1. Introduction

A narrow coastal shelf off Oregon, along the U.S. west coast, is an area of energetic baroclinic alongshore current and coastal upwelling driven by predominantly southward wind in summer. The associated surface alongshore velocity is on average near 0.5 m s⁻¹ and can reach 1 m s⁻¹ during events of intensified wind. Due to upwelling, a horizontal cross-shore density gradient develops in shelf waters. In agreement with the thermal wind balance, the alongshore jet is vertically sheared, with a weaker current near the bottom. While the fundamentals of upwelling dynamics are well understood (Allen et al. 1995), the potential effects of the internal tide on the coastal ocean circulation are relatively less studied. The dominant tidal constituent off of Oregon is M₂ with surface elevation amplitudes near 0.8 m and barotropic tidal currents near 5 cm s⁻¹ on the shelf. Vertical motions associated with the barotropic tidal flow over sloping bathymetry cause baroclinic displacements of isopycnal surfaces at the tidal frequency, forcing the internal tide (e.g., Gill 1982; Baines 1982). Analysis of moored velocity time series data suggests that the M₂ internal tide on the Oregon shelf has a vertical structure related to the first baroclinic mode and is often surface amplified, with baroclinic velocities near 10–15 cm s⁻¹ (Hayes and Halpern 1976; Torgrimson and Hickey 1979; Erofeeva et al. 2003). Up to 50% of the horizontal velocity variance can be contained in the tidal frequency band (see Erofeeva et al. 2003). Thus, near the surface, the internal tide motions can provide a significant fraction of the ocean energetics. Near the bottom, where currents associated with wind forcing are relatively lower, the internal tide can possibly dominate the current variability.

Historically, wind-driven and tidal flows on the Oregon shelf have been studied separately. Using nonlinear primitive equation hydrostatic models, different aspects of the wind-driven flows relevant to the Oregon shelf have been investigated (e.g., Allen et al. 1995; Federiuk and Allen 1995; Allen and Newberger 1996; Oke et al. 2002; Wijesekera et al. 2003; Gan et al. 2005; Kurapov et al. 2005; Springer et al. 2009). Barotropic tides in this area have been modeled by Erofeeva et al. (2003) with assimilation of satellite and in situ observations in a linear shallow water model. To obtain estimates of the M₂ internal tide on the central Oregon shelf, Kurapov et al. (2003) assimilated surface velocity observations from high-frequency (HF) radars in a model that described harmonic linear oscillations of the stratified ocean with respect to the state of rest (no background currents, horizontally uniform background stratification). Estimates of tidal harmonic constants obtained in a series of overlapped time windows revealed internal tide intermittency that was consistent with that observed at a midshelf mooring. In that study, it was noted that during a period of intensified alongshore wind, results were more accurate if the tidal velocity estimates from surface HF radar data at distances <20 km from the coast were excluded from the assimilation. It was hypothesized that those observations could have provided useful information if the model were improved by allowing horizontally variable background stratification and the sheared alongshore current, associated with upwelling.

Surprisingly little research has been done on how wind-driven and internal tidal shelf flows influence each other. One of the most focused studies on this subject dates back to the Ph.D. thesis by Mooers (1970), who used the method of characteristics to solve a wave equation in a two-dimensional (2D) setup (cross-shore and vertical coordinates) to study internal wave propagation through a frontal zone in a coastal upwelling region (Mooers 1975a,b). In particular, he found that a significant asymmetry in the upgoing and downgoing characteristics is introduced when the slope of the isopycnals is comparable to the slope that the characteristics would have without the frontal interaction. Among more recent relevant studies, Chen et al. (2003) modeled and analyzed the internal tide in the presence of a tidal front, in an idealized, cross-shore, configuration relevant to the U.S. east coast. Park and Watts (2006) used data from an array of pressure-sensor-equipped inverted echo sounders and found that changes in background ocean conditions can cause variations in the internal tide energy levels. Pereira et al. (2007) utilized a three-dimensional (3D) model to study the dynamics in the South Brazil Bight and found that the Brazil Current affects the structure of the internal tides in the area, acting in part as a barrier to their onshore propagation. They also found that internal tides enhanced mixing on the inshore side of the coastal current. Hall and Davies (2007) simulated internal tides off the west coast of Scotland and found that both upwelling- and downwelling-favorable winds change the energetics of internal tides over the shelf break area. They suggested that those changes could be associated with changes in stratification along the continental slope and in the surface boundary layer. Observational analysis by Avicola et al. (2007) suggested that turbulence patches found over the shelf in the middle of the water column, away from surface and bottom boundary layers, can be caused by the combined shear from the wind-driven alongshore current and the internal tide.

