1. Introduction
Models and, to some extent, observations show that isolated islands and seamounts can be the site of resonant trapped wave responses at subinertial frequencies (e.g., Longuet-Higgins 1969; Hogg 1980; Brink 1989; Brink 1999). The continental shelf literature (e.g., Dale and Sherwin 1996) makes it natural to ask whether there might also be resonances, or leaky resonances, at superinertial frequencies [e.g., Chambers (1965); Longuet-Higgins (1967) for very idealized topography] as well. If so, these might have interesting implications for semidiurnal tides and tidal currents at islands. In this light, it is interesting that Baines (2007), under some circumstances, found leaky superinertial resonances in his model of a very idealized seamount.
The goal of this contribution is to understand how the waters around an island or seamount respond to superinertial tidal forcing. Specific issues include how strong the near-island response might be, what factors affect the near-resonant frequencies, and how enhanced responses might be observed. Thus, a linear model using idealized, circular, island and seamount geometries is applied to a stratified, rotating ocean with bottom friction. In particular, the roles of reef and island geometry are considered.
One can ask why this problem is worth considering. It is known that the biological productivity of a coral reef depends on the regular flushing (due to waves, winds, or tides) of oceanic water across the reef. This flushing provides dissolved nutrients to the reef, even if the nutrient concentrations are low (e.g., Atkinson 2011). Thus, if tidal near-resonance enhances cross-reef volume fluxes, one might expect that nutrient supplies are enhanced and that a reef’s productivity would then be enhanced. Given that the causes for inter-island productivity variations are very poorly known (Gove et al. 2016), it appears that tidal enhancement is worth consideration as a potential factor.
As a starting point for appreciating the physics, it is helpful to explore the straight-coast analogy. Consider a flat-bottom, stratified, rotating ocean, where the only coastal-trapped wave modes are the Kelvin waves (one for each baroclinic mode), which can exist both above and below the inertial frequency f. When the bottom is flat, none of these Kelvin modes couple to the internal wave continua (one continuum for each vertical mode). If some modest cross-shelf topography is included, the modified Kelvin waves’ dispersion curves can still cross from subinertial to superinertial frequencies, but the bottom slope allows coupling between baroclinic modes. Thus, superinertial near-Kelvin wave modes with modest topography lose energy into the internal wave continua and the “leaky” nearly trapped waves thus decay as they propagate alongshore.
For cases with stratification and substantial shelf-slope topography, the coastal-trapped wave dispersion curves can cross f to reach superinertial frequencies provided that the maximum of αN [where α is the bottom slope and N(z) is the buoyancy frequency], evaluated along the bottom, exceeds f (Huthnance 1978). Huthnance also shows that the subinertial waves behave like internal Kelvin waves when stratification is strong in the sense of a large Burger number
Actually calculating the properties of superinertial, leaky coastal-trapped waves is not a simple matter, but the problem is treated thoroughly by Dale and Sherwin (1996) and by Dale et al. (2001). Among the complications are the presence (depending on formulation) of a spurious solution at the inertial frequency, sometimes-ambiguous offshore boundary conditions, and the multiple apparent superinertial solutions.
One might expect that an island would allow wave behavior that is generally similar, albeit discretized azimuthally, to that for a straight coast. Specifically, if the margins of an island are sufficiently steep, there would presumably exist superinertial trapped wave modes with properties not unlike Kelvin waves. In this case, one might well expect leaky resonances of Kelvin-like waves at superinertial frequencies, such as those of semidiurnal tides. The situation is not altogether straightforward, though, since, as Dyke (2005) points out, trapped waves around cylindrical islands are not really Kelvin waves (even though they clearly must become such as the island becomes large relative to the Rossby radius). Specifically, Longuet-Higgins (1969) thoroughly treats waves around an island with vertical sides. For subinertial frequencies, the trapped waves (when they exist, since the island must be large enough) are solutions that die off monotonically offshore, much like a Kelvin wave. For superinertial frequencies, the wave modes have a sequence of radial nodes and are not really trapped, although for large azimuthal wavenumbers (wavelength short compared to radius), the high-mode Bessel function solutions behave qualitatively more like a decaying exponential (Abramowitz and Stegun 1964).
