## Abstract

The Regional Oceanic Modeling System (ROMS) is applied in a nested configuration with realistic forcing to the Southern California Bight (SCB) to analyze the variability in semidiurnal internal wave generation and propagation. The SCB has a complex topography with supercritical slopes that generate linear internal waves at the forcing frequency. The model predicts the observed barotropic and baroclinic tides reasonably well, although the observed baroclinic tides feature slightly larger amplitudes. The strongest semidiurnal barotropic to baroclinic energy conversion occurs on a steep sill slope of the 1900-m-deep Santa Cruz Basin. This causes a forced, near-resonant, semidiurnal Poincaré wave that rotates clockwise in the basin and is of the first mode along the radial, azimuthal, and vertical directions. The associated tidal-mean, depth-integrated energy fluxes and isotherm oscillation amplitudes in the basin reach maximum values of about 5 kW m^{−1} and 100 m and are strongly modulated by the spring–neap cycle. Most energy is locally dissipated, and only 10% escapes the basin. The baroclinic energy in the remaining basins is orders of magnitudes smaller. High-resolution coastal models are important in locating overlooked mixing hotspots such as the Santa Cruz Basin. These mixing hotspots may be important for ocean mixing and the overturning circulation.

## 1. Introduction

The Southern California Bight (SCB) on the southwestern U.S. West Coast features a complex topography of islands, ridges, sills, deep basins, headlands, bays, and shelves (Fig. 1). In the SCB, the southward-flowing California Current interacts with the northward-flowing Southern California Countercurrent, local wind-driven currents, and the topography. These subinertial dynamics have been extensively measured and discussed by Hickey et al. (2003, and references therein) and modeled by Marchesiello et al. (2003) and Dong et al. (2009).

In contrast, the regional semidiurnal baroclinic wave dynamics are still poorly understood. Yet, the complex topography may be conducive to the generation of semidiurnal internal tides. So far, internal waves have mainly been studied on the mainland shelves of the SCB. There are few studies that have looked at semidiurnal internal wave generation, propagation, and dissipation in the interior of the SCB. In this paper, the Regional Oceanic Modeling System (ROMS; Shchepetkin and McWilliams 2005) is applied to fill in this hiatus. The paper builds on work by Hill-Lindsay (2009) that compared an older ROMS configuration of the SCB with measurements of barotropic and baroclinic tidal currents and temperature fields. The present ROMS configuration comprises three nested grids and is forced with realistic boundary conditions such as wind, tides, solar heating, subtidal currents, and stratification.

Of the few interior Bight studies, Emery (1956) observed large isotherm oscillations of up to 200 m on the west side of Santa Catalina Basin (SaCaB), some of them with a semidiurnal period. He attributed these oscillations to a deep standing internal wave. In a multiple-ship surveys in the SCB, Summers and Emery (1963) observed onshore propagating semidiurnal internal waves with speeds of about 3.5 m s^{−1}, wave lengths of about 150 km, vertical amplitudes of 30 m, and crest lengths on the scale of the SCB. They speculated that these waves may be generated at the continental shelf.

The semidiurnal internal waves on the mainland (and, to a lesser extent, island) shelves, with widths up to 10 km, are well documented. They can be partially progressive first-mode internal waves that propagate upslope (Lerczak et al. 2003; Noble et al. 2009) or more like standing waves (Winant and Bratkovich 1981). The strength of these internal waves varies with the season: stronger (weaker) internal waves occur during the more stratified summer (well-mixed winter) months (Cairns and Nelson 1970; Winant and Bratkovich 1981; Gelpi and Norris 2005). In some cases the semidiurnal internal waves on the shelf portray a distinct spring–neap variability, with stronger waves during spring tides (Cairns 1967; Pineda 1994), whereas in other cases this is less clear (Lerczak et al. 2003; Noble et al. 2009). The shoreward shallowing of the shelf causes the internal waves to become nonlinear, form into bores, and/or break into trains of high-frequency solitary waves (Emery and Gunnerson 1973; Pineda 1994; Noble and Xu 2003).

Mixing studies provide hints that the internal waves dissipate near the boundaries of the deep basins. Gregg and Kunze (1991) found strong semidiurnal currents of 5–7 cm s^{−1} and strain and shear spectra above the Garrett and Munk 1976 model (GM76; Garrett and Munk 1975; Cairns and Williams 1976) near the boundaries in Santa Monica Basin (SMB). They speculated that the boundary mixing is due to the semidiurnal barotropic tide interacting with the bathymetry or due to the critically reflected internal waves. In tracer dispersion studies in Santa Monica Basin by Ledwell and Hickey (1995) and in Santa Cruz Basin (SaCrB) by Ledwell and Bratkovich (1995), interior mixing was found to be smaller than boundary mixing. Similarly, they mentioned that the enhanced mixing is possibly due to the critical reflection of internal waves.

Questions that arise from the existing literature are as follows: 1) What are the semidiurnal internal wave generation sites (e.g., mainland shelf breaks, sills and ridges, the continental shelf break, or beyond the continental shelf break)? 2) What are the internal wave pathways? 3) What is the fate of the internal waves? 4) What affects their temporal variability? The aim of this paper is to address these questions by applying the three-dimensional ROMS model to the SCB in a realistic high-resolution setup.

In the next section, the ROMS model, observations, and analysis techniques are presented. The model performance is assessed for the barotropic and baroclinic tides in the subsequent section. The fourth section analyzes the internal wave parameter regime, the spatial patterns in the semidiurnal baroclinic energy fluxes and energy, and the most important terms in the tidally averaged baroclinic energy balance equation. In the discussion section, the first-mode basin wave in SaCrB is further analyzed. The last section presents the conclusions. The abbreviations of many topographic names used in the text are defined in the caption of Fig. 1.

