Numerical Simulations of Internal Tide Dynamics in a Steep Submarine Canyon

Eiji Masunaga aGlobal and Local Environment Co-creation Institute, Ibaraki University, Ibaraki, Japan
bScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Matthew H. Alford bScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Andrew J. Lucas bScripps Institution of Oceanography, University of California, San Diego, La Jolla, California
cDepartment of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California

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Andrea Rodriguez-Marin Freudmann bScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Abstract

This study investigates three-dimensional semidiurnal internal tide (IT) energetics in the vicinity of La Jolla Canyon, a steep shelf submarine canyon off the Southern California coast, with the Stanford Unstructured Nonhydrostatic Terrain-Following Adaptive Navier–Stokes Simulator (SUNTANS) numerical simulator. Numerical simulations show vertical structure and temporal phasing consistent with detailed field observations. ITs induce large (approximately 34 m from peak to peak) isotherm displacements and net onshore IT energy flux up to 200 W m−1. Although the net IT energy flux is onshore, the steep supercritical slope around the canyon results in strong reflection. The model provides the full life span of internal tides around the canyon, including internal tide generation, propagation, and dissipation. ITs propagate into the canyon from the south and are reflected back toward offshore from the canyon’s north side. In the inner part of the canyon, elevated mixing occurs in the middle layer due to an interaction between incident mode-1 ITs and reflected higher-mode ITs. The magnitude of IT flux, generation, and dissipation on the south side of the canyon are higher than those on the north side. An interference pattern in horizontal kinetic energy and available potential energy with a scale of approximately 20–50 km arises due to low-mode wave reflections. Our results provide new insight into IT dynamics associated with a small-scale canyon topography.

Significance Statement

Internal waves play an important role in ocean circulations and ecosystems. In particular, internal waves with frequencies of tides, known as internal tides, strongly enhance energy, heat, and mass transport in coastal oceans. This study presents internal tide dynamics in La Jolla Canyon, California, using a high-resolution numerical model. Model results show energy convergence in the canyon leading to internal tide energy dissipation and mixing. Some parts of internal tide energy reflect back offshore resulting in standing internal waves off California. This study provides new insights into internal tide dynamics and energy budgets in submarine canyons.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Eiji Masunaga, eiji.masunaga.office@vc.ibaraki.ac.jp

Abstract

This study investigates three-dimensional semidiurnal internal tide (IT) energetics in the vicinity of La Jolla Canyon, a steep shelf submarine canyon off the Southern California coast, with the Stanford Unstructured Nonhydrostatic Terrain-Following Adaptive Navier–Stokes Simulator (SUNTANS) numerical simulator. Numerical simulations show vertical structure and temporal phasing consistent with detailed field observations. ITs induce large (approximately 34 m from peak to peak) isotherm displacements and net onshore IT energy flux up to 200 W m−1. Although the net IT energy flux is onshore, the steep supercritical slope around the canyon results in strong reflection. The model provides the full life span of internal tides around the canyon, including internal tide generation, propagation, and dissipation. ITs propagate into the canyon from the south and are reflected back toward offshore from the canyon’s north side. In the inner part of the canyon, elevated mixing occurs in the middle layer due to an interaction between incident mode-1 ITs and reflected higher-mode ITs. The magnitude of IT flux, generation, and dissipation on the south side of the canyon are higher than those on the north side. An interference pattern in horizontal kinetic energy and available potential energy with a scale of approximately 20–50 km arises due to low-mode wave reflections. Our results provide new insight into IT dynamics associated with a small-scale canyon topography.

Significance Statement

Internal waves play an important role in ocean circulations and ecosystems. In particular, internal waves with frequencies of tides, known as internal tides, strongly enhance energy, heat, and mass transport in coastal oceans. This study presents internal tide dynamics in La Jolla Canyon, California, using a high-resolution numerical model. Model results show energy convergence in the canyon leading to internal tide energy dissipation and mixing. Some parts of internal tide energy reflect back offshore resulting in standing internal waves off California. This study provides new insights into internal tide dynamics and energy budgets in submarine canyons.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Eiji Masunaga, eiji.masunaga.office@vc.ibaraki.ac.jp

1. Introduction

Internal waves are ubiquitous oceanic phenomena that play an important role in ocean energy, heat, and mass transport (e.g., Munk and Wunsch 1998), as well as in nutrient supply for ecosystems (e.g., Woodson 2018). Tidal flow over seafloor topography generates tidal-frequency internal waves known as internal tides (ITs). Major generation sites of intensified ITs are continental slopes/shelves (e.g., Cacchione et al. 2002; Masunaga et al. 2017) and shallow ridges/sills (e.g., Alford et al. 2015; Masunaga et al. 2019). ITs are mainly generated in oceans where the depth is less than roughly 3000 m (Kang and Fringer 2012; Masunaga et al. 2017). Even though they often dissipate or reflect on continental slopes, even a small fraction of energy transmitted onto the shelf can contribute significantly to coastal ocean dynamics (e.g., Lucas et al. 2011; Walter et al. 2012; Masunaga et al. 2016; Hamann et al. 2021).

Submarine canyons are known to be energetic IT sites and play an important role in ocean circulation and mixing as well as coastal dynamics (e.g., Kunze et al. 2002; Nazarian et al. 2021). Previous field observations have revealed the dynamics of ITs and the associated mixing processes in canyons (Gregg et al. 2011; Zhao et al. 2012; Wain et al. 2013; Alford and MacCready 2014; Walter et al. 2012; Waterhouse et al. 2017; Hamann et al. 2021). ITs propagate into shallow and narrow canyons, eventually breaking and dissipating, leading to enhanced mixing (e.g., Hamann et al. 2021). Masunaga et al. (2016) observed highly enhanced turbulent kinetic energy dissipation caused by IT breaking in a narrow and shallow bay. Waterhouse et al. (2017) observed elevated turbulent mixing due to the superposition of semidiurnal ITs and bottom-trapped diurnal ITs in Eel Canyon, California. Numerical simulations also show dynamics associated with ITs in canyons (e.g., Hall and Carter 2011; Kang and Fringer 2012). Hall and Carter (2011) showed strong IT energy convergence along the canyon trench using a high-resolution numerical model. Kang and Fringer (2012) reported details of IT energy budgets using a nonhydrostatic numerical simulator.

The dynamics of internal tides in submarine canyons are interesting because they involve a number of distinct processes, all operating in close proximity. Topographic slope is an important parameter to describe IT dynamics (e.g., Cacchione et al. 2002). IT energy can propagate shoreward over a gentle or “subcritical” slope, inducing energy flux convergence in the shallow waters of the continental shelf (e.g., Hall et al. 2013). Transmitted IT energy toward the coast results in internal wave breaking accompanied by intensified mixing in shallow waters (e.g., Lucas et al. 2011; Walter et al. 2012; Masunaga et al. 2016). On the other hand, over a steep slope (supercritical slope), ITs cannot propagate onshore and so they reflect back offshore. This reflection is rarely complete, with some fractions of IT energy dissipating over slopes even in supercritical conditions (Alberty et al. 2017; Hamann et al. 2021). Because of the shallow depths over continental shelves, even a small net flux can result in a large energy density that can dominate over other coastal processes.