Thus, there is reason to expect the existence of superinertial, nearly trapped waves at islands and seamounts, provided that S is large enough. In the following, the linear response of waters near islands and seamounts to tidal-type forcing is considered. This is an approach to the questions of whether there might be leaky superinertial modes, and how their properties depend upon ambient conditions. The problem is formulated here, solved numerically and the properties of the solutions considered.
2. Formulation
At the edge of the grid, h0 is the (constant) water depth at the outer boundary and δnm is the Kronecker delta. Developing this boundary condition led to some insight (see appendix A) on the offshore boundary condition appropriate to a free wave (unforced) model of the sort considered by Dale and Sherwin (1996).
For each (±iωt ± θ) sign combination, the (r, z) boundary value problem (3)–(6) with either (7) or (8) is discretized onto a terrain-following vertically stretched grid with typically 1800 radial grid points and 100 vertical grid points. It is then solved by a straightforward matrix inversion. The grid resolution is a balance among several competing considerations: 1) the total number of grid points is a computational constraint, 2) it is desirable to resolve internal waves (which can have fairly short horizontal and vertical scales, as seen below), and 3) “hydrostatic consistency” (e.g., Haney 1991) requires that hrΔr/Δz not be too large. This last consideration, over the steep topography typically found at midocean islands or seamounts, allows for very fine, O(50) m horizontal resolution, but does not allow Δz to become too small. Tests show that this resolution allows accurate detection of critical frequencies (where internal wave phase propagation is perpendicular to the bottom slope, and particle motions are parallel to the slope) over the parameter range reported here. Given the limitations of the present linear model with an infinitesimal bottom boundary layer, many interesting phenomena (e.g., mixing, wave steepening, and nonlinear interactions; see, e.g., Liang and Wunsch 2015) do not occur, but these are not central to the present objective of seeking near-resonant phenomena.
Various configurations (Table 1, appendix B) of topography, latitude, stratification or bottom friction are treated. Case 1 is the basic case, and the other cases generally only differ in one regard (such as doubling N2 everywhere), and these differences are noted in the table and described in more detail in appendix B. Any property not noted in the table is the same as case 1. In all cases, u0 = 10 cm s−1. The Coriolis parameter f is usually kept at 0.3 × 10−4 s−1 (this should be understood as angular frequency, i.e., radians per second throughout the following) or less in order to make sure the results are not contaminated by spurious effects near the inertial frequency (e.g., Dale et al. 2001). In most cases, the forcing frequency is varied over a range, from 0.40 × 10−4 to 4.00 × 10−4 s−1, that includes the diurnal and semidiurnal tidal bands. The exceptions are two cases (12 and 13) with f = 0.779 × 10−4 s−1, where response is only studied for ω ≥ 0.90 × 10−4 s−1. Response peaks (“leaky resonances”) are resolved to a frequency resolution of 0.01 × 10−4 s−1. Response is measured in terms of azimuthally averaged quantities: total kinetic energy averaged over the entire domain (eke), variance of coastal sea level response ζC, or variance of total depth-averaged flow
Summary of island cases.
In reality, at a leaky resonance, one would expect large amplitudes so that nonlinearities come into play. Further, dissipation, both near the bottom and within the water column, will become more complicated than allowed by the present simple representation of bottom friction. These difficulties do not detract from the two objectives of 1) showing that leaky resonances can be expected, and 2) examining parameter dependencies of the “resonant” frequencies. Thus, little emphasis is placed on the response magnitude at resonant peaks, since the computed peak height is likely not very meaningful. In addition, the peaks themselves are generally so sharp that it would call for a good deal more computation to define their exact maximum frequencies much more closely than the 0.01 × 10−4 s−1 resolution used in the following. In contrast, the calculated responses at off-resonant frequencies appear to be locally (in ω) less sensitive, and thus more nearly reliable than the magnitudes of resonant peaks.
