1. Introduction
Near-inertial internal waves (NIWs) are a ubiquitous feature in the ocean (Garrett 2001). As a natural resonant frequency of fluids on a rotating planet, NIWs are efficiently generated by time-varying wind stresses associated with the passage of atmospheric fronts, tropical cyclones, and synoptic storms (D’Asaro 1985). Global estimates of the wind work on NIWs range from 0.5 to 1.4 TW (e.g., Alford 2003; Jiang et al. 2005; Rimac et al. 2013), comparable to the wind work on large-scale ocean general circulations (Wunsch 1998).
NIWs are of central importance to a variety of ocean processes. Numerical model studies reveal an increase of vertical velocity variance if the temporal resolution of wind fields is high enough to resolve the near-inertial frequency (Danioux et al. 2008; Cardona and Bracco 2012). This increase is evident throughout the water column and is even more pronounced in the deep ocean (Danioux et al. 2008). The deep extension of near-inertial vertical velocity is mainly ascribed to the refraction of NIWs by the vorticity of mesoscale and submesoscale eddies, which significantly shortens the horizontal scale of NIWs and accelerates their downward propagation (Gill 1984; Young and Ben Jelloul 1997; Lee and Niiler 1998; Klein and Smith 2001; Klein et al. 2004). Furthermore, previous theoretical studies suggest that NIWs can transfer energy to their harmonics through triad wave resonance (Niwa and Hibiya 1997; Danioux and Klein 2008; Danioux et al. 2011). This is confirmed by simulated vertical velocity frequency spectra that exhibit pronounced peaks not only at the inertial frequency but also its harmonic frequencies (e.g., Niwa and Hibiya 1997; Komori et al. 2008; Danioux et al. 2011). As the horizontal/vertical aspect ratio of internal waves becomes smaller with the increasing wave frequency, this resonance mechanism provides an efficient way in intensifying small-scale vertical velocity.
A recent high-resolution numerical model study by Zhong and Bracco (2013) reported energetic small-scale (<10 km) vertical velocity in the deep Gulf of Mexico. Their appearance tends to be collocated with the energetic NIWs, suggesting a dynamic linkage between them. However, their simulated small-scale vertical velocity variance does not only reside in the inertial frequency and its harmonic frequencies but also spreads over all the superinertial frequencies. This broadband feature cannot be explained by the triad wave resonance proposed in previous studies (Niwa and Hibiya 1997; Danioux and Klein 2008). Instead, it implies that the underlying dynamics might be understood from the perspective of wave–mean flow interactions, where the “wave” and “mean flow” correspond to the small-scale superinertial internal waves (SSIWs) and relatively large-scale NIWs, respectively. Unlike the triad wave resonance where NIWs can only interact with their harmonics, the wave–mean flow interactions allow NIWs to exert an influence on SSIWs with a broad range of frequencies. In this study, we analyze the validity of this conjecture based on theoretical and numerical models.
The paper is organized as follows. An idealized high-resolution numerical simulation designed to illustrate the relationship between SSIWs and NIWs is first presented in section 2. Theoretical solutions governing SSIW–NIW interactions are developed in section 3 and are validated against the idealized numerical simulation. A further validation of the theoretical results using a realistic simulation of circulation in the Gulf of Mexico is made in section 4. Conclusions and a discussion are finally given in section 5.
2. An idealized high-resolution numerical simulation
a. Model description and experiment design
To explore the underlying dynamics for SSIW–NIW interaction, an idealized high-resolution numerical experiment is performed using the Regional Ocean Modeling System (ROMS), which is a free surface 3D primitive equation model based on hydrostatic and Boussinesq approximations (Shchepetkin and McWilliams 2005). The model is configured over a 20° × 20° domain with a uniform depth of 2000 m. Fifty vertical layers are used, with 19 layers concentrated in the upper 100 m. The horizontal grid size is set at 1 km × 1 km. The nonlocal K-profile parameterization (Large et al. 1994) is used to parameterize vertical mixing, but no horizontal eddy viscosity and diffusivity are used since at 1-km resolution the model should be able to explicitly resolve mesoscale and submesoscale eddies. The horizontal diffusivity and viscosity are thus taken as their molecular values. Finally, a radiation boundary condition is used at lateral boundaries of the model domain.
