## 1. Introduction

In summer, currents over the Oregon shelf (U.S. West Coast) are predominantly wind driven, varying on the temporal scale of several days. The wind-driven alongshore current is predominantly to the south, vertically sheared, and associated with upwelling of cold waters near the coast (Allen et al. 1995). At the surface, summer-average magnitude of the wind-driven jet is near 0.5 m s^{−1}. Tides are relatively moderate. The dominant barotropic tidal constituent off Oregon is *M*_{2}. Although the *M*_{2} tidal depth-averaged currents are rather small, less than 0.05 m s^{−1} (Erofeeva et al. 2003), they flow over variable bathymetry and generate baroclinic motions near the *M*_{2} tidal period (12.4 h; Baines 1982), resulting in an internal tide that can be 0.1–0.15 m s^{−1}, based on mooring observations (Hayes and Halpern 1976; Torgrimson and Hickey 1979; Erofeeva et al. 2003). The internal tide thus can be an important contributor to current variability at the surface. Near the bottom, where the wind-driven current is weaker, internal tide currents can potentially dominate.

The characteristic horizontal length scale of the *M*_{2} internal tide over the shelf is *O*(10 km). Internal tide generation and propagation are sensitive to variability in background currents and hydrographic conditions associated with wind-driven upwelling and mesoscale eddies (Mooers 1970, 1975b,a; Chavanne et al. 2010; Chen et al. 2003; Park and Watts 2006; Pereira et al. 2007; Hall and Davies 2007; Xing and Davies 1997, 1998, 2005; for a brief review, see Kurapov et al. 2010). In particular, Xing and Davies (1997), using an idealized model, demonstrated that background changes associated with upwelling can increase the transfer of tidal energy to higher harmonics. Xing and Davies (2005) showed that a cold water dome can trap energy of propagating internal waves. Observations available in the coastal area are generally too sparse to resolve the resulting spatial and temporal intermittency of the internal tide. To describe this variability, provide estimates of the *M*_{2} tidal energetics on the slope and shelf, and ultimately understand how the wind-driven and tidal flows influence each other, high-resolution model simulations seem to be the most promising approach.

In modeling studies off Oregon, wind-driven currents and internal tides have traditionally been studied separately (e.g., Kurapov et al. 2003; Allen et al. 1995; Federiuk and Allen 1995; Kurapov et al. 2005; Oke et al. 2002a,b; Springer et al. 2009; Koch et al. 2010). Recently, Kurapov et al. (2010) used the hydrostatic Regional Ocean Modeling System (ROMS; http://www.myroms.org) to study influences of the wind-driven upwelling and internal tide in an idealized, two-dimensional (2D; cross-shore and vertical coordinates) setup. They found that internal tides with magnitudes comparable to those measured off Oregon affect subtidal cross-shore and alongshore transports. The utility of this 2D approach is limited, however, because the *M*_{2} internal tidal motions are superinertial and freely propagate in three dimensions. Besides, the 2D approach may provide a misleading picture, because internal tides in such a model are only generated by the cross-shore barotropic tidal current flowing up and down the continental slope. In fact, off Oregon and in many other coastal environments, the *M*_{2} barotropic tide propagates as a shelf-modified Kelvin wave, in which the *M*_{2} tidal current is strongly polarized and aligned with the continental slope bathymetry. Alongshore small-scale bathymetric variations can thus be important for internal tide generation, potentially resulting in an internal tide of greatly varying intensity along the coast.

In this paper, we analyze the output of a 3D model (described in section 2) that realistically represents both wind-driven and *M*_{2} tidal flows over the Oregon shelf and slope (as verified against observations in section 3). We demonstrate strong nonuniformity of the *M*_{2} internal tide generation along the Oregon slope and describe its intermittency (section 4). Some effects of bathymetry on model estimates of internal tide generation and on-shelf propagation are discussed in section 5. Section 6 provides a summary.

## 2. Model description

We use ROMS, a free-surface, hydrostatic, Boussinesq, terrain-following primitive-equation model featuring advanced numerics (Shchepetkin and McWilliams 2005). The model domain is approximately 300 km × 540 km (Fig. 1). The study period is 1 April through 31 August 2002. A thoroughly verified model solution of subtidal circulation off Oregon is available during the same period of time (Koch et al. 2010). It provides boundary conditions for our higher-resolution study. Data from the Global Ocean Ecosystem Dynamics (GLOBEC) field study (Batchelder et al. 2002) are available during this period, including mooring velocity profile measurements on the shelf. Note that internal tides on the Oregon shelf are generally stronger during summer than winter (Erofeeva et al. 2003) because the winter waters are well mixed because of seasonal downwelling and storms.

The model horizontal resolution is approximately 1 km. A total of 40 terrain-following *s* layers are used in the vertical, with an emphasis on resolving the surface and bottom boundary layers (using ROMS terminology, *θ _{s}* = 5,

*θ*= 0.4, and thermocline = 50 m). Subgrid-scale turbulence is handled using the Mellor–Yamada 2.5 scheme (Mellor and Yamada 1982) modified by Galperin et al. (1988). The horizontal eddy diffusion and viscosity coefficients are set to 2 m

_{b}^{2}s

^{−1}in the interior and increase over a 40-km sponge layer (via a sinusoidal ramp) to 15 m

^{2}s

^{−1}at the three open boundaries. Bathymetry is a combination of the 5-minute gridded elevations/bathymetry for the world (ETOPO5; NOAA 1988) and the higher-resolution 12″ National Oceanic and Atmospheric Administration (NOAA)–National Geophysical Data Center bathymetry datasets. To investigate the effects of bathymetric resolution on internal tide energetics, cases using different degrees of bathymetric smoothing (Figs. 1a,b) have been considered. Smoothing is performed solving a pseudoheat equation with fixed values along the boundaries to preserve the shape of the coastline (Bennett 1992, chapter 2). The minimum depth is set to 10 m along the coast. Unless specifically mentioned, results for the case with rougher bathymetry are presented.

Subtidal boundary values (free-surface elevation, velocities, temperature, and salinity) are obtained by interpolation between snapshots from a larger-scale, 3-km-horizontal-resolution ROMS simulation forced by winds and heat fluxes but not tides (Koch et al. 2010). These boundary fields are provided every 48 h. Initial conditions are obtained from the same model. Flather (1976) and Chapman (1985) boundary conditions are used for normal barotropic velocities and the free surface, respectively. Passive/active radiation boundary conditions are applied for baroclinic velocities, temperature, and salinity (Marchesiello et al. 2001). Tidal forcing is added to the subtidal boundary conditions every time step using *M*_{2} harmonic constant estimates of sea surface elevation amplitude and depth-averaged currents from a data-assimilating 1/30°-resolution barotropic tidal model (Egbert et al. 1994; Egbert and Erofeeva 2002). Although this adds barotropic tides to the boundary (which then propagate into the domain), data for internal tides at the boundary are unavailable and their omission is a potential source of error. To simplify analysis, only the dominant *M*_{2} tidal constituent is included in tidal forcing.

