and large-scale/small-scale NIWs) may be nonnegligible. Furthermore, following the different filtering steps (section 3), the vertical dissipation and buoyancy flux cannot be consistently calculated for the respective down-going large-scale and small-scale NIWs. We thus compute the remaining energy terms of NIWs KE equation and attribute the residual to the vertical dissipation, buoyancy flux, and the cross terms that might be significant due to the spatial averaging within the open time-varying boundaries

A different set of threshold values for Ro and Ri, to define areas of negative vorticity and high shear, modifies the quantitative results (shown in sections 5b and 5c) obtained following masking and horizontal averaging of the terms in Eq. (1), but qualitative results remain unchanged. Furthermore, we acknowledge that certain scales of vorticity or shear would have a larger influence on NIWs of particular wavelength. Nevertheless, the study of the effect of certain scales of vorticity and shear on NIWs of varying wavelengths requires multiple bandpass filters (as opposed to a single high-pass filter described in section 3c) and is thus beyond the scope of the present study. We note that all analyses and discussions that ensue will focus on downward-propagating NIWs, either for the large- or small-scale NIWs.

b. Downward propagation of large-scale (low horizontal wavenumber) NIWs

Figure 7 shows the evolution of the terms that form Eq. (1) (omitting the advection and the buoyancy flux terms) for the large-scale NIWs, horizontally averaged over negative vorticity areas. The Hovmöller plots for the different energy terms are used to track the sources and sinks of NIE as the large-scale NIWs propagate into the ocean’s interior.

Fig. 7.
Fig. 7.

Horizontally averaged terms for the large-scale NIWs: cumulative sum normalized by the time series for (a),(b) wind work; (c),(d) KE; (e),(f) vertical flux; (g),(h) mean to wave conversion; (i),(j) convergence; (k),(l) horizontal dissipation; and (m),(n) vertical dissipation. The hatching delimits the mixed layer where the estimated vertical dissipation is erroneous. (left) Terms averaged in negative vorticity areas (Ro < −0.04) and (right) terms averaged in remaining areas (Ro < −0.04). The vertical dotted line shows the start of the main wind event.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

Following the main wind event (Fig. 1b), there is a sudden increase in the overall wind work on near-inertial motions as shown in Fig. 1c. A similar increase is observed in the wind work on down-going large-scale NIWs (Figs. 7a,b) and in the associated peak in vertical energy flux for the down-going large NIWs in negative vorticities on 7 April 2006. As indicated in Fig. 7e, a maximum in the vertical energy flux appears near the surface and propagates to depth over time, thereby demonstrating a rapid propagation of large-scale NIWs to a depth of roughly 700 m in negative vorticity areas. The maximum in energy flux is less prominent in nonnegative vorticity areas (Fig. 7f). Furthermore, the horizontal convergence of the energy flux is much higher in negative vorticity areas (Fig. 7i). The rapid propagation of the NIWs into the interior and the high level of convergence of the NIWs in negative vorticity areas, corroborate the view of inertial chimneys, whereby NIWs are trapped by strong eddies and rapidly fluxed into the interior (Lee and Niiler 1998; Asselin and Young 2020).

The large-scale NIWs KE in negative vorticity regions (Fig. 7c) are consistent with the structure of the vertical energy flux, where the maximum in KE starts at the surface and shifts to deeper levels in the water column. Nevertheless, the Hovmöller plot of KE depicts a clearer picture of the fate of the large-scale NIWs as they presumably reach the bottom of the negative vorticity structures. The elevated KE in the negative vorticity structures persist at depths of roughly 600 m for an extended period of time (nearly 2 weeks), suggesting that the downward-propagating NIWs have stalled. The waves are trapped in the inertial chimney, propagate vertically down, and ultimately reach a critical level at the base of the eddy, where the intensity of the negative vorticity is reduced. The required condition for subinertial propagation of NIWs is provided by the negative vorticity, which implies that beyond the anticyclonic eddy core, the NIWs can no longer exist. The potential stalling signal is predominantly from the large eddy north of the extension of the Kuroshio, at 36.6°N, 151°E (figure not shown). Furthermore, a fraction of the NIWs is leaking from the base of the negative vorticity structures as suggested by the faint divergence of the horizontal flux below 600 m depth (Fig. 7i).