While the effects of the relatively slowly changing background hydrographic conditions on internal tide propagation have been documented, it remains largely unknown whether and how the internal tide affects coastal transports on relatively slower temporal scales.

Here, we present the results of simulations of the wind-driven coastal upwelling and internal tide in combination, run in a 2D, cross-shore model. The main focus is on whether and how the internal tide changes alongshore, cross-shore, and vertical transports as well as the variability in the turbulence fields, in comparison to previously studied cases without tides (see Allen et al. 1995, Wijesekera et al. 2003). We also provide analysis of the internal tide energetics over the continental slope and shelf to better understand intermittency in the internal tide.

2. Model

Computations are performed with the Regional Ocean Modeling System (ROMS; information online at www.myroms.org), based on free-surface, hydrostatic, Boussinesq primitive equations and utilizing efficient and accurate numerical algorithms (Shchepetkin and McWilliams 2005). To represent the physics on subgrid scales, the Mellor–Yamada 2.5 turbulence scheme is used (Mellor and Yamada 1982; see also Wijesekera et al. 2003). A quadratic bottom stress formulation is utilized. The 2D solutions (varying in the cross-shore and vertical directions, and uniform in the alongshore direction) are obtained by running the model in a short south-to-north periodic channel with alongshore uniform bathymetry, forcing, and other conditions. Cases discussed in the following sections are designed to be relevant to summer wind-driven upwelling off Oregon, with the Coriolis parameter f = 10⁻⁴ s⁻¹. Cartesian coordinates are introduced with the x axis directed toward the coast, y axis north, and z axis upward. The cross-shore u and alongshore υ components of the horizontal velocity are projections in the x and y directions, respectively. Note that ROMS equations are derived by transformation from Cartesian z to terrain-following s coordinates, which allows for the resolution of the surface and bottom boundary layers over the shelf and slope. After discretization, the lowest s surface is defined at the bottom, z = −H(x), and the upper s surface at the free surface, z = ζ(x, t); the positions and thicknesses of the s layers vary with time (see Shchepetkin and McWilliams 2005).

The ocean bottom profile, H(x), is obtained as the alongshore average of the bathymetric data off the Oregon coast between 45.05° and 45.20°N, with a maximum depth of 1000 m (Fig. 1). We deliberately included several small-scale bathymetric features on the slope to allow more effective baroclinic tide generation by the cross-shore barotropic tidal flow. The resolution is 250 m in the horizontal and 120 (section 3) or 60 (section 4) layers in the vertical. Near-surface and bottom layers are resolved relatively better. All analyses have been performed using hourly model outputs.

The depth-averaged currents, denoted with an overbar, are referred to as barotropic and deviations from the depth-averaged currents as baroclinic; for example,

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The time series fields are split into the subtidal (low pass filtered) and high-frequency (high-pass filtered) components; for example,

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where the subtidal component 〈u〉 is obtained by applying a 40-h half-amplitude filter.

No-normal-flow and free-slip boundary conditions are applied at the coast (x = 0). At the offshore boundary (x = −200 km), the Flather condition (Flather 1976) is utilized for the depth-averaged cross-shore velocity u:

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where ω = 2π/12.42 h⁻¹ is the M₂ frequency, g the gravitational acceleration, and t₀ the initial time. Passive radiation boundary conditions are applied for all the other variables, including ζ, u, υ, temperature T, and salinity S.

