Editorial Type:
Article
Article Type:
Research Article

Energetics of Barotropic and Baroclinic Tides in the Monterey Bay Area

Dujuan Kang Department of Civil and Environmental Engineering, Stanford University, Stanford, California

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Oliver Fringer Department of Civil and Environmental Engineering, Stanford University, Stanford, California

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Print Publication:
01 Feb 2012
DOI:
https://doi.org/10.1175/JPO-D-11-039.1
Page(s):
272–290
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Abstract

A detailed energy analysis of the barotropic and baroclinic M2 tides in the Monterey Bay area is performed. The authors first derive a theoretical framework for analyzing internal tide energetics based on the complete form of the barotropic and baroclinic energy equations, which include the full nonlinear and nonhydrostatic energy flux contributions as well as an improved evaluation of the available potential energy. This approach is implemented in the Stanford Unstructured Nonhydrostatic Terrain-Following Adaptive Navier–Stokes Simulator (SUNTANS). Results from three-dimensional, high-resolution SUNTANS simulations are analyzed to estimate the tidal energy partitioning among generation, radiation, and dissipation. A 200 km × 230 km domain including all typical topographic features in this region is used to represent the Monterey Bay area. Of the 152-MW energy lost from the barotropic tide, approximately 133 MW (88%) is converted into baroclinic energy through internal tide generation, and 42% (56 MW) of this baroclinic energy radiates away into the open ocean. The tidal energy partitioning depends greatly on the topographic features. The Davidson Seamount is most efficient at baroclinic energy generation and radiation, whereas the Monterey Submarine Canyon acts as an energy sink. Energy flux contributions from nonlinear and nonhydrostatic effects are also examined. In the Monterey Bay area, the nonlinear and nonhydrostatic contributions are quite small. Moreover, the authors investigate the character of internal tide generation and find that in the Monterey Bay area the generated baroclinic tides are mainly linear and in the form of internal tidal beams. Comparison of the modeled tidal conversion to previous theoretical estimates shows that they are consistent with one another.

Corresponding author address: Dujuan Kang, Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305. E-mail: kangdj@stanford.edu

Abstract

A detailed energy analysis of the barotropic and baroclinic M2 tides in the Monterey Bay area is performed. The authors first derive a theoretical framework for analyzing internal tide energetics based on the complete form of the barotropic and baroclinic energy equations, which include the full nonlinear and nonhydrostatic energy flux contributions as well as an improved evaluation of the available potential energy. This approach is implemented in the Stanford Unstructured Nonhydrostatic Terrain-Following Adaptive Navier–Stokes Simulator (SUNTANS). Results from three-dimensional, high-resolution SUNTANS simulations are analyzed to estimate the tidal energy partitioning among generation, radiation, and dissipation. A 200 km × 230 km domain including all typical topographic features in this region is used to represent the Monterey Bay area. Of the 152-MW energy lost from the barotropic tide, approximately 133 MW (88%) is converted into baroclinic energy through internal tide generation, and 42% (56 MW) of this baroclinic energy radiates away into the open ocean. The tidal energy partitioning depends greatly on the topographic features. The Davidson Seamount is most efficient at baroclinic energy generation and radiation, whereas the Monterey Submarine Canyon acts as an energy sink. Energy flux contributions from nonlinear and nonhydrostatic effects are also examined. In the Monterey Bay area, the nonlinear and nonhydrostatic contributions are quite small. Moreover, the authors investigate the character of internal tide generation and find that in the Monterey Bay area the generated baroclinic tides are mainly linear and in the form of internal tidal beams. Comparison of the modeled tidal conversion to previous theoretical estimates shows that they are consistent with one another.

Corresponding author address: Dujuan Kang, Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305. E-mail: kangdj@stanford.edu
Keywords: Baroclinic flows; Barotropic flows; Internal waves; Nonhydrostatic models; Numerical analysis/modeling; Ocean models

1. Introduction

The ocean is a key component of the earth’s climate system, and mixing processes are critical in determining its distribution of salt, heat, and energy. The tides are one of the major sources of energy to mix the ocean. Studies have shown that 25%–30% of the global barotropic tidal energy is lost in the deep ocean (Munk and Wunsch 1998; Egbert and Ray 2000, 2001). Internal tides are believed to play an important role in transferring this energy into deep ocean turbulence (Fig. 1). When the barotropic tide flows over rough topographic features, a portion of the barotropic energy is lost directly through local dissipation and mixing, while the other portion is lost to the generation of internal [baroclinic)] tides. This generated baroclinic energy either dissipates locally or radiates into the open ocean.

Fig. 1.
Fig. 1.

Processes that transfer barotropic (BT) tidal energy to heat in the ocean. The associated energy distributions are indicated by the energy terms in the barotropic and baroclinic (BC) energy equations.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

In the past few years significant effort has been made to estimate regional internal tide energetics using numerical simulations. Regions studied include the northern British Columbia coast (Cummins and Oey 1997), the Hawaiian Ridge (Merrifield et al. 2001; Merrifield and Holloway 2002; Carter et al. 2008), the East China Sea (Niwa and Hibiya 2004), the Monterey Bay region (Jachec et al. 2006; Carter 2010; Hall and Carter 2011), and the Mid-Atlantic Ridge (Zilberman et al. 2009). All of these studies employed the hydrostatic Princeton Ocean Model (POM) (Blumberg and Mellor 1987), except for the work by Jachec et al., which employed the Stanford Unstructured Nonhydrostatic Terrain-Following Adaptive Navier–Stokes Simulator (SUNTANS) (Fringer et al. 2006). Some authors estimated the barotropic-to-baroclinic energy conversion using the hydrostatic portion of baroclinic energy flux divergence (Cummins and Oey 1997; Merrifield et al. 2001; Merrifield and Holloway 2002; Jachec et al. 2006), while others used the conversion term derived from barotropic and baroclinic energy equations (Niwa and Hibiya 2004; Carter et al. 2008).

Monterey Bay lies along the central U.S. West Coast. It consists of the prominent Monterey Submarine Canyon (MSC), numerous ridges, and smaller canyons to the north and south and a continental slope and break region (Fig. 2). This area is exposed to the large-scale and mesoscale variations of the California Current System as well as tidal currents. The complex bathymetry is favorable for internal tide generation, and energetic internal wave activity has been observed in the submarine canyon (Petruncio et al. 1998; Kunze et al. 2002; Carter and Gregg 2002). Jachec et al. (2006) performed simulations using the nonhydrostatic SUNTANS model to simulate internal tides in the Monterey Bay area. They determined that the Sur Platform region is the primary source for the M2 internal tidal energy flux observed within MSC. The total baroclinic energy generated within their domain (outlined by black lines in Fig. 2) is approximately 52 MW.

Fig. 2.
Fig. 2.

