Saturation of the Internal Tide over the Inner Continental Shelf. Part II: Parameterization

Johannes Becherer aCollege of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon
bInstitute of Coastal Research, Helmholtz-Zentrum Hereon, Geesthacht, Germany

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James N. Moum aCollege of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Joseph Calantoni cOcean Sciences Division, U.S. Naval Research Laboratory, Stennis Space Center, Mississippi

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John A. Colosi dDepartment of Oceanography, Naval Postgraduate School, Monterey Bay, California

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John A. Barth aCollege of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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James A. Lerczak aCollege of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Jacqueline M. McSweeney aCollege of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Jennifer A. MacKinnon eScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Amy F. Waterhouse eScripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Abstract

Here, we develop a framework for understanding the observations presented in Part I. In this framework, the internal tide saturates as it shoals as a result of amplitude limitation with decreasing water depth H. From this framework evolves estimates of averaged energetics of the internal tide; specifically, energy ⟨APE⟩, energy flux ⟨FE⟩, and energy flux divergence ∂xFE⟩. Since we observe that dissipation ⟨D⟩ ≈ ∂xFE⟩, we also interpret our estimate of ∂xFE⟩ as ⟨D⟩. These estimates represent a parameterization of the energy in the internal tide as it saturates over the inner continental shelf. The parameterization depends solely on depth-mean stratification and bathymetry. A summary result is that the cross-shelf depth dependencies of ⟨APE⟩, ⟨FE⟩, and ∂xFE⟩ are analogous to those for shoaling surface gravity waves in the surf zone, suggesting that the inner shelf is the surf zone for the internal tide. A test of our simple parameterization against a range of datasets suggests that it is broadly applicable.

Significance Statement

Observational results from Part I suggest a new approach to understanding the complicated breakdown of the internal tide over continental shelves. By defocusing on the details in favor of focusing on the integral energy of the internal tide, we develop a new framework analogous to breaking surface waves on beaches. This framework leads to a simple parameterization that depends solely on the surface-to-seafloor temperature difference and the bathymetry of the continental shelf. A test of this parameterization against a range of datasets shows broad applicability. At the same time, it provides a means of determining the net mixing of continental shelf waters and suggests that the inner continental shelf is the surf zone for the internal tide.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Johannes Becherer, johannes.becherer@hereon.de

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JPO-D-20-0264.1.

Abstract

Here, we develop a framework for understanding the observations presented in Part I. In this framework, the internal tide saturates as it shoals as a result of amplitude limitation with decreasing water depth H. From this framework evolves estimates of averaged energetics of the internal tide; specifically, energy ⟨APE⟩, energy flux ⟨FE⟩, and energy flux divergence ∂xFE⟩. Since we observe that dissipation ⟨D⟩ ≈ ∂xFE⟩, we also interpret our estimate of ∂xFE⟩ as ⟨D⟩. These estimates represent a parameterization of the energy in the internal tide as it saturates over the inner continental shelf. The parameterization depends solely on depth-mean stratification and bathymetry. A summary result is that the cross-shelf depth dependencies of ⟨APE⟩, ⟨FE⟩, and ∂xFE⟩ are analogous to those for shoaling surface gravity waves in the surf zone, suggesting that the inner shelf is the surf zone for the internal tide. A test of our simple parameterization against a range of datasets suggests that it is broadly applicable.

Significance Statement

Observational results from Part I suggest a new approach to understanding the complicated breakdown of the internal tide over continental shelves. By defocusing on the details in favor of focusing on the integral energy of the internal tide, we develop a new framework analogous to breaking surface waves on beaches. This framework leads to a simple parameterization that depends solely on the surface-to-seafloor temperature difference and the bathymetry of the continental shelf. A test of this parameterization against a range of datasets shows broad applicability. At the same time, it provides a means of determining the net mixing of continental shelf waters and suggests that the inner continental shelf is the surf zone for the internal tide.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Johannes Becherer, johannes.becherer@hereon.de

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JPO-D-20-0264.1.

1. Introduction

The inner continental shelf, the transition zone between midshelf and surf zone, can be characterized as the region where surface and bottom boundary layer overlap (Lentz 1994). Alternatively, it is the region where cross-shore winds significantly contribute to cross-shelf transport; this is in contrast to the unbound Ekman dynamics of the midshelf (Fewings et al. 2008). However, it is also the place where the internal tide dissipates almost all of its energy, a point that has found less attention as a defining feature of the inner shelf. Here we focus on this particular aspect and develop a conceptual framework that suggests a new perspective on the inner shelf’s role as the surf zone for the internal tide (section 5a). This internal surf zone, in which the bulk-averaged internal tide exists in a saturated state confined by the water depth, has features analogous to the surf zone for surface gravity waves (Thornton and Guza 1983; Sallenger and Holman 1985; Battjes 1988).

Generation of the internal tide happens both locally at the shelf break (Sharples et al. 2001; Duda and Rainville 2008; Kang and Fringer 2010) and at remote locations—hundreds to thousands of kilometers away (Alford et al. 2007; Nash et al. 2012; Kumar et al. 2019). The shoaling internal tide can provide a significant fraction of the inner shelf’s energy (Moum et al. 2007b; Kang and Fringer 2012). Here the internal tide’s energy is dissipated causing turbulence and diapycnal mixing. Inside the inner shelf region the internal tide significantly contributes to cross-shelf transport of energy (Moum et al. 2007a), heat (Gough et al. 2020), mass (Shroyer et al. 2010b), sediment (Butman et al. 2006; Pomar et al. 2012; Boegman and Stastna 2019; Becherer et al. 2020), nutrients (Sandstrom and Elliott 1984; Sharples et al. 2007), and biomass (Scotti and Pineda 2007).

Understanding the shoaling dynamics and eventual dissipation of the internal tide on the shelf is a challenging research question and the focus of many past studies (Helfrich and Melville 2006). Because of nonlinear interactions, the internal tide usually appears on the inner shelf as a complex superposition of a multitude of different nonlinear internal wave (NLIW) forms with frequencies ranging from the tidal forcing frequency to the buoyancy frequency N (McSweeney et al. 2020b). Much progress has been made in understanding the shoaling behavior of individual NLIWs in idealized theoretical (e.g., Holloway et al. 1999; Grimshaw et al. 2004), laboratory (Boegman et al. 2005; Boegman and Ivey 2009; Carr et al. 2008), and numerical studies (e.g., Vlasenko and Hutter 2002; Kang and Fringer 2010; Venayagamoorthy and Fringer 2007; Lamb 2014) as well as detailed process-orientated observational studies (e.g., Inall et al. 2000; Klymak and Moum 2003; Moum et al. 2003, 2007a; Scotti et al. 2008; Bourgault et al. 2008; Shroyer et al. 2010a; Walter et al. 2014; Becherer et al. 2020).