Cases driven by the tide and wind stress in combination (TW) and by the wind stress only (WO) will be considered. Periods of time at the beginning of the TW runs, when only tidal forcing is applied, will be referred to as case TO.

The forcing amplitude U₀ = 0.2 m s⁻¹ is chosen to be large enough to produce internal tide currents on the shelf with maximum amplitudes near observed values of 0.1–0.15 m s⁻¹. Note that ζ in (3), which is determined by the interior dynamics, is near 2 m and the typical barotropic tidal currents in the domain interior are near 0.02 m s⁻¹ ≪ U₀. Although these barotropic tidal features differ from those actually observed off Oregon, we will show that energetics and variability in the baroclinic tide are representative of this area. Note that in a more realistic, 3D case the internal tide would be generated as a result of barotropic flow over variable bathymetry in both the alongshore and cross-shore directions. A larger tidal sea surface elevation variation is needed for our study in part because in the 2D case the internal tide is generated only by the cross-shore component of the barotropic tidal current.

3. Cases with constant wind stress

Case TW begins at t₀ = −10 days and runs for a total of 30 days. It starts from a state of rest, with horizontally uniform initial T and S corresponding to the average conditions off the mid–Oregon shelf in April (Smith et al. 2001). During the first 10 days, t = (−10, 0), only tidal forcing is applied. During the first 7 days of simulation, the magnitude of the internal tide is increased gradually until the internal tide reaches a quasi-equilibrium state (Fig. 2). The exact equilibrium cannot be achieved in part because the internal tide mixes water near bottom on the shelf, changing the background stratification. However, after t = −3 days, the intensity of the internal tide and the beamlike structure of the zones of increased tidal kinetic energy (see section 3a) remain very similar, as verified in a run for an extended period of 20 days forced by tides alone.

Beginning at t = 0, spatially uniform atmospheric forcing is applied. Both the wind stress and heat flux are ramped up over a 2-day period (using a cosine-shape function) from 0 to −0.05 N m⁻² and 200 W m⁻², respectively, and then held constant. Case WO starts at t = 0 from rest, using the same initial stratification, and is run for 20 days with the same wind stress and heat flux as in the TW case.

Figure 3 shows cross-shore sections of the daily averaged alongshore velocity and potential density σ_θ for the TW case, with averaging over intervals of t = (−1, 0), (4, 5), (9, 10), (14, 15), and (19, 20), referred to as days 0, 5, 10, 15, and 20, respectively. The wind stress forces the development of a sheared southward coastal jet with the core moving progressively offshore. The offshore Ekman transport in the surface boundary layer is balanced by the onshore transport of near-bottom denser water. For instance, contour σ_θ = 26 kg m⁻³, initially at a depth of 120 m, is about to outcrop near the surface by day 20. Next, we analyze the effects associated with the internal tide evolving in the presence of this time-variable background.

a. Internal tide variability

To show the spatial distribution of the zones of the intensified internal tide, we compute estimates of the tidal baroclinic kinetic energy (KE) per unit mass averaged over a tidal cycle:

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where u_bc(x, z) and v_bc(x, z) are complex amplitudes of M₂ tidal harmonic oscillations of the two components of the baroclinic velocity, obtained by harmonic analysis of the corresponding high-pass-filtered time series u′_bc(x, z, t) and υ′_bc(x, z, t) in a given interval. To analyze the internal tide intermittency, we would like to perform a harmonic analysis over a series of short time intervals. However, to allow separation of the M₂ tidal and inertial (period of 17 h) harmonics, these intervals are chosen to be ≥3 days. In cases described in this section, the inertial motions are very low. However, in cases with a realistically varying wind stress (section 4), the tidal and inertial motions can be of comparable magnitude.