The principal sink for the wave kinetic energy is dissipation. The horizontal dissipation (Figs. 7k,l) is negligible for large (low k) NIWs for both negative and nonnegative vorticities. We note that the vertical dissipation is included in Fig. 5 despite the large bias introduced when KPP is used on the filtered down-going NIWs (section 4c). The hatched area (Figs. 7m,n) delimits the upper mixed layer where the estimates of vertical dissipation are erroneous; the estimated values of ϵυ are correct below the mixed layer. Although we cannot accurately separate the vertical dissipation of the down-going large-scale NIWs in the mixed layer, Fig. 4 supports the view that the main sink of energy for the latter is vertical dissipation (given that ϵυ is a major energy sink for the total NIWs, it should also represent a major fraction of energy sink for both large-scale and small-scale NIWs). In addition, some elevated levels of vertical dissipation can be observed at 600 m from mid-April only in the negative vorticity regions (Fig. 7m); the depth and time occurrence of the submaximum in ϵυ supports the hypothesis of NIWs stalling at the base of anticyclonic eddies (Kunze et al. 1995).

To compare the magnitude of the terms of the wave KE equation [Eq. (1)], also shown on the Hovmöller plots of Fig. 7, all terms are integrated over depth [Eq. (2)] from the surface to 500-m depth. Figure 8 shows the terms of wave KE equation as depth-integrated sums and divided by a total time of 25 days to convert all terms to units of watts per square meter. The energy terms are horizontally averaged in negative vorticity regions (hatched bars) and compared with the energy terms averaged over the study region (solid bars). There are two main contributing sources for the large-scale (low k) NIWs, namely, the wind work and the MTW interactions. The wind work triggers the generation of the NIWs, which are further amplified by the MTW conversions. As observed from Fig. 7g, the MTW is relatively high after the wind event and follows the same trend as the vertical energy flux, showing that as the large-scale NIWs propagate down the inertial chimneys, they interact and draw energy from the eddies. On the other hand, the dominant sinks for the large-scale NIWs are the vertical energy flux penetrating through 500 m (Figs. 6, 7e, and 8), which acts as a transmission depth, and the vertical dissipation (deduced from Fig. 4). The absence of high vertical energy flux (Fig. 7f) and convergence (Fig. 7j) in nonnegative vorticity areas supports the theory that NIWs are only trapped in negative vorticity areas. In addition, Figs. 7 and 8 provide evidence that, in the absence of NIW trapping, there is minimal propagation into the interior and weak energy transfer from the eddies to the NIWs (Fig. 7h).

Fig. 8.
Fig. 8.

Cumulative sum of the terms from the wave KE budget, integrated in the vertical from 0 to 500 m, for large-scale (low k) NIWs horizontally averaged in the study region and in the negative vorticity regions (Ro < −0.04), respectively. The last value of the respective cumulative sums is shown on the bar plots and is in W m−2 as the time integrated value of each term has been divided by the total time. The labels of the energy terms are the same as in Fig. 4.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

Based on Fig. 8, all energy terms are of larger amplitude in the negative vorticity areas (hatched bars), except for the wind work and the wave-to-wave conversion (WTW). The WTW term is included in Fig. 8 to account for the conversion of energy between the large-scale and small-scale NIWs. We compute the WTW term using the same formula as the MTW [uh(u)u¯h], but in this case u′ represent the small-scale NIW velocity while u¯h is the large-scale NIW velocity field. We observe that energy cascades from the large-scale to the small-scale NIWs. The WTW conversion is smaller in negative vorticity regions given that the large-scale and small-scale NIWs are under the influence of selective trapping. Large-scale NIWs are trapped in anticyclonic eddies while their smaller-scale counterparts are at the edge of the vortices, thus inhibiting any interaction between the two groups of NIWs when Ro < −0.04.