Several different mechanisms have been identified by which the internal tide and NLIWs dissipate their energy while propagating across the shelf (Lamb 2014). These include direct breaking at steep slopes (Klymak et al. 2011; Arthur and Fringer 2014) and at the point where the waveguide intersects the bottom (Vlasenko and Hutter 2002; Aghsaee et al. 2010; Lamb 2014). The breaking mechanisms can differ depending (for instance) on the ratio between bottom and isopycnal slope, quantified by the Iribarren number ξ (Vlasenko and Hutter 2002; Boegman et al. 2005). Four different breaking regimes are identified: surging, collapsing, plunging, and fission. Aghsaee et al. (2010) argues that for most oceanic shelves with their comparably shallow slopes (<0.01) fission is the most important process. After breaking, some part of the NLIWs can potentially propagate beyond the point where the pycnocline intersects with the bottom in small wave packets of elevation with trapped cores or boluses (Venayagamoorthy and Fringer 2007; Bourgault et al. 2008; Walter et al. 2012, 2014). In addition to catastrophic breaking, the NLIWs associated with the internal tide also gradually dissipate energy while shoaling. This can either happen at the pycnocline due to shear instabilities (Bogucki and Garrett 1993; Sandstrom and Oakey 1995; Moum et al. 2003; Barad and Fringer 2010; Lamb and Farmer 2011) or at the bottom due to frictional interaction with the bed (Bogucki and Redekopp 1999; Stastna and Lamb 2002; Rippeth and Inall 2002; Boegman and Stastna 2019; Becherer et al. 2020).

In Figs. 4 and 5 of Becherer et al. (2021, hereinafter Part I) we showed that the shoaling internal tide appears in many different wave configurations, ranging from trains of high-frequency internal waves of both elevation and depression over almost rectangular-shaped bore fronts, to gentle low-frequency waves. A few randomly chosen examples of internal tide features that can be traced across several moorings as the tidal signal propagates from the 100-m isobath through 50 m to 25 m are illustrated in Fig. 1. More examples can be found in McSweeney et al. (2020a,b), Kumar et al. (2021), and Becherer et al. (2020). Besides the energy in and shape of the incident tidal signal, its evolution across the shelf is dictated by bathymetry, stratification and background currents. Except for bathymetry these factors can change on relatively short time scales, where preceding bore fronts influence the background state trailing bores propagating into (McSweeney et al. 2020b), which can also substantially change with along-shelf location (McSweeney et al. 2020a). This chaotic interaction between successive wave fronts makes it potentially impossible to predict the exact development of a particular tidal signal across the shelf.

Fig. 1.
Fig. 1.

Three examples of shoaling internal tide signals from the ISDE that can be tracked onshore from data at three moorings at isobaths 100, 50, and 25 m across the shelf. Corresponding time series of velocity, temperature, and dissipation rate can be found in Part I (E1 and E2 in Fig. 4 of Part I; E3 in Fig. 5 of Part I). The onshore component of the current velocity is image colored; the black/white lines show the isotherms closest to the vertical position of peak stratification in the water column. Measured time series at moorings are converted to space with x = xmoorc0(tt0), where xmoor is the mooring’s position on a cross-shelf transect, c0 the linear mode-1 phase speed [see (9)], and t0 is the start time of each event. For each event, t0 is written to the right of the dashed vertical line that marks xmoor. Because of the convention used here, x and t increase in opposite directions as indicated by the black arrows in (c). The blue vertical line ahead of each event indicates the maximum isothermal displacement.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-21-0047.1

Despite the complexities of interaction and evolution that compose the shoaling internal tide, we were able to extract several major findings in Part I. By studying the internal tide and its properties in an average sense, where our averaging window (36 h) was chosen such that it contains an ensemble of internal tide forms (see Figs. 4 and 5 in Part I) we find that

  • the internal tide dissipates most of its energy between the 100 and 20-m isobath;

  • as the internal tide shoals, the energy flux at shallower sites becomes progressively decorrelated from the offshore incident wave energy;

  • stratification is the controlling factor in determining the relative magnitude of internal tidal energy tunneled to shallower sites; and

  • internal tide energy levels over the inner shelf can be completely independent of offshore energy levels, thus indicating that the internal tide loses the memory of its initial strength as it shoals.

This latter point suggests that it is possible to predict internal tide energy levels over the inner shelf with little, and mainly local, information. Testing this hypothesis is the primary objective of this paper.

Our strategy here is to neglect the hierarchical complexities of the internal tide propagating across the inner shelf and develop a practical framework to describe the internal tide’s shoaling behavior in a bulk-averaged sense. We use averaged energetic considerations that are largely independent of wave type or frequency. By doing so we employ a very different approach than previous process-oriented studies that concentrated on understanding the important problems associated with the shoaling behaviors of individual waves. This bulk-averaged approach leads to a holistic description of the internal tide’s dissipation over the shelf.

To this end we introduce a new conceptual framework that assumes the internal tide reaches a state of energy saturation over the inner shelf (section 2). Based on this framework we construct simple parameterizations for energy, energy flux, and flux divergence (that we take to be equivalent to the energy dissipated by turbulence) of the internal tide over the inner shelf that depends only on mean background stratification and bathymetry (section 2d). An evaluation of this parameterization by comparison with our direct measurements is done in section 2g. We consider implications of this result in section 3 and demonstrate that our new parameterization quantitatively reproduces results from a range of previous studies (section 4), thus suggesting general applicability. It also suggests a new additional definition of the inner shelf as the region of the saturated internal tide.

The saturation framework provides an accounting for the interesting correlations found in Part I, mentioned above (sections 5b and 5c) and further insights into internal tide dynamics over the inner shelf, including a prediction of the cross-shore location of maximum dissipation (section 5e), a formulation of the dissipation length scale (section 5f), and hints toward a mixing feedback mechanism (section 5d). A summary and conclusions follow (section 6).

2. Saturated internal tide

a. Concept

Figure 2 conceptually illustrates the shoaling process of an idealized first-mode internal wave. The internal wave propagates shoreward with isopycnal displacement η. As the wave shoals, the amplitude η0 becomes comparable to the water depth H. At some point H imposes an effective upper bound on η0 such that η0 must decrease to propagate into shallower waters (Fig. 2a).

Fig. 2.
Fig. 2.

Schematic of an idealized shoaling internal wave. (a) During shoaling the water depth decreases. At some point HS the water depth becomes comparable to the amplitude of the internal wave. Inshore from this point the water depth is the effective limit for the amplitude of the internal wave, such that the amplitude has to decrease in order for the internal wave to be able to propagate into the saturated range. (b) Since the amplitude of the internal wave is related to the APE through (4), the upper bound for η0 translates into an upper bound for the available potential energy, APEc, that is proportional to H3 [see (5)].

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-21-0047.1

The magnitude of η largely defines the local available potential energy (APE; Winters et al. 1995). Water depth and stratification limit the maximum available potential energy APEc. This maximum capacity decreases rapidly with water depth (∝H3). A shoaling internal tide must lose energy to follow the restrictions set by APEc (Fig. 2b).

It is important to note that the schematic wave depicted in Fig. 2a does not show the internal tide’s complexity seen in Fig. 1. It is intended as a statement that the common feature of the wave hierarchy is that their individual amplitudes determine their maximum APE. The schematic serves as guidance for describing the shoaling of the internal tide in a bulk-averaged sense, while disregarding its complexity.