Supercritical slopes scatter IT energy to higher modes (Müller and Liu 2000). It has been found that reflected ITs contain higher-wavenumber mode energy than incoming ITs (Nash et al. 2004; Kelly et al. 2013), leading to enhanced vertical shear and strain in the water column. Highly reflective conditions lead to standing internal waves. Nash et al. (2004) observed that shoreward ITs interact with reflected ITs from a steep slope leading to standing internal waves off Virginia. Martini et al. (2007) reported that mode-1 standing internal waves generated by ITs propagated from two generation sources in Mamala Bay, Hawaii, from shipboard observations and numerical simulations. Zhao et al. (2012) and Hall et al. (2014) described a transition between a standing wave mode and progressive wave mode modulated by background stratification in Monterey Bay.

The Southern California coast contains steep slopes in its near shore regions and is known to be a site generating IT reflections and standing internal waves (e.g., Emery 1956; Buijsman et al. 2012; Ponte and Cornuelle 2013; Waterhouse et al. 2017; Hamann et al. 2021). An early study of Emery (1956) hypothesized that temperature fluctuations observed in the Catalina Basin imply basin-scale standing internal waves. High-resolution numerical simulations conducted by Buijsman et al. (2012) showed near-resonant standing internal waves in the Southern California Bight surrounded by steep supercritical slopes.

ITs and associated processes have been well studied in Monterey Canyon located along the central coast of California (e.g., Petruncio et al. 1998; Hall and Carter 2011; Zhao et al. 2012; Walter et al. 2012; Kang and Fringer 2012; Wain et al. 2013). Numerical studies conducted by Hall and Carter (2011) and Kang and Fringer (2012) showed that IT energy converges into the shallow canyon trench. Walter et al. (2012) reported IT breaking and mixing controlled by the angle of the canyon slope. Monterey Canyon incises the continental slope that is largely connected to deep open oceans rather than coastal oceans. In contrast, the La Jolla Canyon is located near the coast, categorized as a “shelf canyon,” and is impactful for regional dynamics between the inner shelf and coastal oceans (Hamann et al. 2021). In addition, La Jolla Canyon is smaller and consists of steeper slopes than Monterey Canyon.

The La Jolla Canyon is one of the energetic submarine canyons located off the coast of Southern California (e.g., Alberty et al. 2017; Lucas et al. 2017; Hamann et al. 2021). Alberty et al. (2017) conducted field campaigns in the canyon and suggested that middepth mixing is enhanced by a superposition of incident low vertical mode ITs and reflected high vertical mode ITs. Mode 2 was particularly strong, giving large strain and inferred turbulence at middepth. Field observations in a recent study of Hamann et al. (2021) showed that first-mode ITs have a standing wave character, while higher vertical mode ITs reflect offshore. The latter study was able to roughly close the energy budget by balancing the measured onshore energy flux and dissipation of turbulent kinetic energy in the water column within the canyon.

These previous observational studies revealed IT dynamics and associated mixing processes in La Jolla Canyon. However, the full life span of ITs from generation to propagation–dissipation should be investigated by observationally validated numerical simulations because it is difficult to obtain large-scale generation and propagation processes from field observations due to limited coarse resolutions. IT generation, propagation, and dissipation have been well studied by using numerical simulations in Monterey Bay (e.g., Hall and Carter 2011; Kang and Fringer 2012). However, ITs in La Jolla Canyon have not been investigated by numerical simulations resolving offshore generation, onshore propagation, and dissipation. To investigate the details of IT dynamics with the full life span of ITs, we employed a high-resolution numerical model, Stanford Unstructured Nonhydrostatic Terrain-Following Adaptive Navier–Stokes Simulator (SUNTANS), for La Jolla Canyon forced by semidiurnal, M2, barotropic (BT) tides. This study focuses on offshore IT generation, propagation of ITs toward the canyon and reflected ITs by the steep canyon wall, and IT energy dissipation processes in the canyon. The model is validated by comparing it to observations to confirm the quality of the model (e.g., Kang and Fringer 2012; Masunaga et al. 2017). We compared model outputs with mooring observations in the study area to validate internal tides reproduced in the model.

The remainder of this paper is organized as follows. Section 2 describes model configurations. Validations comparing model outputs and observations are shown in section 3. Section 4 presents model outputs and discussion associated with IT dynamics, wave energy budgets, and standing waves. Conclusions are given in section 5.

2. Model configurations

This study employed a numerical simulator, SUNTANS (Fringer et al. 2006), to investigate ITs in the vicinity of the La Jolla Canyon. The model can solve for the nonhydrostatic pressure term; however, the effects of nonhydrostatic pressure are much smaller than those of hydrostatic pressure on canyon scales (Kang and Fringer 2012). Thus, we disabled the nonhydrostatic solver in order to reduce computation time. The model domain covers Southern California with major and minor axis lengths of approximately 1400 and 600 km, respectively (Fig. 1a). The model domain consists of unstructured triangle grids. The horizontal grid resolution is refined near the La Jolla Canyon with a resolution of 100 m and is stretched offshore with a resolution of 50 km near the open boundaries (Figs. 1a,b). The bathymetry is taken from the ETOPO1 Global Relief Model (NOAA) and from high-resolution bathymetry observations around the La Jolla Canyon by Scripps Institution of Oceanography. Far offshore, the bathymetry is simplified to a constant 3000-m depth to reduce computational efforts (Fig. 1c; Masunaga et al. 2017). The bathymetry of the La Jolla Canyon is well resolved in the numerical domain with the resolution of 100 m horizontally (Figs. 1d–f). This study presents numerical results in three subdomains, A1, A2, and A3 (Figs. 1d–f), for visualization purposes. The vertical grid coordinate is a 100-layer z level with greater resolution toward the surface. A vertical sigma-layer coordinate was not employed for our model because the sigma pressure error would cause artificial/error flows on the steep submarine canyon (e.g., Mellor et al. 1994). The minimum and maximum vertical grid spacing at surface and bottom layers are 2.4 and 117 m, respectively.

Fig. 1.
Fig. 1.

(a),(b) The unstructured triangle grid of the model domain, (c) bathymetry in the whole computational domain, and (d)–(f) bathymetry in the three subdomains (A1–A3).