Atmospheric momentum and heat flux are computed using the bulk flux formulation adapted in ROMS (Fairall et al. 1996a,b). The daily averaged wind speed is obtained from the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS; Hodur 1997) and inputs for heat flux computations (air temperature and pressure, relative humidity, and solar shortwave radiation; all monthly averaged) from the National Centers for Environmental Prediction (NCEP) reanalysis (Kalnay et al. 1996). The meridional wind stress is well correlated along the Oregon coast: for example, as can be seen in the time series at two midshelf locations near 44.7° and 42.4°N [near the NH10 and Rogue River (RR) moorings, described below; Fig. 3a]. The wind stress is substantially stronger south of Cape Blanco (42.8°N) because of orographic effects (Samelson et al. 2002).

## 3. Model verification

We first verify the accuracy of the subinertial wind-driven circulation and the tidally forced flows. To separate subinertial (mostly wind driven) and superinertial (mostly tidal) variability for this analysis, a 40-h half-amplitude low-pass filter is applied to both observed and modeled time series. To describe tidal intermittency, high-pass-filtered time series are harmonically analyzed in a series of relatively short overlapping time windows. In each window, the high-pass-filtered signal is assumed to be harmonic: for example, *ω* is the *M*_{2} angular frequency (1.405 × 10^{−3} rad s^{−1}) and *M*_{2} and *S*_{2} tidal constituents in the data and thus facilitate model–data comparisons. Also, for the purposes of this study, we define barotropic current as depth-averaged current and baroclinic current as the deviation from the depth average. Note that separation of the flow into barotropic and baroclinic components over varying bathymetry is not trivial. For instance, the bottom boundary layer would introduce vertical variability in barotropic tidal flows. Variability in the bottom boundary layer associated with wind-driven processes can introduce some intermittency in estimates of barotropic (depth averaged) tidal currents. The definition adopted here allows clear separation of equations for the barotropic and baroclinic tidal energy but is not without shortcomings (for further details, see Kurapov et al. 2003).

### a. Wind-driven circulation

Maps of monthly averaged SST from the 5.5-km-resolution Geostationary Operational Environmental Satellite (GOES; Maturi et al. 2008), the 1-km model solution, and the 3-km solution that provided subtidal boundary conditions are compared in Fig. 2. In both models, the geometry of the SST front is approximately correct, particularly the location of the coastal jets separated from the coast near 45° and 42°N in July and August.

Monthly averaged SST: (top) GOES (5.5-km resolution) satellite observations, (middle) the 1-km-resolution ROMS model, and (bottom) the 3-km resolution ROMS model that provided subtidal boundary conditions. Locations of ADCP moorings analyzed in this study are shown as red circles. The black rectangle is the domain of the 1-km model. The white contour is the 200-m isobath.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Monthly averaged SST: (top) GOES (5.5-km resolution) satellite observations, (middle) the 1-km-resolution ROMS model, and (bottom) the 3-km resolution ROMS model that provided subtidal boundary conditions. Locations of ADCP moorings analyzed in this study are shown as red circles. The black rectangle is the domain of the 1-km model. The white contour is the 200-m isobath.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Monthly averaged SST: (top) GOES (5.5-km resolution) satellite observations, (middle) the 1-km-resolution ROMS model, and (bottom) the 3-km resolution ROMS model that provided subtidal boundary conditions. Locations of ADCP moorings analyzed in this study are shown as red circles. The black rectangle is the domain of the 1-km model. The white contour is the 200-m isobath.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

*α*(Kundu 1976),

*u*and

*υ*denote zonal and meridional velocity components, respectively; and 〈·〉 denotes time averaging. To compute correlations, time-averaged values are subtracted. The phase angle is a measure of the average veering between the two vector time series. Based on these comparisons, both models reproduce velocity variability on temporal scales of several days and longer qualitatively correctly at all three locations. One aspect of the observed flow at the Rogue River site, south of Cape Blanco, is large-amplitude variability on time scales of 3–5 days, particularly during the first half of the study period (see Fig. 3d). This is not reproduced in the 1-km model runs, nor is it reproduced in the 3-km model used to provide subtidal boundary conditions. Koch et al. (2010) note that this variability may be due to remote forcing south of the 3-km model domain (40.6°N). Model–data discrepancies over days 210–230 (Coos Bay; Fig. 3c) and 185–210 (Rogue River; Fig. 3d) can be associated with large-scale eddies in the adjacent interior ocean affecting the offshore displacement of the upwelling front.

Model–data statistics for the depth-averaged currents at mooring locations.

(a) Meridional wind stress at the NH10 and Rogue River mooring locations. (b)–(d) 40-h low-pass-filtered, depth-averaged observed and model meridional velocities at the NH10, CB, and RR mooring locations, respectively.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

(a) Meridional wind stress at the NH10 and Rogue River mooring locations. (b)–(d) 40-h low-pass-filtered, depth-averaged observed and model meridional velocities at the NH10, CB, and RR mooring locations, respectively.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

(a) Meridional wind stress at the NH10 and Rogue River mooring locations. (b)–(d) 40-h low-pass-filtered, depth-averaged observed and model meridional velocities at the NH10, CB, and RR mooring locations, respectively.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

### b. Tidally forced flows

To ensure that the forcing of baroclinic flow is correct, barotropic tides from the three-dimensional 1-km model are compared to the solution from the shallow-water tidal model used to provide boundary conditions. The *M*_{2} sea surface elevation tidal amplitude and phase in both models are consistent over the entire 1-km model domain (Figs. 4a,b). The barotropic *M*_{2} tide propagates from south to north along the coast as a shelf-modified Kelvin wave. It takes approximately 1 h for the barotropic *M*_{2} tide to propagate along the Oregon coast. The sea surface elevation amplitude increases from about 0.6 m in the south to 0.85 m in the north. Figure 4c shows a close-up of the amplitude and phase map over a part of the shelf region and also includes barotropic tidal current ellipses. Here and throughout the entire paper, shaded (clear) ellipses indicate counterclockwise (clockwise) rotation. In the deep water, the tidal current ellipses are strongly polarized and roughly parallel to the coast, with a maximum current velocity of about 0.05 m s^{−1}. Over the slope and shelf (defined here as areas shallower that 200 m; see black contour in the figure), the current ellipses can cross isobaths. In particular, the barotropic current flows over the southern flank of Heceta Bank (43.8°N), suggesting that this might be an area of intensified internal tide generation.