Last, the residual term shown in Fig. 8 comprises the vertical dissipation, buoyancy flux term, and the cross terms (between wave and nonwave, up-going and down-going, low and high vertical wavenumber NIWs) that are nonnegligible when averaging over a time varying open boundary. The vertical dissipation is a definite sink and predominantly balances the residual (based on Fig. 4). Conversely, the buoyancy flux can either be a source or sink of KE; nevertheless, the contribution of the buoyancy flux is relatively minor (figure not shown) compared to the other energy terms in Eq. (2). We recall that Fig. 8 is not used to close the NIWs’ energy budget, but rather to showcase which terms, among the energy terms we can properly estimate following the filtering steps, act as important KE contributors or sinks.

c. Propagation of small-scale (high horizontal wavenumber) NIWs

Figure 9 shows the evolution of the energy terms [Eq. (1)] for the small-scale NIWs, horizontally averaged over high-shear regions. The small-scale (high k) NIWs in high-shear regions are energized by two rapid increases in wind work, namely, on 7 April, following the main storm event, and on 21 April (Fig. 9a). The second increase in wind work on the small NIWs is not observed in the wind work on large NIWs (Fig. 7a), nor in the wind work on the overall inertial motions (Fig. 1c). Both increases in wind work on small NIWs are followed by elevated levels in the small-scale NIW KE (Fig. 9c) and vertical energy flux (Fig. 9e), suggesting that small changes in the wind can independently trigger small-scale NIWs. The second peak (on 21 April) in the small-scale NIW vertical energy flux could be caused by a small increase in wind stress or by an alignment of the winds with the surface currents in the high-shear region, or through a combination of both effects. In this instance, analysis of the time series of wind stress and wind work (figure not shown) shows that it is the alignment of the wind with mesoscale currents that primarily drives the enhanced generation of purely small-scale NIWs. In addition, the effects of the MTW conversion are important for the small-scale NIWs amplification in high-shear regions during both wind events. Figure 9g shows an increased energy transfer from the mean flow to the small NIWs, in the upper 250 m, as the wind work injects energy into the waves. A similar energy pathway to large-scale NIWs can be inferred, whereby the wind triggers the formation of small-scale NIWs; the high k waves subsequently draw energy from the mean flow as they propagate.

Fig. 9.
Fig. 9.

Horizontally averaged terms for the small-scale NIWs: cumulative sum normalized by the time series for (a),(b) wind work; (c),(d) KE; (e),(f) vertical flux; (g),(h) mean to wave conversion; (i),(j) convergence; (k),(l) horizontal dissipation; and (m),(n) vertical dissipation. The hatching delimits the mixed layer where the estimated vertical dissipation is erroneous. (left) Terms averaged in high-shear areas [log(Ri) < 2], and terms averaged in remaining areas [log(Ri) < 2]. The vertical dotted lines show the wind events that trigger the small-scale waves.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

The elevated levels of KE (Fig. 9c) and vertical energy flux (Fig. 9e) for the small-scale NIWs are restricted to the upper 500 m, suggesting that small-scale NIWs cannot propagate as deep into the ocean interior in high-shear regions, as opposed to their large-scale counterpart in negative vorticity areas. Furthermore, the small-scale NIWs’ energy has a long residence time in the upper layer, which is expected given the slow vertical propagation speed of small-scale NIWs (Alford et al. 2016). However, the elevated levels in both KE (Fig. 9c) and vertical energy flux (Fig. 9e) do not shift to deeper levels, even after a prolonged period. Instead, the small-scale NIW vertical energy flux and KE slowly decay over time. The correlation between elevated horizontal dissipation rates (Fig. 9k) and high levels of KE (Fig. 9c) points to dissipation as the primary driver of small-scale wave energy loss.