## 2. Methodology

### a. ROMS

The ROMS is a general circulation model ideal for coastal modeling (Shchepetkin and McWilliams 2003, 2005). It features *K*-profile parameterization (KPP; Large et al. 1994; Durski et al. 2004), a nonlocal turbulent closure model that diagnoses vertical momentum dissipation and tracer mixing in the surface and bottom boundary layers and in the interior of the fluid. The present ROMS configuration consists of triply nested model domains with an offline, one-way nesting technique that downscales from 5-km horizontal resolution for the U.S. West Coast (L0 with 514 × 402 grid cells; Fig. 2), to 1-km resolution for the SCB (L1 with 402 × 514 grid cells), and then to 250-m horizontal resolution for the interior of the SCB with the Channel Islands and the deep basins (L2 with 562 × 1122 grid cells). Each domain has 40 topography-following levels, vertically stretched such that gridcell refinement occurs most strongly near the surface and the bottom. All runs are performed on an in-house cluster with 256 cores. It takes about 1 day wall-clock time to compute 2 yr on L0, 3 months on L1, and 1 month on L2. The model topography is based on the 30-arc second resolution global topography/bathymetry grid (SRTM30; Becker et al. 2009) in general and on the 3-s National Oceanic and Atmospheric Administration/National Geophysical Data Center (NOAA/NGDC) coastal relief dataset (http://www.ngdc.noaa.gov/mgg/coastal/crm.html) for the nearshore regions depending on data availability. The minimum water depths in the L0, L1, and L2 domains are 50, 3, and 3 m, respectively.

In all three domains, modified Flather-type radiation boundary conditions for the barotropic fields and Orlanski-type radiation conditions for the baroclinic fields are used. Moreover, the boundaries feature weak sponge layers of 50 grid cells wide with a linearly decreasing lateral harmonic viscosity from 2 m^{2} s^{−1} at the boundaries to 0 m^{2} s^{−1} in the interior. In this way, differences in prognostic variables associated with two different model resolutions are absorbed. The Orlanski-type condition is further combined with a weak nudging toward the parent solution at a time scale of 30 day^{−1} for outgoing baroclinic momenta, 120 day^{−1} for outgoing tracers, and 1 day^{−1} for incoming baroclinic momenta and tracers in order to constrain the interior solution to the parent one. This procedure effectively suppresses reflection of outgoing waves while bringing the parent information into the child domain. In addition, it significantly reduces boundary artifacts (Mason et al. 2010).

The outermost L0 domain is initialized with spatially interpolated climatological Simple Ocean Data Assimilation (SODA) data [the assimilated global oceanic dataset based on the Parallel Ocean Program (POP) model, version 2.0.4; Carton and Giese 2008] and is spun up for 15 yr to reach an equilibrium state. Then L0 is integrated without tides from 1996 to 2007. The barotropic and baroclinic time steps are 11.5 and 600 s. For all domains, the barotropic time steps are chosen to satisfy the barotropic Courant–Friedrichs–Lewy (CFL) condition, whereas the baroclinic time steps are determined experimentally to prevent numerical instability. L0 is forced with the monthly averaged SODA data as a lateral boundary condition; the monthly averaged Quick Scatterometer (QuikSCAT)–European Centre for Medium-Range Weather Forecasts (ECMWF) blended wind dataset (http://cersat.ifremer.fr/data/discovery/by_product_type/gridded_products/mwf_blended) as a surface momentum stress; the monthly average Advanced Very High Resolution Radiometer (AVHRR) pathfinder satellite sea surface temperature dataset (http://www.nodc.noaa.gov/SatelliteData/pathfinder4km/); and the COADS climatological dataset (http://www.ncdc.noaa.gov/oa/climate/coads/) for the other surface fluxes. On L0, the monthly climatology of runoff from major rivers (Dai and Trenberth 2002) is taken into account. These forcings are linearly interpolated on the baroclinic time steps. The L0 output fields are time averaged each 24 h and saved.

The initial condition for L1 is a spatially interpolated L0 solution on 1 January 2006. The L1 model is integrated for the entire year of 2006. At its boundaries, it is forced with a daily averaged solution of L0 that is linearly interpolated on the baroclinic L1 time step of 100 s. The barotropic time step is 2.6 s. Moreover, L1 is forced at its ocean boundaries with tidal water levels and currents. These are computed from amplitudes and phases for 10 tidal constituents (*M*_{2}, *S*_{2}, *N*_{2}, *K*_{2}, *K*_{1}, *O*_{1}, *P*_{1}, *Q*_{1}, Mf, and Mm), extracted from the inverse solution of the Topex/POSEIDON altimetric data (TPXO7.1; Egbert et al. 1994). Every L1 baroclinic time step, the tidal water levels and velocities are superposed on the linearly interpolated L0 solutions. All the atmospheric surface boundary conditions for L1 and L2 are provided every hour by the doubly nested Weather Research and Forecasting model (WRF; Michalakes et al. 1998) on 18- and 6-km horizontal grid spacings. The 6-km solution is used to force the ROMS L1 and L2 models with a one-way coupling approach. The L1 solutions are time averaged and stored every two hours. The two hours are sufficient to resolve the semidiurnal tides on L2, while not exceeding the file storage capacity.

The L2 domain is initialized with a spatially interpolated L1 solution on 1 July 2006 and is integrated for 5 months until 1 December 2006. The L1 and L2 model results shown in this paper are for the period from 1 August to 1 December 2006. The L2 boundary forcing comprises 2-hourly averaged L1 solutions that are linearly interpolated on the baroclinic L2 time step of 20–30 s. The barotropic time step is 0.75 s.