Barotropic *M*_{2} sea surface elevation tidal amplitude and phase: (a) 1/30° shallow-water-equation model (Egbert et al. 1994; Egbert and Erofeeva 2002) providing tidal boundary conditions for our 1-km ROMS solution; (b) 1-km ROMS for the entire domain; and (c) 1-km ROMS for the close-up on the area shown as the black rectangle in (b), with horizontal barotropic tidal current ellipses added. Shaded (clear) ellipses indicate counterclockwise (clockwise) rotation. Black diamonds mark the NH10, CB, and RR moorings. White phase lines are 5° apart.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Barotropic *M*_{2} sea surface elevation tidal amplitude and phase: (a) 1/30° shallow-water-equation model (Egbert et al. 1994; Egbert and Erofeeva 2002) providing tidal boundary conditions for our 1-km ROMS solution; (b) 1-km ROMS for the entire domain; and (c) 1-km ROMS for the close-up on the area shown as the black rectangle in (b), with horizontal barotropic tidal current ellipses added. Shaded (clear) ellipses indicate counterclockwise (clockwise) rotation. Black diamonds mark the NH10, CB, and RR moorings. White phase lines are 5° apart.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Barotropic *M*_{2} sea surface elevation tidal amplitude and phase: (a) 1/30° shallow-water-equation model (Egbert et al. 1994; Egbert and Erofeeva 2002) providing tidal boundary conditions for our 1-km ROMS solution; (b) 1-km ROMS for the entire domain; and (c) 1-km ROMS for the close-up on the area shown as the black rectangle in (b), with horizontal barotropic tidal current ellipses added. Shaded (clear) ellipses indicate counterclockwise (clockwise) rotation. Black diamonds mark the NH10, CB, and RR moorings. White phase lines are 5° apart.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Next, we verify the accuracy of the modeled baroclinic *M*_{2} internal tide. A point-by-point match between observed and modeled internal tide phase and amplitude can be hard to achieve because of the short horizontal scales associated with internal waves and sensitivity to many model aspects. Here, we are primarily interested in whether modeled internal tide current amplitudes and vertical structures are comparable to the observations on average over the season. For the 2002 study period, 1-h-temporal-resolution velocity profiles, suitable for tidal analysis, are available from the NH10, Coos Bay, and Rogue River moorings. To provide additional evidence about baroclinic tidal variability, we will also use ADCP data from the 2001 Coastal Ocean Advances in Shelf Transport (COAST; Boyd et al. 2002) field program. Data are not available at distances of 10–20 m from the surface and the bottom because of ADCP limitations. For consistency with the observations, model currents are sampled at the same depths and the barotropic velocity is computed as the average over these depths. Tidal harmonic constants are estimated in a series of overlapping 16-day windows offset by 4 days and results are presented as horizontal baroclinic tidal current ellipses (Figs. 5, 7, 9, 11). The upward (rightward) direction in the plots corresponds to the northward (eastward) velocity direction; the line from an ellipse center shows velocity direction at zero phase.

(top) Observed and (bottom) modeled barotropic and baroclinic tidal ellipses at NH10 (*h* = 81 m). The vertical axis represents the depth of each tidal ellipse and the horizontal axis time (center of each 16-day analysis window). Ellipses above 10-m depth correspond to barotropic tides. Northward velocity is directed up and eastward velocity is directed to the right. Shaded (unshaded) ellipses indicate counterclockwise (clockwise) rotation.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

(top) Observed and (bottom) modeled barotropic and baroclinic tidal ellipses at NH10 (*h* = 81 m). The vertical axis represents the depth of each tidal ellipse and the horizontal axis time (center of each 16-day analysis window). Ellipses above 10-m depth correspond to barotropic tides. Northward velocity is directed up and eastward velocity is directed to the right. Shaded (unshaded) ellipses indicate counterclockwise (clockwise) rotation.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

(top) Observed and (bottom) modeled barotropic and baroclinic tidal ellipses at NH10 (*h* = 81 m). The vertical axis represents the depth of each tidal ellipse and the horizontal axis time (center of each 16-day analysis window). Ellipses above 10-m depth correspond to barotropic tides. Northward velocity is directed up and eastward velocity is directed to the right. Shaded (unshaded) ellipses indicate counterclockwise (clockwise) rotation.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

At NH10 (Fig. 5), observed and modeled horizontal baroclinic tidal ellipses are plotted every 4 m in the vertical, at every other observation depth. Barotropic currents are plotted at the top of each panel. They vary in time, but not as much as baroclinic currents. Means and standard deviations of modeled and observed barotropic tidal currents are similar (Table 2). First-mode baroclinic structure is apparent in both the observations (Fig. 5, top) and the model (Fig. 5, bottom), with relatively larger currents and opposite phases near the surface and bottom. The largest observed (modeled) baroclinic current is 0.10 m s^{−1} (0.06 m s^{−1}). Both the modeled and observed series exhibit similar intermittency. There could be several reasons for our model estimates of baroclinic tides to be smaller than those observed. For instance, observed current variability in the tidal frequency band may be partially driven by high-frequency atmospheric forcing, which was filtered in the model forcing. Also, the model does not account for the internal tide possibly propagating into our domain.

The mean of the major axis amplitude (m s^{−1}) and mean and standard deviation (*μ* ± *σ*) of the angle of inclination (relative to due east) of observed and modeled depth-averaged *M*_{2} tidal currents at the NH10, CB, and RR moorings. Standard deviation of the major axis amplitude is small (<0.01 m s^{−1}). For both observed and modeled velocities, the depth average is taken only over the depths at which observations are available. Statistics are computed from harmonic constants from 29 time windows.

To further investigate the strength of the baroclinic tide near NH10, a series of surface maps of model baroclinic tidal current ellipses are shown in Fig. 6. For clarity, the ellipses are plotted every 4 km. The black diamond symbol near the center of each panel marks the NH10 mooring. The time period analyzed (days 189–237) corresponds to a period of intensified observed internal tide at NH10 (cf. Fig. 5). During this time period, a region of intensified baroclinic surface tide (with surface current amplitudes > 0.10 m s^{−1}) is found in the model at distances of only 4–10 km north and west of NH10. It is possible that small changes in bathymetry or other model details could bring the area of the intensified internal tide closer to the NH10 location.

Modeled surface baroclinic *M*_{2} tidal ellipses near NH10, analyzed in nine consecutive overlapping 16-day windows, yeardays 189–237, a period of observed intensified internal tide near NH10. The scale circle, plotted over land, is 0.1 m s^{−1}. The 200-m isobath is plotted. The black diamonds indicate the location of the NH10 mooring and three moorings from the 2001 COAST experiment. (e) The black star marks the mooring location in Fig. 7. Shaded (unshaded) ellipses indicate counterclockwise (clockwise) rotation. Ellipses are plotted with 4-km horizontal resolution.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Modeled surface baroclinic *M*_{2} tidal ellipses near NH10, analyzed in nine consecutive overlapping 16-day windows, yeardays 189–237, a period of observed intensified internal tide near NH10. The scale circle, plotted over land, is 0.1 m s^{−1}. The 200-m isobath is plotted. The black diamonds indicate the location of the NH10 mooring and three moorings from the 2001 COAST experiment. (e) The black star marks the mooring location in Fig. 7. Shaded (unshaded) ellipses indicate counterclockwise (clockwise) rotation. Ellipses are plotted with 4-km horizontal resolution.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Modeled surface baroclinic *M*_{2} tidal ellipses near NH10, analyzed in nine consecutive overlapping 16-day windows, yeardays 189–237, a period of observed intensified internal tide near NH10. The scale circle, plotted over land, is 0.1 m s^{−1}. The 200-m isobath is plotted. The black diamonds indicate the location of the NH10 mooring and three moorings from the 2001 COAST experiment. (e) The black star marks the mooring location in Fig. 7. Shaded (unshaded) ellipses indicate counterclockwise (clockwise) rotation. Ellipses are plotted with 4-km horizontal resolution.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

To see if the internal tide in the region of intensification is similar to observations at NH10, baroclinic tidal ellipses from a location 10 km north of NH10 (marked with a star in Fig. 6e) are shown in Fig. 7. The depth of this location (83 m) is close to that of NH10 (81 m). During days 200–240, both modeled and observed baroclinic currents show first-mode structure with tidal velocities of similar magnitude, roughly 0.10 m s^{−1}. Overall, we conclude that the model solution produces internal tide currents in the NH10 region at magnitudes similar to those observed at the mooring (up to 0.10 m s^{−1}), though not necessarily at the same time.