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

The vertical temperature and salinity distributions are initialized with World Ocean Atlas 2009 (WOA09; National Oceanographic Data Center). The initial profiles are horizontally uniform and are taken from the summer season at longitude of 120.5°W and latitude of 33.5°N (Figs. 2a,b). The initial density and squared buoyancy frequency are shown in Figs. 2c and 2d. The buoyancy frequency is expressed as
N=gρ0ρz,
where g is the gravitational acceleration (9.8 m s−2), ρ is the potential water density, z is the vertical coordinate, and ρ0 is the reference density (1024 kg m−3). The maximum density gradient is located at approximately 38-m depth with N2 of 2.4 × 10−4 s−2 (Fig. 2d). Although some background physical processes, e.g., geostrophic flows and eddies, cannot be considered in the horizontally uniform stratification condition, it is known that this condition is appropriate to investigate internal tides in local coastal regions (e.g., Kang and Fringer 2012; Masunaga et al. 2017). The model is forced by BT tides propagating from the southeast (SE) boundary to the northwest (NW) boundary, viz., BT tides propagate parallel to the major axis of the model domain. The M2 tidal forcing is selected for the tidal frequency because semidiurnal, M2, tides/ITs dominate in the study area. The SE and NW boundaries are forced by the BT tidal velocity represented as
uB=AM2cos(kM2xmajωM2t),
where uB is the boundary velocity along the major axis of the model domain (SE–NW), AM2 is the velocity amplitude, kM2 is the horizontal wavenumber, xmaj is the distance along the major axis of the model domain, ωM2 is the M2 frequency (∼1.4 × 10−4 rad s−1), and t is the time. The amplitude of the forcing BT velocity, AM2, is adjusted to produce the surface tidal elevation amplitude of approximately 0.5 m in the La Jolla Canyon and set to 2.6 × 10−3 m s−1 [see the appendix in Masunaga et al. (2017) for more details]. The horizontal wavenumber of the BT tide kM2 is determined by the ratio of the wave frequency (ωM2) and surface wave speed (gH, where H is the depth, 3000 m, in the offshore regions) and is set as approximately 8.2 × 10−7 rad m−1. The boundary velocity forcing is spun up with a function of uB[1 − exp(−t/τ)], where τ is the spinup time scale, 24 h. A sponge boundary layer is employed at the offshore boundaries to avoid internal tide energy reflections at the boundaries (Kang and Fringer 2012). The model boundaries are forced only by barotropic flows and no other forcings, such as mean flows and heat/buoyancy flux.
Fig. 2.
Fig. 2.

Initial conditions for (a) temperature, (b) salinity, (c) density (σθ), and (d) squared buoyancy frequency (N2). The dots show z-level grid points.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

Local dynamics of ITs are known to be influenced by remotely generated ITs far from the study area (e.g., Ponte and Cornuelle 2013; Mazloff et al. 2020). Ponte and Cornuelle (2013) conducted a sensitive test of IT dynamics influenced by the modeled domain size off the Southern California coast. According to their results, the model domain with a size of approximately 500 km would be appropriate to simulate ITs near the coast. The size of our model domain, approximately 1400 km × 600 km (500 km × 300 km with a realistic topography), should be enough to take into account remotely generated ITs (Fig. 1). One of the major IT generation sites in the North Pacific is Cape Mendocino along the Northern California coast (Alford et al. 2019; Mazloff et al. 2020). The magnitude of the southward IT energy flux from the cape reaches up to O(103) W m−1. However, flux from Cape Mendocino, which is approximately 1000 km away from La Jolla Canyon, largely goes offshore of our study site (Alford et al. 2019). Hence, we do not have to consider the strong IT energy flux from Cape Mendocino in our model.

The model time step is 30 s. The vertical eddy viscosity and diffusivities are estimated by the Mellor–Yamada turbulent closure scheme (Mellor and Yamada 1982). A previous study has presented that turbulent kinetic energy dissipation ratio is well parameterized by the Mellor–Yamada turbulent closure scheme, where internal tides break on a shallow slope (Masunaga et al. 2016). We assume that mixing inferred from the turbulent closure scheme is reasonable to discuss internal tide dynamics in our study area. The horizontal viscosity and scalar diffusivity are set to 1.0 and 0.0 m2 s−1, respectively. The bottom drag coefficient is set to a constant value of 0.0025 throughout the model domain. A constant Coriolis frequency of 7.9 × 10−5 rad s−1 (corresponding to a latitude of 33°E) is used. The analysis period of model outputs is 10 M2 tidal cycles (∼124 h) after a spinup of 10 M2 tidal cycles.

3. Model validation

Surface tidal elevations are compared with the TPXO9 dataset provided by Egbert and Erofeeva (2002) to validate BT tides in the model. Tidal amplitude and phase from our model and TPXO9, computed using the T-TIDE harmonic analysis package developed by Pawlowicz et al. (2002), are in good agreement (Fig. 3). The amplitude gradually increases from approximately 0.45 to 0.50 m toward the coast; cotidal phase shows tidal propagations from the SE to the NW direction. The modeled cotidal phase contours for our model are more variable than from TPXO owing to our model’s inclusion of ITs, which are absent in TPXO (see Carter 2010; Ponte and Cornuelle 2013).

Fig. 3.
Fig. 3.

Comparison of M2 tidal elevations between (a) our model and (b) TPXO9 model. Color shows the tidal elevation amplitude, and black contours indicate phase with an interval of 1°.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

In addition to surface tidal elevations, vertical structures from the model are compared with field observation data obtained during an intense field observation effort detailed in Hamann et al. (2021). Here, we use velocity measurements from three mooring locations: WW (a Wirewalker wave–powered profiling mooring), T1 (an ADCP mooring), and MP1 (a McLane profiling mooring; Fig. 1f), and stratification observations from the WW sampling location. Observed data contain numerous other physical phenomena, such as wind-induced currents, tidal flows other than the M2 frequency, and background geophysical flows. To address this issue, composite averaged plots for the M2 cycle are created to compare with model results. Data within seven M2 tidal cycles on 25–29 October 2016 are extracted to create M2 composite averaging for WW observations (Figs. 4a–d). For the T1 and MP1 moorings, 25 M2 tidal cycle data on 11–24 September 2016 are used for M2 composite averaging. Model results within one M2 cycle after a 10 M2 cycle period are used to compare with field observations.

Fig. 4.
Fig. 4.

(a) Temperature and (b) velocity obtained from Wirewalker observation in October 2016, M2 composite averaged (c) temperature and (d) velocity along the major axis from Wirewalker observations, and modeled results of (e) temperature and (f) velocity along the major axis. The black isotherms in (e) indicate isotherms of 14.5° and 11.0°C. The black contour lines in (b) and in (d) and (f) show isotherms with intervals of 1.0° and 0.4°C, respectively.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

The major axis of velocity in the model is within 8° of the observations (Table 1). These differences in the major axis are likely attributed to small-scale topography (less than 100 m) that is not resolved in our model. A striking feature of the WW observations is the presence of second-vertical-mode ITs; these are reproduced in the model results (Figs. 4c–f). Intensified velocity in the intermediate layer, at a depth of ∼40 m, is caused by second-mode ITs reflected from the La Jolla Canyon (Hamann et al. 2021). Modeled velocities at the T1 and MP1 sites also show multi-vertical-mode structures (Fig. 5).

Table 1.