### b. Data

The datasets used to validate the model results are listed in Table 1, and their locations are in Fig. 1. The datasets comprise the ADCP velocity and temperature measurements taken at the Palos Verdes Shelf (PV01), Huntington Beach on the San Pedro Shelf (HB01 and HB06; http://www.sccoos.org; Noble et al. 2009), and in Santa Monica Bay [Santa Monica Bay Observatory Oceanographic Mooring (SMBO06); http://www.ioe.ucla.edu/mucla]. The HB01 and HB06 datasets comprise a number of moorings along a cross-shore transect on the shelf. The PV01 dataset features moorings placed along the shore on the Palos Verdes Shelf. Bottom mounted thermistors around Catalina Island (CAT99 and CAT00; Gelpi and Norris 2005) are also used. Some datasets coincide with the time of the model runs and some are in a different year. However, the latter data can be used in a statistical comparison by using time-mean values, standard deviations, covariances, and spectra.

### c. Analysis techniques

#### 1) Model assessment

When comparing model results with observations, the time series are regridded on the time axis of the model output, which has a time step of 2 h. The time series may be bandpass filtered using a Butterworth filter with cutoff frequencies of 3, 1.5, and 0.6667 cycles per day (cpd), separating the high-passed, semidiurnal, diurnal, and subtidal frequency bands.

#### 2) Energy balance

The baroclinic energy balance is used to evaluate baroclinic energy generation, pathways, and dissipation. Averaged over a semidiurnal tidal cycle and integrated over depth (Carter et al. 2008; Kurapov et al. 2010),

where **∇** · **F** is the divergence of the depth-integrated baroclinic energy flux, *D* is dissipation, *C* is barotropic to baroclinic energy conversion, and 〈〉 is tidal averaging. Carter et al. (2008) computed the terms of the full baroclinic energy equation for the Hawaiian Islands and found that, when averaged over several tidal cycles, the tendency and nonlinear advection terms are small compared to the terms in (1). The University of California, Los Angeles (UCLA) ROMS relies on an upstream-biased advection scheme to provide implicit diffusivity and viscosity for solution smoothness. This makes it much more difficult to compute the dissipation term. Hence, 〈*D*〉 is a residual term computed as 〈*D*〉 = 〈*C*〉 − 〈**∇** · **F**〉. The depth-integrated conversion *C* is computed as

where *p*′(*z* = −*h*) is the baroclinic perturbation pressure at the bottom; **U** is the barotropic velocity vector; the depth mean of the total velocity **u** = (*u*, *υ*), with *u* along the *x* axis and *υ* along the *y* axis; *h* is the positive water depth; and **U** · **∇**(−*h*) is the vertical barotropic velocity *W*. The depth-integrated fluxes can be computed as

where the baroclinic perturbation velocity is

Instead of computing instantaneous fluxes and conversion and time averaging, mean fluxes and conversion are computed based on harmonic constants for velocity and density (Cummins and Oey 1997; Zilberman et al. 2009; Kurapov et al. 2010). In this way, a cumbersome integration over multiple tidal cycles is not necessary. For a moving window of four *M*_{2} tidal cycles, the velocity **u**′, **U**, and potential density *ρ* time series are high-pass filtered to retain the semidiurnal band. The harmonic constants are extracted by harmonic fit with the *M*_{2} frequency *ω*_{2}; for example, for perturbation velocity this is

where *α _{u}*

_{′}and

*β*

_{u}_{′}are the harmonic constants. The amplitude and phase are and

*φ*

_{u}_{′}= arctan(

*β*

_{u}_{′}/

*α*

_{u}_{′}). The harmonic constants represent a mix of all four semidiurnal tidal constituents used in the model forcing, because they cannot be effectively separated over such a short time period. As a consequence, the amplitudes reflect variability due to spring–neap cycles, as well as subtidal changes in stratification. The harmonic constants of perturbation density

*ρ*′ are integrated over depth and multiplied with the gravitational acceleration

*g*to obtain

*α*

_{p}_{′}and

*β*

_{p}_{′}. In a final step, their depth-mean values are removed (Kurapov et al. 2010). The tidally averaged and depth-integrated flux along

*x*is then

Similar expressions can be derived for 〈*F _{y}*〉 and 〈

*C*〉. Because ROMS has sigma levels that follow the water level,

*ρ*′ at sigma coordinates is only affected by the baroclinic field and not by surface heaving (Kelly et al. 2010).

For the moving window, the time-mean density 〈*ρ*(*z*)〉 and buoyancy frequency, 〈*N*^{2}(*z*)〉 = −(*g*/*ρ*_{0})[∂〈*ρ*(*z*)〉/∂*z*], are computed, where *g* is the gravitational acceleration and *ρ*_{0} is a reference density. Vertical eigenfunctions and eigenvalues are determined by solving the hydrostatic Stürm–Liouville equation with 〈*N*^{2}(*z*)〉, and these eigenfunctions are linearly least squares fitted to the harmonic constants to determine the modal amplitudes and phases for **u**′ and *p*′ (Zilberman et al. 2009; Buijsman et al. 2010). In this way, fluxes can be computed per baroclinic mode. The text refers to “total” fluxes, divergence, dissipation, and conversion if these terms are computed with baroclinic velocities and pressures that are not separated into modes.

## 3. Model validation

### a. Barotropic tides

The tidal barotropic Kelvin waves travel northward along the U.S. West Coast. Figure 3a shows the L1 semidiurnal water level phase for the spring tide at 2213 UTC 10 September 2006. This is the average time of the period of four *M*_{2} tidal cycles, for which the tidal amplitudes and phases are computed. The strongest baroclinic energy fluxes occur during spring tides, and throughout the paper the model data for 10 September 2006 are used to evaluate the fluxes and energy balance. The semidiurnal barotropic tidal wave enters the 1900-m-deep SaCrB across the 1100-m-deep south sill and the 700-m-deep east sill and leaves the basin across the 200-m-deep west sill. Hence, the main axes of the tidal ellipses are perpendicular to the east slope of the west sill (Fig. 3c) and reach maximum values of *U*_{a} = 0.5 m s^{−1} on top of the west sill just south of Santa Rosa Island (Fig. 3b). These along-slope velocities force strong vertical isotherm oscillations in SaCrB. The semidiurnal barotropic velocities on the shallow mainland shelves of the SMB and San Pedro Basin (SPB) are generally larger than in the adjacent basins but still smaller than around the islands in the central SCB.