We term the water depth at which the shoaling internal tide is, on average, first influenced by the water column’s energy restriction the saturation water depth Hs and the region inshore of Hs the saturated range (SR). In the following, we develop an expression for APEc that is then used to develop a parameterization for energy, energy flux, and flux divergence of the bulk-averaged internal tide within the SR.

b. Available potential energy

The available potential energy density of internal waves can be expressed as (Nash et al. 2005; Kang and Fringer 2010)
ape=g22ρ0ρ2N2=12ρ0N2η2(Jm3),
where angle brackets denote a 36-h low-pass filter, ρ′ = ρ − ⟨ρ⟩ is a density perturbation, ρ0 is a constant reference density, and g = 9.81 m s−2 the gravitational acceleration.
The background stratification is defined as
N2=gρ0dρdz(s2),
and the isopycnal displacement of the internal wave field is
η=gρ0N2ρ.
The water column–integrated available potential energy is
APE=0Hapedz=12ρ00HN2η2dz(Jm2).
The total depth-integrated energy is then defined as E = APE + KE, where KE is the kinetic energy component.

c. Energy capacity

The water depth and stratification determine the upper bound to APE as follows. The maximum downward isopycnal displacement is restricted by the distance to the bottom, ηmaxD = −z, and the corresponding maximum elevation is restricted by the distance to the surface, ηmaxE = Hz.

Using either ηmaxD or ηmaxE in (4) yields the identical expression for the maximum capacity:
APEc=12ρ0N2¯0Hηmax2dz=16ρ0N2¯H3,
with
N2¯=1H0HN2dz=gρ0ΔρH,
where ⟨Δρ⟩ denotes the mean top-to-bottom density difference.

Using N2¯ in (5) instead of ⟨N2⟩ appears as a simplification at first, but it implicitly incorporates large-amplitude corrections for (4), because it corresponds to an average of ⟨N2⟩ over the full extent of maximum isopycnal displacement (i.e., the entire water column). Analogous to (5) it is possible to derive a maximum capacity for the kinetic energy, which is demonstrated in the appendix.

d. Parameterization

From (5), we develop a parameterization for the mean available potential energy of the internal tide inside the SR:
APEpar=CAAPEc=CA6ρ0N2¯H3,
where CA is a proportionally factor. The available potential energy is related to the baroclinic energy flux (Moum et al. 2007b),
FE=CFcgAPE,
where CF is a proportionally factor. For long waves in shallow water the group speed cg is approximated to first order by the linear phase speed c0. The linear phase speed for first-vertical-mode waves in constant stratification is
c0=1πN2¯1/2H.
Equations (7) and (9) lead to a parameterization for ⟨FE⟩ inside the SR,
FEpar=CFCAc0APEc=CFCA6πρ0N2¯3/2H4.
The proportionality factors CA and CF, are yet to be determined.
The x derivative of (10) yields an expression for the flux divergence,
xFEpar=CFx2CFCA3πρ0N2¯3/2H3xH(Wm2),
with
CFx=1+38HN2¯xN2¯xH.
The second term on the right-hand-side of (12) factors in a potential x dependence of N2¯ into (11).

To estimate CFx we consider two limiting cases:

  1. for spatially uniform stratification, xN2¯=0, and CFx = 1;

  2. in the case of a pycnocline sufficiently far away from the seafloor so that ∂xΔρ = 0 in (6), xN2¯=N2¯H1xH, which yields CFx = 5/8.

For most scenarios we expect 5/8 < CFx < 1, where CFx is likely to be smaller in deep waters than in shallow waters, where the pycnocline is closer to the seafloor.

e. Free parameters

To use the parameterizations (7)(11), we require estimates of the proportionality factors CA and CF. Factor CA is the ratio of low-pass-filtered (36 h) available potential energy of the internal tide ⟨APE⟩ to APEc, which can be related as
CA=APEmaxAPEcAPEAPEmax.
The first factor in (13), the ratio of the maximum available potential energy in a tidal cycle APEmax to APEc, is a measure of the level of saturation. The saturation level ranges from ≈0.2 at H = 50 m to 0.4 at the shallow moorings (Fig. 3a). Excluding mooring MS100, the average saturation level is 0.27 (Table 1).
Fig. 3.
Fig. 3.

Estimates of proportionality factors that appear in (7)(11): (a) saturation level, (b) shape factor, (c) CA, (d) CF, and (e) the product CACF. The symbols indicate the 2-month average for each mooring with corresponding error bars (for mooring positions, see Fig. 1 in Part I). The red lines are average values across the entire mooring array, excluding MS100 (leftmost symbol). The two panels at upper right show idealized wave shapes and corresponding maximum and mean APE in relation to APEc.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-21-0047.1

Table 1.

List of parameters evaluated from the observations (Fig. 3) as input to the parameterizations (7)(11).

Table 1.
The second factor in (13), ⟨APE⟩/APEmax, is related to the shape of the internal tide. For rectangular waves, the shape factor is 1, for sine waves it is 0.5, and for intermittent more-complicated asymmetric wave forms it is < 0.5 (Fig. 3). In general, the shape factor is approximated by
APEAPEmax=1/TTη2dtηmax2.
Excluding MS100, the mean value of the shape factor is 0.26 (Fig. 3b, Table 1).

The product of the saturation level and the shape factor yields CA = 0.07 (Fig. 3c), implying that inside the SR the ⟨APE⟩ is roughly 7% of the maximum capacity APEc.

Factor CF is the proportionality factor in (8). It is difficult to predict CF theoretically. For freely propagating linear waves, the kinetic and available potential energy relate as ⟨KE⟩/⟨APE⟩ = (ω2 + f2)/ (ω2f2), where the wave’s frequency is ω and the Coriolis frequency is f. For a monochromatic wave with M2 tidal frequency at our latitude, we get ⟨KE⟩/⟨APE⟩ = 2.2, corresponding to CF = 1 + ⟨KE⟩/⟨APE⟩ = 3.2. For waves that are close to a lateral boundary (steep slope or coast), the ratio will be very different because of backward reflection of the incident wave (Nash et al. 2004; Waterhouse et al. 2017). In addition, during the shoaling process the internal tide steepens and becomes increasingly nonlinear, which results in higher harmonics as well as the generation of high-frequency undular bores. Even the first harmonic M4 yields with the above formula ⟨KE⟩/⟨APE⟩ ≈ 1, and Moum et al. (2007b) show that packets of high-frequency internal solitons have an equal partitioning between available potential and kinetic energy, which would result in CF = 2.