Comparison of velocity between the observations and model at the WW, T1, and MP1 mooring locations.

Table 1.
Fig. 5.
Fig. 5.

Comparison of M2 composite velocity along the major axis between the (a),(b) mooring observations and (c),(d) model for the (left) T1 and (right) MP1 mooring sites. Black horizontal lines in (c) and (d) show the lower and upper limits of the field observations.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

The correlation coefficient, skill score (SS), and root-mean-square error (RMSE) between the modeled and observed velocities are listed in Table 1. Skill score is given by
SS=1(uOuM)2(uOuO¯)2,
where uO and uM are the observed and modeled velocities, respectively, and the overbar denotes the time averaging operator. The correlation coefficients range between 0.79 and 0.82 (p ≪ 0.01). RMSE is O(10−2) m s−1, which is one order smaller than the magnitude of velocity. Performance levels of SS are categorized as SS > 0.65, excellent; 0.65 < SS < 0.5, very good; 0.2 < SS < 0.5, good; and SS < 0.2, poor (Allen et al. 2007). Our model results are categorized as “very good” for the T1 site and “good” for the MP1 and WW locations. According to validation results shown above, modeled ITs agree with observations.

4. Results and discussion

a. Internal tides in La Jolla Canyon

Energetic ITs are generated in the La Jolla Canyon by offshore BT forcing (Figs. 4 and 5). The amplitude of M2 isotherm displacement is plotted in Fig. 6 for two isotherms of 14.5° and 11.0°C (shown as black solid lines in Fig. 4e). The amplitude of the 14.5°C isotherm increases markedly as ITs propagate into the canyon, reaching ∼17 m at the head of the canyon at a depth of 71.7 m (Fig. 6e), causing total isotherm displacements of 34 m, or roughly half of the total water column. Although the amplitude of 11.0°C isotherm displacement is smaller than that of 14.5°C isotherms, it too increases toward the canyon head (Fig. 6b), as well as on the north and south sides of the canyon (indicated by black arrows in Fig. 6b). The phase of the two isotherms is opposite in the inner part of the canyon (black contour lines in Fig. 4e), indicative of a mode 2 signal as observed by Alberty et al. (2017) and Hamann et al. (2021). The large isotherm displacements in the canyon lead to intensified available potential energy in the canyon (Alberty et al. 2017; Hamann et al. 2021), which is discussed below.

Fig. 6.
Fig. 6.

Isotherm displacements of two isotherms, (a) 14.5° and (b) 11.0°C, with a frequency of M2. Gray bathymetry contours are spaced 50 m apart. Black arrows in (b) indicate enhanced isotherm displacement on the south side and north side of the canyon.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

The depth-integrated IT energy flux (BC pressure flux) is represented as
FIW=dηpuBCdz,
where p′ is the pressure perturbation, uBC is the BC velocity vector, η is the free surface height, and d is the depth (η + d is the total water height). The BC velocity is given by
uBC=uuBT,
uBT=1Hdηudz,
where uBT is the BT velocity vector. Time-averaged FIW over 10 tidal cycles (Fig. 7) is enhanced in offshore regions, FIW exceeding 1000 W m−1 (Fig. 7a). Regions in offshore areas of higher flux have lateral scales of several tens of kilometers, which appear to be influenced by rough topographic features off the Southern California coast (Fig. 1d). Meanwhile, the IT energy flux is weaker (less than approximately 200 W m−1) in inshore regions and in the La Jolla Canyon than offshore (Figs. 7b,c). Although the net IT energy flux is relatively weak in the canyon, the wave energy converges toward the inner part of the canyon along the trench, consistent with Hamann et al. (2021). The maximum IT energy flux in the canyon is approximately 200 W m−1, consistent with observed values (Alberty et al. 2017; Hamann et al. 2021) (Fig. 7c). The direction of FIW is onshore and offshore on the south and north sides of the canyon mouth, respectively, which implies ITs propagate into the canyon on the south side and reflect back offshore on the north side. Red arrows in Fig. 7d are the IT energy flux estimated from the field observations (Fig. 4a in Hamann et al. 2021) and are compared with our numerical results (red arrows). This IT flux pattern has also been found in field observations by Hamann et al. (2021) (Fig. 7d). It is worth noting that the IT energy flux is broadly reproduced in the model without considering remotely generated ITs outside of the model domain, implying that strong IT energy flux from Cape Mendocino (Alford et al. 2019) has only a minor impact on the region’s energetics as suggested above. The IT flux near coastal regions of depths shallower than 100 m (black contour line in Fig. 7c) is much smaller than that in the canyon. The magnitude of FIW in these shallow regions is <10 W m−1, indicating that IT energy dissipates or reflects offshore before ITs reach nearshore shallow regions. Some fractions of IT energy radiate from the A1 subdomain toward more offshore regions (the whole model domain is shown in Fig. 1), and finally, IT energy dissipates in offshore regions or radiates from the far offshore boundaries.
Fig. 7.
Fig. 7.

IT energy flux in the three subdomains (a)–(c) A1–A3 and (d) comparison of the IT energy flux from our model (blue) and field observations obtained from Hamann et al. (2021; red). Gray bathymetry contours in (a) and (b)–(d) are spaced with intervals of 250 and 50 m, respectively. The 100-m isobath is plotted in black in (c) and (d).

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

The depth integrated energy conversion rate from BT tides into BC (internal) tides is (Kang and Fringer 2012)
CBT-BC=dηρgWdz,
where the vertical BT velocity, W = −∇H ⋅ [(d + z)uBT], and ρ′ is the density perturbation. Positive values of CBT-BC imply IT generation. If the advection and diffusion terms are negligible in BC kinetic energy budget, the difference between the FIW divergence and CBT-BC is balanced with the dissipation ε as follows (Kang and Fringer 2012; Masunaga et al. 2017; Jithin et al. 2019):
CBT-BC¯HFIW¯=ε¯.
Here, the overbar denotes time averaging and the negative sign of dissipation is an energy sink. The dissipation term is the residual of the BT–BC energy conversion and IT energy flux divergence, namely, ε¯=HFIW¯CBT-BC¯.

The IT energy budget terms CBT-BC¯, FIW¯, and ε¯ in the A2 and A3 subdomains are shown in Fig. 8. The generation of ITs is high on the south side of the canyon trench (Figs. 8a,d). On the other hand, the generation is negative on the north side of the canyon. Negative BT–BC conversion, implying the BT tide gains energy from ITs, can arise from an interaction between the locally generated ITs and remotely generated ITs propagating across the canyon (Zilberman et al. 2009; Kang and Fringer 2012; Masunaga et al. 2017). The divergence of the IT energy flux, FIW¯, shows a similar pattern to the generation (Fig. 8). Hence, the divergence is explained by the conversion between BT and BC tides. The south and north sides of the canyon trench work in the opposite way for IT generation: 1) FIW¯ divergence occurs due to IT generation caused by barotropic tides on the south side and 2) FIW¯ is suppressed by the negative BT–BC conversion on the north side. The magnitude of dissipation ε¯ is high in the inner part of the La Jolla Canyon, especially on the south side for the canyon trench.