It should be noted that the hydrostatic approximation made by ROMS tends to overestimate the vertical velocity of small-scale flow because of its much reduced horizontal/vertical aspect ratio (Vitousek and Fringer 2011). This tendency is counterbalanced to some extent by numerical dispersion that mimics the missing physical dispersion due to nonhydrostaticity. In fact, previous numerical studies indicated that the numerical dispersion can be tuned to replicate the nonhydrostatic dispersion not resolved in a hydrostatic model (Shuto 1991; Burwell et al. 2007). Therefore, the hydrostatic approximation is acceptable for our qualitative analysis here.
The initial profiles in IHS for (a) potential temperature, (b) salinity, and (c) buoyancy frequency.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
b. IHS result
Both large-scale and small-scale vertical motions emerge in the wake of the idealized hurricane (Fig. 2). Here the large-scale (small-scale) signals are attained by a low-pass (high-pass) filter with a cutoff wavelength of 30 km (15 km). The large-scale vertical velocity is primarily associated with the hurricane-generated NIWs, as evidenced by the pronounced peak around the Coriolis frequency f in its frequency spectrum (Fig. 3a). There is a marked cross-track asymmetry for the large-scale vertical velocity with stronger amplitude on the right of the moving hurricane (Fig. 3a). This is mainly because the wind stress vector rotates anticyclonically on the right of the track and thus is able to resonate with the inertial oscillations there (Price 1981, 1983). Far away from the hurricane track, the vertical structure of large-scale vertical velocity is characterized by the first baroclinic mode (Fig. 2c). The projection coefficient of the first baroclinic mode is an order of magnitude larger than those of higher modes. The dominance of first baroclinic mode in the far field is mainly due to its fastest group velocity. The vertical structure of large-scale vertical velocity becomes more complex near the hurricane track where both the first and second baroclinic modes make important contributions.
(a) Vertical section of large-scale (>30 km) vertical velocity (m s−1) along the longitude 7.4° on 6d0h in IHS and (c) its projection coefficients on different vertical modes. (b),(d) As in (a) and (c), but for the small-scale (<15 km) vertical velocity. The dotted vertical line denotes the center of the hurricane.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
(a) Latitudinal distribution of frequency spectra in logarithm scale (log10) of (a) large-scale (>30 km) and (b) small-scale (<15 km) vertical velocity at 625 m along the longitude 7.4° in IHS. The dotted vertical line denotes the center of the hurricane. Here only the time period between 4d1h and 7d0h in the wake of hurricane is used to compute the frequency spectra. The broad peak in (a) could partially result from the insufficient spectral resolution due to the short time record.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
The small-scale vertical velocity signals are dominated by a few isolated wavelike fronts associated with vigorous vertical velocity of O(0.01) m s−1 (Fig. 2b). The wavelength of the fronts is on the order of 10 km, and their vertical structure is well represented by the first baroclinic mode (Fig. 2d). The frequency spectrum of small-scale vertical velocity exhibits a pronounced peak at frequencies much higher than f (Fig. 3b), suggesting that the fronts of strong small-scale vertical velocity are SSIWs. Indeed, their wavenumber–frequency relationship agrees well with the short internal gravity wave dispersion relationship, ω = c1k, where c1 = 2 m s−1 is the gravity wave speed of the first baroclinic mode (Fig. 4).
The dispersion relation curve for the first baroclinic mode of internal waves (
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
There are several possible mechanisms responsible for the generation of the SSIWs in IHS. One possible candidate is disintegration of large-scale NIWs into SSIWs because of nonlinear steepening of NIWs (Vlasenko et al. 2005). Other candidates include direct small-scale superinertial wind forcing by the hurricane, geostrophic adjustment, or even perturbations resulting from the numerical discretization. However, the focus of this study is not on the generation mechanism of SSIWs. Instead, we are interested in how SSIWs are energized by NIWs after their generation. To illustrate the evolution of SSIWs in the presence of NIWs, we show a series of zoom-in snapshots following the strongest SSIW front in IHS (Fig. 5). It is found that the SSIW front propagates at a speed close to c1 and its phase contour is roughly aligned with that of NIWs. When entering a convergence (downwelling) zone of NIWs, the SSIW front is intensified rapidly before it becomes saturated. In contrast, the intensity of the SSIW front rapidly decreases when it goes into a divergence (upwelling) zone of NIWs. This suggests that NIWs are able to modulate the strength of SSIWs. We will examine the underlying dynamics of this modulation in the next section.