The plot of KE^1/2, or the RMS tidal velocity, corresponding to an interval of t = (−3, 0) (see Fig. 1) provides information on the spatial structure of the internal tide not influenced by wind-driven flow (i.e., case TO). Zones of higher baroclinic tidal velocities are organized in beams that originate at topographic features with a bathymetric slope of dH/dx that is nearly critical (i.e., close to the slope of the internal wave characteristics; see, e.g., Gill 1982):

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Here, φ is the angle that the characteristics make with the horizontal, and N is the background buoyancy frequency. The beams are less steep at shallower depths, where the background stratification is stronger (Holloway 2001). Some internal tide energy propagates into the open ocean, and some onto the shelf, where the baroclinic tide variability is influenced by beams originated over several topographic features and reflected at the surface.

Another way to look at the baroclinic tide’s spatial variability is to plot M₂ barotropic and baroclinic tidal horizontal velocity ellipses (e.g., as shown over the shelf in Fig. 4), for the same interval, t = (−3, 0). The barotropic tidal current is approximately 0.02 m s⁻¹. Note that off the Oregon coast, the M₂ barotropic tide propagates as a Kelvin wave, with ellipses strongly polarized in the alongshore direction (see Erofeeva et al. 2003). These dynamics are not represented in our 2D cross-shore solution and the barotropic tidal ellipses, shown in Fig. 4, are not a realistic representation of the barotropic tide off Oregon. However, the structure of the baroclinic tide on the shelf is qualitatively similar to that found in the observations on the Oregon shelf (see Erofeeva et al. 2003; Kurapov et al. 2003), with weakly polarized ellipses, predominantly first mode structure (higher amplitudes and opposite phases near the surface and bottom), clockwise velocity vector rotation (when looking from the above), and magnitudes on the order of 0.1 m s⁻¹ or weaker. Because of the intermittency of the internal tide, harmonic analysis can underestimate the tidal velocity magnitudes. For that reason, we have also looked at the time series of u′_bc and υ′_bc at locations of intensified internal tides to verify that the harmonic estimates are consistent with the instantaneous high-pass-filtered velocity values (see Fig. 2).

The close-up plot of KE^1/2 in the shelf area, for the time interval t = (−3, 0), is shown in Fig. 5a. The maximum RMS baroclinic tidal velocities are close to 0.1 m s⁻¹ near the surface at a distance of 20–30 km from the coast, where the energetic beam that originated over the continental slope is reflected from the surface, and near the bottom on the midshelf (H < 100 m), where the beam is reflected upward and onshore, nearly parallel to the bottom slope. After the internal tide reaches a quasi-equilibrium state and before the wind forcing is applied (case TO, −3 < t < 0 days), isopycnal surfaces obtain approximately 5-m displacements that propagate onshore as internal waves and become amplified in the vicinity of the shelf slope (Fig. 6a).

The changes in the subtidal hydrographic conditions affect the baroclinic tidal KE (Figs. 5b–d). For instance, with the development of upwelling, the tidal KE is reduced at shallow depths where the tidal beam is reflected from the surface (in the vicinity of x = −25 km). The velocity time series analysis (see Fig. 2a) also shows a decrease in tidal amplitudes at that location associated with the onset of upwelling, although this effect is not uniformly extended to all near-surface locations (Fig. 2b). In both time series plots, irregular low-frequency modulations in the tidal velocity magnitudes are apparent. The reduction in the near-surface baroclinic tidal KE can be associated with increased dissipation in the surface boundary layer as well as changes in the level of baroclinic internal tide energy generated over the slope and propagated onto the shelf, as will be discussed in section 4.

b. Tidal effect on the alongshore transport

The internal tide influences the coastal dynamics on subinertial time scales. For instance, Figs. 7a–d shows cross-shore plots of the barotropic alongshore current υ averaged in time over 5, 10, 15, and 20 days. The speed of the jet core can be smaller in the TW case than in WO by as much as 5 cm s⁻¹ (or 10% of the maximum speed). Time series of subtidal 〈υ〉 at three shelf locations (Figs. 8a–c) also show a weaker current in the TW case throughout the study period. The reason for this reduction can be understood by analyzing the terms in the depth-integrated, low-pass-filtered alongshore momentum equation:

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where D = ζ + H is the total depth; τ^s and τ^b are the surface and bottom stress per unit mass, respectively; and subscripts denote partial derivatives. In both cases, τ^s is the same. The Coriolis term is small, since 〈Du〉 ≈ 0 near the coast. It might be anticipated that the subtidal bottom stress 〈τ^b〉 would be relatively larger in the TW case (Rooth 1972), causing a smaller negative tendency and a lower current in the upwelling spinup simulation. In Figs. 8d–f, we show, at the same three locations, time series of the tendency terms 〈Dυ〉_t and in Figs. 8g–i we show the bottom stress for the two cases, all normalized by the local H. Apparently, periods of a relatively larger bottom stress correlate with those of smaller southward acceleration. To emphasize this effect, in Figs. 8j–l, we plot the time series of the differences in the tendency terms, the WO case minus TW (solid line) and the difference in 〈τ^b〉, the TW case minus WO, all divided by local H. Indeed, these differences are positively correlated. Also, the magnitudes of the differences are comparable, suggesting that the increased bottom friction contributes to the slower current in the TW case. We will see later that the advection term can also contribute to the subtidal alongshore momentum balance (section 3c).

To further explore the effects of the internal tide on subtidal alongshore transport, we analyze vertical profiles of daily averaged υ, (∂υ/∂z)², and N ² at the same three shelf locations (Fig. 9). On the midshelf (profiles at 10 and 6 km from the coast), the difference in subtidal υ is mostly due to the differences in ∂υ/∂z near the bottom. As seen both in profiles of υ (Figs. 9a–d) and (∂υ/∂z)² (Figs. 9e–h), the magnitude of the vertical shear is larger at the vertical distance of 0–20 m from the bottom in the WO case (halftone lines) than in the TW case (black lines). At the same time, the vertical extent of the layer of increased shear (the depth of the bottom boundary layer) is larger in the TW case. Profiles of daily averaged N ² (Figs. 9i–l) are also influenced by the tides. In the WO case, the vertical stratification near bottom increases with the development of upwelling as an effect of the cross-shore Ekman buoyancy transport (Lentz and Trowbridge 1991; MacCready and Rhines 1993). In the TW case, this effect is distributed over a larger layer. In the lower (upper) part of that layer, the strength of the stratification is relatively weaker (stronger) in the TW case. These results suggest that tides introduce additional mixing and reduce stratification near the bottom. At the same time, the upslope Ekman buoyancy transport is distributed over a relatively thicker layer in the TW case. These conclusions are additionally supported by the analysis of profiles of the daily averaged turbulent kinetic energy (TKE) and the vertical eddy dissipation, K_M (Fig. 10).

c. Tidal effects on the cross-shore transport

To verify that the internal tide affects the cross-shore and vertical transport on the subtidal temporal scale, we compute the streamfunction ψ_bc of the baroclinic velocity field:

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To obtain time-averaged ψ_bc corresponding to subtidal flow, we compute the integral in (7) using u_bc at instantaneous depths of vertical levels. Every snapshot of ψ_bc is then interpolated to the neutral z levels, corresponding to ζ = 0, and daily averages of the interpolated values are computed.

These are shown for selected days in Fig. 11. Plots on the left correspond to the WO case and are similar to those in Allen et al. (1995). As upwelling progresses, the surface Ekman transport is distributed over a surface boundary layer of increasing thickness. The compensating onshore shelf transport is distributed over the entire water column during the early stages (Fig. 11a). At later stages of upwelling (Figs. 11b–d), the compensating onshore shelf transport becomes more and more concentrated in a thin boundary layer near the bottom. In the TW case (plots on the right in Fig. 11), the cross-shore and vertical transports are affected by the presence of the internal tide. In particular, in the near-bottom area on the midshelf, H < 100 m, where the internal tide is intensified, the subtidal onshore transport is distributed over a larger layer. Also, local changes in the time-averaged vertical transport are evident along the paths of energetic internal tide beams (particularly apparent between x = −30 and −10 km).