Figure 10 showcases the small-scale wave KE terms [Eq. (2)] temporally and horizontally averaged in high shear regions (hatched bars), and integrated over the upper 500 m. The energy terms of the small-scale NIWs averaged over the study region (solid bars) are also included in Fig. 10. The averaged energy terms (except for the WTW) are of higher magnitude when log(Ri) < 2. The major energy sources for small-scale NIWs in high shear are the MTW conversion followed by wind energy input and WTW conversion (from the down-going large-scale NIWs). We highlight that the MTW conversion term is larger than (almost double) the wind work for small-scale NIWs, suggesting that MTW conversion is an important energy contributor in high-shear regions.

Fig. 10.
Fig. 10.

Cumulative sum of the terms from the wave KE budget, integrated in the vertical from 0 to 500 m, for small-scale (high k) NIWs horizontally averaged in the study region and in high-shear areas [log(Ri) < 2], respectively. The last value of the respective cumulative sums is shown on the bar plots and is in W m−2 as the time integrated value of each term has been divided by the total time. The labels of the energy terms are the same as in Fig. 4.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

Advective processes are important in the extension of the Kuroshio (Zhai et al. 2004), and in this case advective processes transport small-scale NIWs out of the defined area of high shear. On the other hand, the divergence of the horizontal wave flux, shown in Fig. 7i, indicates that the small NIWs are able to propagate horizontally out of the frontal regions, a process that differs from NIWs created by spontaneous generation whereby the waves are trapped in the front (Nagai et al. 2015). While divergence of small-scale NIWs can represent a loss of near-inertial energy from the high-shear region, the principal sink for the small-scale NIWs remains dissipation (Figs. 9k,m), with vertical dissipation likely accounting for the majority loss of NIE (deduced from Fig. 4).

As in the previous section, the residual from Fig. 10 contains the vertical dissipation, the buoyancy flux, and the cross terms that are nonnegligible given the time-varying open boundary [log(Ri) < 2] over which each energy term is averaged. We note that the residual from both sets of energy terms averaged over the study region and over the high-shear area, respectively, is negative. Given that the large negative residual is also present in the energy terms averaged over the study region (solid bar plots in Fig. 10), we discard the hypothesis that the spatial mean over a time-varying open boundary introduces large inconsistencies in the KE budget. The large negative residual instead suggests that a KE contributor is missing from the small-scale NIW KE equation. The vertical dissipation is a sink of wave energy and would only contribute to making the residual even more negative. We instead trace back the missing source of KE to the cross terms between the large-scale and small-scale NIWs in the buoyancy flux term (appendix C). The cross terms in the buoyancy flux provide a sizeable source of KE to the small-scale NIWs and are consistent with the magnitude of the residual. However, no physical mechanism can be attributed to cross terms between the low k and high k waves in wb′. We argue that the large magnitude of the cross terms is rather an artifact from the wavelength separation of the buoyancy term.

6. Discussion

The separation of NIWs into large-scale and small-scale waves, through the advanced filtering methods used in this study, allows us to investigate the distinct dynamics of the NIWs of different wavelengths. In this model simulation, small-scale NIWs are primarily found in high-shear regions and propagate slowly to shallow depths where they are dissipated. These small-scale waves are generated directly from wind forcing and have their energy amplified by the mean flow; the latter do not rely purely on the forward energy cascade from the large-scale waves. On the other hand, large-scale NIWs propagate to greater depth, especially in regions of strong negative relative vorticity. These distinct characteristics, as well as the generation mechanisms, are examined in more detail in the following subsections.