Model solution baroclinic tidal ellipses at (44.74°N, 124.26°W), approximately 10 km north of the NH10 mooring. Depth is 83 m. Details as in Fig. 5.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Model solution baroclinic tidal ellipses at (44.74°N, 124.26°W), approximately 10 km north of the NH10 mooring. Depth is 83 m. Details as in Fig. 5.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Model solution baroclinic tidal ellipses at (44.74°N, 124.26°W), approximately 10 km north of the NH10 mooring. Depth is 83 m. Details as in Fig. 5.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Analysis of tidal currents at the Coos Bay mooring leads to similar conclusions about the strength of internal tides on the Oregon shelf. In Fig. 8, horizontal tidal ellipses at the Coos Bay mooring are plotted every 6 m in the vertical, one-third the observed resolution. Observed and modeled depth-averaged currents are similar (Table 2). The largest observed (0.12 m s^{−1}) and modeled (0.10 m s^{−1}) internal tide currents are also similar. First-mode baroclinic structure is apparent.

(top) Observed and (bottom) modeled solution baroclinic tidal ellipses at the CB mooring (*h* = 100 m). Details as in Fig. 5.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

(top) Observed and (bottom) modeled solution baroclinic tidal ellipses at the CB mooring (*h* = 100 m). Details as in Fig. 5.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

(top) Observed and (bottom) modeled solution baroclinic tidal ellipses at the CB mooring (*h* = 100 m). Details as in Fig. 5.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Observed and modeled baroclinic tidal ellipses at the Rogue River mooring location are shown in Fig. 9. Ellipses are plotted every 4 m in the vertical, at the observed resolution. Modeled *M*_{2} internal tide currents (maximum amplitude of 0.06 m s^{−1}) are also generally a bit stronger than observed (maximum amplitude of 0.04 m s^{−1}), though both are weak compared to barotropic currents (Table 2). Figure 10 shows model baroclinic surface tidal current ellipses in the region around the Rogue River mooring. The mooring location, marked with a black diamond symbol, is in a region of generally weak surface baroclinic *M*_{2} tides, whereas regions of stronger baroclinic tide activity (with currents > 0.10 m s^{−1}) are found to the north and south. Explanations for this pattern are given later, in section 5a.

(top) Observed and (bottom) modeled solution baroclinic tidal ellipses at the RR mooring (*h* = 76 m). Details as in Fig. 5.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

(top) Observed and (bottom) modeled solution baroclinic tidal ellipses at the RR mooring (*h* = 76 m). Details as in Fig. 5.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

(top) Observed and (bottom) modeled solution baroclinic tidal ellipses at the RR mooring (*h* = 76 m). Details as in Fig. 5.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Surface baroclinic *M*_{2} tidal ellipses near Rogue River, analyzed in nine overlapping 16-day windows (offset by 4 days), yeardays 165–213. The scale circle, plotted over land, is 0.1 m s^{−1}. Black contours are 200-, 1000-, and 2000-m isobaths. The diamond marker is the location of the RR mooring. Shaded (unshaded) ellipses indicate counterclockwise (clockwise) rotation. Ellipses are plotted with 4-km horizontal resolution.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Surface baroclinic *M*_{2} tidal ellipses near Rogue River, analyzed in nine overlapping 16-day windows (offset by 4 days), yeardays 165–213. The scale circle, plotted over land, is 0.1 m s^{−1}. Black contours are 200-, 1000-, and 2000-m isobaths. The diamond marker is the location of the RR mooring. Shaded (unshaded) ellipses indicate counterclockwise (clockwise) rotation. Ellipses are plotted with 4-km horizontal resolution.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Surface baroclinic *M*_{2} tidal ellipses near Rogue River, analyzed in nine overlapping 16-day windows (offset by 4 days), yeardays 165–213. The scale circle, plotted over land, is 0.1 m s^{−1}. Black contours are 200-, 1000-, and 2000-m isobaths. The diamond marker is the location of the RR mooring. Shaded (unshaded) ellipses indicate counterclockwise (clockwise) rotation. Ellipses are plotted with 4-km horizontal resolution.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

To develop better understanding of internal tide energetics in the area, we also provide additional analysis of mooring observations from the 2001 COAST experiment (Boyd et al. 2002; Fig. 11). Mooring locations are shown in Fig. 6 as black diamonds along 45°N. At the shallowest and middepth moorings, observed currents are predominately first mode and surface intensified, reaching 0.15 m s^{−1}. At the deepest mooring, first-mode structure is occasionally observed. Measured currents reach 0.10 m s^{−1} at that location. Although background conditions in 2001 and 2002 are likely different, these additional data provide further evidence that *M*_{2} internal tide currents with magnitudes above 0.10 m s^{−1} are not uncommon on the Oregon shelf.

Observed baroclinic tidal ellipses during 2001 at the three COAST moorings along 45°N: depths of (top)–(bottom) 50, 81, and 130 m.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Observed baroclinic tidal ellipses during 2001 at the three COAST moorings along 45°N: depths of (top)–(bottom) 50, 81, and 130 m.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Observed baroclinic tidal ellipses during 2001 at the three COAST moorings along 45°N: depths of (top)–(bottom) 50, 81, and 130 m.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Based on these comparisons, we conclude that the magnitude and variability of the modeled internal tide are qualitatively similar to the observed internal tide. At the same time, the exact timing and location of areas of intensified internal tide are difficult to predict. Further, conclusions about the strength and variability of the *M*_{2} internal tide in a given area based on data from a single mooring may be inaccurate. For a rigorous, point-by-point analysis of the internal tide, a large number of observations is needed over both the shelf and slope. Analysis of high-frequency radar surface currents (O’Keefe 2005; Kurapov et al. 2003) may provide additional insights and is planned for the future.

## 4. Baroclinic *M*_{2} tide energetics

### a. Stable and intermittent features of M_{2} internal tide energetics

**EF**) vector. In (2)–(4), the tilde denotes complex harmonic constants, the asterisk complex conjugates,

*w*

_{1}= −

**u**

_{1}·

**∇**

*h*is the bottom vertical velocity associated with barotropic tidal flow over bathymetry,

**u**

_{1}= (

*u*

_{1},

*υ*

_{1}) is the depth-averaged horizontal current vector,

*h*(

*x*,

*y*) is bottom topography, and

*p*

_{2}is baroclinic tidal pressure, computed as the deviation from the depth-averaged pressure, associated with tidal motions. The residual term accounts for the effects of bottom friction, eddy dissipation throughout the water column, and nonlinear advection effects. In our study, it is not directly computed, but rather estimated as TEC −

**∇**·

**EF**.

Tidal harmonic constant estimates of **EF** and TEC at each horizontal location. To provide an illustration of *M*_{2} internal tide intermittency, **EF** vectors are shown over a portion of the Oregon shelf from three partially overlapping time windows, yeardays 193–225 (Fig. 12). The onshore flux near 45°N increases over this period. The rectangle shown in each panel of Fig. 12 marks the computational domain in Kurapov et al. (2003). In that study, the *M*_{2} internal tide was modeled using a linear baroclinic model and variational assimilation of high-frequency radar surface currents, with correction of baroclinic tidal fluxes along the open boundary. Both studies find that the *M*_{2} baroclinic tidal energy flux into the smaller domain is from the northwest on average and is *O*(10 W m^{−1}). Note that internal tides in the 2003 study were driven by the assimilated data, and the sources of the internal tide propagating into this area were unknown. Here, we can identify the sources within our larger study area.