#### 1) Water levels

Observed water levels *η* at the stations at La Jolla, Los Angeles, Santa Monica, and Oil Platform Harvest are compared with modeled water levels extracted at the same locations for the L1 results. The results are in Table 2 and Fig. 4a. The agreement between model and data in the tidal bands is quite good (correlation coefficient *r*^{2} > 98%). The subtidal band has much smaller *r*^{2} because of eddy variability and coastal trapped waves. The results are similar for all stations. The good agreement in the semidiurnal and diurnal bands is also evident in the water level spectra for Los Angeles in Fig. 4a. The most dominant frequency is *M*_{2}, followed by *K*_{1}, *S*_{2}, *O*_{1}, *N*_{2}, and *Q*_{1}. The data have more energy in the subtidal and supratidal frequencies than the model.

#### 2) Currents

The spectra of the alongshore barotropic currents (*V _{r}*) for both model and data on the Palos Verdes Shelf at mooring lca5 of dataset PV01 (

*h*= 65 m) are in Fig. 4b and at Huntington Beach at mooring h6–8 of dataset HB01 (

*h*= 35 m) in Fig. 4c. The barotropic velocities are rotated such that the alongshore velocity

*V*has maximum variance and the cross-shore velocity

_{r}*U*minimum variance. Similar to

_{r}*η*, the data have slightly more variance than the model. The agreement between the model and the data is best at PV. The tidal frequencies at PV contain more energy than at HB. Mooring lca5 is on a 5-km wide shelf, whereas mooring h6–8 is on a 10-km wide shelf. The flow contraction due to the PV headland increases the barotropic velocities and makes the ellipses more eccentric than at HB. The semidiurnal

**U**of HB06 also have larger amplitudes than the model in Fig. 5 (top).

### b. Baroclinic tides

#### 1) Huntington Beach

The comparison between model and HB06 data for semidiurnal temperature and baroclinic velocities that are rotated to the cross-shore and alongshore directions is in Fig. 5. The model and data standard deviations of and resemble a first baroclinic mode, with *σ _{u}*

_{′,r}of the data a little larger than

*σ*

_{u}_{′,r}of the model. The

*σ*features its maximum at middepth in deep water and near the bottom in shallow water. This is a characteristic of shoaling first-mode waves as they propagate toward the coast: they become nonlinear (Noble et al. 2009). The moorings are bottom anchored and measured temperature may be affected by tidal heaving. This is not corrected for because isotherm oscillations due to internal waves are much larger. Similar to

_{T}*σ*

_{u}_{′,r},

*σ*of the data is also larger. In addition, its maximum is higher in the water column. These model-data differences can be attributed to a difference in 〈

_{T}*T*〉. The 〈

*T*〉 of the data features a shallower surface mixed layer and is also cooler with depth. The weaker stratification in the model likely causes the smaller

*σ*in

**u**′ and

*T*.

#### 2) Santa Monica Bay

SMBO comprises an anchored buoy in ~400 m depth (Fig. 1). It measures velocities and temperature only in the top ~100 m. The temperature is measured relative to the surface, and the impact of tidal heaving is small. In both the data and the model, the thermocline deepens as the season progresses from summer to winter (results not shown). Similar to Huntington Beach, the surface mixed layer in the model is deeper than in the data, and the mean temperature over the top 100 m is higher in the model (〈*T _{d}*〉 = 13.54°C and 〈

*T*〉 = 14.87°C). The depth-mean standard deviation in the semidiurnal band has a maximum in the thermocline and is only slightly larger in the data (

_{m}*σ*

_{T}_{,d}= 0.13°C and

*σ*

_{T}_{,m}= 0.12°C). The depth-mean standard deviations of the total semidiurnal

*u*and

*υ*velocities are larger in the data than the model (

*σ*

_{u}_{,d}=

*σ*

_{υ}_{,d}= 0.024 m s

^{−1}and

*σ*

_{u}_{,m}=

*σ*

_{υ}_{,m}= 0.019 m s

^{−1}).

#### 3) Santa Catalina Island

Bottom-mounted thermograph time series collected by the Catalina Island Conservancy (CCD) around Santa Catalina Island and model temperature interpolated at the same depths show a clear semidiurnal variability with a not very coherent spring–neap variability (results not shown). The semidiurnal temperature variability in Table 3 is larger for the northeastern side than for the southwestern side of the island, and the data have larger variability than the model. Interestingly, lagged cross correlations between the time series of adjacent stations on the northeastern side of Catalina Island reveal a southeastward-propagating baroclinic wave with phase speeds between 0.37 and 0.92 m s^{−1} (Table 4), with reasonable agreement between model and data. At this location, the model features southward-propagating baroclinic Kelvin waves that cause southward energy fluxes.

In summary, the model is doing a reasonable job of predicting the magnitudes and trends in the variance of the tidal water levels, currents, and temperature. Generally, the variance in the model is smaller than in the data, and the modeled thermocline is deeper. The latter may be due to stronger surface mixing and/or the inability of the model to advect colder surface water from the north southward. This deeper thermocline affects amplitudes of the baroclinic tides on the shelf in shallower water, but it has less of an effect on the propagation of the internal tides and the barotropic to baroclinic energy conversion in deeper water. This follows from a sensitivity study, in which first-mode phase speeds *c*_{1} are computed with the Stürm–Liouville equation as a function of time in 1800-m-deep water and on the Huntington Beach shelf in 60-m-deep water. As the stratification weakens from August to December, *c*_{1} only varies by ±2.5% in deep water, whereas it decreases by 70% in shallow water.