a. Energeics of large-scale NIWs

The behavior of large-scale NIWs in negative vorticity regions corroborates the phenomena observed in inertial chimneys, characterized by high convergence (Kunze 1985) and rapid downward propagation of NIWs (Zhai et al. 2005). This study estimates that ∼30% of the wind work on large-scale NIWs converges into anticyclonic structures, and fluxes downward into the interior (down to 1000 m within 2 weeks). The Lagrangian filtering enabled the direct computation of the energy fluxes (both lateral and vertical), which were then compared with the wind energy input, and the other terms in the wave KE equation. Both the wind work and MTW conversion were shown to represent significant sources of energy (Fig. 8). Over the period of April 2006, the wind work injects on average 1.8 mW m−2 of NIE into large-scale NIWs while the MTW conversion contributes to a comparable 1.5 mW m−2 of NIE as the large-scale NIWs draw energy from the anticyclonic eddies upon their downward propagation, supporting the findings from Barkan et al. (2017). The vertical flux from large-scale NIWs transiting through 500-m depth is 1 mW m−2, which is 60% of the wind work, but we underline that the MTW conversion is also an important source term in anticyclonic structures. Therefore, the NIE fluxing through 500-m depth is roughly 30% of the combined energy input from both the wind work and MTW conversion, while the rest of energy is lost to dissipation. The trapping and amplification of large-scale NIWs has been investigated by earlier studies, but a novelty of this study is that we quantify the temporal evolution, with depth, of the different sources and sinks of large-scale NIW energy as the waves travel within the eddy structure.

b. Independent forcing of small-scale NIWs

A second outcome of this study concerns the direct energy input from the winds to the small-scale NIWs (Figs. 9a,c,e). Previous studies, based on Gill’s formalism (Gill 1984), described an energetic pathway whereby synoptic winds imprint their scales on the surface near-inertial motions, which are subsequently distorted and aggregated to smaller scales by the mesoscale eddy field (Mooers 1975; van Meurs 1998). In this view, wind-generated near-inertial motions experience a forward energy cascade to smaller scales, through the reorganizing contribution of mesoscale structures, rather than near-inertial energy being directly injected into small-scale NIWs.

These previous theories predict that the generation of small-scale NIWs will only occur following the larger-scale NIWs generation; however, here we have shown that small-scale NIWs can be generated independently. The second peak in the wind work that acts on the small-scale NIWs (Fig. 9a) is not observed in the wind work on large NIWs (Fig. 7a), further validating the theory that small NIWs are not always an end product of the forward energy cascade from their large-scale counterparts. Recent numerical studies (Rimac et al. 2013) highlighted the need for high-resolution wind forcing, both spatially (Jiang et al. 2005) and temporally (Klein et al. 2004a), to avoid an underestimation of surface NIE. Those studies hinted toward near-inertial ringing at smaller scales by demonstrating enhanced wind work when a finer resolution of the wind forcing is imposed on their respective model, but no study has yet investigated the significance of the direct energy input into small-scale NIWs nor tested whether such pathway was possible. The immediate increase in vertical energy flux (Fig. 9e) and KE (Fig. 9c) for small-scale NIWs, following increases in wind work on small inertial motions (Fig. 9a), supports the view of direct energy input into small-scale waves. A sudden alignment of the winds with smaller-scale surface motions is sufficient to trigger the direct generation of small-scale NIWs; the waves are further amplified by the MTW interactions (Fig. 9g, Fig. 10), with the MTW contribution being twice as large as the wind work in our model.