3. Results
a. A basic case
A basic example uses a circular island with ambient stratification and topography representative of Bermuda, and with an idealized 10-m-deep, 10-km-wide bounding reef at the outer edge of the shelf (Fig. 1). The bottom friction coefficient R = 0.05 cm s−1 everywhere (except very close to the offshore boundary where it vanishes), the Coriolis parameter is f = 0.2 × 10−4 s−1 (corresponding to 8° latitude) and the ambient motion is rectilinear (δ = 0). A smoothed observed N2 profile from near Bermuda is used. The Burger number S is 15 (when L = 38 km is taken to be the distance from the outer edge of the reef offshore to where the bottom becomes flat). Thus, the insular slope (the equivalent of the continental slope along a straight coastline) topography is narrow relative to the first mode Rossby radius, 139 km. The azimuthally averaged variance of the total (forcing plus response) depth-integrated radial flow
Island topography for the basic case. Only the uppermost 500 m are shown.
Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0298.1
Frequency dependence of azimuthally averaged variance of depth-averaged flow at r = 20 km, i.e., at the center of the reef for the basic case (case 1: heavy line), case 7 (light dotted line), and case 13 (light solid line). Vertical dashed lines represent well-known tidal frequencies and the inertial frequency. The horizontal dashed line is the averaged variance of the ambient radial flow, i.e., what the variance would be if there were no island.
Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0298.1
Variance of total depth-averaged radial velocity component for case 1, ω = 1.21 × 10−4 s−1. The cross-hatched area in the center is land. The inner and outer boundaries of the reef are shown by dashed black–white contours: the outermost one is 150 m, and all others are 15 m.
Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0298.1
Some insight can be gained by considering the spatial structure of the variance for uT, υT, and p at the strongest peak, 1.21 × 10−4 s−1. Pressure has the simplest and smoothest spatial structure. Its maximum variance is near the surface offshore of the reef (Figs. 4 and 5), and the maxima, in plan view, occur at an angular offset relative to the forcing. Inside the reef, pressure variance is far weaker than just outside, consistent with the large cross-reef pressure differences associated with passing water above a shallow reef where frictional forces are particularly effective (the inverse frictional decay time R/h here is 0.5 × 10−4 s−1, comparable to the frequency). Pressure variability offshore of the reef decreases with depth (Fig. 5), but passes through a near-node at depth z = −1400 m, consistent with the structure of an m = 1 baroclinic mode [Eq. (6)]. In contrast, radial velocity uT variance structure (Fig. 6) is dominated by internal wave rays, and it is difficult to discern any larger scale maximum of the sort so evident in pressure. This structure is reminiscent of the internal wave physics of interest to Eriksen (1998) on the flanks of a seamount. Azimuthal velocity υT variance (Fig. 7) has lower amplitude than uT variance (by about 20% for maxima in this example) and is structurally intermediate between u and p: while internal wave rays are clearly visible, there is also a broad upper-ocean variance enhancement similar to that in pressure. Density fluctuations have a structure (not shown) comparable to that for υT, in that maximum variations occur just offshore of the reef at 5–10-m depth, but with minimal variations at the surface. The peaks that closely neighbor 1.21 × 10−4 s−1 all have structures similar to Figs. 5–7, but with minor deviations.
Variance of surface response-only pressure for case 1, ω = 1.21 × 10−4 s−1. The central cross-hatched area is land, and the rectilinear ambient current ⟨u⟩ oscillates in the x direction. The inner and outer boundaries of the reef are shown by dashed black–white contours: the outermost one is 150 m, and all others are 15 m.
Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0298.1
Azimuthally averaged variance of response pressure for case 1, ω = 1.21 × 10−4 s−1.
Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0298.1
Logarithm of azimuthally averaged variance of total (forcing plus response) radial velocity for case 1, ω = 1.21 × 10−4 s−1.
Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0298.1
Logarithm of azimuthally averaged variance of total (forcing plus response) azimuthal velocity for case 1, ω = 1.21 × 10−4 s−1.
Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0298.1
It is useful to explore the magnitude of the tidal response at this island. One measure is the horizontal dashed line in Fig. 2 that shows the variance associated with the ambient flow. For most frequencies in this example, the total
The groups of peaks at ω = 0.45 × 10−4 and 0.58 × 10−4 s−1 are tied to spatial structures similar to those for ω = 1.21 × 10−4 s−1 except that they are evidently related to higher baroclinic (vertical) modes. Specifically, the grouping near ω = 0.45 × 10−4 s−1 is characterized by pressure structures that have three nodes in the vertical (like an m = 3 baroclinic mode), while near ω = 0.58 × 10−4 s−1 the pressure structures have two nodes in the vertical (Fig. 8). The higher-frequency grouping (near ω = 2.64 × 10−4 s−1; Fig. 9) shows clear surface intensification in pressure structures, but no consistency about the presence of vertical modes. On the other hand, all of the members of this group show a second pressure variance maximum at the surface at a radius of 60–70 km. This secondary maximum is not unexpected, given results for a circular island with vertical walls (e.g., Chambers 1965; Longuet-Higgins 1967) where the superinertial modes have comparable radial nodes with an increasing number of zero crossings as radial modal number (and thus frequency) increases.
Azimuthally averaged variance of response pressure for case 1, ω = 0.58 × 10−4 s−1.
Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0298.1
Azimuthally averaged variance of response pressure for case 1, ω = 2.64 × 10−4 s−1.
Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-20-0298.1
Table 1, which summarizes the prominent resonant peaks, attempts to sort the peaks in terms of their pressure “modal” structures. Specifically, peaks with simple radial structures (decaying offshore of the reef), and with clear vertical modal structures, are identified with baroclinic modes m = 1, 2, or 3. Also, peaks associated with pressure structures that have a clear second variance peak at a larger radius (as in Fig. 9) are identified as having multiple maxima. Finally, there is a category for response peaks that do not readily fit into one of the previous descriptions, including the situation (typified by case 13) where pressure variance is nearly depth independent but monotonically decreasing offshore. In the following, all of the cases involve configurations identical to the base case (case 1), except for whatever aspect is noted in the last column of Table 1. Appendix B has more complete descriptions of configurations.
b. Different topographies
One immediate question is the extent to which the reef, with its substantial frictional impedance, acts as a vertical wall. This is tested with case 2, where a vertical wall is placed at the outer edge of the reef. Similar to case 1, response peaks are found at ω = 0.54–0.57 × 10−4, 1.2–1.3 × 10−4, 2.41 × 10−4 s−1 (Table 1), and these have associated spatial structures similar to those for the basic case (Figs. 5–7) in the same frequency ranges. This is not to say that the reef is impermeable, but far from it: there is always an active exchange across the reef as long as it is not completely closed off.
In contrast, if the reef is removed, so that there is a constant, relatively gently sloping “shelf” out to the 200-m isobath (case 3), the resonant frequencies are also similar to the basic case (Table 1). It thus seems that for this S, for the purposes of determining the resonant modal frequency, the outer edge of the reef (the “shelf break”) can, as a first approximation, be replaced effectively by a vertical wall. Some caution is required about this conclusion, however, since it is possible that it may not extend to cases with wider topography relative to Rossby radius. Indeed, making the same comparison with two cases having wider “continental slopes” (cases 4 and 5: S = 6) shows that the m = 1 (first baroclinic mode) response peaks agree fairly well (1.20 versus 1.27 × 10−4 s−1) between cases with a reef and a wall at the reef edge. Also, there is some agreement for one higher-frequency (3.62 × 10−4 s−1), radially nodal, peak. However, the agreement in terms of peak frequencies is poor otherwise. The conclusion appears to be that the details of the topography are more important as S gets smaller, even with S > 1.