Sequential snapshots of small-scale (<15km) vertical velocity (m s−1) at 625 m in IHS. The arrows represent the surface horizontal velocity associated with large-scale (>30 km) NIWs with the gray lines denoting the phase contours.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
3. Analytical analysis
In this section, we present analytical analyses to understand the mechanism governing the SSIW–NIW interaction revealed by IHS described above. The fact that both the SSIWs and NIWs project strongly onto the first baroclinic mode allows us to reduce the complexity of the problem by considering only one vertical mode. In the following, we first present the analysis in a reduced gravity model framework represented by the first baroclinic mode and then in a more generalized dynamic model framework.
a. SSIW–NIW interaction in a reduced gravity model
The beta effect is unlikely to play a major role here because of the small horizontal scale of the SSIWs. Therefore, an f-plane approximation is applied. Furthermore, the system is isotropic horizontally under f-plane approximation, so that the Cartesian coordinate can be rotated to make the y axis parallel with the SSIW front. As both the SSIW and NIW exhibit much less variation in the alongfront direction than in the cross-front direction (Fig. 5), the system at hand can be reduced to a unidirectional wave equation by setting all the y derivatives to zero. Note that the neglect of y derivatives does not cause a loss of generality for the following derivations. The conclusions in this section are also valid in a more generalized case where the phase contours of SSIWs do not necessarily align with those of NIWs (see appendix A for details).
The validity of solutions (14)–(16) is demonstrated in appendix B by comparing them to numerical simulations of the reduced gravity model with all nonlinear terms included. With the aid of (16), the modulation dynamics of SSIWs by NIWs becomes quite clear. Following the SSIWs, 〈n〉 is amplified in convergence (downwelling) zones of NIWs because of the converging wave action flux of SSIWs, as indicated by (14a) and (16). Furthermore, the converging currents of NIWs can cause an increase in k and thus ωi by squeezing the phase contours of SSIWs. These two effects work in concert to result in a rapid amplification of 〈e〉, and thus 
The above analysis indicates that SSIWs are enhanced and damped in convergence and divergence regions of NIWs, respectively. However, this does not mean that NIWs have no net effects on the intensity of SSIWs. Imagine that there are two SSIWs with the same initial energy density 〈e0〉: one occurs in a convergence region with a growth rate σ > 0 and the other arrives at a divergence region with a decay rate of −σ < 0. Then their total energy density, 〈e0〉(eσt + e−σt), will increase with time because eσt + e−σt is an increasing function of t. This suggests that SSIWs in general tend to be more energetic in the presence of NIWs.
Wave action conservation of SSIWs in the presence of NIWs suggests that the modulation mechanism described above can be understood from the perspective of wave–mean flow interactions, except that here the “wave” and “mean flow” correspond to SSIWs and NIWs, respectively. This makes the modulation mechanism distinct from the previously proposed triad wave resonance mechanisms (Niwa and Hibiya 1997; Danioux and Klein 2008). Unlike the triad wave resonance where NIWs only interact with their harmonics, the modulation mechanism proposed here allows NIWs to interact with SSIWs in a broad range of frequencies.
b. SSIW–NIW interaction in a 3D primitive equation system
The above analysis based on the reduced gravity model suggests that the underlying dynamics for the modulation of SSIWs by NIWs can be understood in terms of wave action conservation of SSIWs in the presence of NIWs. In this section, we extend these solutions to a 3D primitive equation system.
If stratification is uniform, that is, N is a constant, F1 will be a cosine function. In this case, s will be exactly zero, so that SSIWs do not interact with NIWs. However, for a realistic oceanic setting where stratification is typically very strong in the upper several hundred meters and becomes much weaker in the deep ocean, the magnitude of F1 is much stronger in the upper ocean according to the Wentzel–Kramers–Brillouin (WKB) theory, leading to a positive value of s. In IHS, s is estimated to be 2.2 because of the sharp thermocline centered around 65 m (Fig. 1c).