*M*_{2} baroclinic **EF** vectors analyzed in three partially overlapping 16-day windows over the central Oregon shelf, yeardays 193–225. The 200-m isobath contour is shown in black. The rectangle shows the extent of the model domain in Kurapov et al. (2003).

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

*M*_{2} baroclinic **EF** vectors analyzed in three partially overlapping 16-day windows over the central Oregon shelf, yeardays 193–225. The 200-m isobath contour is shown in black. The rectangle shows the extent of the model domain in Kurapov et al. (2003).

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

*M*_{2} baroclinic **EF** vectors analyzed in three partially overlapping 16-day windows over the central Oregon shelf, yeardays 193–225. The 200-m isobath contour is shown in black. The rectangle shows the extent of the model domain in Kurapov et al. (2003).

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Stable features and intermittency in **EF**, TEC, ∇ · **EF**, and the residual are described next using their time-mean and standard deviation maps. Over the slope, the magnitude of the depth-integrated *M*_{2} baroclinic energy flux is *O*(100 W m^{−1}) (Fig. 13a; vectors shown every 6 km). Zones of strong **EF** divergence are apparent. At (42.6°N, 124.75°W), in the area near Cape Blanco, the **EF** magnitude approaches 700 W m^{−1}.

Time-average and standard deviation ellipses of the *M*_{2} baroclinic **EF**, where color shows **EF** magnitude: (a) **EF** shown every 6 km, with the vector scale and color range (0–300 W m^{−1}) chosen to emphasize **EF** on the slope; (b) **EF** on the shelf shown using a different vector and color scales (0–100 W m^{−1}), for every 4 km; and (c) standard deviation ellipses on the shelf, for every 8 km. Black contours are the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time-average and standard deviation ellipses of the *M*_{2} baroclinic **EF**, where color shows **EF** magnitude: (a) **EF** shown every 6 km, with the vector scale and color range (0–300 W m^{−1}) chosen to emphasize **EF** on the slope; (b) **EF** on the shelf shown using a different vector and color scales (0–100 W m^{−1}), for every 4 km; and (c) standard deviation ellipses on the shelf, for every 8 km. Black contours are the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time-average and standard deviation ellipses of the *M*_{2} baroclinic **EF**, where color shows **EF** magnitude: (a) **EF** shown every 6 km, with the vector scale and color range (0–300 W m^{−1}) chosen to emphasize **EF** on the slope; (b) **EF** on the shelf shown using a different vector and color scales (0–100 W m^{−1}), for every 4 km; and (c) standard deviation ellipses on the shelf, for every 8 km. Black contours are the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

To better see smaller **EF** vectors over the shelf (*h* < 200 m), the scale in Fig. 13b is adjusted and vectors are shown every 4 km. At the shelf break (*h* = 200 m), the time-averaged **EF** is directed onshore everywhere, reaching 40–60 W m^{−1} in some places and decreasing toward the coast. Areas of relatively larger onshore **EF** occur near Cape Blanco (42.8°–43°N), along the southern flank of Heceta Bank (43.5°–44°N) and over a wider shelf portion north of 45.2°N. The shelf between 44.2° and 45°N, studied by Kurapov et al. (2003), turns out to be an area of relatively weak onshore **EF**. **EF** standard deviation ellipses over the shelf are shown at 8-km resolution (Fig. 13c). They are polarized, showing maximum variability in the direction of the mean flux. Standard deviation maxima are close to the mean values at the corresponding locations, indicating that the baroclinic tidal flux (mean ± standard deviation) is mostly onshore. We also find that **EF** over the shelf is weak within about 20 km of the Rogue River mooring (42.44°N) but is much stronger north and south of this shadow zone, similar to the pattern of baroclinic surface tides in Fig. 10.

The map of seasonally averaged TEC (Fig. 14a; color bar limits set at ±0.04 W m^{−2}) reveals a few hot spots of strong positive TEC. Several small areas of relatively weak negative TEC can also be seen. Formally, this implies that energy is transferred from baroclinic to barotropic tides, although areas of negative TEC may also reflect the difficulty in choosing appropriate definitions of barotropic and baroclinic motions (see Kurapov et al. 2003). Contrary to our expectations, the area south of Heceta Bank (43.8°N), where bathymetry is nearly perpendicular to the general direction of barotropic tide propagation (cf. Fig. 4), is not the largest hot spot of TEC. Comparison of Figs. 14a and 13a suggests that the increased onshore **EF** in that area is partially due to the **EF** propagation from the hot spot at 44.4°N, going around the southern flank of Heceta Bank.

Time average and standard deviation of TEC: (a) time average, where the color range (±0.04 W m^{−2}) is chosen to emphasize hot spots; (b) time average at the finer range (±0.01 W m^{−2}); and (c) standard deviation. Black contours indicate the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time average and standard deviation of TEC: (a) time average, where the color range (±0.04 W m^{−2}) is chosen to emphasize hot spots; (b) time average at the finer range (±0.01 W m^{−2}); and (c) standard deviation. Black contours indicate the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time average and standard deviation of TEC: (a) time average, where the color range (±0.04 W m^{−2}) is chosen to emphasize hot spots; (b) time average at the finer range (±0.01 W m^{−2}); and (c) standard deviation. Black contours indicate the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

*P*

_{TEC}(

*υ*), the proportion of the total area-integrated TEC generated by grid points with 0 < TEC <

*υ*. The total TEC is determined by integration over the area of Fig. 14a, with negative TEC spots excluded. More formally,

^{−2}contribute approximately 20% of the total TEC (Fig. 15). Interestingly, the combined area of grid cells with TEC ≥ 0.04 W m

^{−2}is only 1.2% of the area of the continental slope (200 <

*h*< 1800 m). At the same time, about 50% of total TEC is provided by low energy areas with mean values between 0 and 0.015 W m

^{−2}. In Fig. 14b, color bar limits are adjusted to ±0.01 W m

^{−2}to emphasize these lower TEC areas. TEC is still patchy and, even at this adjusted scale, is concentrated over selected portions of the slope, with very little baroclinic tidal energy generated over the shelf. TEC standard deviation (Fig. 14c) is also largest over the slope, although its magnitude is generally smaller than the mean. Areas of large TEC standard deviation over the slope are typically associated with large mean values.