From the mooring data, we find on average CF = 1.23 (Table 1), with some apparent depth dependence, where CF decreases from about 1.5 at H = 50 m to about 1.0 at H = 25 m (Fig. 3d). Despite being on average shoreward directed, we find brief periods during each tidal cycle in which the instantaneous flux is offshore directed (see Fig. 6 in Part I). This indicates that reflection may play a significant role in altering CF. If we ignore direction and just average the sole magnitude of |FE|, we find an average ratio ⟨|FE|⟩/(c0⟨APE⟩) ≈ 1.8 (red dots in Fig. 4), which is closer to 2—the value expected for a wave of equally partitioned KE and APE traveling in one direction.

f. Saturation depth Hs

The parameterization (7)(11) is only valid inside the SR (Fig. 2). The SR is defined by a characteristic water depth Hs inshore of which the internal tide is saturated. To calculate Hs, we set the incoming baroclinic energy flux FEin equal to FEpar in (10) and solve for H, which yields
Hs=[6πCACFρ0]1/4N2¯(3/8)FEin1/4.
Using the mean baroclinic energy flux at MS100 as a measure of FEin=FE|MS100, we find that Hs varies between 40 and 80 m, with an average value of Hs = 58.4 (Fig. 5). This indicates that all moorings, except for MS100, are inside the SR most of the time. MS100, on the other hand, is always outside the SR during the 2-month-long observation period.
Fig. 4.
Fig. 4.

Scatterplot of the flux coefficient CF for every 36-h window at all mooring sites, excluding MS100. Black dots represent averages of the onshore component of ⟨FE⟩, and red dots are averages of the magnitude ⟨|FE|⟩ regardless of direction. The black line corresponds to CF = 1.23, which is the best fit for black dots. The best fit for red dots (red line) has a slope of 1.8. As reference, we plot a slope of 2 (green dashed line), corresponding to a unidirectional wave with an equal partition between APE and KE, and a slope of 3.2 (blue dashed line), corresponding to a freely propagating M2 tide at our latitude.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-21-0047.1

Fig. 5.
Fig. 5.

Saturation depth Hs (purple; left axis), calculated on the basis of the mean baroclinic energy flux FEin (blue; right axis), and the mean stratification in the upper 40 m N2¯ (red; right axis) at MS100. The data gap in FEin and thus Hs in the middle of the record is caused by a recovery and redeployment of the corresponding ADCP. The experiment-long average of the saturation depth is ⟨Hsex = 58.4 m.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-21-0047.1

Depth Hs is a crucial parameter to determine whether the parameterization (7)(11) can be applied to a particular dataset. If (7)(11) strongly overestimate the average measurement at a particular mooring site, it suggests that the mooring lies outside the SR. In this case locally measured ⟨FE⟩ can be used as an input for (15) to determine the extent of the saturated range. On the other hand, if (7)(11) do not overestimate local measurements, it can be assumed that the corresponding mooring already lies within the SR. In this case, it is not possible to determine the offshore extend of the SR solely on the basis of measurements of this single mooring. It is fair, however, to assume that (7)(11) are a valid parameterization inshore of the given mooring location.

g. Validation

To validate the parameterization (7)(11), we use all available energy (⟨APE⟩), flux (⟨FE⟩) and dissipation (⟨D⟩) measurements from every mooring deployed deeper than 20 m during the Inner Shelf Dynamics Experiment (ISDE). Since the parameterization is intended to predict mean quantities rather than instantaneous values, we compare ⟨APE⟩, ⟨FE⟩, and ⟨D⟩ with APEpar, FEpar, and xFEpar, respectively (purple dots in Fig. 6). As input for the parameterization, we use N2¯.

Fig. 6.
Fig. 6.

Validation of parameterization (7)(11). Shown is a comparison of (a) available potential energy, (b) baroclinic flux, and (c) depth-integrated dissipation rate between in situ measurements (x axis) and parameterized values (y axis). The correlation coefficients r at the bottom of (a)–(c) correspond to the colored symbols (⟨⟩ex), excluding MS100 (pink square), where values in the square brackets are error bounds on r. The solid black lines in (a)–(f) indicate a one-to-one relation, dotted lines indicate a factor-of-three difference, and dashed lines indicate a factor-of-10 difference. Also shown is (d)–(f) a nondimensional comparison. The y axis is the ratio between parameterization and in situ observations, where values > 1 or < 1 indicate an over- or underprediction, respectively. The x axis is the saturation water depth Hs nondimensionalized by the respective mooring water depth H such that values > 1 or < 1 respectively lie inside or outside the saturated range. The vertical pink line indicates the edge between the saturated range and unsaturated range. Purple and gray dots are 36-h averages (⟨⟩), and colored symbols are averages over the entire deployment period (⟨⟩ex).

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-21-0047.1

For both ⟨APE⟩ and ⟨FE⟩, most of the 36-h windows are close to the one-to-one line (purple dots; black line in Figs. 6a, b), indicating a good agreement between parameterization and in situ observations over several orders of magnitude. At the smallest values of ⟨APE⟩ and ⟨FE⟩, there is a slight overprediction, possibly associated with times when internal tide energy is simply too small to reach saturation. The cloud of points in Figs. 6a–c at large values corresponds to measurements made at mooring MS100, which lies outside the SR all of the time. Excluding MS100 (gray dots Fig. 6), the correlation between parameterized values and in situ observations, as based on the two month averages, is r = 0.82 for ⟨APE⟩ex and r = 0.92 for ⟨FEex.

The proposed parameterization applies only within the saturated range. This is clear in the nondimensionalized depiction of Figs. 6d–f. Outside the SR (Hs/H < 1), the parameterization strongly overpredicts, but as HHs the ratios APEpar/⟨APE⟩ and FEpar/FE approach 1 (Figs. 6d, e), and these ratios scatter about 1 within the SR.

We have compared xFEpar with ⟨D⟩ and find them be highly correlated but find that xFEpar is larger than the in situ measurements (⟨D⟩) by a factor of 2–3 in the SR (Figs. 6c, f). Either we have not sufficiently measured ⟨D⟩ or a significant fraction of xFEpar goes to processes other than turbulence. Our moored dissipation measurements do not include data from above 12 m, which means that we potentially miss a substantial part of the actual depth-integrated dissipation. At the same time, however, this also helps to reduce the effects of surface-forced turbulence helping to isolate the flux divergence-forced turbulence. One aspect that we have not investigated (and will in a separate study) is the possibility that the incoming tidal energy flux contributes to an internal wave setup (Umeyama and Shintani 2006), similar to surface wave setup on the beach. In this case, we would expect to see an increase in cross-shelf pressure gradient (which is well measured by high-resolution pressure sensors on many of the seafloor landers) and potentially an inertial response to that pressure gradient in the form of a northward-flowing current.

In general we find that, despite its simplicity, the parameterization (7)(11) reproduces our in situ observations of energy and energy flux surprisingly well, producing a highly correlated though overestimated representation of dissipation (within a factor of 2–3) inside the saturated range.