The SSIW energy equations (23) and (24) in the 3D primitive equation system are similar to their counterparts in the reduced gravity model, that is, (13) and (14). Aside from the projection coefficient s, there is only one difference between the growth rate σ in the primitive equation system [(24b)] and reduced gravity system [(14b)]. Compared to (14b), an additional term sV1/2 arises in (24b). This term corresponds to 
(a) Propagation speed of the SSIW front and (b) time evolution of its energy density normalized by the initial value. The solid and dashed lines denote the values computed from the numerical simulation in HIS (blue) and analytical solutions (red), respectively.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
The above analysis suggests that the fundamental conclusions derived from the reduced gravity model hold to a large extent in the 3D primitive equation system. The modulation of SSIWs by NIWs can be primarily understood in terms of wave action conservation of SSIWs in the presence of NIWs. SSIWs are enhanced in convergence (downwelling) regions of NIWs and damped in divergence (upwelling) regions.
c. Validation of analytical analysis using IHS
1) Comparison of group velocity and growth rate of SSIWs
The validity of (23) and (24) can be assessed based on the comparisons with the numerical solutions in IHS. Here we track the SSIW front shown in Fig. 5 from its initial emergence until it propagates out of the model domain. To separate the SSIW from the NIW, a spatial low-pass filter is used to isolate the NIW with a cutoff wavelength of 30 km while the SSIW is attained by using a spatial high-pass filter with a cutoff wavelength of 15 km. As both the NIWs and SSIWs exhibit little variability in the alongfront direction, we take a vertical section perpendicular to the front to analyze their interactions. As in the theoretical analysis in section 3b, the axes are rotated so that the x axis is perpendicular to the front. Sensitivity tests suggest that choosing different vertical sections perpendicular to the front does not have any substantial impact on the following conclusions.
The current u (U) and buoyancy b (B) associated with the SSIWs (NIWs) are projected onto the first baroclinic mode to obtain the projection coefficients u1 (U1) and η1 (Z1) based on (18). The center of the SSIW front in IHS, xc, is defined as (xd + xu)/2, where xd and xu correspond to the locations of the largest negative and positive values of w1. The energy density of the SSIW front in IHS, e1, is computed following (22), and its mean value within the front 〈e1〉 is measured as 
The theoretical solutions of the group velocity and growth rate of the SSIW front are derived by substituting into (24) the values of U1, V1, Z1, ∂U1/∂x, and ∂Z1/∂x averaged within [xzd, xzu]. The group velocity derived from (24a) is consistent with cg,IHS (Fig. 6a). Particularly, they agree well with each other during the initial growing stage of the SSIW front (before 5d10h) with a discrepancy less than 10%. The larger error afterward might be attributed to the nonlinear effects as the amplitude of the SSIW front has undergone a substantial increase (Fig. 6b). As u1 within the front is of the same direction as the group velocity (not shown), the nonlinear advection would contribute to a faster propagation in the numerical experiment than the linear model solution predicts.
The growth rate derived from (24b) is also consistent with σIHS (Fig. 6b). Here the evolution of the SSIW front is divided into three stages. The first stage is the growing stage (before 5d10h) when its magnitude increases rapidly. The second is the saturation stage (5d10h–6d6h) with little magnitude variation. The final one is the decay stage (after 6d6h) when the magnitude starts to attenuate. The SSIW front resides in a convergence zone of NIWs during the first two stages but goes into a divergence zone in the final stage. The theoretical model [(24b)] shows good skill at the growing stage when the nonlinearity of the SSIW front is weak. Furthermore, it qualitatively reproduces the attenuation of the SSIW front at the decay stage because of the reversed sign of σIHS. Not surprisingly, (24b) overestimates the growth rate during the saturation stage when the nonlinear effects become important.
A notable feature is that the time for the SSIW front to reside in the convergence zone of NIWs (i.e., the growing and saturation stages) is considerably longer than the half-wave period of NIWs (Fig. 6b). Around the location (~25°N) of the SSIW front, the NIWs are characterized by a wave frequency Ω around 1.6f (Fig. 3a), corresponding to a half-wave period of 0.37 days. In contrast, the time for the SSIW front to reside in the convergence zone of NIWs reaches up to 1.8 days. This difference results mainly from the fact that the SSIW front does not stand still but propagates at a speed of cg,IHS ≈ 2 m s−1 (Fig. 6a). The frequency of NIWs in a reference frame following the SSIW front becomes Ωr = Ω − kNcg, where 
Finally, it should be noted that the energy exchange due to the vertical shear of NIWs, that is, sV1/2 in (24b), plays a negligible role compared to the remaining terms in (24b). The growth rates computed with and without the term sV1/2 agree well with each other with a discrepancy less than 5% (Fig. 6b). Therefore, SSIW wave action in the 3D primitive equation system can still be treated as a conserved quantity in practice.