c. Importance of the filtering techniques for analyzing NIWs’ energetics

The filtering employed in our study allows the estimation of the multiple sources and sinks of NIE, including the energy exchange with the mean flow. Numerous past studies have considered the wind work as the sole energy source for the wind-generated NIWs and have subsequently compared it to the propagating NIW energy. Those studies (Zhai et al. 2005; Furuichi et al. 2008) assumed that the difference between the wind energy input and the wave energy is turbulent dissipation, which results in mixed layer entertainment and deepening (Plueddemann and Farrar 2006). However, our results show that the background mean flow is important in feeding energy to the wind-generated NIWs, and that wave–mean flow energy exchanges have to be taken into account when analyzing the NIW energetics. The processes involved in the NIW propagation and interaction with the background flow are complex and remain to be completely understood. While theoretical studies (Young and Jelloul 1997; Balmforth et al. 1998) and idealized models (Whitt and Thomas 2013; Rocha et al. 2018; Asselin et al. 2020) have explained some of the underlying physical processes driving the NIW dynamics, numerical simulations with more realistic conditions are warranted to support the hypotheses posited by theoretical works. This study highlights the ability of our current filtering methodology to validate some of the proposed theories, such as the wave–mean flow KE exchanges. On the other hand, estimation of the energy terms such as the buoyancy flux is biased when evaluated separately for large-scale and small-scale NIWs, as in the approach used in our study. The wavelength separation is cleaner when filtering fields of velocity and pressure but seems to fail when filtering buoyancy. To avoid the biases associated with the wavelength filtering of buoyancy fields, we propose spectral transfers (Arbic et al. 2012) as an alternate method to study the transfer of energy between NIWs of different scales. As spatial scales between mesoscale motions and NIWs are not always distinguishable (Barkan et al. 2017), we suggest extracting the wave component using the Lagrangian filtering and applying the filtering in terms of the wave propagation, and then using spectral transfers in the wavenumber space. The spectral transfer in the wavenumber space would thus show the KE and APE fluxes across NIWs of different wavelengths. Alternatively, the coarse graining method (Aluie et al. 2018; Barkan et al. 2021) could be extended to investigate both the KE and APE exchanges across the different scales of NIWs.

Despite the caveats in our methodology (discussed in section 4 and above), our results agree with previous studies with respect to the trapping and amplification of NIWs through MTW conversions in regions of high background vorticity (Polzin 2008) and shear (Whitt and Thomas 2013), respectively. We hence argue that the wave filtering and analyses presented here are powerful tools in understanding and validating the energetics as well as dynamics of NIWs.

Acknowledgments.

JR acknowledges support from the Australian Government’s Research Training Program (AGRTP). CJS acknowledges support from an ARC Discovery Early Career Researcher Award DE180100087 and an Australian National University Futures Scheme award. The authors would also like to thank two anonymous reviewers. Their comments greatly helped in improving the initial manuscript. Numerical simulations and analyses were conducted on the National Computational Infrastructure (NCI) facility, Canberra, Australia.

Data availability statement.

This work is based on output from high‐resolution numerical models; model configuration files can be accessed on https://github.com/Jem-R/kdependence_NIWs and the Lagrangian filtering library can be accessed on https://github.com/angus-g/lagrangian-filtering.

APPENDIX A

Filtering of the Pressure Field into Its Wave and Nonwave Component

The pressure field on an Eulerian frequency spectrum does not show a clear minimum just below the inertial frequency, and the absence of this minimum complicates the partitioning of the pressure field into its wave and nonwave component. As opposed to the velocity field, which on the large-scale averages out to zero, the pressure field possesses a significant background component that can also undergo advection by the mean flow. The background component of the pressure field translates into high levels at low frequencies, and the advective “smearing” results in a continuous roll off from low to higher frequencies on the Eulerian frequency spectrum (Fig. A1). Consequently, any Fourier filtering of the pressure signal on an Eulerian frame results in substantial ringing. Gibbs’ phenomenon can be attenuated through tapering of the frequency filter but at the expense of unwanted spectral leakage.

Fig. A1.
Fig. A1.

Power spectral density for the total waves’ (a) zonal velocity (u′) and (b) pressure (p′). The spectra are computed from Eulerian and Lagrangian data, respectively, at 200 m, and the spectra are horizontally averaged over a section defined by 23°–30.5°N and 155°E–180°. The vertical line is the near-inertial frequency cutoff used in the Lagrangian filtering as defined in section 3a.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

The Lagrangian filter avoids the bias introduced by the advection of the pressure signals and hence prevents any ringing during the filtering process. Given that the Lagrangian filter follows the water parcels, it removes the smeared out lower frequencies, observed on the Eulerian spectrum, and uncovers the minimum present at the near-inertial frequency. The Lagrangian filter thus facilitates the separation into the wave and nonwave component.