3. Application of FEpar and xFEpar to the experimental site

We envision several useful application of the parameterization (7)(11). It might be used, for instance, to generate maps of energy flux and dissipation over the inner shelf. To this end, we first calculate the gradient of the bathymetry of the entire region in which the ISDE has been conducted (Fig. 7a) to get H and ∂xH. In a second step we interpolate the depth mean stratification between mooring locations (Fig. 7b) to get N2¯ex (Fig. 7b), where ⟨⟩ex denotes an experiment-long averages. There are a number of different ways to obtain N2¯ex . Here we assume that stratification measured at MS100 is representative for N2¯ex along the 100 m isobath to interpolate along isobaths between different moorings and in the last step in across-shore direction to get a regional coverage of N2¯ex. Depending on the available data it is also possible to make different assumptions; for instance, if only one mooring is available one might assume horizontally uniform stratification. For the dataset presented here this is a reasonable assumption, with depth-mean stratification changing across the shelf by less than a factor of 2 (Fig. 7b). Either way, once H, ∂xH, and N2¯ex are determined we can use (10) and (11) to calculated FEpar and xFEpar across the shelf (Figs. 7c, d).

Fig. 7.
Fig. 7.

Map of parameterized flux and flux divergence in comparison with in situ measurements of flux and dissipation: (a) bottom slope ∂xH, (b) interpolation of depth averaged stratification N2¯ex, (c) baroclinic energy flux FEpar, and (d) depth-integrated dissipation rate xFEpar. The colored circles in each panel show corresponding in situ observations at each mooring location. The white line in (c) and (d) illustrates the saturation depth ⟨Hsex, as based on (15). The maps correspond to average values (⟨⟩ex) over the entire deployment length from 9 Sep to 1 Nov 2017.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-21-0047.1

A comparison between in situ observations (circles) and parameterized values suggests considerable similarity in the alongshore variability inside the SR (inshore of the white line marking ⟨Hsex in Figs. 7c, d). Besides the general trend of decreasing flux, flux divergence and dissipation toward shore there can be identified some interesting local variability. In a region just north of Point Sal with steeper bathymetry (∂xH ≈ 7 × 10−3) we find enhanced dissipation between 40- and 50-m water depth relative to other places along the same isobaths with smaller slopes ((∂xH ≈ 5 × 10−3 at OC and VB array). More dissipation can potentially yield more vertical mixing, which might be an explanation why N2¯ex is smaller directly off Point Sal relative to other regions farther north or south (Fig. 7b). Alternatively, this could be the result of headland related mixing processes, which are further investigated in Kovatch et al. (2021).

Inshore of H = 30 m, however, the slope just north of Point Sal becomes shallower, which results in a localized region of decreased dissipation, consistent in both parameterization and observation (small blue region in Fig. 7d). Similarly, south of Point Sal we find an extended region of small dissipation [xFEpar=O(103) W m−2] associated with a relatively shallow slope ((∂xH ≈ 2 × 10−3; cf. the yellow region in Fig. 7a and blue region in Fig. 7d south of Point Sal).

This demonstrates that the parameterization can predict, in addition to general cross-shore variations, some aspects of alongshore variability in ⟨FEex and ⟨Dex that are due to variations in bathymetry and stratification.

4. General applicability

In the following we test if the parameterization (7)(11) is applicable beyond the dataset obtained during the ISDE. To this end we employ representative studies that list energy, energy flux or dissipation measurements on continental shelves at a range of worldwide locations. We then compare the direct measurements from these experiments with estimates that are based on our parameterization (7)(11). As input for these calculations, we need bathymetry and stratification. Where not directly provided we make best guesses of N2¯ and H as well as ⟨E⟩, ⟨FE⟩, and ⟨D⟩ based on figures and tables from the respective publications and list these together with our corresponding parameterized values in Table 2. These studies cover a broad range of parameter space in terms of incoming internal tide energy and stratification (Fig. 8).

Table 2.

Comparison between different observational studies and parameterized values from the saturation framework. The input parameters H, ∂xH, and N2¯ as well as the energy estimates ⟨E⟩, ⟨FE⟩, and ⟨D⟩ are either directly given or are guessed on the basis of figure and tables of the respective studies. The parameterized values APEpar, FEpar, and xFEpar are calculated from (7), (10), and (11) with the given input parameters. The saturation depth Hs is either calculated by (15) with the corresponding ⟨FE⟩ as FEin or, where not feasible, as a guess on the basis of the ratio FEpar/FE.

Table 2.
Fig. 8.
Fig. 8.

Parameter space. Contour lines show Hs according to (15). The boxes represent different studies around the world: ISDE off the coast of California is described in this paper, S01 (Sharples et al. 2001) on the New Zealand shelf, S88 (Sherwin 1988) on Marlin shelf, DR08 (Duda and Rainville 2008) in the South China Sea, H01 (Holloway et al. 2001) on the Australian shelf, M07 (Moum et al. 2007a) on the Oregon shelf, S10 (Shroyer et al. 2010a) on the New Jersey shelf, and MG (MacKinnon and Gregg 2003b) on the New England shelf. The FEin represent long-term averages of the incoming baroclinic energy flux for all studies except for Moum et al. (2007a), where it corresponds to a short-term average over a single wave train. Note that the value of FEin from Sharples et al. (2001) is likely underestimated, since their deepest flux measurements was taken already inside the saturated range.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-21-0047.1

Colosi et al. (2018) present data from a pilot study to the ISDE described here, conducted close to our Point Sal array (see purple symbols in Fig. 1 in Part I) from mid-June to end of July 2015. Their energy flux measurements are well represented by (10) (see Table 2). Furthermore, the theory behind our parameterization is able to explain the finding of Colosi et al. (2018) that FE is correlated to N2 but not to the neap–spring cycle (see sections 5b and 5c). Based on an onshore decrease of ⟨FE⟩, Colosi et al. (2018) calculate a dissipation length scale of roughly LD ≈ 2 km for both internal bores and high-frequency internal solitary waves. Our framework reproduces this result, with the surprising finding that LD depends on bathymetry only (see section 5f).

Duda and Rainville (2008) describe data from four moorings deployed in the South China Sea during spring 2001. They find ⟨FE⟩ to be dominated by diurnal frequencies. A comparison of ⟨FE⟩ with FEpar shows good agreement for the two shallower (85 and 120 m) moorings (Table 2). Using measured ⟨FE⟩ values from the deeper (200 and 350 m) two moorings as input for (15) both predict saturation to be reached at Hs ≈ 140 m.

This seems to be consistent with average dissipation data obtained by nearby microstructure measurements in spring 2005 (St. Laurent 2008), where average values of ⟨D⟩ are remarkably close to our prediction xFEpar for water depths of H = 50–110 m. For H > 190 m, xFEpar strongly overpredicts ⟨D⟩, which implies that 110 < Hs < 190 m. Maximum dissipation is apparently reached near Hs (section 5e).

Shroyer et al. (2010a) describe data from the relatively shallow slope of the New Jersey shelf. They average three moorings (62, 72, and 82 m) together to provide a time series of daily averaged internal tidal energy (see Fig. 5 in Shroyer et al. 2010a). We average this time series in three consecutive periods (1–9 August, 9–15 August, and 15–25 August) for which average stratification data are available from Shroyer et al. (2010b). Using average wave speed estimates for these three periods [c = 0.8, 0.6, and 0.7 from Shroyer et al. (2011)], we are able to estimate ⟨FE⟩ = cgE⟩. A comparison of ⟨FE⟩ values from Shroyer et al. (2010a) with FEpar estimates implies that during the first two periods the moorings lie just outside the saturated range (Hs ≈ 60), whereas in the last higher energy period the moorings lie well within the saturated range.