2) Comparison of energy transfer rate from NIWs to SSIWs
Figure 7 displays several snapshots of TRIHS during the growing stage of SSIW front. Consistent with our theoretical solutions, there is a pronounced energy transfer from the NIWs to the SSIW front. In particular, both the spatial distribution and magnitude of TRIHS agree well with those derived from the theoretical solutions. This provides further evidence for the validity of theoretical solutions.
Sequential snapshots of vertically integrated energy transfer rate (m3 s−3) from NIWs to SSIWs computed from (top) the numerical simulation in IHS based on (25) and (bottom) the theoretical solution from (26).
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
4. Validation of analytical analysis using realistic numerical simulations
a. Model configurations
In sections 2 and 3, we proposed a new mechanism for modulation of SSIWs through SSIW–NIW interaction based on the idealized numerical simulation and theoretical analyses. To test whether the theory has any relevance to reality, we perform two more realistic simulations using ROMS configured for the entire Gulf of Mexico and forced by reanalysis atmospheric forcing (Fig. 8). The horizontal resolution of ROMS is set at 3 km with 60 vertical layers. The nonlocal K-profile parameterization (Large et al. 1994) is again used to parameterize vertical mixing.
Model domain in GoM-C and GoM-F. The contours are the topography. Only the region within the gray box (23.5°–27.5°N, 95°–85°W) is used for analysis in this study.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
Both simulations start from 21 March 2010 using Hybrid Coordinate Ocean Model (HYCOM) data assimilation (Chassignet et al. 2007) as the initial and boundary conditions and last for 90 days. Only the data in the last 20 days are used for the analysis shown below. One simulation, which is referred to as control run (hereinafter GoM-C), is forced by the 6-hourly and 0.25°-resolution atmospheric surface variables (e.g., wind and surface air temperature) obtained from the ERA-Interim reanalysis dataset (Dee et al. 2011). The other simulation, which is referred to as the filtered run (hereinafter GoM-F), uses the same atmospheric variables as in GoM-C except that the winds are daily averaged. As the inertial period in the Gulf of Mexico ranges from 1.6 days at 18°N to 1 day at 30°N, the daily averaged winds contain little variance at inertial and higher frequencies, leading to much suppressed NIWs in GoM-F. Therefore, a comparison between these two experiments with identical model resolutions and physics parameterizations can provide a useful evaluation for the influence of NIWs on SSIWs. To minimize topographic effects, we applied the analysis to the central Gulf of Mexico (23.5°–27.5°N, 95°–85°W) where the water depth is greater than 1000 m.
b. SSIWs in GoM-C and GoM-F
In GoM-C, energetic NIWs are excited by the 6-hourly wind forcing, as evidenced by a pronounced peak around f in the frequency spectrum of large-scale (>30 km) horizontal convergence/divergence ∇HU at 100 m (Fig. 9a). More than 60% of ∇HU variance comes from the near-inertial [(0.8–2)f] band, suggesting that NIWs make a dominant contribution to large-scale convergence/divergence. The energetic NIWs may be linked to the sea breeze whose period is close to the inertial period in the northern Gulf of Mexico (Zhang et al. 2009).