It is also useful to compare the base case (1, S = 15) to a similar case having a wider, gentler insular slope (4, S = 6). Again, agreement is reasonable for the frequency peak having p structure resembling a radially trapped, m = 1 baroclinic mode (1.21 × 10−4 s−1), but the trapped, higher baroclinic modes are not obviously present, and there are more, and more energetic, higher-frequency (ω > 2.44 × 10−4 s−1) peaks that do not match up with the base case. It is tempting to conclude that, for a given vertical mode, if its internal Rossby radius (139, 65, and 48 km for modes 1, 2, 3) is substantially larger than the insular slope width (55 km for case 4 and 38 km for case 1), that there will be a near-resonant peak in that mode corresponding to a radially trapped pressure structure.
c. Changing internal wave speeds
Three cases effectively change the internal wave modal speeds in the ambient ocean. In case 6 (see Table 1), the thermocline stratification is strengthened somewhat, so that c1 changes from 277 to 287 cm s−1. This leads to a moderate increase in the resonant frequencies, e.g., from 1.21 to 1.26 × 10−4 s−1 for the strongest (m = 1) peak. Increasing N2 by a factor of 2 everywhere (case 7: Fig. 2) increases the ambient, flat-bottom internal wave speeds cn by 41% for all modes, and it also increases the m = 1 and m = 2 resonant frequencies by the same fractional amount. For case 8 (maximum ocean depth hMax is decreased to 3000 m, compared to 4400 m for case 1), the first two internal gravity wave speeds decrease by about 13%, and the resonant frequencies are lowered by 14% and 34% (for m = 1 and m = 2, respectively). Using a 50% larger Coriolis parameter (case 9, which is like case 1 otherwise) does not make much difference in the resonant frequencies. The tentative conclusion is that, as in the case of a cylindrical island (e.g., Longuet-Higgins 1969), the ambient long internal gravity wave speed is a key parameter in setting the resonant frequencies. This outcome should not be surprising since the cases considered all have Burger numbers of about 6–15: well into the range where wave frequencies ought to be determined more by stratification than by topography or rotation (e.g., Huthnance 1978).
d. Island radius
Increasing the island’s radius without changing the width of the topography (case 10, which is like case 1, except that all shoreline and isobath radii are increased by 10 km, thus leaving bottom slopes unchanged) lowers the principal m = 1 resonant frequencies (from 1.21–1.31 × 10−4 to 1.12–1.21 × 10−4 s−1), consistent with the notion that it would take more time for a wave to propagate around an island. However, the decrease (about 7%) is not consistent with the change in the island’s outer reef circumference (38%). For the second vertical mode, the larger circumference appears to be associated with a higher (!) resonant frequency than in the base case (Table 1). It is not obvious why this might be.
e. Latitude
The Coriolis parameter does seem to make some difference in terms of the amplitude of the tidal response (Table 1: case 9 versus 1). Comparing
f. Elliptical ambient currents
Two cases use δ = 1, i.e., the ambient tidal flow traces an elliptical hodograph rather than a strictly rectilinear one. When conditions are identical to case 1, only with δ = 1 (case 11), very little changes in terms of the primary response peaks over the entire frequency range, although amplitudes increase slightly. One minor variation is that with elliptical flow, the 1.31 × 10−4 s−1 peak corresponds to a pressure structure that is not so readily associated with a baroclinic mode as in case 1. Rather, it has a radial structure similar to that of case 1 with ω = 1.21 × 10−4 s−1 (Fig. 5), except that it is surface-intensified without having a secondary pressure variance maximum at depth.
Case 12 is meant to represent an idealized version of Bermuda, thus f = 0.779 × 10−4 s−1 (c1/f = 36 km, S = 0.9). A narrower range of frequencies (0.90–4.0 × 10−4 s−1) is considered in order to avoid the spurious mode at ω = f. The frequencies for the m = 1, 2, and 3 peaks are all higher (by 6%–53%) than in the basic case, while the peaks associated with multiple radial pressure extrema are almost unchanged (2.68 × 10−4 versus 2.64 × 10−4 s−1). Often, for subinertial coastal-trapped waves along a straight coast, wave frequency increases when S is not large and |f| increases (e.g., Brink 1982), but the relevance of this finding to superinertial fluctuations near an island is not obvious. It is worth noting that Hogg (1980) sought coherences among records from moorings on about the 1100-m isobath around Bermuda. He found a faint (but significant) temperature coherence peak at ω = 1.25 × 10−4 s−1 (14.0 h period) that corresponds with a model m = 1 peak at 1.28 × 10−4 s−1 (Table 1, case 12). Further, the phase difference (21°–97°) is consistent in sign and magnitude with what would be expected for a Kelvin-like wave.