Also, dissipation rates estimated by Shroyer et al. (2010a) based on shelfwide decay rates agree to within a factor of two with xFEpar during the last two periods (Table 2). Shroyer et al. (2010a) developed a simple parameterization of dissipation based on the maximum energy, ⟨D⟩ = μEmax. We can use our parameterization (7) and (11) to develop a direct formula for the regression slope:
μ=APEAPEmaxxFEparAPEpar=4CFAPEAPEmaxN2¯1/2xH.
Using the parameters from Table 1 and the stratification and bottom slope values provided in Shroyer et al. (2010a), (16) gives μ = 26.7–29.2 × 10−6 s−1, which is close to the value μ = 24 × 10−6 s−1 found by Shroyer et al. (2010a) using linear data fit.

Sharples et al. (2001) calculate total internal tidal energy and flux at two moorings in relatively deep water (110 and 150 m) with a relatively weak mean stratification on the New Zealand shelf. Agreement of ⟨FE⟩ and FEpar to within a factor of ~2 for both sites suggests that despite their depth both sites are inside the saturated regime where our parameterization is applicable. A similar agreement is found between ∂xFE⟩ and xFEpar (Table 2).

A comparison with data provided by Holloway et al. (2001) over the Australian shelf show mixed agreement with our parameterization. We can use flux estimates from the deeper station to predict a saturation depth of roughly Hs = 105–120 m. Consistent with this we find that the ratio FEpar/FE is within a factor of 4 inshore of H = 125 m (ignoring the station at 70 m). The big spread in FEpar/FE at different stations can potentially be explained by the type of dataset. Holloway et al. (2001) provide data from three mooring stations (label m) with very sparse vertical resolution and no density measurement closer than 20 m to the surface and several full depth shipboard CTD stations (label c), which have been occupied for about 12 h each. The internal tide is therefore potentially underresolved either in time or space depending on the type of station, which might explain some of the mixed agreement to our parameterization.

A different type of dataset comes from Moum et al. (2007a) where single high-frequency solitons were tracked as they shoaled over the Oregon shelf. The ratio FEpar/FE<1.5 inshore of H = 122 m and larger offshore, suggesting a saturated regime inshore from this point. Consistent with previous considerations we find maximum dissipation in the data to occur close to the saturation depth of Hs ≈ 120 m. However, the predicted dissipation levels are larger than the measurements by between a factor of 2 and 20, which might suggest that high-frequency solitons are less dissipative than low-frequency internal tidal bores.

Other studies like MacKinnon and Gregg (2003b) from the New England shelf or Sherwin (1988) from the Malin shelf describe internal tides in places that appear to be outside the saturated range (H > Hs). This is indicated by a strong overprediction of our parameterization relative to the measured energy and dissipation levels of the internal tides.

In general, our parameterization seems to apply to other datasets with very different bottom slopes, stratification and over a range of energy levels, so long as the data lie within the saturated range. This is surprising given that we did not fine tune any of the free parameters but simply applied the parameters found during the ISDE (Table 1). Furthermore, it appears that we can apply this parameterization to waves of different frequencies from low-frequency diurnal tides (Duda and Rainville 2008) to high-frequency solitons (Moum et al. 2007a). Since we make no assumptions about frequency in the derivation of (7)(11), it is possible that this framework might be used for other shoaling internal waves with amplitudes large enough to be restricted by the water depth.

It is interesting to note that, where FEpar agrees well with ⟨FE⟩, APEpar is close to the measured total energy ⟨E⟩ and is greater than the APE fraction [see Duda and Rainville (2008), Sharples et al. (2001), Moum et al. (2007a), and Shroyer et al. (2010a) in Table 2]. This suggests a slightly different parameter combination at those studies with a smaller CA and a larger CF than what is observed here (Table 1). It is not surprising, however, that ⟨E⟩ follows a similar relation as APEpar, since, analogous to ⟨APE⟩, we expect KEN2¯H3 (see the appendix).

5. Discussion

a. Internal surf zone

To our knowledge, the scaling outlined in section 2d to assess the evolution of the energetics of shoaling internal waves over the inner shelf is novel. However, analog scalings describe surf-zone dynamics (Thornton and Guza 1983; Sallenger and Holman 1985; Battjes 1988; Feddersen and Trowbridge 2005). Similar to our approach, it is assumed that breaking waves in the surf zone are saturated (Sallenger and Holman 1985; Battjes 1988) or self-similar (Feddersen and Trowbridge 2005), and the wave height (Hrms) is modeled as a constant fraction γ of the water depth, Hrms = γH. In the framework described in section 2 we look at saturation from an energetic point of view with the characteristic ratio APEmax/APEc, which corresponds to γ2 in surf-zone dynamics given that the energy EHrms2.

Scaling equations for energy, energy flux, and dissipation in the surf zone analogous to those in section 2d have been described in the literature (Thornton and Guza 1983; Stive 1984; Battjes 1988; Feddersen and Trowbridge 2005; Feddersen 2012). Vertical stratification is the major additional complication that comes with internal waves when compared with surface waves. However, if we express stratification in terms of reduced gravity g=HN2¯, Eqs. (7), (10), and (11) can be formulated in a way that is equivalent to the surf-zone scalings (Table 3 presents a comparison). Note that despite very different scales in terms of wave speed, amplitude, period, and wavelength, the nondimensional parameter ξ is comparable between surface waves in the surf zone and internal waves on the inner shelf (Table 3).

Table 3.

Analogy between inner shelf and surf zone. Reference key: TG83 (Thornton and Guza 1983), S84 (Stive 1984), B88 (Battjes 1988), FT05 (Feddersen and Trowbridge 2005), and F12 (Feddersen 2012).

Table 3.

The term internal surf zone has been used in the literature before, but in different contexts. The first example comes from the nature of internal waves propagating at angles critical to the local bottom slope. The group velocity of internal waves in a continuously stratified medium has a characteristic angle to the horizontal plane β that depends on the wave frequency and stratification. Eriksen (1985) pointed out the importance of locations on the continental slope where the bottom slope αs is critical with respect to the group velocity angle of internal waves, βαs. It was found that regions with critical slopes are often associated with elevated turbulence and mixing levels near the bottom (Ivey and Nokes 1989; Moum et al. 2002; Nash et al. 2004). This critical slope region was termed an internal surf zone by Thorpe (1997). This definition has little to do with our definition since it could potentially refer to relatively deep sections of the continental shelf subject to different processes than the inner shelf.

