Editorial Type:
Article
Article Type:
Research Article

Note on the Beam Structure in Step-Trapped Internal Tides

Paul A. Sanders aDepartment of Applied Physics, Eindhoven University of Technology, Eindhoven, Netherlands

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Martijn D. Dorrestijn aDepartment of Applied Physics, Eindhoven University of Technology, Eindhoven, Netherlands

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Theo Gerkema bDepartment of Estuarine and Delta Systems, NIOZ Royal Netherlands Institute for Sea Research, Yerseke, Netherlands

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https://orcid.org/0000-0002-7702-1788
Online Publication:
29 Feb 2024
Print Publication:
01 Mar 2024
DOI:
https://doi.org/10.1175/JPO-D-23-0084.1
Page(s):
871–875
Free access

Abstract

The along-slope propagation of subinertial trapped internal tides is studied for the configuration of a simple step. It is revealed that they form a beam structure in the along-slope direction that is evanescent above the top of the step; these beams lack strict periodicity in the along-slope direction. As in classical internal Kelvin waves, they become less sharp away from the step, as higher modes decay more rapidly in the cross-slope direction. We discuss implications for abyssal mixing and outline the necessary ingredients for their generation.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Theo Gerkema, theo.gerkema@nioz.nl

Abstract

The along-slope propagation of subinertial trapped internal tides is studied for the configuration of a simple step. It is revealed that they form a beam structure in the along-slope direction that is evanescent above the top of the step; these beams lack strict periodicity in the along-slope direction. As in classical internal Kelvin waves, they become less sharp away from the step, as higher modes decay more rapidly in the cross-slope direction. We discuss implications for abyssal mixing and outline the necessary ingredients for their generation.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Theo Gerkema, theo.gerkema@nioz.nl
Keywords: Baroclinic flows; Internal waves; Kelvin waves; Tides

1. Introduction

Internal tides in the ocean are mostly superinertial (i.e., their frequency exceeds the local Coriolis parameter) and hence can propagate as free waves. This is an immediate consequence of the fact that the dominant tidal constituent is in most places the principal semidiurnal M2, which is superinertial up to latitudes of 74.5°. Still, subinertial frequencies are of interest; notably, diurnal constituents are important in some midlatitude areas in the Pacific (O1 and K1 become subinertial poleward of 27.6° and 30.0°, respectively). An example is the Hawaiian Ridge (Smith et al. 2017).

A well-studied aspect of subinertial internal tides is the conversion rate from barotropic to baroclinic tidal energy, and associated energy fluxes. Falahat and Nycander (2015) carried out a numerical global study, assuming a small-amplitude topography. Subinertial tides, despite their relatively small contribution to the conversion globally, were shown to be significant in some regions, notably in the Pacific Ocean.

The characteristics of superinertial internal tides are well understood: the waves propagate away from the topography where they have been generated and the waveform can be regarded as a superposition of modes that together form a beam. For subinertial tides, however, a clear picture of the way of propagation is still lacking. The present paper aims to address this gap. What is clear, though, is that the waves must be trapped at a slope along which they propagate. But is this still a beam-like propagation, and if so, exactly how?

More generally, coastally trapped waves in the ocean have been extensively studied (see, e.g., the review by Mysak 1980). In the atmosphere, too, subinertial internal waves have been studied, as “boundary disturbances” propagating along coastal escarpments, following Gill (1977). In a later study, Durran (2000) studied their modal structure; he assumed a step-like topography and introduced the term “step-trapped Kelvin waves,” which we shall adopt, too. Constructing the wave structure is not straightforward as the solution appears to be nonseparable in the cross-slope and vertical directions. Durran (2000), following Chapman (1982), developed a numerical procedure, recently adapted by Hughes and Klymak (2019), who made their Python codes publicly available. We gratefully use those codes in this paper.

Durran (2000) identified key elements of the wave structure: he found that it resembles a classical internal Kelvin wave (i.e., an internal Kelvin wave propagating along a wall) in the deeper side below the step, but is evanescent higher up. So, as a starting point, it is useful to go back to the classical case of internal Kelvin waves propagating along a wall; these solutions were already derived by Krauss (1966), and also discussed in more recent literature (e.g., Hughes and Klymak 2019). We recap the solution in section 2 and infer the principal characteristics of the beam-like structure, which (as far as we are aware) have not been explored before, even though those properties follow directly from the known solution. One may expect that those characteristics are, at least partly, carried over to a setting where we have an open shallow shelf attached to the deep ocean, i.e., to step-trapped internal Kelvin waves. This aspect, in particular the beam characteristics, is explored in section 3, followed by a discussion and conclusion in sections 4 and 5.