(a) The area-mean (23.5°–27.5°N, 95°–85°W) frequency spectrum of ∇HU at 100 m in GoM-C (red) and GoM-F (blue). (b) The cumulated contribution C(ω) to the difference of ∇HU variance between GoM-C and GoM-F. The dashed gray lines denote the near-inertial (0.8–2f) frequency band.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
Figure 10 displays the area-mean (23.5°–27.5°N, 95°–85°W) frequency spectra of large-scale (>30 km) and small-scale (<15 km) vertical velocity at various depths in GoM-C. For the large-scale vertical velocity, most of its variance comes from the subinertial (<0.8f) and near-inertial [(0.8–2)f] frequency bands. For instance, the contribution from the superinertial frequency band (>2f) is less than 8%, 7%, and 7% at 100, 300, and 500 m, respectively. Consistent with the previous numerical studies (e.g., Danioux et al. 2008; Komori et al. 2008), the frequency spectrum of large-scale vertical velocity exhibits a pronounced peak at f and a secondary peak at 2f (Fig. 10a). The former is directly related to wind-generated NIWs while the latter may result from the triad wave resonance transferring energy from wind-generated NIWs to their harmonics (Danioux and Klein 2008). In contrast, the superinertial frequency band accounts for more than 50% of small-scale vertical velocity variance at various depths (Fig. 10b), revealing the dominant role of SSIWs in generating small-scale vertical velocity in the ocean interior. In particular, there are no notable peaks at f and its harmonic frequencies in the frequency spectrum of small-scale vertical velocity.
The area-mean (23.5°–27.5°N, 95°–85°W) frequency spectra of (a) large-scale (>30 km) and (b) small-scale (<15 km) vertical velocity at various depths in GoM-C. The vertical dashed lines denote (left–right) f, 2f, and 3f.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
To test the theory developed in sections 2 and 3 in a realistic setting, we compare the superinertial (>2f) small-scale (<15 km) vertical velocity wss in convergence (∇HU < 0) and divergence (∇HU > 0) zones of large-scale flows in GoM-C. Here ∇HU is computed from the large-scale horizontal velocity at 100 m. Using velocity at different depths between the surface and 200 m does not change the results presented below significantly. Consistent with the analytical solutions, wss in the ocean interior (at 300 m) exhibits marked asymmetry between convergence (∇HU < 0) and divergence (∇HU > 0) zones of large-scale flows (Fig. 11a). The mean 
(a) PDFs of 
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
In spite of the fact that both GoM-C and GoM-F are conducted at the same resolution of 3 km, the small-scale vertical velocity becomes significantly weaker in GoM-F than in GoM-C (Fig. 12a). For instance, the area-mean (23.5°–27.5°N, 95°–85°W) small-scale vertical velocity variance at 300 m is 1.5 × 10−9 m2 s−2 in GoM-C, while it decreases to 0.8 × 10−9 m2 s−2 in GoM-F (the difference is statistically significant at 95% significance level). We note that about 55% of small-scale vertical velocity variance difference at 300 m between GoM-C and GoM-F is due to the superinertial frequency band while the near-inertial and subinertial frequency bands only account for 31% and 14% of the difference, respectively (Fig. 12b). The situation is similar at other depths. This suggests that the stronger small-scale vertical velocity in GoM-C than in GoM-F is primarily due to the intensified SSIWs.
(a) Ratios of area-mean (23.5°–27.5°N, 95°–85°W) 
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
It should be noted that the 6-hourly winds used in GoM-C do not contain any energy in the superinertial frequency band, so the stronger SSIWs in GoM-C are unlikely to be directly caused by the winds. Furthermore, as the enhanced small-scale superinertial vertical velocity variance in GoM-C is not confined to the harmonic frequencies of f but spreads over all the superinertial frequencies (Fig. 12b), it cannot be simply ascribed to the triad wave resonance mechanism (Danioux and Klein 2008), either. However, this broadband enhancement is consistent with our modulation mechanism that allows NIWs to exert an influence on SSIWs with a broad range of frequencies. This lends support to the important role of the modulation mechanism in intensifying SSIWs in the reality.
5. Summary and discussion
In this study, the influence of NIWs on SSIWs and their associated vertical velocity is theoretically and numerically analyzed using high-resolution ROMS simulations. We present a dynamic mechanism for modulating SSIW strength by background NIWs. It shows that in convergence (downwelling) regions of NIWs, energy flux of SSIWs converges and energy is transferred from NIWs to SSIWs, leading to enhanced SSIWs. The opposite is true in divergence (upwelling) zones of NIWs. The underlying dynamics can be understood in terms of wave action conservation of SSIWs in the presence of background NIWs.
The theoretical solution is validated by high-resolution ROMS simulations forced by realistic atmospheric forcing in the Gulf of Mexico. The simulations show that the strengths of SSIWs within convergence and divergence regions of NIWs exhibit a marked asymmetry with significantly stronger small-scale superinertial vertical velocity in convergence regions. By removing near-inertial wind forcing, the NIWs are substantially suppressed. This further leads to a significant decrease in the small-scale vertical velocity variance throughout the superinertial frequency band.