More closely related to our definition, Bourgault and Kelley (2003) use the term internal surf zone to describe the shelf region beyond the breaking point of individual internal waves [see also Bourgault et al. (2008), where it is referred to as internal beach]. This break point is defined by the breaking water depth Hb (Vlasenko and Hutter 2002); Hb is calculated for individual waves by using an empirical formula by Vlasenko and Hutter (2002),
Hb=η00.8°αs+0.4+Hpyc,
with the slope angle in degrees, αs, and the pycnocline depth Hpyc. In comparison, we identify the internal surf zone as a saturation range, defined as the region inshore of the saturation depth, Hs [see (15)]. The fact that these regimes are both defined by characteristic water depths is an interesting parallel. However, the critical distinction is that Hb refers to individual waves, Hs to the incident energy flux of the ensemble of waves associated with the internal tide. The Hs is determined by (15), a formula derived from simple energy considerations (see section 2). Using (17) for calculating Hb for typical waves found during the ISDE (αs ≈ 0.3°, Hpyc = 15 m, and a range of maximum isopycnal displacement 10 < η0 < 40 m) gives a range 18 < Hb < 28 m, which is usually smaller than Hs. This indicates that direct breaking occurs closer to shore than first amplitude restriction by water depth, or in other words that our definition of the internal surf zone extends farther offshore than the definition of Bourgault and Kelley (2003). Our reference here to internal surf zone derives from the direct analogy of our scaling (see section 2) to commonly used surf-zone scalings (Thornton and Guza 1983; Sallenger and Holman 1985; see Table 3).

b. The internal tide loses its memory

An important finding from Part I is that the energy of the internal tide inshore of 40-m water depth is independent of the incoming energy flux FEin (see Fig. 9a in Part I). The concept of a saturated internal tide provides an explanation. From (10) we see that inside the SR the energy flux depends only on water depth and stratification and is independent of the incoming energy flux. Internal tide energy inside the SR is decoupled from the energy at the generation site. The internal tide loses memory of its initial strength during saturation.

At the same time, FEin influences Hs via (15). So although internal tide energy within the SR is independent of incoming energy levels, FEin determines the extent of the SR across the shelf.

c. Stratification channels energy onto the inner shelf

Another important finding from Part I is that, at the shallow moorings (<40 m), ⟨FE⟩ is most strongly correlated to stratification (see Fig. 9b in Part I). The parameterization (10) implies that besides bathymetry ⟨FE⟩ depends only on stratification inside the SR. This explains why we observe more energy close to shore during strong stratification periods than during weak stratification periods (see Fig. 7c in Part I).

Stratification also influences the extent of the SR across the shelf [(15)]. Stronger stratification leads to a smaller value of Hs and thus to a smaller cross-shore extent of the SR. On the other hand, when stratification is weak across the shelf, saturation occurs in deeper waters so that the internal tide loses its energy farther offshore.

Shelf stratification acts as a control that locates internal tide energy dissipation. Under strong stratification the energy is dissipated farther inshore and vice versa during periods of weak stratification.

d. Mixing feedback

Equation (11) implies that stronger stratification yields greater dissipation in the SR. A certain fraction of the dissipation goes to diapycnal mixing. Mixing causes ⟨N2⟩ to decrease, which via (11) causes dissipation/mixing to decrease, thus creating negative feedback. One potential consequence is a relatively constant background stratification over the inner shelf for extended periods. In this context one might ask how much of the internal tidal energy does contribute to diapycnal mixing on the shelf? Given our observation that a major fraction of dissipation is happening in the well-mixed bottom region as opposed to the stratified interior, the overall mixing efficiency may be small. The feedback loop described above as well as the overall mixing efficiency of the internal tide on the inner shelf will be investigated in more detail in a following study.

e. Cross-shelf structure of dissipation

Equation (11) also indicates that, given moderate shelf slopes and cross-shelf changes in N2¯, the tendency ⟨D⟩ ∝ H3 ought to govern the cross-shelf structure in the SR. In addition, because internal tide propagation is not restricted by depth offshore of Hs, there ought to be a maximum in ⟨D⟩ near Hs.

The cross-shelf distribution of ⟨D⟩ normalized by its shelfwide mean value shows a broad peak near and offshore of Hs/H ≈ 1 (Fig. 9). This suggests that most internal tide dissipation on the shelf occurs as the internal tide begins to saturate. Therefore, (15) predicts the location of peak dissipation for a given incoming flux FEin and shelf stratification N2¯. This cross-shelf distribution also roughly conforms to ⟨D⟩ ∝ H3 (blue line in Fig. 9).

Fig. 9.
Fig. 9.

Nondimensional cross-shelf distribution of energy dissipation. The gray dots show 36-h hour averages of depth-integrated dissipation ⟨D⟩ for each mooring, normalized by the respective shelfwide-averaged dissipation rate ⟨Dshelf, vs nondimensional water depth Hs/H; Hs/H > 1 represents the saturated inner shelf (shading). The black circles are bin averages with respective 95% bootstrap uncertainty (vertical black lines) The blue line corresponds to a ⟨D⟩ ∝ H3 relation as suggested by (11) with an arbitrary background dissipation.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-21-0047.1

f. Dissipation length scale

From the ISDE pilot experiment in 2015, Colosi et al. (2018) estimated the length scale over which the internal tide dissipates its energy by fitting an exponential to ⟨FE⟩ measured at moorings distributed across the inner shelf. These mooring locations roughly correspond to the Point Sal array shown here (magenta symbols in Fig. 1 in Part I). They find this e-folding length scale, which can be interpreted as a dissipation length scale for internal waves, to be relatively insensitive to environmental condition and/or frequency. Their estimates were based on independent evaluations of internal tidal bores and higher-frequency, large-amplitude wave packets and was 2.0–2.4 km in both cases.

From the framework developed in section 2 we derive an expression for the dissipation length scale by dividing (10) by (11),
LD=FEparxFEpar14HxH(m).
That is, LD depends only on the bathymetry. Approximating the bottom slope ∂xHH/Lshore, where Lshore is simply distance to shore, LDLshore. The typical dissipation length scale is proportional to the distance to shore. From bathymetry data along the mooring locations used by Colosi et al. (2018) in (18) we find LD =1.2–2.5 km, which corresponds to the e-folding scale found independently by Colosi et al. (2018) of 2.0–2.4 km. Since LD is independent of wave frequency, (18) ought to apply to all large-amplitude shoaling internal waves and not just to the low-frequency internal tides.

g. Limitations

Inside the SR, the predictions of (7)(11) agree reasonably well with data, but ⟨D⟩ ∝ H3 creates a large overprediction outside the SR. Hence a clear determination of Hs, which varies with time, is important.

Another limitation arises from the fact that we implicitly assume a cross-shore bathymetry profile with monotonically decreasing water depth toward shore. If we imagine more complex bathymetries with local regions of very shallow or even negative slope our parameterization yields nonphysical results. Equation (11), for instance, predicts negative dissipation rates where xH < 0. The problem is that in such regions the assumption of local saturation is not satisfied. Therefore, the parameterization likely fails to describe shelves, which are very wide; for example, the Northwest European shelf with mean slopes of xH < 10−4. The saturation constraint imposed by the sloping bottom in these regions is probably only valid in localized places with steeper topography. Propagating over extended regions of flat bathymetry, internal waves will lose considerable and potentially all energy to bottom friction, which may result in an undersaturated state that cannot accurately be described by our parameterization.