APPENDIX B

Bias in the Separation of the Up-Going and Down-Going Component

The Doppler shift can be large enough to change the sign of the waves’ Eulerian frequency and hence introduce a bias in the separation of the NIWs into their respective up-going and down-going component. The Doppler shift is defined as ku¯ in ωo=ωi+ku¯. The ωi is the waves’ intrinsic frequency, k is the wavenumber, and u¯ the speed of the mean flow. The up–down separation uses Lagrangian filtered fields, thus implying that ωi > f, where f is the inertial frequency. Consequently the only scenario for ωo to change sign is when U>f/k and when the waves are propagating against the mean flow. Given the smallest wavelength (resolved in our model configuration) of 30 km and f = 5.7 × 10−5 s−1, the mean flow U needs to exceed 0.16 m s−1 for the up–down separation to break down. Failures in the up–down separation (Fig. 5) are observed in regions where U > 0.16 m s−1, which is in some sections of the extension of the Kuroshio and at the outer edge of eddies. Nevertheless, we note that such bias is not significant and that the overall separation into the up-going and down-going component holds.

APPENDIX C

Cross Terms in the NIW KE Equation

We verify the different biases introduced in the estimation of the energy budget following the different filtering steps described in section 3. For instance, filtering in terms of the propagating direction introduces cross terms between the up-going and down-going NIWs, while the wavelength separation entails the cross terms between the large-scale and small-scale NIWs. We first identify the biases that come into play when filtering the NIWs as per their respective propagating direction. Each term of the KE budget [Eq. (2)] is computed separately using the respective filtered fields of velocity and pressure of the downward- and upward-propagating NIWs. The different energy terms are horizontally averaged in the study region and vertically integrated from the surface to 500 m. In Fig. C1, the respective pair of down-going and up-going NIW energy terms are compared with their corresponding total NIW energy term from Fig. 4. The contribution from the up-going and down-going wave energy should ideally add up to their total wave counterpart. However, the filtering of the NIWs into their propagating direction is imperfect and any missing fraction of the total wave energy lies in the cross terms between the up-going and down-going waves. For example, expanding the total wave wind work into its up-going and down-going fields, we notice the absence of cross terms [(uτ)total=udownτ+uupτ], which is reflected in Fig. C1. On the other hand, expanding the total waves vertical flux into its up-going and down-going component [(wp)total=wdownpdown+wdownpup+wuppdown+wuppup] demonstrates the presence of cross terms that are not considered when adding the upward and downward vertical flux of the NIWs.

Fig. C1.
Fig. C1.

Energy terms computed from the upward-propagating and downward-propagating waves’ fields, horizontally averaged over the study region and integrated over the depth range 0–500 m. Similar to Fig. 4, the last value of the time cumulative sum of each energy term (divided by the total time) is recorded for both the up-going and down-going signal, and is normalized against their corresponding values computed from the total NIWs from Fig. 4. The buoyancy flux term (wb) has been neglected given its large magnitude from both the down-going and up-going NIWs (opposing signs) with respect to wb of small value when computed from the total NIWs.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

Based on Fig. C1, most cross terms between the up-going and down-going waves are relatively minor (<35% of |down-going| + |up-going|) for most of the energy terms; the cross terms are thus considered to not be crucial terms in the KE budget. Nevertheless, Fig. C1 shows that the vertical dissipation (ϵυ) computed for the respective up-going and down-going NIWs is erroneously high. We suspect that a small bias is introduced when performing the filtering in terms of the propagating direction; the bias is further amplified when estimating ϵυ using viscosities from the KPP scheme in the mixed layer. We note that below the mixed layer, ϵυ is correctly estimated for both up-going and down-going NIWs since the latter components approximately add up to ϵυ computed from the total NIWs’ fields. We thus conclude that the computation of ϵυ for the filtered NIWs in terms of their propagating direction is biased in the surface mixed layer.