The modulation mechanism proposed in this study is analogous to the modulation of surface gravity wave strength by internal waves proposed by Alpers (1985). Its dynamics can be understood from the perspective of wave–mean flow interactions where the “wave” and “mean flow” correspond to SSIWs and NIWs, respectively. This makes the modulation mechanism proposed here distinct from the triad wave resonance mechanisms proposed in previous studies (Niwa and Hibiya 1997; Danioux and Klein 2008). While NIWs can interact only with their harmonics through the triad wave resonance, the modulation mechanism allows NIWs to exert an influence on SSIWs with a broad range of frequencies. Furthermore, the modulation mechanism proposed in this study should be distinguished from frontogenesis (Hoskins and Bretherton 1972) and filamentary intensification (McWilliams et al. 2009). Although frontogenesis and filamentary intensification could also give rise to stronger small-scale vertical velocity in convergent large-scale background flows, the small-scale vertical motions in these mechanisms are, to a large extent, balanced and have a frequency smaller or comparable to f (Hoskins and Bretherton 1972; Hoskins 1975; Thomas et al. 2008), making them distinct from SSIWs. In addition, the strong horizontal density gradient and presence of a horizontal boundary are key ingredients for the frontogenesis (Hoskins and Bretherton 1972) and filamentary intensification (McWilliams et al. 2009), so that small-scale vertical motions in these mechanisms are mainly confined to the surface mixed layer (Thomas et al. 2008). In contrast, the SSIWs have a much deeper vertical structure and can generate strong vertical velocity in the ocean interior.
Finally, the modulation mechanism proposed in this study provides a way of transferring energy from NIWs to SSIWs. The energy transfer is transient as the energy is transferred from NIWs to SSIWs in convergence regions of NIWs with reversed energy transfer when SSIWs propagate through divergence regions. However, this energy transfer could become permanent in the case that intensified SSIWs in the convergence region of NIWs become highly nonlinear and break. Such a scenario deserves further investigations in future studies, as it could potentially contribute to turbulent mixing in the oceans.
Acknowledgments
This research was made possible in part by a grant from BP/The Gulf of Mexico Research Initiative through the Gulf Integrated Spill Response Consortium. Z. Jing is partially supported by the Chinese Scholarship Council. P. Chang acknowledges the support from the National Program on Key Basic Research Project (973 Program) Award 2014CB745000 and the China National Global Change Major Research Project Award 2013CB956204. We thank Dr. Richard Greatbatch for his insightful discussion during the course of this work.
APPENDIX A
Wave Action Conservation for the Two-Dimensional Case
APPENDIX B
Validation of the Analytical Solutions (14)–(16)
Initial condition for Z + η.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
We use a high-pass (low-pass) filter with a cutoff wavelength of 15 (30) km to attain signals associated with the SSIW (NIW). As η(x, t) is modeled as a Gaussian wave packet, a Hilbert transform is used to estimate its envelope Λ(x, t) and phase θ(x, t). Then the center of SSIW, xc(t), is defined as the location where Λ(x, t) reaches its maximum. In this case, the group velocity can be computed as dxc/dt. The horizontal wavenumber of SSIW, k, is estimated based on a least squares fit to θ(x, t). The energy density of SSIW, 〈e〉, is computed following (4) as 
Figure B2 shows the evolution of 〈e〉, 〈n〉, cg, and k computed from the numerical simulations and analytical solutions (14)–(16). The analytical solutions agree well with numerical simulations. The difference in 〈e〉, 〈n〉, cg, and k between the analytical solutions and numerical simulations is within 10% of their initial values. We conclude that the analytical solutions are valid.
The evolution of (a) energy density 〈e〉, (b) action density 〈n〉, (c) horizontal wavenumber k, and (d) group velocity cg of SSIW derived from the reduced gravity model simulations (solid red) and theoretical solutions (14)–(16) (dashed blue). Here 〈e〉, 〈n〉, and k have been normalized by their initial values.
Citation: Journal of Physical Oceanography 46, 12; 10.1175/JPO-D-15-0239.1
APPENDIX C
Perturbation Analysis in a 3D Primitive Equation System
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