## 4. Internal wave energetics

### a. Internal wave classification

The internal wave regime is often classified using several nondimensional parameters (Garrett and Kunze 2007): 1) the tidal excursion length *δ* = *U*_{0}/(*Lω*), where *L* is a horizontal topographic length scale, *ω* is the tidal frequency, and *U*_{0} is the amplitude of the far-field barotropic current; 2) the criticality parameter

which is the ratio between the local slope and the internal wave beam angle, where ; 3) the topographic Froude number Fr* _{t}* =

*U*

_{0}/(

*N*), where

_{r}h_{r}*h*is the height of the ridge and

_{r}*N*is the value of

_{r}*N*(

*z*) at the crest; and 4) the frequency ratios

*ω*/

*f*and

*ω*/

*N*.

For *δ* ≫ 1 stationary lee waves become important, whereas for *δ* ≪ 1 the regime comprises linear internal tides at the forcing frequency. The strongest internal waves in the SCB originate from the west sill of SaCrB. The sill features *L* = 20 km, *h _{r}* = 1700 m,

*U*

_{0}= 0.018 m s

^{−1},

*N*= 0.005 rad s

_{r}^{−1},

*ω*=

*ω*

_{2}= 1.4 × 10

^{−4}rad s

^{−1}, and

*f*= 0.8 × 10

^{−4}rad s

^{−1}. This yields

*δ*= 0.007 at the sill and can be considered a maximum value in the SCB. In the SCB, linear internal tides are generated at the forcing frequency.

For *γ* < 1 (subcritical slopes) generated beams go upward and incoming energy is transmitted, whereas for *γ* > 1 (supercritical slopes) generated beams go downward and incoming energy is reflected and scattered into higher modes (Müller and Liu 2000). The *γ* computed for the mean stratification near the bottom on L2 on 10 September 2006 is shown in Fig. 6. The shelf slopes below the island and mainland shelf breaks are highly supercritical, featuring *γ* > 5. These steep shelf slopes send baroclinic energy into the basins and keep it there through reflection and scattering.

For Fr* _{t}* > 1 the flow is relatively unaffected by the topography, whereas for Fr

*< 1 small-scale hydraulic effects like nonlinear breaking lee waves occur (Legg and Klymak 2008). At the sill, Fr*

_{t}*= 0.002. Indeed, the model reveals small nonlinear breaking lee waves with horizontal scales of*

_{t}*O*(1 km) at the supercritical slopes near the sill crest.

In the SCB, semidiurnal frequencies are >*f* and <*N*, whereas diurnal frequencies *ω*_{1} ≈ 0.7 × 10^{−4} < *f*; that is, the diurnal internal waves are evanescent. The SCB is in internal wave regime 5 of Garrett and Kunze (2007).

### b. Fluxes and energy

The total depth-integrated baroclinic fluxes in the central SCB for 10 September 2006 on the L2 grid are in Fig. 7a. They are quite representative of the fluxes for other spring tides. The L2 fluxes in the central SCB have similar patterns and magnitudes as the fluxes computed on the larger L1 grid (not shown) but are more detailed because of the higher-resolution topography of L2. The continental shelf break present in the L1 domain (−2500 m contour in Fig. 1) does not generate significant onshore fluxes.

A striking feature in the SCB is the clockwise (CW) flux pattern in the 1900-m-deep SaCrB (the basin marked by A in Fig. 7c). This feature has the largest fluxes of about 5 kW m^{−1}. The clockwise flux pathway in SaCrB originates from the east slope of the northern part of the west sill. It remains circular during spring–neap cycles and in the period from summer to winter. The fluxes in the SMB, SPB, and SaCaB are generated on the slopes on the northwest sides of these basins and are mainly directed southeastward over tens of kilometers. Their magnitudes decrease in this direction because of dissipation. The deep south and east sills of the SaCrB allow fluxes to leave the basin. The southeastward fluxes to the northeast of Santa Catalina Island are due to a progressive baroclinic Kelvin wave that originates from the ridge that extends an island length northward from Santa Catalina Island. The baroclinic energy fluxes in the SBB and on the mainland shelves are relatively small.

The circular flux pattern in the SaCrB is mainly due to the first vertical baroclinic mode in Fig. 7b. There is little first-mode flux that leaves the basin, because the sidewalls are supercritical and the gaps are too small compared to the first-mode wavelength. There is also some first-mode conversion on the slope south of Anacapa Island. Most of this energy propagates southward before it dissipates in the SMB and SPB. The fluxes due to the sum of modes 2–20 in Fig. 7c are largest in SaCrB, generally smaller than the first-mode fluxes, and their patterns and directions are different. There appears to be a beam originating from the saddle in the middle of the west sill of the SaCrB that is directed northeastward. In this area, higher-mode flux divergence coincides with mode-1 flux convergence (not shown). This is indicative of the scattering of the mode 1 into higher modes near critical topography (Johnston et al. 2003). In contrast to the first modes, the higher modes make it across the east sill and propagate along the shelf south of Anacapa.

The tidal-mean horizontal kinetic energy ; available potential energy ; and isotherm oscillation amplitudes *A _{ζ}*(

*z*) =

*A*(

_{b}*z*)/〈

*N*

^{2}(

*z*)〉, where buoyancy

*b*= −(

*g*/

*ρ*

_{0})

*ρ*′, along transects T1 in SaCrB and T2 in SMB reflect beam-like structures (Fig. 8; the transect locations are in Fig. 6). High 〈HKE〉 is concentrated at the surface where

*N*is large, whereas 〈APE〉 has also higher values in deeper water. In contrast, the largest isotherm oscillations occur near the bottom, where

*N*is weak. Similar to the fluxes, 〈HKE〉, 〈APE〉, and

*A*are much stronger in SaCrB than elsewhere in the SCB. In SaCrB,

_{ζ}*A*> 25 m below the sills and >100 m near the walls (e.g., near

_{ζ}*x*= 37 km and

_{l}*z*= −1400 m in Fig. 8b). The flux of energy from SaCrB to SMB is also reflected in the higher energetics on the west side of transect T2.