We then compute each energy term from Eq. (2) but using large-scale (low k) and small-scale (high k) waves; each energy term is horizontally average over the study region and integrated over the upper 500 m of the water column. Low k and high k NIWs in this instance each comprise both the up-going and down-going constituents. The respective pair of energy terms from the large-scale and small-scale NIWs are combined and then compared with the total NIW energy terms (Fig. C2). The combined components fall short of the total NIW energy and the cross terms between the large-scale and small-scale NIWs account for the missing fraction of energy. Most cross terms between low k and high k waves are relatively minor but we note that the cross terms in the buoyancy flux (Fig. C2f) is significant, representing almost 40% of total NIW buoyancy flux.

Fig. C2.
Fig. C2.

Energy terms computed from the large-scale (low k) waves (solid blue line), small-scale (high k) waves (solid red line), and total waves (dashed black line) horizontally averaged over the study region and integrated over the depth range 0–500 m. The percentage difference between the combined energy component from the large-scale and small-scale waves (dotted black line), and their corresponding total wave energy value is shown in each subplot. All energy terms are cumulatively time integrated and as opposed to Fig. C1, the terms are not normalized with respect to the total wave energy values.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

The wavelength separation (last filtering step in Fig. 2) works well on wave velocity (u′, υ′, w′) and pressure fields (p′), but fails when applied to the wave buoyancy field (b′). The imperfect wavelength separation of b′ introduces nonnegligible cross terms in the buoyancy flux between the large-scale and small-scale waves. Figure C3 shows the buoyancy flux terms but for the down-going low k and down-going high k NIWs, as well as the cross terms between the two components, horizontally averaged in the study region. The terms in Fig. C3 are positive and act as energy sources for the respective NIWs, converting APE to KE for the down-going large-scale NIWs [(wb)lowK] and small-scale NIWs [(wb)highK], respectively. The combined contribution from the down-going high k NIWs and the cross terms (whighKblowK, wlowKbhighK) provide a sizeable source of KE (≈6 × 10−5 W m−2), which is consistent with the magnitude of residual of the down-going small-scale NIWs in Fig. 10 (solid bar that represents residual from energy terms horizontally averaged in the study region). Nevertheless, no physical mechanism can be attributed to the cross terms in the buoyancy flux. We argue that the cross terms are artifacts from the wavelength separation of the buoyancy term and suggest other methods (see section 6c) to study the transfer of available potential energy (APE) across the different scales of NIWs.

Fig. C3.
Fig. C3.

The buoyancy flux integrated from 0 to 500 m, computed for the down-going waves [(wb)down], down-going large-scale waves [(wb)lowK], down-going small-scale waves [(wb)highK], and the cross terms between down-going low k and high k fields (whighKblowK, wlowKbhighK). All the individual components are horizontally averaged in the study region. Each term is time integrated and divided by the total time for better comparison with the residual in Fig. 10.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0266.1

To determine whether the respective energy terms from Eq. (2) are adequately estimated when using the filtered (down-going low k and high low k) wave fields, we define a threshold of less than 35% of the total NIWs’ energy for the biases and cross terms that result from the filtering of the waves. In instances when the total (net) NIW energy is smaller than the contributing components (e.g., the vertical flux from the up-going and down-going components is of opposite signs), we take the absolute value of each contributing component as benchmark for our threshold. Based on our defined threshold and considering Figs. C1 and C2, most energy terms from Eq. (2) are reasonably estimated when using the filtered wave fields, except for the vertical dissipation (ϵυ) and the buoyancy flux (wb′). A very large bias is introduced when estimating the vertical dissipation of the filtered up-going and down-going NIWs through KPP (Fig. C1) while the cross terms between the high k and low k NIWs are relatively large for the buoyancy flux (Fig. C2). We thus establish that the vertical dissipation (within the mixed layer) and the buoyancy flux cannot be correctly estimated when using the wave fields filtered in terms of propagating direction and wavelength.

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