Bathymetry map of Monterey Bay and the surrounding open ocean. The location of station K is indicated by a black asterisk. The domain outside the white box indicates the area affected by the sponge layers in the simulations. The domain of Jachec et al. (2006) is outlined by a solid black line.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

In this paper, we extend the work of Jachec et al. and perform a detailed analysis of the energetics in the Monterey Bay area by understanding the relative contribution of different regions in the bay to the overall energetics. We provide a theoretical framework for accurate evaluation of the tidal energy flux budget by including nonlinear and nonhydrostatic contributions. We conduct numerical simulations of internal tides in the Monterey Bay area and estimate the tidal energy budget based on the theoretical framework. A brief derivation of the barotropic and baroclinic energy equations is presented in section 2. Subsequent sections focus on the numerical simulations, which include the model setup and validation in section 3 and the energetics analysis in section 4. Section 5 examines the generation characteristics and compares the model estimate with previous theoretical results. Finally, conclusions are summarized in section 6.

2. Theoretical framework

We derive the barotropic (BT) and baroclinic (BC) energy equations to study the energetics of barotropic and baroclinic tides. These equations provide a theoretical framework for the numerical evaluation of the tidal energy budget in subsequent sections. Here we provide a brief derivation of the equations. More detail is provided in Kang (2010). The derivation is based on the governing equations in SUNTANS, which simulates the three-dimensional Reynolds-averaged Navier–Stokes equations under the Boussinesq approximation, along with the density transport equation and the continuity equation,
e1
e2
and
e3
where u = (u, υ, w) is the velocity vector and Ω is the earth’s angular velocity vector; ν and κ, in units of meters squared per second, are the eddy viscosity and eddy diffusivity, respectively, and (···)H and (···)V are the horizontal and vertical components of a variable or operator. In SUNTANS, transport equations for temperature and salinity are solved individually, and a state equation is employed to compute density from temperature, salinity, and pressure (Millero et al. 1980; Millero and Poisson 1981). Here we only list the density transport equation (2) to make the derivation clearer. The total density is given by
e4
where ρ0 is the constant reference density, ρb is the background density, and ρ′ is the perturbation density due to wave motions. The pressure is split into its hydrostatic (ph) and nonhydrostatic (q) parts with p = ph + q, where the hydrostatic pressure can be further decomposed with
e5
in which η is the free surface elevation. The reference pressure p0 and the background pressure pb include the part due to barotropic heaving by free surface movement since they are integrated to the free surface in our definition. In SUNTANS, the nonhydrostatic pressure q is solved by employing the pressure correction method. In the computation, a three-dimensional Poisson equation for the nonhydrostatic pressure arises. This equation is the most computational demanding portion of a nonhydrostatic solver (Fringer et al. 2006).
To obtain the barotropic and baroclinic equations, we first split the velocity into its barotropic and baroclinic parts as u = U + u′. The barotropic velocities are defined as
e6
and
e7
where is the depth integration of a quantity from the bottom z = −d(x, y) to the surface z = η(x, y, t) and the total water depth is H = η + d. Based on the velocity decomposition, the kinetic energy density, in joules per cubic meter, is decomposed as , where
e8
e9
and
e10
Here Ek0 is the barotropic horizontal kinetic energy density; is the baroclinic kinetic energy density; and is the cross term, which vanishes upon depth integration. Following Gill (1982), the perturbation potential energy due to surface elevation, in units of joules per square meter, is given by
e11
The available potential energy (APE) density (joules per cubic meter) is defined as
e12
where ζ is the vertical displacement of a fluid particle due to wave motions. Kang and Fringer (2010) provided a clear graphical interpretation of this and two other APE definitions (their Fig. 1). This definition is an exact expression of the local APE because it computes the true active potential energy between the perturbed and unperturbed density profiles. It has been employed in analyzing internal wave energetics by numerous authors (Scotti et al. 2006; Lamb 2007; Lamb and Nguyen 2009; Kang and Fringer 2010).
Applying the variable decompositions and the boundary conditions, we obtain the depth-integrated barotropic and baroclinic energy equations as
e13
e14
where the depth-integrated barotropic and baroclinic energy flux terms, with the small unclosed terms neglected (see Kang 2010), are given by
e15
and
e16
where the contributions from energy advection, pressure work, and diffusion have been labeled. The barotropic-to-baroclinic conversion rate, the dissipation rates, and the bottom drag terms are given by
e17
e18
e19
e20
e21
e22
and
e23
where Cd is the bottom drag coefficient.
The time-averaged forms of Eqs. (13) and (14) are given by
e24
e25
where is the time average of a quantity over a time interval T. Therefore, net changes of the depth-integrated barotropic and baroclinic total energy are given by
e26
and
e27
For a periodic system with period T, and tend to zero and thus the first term in Eqs. (24) and (25) vanishes. The remaining terms describe the energy balance associated with tidal dissipation processes. The term represents the total barotropic energy available for conversion to baroclinic energy, represents the portion of the barotropic energy converted into baroclinic energy, and the term represents the portion of the converted baroclinic energy that radiates from the conversion site. Local dissipation occurs along with the conversion and radiation processes, and they are measured by the barotropic and baroclinic dissipation terms, respectively. Figure 1 illustrates the tidal energy budget using these terms.

This approach presents an exact measure of the barotropic-to-baroclinic tidal energy conversion and highlights its relation to the total convertible barotropic energy and the radiated baroclinic energy. The conversion term includes two parts representing the hydrostatic and nonhydrostatic contributions, respectively. Furthermore, we consider the contributions to the energy flux budget from the available potential energy flux and the kinetic energy fluxes due to nonlinear and nonhydrostatic effects. In many previous studies, only the dominant hydrostatic baroclinic energy term, , was calculated to represent the conversion (Cummins and Oey 1997; Merrifield et al. 2001; Merrifield and Holloway 2002; Jachec et al. 2006). Niwa and Hibiya (2004) evaluated the conversion using a similar term as , which is the hydrostatic part of our conversion term (17). Carter et al. (2008) derived barotropic and baroclinic equations from the POM hydrostatic governing equations and thus did not include the nonhydrostatic contribution. Their approach distinguishes between the conversion from the barotropic input and the baroclinic radiation. However, they used a linear APE definition, which may cause error in the presence of strongly nonlinear stratification (Kang and Fringer 2010). MacCready et al. (2009) provided the kinetic and potential energy equations for the hydrostatic Regional Ocean Modeling System (ROMS) (Haidvogel et al. 2000). They did not derive the barotropic and baroclinic energy equations individually and thus were unable to estimate the barotropic-to-baroclinic tidal conversion. However, they split the hydrostatic pressure into barotropic and baroclinic parts and therefore were able to evaluate the corresponding tidal energy flux budget. Moreover, their approach can estimate the contribution of the winds to the total energy budget because a wind stress term was included in their energy equations.