Fig. 8.
Fig. 8.

(a),(d) Barotropic–baroclinic conversion rate, CBT-BC¯; (b),(e) divergence of the internal wave energy flux, FIW¯; and (c),(f) dissipation, ε¯=FIW¯CBT-BC¯, in the (top) A2 and (bottom) A3 subdomains. Black vectors indicate IT energy flux. Gray bathymetry contours are spaced with 50 m.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

According to these results, ITs are more energetic on the south side of the canyon than those on the north side. The energetic ITs on the south side of the canyon may be due to incoming FIW from the southwest and enhanced CBT-BC on the south side and are also evident in previous field observations (Hamann et al. 2021). The same north–south (or asymmetric) pattern also appears in other studies, in the Monterey Canyon (Kang and Fringer 2012), in the Eel Canyon (Waterhouse et al. 2017), and bays located in the middle of Japan mainland (Masunaga et al. 2017). One of the possible reasons for the asymmetric patterns of FIW may be Earth’s rotational effects. Although semidiurnal ITs in midlatitudes (where ωM2 > f, where f is the Coriolis frequency) are not trapped by Earth’s rotation as Kelvin waves, they are influenced by Earth’s rotation as Poincare waves (Gill 1982; Zhao 2017). It is known that Poincare waves propagate with the coast to their right near the coastal boundary in the Northern Hemisphere (e.g., Shimizu et al. 2007) similar to Kelvin waves. Additionally, a previous study of Hall and Carter (2011) reported that IT energy flux on the south side of Monterey Canyon is enhanced by Earth’s rotation.

In our simulations, the direction of IT energy flux is approximately northward along the coast (Figs. 7b,c), as expected for Poincare waves. Incident ITs propagate into the canyon along the south side of the canyon mouth and lose a part of their energy due to dissipation in the inner part of the canyon. Then, residual IT energy radiates offshore from the north side of the canyon. Thus, the differences in IT energetics between the south and north sides of the La Jolla Canyon are caused by both rotation and dissipation.

b. Internal tide reflections

The dispersion relation of linear internal waves is given by
sIT=ω2f2N2ω2,
where s is the internal wave slope, known as the wave beam angle, and ω is the wave frequency (M2 tide frequency in this study), and one of the important parameters associated with the dispersion relation is the ratio of sIT and topographic slope, stopo/sIT,
stopo=(dx)2+(dy)2,
where d is the local depth and x and y are the horizontal coordinates. When stopo/sIT is higher than unity, internal wave energy is reflected back offshore and cannot propagate toward the coast; this is referred to as a “super critical slope.” Under conditions of stopo/sIT less than unity, internal wave energy can propagate and be transmitted toward shallow regions (subcritical slope). When stopo is equal to sIT, strong internal wave energy convergence occurs on the sloping bottom (critical slope, e.g., Cacchione et al. 2002).

The distribution of stopo/sIT is computed from the initial conditions and bathymetry used in the model and is shown in Figs. 9a,b. Deep regions are surrounded by steep slopes, sITO(0.1–1), near the coasts with depths approximately 100–300 m, which results in super critical slopes [stopo/sITO(10)] (Figs. 9a,b). The probability density function (PDF) of stopo/sIT shows that topography is dominantly supercritical near shallow regions where the depth is less than 500 m (Fig. 9c, 83.5% is characterized by the super critical condition). On the other hand, the peak in the PDF of stopo/sIT is approximately unity in deep regions (depth > 500 m, Fig. 9d), which results in enhanced IT generation and dissipation, known as the critical slope condition (e.g., Cacchione et al. 2002). Therefore, ITs are generated by a topography–tide interaction under near critical conditions in deep regions and are reflected by shallow supercritical slopes. The reflective condition in shallow regions could be a cause of the offshoreward IT energy flux on the north side of the mouth of the La Jolla Canyon (Fig. 7c). Some fraction of IT energy dissipates at the mouth of the canyon, viz., imperfect IT reflections occur due to internal wave energy dissipation in the canyon.

Fig. 9.
Fig. 9.

(a),(b) Ratio between the internal wave bean angle (stopo) and topographic angle (sIT), stopo/sIT, in the subdomains of A1 and A2, respectively, and (c),(d) probability density functions (PDFs) of stopo/sIT in the A1 subdomain depth shallower than 500 m in (c) and deeper than 500 m in (d). Gray bathymetry contours in (a) and (b) are spaced with intervals of 250 and 50 m, respectively. Black bathymetry contours in (a) show isobaths of 500 m.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

In addition to the ratio between the linear internal wave beam slope and topographic slope (stopo/sIT), reflection/dissipation of interfacial internal waves over shallow slopes can be evaluated by the internal Iribarren number given by (Boegman et al. 2005; Walter et al. 2012; Masunaga et al. 2016)
ξ=stopoa/λ,
where a is the vertical amplitude of the wave interface and λ is the horizontal wavelength. This parameter is developed for evaluating interfacial wave breaking and dissipation via nonlinear processes; thus, it is a completely different parameter from stopo/sIT derived from a linear theory with a continuous stratification. Assuming that s is 0.04 along the canyon trench, a is 17.3 m (Fig. 6a), and λ is 50 km, ξ is calculated as approximately 2.2 ∼ O(1) (details for the wavelength are described in appendix B). Previous laboratory and numerical experiments have reported that approximately 60%–80% of the total incoming wave energy is reflected back toward the offshore when ξ = O(1) (Boegman et al. 2005; Masunaga et al. 2016). Thus, most of incoming wave energy would likely be reflected by the steep La Jolla Canyon with a slope with ξ of O(1). Although internal wave breaking at the subgrid scale is not directly resolved in our model, mixing and energy dissipation due to wave breaking occur through the model’s turbulent closure scheme (e.g., Wang et al. 2011; Masunaga et al. 2016, 2020).

c. Vertical structure of mixing

IT radiation on the north side of the canyon mouth can be explained by stopo/sIT or ξ as described above; however, strong dissipation in the inner part the canyon cannot be simply explained by either stopo/sIT or ξ (Figs. 7c and 8f). To further investigate IT dissipation processes in the canyon, vertical structures at the T1 mooring site are plotted in Fig. 10. ITs intensify vertical shear, S2 = (du/dz)2 + (/dz)2, and stretch vertical density gradient in the middle layer, from z ∼ −50 to −150 m (Figs. 10a,b). The strain ratio γ clearly shows highly strained vertical structures in the middle layer (γ=Ninit2/N2, where Ninit2 is the initial conditions of N2). An example of the high-strained areas is indicated by a black-outlined box in Fig. 10. The high strain ratio and enhanced vertical shear result in a low Richardson number, Ri = N2/S2 (Fig. 10d), which owes to incident mode-1 ITs and reflected higher mode ITs (Hamann et al. 2021). Low Richardson number regions also occur under conditions of high shear with moderate/low strain ratio. The range of S2 is one order higher than N2 (Figs. 10a,b); S2 and N2 range from approximately −7 to −4.5 and from −4.5 to −3.5 in log10 scale, respectively. Therefore, in addition to the high strain ratio suggested by the previous study (Hamann et al. 2021), the interplay between vertical shear and strain may play an important role in water column instability in the study area.