g. Very weak stratification
Huthnance (1978) shows that, for a straight coast, the maximum frequency of a coastal-trapped wave is ωM, the maximum of (hxN) evaluated at the bottom (where hx is the offshore bottom slope). Thus, if ωM/f > 0, the wave’s dispersion curve will reach and cross ω = f, although the superinertial modes are expected to leak energy into internal waves (Dale and Sherwin 1996). For model runs 1–12 and 14–16, the maximum of (hrN/f) is greater than 10, so that it seems highly probable that subinertial island-trapped waves will have their azimuthally discretized dispersion “curves” reach up to, and cross ω = f. Thus, it is not surprising that many of the superinertial resonant peaks found in the above calculations have vertical structures that resemble internal Kelvin waves. One might then ask how the system responds when wave dispersion curves are not expected to cross f. Case 13 (Table 1, appendix B) deals with this possibility by decreasing N2 everywhere by a factor of 200, and by using f = 0.779 × 10−4 s−1 so that the maximum of (hrN/f) is 0.9. In this instance, only barotropic superinertial resonances are expected. Not surprisingly, then, case 13 (Fig. 2) shows only very weak, broad response maxima (it is hard to call these “peaks”) and all of them display a pressure modal structure that is essentially barotropic offshore of the reef. This outcome is consistent with the idea that internal wave modal structures play a dominant role in determining superinertial resonances around islands with steep (in the sense of large Burger number) sides.
h. Secondary radial maxima
Most cases have a clustering of resonant peaks (analogous to those near ω = 2.64 × 10−4 s−1 in case 1) that have one or more secondary pressure variance maxima as a function of radius (e.g., Fig. 9). These frequencies (greater than about 1.65–3.64 × 10−4 s−1, depending on conditions) are recorded in the fifth column of Table 1 as having multiple p variance maxima offshore of the reef. The one example with no secondary radial maximum is case 13, which has very small N2 (decreased by a factor of 200 compared to case 1), and where internal Kelvin wave-like responses are not to be expected. For the grouping of resonant peaks having multiple radial maxima, the resonant frequencies increase proportionally to the internal wave speed c1 (e.g., cases 6 or 7; see also cases 1 versus 7 in Fig. 2 at higher frequencies). This frequency range does not appear to depend on island radius when stratification and bottom slope are held constant (cases 10 versus 1). In contrast, when topography is changed substantially without changing stratification [with no reef (case 3) or with a wide slope (case 4)], the band’s frequency (relative to case 1) can change substantially. It is natural to compare these solutions to Chambers’ (1965) solutions having radial nodes in a flat-bottom ocean with a vertical coastal wall. Given c1 = 277 cm s−1, the first such mode occurs at frequency 2.1 × 10−4 s−1 for a cylindrical island of radius 50 km, and at 3.8 × 10−4 s−1 for radius 27 km (i.e., the outer edge of the reef). Higher vertical modes lead to lower frequencies. Thus, the Chambers theory predicts a frequency for radially nodal waves that is reasonable, although his result depends more strongly on island radius than found here with more realistic topography. Given the structure of the pressure variance and the frequency prediction, it thus seems likely that, to some extent, the higher-frequency peaks (e.g., around 2.64 × 10−4 s−1 for case 1) may be associated with these baroclinic radially nodal (albeit leaky over topography) types of solutions.