### c. Temporal variability

The temporal variability in the water level amplitude *A _{η}*, the tidal-mean stratification averaged over the top 50 m 〈

*N*〉 in station A, and the absolute depth-integrated energy fluxes at stations A–D are in Fig. 9. The station locations are in Fig. 7c. The value of 〈|

**F**|〉 at A in SaCrB is largest, mainly mode 1, and well correlated with

*A*. Even the monthly beat in the spring–neap cycle is present. The value of 〈

_{η}*N*〉 decreases by 50% from summer to winter, but it has little effect on the deep-water fluxes in A–C. The maximum (minimum) fluxes occur 1–3 days after the spring (neap) tides in

*η*because the barotropic currents reach their maximum (minimum) a few days after the springs (neaps). Outside the SaCrB, the fluxes are orders of magnitude smaller; the correlation with the spring–neap cycle is weaker; and the higher vertical modes become more important, except on the shelf (station D). In station D, the flux is mainly first mode, and the predicted 〈|

**F**|〉 compares well with the fluxes based on the data at station HB-MA-10 of HB06. In agreement with Noble et al. (2009), these fluxes are not generated at the supercritical mainland shelf break but a few kilometers offshore on the deeper and more critical shelf slope. At this location, conversion is affected by remotely generated internal waves and submesoscale variability (Kelly and Nash 2010). The spring–neap variability is further modulated by the stratification on the shelf, which, if nonexistent, can shut down the fluxes completely. This has occurred during the last 2 weeks in Fig. 9f.

### d. Energy budget

The barotropic to baroclinic energy conversion 〈*C*〉, the flux divergence 〈**∇** · **F**〉, and negative dissipation −〈*D*〉 for L2 and the spring tide on 10 September are in Fig. 10. Here, 〈*D*〉 is plotted with a minus sign, so that regions of convergence and dissipation have the same color. The occurrence of small areas with 〈*D*〉 < 0 may be attributed to small nonlinear terms that are ignored and uncertainties in the filtering and the harmonic analyses used to compute the terms. Note that the colors are saturated in Figs. 10a,b, and absolute values >0.2 W m^{−2} occur. Areas of large positive and negative 〈*C*〉 occur in SaCrB, and they coincide with large positive and negative 〈**∇** · **F**〉. The northern part of the west sill features the largest conversion of ~1 W m^{−2}, and it is the origin of the circular flux pattern in SaCrB (Figs. 7a,b). The strong positive 〈*C*〉 on the northeastern slopes of SMB and SaCaB also correlate well with the southeastward fluxes in these basins. The areas with negative 〈**∇** · **F**〉 indicate dissipation of the semidiurnal internal waves and baroclinic to barotropic energy conversion. The largest dissipation occurs near bathymetry with the strongest generation: that is, in the SaCrB and northern SMB. The strong dissipation near (*x*, *y*) = (140, 70) km and *z* = −1000 m in Fig. 10c coincides with large predicted semidiurnal bottom currents with maximum values of 0.2 m s^{−1} and viscosity fluctuations above the background value of 10^{−4} m^{2} s^{−1}.

The terms 〈**∇** · **F**〉, 〈*C*〉, and 〈*D*〉 are computed using the total baroclinic pressures and velocities and are integrated over areas A1–A8 (Table 5). The areas A1–A8 and the fluxes integrated over the area boundaries are in Fig. 11. The terms in, out, in–out in Table 5 are integrated fluxes perpendicular to the area boundaries. The largest of 121.17 MW coincides with the largest of 107.53 MW in A1. This implies that most of the energy flux generated in the basin is dissipated in the basin. The fluxes across the boundaries of A1 are an order of magnitude smaller and are mainly higher modes. The conversion, fluxes, and dissipation are much smaller in the other basins and are smallest at the shelf areas A5–A7. However, the volume averaged dissipation 〈*ε*〉 is relatively large on the shelves, because the shelves have a small volume.

Of the basins, 〈*ε*〉 is largest in SaCrB, followed by the SMB. This is in qualitative agreement with findings by Ledwell and Hickey (1995) and Ledwell and Bratkovich (1995), who estimated the effective diffusivity *K*_{eff} ≈ 10 cm^{2} s^{−1} for the SaCrB and *K*_{eff} ≈ 1.5 cm^{2} s^{−1} in SMB and SPB combined. The relation between the diffusivity and dissipation is *K _{ρ}* ≈ 0.2

*εN*

^{−2}. If it is assumed that most of the dissipation occurs in the first 100 m near the bottom and that 〈

*ε*〉 is horizontally invariable,

*K*≈ 36 cm

_{ρ}^{2}s

^{−1}for the SaCrB and 11 cm

^{2}s

^{−1}for the SMB and SPB combined. In the case where 〈

*ε*〉 is determined over the entire basin volume,

*K*≈ 1 cm

_{ρ}^{2}s

^{−1}for SaCrB and 0.3 cm

^{2}s

^{−1}for the SMB and SPB combined. The

*K*

_{eff}determined in the literature is within this broad range. In the model, the dissipation and estimated diffusivity in the SaCrB are larger than in the SMB and SPB combined. Both the model and the data indicate that the dissipation is stronger in SaCrB. However, in the model the diffusivity in SaCrB is only a factor of 3 larger than in the SMB and SPB combined, whereas in Ledwell and Bratkovich (1995) it is a factor of 10 larger.