2. Beam structure in classical internal Kelvin waves

Here we consider a setting in which a vertical wall is placed along the y axis, at x = 0. Thus, x is the cross-slope (or lateral) direction and y the direction of wave propagation; the vertical coordinate is z, positive upward. The upper surface is regarded as a rigid-lid at z = 0. The corresponding current velocities are u, υ, and w. Additional variables are the perturbation pressure p′ (i.e., the departure from the static pressure), the reference density ρ∗  and buoyancy b. We adopt the standard linear equations for internal waves on an f plane (see, e.g., Gerkema 2019, chapter 7), under the hydrostatic approximation:
∂u∂t−fυ=−1ρ∗∂p′∂x,
∂υ∂t+fu=−1ρ∗∂p′∂y,
1ρ∗∂p′∂z=b,
∂b∂t+N2w=0,
∂u∂x+∂υ∂y+∂w∂z=0.
These equations feature the Coriolis parameter f and buoyancy frequency N, both considered constant in this study.
We seek solutions for which the normal current velocity is everywhere zero, by analogy with barotropic Kelvin waves; thus u = 0. We anticipate that the fields have a modal structure, so e.g., for vertical velocity w:
w=∑nanexp⁡(knx)exp⁡i(lny−ωt)sin⁡mnz ,
with arbitrary constants an, vertical wavenumber mn = nπ/H (uniform water depth H), and frequency ω. For given frequency, kn and ln are to be determined; the former specifies the decay rate away from the wall, the latter the wavenumber along the wall. By substituting (6) into (4) and (5), one readily obtains expressions for b and υ, respectively. From the momentum equations, (1)–(3), one then obtains three results. First, an expression for p′; second, the dispersion relation, in terms of phase speed cn,
cn≡ωln=NHnπ,
which is the familiar expression for the phase speed of long internal waves without Coriolis effects, again analogous to barotropic Kelvin waves. Third, we obtain the lateral decay rate, which does depend on the Coriolis parameter,
kn=fnπNH,
implying that higher modes (i.e., higher n) decay more rapidly.

A beam structure in the along-slope direction arises from a superposition of modes. Adding more modes sharpens the structure of the beam, as illustrated in Fig. 1 (left panels). However, farther from the wall (right panels), the higher modes lose their significance compared to low modes, a consequence of (8). As a result, adding more modes does not change the solution anymore (cf. middle and bottom panels in Fig. 1).

Fig. 1.
Fig. 1.

Example of a classical internal Kelvin wave, with snapshots showing the along-slope current velocity υ at t = 3T/4 (where T is the tidal period of K1) for an increasing number of modes and at two horizontal positions. Red indicates positive values, and blue indicates negative values. (left) At the wall, x = 0 km; (right) farther into the ocean, at x = −5 km. (top) The first mode; (middle) the first three modes; (bottom) the first eight modes. Parameter values are water depth H = 4 km, buoyancy frequency N = 1 × 10−3 rad s−1, latitude ϕ = 60°N, hence Coriolis parameter f = 1.263 × 10−4 rad s−1, and tidal frequency ω = 7.292 × 10−5 rad s−1 (diurnal constituent K1). The vertical and horizontal axes are in kilometers. Modal amplitudes were chosen to be an = 1/n.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0084.1

Finally, we note that the ratio
mnln=Nω
is independent of mode number n. In other words, one and the same angle pervades all modes: the ratio represents the tangent of the angle of the beam with the vertical.

3. Beam structure of step-trapped Kelvin waves

We now consider a step-topography with an ocean of uniform depth Ho for x < 0 (equivalent to H from the previous section), and a shallow shelf of uniform depth Hs for x > 0. We use subscript ‘o’ for quantities pertaining to the ocean, and ‘s’ for those to the shelf. Buoyancy frequency N is again assumed constant throughout.