3. Numerical simulations

a. Model setup

We employ the SUNTANS model of Fringer et al. (2006) with modifications by Zhang et al. (2011) to incorporate high-resolution scalar advection. The simulation domain extends approximately 200 km north and south of Moss Landing and 400 km offshore (Fig. 2). The domain of Jachec et al. (2006) (~200 km alongshore × 90 km offshore) is outlined by black lines in Fig. 2. Compared to their domain, ours is larger and allows the evolution of offshore-propagating waves. The horizontal unstructured grid for our simulations is depicted in Fig. 3. The grid resolution smoothly transitions from roughly 80 m within the bay to 11 km along the offshore boundary. Approximately 60% of the grid cells have a resolution smaller than 1000 m, and 80% of the grid cells have a resolution smaller than 1600 m. In the vertical, there are 120 z levels with thickness stretching from roughly 6.6 m at the surface to 124 m in the deepest location, which provides better vertical resolution in the shallow regions. The vertical locations of grid centers are indicated by the black dots in Fig. 4. In total, the mesh consists of approximately six million grid cells in 3D.

Fig. 3.
Fig. 3.

(left) The unstructured grid of the computational domain and zoomed-in views for (middle) subdomain (a) indicated in Fig. 6 and (right) Monterey Bay. In left and middle plots, only cell centers are shown for clarify, while in the right plot cell edges are shown.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

Fig. 4.
Fig. 4.

Initial temperature and salinity profiles for the simulations. The black dots represent the vertical grid spacing in the simulations.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

The initial free surface and velocity field are initialized as quiescent throughout the domain. As shown in Fig. 4, the initial stratification is specified with horizontally homogeneous temperature and salinity profiles obtained from the 2006 Assessing the Effectiveness of Submesoscale Ocean Parameterizations (AESOP) field experiment (Terker et al. 2011, manuscript submitted to Cont. Shelf Res.). We use linear extrapolation to extend the observed profiles from 2000 m to 4800 m.

At the coastline, we apply the no-flow condition, whereas at the three open boundaries the barotropic velocities are specified as
e28
where ω is the M2 tidal frequency and the corresponding amplitude U0 and phase ϕ are obtained with the OTIS global tidal model (Egbert and Erofeeva 2002). Rather than employ a partially clamped condition (Blayo and Debreu 2005), we found that a clamped condition was suitable because the transient barotropic wave decays after two tidal cycles and little barotropic energy reflects off of the boundaries during the simulations. To prevent transient oscillations associated with impulsively starting the tidal forcing, the boundary velocities are spun up over a time scale of τr to approach the imposed forcing with
e29
where τr is set to one day in the simulations. Furthermore, at all three open boundaries a sponge layer is imposed to absorb the internal waves and minimize the reflection of baroclinic energy into the domain. Following Zhang (2010), the sponge layer is applied as a damping term on the right-hand side of the horizontal momentum equation of the form
e30
in which r is the distance to the closest open boundary and Ls is the width of the sponge layer. The damping time scale τs is determined with
e31
in which c is the internal wave speed and α is the decay rate. The first mode M2 internal wave speed in a depth of 3000 m is c1 = 1.98 m s−1 and the corresponding wavelength is approximately 88 km. In our simulations this value is used as the width of the sponge layer, which absorbs α = 99.99% of the first-mode baroclinic wave energy over a time scale of τs = 2750 s.

The simulation begins on 18 August 2006 (yearday 229) and is run for 18 semidiurnal M2 tidal cycles. A time step size of Δt = 18 s is used to ensure stability. In the simulation no turbulence model is employed. Diffusion of scalars is ignored by setting κH = κV = 0. A horizontal eddy viscosity of νH = 1 m2 s−1 and a vertical viscosity of νV = 5 × 10−3 m2 s−1 are applied uniformly throughout the domain. In a sensitivity test we found that the energetics are weakly affected by changing the viscosities by one order of magnitude (see the appendix). A constant bottom drag coefficient of CD = 0.0025 and a constant Coriolis frequency of f = 8.7 × 10−5 rad s−1 are specified. For the 18-M2-cycle simulation, the model runs in 1.25 wall-clock days, or 3830 CPU hours using 128 processors on the MJM Linux Networx Intel Xeon EM64T Cluster at the Department of Defense/Army Research Laboratory Supercomputing Resource Center (ARL DSRC).

b. Model validation

In previous numerical studies of internal tides in Monterey Bay, SUNTANS has shown a high level of skill in predicting the water surface and velocities in the canyon (Jachec et al. 2006; Jachec 2007). Here we do two more comparisons.

First, the sea surface predictions are compared with the observations at two water-level gauges (Monterey and Point Reyes) maintained by the National Oceanic and Atmospheric Administration. The water level data at these two gauges cover our simulation period (available at http://tidesandcurrents.noaa.gov/). We employ the T-TIDE package (Pawlowicz et al. 2002) to estimate the M2 surface amplitudes and phases. For the observations, 6-min NOAA data for 15–29 August 2006 are used for the harmonic analysis, while for the predictions 6-min model outputs of the last 10 M2 cycles are used. To quantitatively evaluate the M2 surface discrepancy between observations and model predictions, we compute the rms error,
e32
and the relative rms error,
e33
following previous studies (Cummins and Oey 1997; Carter et al. 2008; Carter 2010). Here subscripts o and m denote observed and model predicted amplitudes A and phases G. For this case, the absolute and relative rms errors are 0.37 cm and 0.8% at Monterey, whereas they are 2.1 cm and 3.9% at Point Reyes.

The model predictions are further compared with the field observations at a R/V FLIP station (station K) during the 2006 AESOP field experiment. The field data was provided by J. Klymak and R. Pinkel. Station K is located north of the Sur Platform, as indicated in Fig. 2. We examine the model performance in predicting the baroclinic features. Figure 5 compares the M2 baroclinic (total minus depth-averaged) velocity profiles between observations and model predictions at station K. An M2 bandpass filter is applied to the original observations to obtain the M2-fit baroclinic velocity profiles (Fig. 5a). We implement the M2 bandpass filtering in this way: the observed time series is first mapped to the frequency space via Fourier transform, then the amplitude of all frequencies are reduced to zero except for the band within 5% of M2, and finally an inverse Fourier transform is employed to obtain the M2-fit time series. Intermittent behavior can be detected in the observed baroclinic velocities. The intermittency of the internal tide has been attributed to the variability in background conditions (Manders et al. 2004; Kurapov et al. 2010). Station K is under the influence of wind-driven upwelling, which may cause variable background stratification and, hence, result in the observed tidal intermittency. The modeled baroclinic velocities do not show intermittent behavior because the large-scale and mesoscale wind-driven features are not considered in this simulation. Due to intermittency, spectral analysis can underestimate the tidal velocity magnitudes (Kurapov et al. 2010). This may explain why the strength of the M2-fit baroclinic velocities (Fig. 5a) are slightly weaker than the model predictions (Fig. 5b). The model predictions capture the vertical structure of the baroclinic velocities. The multimode feature can be seen in both field observations and model predictions, which indicates the existence of higher-mode baroclinic tides at this location. We finally compute the relative rms errors to quantitatively assess the differences between observed and modeled M2 baroclinic velocities. The average relative rms errors for u′ and υ′ are 69.4% and 34.6%, respectively.