Fig. 10.
Fig. 10.

Time series of (a) squared vertical shear (S2), (b) squared buoyancy frequency (N2), (c) strain ratio (γ), (d) the Richardson number (Ri), and (e) dissipation of TKE (εTKE) at the T1 mooring site. The location of T1 is shown in Fig. 1f. A black-outlined box indicates the high-strained regions.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

Previous observational studies reported that high-strain events lead to Ri less than the critical value (0.25) coinciding with elevated mixing (Alberty et al. 2017; Hamann et al. 2021). On the other hand, our model results do not show Ri less than 0.25 because instability and turbulent overturning in subgrid scales are not resolved in the model. Instead of using the Richardson number, we can evaluate turbulent mixing by using the mixing parameters estimated from the Mellor–Yamada turbulent closure scheme (Mellor and Yamada 1982). The dissipation rate of turbulent kinetic energy εTKE is also calculated by using a conventional formula, εTKE = KρN2/Γ (Osborn 1980). The term is the vertical eddy diffusivity computed from the turbulent closure, and Γ is the mixing efficiency and is assumed as the commonly used value of 0.2. Energy dissipation from the turbulent closure model is lower than that from the IT energy budget analysis from Eq. (8) (see appendix A). The εTKE is elevated to values of O(10−8–10−7) W kg−1 in the middle of the water column (Fig. 10e), which is consistent with field measurements (Alberty et al. 2017; Hamann et al. 2021). High εTKE layers propagate downward which is also consistent with downward high dissipation layers attributed to mode-2 ITs shown by Hamann et al. (2021, see their Fig. 2e) and Alberty et al. (2017, see their Fig. 5).

Time-averaged εTKE along two transects (along trench and cross canyon) and time–depth-averaged εTKE between 100- and 200-m depth are shown in Fig. 11. Highly enhanced mixing in the middle layer extends approximately 5–10 km from the head of the canyon with an amplitude of O(10−8–10−7) W kg−1, which corresponds with a bottom depth of ∼400 m (Fig. 11). Although the mixing intensity is also high in shallow regions along isobaths of ∼100 m far from the canyon (e.g., A2 domain except A3), mixing is not highly enhanced in deeper regions, e.g., isobath of ∼400 m, far from the canyon (Fig. 11c, isobaths of 100 and 400 m are shown as black contours in Figs. 11c,d). The dissipation far away from the canyon is low O(10−10–10−9) W kg−1; thus, elevated mixing in the canyon is approximately 100 times higher than the background value. The mixing intensity is higher on the south side of the canyon than that on the north side (Fig. 11d, highly enhanced mixing area on the south side is indicated by a black arrow). This north–south variation can be explained by the north–south variation of IT energy owing to Earth’s rotation mentioned earlier and is also evident in field measurements by Hamann et al. (2021, see their Figs. 4c and 5c, transect SL6). Hamann et al. (2021) observed intensified dissipation near the region indicated by the black arrow in Fig. 11d.

Fig. 11.
Fig. 11.

Time-averaged εTKE (a) along canyon trench and (b) cross canyon transects and (c),(d) time- and depth-averaged εTKE within depths of 100–200 m. The along and crossing canyon transect are indicated in (c) and (d) as blue solid and blue dashed–dotted lines, respectively. Black contour lines in (c) and (d) are isobaths of 100 and 400 m. A black arrow in (d) indicates the enhanced mixing region on the south side of the canyon.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

d. Internal tide energy budget

To investigate the IT energy budget in the La Jolla Canyon, cross-sectional integrated shoreward BC energy flux, BT–BC energy conversion (CBT-BC¯), divergence of BC energy flux (FIW¯), and dissipation (ε¯) are estimated as a function of the distance from the mouth of the canyon with an along canyon bin of 500 m (Fig. 12). In steady state, the model energy equation gives a three-term balance between conversion, flux divergence, and dissipation [see Eq. (8)].The zero location is set at 5.5 km from the head of the canyon (WW mooring location). Areas with a water depth < 100 m are ignored in the integration. The integration area is shown in Fig. 12d. The shoreward BC energy flux increases toward the mouth of the canyon (Fig. 12a, distance = 0–2 km) and decreases toward the canyon head in the inner canyon (distance = 2–5.5 km). This energy flux variation can be explained by the BT–BC energy conversion and energy dissipation via [Eqs. (7) and (8)]. At the canyon mouth, flux is divergent because BT–BC energy conversion is much higher than dissipation (Fig. 12b). Inshore of this, IT energy flux is convergent because the magnitude of dissipation is higher than IT generation. The IT energy flux off the canyon mouth (distance less than zero) is offshore. This offshore IT energy radiation, which appears in the map of FIW (Fig. 7c), owes both to conversion between 0 and 2 km and reflected energy from the canyon head (Fig. 9).

Fig. 12.
Fig. 12.

Cross-sectional integrated (a) IT energy flux toward the inner canyon (negative is offshore flux) and (b) IT energy budgets of FIW divergence (white), BT–BC conversion (gray), and dissipation (black) between the canyon month and the shallow edge of the canyon. (c) Bathymetry along the canyon trench and (d) map of the cross-sectional integration area.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

Based on these analyses, the canyon can be conceptually separated into two regimes: 1) generation regime in the mouth of the canyon where generation dominates and 2) dissipation regime in the inner part of the canyon where dissipation is higher than IT generation. We define these two regions based on the tendency of the cross-sectional integrated IT energy flux (Fig. 12a), namely, 1) generation regime is the canyon mouth where the IT energy flux increases toward shore (distance between 0 and 2 km) and 2) dissipation regime is the inner part of the canyon where the IT energy flux decreases as the distance increases (distance > 2 km).

The integrated IT energy budget in the two regimes is computed and shown in Fig. 13. IT generation (C) is approximately 3 times higher than the magnitude of dissipation (D) in the mouth of the canyon (generation regime), allowing IT energy radiation both away from shore and toward the inner canyon. The IT energy radiated from the mouth to the inner canyon is mostly dissipated locally, but some fraction of the IT energy radiates toward the coast out of our domain into water depths of <100 m. This shoreward IT energy radiation, which is also observed by Alberty et al. (2017), Hamann et al. (2021), and Lucas and Pinkel (2022), contributes to shallow water mixing around the canyon. IT energy radiation at the mouth of the canyon is not unidirectional, as shown in the IT energy flux map (Fig. 7c). The direction of the IT energy flux at the offshore mouth boundary is shoreward at the south end and offshore at the north end of the canyon mouth. The north offshore radiation (106 kW) is greater than the south onshore radiation (67 kW), resulting in a net offshore flux (39 kW) due to IT reflections further inshore. The radiation along the shallow edge (100 m isobath) in the mouth region is not negligible (25 kW) and the direction of the shallow edge ITs is into the mouth (“Radiation from the edge” in Fig. 13) (the shallow edge is shown as a black contour line in Fig. 12d).