Furthermore, it is important to note that the parameterization is only meant to predict the inner shelf dissipation due to shoaling internal waves. There are inner shelf regions in the world where other forcing mechanisms, including winds and barotropic tides, are more important than internal waves in terms of energy dissipation. But we expect that, at these locations, the parameterization can be used to estimate the internal tide’s contribution to dissipation even if small relative to other sources. In this context it is worth noting that even on shelves where barotropic tidal currents are the dominant source for dissipation via turbulence, internal waves are important contributors to diapycnal mixing. Since barotropic tides dissipate most of their energy at the seafloor, they are likely associated with smaller bulk mixing efficiencies (Simpson and Hunter 1974) than internal waves (Stigebrandt and Aure 1989), which often generate shear instabilities in the pycnocline itself (Moum et al. 2003).

h. Variability of model parameters

An issue with the applicability of our parameterization (7)(11) is the value of the free parameters CA and CF may change in different systems. Parameter CF depends on wave reflectivity as well as the ratio between ⟨APE⟩ and ⟨KE⟩. The latter is known to change for freely propagating waves depending on frequency and latitude (see also section 2e). However, as we demonstrate in the appendix, the wave energy in the saturated range is simultaneously limited by APEc and KEc, which suggests that the ratio ⟨APE⟩/⟨KE⟩ does not substantially change once a system is in a saturated state.

The reflection rate on the other hand will strongly depend on the steepness of the bottom slope. Bourgault and Kelley (2007) suggest that the reflection rate of individual waves can be parameterized with ξ. Using their parameterization for typical waves found during the ISDE (see Fig. 1) the reflection rates lies between 10% and 30%. This is in agreement with our considerations in section 2e in the context of Fig. 4. How and whether CF depends on ξ and other possible parameters would be an interesting subject for further investigations. In general, we expect smaller CF for steeper slopes. However, also highly nonlinear waves will have a tendency to radiate energy away from steep fronts in both direction, thus potentially causing a fraction of the energy to radiated against the propagation direction of the main wave.

Parameter CA is composed of the maximum saturation APEmax/APEc and the wave shape parameter ⟨APE⟩/APEmax. It can be expected that the maximum saturation is not varying by much in different saturated systems. The wave shape, however, can potentially be very different. Here we observe very frequent strong internal tidal bores every 6–12 h. This means that we have a relatively strong isopycnal displacement from a potential rest state most of the time. The value of ⟨APE⟩/APEmax = 0.26 measured here is therefore probably on the high end for long-term averages. If strong displacements occur, more intermittent ⟨APE⟩/APEmax is likely found to be smaller than here.

6. Summary and conclusions

From Part I we made several relevant observations about the averaged internal tides shoaling over the inner shelf of California:

  1. the internal tide dissipates most of its energy between the 100- and 20-m isobaths;

  2. approaching shallower water, the internal tide’s energy flux becomes progressively decorrelated from the offshore incident wave’s energy flux; and

  3. internal tide stratification controls how much internal tide energy is tunneled to shallower sites such that

  4. inshore energy levels of the internal tide are entirely independent of the incoming internal tide.

Importantly, we find that the internal tide loses memory of its initial strength as it shoals.

We consider these observations in terms of a conceptual framework in which the internal tide saturates as is shoals. Simply, the depth of the water column limits the amplitude η0 of the internal tidal wave, both of which decrease toward shallow water. Since the tidal wave energy is proportional to η2 [(4)], the internal tide loses energy as it propagates onshore. We show that the maximum capacity of the vertically integrated available potential energy (APEc) of the internal tide depends on the depth-averaged stratification and H3 [(5)]. For an internal wave to propagate shoreward past the point of saturation, where its energy is a certain fraction of the maximum capacity, it must lose energy at a rate dictated by the decreasing energy capacity of the water column. This concept of wave saturation in shallow waters is a well-known phenomenon for surface waves, where inside the surf zone the wave height is found to be a certain fraction of the water depth.

We then use the idea of a saturated internal tide to develop simple parameterizations for ⟨APE⟩, ⟨FE⟩, and ∂xFE⟩ (which we interpret as ⟨D⟩) inside the saturated range (7)(11). The parameterization requires only the depth-mean background stratification and bathymetry, permitting application based on a minimal amount of information of the system. A test of the parameterization using the data from our experiment shows surprisingly good agreement within the saturated range.

The theoretical framework of the saturated internal tide makes additional points:

  1. it explains the memory loss of the internal tide and the controlling role of stratification on internal tide energy on the shelf;

  2. it predicts that peak dissipation rates on the shelf are found close to the location of the saturation depth Hs—a point confirmed by our measurements;

  3. an expression is derived for the dissipation length scale of the internal tide inside the saturated range that depends only on bathymetry; and

  4. an empirical parameterization of dissipation that is based on internal tide peak energies found by Shroyer et al. (2010a) is confirmed and is theoretically derived.

We applied our parameterization to a range of shelf sites with varying energies and stratifications where experiments with measurements of ⟨APE⟩, ⟨FE⟩, and/or dissipation have been undertaken (Table 2). The resultant predictions suggests that our parameterization is broadly applicable, including application to high-frequency solitary waves and lower-frequency diurnal tides, beyond that of semidiurnal internal tides examined in detail here.

The inner shelf has been previously defined in terms of its response to wind forcing (Lentz 1994; Fewings et al. 2008). The applicability of our parameterization at a range of sites suggests that a saturated range for the internal tide is an intrinsic feature of the inner shelf. This might yield a new or additional definition of the inner shelf from an internal tide perspective: the inner shelf is that part of the continental shelf where the internal tide saturates. The analogy of our framework to surface waves approaching the beach suggests that the inner shelf is the surf zone for the internal tide.

Acknowledgments

The work described here was fully funded by the Office of Naval Research through the Inner Shelf DRI. We are grateful to Falk Fedderson, Jim Thomson, and Matthew Spydell for pointing out the connection of our framework to surf-zone dynamics. We are grateful to two anonymous reviewers and editor Jody Klymak for their time and valuable comments.

Data availability statement

All data used in this paper are achieved and publicly available (https://doi.org/10.6075/J0WD3Z3Q).

APPENDIX

Kinetic Energy Capacity

Analogous to (5), it is possible to derive a maximum capacity for the kinetic energy of a given water column. The depth-integrated kinetic energy is
KE=12ρ00Hu2dz(J m2).
We assume that the flow is subcritical on average (Fr = u/c0 < 1). Although this may not be the case for highly nonlinear waves where internal recirculation occurs, we expect it to be the typical case. We consider u = c0 to be the upper bound in (A1), which yields an expression for the maximum capacity of kinetic energy,
KEc=12ρ0c020Hdz=12π2ρ0N2¯H3,
using (9).

Aside from the prefactor, (A2) is equivalent to (5). That is, the framework developed in section 2d could alternatively be based on KEc rather than on APEc. Since both KEc and APEc change simultaneously with energy restrictions on the shoaling internal waves we expect that the ratio ⟨APE⟩/⟨KE⟩ does not change inside the saturated range.

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