Fig. 5.
Fig. 5.

Time series of the M2 baroclinic east–west (u′) and north–south (υ′) velocities at station K: (a) M2 filtered field observations and (b) SUNTANS predictions. The white areas indicate regions without observational data.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

4. Energetics

We evaluate the depth-integrated, time-averaged barotropic and baroclinic energy, (24) and (25), for the energy analysis in this section. They are averaged over the last six M2 tidal cycles of the 18-M2-cycle simulation. Because the system is periodic, the first term in Eqs. (24) and (25) tends to zero upon period averaging. We therefore obtain the balance relations
e34
and
e35
The model computes all the energy terms in the barotropic and baroclinic equations. In the following analysis, the conversion rate and the energy flux divergence terms, and are from direct model outputs. However, the barotropic and baroclinic dissipation rates are inferred from the above balance relations as in Niwa and Hibiya (2004). Carter et al. (2008) analyzed the domain-integrated energy budget in the same way. A set of sensitivity tests have shown that the conversion, divergence, and inferred dissipation terms are nearly independent of the dissipation parameters (νH, νV, and Cd), whereas the directly computed dissipation depends greatly on the dissipation parameters (see the appendix for details). In the simulations, the values of the diffusion parameters are not the realistic ocean values but instead are based on the stability requirements of numerical differencing. Moreover, in the real ocean the eddy viscosities and bottom drag are functions of location and time, whereas in the simulations we only apply constant values throughout the domain. Therefore, even though we are able to compute the dissipation directly, it may not truly represent the physical dissipation in the real ocean. For this reason, the inferred dissipation, which is nearly independent of the dissipation parameters, is used for the energy analysis in this section. Please note that this inferred dissipation includes both the physical portion and the numerical portion, as described in the appendix.

a. Horizontal structure

Figure 6 illustrates the horizontal distribution of the depth-integrated baroclinic energy flux vectors . Large fluxes are seen in the vicinity of four topographical features, which include a northern shelf–slope region (b), the MSC (c), the Sur Ridge/Platform region (d), and the Davidson Seamount (e). The energy budget within each subdomain is discussed in the next section. Figure 7 shows the horizontal distribution of the depth-integrated barotropic-to-baroclinic conversion rate . Red color represents positive energy conversion rate, which implies generation of internal tides, and the figure shows that most of the generation is contained within the 200-m and 3000-m isobaths. Negative energy conversion rate (blue color) represents energy transfer from the baroclinic tide to the barotropic tide. This is due to the phase difference between locally and remotely generated baroclinic tides and therefore indicates multiple generation sites (Zilberman et al. 2009). Significant negative conversion occurs within the MSC because the locally generated baroclinic tides interact with those generated at the North Sur Platform region. Large baroclinic energy can be seen radiating from North Sur Platform into the MSC following the canyon bathymetry (Fig. 6). This effect was also demonstrated by Jachec et al. (2006), Carter (2010), and Hall and Carter (2011). Figure 8 illustrates the divergence of the depth-integrated baroclinic energy flux . The difference between Figs. 7 and 8, which represents the baroclinic dissipation rate , is shown in Fig. 9. Large baroclinic energy dissipation occurs near the locations of strong internal tide generation.

Fig. 6.
Fig. 6.

Depth-integrated, period-averaged baroclinic energy flux, . The five boxes (a–e) indicate the subdomains in which energetics are investigated in detail. Bathymetry contours are spaced at −200, −500, −1000, −1500, −2000, −2500, −3000, and −3500 m.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

Fig. 7.
Fig. 7.

Depth-integrated, period-averaged barotropic-to-baroclinic conversion rate . Bathymetry contours are spaced at −200, −500, −1000, −1500, −2000, −2500, −3000, and −3500 m. Darker contours are −200 and −3000 m.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

Fig. 8.
Fig. 8.

Depth-integrated, period-averaged baroclinic energy flux divergence . Bathymetry contours are spaced at −500, −1000, −2000, −3000, and −3500 m.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

Fig. 9.
Fig. 9.

Depth-integrated, period-averaged baroclinic dissipation rate : bathymetry contour spacing as in Fig. 8.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

b. Energy flux budget

The total power within a region is obtained by area integrating the period-averaged and depth-integrated energy terms to give
e36
e37
e38
e39
e40
and
e41
in which ∑ implies summation of the grid cells within a particular region and ΔA is the area of each grid cell. We first study the energy distribution as a function of bottom depth. Each term in Eqs. (36)–(41) is summed over cells bounded by isobaths at increments of 200 m. The bottom panel of Fig. 10 compares the energy distribution in 200-m isobath bounded bins, while the top panel compares the cumulative sum. In the region shallower than 200 m only a small portion of the input barotropic (BT) energy is converted into baroclinic (BC) energy, and the negative baroclinic energy radiation implies that baroclinic energy generated in deeper regions flows into this shallow region and is then dissipated. In the region deeper than 2200 m, nearly all of the input barotropic energy is converted to baroclinic energy. The radiated baroclinic energy is large in the region between the 1400-m and 2600-m isobaths and is much smaller in other regions. Most of the barotropic energy dissipation occurs in the region shallower than 2000 m, while the baroclinic energy dissipation occurs at all depths, with two peaks near 1200 m and within 200 m. Figure 11 illustrates a schematic of the energy budget for the shelf and slope regions bounded by the 200-m and 3000-m isobaths. Barotropic energy is lost at a rate of 147 MW to the slope region, and approximately 87% of this energy is converted into baroclinic energy. Most of this generated baroclinic energy is dissipated locally, whereas the remaining portion (38%) is radiated. The shelf region acts as a baroclinic energy sink because it dissipates both the energy generated locally and the portion flowing into it from the slope region.
Fig. 10.
Fig. 10.

Energy distribution as a function of bottom depth: (bottom) energy terms (36)–(40) in 200-m isobath bounded bins and (top) cumulative sum.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

Fig. 11.
Fig. 11.