Fig. 13.
Fig. 13.

A schematic cartoon for the IT energy budgets in the La Jolla Canyon. The terms C, D, and R indicate CBT-BC¯, ε¯, and IT energy radiation, respectively.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

e. Standing waves

The vertically integrated horizontal kinetic energy (HKE) and available potential energy (APE) are estimated by the following formulas:
HKE=dη12ρ0(u2+υ2)dz,
APE=dηg2ρ22ρ0N2dz,
where u and υ are the zonal and meridional velocities, respectively, and ρ′ is the density perturbation. APE expressed by Eq. (13) is not an exact value of the APE and it contains the third-order error (Kang and Fringer 2010). We assume that the third-order error is much smaller than APE computed from Eq. (13) and can be neglected. The kinetic energy, HKE, can be separated into the BT and BC parts as follows:
KEBT=dη12ρ0(uBT2+υBT2)dz,
KEBC=dη12ρ0(uBC2+υBC2)dz.
The cross term of kinetic energy, ρ0(uBTuBC + υBCυBC), is zero after vertical integration. The estimated HKE increases in the trench of the canyon and is dominated by the BC component (Figs. 14a–c). The magnitude of KEBC is roughly one order higher than that of KEBT except in shallow areas where depths are less than 100 m. APE is also enhanced in the canyon and appears to be roughly O(10) higher than HKE (Figs. 14a,d). These results indicate that ITs propagating into the canyon strongly enhance both HKE and APE. For progressive waves, the ratio of HKE and APE, r = HKE/APE, is a function of the inertial frequency f and is theoretically given by (Gill 1982)
r=HKEAPE=ω2+f2ω2f2=1.92.
The ratio of HKE and APE is normalized by the theoretical value of 1.92, represented as
rn=HKE1.92APE.
This normalized indicator rn is unity for progressive waves and varies for standing waves. For standing waves, the local maxima and minima of rn appear at antinodes and nodes, respectively, in wavelength scales (Nash et al. 2004). A schematic cartoon of rn for standing waves is shown in Fig. 15e.
Fig. 14.
Fig. 14.

(a) Horizontal kinetic energy (HKE), (b) barotropic kinetic energy (KEBT), (c) baroclinic kinetic energy (KEBC), and (d) available potential energy (APE) in the subdomain of A3.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

Fig. 15.
Fig. 15.

(a)–(c) HKE/APE normalized by (ω2 + f2)/(ω2f2) = 1.92, rn, in the subdomains of A1–A3, (d) probability density function (PDF) of rn, and (e) schematic cartoon of standing waves. The black dotted line and gray shaded area in (d) represent the median and range of 95% (median ± 42.5%) of rn. Black solid and chain-dotted lines along the x–z plane in (e) depict isotherm displacements at two opposite phases.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

APE is much higher than HKE at the head of the La Jolla Canyon with a minimum rn value of 0.014 (Fig. 15c). APE is approximately 37 times higher than HKE under this condition. This small value of rn is induced by the superposition of incoming and reflected internal waves generated at the wall of the steep La Jolla Canyon. A high value of rn (∼5) appears in offshore areas approximately 20 km away from the canyon (Fig. 15b, x ∼ 1890 km and y ∼ 565–590 km). Although rn becomes zero (infinity) at nodes (antinodes) in the standing wave theory, the lowest (highest) rn does not reach to zero (infinity) because imperfect IT reflections occur due to energy dissipation. High and low rn areas distribute as a coherent pattern with horizontal scales of roughly 20–50 km in the large A1 subdomain (Fig. 15a). Extremely high or low rn appears near the coast (Fig. 13c); however, 95% of the total rn is in a range between 0.3 and 3.0 with a median value of 1.1 in the A1 subdomain (Fig. 11d). It is worth noting that the distribution of rn follows a lognormal-like distribution with a peak of approximately unity. Here, the unity of rn indicates HKE/APE = (ω2 + f2)/(ω2f2) = 1.92.

The wave speed of mode-1 ITs goes up to approximately 1.8 m s−1 with an average of 1.6 m s−1 in the A1 subdomain (see appendix B). This implies that the averaged wavelength scale is approximately 52 km. If standing waves occur with the wavelength of 52 km, low and high rn peaks distribute with an interval of approximately 26 km (two high/low rn patterns appear within one wavelength scale, see Fig. 15e). Although distances between high/low rn peaks in our model are not clear due to complicated scattering ITs in a 3D field, the estimated averaged rn scale is roughly consistent with the coherent pattern of the modeled rn shown in Figs. 15a–c. Therefore, the coherent pattern of rn is generated by standing waves owing to reflected ITs off the La Jolla Canyon and steep coastal topography. The IT reflections and associated standing waves in this area are consistent with numerical simulations for the Southern California Bight conducted by Buijsman et al. (2012).

5. Conclusions

This study investigated internal tide (IT) dynamics in and off the La Jolla Canyon, California, using a high-resolution oceanic numerical simulator, SUNTANS. Model results showed good agreement with field observations at three mooring locations. The steep slope bathymetry generates reflections of ITs leading to the second-mode vertical internal wave structure. The La Jolla Canyon is separated into two regimes based on IT energy budgets: 1) at the mouth of the canyon, IT generation and radiation is much higher than dissipation leading to IT energy radiation (referred to as a generation regime) and 2) in the inner part of the canyon, energy dissipation is higher than IT generation and wave energy radiation (referred to as a dissipation regime). The dissipation regime in the inner part of the canyon is attributed to mixing caused by highly enhanced vertical strain due to the superposition of incident mode-1 ITs and reflected higher mode ITs. In addition, variations in the cross-canyon direction are caused by Earth’s rotation. The magnitudes of IT energy flux and energy dissipation are higher on the south side of the canyon than those on the north side, owing to incoming IT energy flux from south of the canyon mouth. Reflected IT energy from coastal regions generates standing internal waves off the Southern California coast. Standing internal waves modulate the ratio of the kinetic energy (HKE) and available potential energy (APE). A coherent pattern of high and low HKE/APE appears within a scale of several tens of kilometers, which is likely determined by the horizontal wavelength scale.

Numerous previous literatures suggested that submarine canyons are important regions for IT dynamics associated with turbulent mixing (e.g., Kunze et al. 2002; Nazarian et al. 2021). Our study region, La Jolla Canyon, is categorized as a shallow shelf canyon located near the coast. ITs increase their nonlinearity on shallow slopes (e.g., Masunaga et al. 2016; Lucas and Pinkel 2022). Thus, it is challenging to reproduce ITs in coastal shallow regions. The model developed in this study successfully reproduces the vertical structures and IT energy flux shown by field observations, which have not been provided by previous numerical works even in other submarine canyons.