## 5. Discussion

### a. Poincaré wave

The dominance of the circular vertical first-mode fluxes in SaCrB in Fig. 7b suggests the presence of a vertical first-mode wave that is progressive in the azimuthal direction and standing in the radial direction. The phase of the first-mode bottom perturbation pressure in Fig. 12b shows a gradual CW increase from −180° to 180°, implying the first-mode wave cycles CW through the basin in one semidiurnal tidal cycle. Its maximum phase speed near the walls is about 3.8 m s^{−1}. This is nearly twice the phase speed of a progressive vertical first-mode wave in 1900-m-deep water (*c*_{1} ≈ 1.8 m s^{−1}). The amphidromic point in the center of the basin coincides with zero fluxes in Fig. 12c. The amplitude of the first-mode bottom perturbation pressure *A _{p}*

_{′}is greatest near the walls of the basin and smallest at the amphidromic point (Fig. 12a). The first-mode bottom velocity ellipses in Fig. 12a are circular, are clockwise polarized, and have the same phase across the basin, and, in contrast to

*A*

_{p}_{′}, the major axes of the ellipses have the greatest amplitudes near the amphidromic point and the smallest amplitudes near the sidewalls. The spatial differences in

*A*

_{p}_{′}and are reflected in the vertical first-mode, depth-integrated, tidal-mean, horizontal kinetic energy 〈HKE

_{1,I}〉 and the available potential energy 〈APE

_{1,I}〉 in Figs. 12c,d, where

_{1,I}refers to first-mode and depth integrated.

This wave does not resemble a vertical first-mode Kelvin wave, because in the Northern Hemisphere a Kelvin wave has the coastal wall to the right of its propagation direction, the velocity ellipses are bidirectional and oriented along the wall, and and *A _{p}*

_{′}have their largest amplitudes near the wall. The baroclinic wave in SaCrB resembles a CW rotating Poincaré wave, because its potential energy is largest near the walls, its kinetic energy is largest at the basin center, and the bottom velocities in the center always have the maximum bottom pressure to their left (Csanady 1967; Antenucci and Imberger 2001). The ratio

*κ*= 〈APE

_{tot}〉/〈HKE

_{tot}〉 < 1 for anticyclonic Poincaré waves and

*κ*> 1 for cyclonic Kelvin waves in circular and elliptic basins (Antenucci and Imberger 2001), where

_{tot}refers to volume integration of the first-mode energy. In SaCrB,

*κ*≈ 0.35. The Poincaré wave in SaCrB is of mode one in the radial and azimuthal directions: that is, the bottom pressure has one maximum and one minimum value along these directions.

Inviscid dispersion relations for naturally occurring baroclinic waves in two layers have been derived for rotating basins that are circular by Csanady (1967) and elliptic by Antenucci and Imberger (2001). The wave characteristics are identical for both basin shapes. They found subinertial counterclockwise (CCW) rotating baroclinic Kelvin waves and superinertial CW rotating baroclinic Poincaré waves. The dispersion relations are a function of the Burger number *S* = *R*/*R*_{0}, where *R*_{0} is the basin radius or the major radius for an elliptic basin and *R* = *c*_{0}/*f* is the baroclinic Rossby radius of deformation and *c*_{0} is the linear phase speed for a two-layer fluid. The Burger number indicates the strength of the stratification versus rotation. SaCrB has an elliptic shape with a major radius of 35 km and a minor radius of 20 km, a mean depth of ~1300 m, *f* = 8.1 × 10^{−5} s^{−1}, *c*_{0} = 1.6 m s^{−1}, and *R* = 19.7 km. Note that the radial crest to trough distance of the Poincaré wave is about half the wavelength of a progressive vertical first-mode wave with *L*_{1} ≈ 80 km. For circular and elliptic basins, *S* = 0.98 and 0.56, respectively. After inserting in the dispersion relations, the ratios between the eigenfrequency and the forcing frequency are *ω _{e}*/

*ω*

_{2}= 1.35 and 1.25. The elliptic basin is closer to resonance. The eigenfrequency decreases for lower

*S*because of larger

*R*

_{0}or a smaller

*c*

_{0}. Significantly higher eigenfrequencies are found for azimuthal and radial modes >1. The seasonal variability in

*ω*/

_{e}*ω*

_{2}due to changing

*N*(

*z*) is less than 5%. SaCrB’s walls are not vertical but steeply sloping, and the baroclinic wave propagation is not inviscid. These factors may affect the basin’s eigenfrequencies and resonance characteristics.

### b. Forcing the Poincaré wave

The Poincaré wave in Fig. 12 does not propagate through the entire basin: the southern part of the basin is omitted. The wave’s flux pathway is not elliptic but circular. However, the center of the Poincaré wave, where the energy fluxes are zero, is not at the basin center but offset to the southwest by a few kilometers (Fig. 7a,b). Hence, the flux pathway does not form a perfect circle and *R*_{0} ~ 27 km in all but the southwestern quadrant, where *R*_{0} ~ 17 km. The wave’s offset position may be due to strong positive and negative conversion on the west sill.

The terms of the total depth-integrated and time-mean energy balance are spatially integrated for boxes I–IV in SaCrB (Fig. 13). The boxes are designed so that they cover the four areas of positive and negative conversion in SaCrB in Fig. 10a. The strongest positive conversion of 121.1 MW occurs in box I on the west sill, followed by 44.3 MW in box III. Box IV has the largest negative conversion of −39.3 MW, followed by −22.4 MW in box II. The Poincaré wave is mainly forced on the east slope of the west sill in box I. It cycles CW through the basin and loses some energy to the barotropic tide and dissipation in box II. Its net energy loss is nearly zero in box III, because the local positive conversion equals dissipation. By the time the wave enters area IV, it still has a flux of 78.5 MW, about 65% of 121.1 MW. In box IV about half of this energy flux is converted to the barotropic tide, and 30.1 MW goes to box I, where it is mainly dissipated .