Schematic of the M2 tidal energy budget for the two subdomains bounded by the 0-m, 200-m, and 3000-m isobaths.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

We list the detailed M2 energy budget (Table 1) for the five subdomains (a)–(e) indicated in Fig. 6. Subdomain (a), a 200 km × 230 km domain, is used to represent the Monterey Bay area because it includes all typical topographic features in this area. For this area, approximately 133 MW (88%) of the 152 MW of barotropic energy is converted into baroclinic energy, and 56 MW (42%) of this generated baroclinic energy radiates away. The baroclinic dissipation (77 MW) is roughly four times as large as the barotropic dissipation (19 MW). The tidal energy budget depends strongly on topographic features as shown in Table 1. The Davidson Seamount and the northern shelf-break region are the most efficient topographic features to convert (~94%) barotropic energy into baroclinic energy and then let it radiate out into the open ocean (>70%). The Sur Platform region also converts a large portion (87.5%) and radiates about half of the barotropic energy as baroclinic energy. The MSC acts as an energy sink because it does not radiate energy but, instead, absorbs the baroclinic energy from the Sur Platform region (Fig. 6). In particular, the energy budget for the Davidson Seamount [subdomain (e)] is quite similar to that for the Hawaiian Islands by Carter et al. (2008), which shows that 85% of the barotropic energy is converted into baroclinic energy and 74% of this baroclinic energy radiates into the open ocean.

Table 1.

Semidiurnal M2 tidal energy budget (MW) for the five subdomains indicated in Fig. 6.

Table 1.

c. Energy flux contributions

As discussed in section 2, our method computes the full energy fluxes and thus allows us to compare the contributions of different components. Here we choose subdomain (a) as our study domain. The baroclinic energy radiation within this region is computed by Eq. (38) as . Based on Eq. (16), this baroclinic energy radiation has five contributions from kinetic energy advection , available potential energy advection , hydrostatic pressure work , nonhydrostatic pressure work , and energy diffusion . Table 2 compares these five contributions to the total baroclinic (BC) energy radiation in subdomain (a). The component due to hydrostatic pressure work (the traditional energy flux) is the dominant term. If we consider the total energy radiation as 100%, the hydrostatic contribution is ~101%, while the other terms account for the remaining −1%. The small advection and nonhydrostatic contributions imply that the internal tides in the Monterey Bay area are mainly linear and hydrostatic.

Table 2.

Contributions of different energy fluxes to the total baroclinic energy flux divergence for subdomain (a). The contributions are estimated in percentages.

Table 2.

Vitousek and Fringer (2011) show that the horizontal grid spacing must be smaller than roughly half of the water depth to begin to resolve nonhydrostatic effects. Therefore, although the contribution of the nonhydrostatic pressure in our simulations is small and may be on the same order as errors in evaluating the overall tidally averaged energy balance, the resolution we employ is sufficient to assess the role the nonhydrostatic pressure may play in the energetics. Figure 12 shows that the hydrostatic and nonhydrostatic energy fluxes oppose one another within MSC. This occurs because the effect of the nonhydrostatic pressure is to restrict the acceleration owing to the impact of vertical inertia. Hydrostatic models therefore tend to overpredict the energy flux, particularly for strongly nonhydrostatic flows.

Fig. 12.
Fig. 12.

Baroclinic energy flux contributions from (a) hydrostatic and (b) nonhydrostatic pressure work in the Monterey Submarine Canyon region. Bathymetry contours are spaced at −100, −500, −1000, −1500, and −2000 m.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

5. Generation characteristics

a. Parameter space

Four nondimensional parameters are generally employed to discuss the character of internal tide generation, which is governed by the dimensional parameters listed in Table 3. The first parameter is the steepness parameter defined by
e42
where the topographic slope is given by
e43
and the internal wave characteristic slope is given by
e44
The steepness parameter is used to distinguish between subcritical (ϵ1 < 1) and supercritical topography (ϵ1 > 1). The topography is referred to as critical when ϵ1 = 1. Many studies have been carried out to understand the barotropic-to-baroclinic conversion over subcritical (Bell 1975; Llewellyn Smith and Young 2002; Balmforth et al. 2002) and supercritical topography (Khatiwala 2003; St. Laurent et al. 2003; Balmforth and Peacock 2009).
Table 3.

Key physical parameters governing internal tide generation and propagation.

Table 3.
The second nondimensional parameter is the tidal excursion parameter defined by
e45
which measures the ratio of the tidal excursion U0/ω to the horizontal scale of the topography . Here U0 is simply the maximum magnitude of the local barotropic flow. A better way might be to include the angle of the principle axis of the barotropic tidal ellipse with respect to the topography, but this produced results that were too noisy and therefore we just keep U0. In general, the excursion parameter is computed for a single ridge or shelf slope. However, since the local topographic scale is not generally known, we estimated it with kb = γ/h0 following Garrett and Kunze (2007), where h0 = H0 − d is the topographic amplitude. In this study, the reference depth H0 is defined by the maximum water depth in the studied domain. The excursion parameter is used to examine the nonlinearity of the waves (Balmforth et al. 2002; St. Laurent and Garrett 2002; Vlasenko et al. 2005; Legg and Huijts 2006; Garrett and Kunze 2007). When ϵ2 ≪ 1, linear internal tides are generated mainly at the forcing frequency ω. Over subcritical topography (ϵ1 < 1), most of the energy generation is in the first-mode internal tide, whereas, over critical or supercritical topography (ϵ1 ≥ 1), higher modes are generated and their superposition creates internal tidal beams. At intermediate excursion (ϵ2 ~ 1), nonlinearity becomes important, and nonlinear internal wave bores, weak unsteady lee waves, and solitary internal waves may be generated depending on the topographic features. It has been found in numerical models that lee waves are also generated for ϵ2 < 1 over tall and steep topography (Legg and Huijts 2006; Legg and Klymak 2008; Buijsman et al. 2010). When ϵ2 > 1, in addition to bores and solitary internal waves, strong unsteady lee waves may form (Vlasenko et al. 2005).
The third nondimensional parameter is the ratio of topographic amplitude to the total water depth, given by
e46
This parameter has been used to investigate the role of finite ocean depth on internal wave generation (Llewellyn Smith and Young 2002; Khatiwala 2003; St. Laurent et al. 2003).
The last nondimensional parameter is the Froude number, defined by
e47
which has been used to examine the obstruction of the flow by the topography (Legg and Huijts 2006; Legg and Klymak 2008). The Froude number and the first two nondimensional parameters are related to one another via
e48
Based on (44), in the midfrequency case (f ≪ ω ≪ N) or the nonrotating and hydrostatic case (f = 0 and k ≪ m), we have k/m = ω/N and hence ϵ2/ϵ1 = Fr. This relation was demonstrated in the parameter space diagram of Garrett and Kunze (2007).

b. Energy distribution versus parameters

Although internal wave generation is a complex process, we can summarize the behavior of the internal wave generation in Monterey Bay by plotting histograms of the conversion and divergence terms as functions of the criticality and excursion parameters. Here we compute the two parameters throughout subdomain (a). Figures 13a,b demonstrate the distribution of conversion and BC radiation as a function of the nondimensional parameters ϵ1 and ϵ2. The energy terms (37) and (38) as functions of the parameters are computed as the total sum of the energy within 20 bins. As seen in Fig. 13, barotropic-to-baroclinic conversion (green bins) occurs predominantly in regions within which ϵ1 = γ/s < 5 and ϵ2 = U0kb/ω < 0.06. Under these conditions, baroclinic tides generated in this region are mainly linear and in the form of internal tidal beams (St. Laurent and Garrett 2002; Vlasenko et al. 2005; Garrett and Kunze 2007).