Proportion of area-integrated positive TEC *P*_{TEC}(*υ*) [Eq. (5)] below a threshold value *υ*.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Proportion of area-integrated positive TEC *P*_{TEC}(*υ*) [Eq. (5)] below a threshold value *υ*.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Proportion of area-integrated positive TEC *P*_{TEC}(*υ*) [Eq. (5)] below a threshold value *υ*.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

To provide further insight about what contributes to spatial variability of TEC, we examine the factors in (2): namely, the time-averaged amplitudes of barotropic tidal vertical velocity at the bottom ^{−2}). The bottom vertical velocity is affected by finescale bathymetric variations and exhibits spatial variability on smaller scales than the bottom baroclinic pressure. The latter can be associated not only with generation but also with propagation of the internal waves. The horizontal wave lengths on the slope are generally larger than the scales of bathymetric variations resolved by the 1-km model, yielding a smoother baroclinic pressure amplitude field. In particular, the amplitude of

Time-averaged amplitudes of harmonic constants determining TEC [Eq. (2)]: (a) vertical barotropic velocity at the bottom and (b) tidal baroclinic pressure at the bottom. Black contours mark the 200-, 1000-, and 2000-m isobaths. Half-tone contours mark TEC = 0.02 W m^{−2}.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time-averaged amplitudes of harmonic constants determining TEC [Eq. (2)]: (a) vertical barotropic velocity at the bottom and (b) tidal baroclinic pressure at the bottom. Black contours mark the 200-, 1000-, and 2000-m isobaths. Half-tone contours mark TEC = 0.02 W m^{−2}.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time-averaged amplitudes of harmonic constants determining TEC [Eq. (2)]: (a) vertical barotropic velocity at the bottom and (b) tidal baroclinic pressure at the bottom. Black contours mark the 200-, 1000-, and 2000-m isobaths. Half-tone contours mark TEC = 0.02 W m^{−2}.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Maps of *M*_{2} baroclinic energy flux divergence (**∇** · **EF**; Fig. 17) are analyzed in similar ways. Time-averaged **∇** · **EF** has a few hot spots over the slope (Fig. 17a), typically collocated with the TEC hot spots. At the adjusted scale (Fig. 17b), areas of large divergence on the slope coincide with areas of large TEC (Fig. 14b). The adjusted scale also reveals areas of energy flux convergence (negative divergence), found both over the slope and shelf, which can be associated with internal tide dissipation.

Time average and standard deviation of the *M*_{2} baroclinic **EF** divergence: (a) time average, where the color range (±0.04 W m^{−2}) is chosen to emphasize hot spots; (b) time average at the finer range (±0.01 W m^{−2}); and (c) standard deviation. Black contours are the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time average and standard deviation of the *M*_{2} baroclinic **EF** divergence: (a) time average, where the color range (±0.04 W m^{−2}) is chosen to emphasize hot spots; (b) time average at the finer range (±0.01 W m^{−2}); and (c) standard deviation. Black contours are the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time average and standard deviation of the *M*_{2} baroclinic **EF** divergence: (a) time average, where the color range (±0.04 W m^{−2}) is chosen to emphasize hot spots; (b) time average at the finer range (±0.01 W m^{−2}); and (c) standard deviation. Black contours are the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

The residual TEC −**∇** · **EF**, which we would like to interpret as dissipation, is predominately positive. Maps of the season-averaged residual (Fig. 18a) reveal that the TEC hot spots (with their edges shown as white contours at the 0.02 W m^{−2} level) are well balanced by **∇** · **EF**. The map at the adjusted color scale (Fig. 18b) shows vast areas of increased residual over the slope and some over the shelf.

Time average and standard deviation of the residual in (1): (a) time average, with the color range ±0.04 W m^{−2}; (b) time average at the finer range (±0.01 W m^{−2}); and (c) standard deviation. Black contours are the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time average and standard deviation of the residual in (1): (a) time average, with the color range ±0.04 W m^{−2}; (b) time average at the finer range (±0.01 W m^{−2}); and (c) standard deviation. Black contours are the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time average and standard deviation of the residual in (1): (a) time average, with the color range ±0.04 W m^{−2}; (b) time average at the finer range (±0.01 W m^{−2}); and (c) standard deviation. Black contours are the 200-, 1000-, and 2000-m isobaths.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

### b. Reasons for internal tide intermittency

Internal tide intermittency can result from a number of local and remote factors, including focusing of internal tide rays in a nonuniform medium (Chavanne et al. 2010), variability in TEC associated with changes in stratification around the hot spots (Kurapov et al. 2010) and constructive/destructive interference of remotely generated internal waves (Kelly and Nash 2010). Chavanne et al. (2010) used a ray tracing technique to consider internal wave energy propagation from an underwater ridge in the Hawaiian archipelago and found that a mesoscale eddy can create zones of ray focusing, resulting in areas of larger internal tide signal near the surface. Without a doubt, variability in background conditions affects internal wave propagation in our study area. However, in our case, not only internal tide propagation but also generation varies with time. Because along the Oregon coast major TEC hot spots are located in close proximity, it is hard to interpret whether changes in TEC over a given spot are due to local changes in stratification or remote influence of nearby spots. This remote influence (constructive interference mentioned above; see Kelly and Nash 2010) can be explained using (2). Although *w*_{1} is determined by the local topography and barotropic tide and is relatively steady, near-bottom *p*_{2} can be influenced by both local generation processes and the baroclinic waves propagating into the area (e.g., from neighboring hot spots). The incoming wave can modify the phase and amplitude of *p*_{2} such that the product

Additional analysis in smaller areas shows that the range of change of TEC can be comparable to the change in the onshore **EF** in the vicinity. However, it would not be the only factor determining the amount of onshore **EF**. For illustration, Figs. 19a,b show maps of **EF** and the near-bottom *p*_{2} amplitude in the area south of Heceta Bank in two windows, centered on days 193 and 229. Figure 20 shows the time series of TEC and residual integrated over the slope area (bounded by the edges of the map in Fig. 19 and the 200-m isobath), as well as the integrated **EF** across the 200-m isobath and the net **EF** through the southern, western, and northern boundaries of this small area. Values corresponding to the selected time windows are marked with circles. The first (second) time window selected in Fig. 19 corresponds to a relatively smaller (larger) onshore **EF**. The onshore **EF** is increased by 2.5 MW between days 193 and 229. This increase is comparable to that in TEC over the same period. A change in the strength of the internal tide is also clearly seen in surface horizontal baroclinic tidal ellipses (Figs. 19c,d). Figures 19a,b suggest that the influence of the hot spot at 44.3°N on *p*_{2} south of Heceta Bank (43.7°N) is smaller in the first time window, days 185–201, than the second window, days 221–237. Remarkably, the residual (dissipation) in the pictured area increases sharply after day 200 (by more than 5 MW), whereas the net outward **EF** through the outer boundaries is reduced during the same period. So, despite TEC and onshore **EF** changing by similar amounts over days 193–230, we cannot claim that local increases in internal tide generation are solely responsible for the increase in the onshore **EF**. Influences of remotely generated internal tide complicate analysis of tidal intermittency, making it difficult to establish a definitive relationship with wind-driven conditions.