From (1) to (5), the following equation for p′ can be derived,
(∂2∂t2+f2)∂2p′∂z2+N2(∂2p′∂x2+∂2p′∂y2)=0.
We assume wave propagation to be sinusoidal in the along-step direction,
p′=p^(x,z)exp⁡[i(ly−ωt)].
Substitution in (10) then gives
(f2−ω2)∂2p^∂z2+N2(∂2∂x2−l2)p^=0.
Following Hughes and Klymak (2019), we impose continuity of pressure and cross-slope current velocity u, over the step, i.e.,
p^o=p^su^o=u^s}atx=0,−Hs<z<0.
The latter condition can readily be cast in terms of p^ by combining (1) and (2). Furthermore, we require that there is no normal flow at the step:
uo=0atx=0,−Ho<z<−Hs.
Finally, we impose that the vertical current velocity vanishes at the bottom and at the rigid-lid, which translates, via (3) and (4), into the condition that ∂p^/∂z has to vanish there.
We seek a solution that decays exponentially away from the step, with M modes involved:
p^o=∑n=0MAnexp⁡(αnxn)cos⁡[nπHo(z+Ho)],x<0
p^s=∑n=0MBnexp⁡(−βnxn)cos⁡[nπHs(z+Hs)],x>0
with
αn2=l2+(nπHo)2,βn2=l2+(nπHs)2.
By this arrangement, the conditions of no normal flow at the bottom and rigid-lid are already satisfied. The boundary conditions at x = 0 still have to be satisfied. As described by Hughes and Klymak (2019), following Chapman (1982), they are first cast into an integral form by multiplying with the vertical modes and integrating over the vertical interval in question. For the number of modes, we used M = 20, similar to Hughes and Klymak (2019). Wavenumber l is found by an iterative method; lateral wavenumbers αn and βn then follow from (17). A detailed description of the method can be found in Hughes and Klymak (2019). Thus, we obtain a sequence of values for l (the numerical procedure generally yields at most M values).

Despite some similarities with the classical internal Kelvin wave of the previous section, the solution already reveals a higher degree of complexity. Notably, the function p^(x,z) in (11) now is a sum of products of functions of x and z, which implies nonseparability in x and z for any particular along-slope wave form exp[i(ly − ωt)], in contrast to the internal Kelvin wave (6) where it was accompanied by a simple product of functions of x and z.

The corresponding modes can be added up creating along-step traveling beams. An example is shown in Figs. 2 and 3, where we have plotted the spatial structures of the amplitude and phase of along-slope current υ, along the step.

Fig. 2.
Fig. 2.

Example of a step-trapped internal Kelvin wave with a shelf depth of 200 m (green dashed line), showing the amplitude of the along-slope current υ. Darker colors indicate higher amplitudes; the lightest color represents a value of zero. (top) Close to the ridge, x = −0.5 km; (bottom) farther into the ocean, at x = −5 km. Note that the ranges of the color bars are different. Other parameter values are as in Fig. 1.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0084.1

Fig. 3.
Fig. 3.

As in Fig. 2, but for the phase of the signal.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0084.1

While beams are still present, their structure departs in a couple of ways from that of a classical internal Kelvin wave.

First, close to the shelf, the beam is confined to the layer below the step, as if it reflects from the virtual level z = −Hs. Importantly, this apparent reflection is not due to any discontinuity in N (recall that N is constant throughout) but is an inherent property of the beams of step-trapped internal waves. Farther away from the step (Fig. 2, bottom), the effect becomes less pronounced, but the signal is still weaker in the upper layer.

Another difference from classical internal Kelvin waves is that there is no longer an exact periodicity in the along-slope direction (as is most clearly seen in Fig. 3, top), which slowly changes the structure of the beams. This has to do with the fact that l no longer obeys a simple equation yielding rational proportions of different l; they now become incommensurable. The ratio ln/mn will no longer be independent of n (after all, mn = nπ/Ho remain commensurable, while ln is not). Yet, to a good approximation, the beams still follow the angle associated with N/ω from (9), as we verified by numerical inspection.

4. Discussion

The rapid decay of the beams above the top of the step (z = −Hs), and hence their apparent reflection from that level, is in line with the finding by Durran (2000) that the modes are evanescent above the top of the step. In that study, an atmosphere was considered of infinite height, in which case it is natural to impose exponential decay in the upper layer. In our case, with a finite height, the solution can remain sinusoidal in z, as expressed in (15) and (16), while still exhibiting the decaying behavior in the top layer (through a superposition of sinusoids).

The beam structure in the along-slope direction, which is nearly (though not strictly) periodic, implies that there are specific spots where the beams reflect from the ocean floor; these are potentially places of enhanced mixing.

In the setting of this paper, no forcing mechanism was included. As a consequence, we could arbitrarily choose the amplitude of the modes. In a real setting with forcing, the amplitudes will be specified by the forcing, which thus determines the relative importance of the various modes, which in turn determines the width and sharpness of the beam. In the end, the width of the beam has to be set by some external spatial scale, in this case presumably in the along-slope direction. The wavelength of the barotropic tide (equally propagating in the along-slope direction) introduces such a scale, but it does not match the much shorter baroclinic scales. A more natural scale is an along-slope irregularity in the topography, e.g., a canyon. Indeed, such spatial structures are expected to be instrumental in reshaping the barotropic tidal currents and hence in the generation of trapped internal tides. As the problem then becomes fully three-dimensional, the analysis would no longer be tractable by analytical methods, as applied in this note, but would require a numerical model, in which the barotropic tide is imposed at the open boundaries.