Fig. 13.
Fig. 13.

Distribution of the conversion and baroclinic (BC) radiation as a function of (a) ϵ1 = γ/s and (b) ϵ2 = U0kb/ω for subdomain (a) in Fig. 6.

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

As expected, conversion of barotropic energy into baroclinic energy peaks for critical topography near ϵ1 ~ 1, as shown in Fig. 13a. More interesting, however, is that there is also a peak in conversion for a particular value of ϵ2 ~ 0.005, as shown in Fig. 13b. A peak in the conversion for a particular value of ϵ2 implies that narrow ridges relative to the tidal excursion (i.e., large ϵ2) do not efficiently convert barotropic energy, and wide ridges are weak converters as well. This may arise because the dissipative effects that are not included in the parameter space strongly impact the generation physics for small and large values of ϵ2 but play a minimal role for the optimal excursion parameter. One possible explanation is that narrow topography with respect to the tidal excursion (large ϵ2) is relatively rough and so dissipates too much energy and is a poor converter of barotropic energy. The same may be true for relatively wide topography (small ϵ2) in that dissipative mechanisms dominate when the topography over which the internal wave action occurs is extensive, even if the associated currents are weak. This behavior was observed by Venayagamoorthy and Fringer (2007) in idealized simulations of internal waves interacting with slopes. They showed that there was an excursion parameter that optimized on-slope internal wave energy propagation. Too small an excursion parameter led to extensive bottom friction relative to the weak internal waves, whereas a too large excursion parameter led to extensive turbulent dissipation and mixing. Although the present simulations do not directly simulate these dissipative and turbulent processes, the model may be accounting for them through elevated numerical dissipation or simply via bottom friction due to the quadratic drag law. This is highlighted by the fact that the BC radiation terms are always negative for small values of ϵ1 and ϵ2. In these regimes, the baroclinic energy is being lost to (modeled) dissipation and mixing.

c. Comparison to linear theory

A number of theoretical estimates of tidal energy conversion have been presented in previous studies (Bell 1975; Llewellyn Smith and Young 2002; Balmforth et al. 2002; St. Laurent et al. 2003; Balmforth and Peacock 2009). Using linear theory, Bell examined the tidal conversion by two-dimensional topography in the limits of ϵ1 ≪ 1 and δ ≪ 1. Considering an ocean of constant N and an isolated ridge represented by the witch of Agnesi profile,
e49
Bell’s estimate of the tidal conversion, in units of watts, is given by
e50
where L is the length of the topography. Note that CBell is independent of the width b of the ridge. Llewellyn Smith and Young (2002) extended Bell’s result to an ocean of finite depth but assumed ϵ1 ≪ 1 and ϵ2 ≪ 1. For the same ridge, they estimated the tidal conversion (watts) as
e51
where . Balmforth et al. (2002) kept the infinite depth assumption (δ ≪ 1) but extended the small slope limit (ϵ1 ≪ 1) to the full subcritical range (0 ≤ ϵ1 < 1). As ϵ1 increases, the conversion rate is enhanced smoothly and modestly above Bell’s estimate. As ϵ1 → 1, the enhancement is 56% for sinusoidal topography, whereas the enhancement is only 14% for a Gaussian bump and roughly 6% for random topography. St. Laurent et al. (2003) further examined the tidal conversion at abrupt topography. For a knife-edge ridge (ϵ1 → ∞), their estimate of the tidal conversion (watts) is given by
e52
in which n is the vertical mode number; an are obtained by solving the matrix problem, Amnan = cm. The formation of matrix and vector c is given by Eqs. (17)–(20) in St. Laurent et al. (2003).

We compare our simulated tidal energy conversion to the theoretical estimates (50)–(52). The Davidson Seamount is a relatively isolated, ridgelike feature (ϵ1 = 2) with its peak ~1500 m below the sea surface. It has a length scale of roughly L = 40 km and an amplitude of h0 = 1500 m (δ = 0.5). The mean barotropic tidal current over this region is approximately 3 cm s−1. At depths of the Davidson Seamount the ocean stratification is close to linear, which results in a nearly constant N = 1 × 10−3 s−1. Using these numbers and constant values of ρ0 = 1000 kg m−3, rad s−1, and f = 8.7 × 10−5 rad s−1, we obtain Bell’s estimate from Eq. (50) as CBell = 24.98 MW and the Llewellyn Smith and Young estimate from Eq. (51) as CLSY = 22.92 MW. The estimate of St. Laurent et al. is obtained from Eq. (52) as CStL = 37.02 MW for 1000 vertical modes. These theoretical results and our model estimate (28.24 MW) are of the same order of magnitude. The first two theoretical estimates are derived in the limit of ϵ1 ≪ 1 and therefore underestimate the supercritical conversion by the Davidson Seamount based on the studies of Balmforth et al. (2002) and St. Laurent et al. (2003). The third theoretical estimate is derived in the limit of ϵ1 → ∞ and thus may overestimate the conversion by the Davidson Seamount.

We do not compare the model and theoretical estimates for the entire domain because δ → 1 on the shelf slope and the stratification of the upper ocean is strongly nonlinear (i.e., N increases by an order of magnitude from a depth of 1500 m to a depth of 25 m). With such variable N, finite ϵ1, and δ → 1, it is difficult to estimate the energy conversion using Eqs. (50)–(52) (Garrett and Kunze 2007).

6. Summary and discussion

We have performed a detailed energy analysis of the barotropic and baroclinic M2 tides in the Monterey Bay area. A theoretical framework for analyzing internal tide energetics is derived based on the complete form of the barotropic and baroclinic energy equations. These equations provide a more accurate and detailed energy analysis because they include the full nonlinear and nonhydrostatic energy flux contributions as well as an improved evaluation of the available potential energy. Three-dimensional, high-resolution simulations of the barotropic and baroclinic tides in the Monterey Bay area are conducted using the hydrodynamic SUNTANS model. Based on the theoretical approach, model results are analyzed to address the question of how the barotropic tidal energy is partitioned between local barotropic dissipation and local generation of baroclinic energy, and then how much of this generated baroclinic energy is lost locally versus how much is radiated away for open-ocean mixing. Subdomain (a), a 200 km × 230 km domain, is used to represent the Monterey Bay area because it includes all typical topographic features in this region. Of the 152-MW energy lost from the barotropic tide, approximately 133 MW (88%) is converted into baroclinic energy through internal tide generation, and 42% (56 MW) of this baroclinic energy radiates away for open-ocean mixing (Fig. 14). The tidal energy partitioning depends greatly on the topographic features. The Davidson Seamount and northern shelf-break region are most efficient at baroclinic energy generation (~94%) and radiation (>70%). The Sur Platform region converts a large portion (~88%) and radiates roughly half of the barotropic energy as baroclinic energy. The MSC acts as an energy sink because it does not radiate energy but instead absorbs the baroclinic energy from the Sur Platform region. The energy flux contributions from nonlinear and nonhydrostatic effects are also examined. The small advection and nonhydrostatic contributions imply that the internal tides in the Monterey Bay area are predominantly linear and hydrostatic.