A closer look at streamfunctions for day 15, with contours shown at smaller intervals (Fig. 12), reveals differences in the cross-shelf transport in the mid- and inner-shelf areas (H < 100 m). For instance, in the TW case the streamfunction lines are relatively more closely spaced in the area −6 < x < −2 km at a distance of approximately 20 m from the bottom, indicating an increased cross-shore flow. To confirm this, profiles of daily averaged 〈u〉, WO (halftone) and TW (black) cases, are shown in Fig. 13a. For this analysis, instantaneous u was projected on neutral (stationary) vertical levels, low-pass filtered, and then averaged over a given time interval. In Fig. 13a, the increased onshore transport in the inner-shelf zone, in the TW case compared to WO, is compensated by the increased offshore transport near the surface. Figure 13b shows the integrated onshore transport, obtained by vertical integration of positive u segments of each profile in Fig. 13a. In the WO case, the value corresponding to the cross-shore Ekman transport (approximately 0.5 m² s⁻¹) is reached at x = −15 km. On average over the shelf area considered, the integrated onshore transport (and consequently the offshore transport) is 25% larger in the TW case than in WO.

The influence of internal tides on the subtidal cross-shore transport is clearly a nonlinear effect. Theoretical studies (e.g., Wunsch 1971) have suggested that the advection associated with baroclinic tidal currents can contribute to the wave-averaged momentum balance over the shelf slope, in the form of the Reynolds stress divergence. To verify the relative importance of this mechanism, we have utilized the ROMS diagnostics and computed the terms on the rhs of the u-momentum equation averaged over t = (10, 15) days (for the technical details of this term balance analysis, see the appendix). The first-order balance is between the Coriolis and pressure gradient terms. In the WO case, as shown in Fig. 14a for the inner shelf location (x = −3 km) and Fig. 14c for the midshelf location at x = −6 km, the ageostrophic term (an algebraic sum of the Coriolis and pressure gradient terms) is nearly 0 in the middle of the water column and is increased in the surface and bottom boundary layers where it is balanced by the vertical dissipation term. In this case the advection term is near 0. In the TW case, at the same locations (Figs. 14b and 14d), the advection term is not negligible throughout the water column. In the inner shelf area (Fig. 14b; x = −3 km) the advection term provides a sizable contribution to the cross-shore momentum balance near the surface, potentially influencing the cross-shore Ekman transport at that location (see Fig. 13). At the midshelf location (Fig. 14d; x = −6 km), the advection term is relatively small near the surface and bottom, although it is important in the middle of the water column.

Away from the boundary layers, the cross-shore advection term is balanced by the ageostrophic term over the entire shelf and the advection term contributes to the cross-shelf dynamics in the TW case, much more so than in case WO, as can be seen in a series of profiles of the advection term (halftone) and the combination of the ageostrophic and dissipation terms (black) in Fig. 15.

An estimate of the Reynolds stress divergence component of the total advection [see the appendix, especially (A1)] was obtained by postprocessing the model output, utilizing the same numerical scheme for advection as that in ROMS. We found that the total advection term in the TW case is almost entirely due to the Reynolds stress divergence.

It can also be instructive to look at alongshore momentum term balance profiles, at the same locations, x = −6 and −3 km, and averaging interval, t = (10, 15). In the WO case (Figs. 16a and 16b), advection plays an important role in the alongshore direction. In the upper part of the water column the southward jet acceleration (circles) is forced by the turbulent dissipation term. At x = −3 km, the advection term works against this force. Note that in this alongshore-uniform case ∂υ/∂y = 0 such that only the cross-shore and vertical terms contribute to the advection. The sign of the alongshore advection term in Fig. 16 is largely determined by the vertical component (−w∂υ/∂z > 0, since w > 0 and ∂υ/∂z < 0). Also note that since ∂p/∂y = 0, the ageostrophic term is identical to the Coriolis term. The tidal current yields quantitative changes in the term profiles throughout the water column (Figs. 16c and 16d). Below the surface boundary layer, modulations in the advection caused by the nonlinear tidal effect tend to be balanced by corresponding variations in the Coriolis term (−fu).