i. Bottom friction
The case 1 bottom resistance coefficient of R = 0.05 cm s−1 (representative of what is used in the continental shelf literature) is very likely far too small for use with a rough coral reef. Thus, other cases (14 and 15) treat stronger bottom friction or a shallower reef (16) with all other parameters as in case 1. Not surprisingly, doubling R everywhere (case 14) decreases the M2
j. Seamounts
Several of the cases summarized in Table 1 are repeated with the island removed (Table 2). These are numbered as “1s,” etc., so that the case 1 island is comparable to the case 1s seamount. In each case, the depth inside of the reef is kept constant at 56 m, which is the greatest depth inshore of the reef in island case 1. At the reef and farther offshore, topography is as in the island case. Given that the reef appears to act as an almost solid boundary in terms of determining island resonant frequencies, one might expect the near-resonant frequencies in the seamount case to be essentially the same as in the island case. This is indeed true. For example, comparing the basic island (1) and seamount (1s) cases (Tables 1 and 2), all five major
Summary of seamount cases.
One can compare the strength of the nonresonant M2 responses. For u0 = 10 cm s−1 and δ = 0, the variance of midreef
4. Estimating near-resonant frequencies
For most cases considered (all but case 13, which has very weak stratification), there are clear response peaks corresponding to an m = 1 vertical mode in the ambient ocean. Very often, this is the strongest resonance, based on the width of the peak or on its variance maximum. Further, there are often m = 2 and m = 3 peaks. The island’s near-resonances are apparently related to the internal wave speeds, at least when the topography is steep. For example, for case 1, the ratio of the internal wave speeds c1:c2:c3 is 1:0.47:0.35, and the ratio of island-trapped (in p) baroclinic modal-like calculated resonances ω1:ω2:ω3 = 1:0.48:0.38. For case 4, where the island’s slope is widened and S is smaller (S = 6, compared to S = 15 for case 1; appendix B), topography plays a stronger role and it is hard to even identify an m = 2 response from the model calculations.
When the island’s flanks are steep in the sense of large S, it is tempting to think of the resonances as reflecting dynamics similar to azimuthal mode 1 internal Kelvin waves, i.e., waves that have a wavelength equal to one circumference of the island. Based on the above results, it seems reasonable to estimate the circumference based on the radius of the outer edge of the reef (rR = 27 km for case 1, where c1 = 277, c2 = 130, and c3 = 97 cm s−1). In that case, one would expect near-resonant periods of 2πrR/c1, 2πrR/c2, 2πrR/c3, …, or ω = cm/rR ≈ 1.03 × 10−4, 0.48 × 10−4, 0.36 × 10−4, … s−1. These frequencies are to be compared (case 1) with 1.21–1.31 × 10−4, 0.58 × 10−4, 0.45 × 10−4 s−1, so that, roughly speaking, ω ≈ 1.2cm/rR for the present steep-sided (i.e., large S) examples (e.g., cases 1, 2, 3, 6, 9, 10).
Semidiurnal tidal leaky resonances for internal Kelvin wave-like modes at a steep-sided island are possible as long as the tidal frequency is equal to or lower than the highest of these frequencies, roughly ωT ≤ 1.2c1/rR., where ωT is the tidal frequency. This, in turn, places a constraint on the size of the island. Specifically, if semidiurnal Kelvin-like m = 1 tidal resonances of this type are to be possible, then (for c1 = 277 cm s−1), a steep-sided island must have rR ≤ 22 km.
Some comments should be made.
It is not obvious how to estimate leaky resonance frequencies when the island’s sides are not steep (in the sense of large S), because the azimuthal phase speed, depending on both topography and stratification, is harder to estimate in general.
While it is apparent that resonances which have multiple pressure extrema as a function of r correspond to radially nodal modes, a straightforward frequency scaling is again not obvious.
The scaling based on Kelvin-like modes is troubling, because it assumes a simple Kelvin wave propagating around the island, but it is clear (e.g., Chambers 1965; Longuet-Higgins 1969) that, for vertical island walls, all superinertial waves should be oscillatory as a function of r, not radially decaying. It is easy enough to show how the governing Eq. (3) becomes its Cartesian equivalent for large island radius, but it is less obvious how the Bessel function solution to (3) can limit to a conventional Kelvin wave (especially for azimuthal mode 1). It appears that a steeply sloping bottom is required in order to obtain a solution that looks like a leaky Kelvin wave.