Fig. 14.
Fig. 14.

Schematic of the M2 tidal energy budget in percentages for subdomain (a) in Fig. 6. The bold percentages are relative to the total input barotropic energy (BT), and the thin italic percentages are relative to the generated baroclinic energy (BC).

Citation: Journal of Physical Oceanography 42, 2; 10.1175/JPO-D-11-039.1

We also investigate the character of internal tide generation by examining the energy distribution as a function of two nondimensional parameters: namely, the steepness parameter (ϵ1 = γ/s) and the excursion parameter (ϵ2 = U0kb/ω). The generation mainly occurs in the regions satisfying ϵ1 < 5 and ϵ2 < 0.06, indicating that baroclinic tides generated in the Monterey Bay area are mainly linear and in the form of internal tidal beams. The results highlight how description of the conversion process with simple nondimensional parameters produces results that are consistent with theory, in that internal wave energy generation peaks at critical topography (ϵ1 ~ 1). The results also indicate that conversion peaks for a particular excursion parameter (ϵ2 ~ 0.005 for this case). This implies that it may be possible to parameterize conversion of barotropic to baroclinic energy in barotropic models with knowledge of ϵ1 and ϵ2. For example, a parameterization of internal wave generation based on the steepness parameter has been widely used in global barotropic tidal models (Jayne and St. Laurent 2001; St. Laurent et al. 2002) and ocean general circulation models (Simmons et al. 2004; Jayne 2009).

To demonstrate that parameterizations of internal wave energy generation produce results that are valid even in complex domains with complex topography and tidal currents, we compare the model estimate of the barotropic-to-baroclinic conversion with three theoretical results. The Davidson Seamount (ϵ1 = 2) is chosen as a comparison region because it mostly satisfies the assumptions under which the theoretical estimates were derived. The theoretical and model estimates are of the same order of magnitude. The model estimate is slightly larger than the first two theoretical estimates that were derived in the limit of ϵ1 ≪ 1 (Bell 1975; Llewellyn Smith and Young 2002), whereas it is smaller than the third theoretical estimate that was derived in the limit of ϵ1 → ∞ (St. Laurent et al. 2003). Moreover, the energy budget for the Davidson Seamount is quite similar to that for the Hawaiian Islands in a previous model study by Carter et al. (2008). They showed that the Hawaiian Ridge converts 85% of the barotropic energy into baroclinic energy and then radiates 74% of this baroclinic energy into the open ocean. In our study, the Davidson Seamount converts 95% of the barotropic energy into baroclinic energy and then radiates 81% of this baroclinic energy away for open-ocean mixing.

This work outlines a systematic approach to analyze internal tide energetics and estimate tidal energy budget regionally and globally. The results draw a picture of how the M2 tidal energy is distributed in the Monterey Bay region. However, the simulation is limited to only one stratification taken in late summer 2006. The stratification is specified as horizontally uniform throughout the domain as an initial condition. Earlier internal tide observations (Petruncio et al. 1998) and simulations (Rosenfeld et al. 2009; Wang et al. 2009) indicate that the internal tides are sensitive to stratification. The Monterey Bay area is exposed to the large-scale California Current System and mesoscale eddies and upwelling. The seasonally varying dynamics may affect the stratification and thus the generation and propagation of internal tides in this area. Therefore, it may be necessary to consider seasonal effects of stratification and to include mesoscale effects by coupling with a larger-scale regional model such as ROMS (Haidvogel et al. 2000; Shchepetkin and McWilliams 2005).

Acknowledgments

The authors gratefully acknowledge the support of ONR Grant N00014-05-1-0294 (Scientific officers: Drs. C. Linwood Vincent, Terri Paluszkiewicz, and Scott Harper). We thank Samantha Terker, Drs. Jody Klymak, Robert Pinkel, James Girton, and Eric Kunze for kindly providing the field data. The helpful discussions with Drs. Robert Street, Stephen Monismith, Leif Thomas, and Rocky Geyer are greatly appreciated. We also thank Dr. Steven Jachec for useful help with simulation setup. Comments and suggestions from four anonymous reviewers greatly helped to improve the manuscript.

APPENDIX

Inferred versus Modeled Dissipation

A set of M2-forced numerical simulations are carried out to examine the sensitivity of model results to the dissipation parameters. The reference simulation (simulation 0) is that discussed in section 4, which employs the constant eddy viscosities (νH = 1 m2 s−1 and νV = 5 × 10−3 m2 s−1) and a constant bottom drag coefficient (Cd = 0.0025). In simulations 1–6, the value of νH, νV, or Cd is changed by one order of magnitude. For each simulation, we calculate the domain-integrated, period-averaged baroclinic energy terms for subdomain (a) in Fig. 6. They are the barotropic-to-baroclinic conversion [(37)], the baroclinic radiation [(38)], the inferred baroclinic dissipation [(40)], and the directly computed baroclinic dissipation . Detailed comparisons are listed in Table A1.

Table A1.

Sensitivity test of dissipation parameters.

Table A1.

If we consider the effect of numerical error, the period-averaged baroclinic energy Eq. (25) gives the following balance relation:
ea1
where numerical dissipation results primarily from the numerical diffusion associated with scalar and momentum advection and error arises from the assumption of periodicity. For a periodic system, error should tend to zero. This is what we saw in the last six M2 tidal cycles of the simulations. Table A1 shows that changing the dissipation parameters by one order of magnitude has a minor effect on conversion, radiation, and hence the inferred dissipation. However, the directly computed dissipation (or modeled dissipation) depends greatly on the dissipation parameters. For example, increasing/decreasing νV by one order of magnitude roughly doubles/halves the modeled dissipation. Similarly, increasing/decreasing Cd by one order of magnitude causes a 74% increase/9% decrease in modeled dissipation. Interestingly, an increase/decrease of νH results in a decrease/increase in modeled dissipation. This may imply that increasing νH smears out gradients so that the impact is to reduce νH ∇Hu′ · ∇Hu′ since ∇Hu′ decreases by more than νH increases. Overall, this set of sensitivity tests show that the inferred dissipation is nearly a constant, whereas the modeled dissipation is a strong function of the dissipation parameters. Based on the balance relation (A1), the numerical dissipation also depends greatly on the dissipation parameters.