The positive (negative) conversion in boxes I and III (II and IV) occurs because the tidal barotropic vertical velocity is in phase (out of phase) with the bottom perturbation pressure of the Poincaré wave (Fig. 14). Boxes I and III are out of phase: when the barotropic flow goes up slope in box I, it goes down slope in box III, and when the bottom pressure is positive in I, it is negative in III. The latter also applies to boxes II and IV, but here {*W*(*t*)} is out of phase with {*p*′(*z* = −*h*)}, where {} is area averaging, causing {*C*(*t*)} < 0 during the entire tidal cycle (Fig. 14c). Clearly, the local generation is affected by the remotely generated internal wave, causing negative conversion. The spatial variability in {*W*(*t*)} is due to the pathway of the tidal wave through SaCrB (Figs. 3a,b). It enters SaCrB across the east and south sills, is blocked by Santa Cruz Island, and is forced out of SaCrB across the northern part of the west sill. The tidal-mean depth-integrated barotropic tidal fluxes during this spring tide in SaCrB are about 200 kW m^{−1}, about 40 times larger than the largest baroclinic fluxes. Therefore, it seems unlikely that the pathway of the tidal wave is affected by the Poincaré wave. Hence, the downslope and upslope barotropic velocities can be regarded as predetermined, and the sign of {*C*(*t*)} is influenced by the Poincaré wave. The phasing difference between the barotropic velocities on mainly the west sill and the near-resonant characteristics of the basin allow the Poincaré wave to exist in its most optimum form. The positive and negative conversion on the west sill “replaces” the southwestern quadrant of the wave. The circular part of the wave with *R*_{0} ~ 27 km yields *S* = 0.73 and *ω _{e}*/

*ω*

_{2}= 1.08, which is closer to resonance.

### c. Standing wave patterns

It has been shown that the summation of energy fluxes from various sources (topography with positive conversion) causes standing wave interference patterns in the observed fluxes (Nash et al. 2006; Martini et al. 2007; Rainville et al. 2010). Nash et al. (2006) showed that the radial fluxes are zero between two opposing sources of equal strength, 〈HKE_{1,I}〉 and 〈APE_{1,I}〉 have alternating maximum values separated by a quarter horizontal wavelength of the vertical first-mode, and the transverse (azimuthal) fluxes have the areas of maximum 〈HKE_{1,I}〉 to their right. If the sources are of unequal strength, the radial fluxes are not zero (Martini et al. 2007).

Similarly in SaCrB, the fluxes rotate CW around the center with maximum 〈HKE_{1,I}〉, and the areas of maximum 〈APE_{1,I}〉 and 〈HKE_{1,I}〉 are separated by a about a quarter first-mode wavelength (*L*_{1} ~ 80 km). Although the source in box I is 3 times as large as in box III in Fig. 13, the radial fluxes between these sources in SaCrB are zero in Figs. 7a,b. In contrast to the studies by Nash et al. (2006) and Martini et al. (2007), the walls in SaCrB steer the fluxes. The reflections off the walls result in essence in multiple sources along the perimeter, causing wave energy patterns that are associated with a radial and azimuthal mode-1 Poincaré wave.

## 6. Conclusions

A triply nested ROMS configuration with realistic forcing is set up for the Southern California Bight (SCB). The high resolution and small horizontal grid size of 250 m allows for a detailed study of the generation, prop-agation, and dissipation of semidiurnal internal waves in the undermeasured SCB. The model predicts the barotropic and baroclinic tides fairly well, although the variance in the data is slightly larger than in the model.

Linear internal wave generation occurs at the forcing frequency. It is strongest in the interior of the SCB, where the barotropic tide is forced through deep basins and across shallow sills.

The largest barotropic to baroclinic conversion of ~1 W m^{−2} occurs on the west sill slope of the 1900-m-deep Santa Cruz Basin (SaCrB) during spring tides. This yields the largest baroclinic fluxes of 5 kW m^{−1}, which are of the same order of magnitude as the fluxes emitted at the Hawaiian Islands Ridge (Carter et al. 2008). In the basin, isotherm oscillation amplitudes are up to 100 m near the walls.

The energy fluxes form a clockwise circular pattern that resembles a forced and near-resonant clockwise Poincaré wave that is of first mode in the vertical, radial, and azimuthal directions. In this wave the kinetic (potential) energy is largest near the center (walls) and smallest near the walls (center).

Most of the energy is dissipated in the basin and only 10% escapes in the form of higher modes. In agreement with dispersal studies by Ledwell and Hickey (1995) and Ledwell and Bratkovich (1995), the dissipation in the SaCrB is larger than in the Santa Monica Basin (SMB) and the San Pedro Basin (SPB).

The fluxes and isotherm oscillations in the basins outside the SaCrB are much smaller, because of weaker barotropic to baroclinic conversion in these basins. In the SMB, SPB, and Santa Catalina Basin (SaCaB), the energy propagates southeastward. Near Santa Catalina Island, the modeled baroclinic first-mode Kelvin wave can also be recognized in the data. The fluxes are smallest on the shelves.

The spring–neap cycle modulation of the semidiurnal depth-integrated fluxes is strongest in the SaCrB, weaker in other basins, and weakest on the shelves. The circular flux pattern and the resonant characteristics of the Poincaré wave are little affected by seasonal changes in stratification.

This study illustrates the need for realistic and regional high-resolution coastal ocean models, because these models may reveal mixing hot spots like the SaCrB. It is not yet clear what the importance of these hot spots is for the general overturning circulation (Munk and Wunsch 1998). A week-long survey campaign is planned to verify the existence of the Poincaré wave.

## Acknowledgments

Alexander Shchepetkin is thanked for assistance with the ROMS numerics, and Craig Gelpi is thanked for his sharing of processed Catalina Island Conservancy thermistor data. The Los Angeles County Sanitation District and the Orange County Sanitation District are acknowledged for sharing their ADCP and thermistor data. The reviewers are thanked for their helpful comments.

## REFERENCES

_{2}barotropic-to-baroclinic tidal conversion at the Hawaiian Islands

_{2}internal tide south of the Hawaiian Ridge

_{2}internal tide generation over Mid-Atlantic Ridge topography

## Footnotes

Current affiliation: Atmospheric and Oceanic Sciences Program, Princeton University, Princeton, New Jersey.