(top) Depth-integrated, tidally averaged *M*_{2} baroclinic **EF** vectors and bottom baroclinic pressure during yeardays (a) 185–201 and (b) 221–237. The 200-, 1000-, and 2000-m isobaths are contoured in gray. Scale vectors (right of each panel) are 100 W m^{−1}. (bottom) Surface baroclinic tidal ellipses for the corresponding time periods. Scale circle at lower right is 0.10 m s^{−1}. Isobaths are contoured at the same intervals.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

(top) Depth-integrated, tidally averaged *M*_{2} baroclinic **EF** vectors and bottom baroclinic pressure during yeardays (a) 185–201 and (b) 221–237. The 200-, 1000-, and 2000-m isobaths are contoured in gray. Scale vectors (right of each panel) are 100 W m^{−1}. (bottom) Surface baroclinic tidal ellipses for the corresponding time periods. Scale circle at lower right is 0.10 m s^{−1}. Isobaths are contoured at the same intervals.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

(top) Depth-integrated, tidally averaged *M*_{2} baroclinic **EF** vectors and bottom baroclinic pressure during yeardays (a) 185–201 and (b) 221–237. The 200-, 1000-, and 2000-m isobaths are contoured in gray. Scale vectors (right of each panel) are 100 W m^{−1}. (bottom) Surface baroclinic tidal ellipses for the corresponding time periods. Scale circle at lower right is 0.10 m s^{−1}. Isobaths are contoured at the same intervals.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time series of energy balance terms for the slope region in Fig. 19 [(125.3°W, 200-m isobath) × (43.5°N, 44.4°N)]. For the flux terms, positive values correspond to energy radiating out of the region. Circles on each line mark yeardays 193 and 229, the midpoints of the windows in Fig. 19.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time series of energy balance terms for the slope region in Fig. 19 [(125.3°W, 200-m isobath) × (43.5°N, 44.4°N)]. For the flux terms, positive values correspond to energy radiating out of the region. Circles on each line mark yeardays 193 and 229, the midpoints of the windows in Fig. 19.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time series of energy balance terms for the slope region in Fig. 19 [(125.3°W, 200-m isobath) × (43.5°N, 44.4°N)]. For the flux terms, positive values correspond to energy radiating out of the region. Circles on each line mark yeardays 193 and 229, the midpoints of the windows in Fig. 19.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

## 5. Bathymetric effects on *M*_{2} internal tide energy fluxes

### a. Bathymetry criticality

*ϕ*that a characteristic makes with the horizontal is determined by

*ω*is the tidal frequency,

*f*is the inertial frequency, and

*N*is the buoyancy frequency. Internal wave beams incident on a supercritical slope (where the bottom slope

**∇**

*h*= tan

*θ*is steeper than the characteristic slope tan

*ϕ*) reflect toward deeper water, whereas those incident on subcritical bathymetry (tan

*θ*< tan

*ϕ*) will propagate into shallower water.

To calculate *ϕ*, the buoyancy frequency is estimated 20 m above the ocean floor using density profiles computed from temperature and salinity averaged over the entire model run. Figure 21 shows a map of the angle difference *θ* − *ϕ*. On the offshore side of the 200-m isobath, bathymetry varies from subcritical (*θ* − *ϕ* < 0; blue) to near critical (*θ* − *ϕ* ≈ 0; green) and supercritical (*θ* − *ϕ* > 0; red). Seasonally averaged **EF** vectors across the 200-m isobath are also plotted in Fig. 21, repeated from Fig. 13. Sections with relatively larger onshore baroclinic tidal **EF** are generally found near areas where bathymetry is subcritical or near critical on the offshore side of the 200-m isobath, such as in the ranges 41.7°–42.2°N and 43.5°–43.9°N. Sections with low onshore energy flux in places where bathymetry is supercritical at the shelf break are also easily identified: for example, in the areas of 41.0°–41.7°N, 42.3°–42.5°N, 42.9°–43.1°N, and 43.9°–44.5°N.

Bathymetric criticality (bathymetric slope angle minus wave characteristic slope angle), shown over the continental slope (color; shelf omitted), and **EF** across the 200-m isobath (vectors). Positive bathymetric criticality (warm colors) corresponds to the steep, supercritical slope. Thick black contours are the coast and 200-m isobaths. Thin black contours are every 500 m.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Bathymetric criticality (bathymetric slope angle minus wave characteristic slope angle), shown over the continental slope (color; shelf omitted), and **EF** across the 200-m isobath (vectors). Positive bathymetric criticality (warm colors) corresponds to the steep, supercritical slope. Thick black contours are the coast and 200-m isobaths. Thin black contours are every 500 m.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Bathymetric criticality (bathymetric slope angle minus wave characteristic slope angle), shown over the continental slope (color; shelf omitted), and **EF** across the 200-m isobath (vectors). Positive bathymetric criticality (warm colors) corresponds to the steep, supercritical slope. Thick black contours are the coast and 200-m isobaths. Thin black contours are every 500 m.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

To provide a more quantitative assessment, we consider the correlation between the magnitude of the onshore **EF** across *h* = 200 m and bathymetric criticality in the 10-km band offshore of this isobath. This relationship is complicated by nonuniformity of internal tide energy generation along the slope. For instance, between 44.5° and 45°N, bathymetry is mostly subcritical just offshore of the 200-m isobath, but **EF** here is relatively low, because little baroclinic energy is generated offshore near this area (see Fig. 14b). In contrast, at 42.7°N, where bathymetry is supercritical just offshore of the 200-m isobath, the onshore **EF** is larger than anywhere else along the 200-m isobath, because there is an extremely powerful source nearby, in the deep water at 42.3°N. To account for alongshore variation in generation, the **EF** magnitude at each point on the 200-m isobath is compared to the value at a point 10 km offshore, in the direction normal to the 200-m isobath. Dividing the two values, we obtain the percent ratio of propagated **EF**, which is then compared to the mean of *θ* − *ϕ* over the 10-km distance between the two points. Results (Fig. 22, where *θ* − *ϕ* > 0° corresponds to supercritical bathymetry) suggest that, in general, relatively less energy propagates onshore through areas of stronger supercritical bathymetry along the shelfbreak rim. Two aspects of the plot necessitate further explanation. First, percentages exceed 100% at several locations where substantial internal tide energy is generated in the 10-km band. Second, strips of the supercritical bathymetry do not reflect all the internal tide energy into the deeper water. It is possible that some internal tide energy can propagate over narrow regions of supercritical bathymetry along characteristics that do not touch the slope (e.g., along characteristics that reflect off the surface in those areas).

Scatterplot of the percent ratio of **EF** passing through the 10-km strip offshore of the 200-m isobath vs the mean angle difference (bathymetric slope minus wave characteristic slope). Averaging is in the direction perpendicular to the 200-m isobath. Positive angle difference corresponds to supercritical bathymetry, and negative difference to subcritical bathymetry.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Scatterplot of the percent ratio of **EF** passing through the 10-km strip offshore of the 200-m isobath vs the mean angle difference (bathymetric slope minus wave characteristic slope). Averaging is in the direction perpendicular to the 200-m isobath. Positive angle difference corresponds to supercritical bathymetry, and negative difference to subcritical bathymetry.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Scatterplot of the percent ratio of **EF** passing through the 10-km strip offshore of the 200-m isobath vs the mean angle difference (bathymetric slope minus wave characteristic slope). Averaging is in the direction perpendicular to the 200-m isobath. Positive angle difference corresponds to supercritical bathymetry, and negative difference to subcritical bathymetry.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

An example of how bathymetric criticality affects the onshore **EF** and resulting tidal currents is seen in the region surrounding the Rogue River mooring. Between 42° and 43°N, the mean onshore **EF** (Fig. 21) and *M*_{2} tidal currents (Fig. 10) are relatively large, except around 42.4°N. Bathymetry offshore of the 200-m isobath at 42.4°N is supercritical in a 35-km-wide region next to the 200-m isobath. While this region of steep bathymetry is generating a large amount of baroclinic tidal energy (Fig. 14a), the energy is either reflected offshore or propagated to the north and the south of 42.4°N (Fig. 13a), creating a “shadow” zone (Fig. 13b) surrounding the Rogue River mooring.