We are not aware of reports in the literature on these along-slope propagating internal tidal beams. A natural place to look for them would be the Pacific Ocean, where several higher-latitude areas exist with a predominantly diurnal tide, notably in the northwest Pacific and around the Pacific Antarctic embayment (see Gerkema 2019, Fig. 1.2). The analysis of this note implies that one would have to carry out measurements very close to the slope, since the beam structure rapidly disappears away from the slope, due to the lateral decay of the higher modes. Another interesting prediction is the nearly periodically spaced spots of higher mixing expected where the beams reflect from the ocean floor.

5. Conclusions

We studied the along-slope propagation of subinertial tides for the configuration of a simple step and demonstrated that they exhibit a beam structure, partly similar to that of internal Kelvin waves along a wall (discussed in section 2). In particular, the beams become less sharp away from the step, as higher modes decay more rapidly in the cross-slope direction.

The beams of step-trapped internal waves manifest themselves mainly below the top of the step, being evanescent in the upper layer, as if they reflect from the virtual level at the top of the step. Furthermore, the signal is no longer purely periodic in the along-slope direction.

Acknowledgments.

We thank Anna S. van der Kaaden for inspiring discussions on the nature of the tides at Rockall Bank (van der Kaaden et al. 2021), which led us to the present study. This work was part of a MSc project by the first two authors, supervised by the third author and co-supervised by Matias Duran-Matute (TU/e).

Data availability statement.

In section 3 we used the Python code developed by Hughes and Klymak (2019), which they made available at github.com/hugke729/RidgeTrappedWave.

REFERENCES

  • Chapman, D. C., 1982: On the failure of Laplace’s tidal equations to model subinertial motions at a discontinuity in depth. Dyn. Atmos. Oceans, 7, 1–16, https://doi.org/10.1016/0377-0265(82)90002-1.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 2000: Small-amplitude coastally trapped disturbances and the reduced-gravity shallow-water approximation. Quart. J. Roy. Meteor. Soc., 126, 2671–2689, https://doi.org/10.1002/qj.49712656904.

    • Search Google Scholar
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  • Falahat, S., and J. Nycander, 2015: On the generation of bottom-trapped internal tides. J. Phys. Oceanogr., 45, 526–545, https://doi.org/10.1175/JPO-D-14-0081.1.

    • Search Google Scholar
    • Export Citation

  • Gerkema, T., 2019: An Introduction to Tides. Cambridge University Press, 214 pp.

  • Gill, A. E., 1977: Coastally trapped waves in the atmosphere. Quart. J. Roy. Meteor. Soc., 103, 431–440, https://doi.org/10.1002/qj.49710343704.

    • Search Google Scholar
    • Export Citation

  • Hughes, K. G., and J. M. Klymak, 2019: Tidal conversion and dissipation at steep topography in a channel poleward of the critical latitude. J. Phys. Oceanogr., 49, 1269–1291, https://doi.org/10.1175/JPO-D-18-0132.1.

    • Search Google Scholar
    • Export Citation

  • Krauss, W., 1966: Interne Wellen. Gebrüder Borntraeger, 248 pp.

  • Mysak, L. A., 1980: Recent advances in shelf wave dynamics. Rev. Geophys., 18, 211–241, https://doi.org/10.1029/RG018i001p00211.

  • Smith, K. A., M. A. Merrifield, and G. S. Carter, 2017: Coastal-trapped behavior of the diurnal internal tide at O‘ahu, Hawai‘i. J. Geophys. Res. Oceans, 122, 4257–4273, https://doi.org/10.1002/2016JC012436.

    • Search Google Scholar
    • Export Citation

  • van der Kaaden, A.-S., and Coauthors, 2021: Feedbacks between hydrodynamics and cold-water coral mound development. Deep-Sea Res. I, 178, 103641, https://doi.org/10.1016/j.dsr.2021.103641.

    • Search Google Scholar
    • Export Citation
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Journal of Physical Oceanography

  • Chapman, D. C., 1982: On the failure of Laplace’s tidal equations to model subinertial motions at a discontinuity in depth. Dyn. Atmos. Oceans, 7, 1–16, https://doi.org/10.1016/0377-0265(82)90002-1.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 2000: Small-amplitude coastally trapped disturbances and the reduced-gravity shallow-water approximation. Quart. J. Roy. Meteor. Soc., 126, 2671–2689, https://doi.org/10.1002/qj.49712656904.