In the numerical simulations, the values of the diffusion parameters are set by the stability requirements of numerical differencing and are not necessarily the realistic ocean values. Moreover, in the real ocean the eddy viscosities and bottom drag are functions of location and time, whereas in the simulations we only apply a constant value for each throughout the domain. Therefore, although we are able to compute the dissipation directly, it may not represent the true physical dissipation in the real ocean. For this reason, the inferred dissipation, which is virtually independent of the dissipation parameters, is used for the energy analysis in section 4.

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  • Zhang, Z., O. B. Fringer, and S. R. Ramp, 2011: Three-dimensional, nonhydrostatic numerical simulation of nonlinear internal wave generation and propagation in the South China Sea. J. Geophys. Res., 116, C05022, doi:10.1029/2010JC006424.

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  • Zilberman, N. V., J. M. Becker, M. A. Merrifield, and G. S. Carter, 2009: Model estimates of M2 internal tide generation over mid-Atlantic Ridge topography. J. Phys. Oceanogr., 39, 2635–2651.

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Processes that transfer barotropic (BT) tidal energy to heat in the ocean. The associated energy distributions are indicated by the energy terms in the barotropic and baroclinic (BC) energy equations.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Bathymetry map of Monterey Bay and the surrounding open ocean. The location of station K is indicated by a black asterisk. The domain outside the white box indicates the area affected by the sponge layers in the simulations. The domain of Jachec et al. (2006) is outlined by a solid black line.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    (left) The unstructured grid of the computational domain and zoomed-in views for (middle) subdomain (a) indicated in Fig. 6 and (right) Monterey Bay. In left and middle plots, only cell centers are shown for clarify, while in the right plot cell edges are shown.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Initial temperature and salinity profiles for the simulations. The black dots represent the vertical grid spacing in the simulations.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Time series of the M2 baroclinic east–west (u′) and north–south (υ′) velocities at station K: (a) M2 filtered field observations and (b) SUNTANS predictions. The white areas indicate regions without observational data.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Depth-integrated, period-averaged baroclinic energy flux, . The five boxes (a–e) indicate the subdomains in which energetics are investigated in detail. Bathymetry contours are spaced at −200, −500, −1000, −1500, −2000, −2500, −3000, and −3500 m.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Depth-integrated, period-averaged barotropic-to-baroclinic conversion rate . Bathymetry contours are spaced at −200, −500, −1000, −1500, −2000, −2500, −3000, and −3500 m. Darker contours are −200 and −3000 m.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Depth-integrated, period-averaged baroclinic energy flux divergence . Bathymetry contours are spaced at −500, −1000, −2000, −3000, and −3500 m.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Depth-integrated, period-averaged baroclinic dissipation rate : bathymetry contour spacing as in Fig. 8.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Energy distribution as a function of bottom depth: (bottom) energy terms (36)–(40) in 200-m isobath bounded bins and (top) cumulative sum.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Schematic of the M2 tidal energy budget for the two subdomains bounded by the 0-m, 200-m, and 3000-m isobaths.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Baroclinic energy flux contributions from (a) hydrostatic and (b) nonhydrostatic pressure work in the Monterey Submarine Canyon region. Bathymetry contours are spaced at −100, −500, −1000, −1500, and −2000 m.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    Distribution of the conversion and baroclinic (BC) radiation as a function of (a) ϵ1 = γ/s and (b) ϵ2 = U0kb/ω for subdomain (a) in Fig. 6.

  • <sc>Fig</sc>. 14.
    View in gallery
    Fig. 14.

    Schematic of the M2 tidal energy budget in percentages for subdomain (a) in Fig. 6. The bold percentages are relative to the total input barotropic energy (BT), and the thin italic percentages are relative to the generated baroclinic energy (BC).

Fig. 1.

Processes that transfer barotropic (BT) tidal energy to heat in the ocean. The associated energy distributions are indicated by the energy terms in the barotropic and baroclinic (BC) energy equations.
Fig. 2.

Bathymetry map of Monterey Bay and the surrounding open ocean. The location of station K is indicated by a black asterisk. The domain outside the white box indicates the area affected by the sponge layers in the simulations. The domain of Jachec et al. (2006) is outlined by a solid black line.
Fig. 3.

(left) The unstructured grid of the computational domain and zoomed-in views for (middle) subdomain (a) indicated in Fig. 6 and (right) Monterey Bay. In left and middle plots, only cell centers are shown for clarify, while in the right plot cell edges are shown.
Fig. 4.

Initial temperature and salinity profiles for the simulations. The black dots represent the vertical grid spacing in the simulations.
Fig. 5.

Time series of the M2 baroclinic east–west (u′) and north–south (υ′) velocities at station K: (a) M2 filtered field observations and (b) SUNTANS predictions. The white areas indicate regions without observational data.
Fig. 6.

Depth-integrated, period-averaged baroclinic energy flux, . The five boxes (a–e) indicate the subdomains in which energetics are investigated in detail. Bathymetry contours are spaced at −200, −500, −1000, −1500, −2000, −2500, −3000, and −3500 m.
Fig. 7.

Depth-integrated, period-averaged barotropic-to-baroclinic conversion rate . Bathymetry contours are spaced at −200, −500, −1000, −1500, −2000, −2500, −3000, and −3500 m. Darker contours are −200 and −3000 m.
Fig. 8.

Depth-integrated, period-averaged baroclinic energy flux divergence . Bathymetry contours are spaced at −500, −1000, −2000, −3000, and −3500 m.
Fig. 9.

Depth-integrated, period-averaged baroclinic dissipation rate : bathymetry contour spacing as in Fig. 8.
Fig. 10.

Energy distribution as a function of bottom depth: (bottom) energy terms (36)–(40) in 200-m isobath bounded bins and (top) cumulative sum.
Fig. 11.

Schematic of the M2 tidal energy budget for the two subdomains bounded by the 0-m, 200-m, and 3000-m isobaths.
Fig. 12.

Baroclinic energy flux contributions from (a) hydrostatic and (b) nonhydrostatic pressure work in the Monterey Submarine Canyon region. Bathymetry contours are spaced at −100, −500, −1000, −1500, and −2000 m.
Fig. 13.

Distribution of the conversion and baroclinic (BC) radiation as a function of (a) ϵ1 = γ/s and (b) ϵ2 = U0kb/ω for subdomain (a) in Fig. 6.
Fig. 14.

Schematic of the M2 tidal energy budget in percentages for subdomain (a) in Fig. 6. The bold percentages are relative to the total input barotropic energy (BT), and the thin italic percentages are relative to the generated baroclinic energy (BC).
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