Our numerical model supports and extends the sparse IT energy flux measurements and rough energy budgets observed by Hamann et al. (2021). Internal tide energy budgets in another submarine canyon, Monterey Canyon, have been reported in previous numerical works within scales from several tens of kilometers to 100 km (e.g., Hall and Carter 2011; Kang and Fringer 2012); however, these previous studies did not focus on small-scale dynamics. The scale of the La Jolla Canyon is much smaller than Monterey Canyon, and the present study provides small-scale IT dynamics with scales less than several kilometers, e.g., the generation in the canyon mouth and the dissipation regime in the inner canyon, and the north–south variations of IT energy likely owed to Earth’s rotation. Internal-wave standing modes are an important issue in describing IT dynamics (e.g., Nash et al. 2004; Alford and Zhao 2007; Martini et al. 2007). It has been known that steep topography along the Southern California coast causes IT reflections leading to standing ITs (e.g., Zhao et al. 2012; Waterhouse et al. 2017; Hamann et al. 2021). However, a spatial pattern of standing ITs has not been reported in previous studies due to limitations in observations. Our model results show a coherent pattern of standing IT structures associated with the horizontal wavenumber, which would be a new insight into understanding IT dynamics near the coast.

ITs significantly contribute to oceanic ecosystems via IT-induced transport of sediments, nutrients, larvae, plankton, etc., in coastal regions (e.g., Bourgault et al. 2014; Leichter et al. 1996; Pineda 1994). A recent observational study showed cross-shore IT-induced nutrient transport contributing to the maintenance of kelp forests in the vicinity of the La Jolla Canyon (Leichter et al. 2023). Indeed, a kelp forest is located on the south side of the canyon where intensified IT flux appears in our model, and there are no kelp forests located on the north side of the canyon (Parnell et al. 2005). Although our numerical model does not directly provide information for oceanic ecosystems, the small-scale IT variations around La Jolla Canyon found in this study can contribute to our understanding of oceanic ecosystems off Southern California coast.

Acknowledgments.

This study was supported by the U.S. Office of Naval research (Grant N00014-22-1-2044) and the Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (Grants 18H03798, 22K18018, and 22H05201). E. M. acknowledge the support from Alpha Hydraulic Engineering Consultants Co., Ltd. A. J. L. was funded by ONR (Grant N00014-22-1-2730).

Data availability statement.

The SUNTANS model used in this study is distributed by its developers and is available at https://github.com/ofringer/suntans. The modeled and observed data presented here are available upon request.

APPENDIX A

Energy Dissipation from the Turbulent Closure Scheme

According to numerical simulations done by previous studies, IT energy dissipation is reasonably provided by numerical diffusion without the turbulent closure scheme (Kang and Fringer 2012; Masunaga et al. 2017). In addition to vertical diffusion/dispersion, horizontal diffusion added by viscosity would contribute to IT energy dissipation (viscosity is set at 1.0 m2 s−1 throughout the model domain to maintain numerical stability). Time-averaged and depth integrated energy dissipations estimated from the IT energy budget analysis via Eq. (8) and the Mellor–Yamada turbulent closure scheme are compared in Fig. A1. To show two dissipations with the same unit (W m−2), εTKE (W kg−1) is multiplied by water density before vertical integration, viz., εTKEρ. Energy dissipation from the IT energy budget, ε, is higher than the turbulent closure model, εTKEρ. The time-averaged and depth-area integrated εTKEρ within the dissipation regime is −14 kW, which is 25% of the total energy dissipation explained by the IT energy budget analysis (see Figs. 12 and 13). Therefore, the total IT energy dissipation is dominated by numerical diffusion or horizontal diffusion.

Fig. A1.
Fig. A1.

Time-averaged and depth integrated energy dissipation (a) from the internal tide energy budgets via Eq. (8) and (b) from the Mellor–Yamada turbulent closure model.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

APPENDIX B

Wavelength of Internal Tides

The wavelength is computed from the wave period (TM2 = 12.42 h) and propagation wave speed I, λ = c/TM2. The propagation speed of waves is computed by solving the following formula:
2z2w(z)+N2(z)c2w(z)=0,
where w is the vertical velocity magnitude or vertical isopycnal displacement with boundary conditions of w = 0 at the surface and bottom. The wavelength is computed from cTM2, viz., the wavelength and propagation speed are proportional to each other. The vertical mode number is assumed to be one since the mode-1 IT flux dominates in the study area (Hamann et al. 2021). A map and PDF of the wave propagation speed/wavelength are shown in Fig. B1.
Fig. B1.
Fig. B1.

(a) Propagation speed of mode-1 internal waves and wavelength and (b) probability density function (PDF) of propagation speed of mode-1 internal waves and wavelength. A vertical dashed–dotted line in (b) shows the average value of the wave propagation speed/wavelength.

Citation: Journal of Physical Oceanography 53, 11; 10.1175/JPO-D-23-0040.1

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Hall, R. A., J. M. Huthnance, and R. G. Williams, 2013: Internal wave reflection on shelf slopes with depth-varying stratification. J. Phys. Oceanogr., 43, 248258, https://doi.org/10.1175/JPO-D-11-0192.1.

    • Search Google Scholar
    • Export Citation
  • Hall, R. A., M. H. Alford, G. S. Carter, M. C. Gregg, R.-C. Lien, D. J. Wain, and Z. Zhao, 2014: Transition from partly standing to progressive internal tides in Monterey Submarine Canyon. Deep-Sea Res. I, 104, 164173, https://doi.org/10.1016/j.dsr2.2013.05.039.

    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Leichter, J. J., L. B. Ladah, P. Parnell, M. D. Stokes, M. Costa, J. Fumo, and P. K. Dayton, 2023: Persistence of southern California giant kelp beds and alongshore variation in nutrient exposure driven by seasonal upwelling and internal waves. Front. Mar. Sci., 10, 1007789, https://doi.org/10.3389/fmars.2023.1007789.

    • Search Google Scholar
    • Export Citation
  • Lucas, A. J., and R. Pinkel, 2022: Observations of coherent transverse wakes in shoaling nonlinear internal waves. J. Phys. Oceanogr., 52, 12771293, https://doi.org/10.1175/JPO-D-21-0059.1.

    • Search Google Scholar
    • Export Citation
  • Lucas, A. J., P. J. S. Franks, and C. L. Dupont, 2011: Horizontal internal-tide fluxes support elevated phytoplankton productivity over the inner continental shelf. Limnol. Oceanogr. Fluids Environ., 1, 5674, https://doi.org/10.1215/21573698-1258185.

    • Search Google Scholar
    • Export Citation
  • Lucas, A. J., R. Pinkel, and M. Alford, 2017: Ocean wave energy for long endurance, broad bandwidth ocean monitoring. Oceanography, 30, 126127, https://doi.org/10.5670/oceanog.2017.232.

    • Search Google Scholar
    • Export Citation