    • Search Google Scholar
    • Export Citation

  • Falahat, S., and J. Nycander, 2015: On the generation of bottom-trapped internal tides. J. Phys. Oceanogr., 45, 526–545, https://doi.org/10.1175/JPO-D-14-0081.1.

    • Search Google Scholar
    • Export Citation

  • Gerkema, T., 2019: An Introduction to Tides. Cambridge University Press, 214 pp.

  • Gill, A. E., 1977: Coastally trapped waves in the atmosphere. Quart. J. Roy. Meteor. Soc., 103, 431–440, https://doi.org/10.1002/qj.49710343704.

    • Search Google Scholar
    • Export Citation

  • Hughes, K. G., and J. M. Klymak, 2019: Tidal conversion and dissipation at steep topography in a channel poleward of the critical latitude. J. Phys. Oceanogr., 49, 1269–1291, https://doi.org/10.1175/JPO-D-18-0132.1.

    • Search Google Scholar
    • Export Citation

  • Krauss, W., 1966: Interne Wellen. Gebrüder Borntraeger, 248 pp.

  • Mysak, L. A., 1980: Recent advances in shelf wave dynamics. Rev. Geophys., 18, 211–241, https://doi.org/10.1029/RG018i001p00211.

  • Smith, K. A., M. A. Merrifield, and G. S. Carter, 2017: Coastal-trapped behavior of the diurnal internal tide at O‘ahu, Hawai‘i. J. Geophys. Res. Oceans, 122, 4257–4273, https://doi.org/10.1002/2016JC012436.

    • Search Google Scholar
    • Export Citation

  • van der Kaaden, A.-S., and Coauthors, 2021: Feedbacks between hydrodynamics and cold-water coral mound development. Deep-Sea Res. I, 178, 103641, https://doi.org/10.1016/j.dsr.2021.103641.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Example of a classical internal Kelvin wave, with snapshots showing the along-slope current velocity υ at t = 3T/4 (where T is the tidal period of K1) for an increasing number of modes and at two horizontal positions. Red indicates positive values, and blue indicates negative values. (left) At the wall, x = 0 km; (right) farther into the ocean, at x = −5 km. (top) The first mode; (middle) the first three modes; (bottom) the first eight modes. Parameter values are water depth H = 4 km, buoyancy frequency N = 1 × 10−3 rad s−1, latitude ϕ = 60°N, hence Coriolis parameter f = 1.263 × 10−4 rad s−1, and tidal frequency ω = 7.292 × 10−5 rad s−1 (diurnal constituent K1). The vertical and horizontal axes are in kilometers. Modal amplitudes were chosen to be an = 1/n.

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

    Example of a step-trapped internal Kelvin wave with a shelf depth of 200 m (green dashed line), showing the amplitude of the along-slope current υ. Darker colors indicate higher amplitudes; the lightest color represents a value of zero. (top) Close to the ridge, x = −0.5 km; (bottom) farther into the ocean, at x = −5 km. Note that the ranges of the color bars are different. Other parameter values are as in Fig. 1.

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

    As in Fig. 2, but for the phase of the signal.

Fig. 1.

Example of a classical internal Kelvin wave, with snapshots showing the along-slope current velocity υ at t = 3T/4 (where T is the tidal period of K1) for an increasing number of modes and at two horizontal positions. Red indicates positive values, and blue indicates negative values. (left) At the wall, x = 0 km; (right) farther into the ocean, at x = −5 km. (top) The first mode; (middle) the first three modes; (bottom) the first eight modes. Parameter values are water depth H = 4 km, buoyancy frequency N = 1 × 10−3 rad s−1, latitude ϕ = 60°N, hence Coriolis parameter f = 1.263 × 10−4 rad s−1, and tidal frequency ω = 7.292 × 10−5 rad s−1 (diurnal constituent K1). The vertical and horizontal axes are in kilometers. Modal amplitudes were chosen to be an = 1/n.
Fig. 2.

Example of a step-trapped internal Kelvin wave with a shelf depth of 200 m (green dashed line), showing the amplitude of the along-slope current υ. Darker colors indicate higher amplitudes; the lightest color represents a value of zero. (top) Close to the ridge, x = −0.5 km; (bottom) farther into the ocean, at x = −5 km. Note that the ranges of the color bars are different. Other parameter values are as in Fig. 1.
Fig. 3.

As in Fig. 2, but for the phase of the signal.
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