It also appears that regions of relatively large TEC are frequently associated with supercritical bathymetry. A total of 99.75% of TEC hot spot (>0.04 W m^{−2}) area is over supercritical bathymetry, and 87.67% of the area with TEC greater than 0.015 W m^{−2} (the threshold separating the upper 50% of area-integrated positive TEC activity from the bottom 50%; see Fig. 15) is also found over supercritical bathymetry.

### b. Bathymetry roughness and area-integrated energy balance

Figure 23 shows time-averaged, depth-integrated baroclinic **EF** vectors and standard deviation ellipses over the slope and shelf for the smoother bathymetry case. **EF** in the smoother bathymetry case is somewhat weaker, although the spatial distribution of **EF** over the 200-m isobath is similar to that in the rougher bathymetry case (cf. Figs. 13b, 23b). These results are consistent with those of Jachec et al. (2007), who studied tidal circulation in Monterey Bay, California, using models of different resolution.

Time-averaged *M*_{2} baroclinic **EF** vectors and standard deviation ellipses from the smoother bathymetry case. Details as in Fig. 13.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time-averaged *M*_{2} baroclinic **EF** vectors and standard deviation ellipses from the smoother bathymetry case. Details as in Fig. 13.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time-averaged *M*_{2} baroclinic **EF** vectors and standard deviation ellipses from the smoother bathymetry case. Details as in Fig. 13.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Another perspective on the energy balance, as well as its sensitivity to bathymetric resolution, is provided by area integration of the terms in (1) over the slope (200 m < *h* < 1800 m) and shelf (*h* < 200 m) regions. The time series of these terms are shown for the rougher bathymetry case in Fig. 24. Integration of **∇** · **EF** over the shelf yields the total energy radiating onto the shelf across the 200-m isobath (fluxes across the southern and northern boundaries are neglected). This value does not vary much with time, despite the substantial temporal and spatial variability in the *M*_{2} internal tide over the shelf. There is very little TEC over the shelf, which is consistent with Figs. 14b,c.

Time series of the area-integrated TEC, ∇ · **EF**, and the residual. Half-tone lines are for the area between the 1800- and 200-m isobaths (the slope), and black lines are for the area between the 200-m isobath and the coast (the shelf).

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time series of the area-integrated TEC, ∇ · **EF**, and the residual. Half-tone lines are for the area between the 1800- and 200-m isobaths (the slope), and black lines are for the area between the 200-m isobath and the coast (the shelf).

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time series of the area-integrated TEC, ∇ · **EF**, and the residual. Half-tone lines are for the area between the 1800- and 200-m isobaths (the slope), and black lines are for the area between the 200-m isobath and the coast (the shelf).

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Time averages of the area-integrated terms are summarized in Fig. 25, for both the rougher (Fig. 25a) and smoother (Fig. 25b) bathymetry cases. In the rougher bathymetry case (Fig. 25a), the integrated TEC over the slope has a mean value of about 81 MW. About 68% of this, approximately 55 MW, is dissipated over the slope. Integrating ∇ · **EF** over the slope yields the energy radiated out of the area, approximately 26 MW, which is nearly the sum of fluxes into the ocean interior (across the 1800-m isobath) and onto the shelf. About 14 MW is radiated from slope to shelf and about 12 MW is radiated to the interior ocean.

Diagrams summarizing time-averaged TEC, dissipation, and **EF** over the slope (half-tone) and the shelf (black): (a) the rougher bathymetry case and (b) the smoother bathymetry case.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Diagrams summarizing time-averaged TEC, dissipation, and **EF** over the slope (half-tone) and the shelf (black): (a) the rougher bathymetry case and (b) the smoother bathymetry case.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

Diagrams summarizing time-averaged TEC, dissipation, and **EF** over the slope (half-tone) and the shelf (black): (a) the rougher bathymetry case and (b) the smoother bathymetry case.

Citation: Journal of Physical Oceanography 41, 11; 10.1175/JPO-D-11-02.1

In the smoother bathymetry case (Fig. 25b), baroclinic energy production over the slope is reduced by 34% to 54 MW. Dissipation is reduced by 38% to 34 MW. Thus, the ratio of dissipated to generated internal tide energy remains about the same over rougher and smoother bathymetry, with about 65% of baroclinic energy generated over the slope also dissipated there. In the case of smoother bathymetry, the integrated divergence of **EF** from the slope to the shelf is also reduced (to a total of 11 MW), but only by 18%. It appears that bathymetric resolution affects the levels of internal tide generation and dissipation more than the integrated value of onshore propagating **EF**.

## 6. Summary

The nested 1-km-resolution model presented here describes coastal ocean circulation off Oregon in both the subinertial (wind driven) and superinertial (semidiurnal) frequency bands quantitatively correctly. Our study has focused on generation (TEC) and propagation of the superinertial *M*_{2} internal tide in this area. We find that most baroclinic tidal energy is generated over the slope with a few hot spots (occupying 1.2% of the area of the continental slope) contributing 20% of the total TEC.

Variability in both space and time is a prominent feature of the internal tide on the Oregon shelf, reflecting variation in background currents and hydrographic conditions. Although the model describes intermittency in the internal tide qualitatively correctly, the exact locations and times of intensified *M*_{2} internal tide are still difficult to predict. Furthermore, although it is clear that internal tide intermittency is associated with variability in wind-driven circulation, no simple relationship has been found between the intensity of the internal tide and characteristics of upwelling/downwelling, despite our attempts to find such a relationship.

Some aspects of baroclinic *M*_{2} energetics are quite stable. For instance, the locations of TEC hot spots are defined primarily by bathymetric details and are relatively weakly affected by background conditions. Despite day-to-day variability in the onshore baroclinic **EF**, the net flux onto the shelf, integrated along the 200-m isobath, remains rather steady during the summer season. About 65% of the internal tide energy converted from the barotropic tide over the slope is dissipated on the slope. The remaining energy is propagated both into the interior ocean and onto the shelf, in roughly equal proportions. Spatial variability in the onshore energy flux is controlled to a great extent by bathymetric criticality offshore of the 200-m isobath, with areas of supercritical bathymetry along the shelf break rim reflecting a large fraction of the onshore-directed **EF** to deeper waters.

This study has helped us identify areas of intensified internal tide generation and propagation on the Oregon shelf, directing future observational and modeling studies toward better understanding the role of the tidal currents in the coastal ocean.

## Acknowledgments

This research was supported by NSF (Grants OCE-0648314 and OCE-1030922), the Office of Naval Research (ONR) Physical Oceanography Program (Grants N000140810942 and N000141010745), NOAA-CIOSS, and NOAA-IOOS (NANOOS). We also appreciate comments provided by two anonymous reviewers, who strengthened the paper in several ways.

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