4. Cases with time variable wind stress

Computations with time-variable wind stress, using observed wind speed time series off the Oregon coast during summer 2001 (Kurapov et al. 2005) were performed to see how robust the results of the previous sections are with a more realistic wind forcing. This also allowed us to check if the intensity of the internal tide, and its effects on subinertial circulation, correlate with the occurrence of upwelling and relaxation events. The tidal forcing is the same as in the previous section. For brevity, we only discuss additional analyses complementary to the previous section.

Time is measured in days since the beginning of the year 2001. The TW case was started from rest at t = 127 days and tidally forced for a period of 10 days. At t = 137 days, the alongshore wind stress is applied, in addition to the barotropic tidal forcing, and is ramped up from 0 to the observed values over 2 days (Fig. 19a), while the heat flux is ramped up to a constant value of 150 W m⁻². The WO case is started from rest at t = 137 days, with the same surface forcing.

Time series of the depth-averaged alongshore velocity υ at locations 10, 6, and, 2 km from the coast are shown in Figs. 19b–d. The internal tide has a relatively larger impact during days 150–170. In particular, at x = −6 and −2 km, the current is reduced by nearly 0.1 m s⁻¹ (or by approximately 25%). As in the previous section, the reduction in υ is caused by the increased subtidal bottom stress in the TW case. However, after day 170, the effects of the tides on the subtidal alongshore current are diminished. Also, no apparent correlation between the wind stress and the tidal effect is apparent.

The decreased effects of the internal tide during the later stages of the computational experiment are clarified by analysis of the baroclinic tidal energy balance, based on linear equations [Kurapov et al. (2003), see Eq. (34)]. On average over the tidal period, the divergence in the depth-integrated baroclinic energy flux (F_int) is balanced by the topographic energy conversion (EC) and dissipation (DISS), the latter including contributions from bottom friction and turbulence throughout the water column:

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where

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where F(x, z) is the cross-shore component of the tidal horizontal baroclinic energy flux averaged over the tidal period, p_bc is the baroclinic (total minus depth averaged) tidal pressure, boldface symbols denote tidal complex amplitudes as before (see section 3a), and the dagger symbol denotes a complex conjugate. To compute p_bc, the high-pass-filtered potential density σ′_θ(x, z, t) is harmonically analyzed; the resulting complex amplitude σ_θ (x, z) is integrated in the vertical from z to 0 and multiplied by g to obtain the contribution of tidal density variations to the total pressure, and each vertical pressure profile is demeaned.

Figure 20 shows cross sections of the tidal baroclinic energy flux F(x, z) (top panels), the vertically integrated flux F_int (middle panels), and—dF_int/dx (black), EC (halftone), and the residual dF_int/dx—EC (red) (bottom panels). The residual is largely negative and can be associated with the term DISS in (8). Plots from left to right correspond to different intervals over which a harmonic analysis was performed. During the period when only tidal forcing was applied, t = (134, 137), the barotropic-to-baroclinic energy conversion occurred mainly over the continental slope, within the range of H = 400–1000 m (Fig. 20c). This conversion is balanced by the energy flux divergence term, with very little residual (dissipation). Approximately the same amount of energy is radiated in both directions, toward the open ocean and onto the shelf (Figs. 20a and 20b). On the shelf (x > −30 km, H < 200 m), the energy flux is decreased toward the coast, due to dissipation. During the period of increased tidal influence on the alongshore transport, t = (150, 160) (Figs. 20d–f), EC at the continental slope is reduced compared to the previously discussed interval, although a significant flux is still directed onshore. Additional topographic energy conversion occurs in the vicinity of the shallow bathymetric feature at x = −30 km. For the interval t = (180, 190) (Figs. 20g–i), when the tidal effect on the alongshore transport is reduced, the energy flux directed from the continental slope onshore is very low. The barotropic-to-baroclinic energy conversion over the shallow topographic feature near x = −30 km is larger than at the earlier time, increasing the onshore flux at depths H < 200 m, although only about half of the baroclinic tidal energy generated in that area is radiated and the rest is dissipated locally. We also note that the magnitude of F_int (see Figs. 20b, 20e, and 20h), ≤20 W m⁻¹, is comparable to the estimates obtained for the mid–Oregon shelf in a 3D model constrained by assimilation of HF radar surface currents (Kurapov et al. 2003).

Note that in the three time windows discussed in the previous paragraph, the magnitude of the onshore integrated flux is more affected by the background conditions than the offshore flux (see Figs. 20b, 20e, and 20h). Also note that the reduction in the integrated flux magnitudes at depths H > 200 m is not associated with dissipation, since in those areas the flux divergence is well balanced by EC. Without the wind forcing, the bathymetric slope at 400 < H < 1000 m is very close to critical, such that M₂ characteristic slopes (5) are nearly parallel to the bathymetric slope (see characteristic slope lines in Fig. 20a). With the development of upwelling, the near-bottom background stratification over the slope is increased such that the characteristic slopes become less steep (see Figs. 20d and 20g). The slope becomes increasingly supercritical, reducing the efficiency of the baroclinic energy conversion on this slope, and radiating a larger fraction of the available baroclinic energy into the open ocean. Note that in this analysis, the buoyancy frequency was computed using density values in the two lowest layers. Of course, in addition to this interpretation, other mechanisms can be suggested to explain the internal tide intermittency.

5. Summary

We have demonstrated that a moderate internal tide, with currents of 10–15 cm s⁻¹ amplitude, can influence alongshore and cross-shore transports on subinertial time scales, in part through modifications of the bottom boundary layer. The divergence in the tidally induced Reynolds stress contributes to the cross-shore momentum balance on the shelf and can affect the subtidal cross-shore transport in the middle of the water column. No simple relationship was found between wind stress fluctuations and tidal energetics or tidal effects on the subtidal flow.

Our study has confirmed that variability in the background conditions associated with wind-driven upwelling can cause substantial intermittency in the internal tide on the continental shelf. Changes in the background stratification over the continental slope associated with upwelling are sufficient to change the classification of the slope to subcritical, critical, or supercritical, providing control on the amount of barotropic-to-baroclinic energy conversion and the preferred direction of baroclinic energy flux (offshore or onshore).

The assumption that flows are alongshore uniform is a limitation of this idealized study. In particular, superinertial waves are essentially 3D features that can propagate at angles oblique to the coastline. Clearly, similar analyses with a high-resolution 3D model should be pursued. However, the effects of the internal tide in decreasing the subtidal alongshore current speed will be more difficult to analyze in the 3D setup because differences between the WO and TW solutions may result from alongshore pressure gradients and other alongshore variable conditions set up in the model interior. In the 2D formulation adopted here, such complications are eliminated and it is clearly demonstrated that the changes in the alongshore momentum are associated with increased bottom friction.

The intermittent internal tide is difficult to analyze using traditional oceanographic observations. For instance, inferences from a single mooring (or a sparse array of moorings) about the internal tide energetics, particularly its dependency on background ocean conditions, can be misleading since the zones of most intense activity move around. HF radar observations provide opportunities to obtain spatial maps of tidal surface velocities (O’Keefe 2005) and these can be utilized to verify model predictions or constrain the model estimates by means of data assimilation (Kurapov et al. 2003; Zaron et al. 2009). The modeling approach provides unique opportunities to understand the structure and dynamical role of the internal tide in the coastal ocean. To adequately represent the variability in the internal tide, these models must include realistically changing background conditions.