5. Conclusions
Results of a linear, stratified island model with realistic, albeit strictly circular, geometry show that indeed, leaky resonances can occur at superinertial frequencies. The pressure modal structures vary substantially, depending on frequency, island geometry, and so forth. In the limit of an island with very steep flanks, there are response peaks that appear to correspond to internal Kelvin waves propagating around the island. This simple picture becomes more complicated when the island sides are not so steep (i.e., when the insular slope width becomes comparable to, or larger than, the internal Rossby radius of deformation). At higher frequencies, resonances are often found that resemble the baroclinic version of the large-scale, radially nodal modes explored by Chambers (1965) in a simple geometry. The radically different response behavior (fewer, far weaker response peaks) in a case with very weak stratification demonstrates that, over the whole frequency band considered, substantial stratification is important for allowing near-resonant responses in the frequency range considered here (periods of 4–44 h with ω > f).
The response peaks correspond to elevated flushing across the island’s bounding reef, and the enhancement appears to be strong enough to increase flushing by about a factor of 5 relative to a nonresonant response. However, this estimate needs to be taken with an enormous grain of salt because the model, being strictly linear, and with dissipation only at the bottom, undoubtedly breaks down to some extent when nonlinearity and interior mixing and dissipation come into play. However, the real point is that the enhancements are possible, although quantitative predictions require a more realistic model. The leaky resonances have biological implications because the productivity of a coral reef is known to depend on the speed of the flow over the reef (e.g., Atkinson 2011). It seems possible that some actual islands exist that are near-resonant at an important tidal frequency and thus have anomalously high reef productivity relative to similar, but nonresonant islands or atolls.
Given the importance of internal waves for ocean mixing, it is interesting to consider the outward energy flux predicted at the M2 frequency. For u0 = 10 cm s−1, the outward depth-integrated internal wave energy flux (estimated near the outer grid boundary) for the present model runs (excepting run 13 which has unrealistically weak stratification) is in the range of 1–28 × 108 W. This loss rate is comparable to the frictional dissipation for most of these runs, but radiation is clearly the stronger loss in stratified runs with no reef. The outward fluxes are large when the M2 response amplitude (Table 1) is large and vice versa. One point of comparison for these flux numbers is the circular seamount and island numerical calculations of Holloway and Merrifield (1999) who get outward fluxes of O(3 × 1011) W when normalized to u0 = 10 cm s−1. They find that if the seamount is elongated in the sense of blocking more of the ambient flow, the internal wave energy radiation increases accordingly. Further comparison with their model is problematic, because of differing geometries and questions about their relatively coarse (by today’s standards) 2-km grid spacing. A second point of comparison is the linear “pillbox” seamount model of Baines (2007), who gives 108 W (scaled to the present forcing) as a representative loss to internal waves at a seamount. Finally, Zhang et al. (2017) treat relatively small Gaussian seamounts, but [scaling with their Eq. (7) to account for radius, height, and forcing amplitude] their M2 energy conversion for a single feature appear to be in the same magnitude range as the present results.
Finally, one might ask how these island near-resonances might be detected in nature. One immediate complication is that the results presented here are almost invariably azimuthal averages. In many cases (e.g., sea level; Fig. 4), there is substantial azimuthal dependence, so that using information from a single location (i.e., a single azimuthal angle) near an island could be very misleading. Add to that the fact that reefs are a very heterogeneous environment: it is hard to find a “typical” place to place a discrete instrument. Related to this is the apparent sensitivity of results at a single point to ray path geometry, which in turn implies that, at a fixed location, resonant peaks could appear to come and go in response to minor changes in stratification or ambient flow. Further, it is difficult to do a controlled experiment: if the cross-reef velocity is x here, how do you compare that to y at another island? It would seem that, in general, one would need to do a detailed, realistic numerical model and compare it to thorough in situ observations. This is a daunting prospect if only because steep-sided islands are very challenging computationally.
Another approach would be to estimate tidal amplitudes strictly via satellite remote sensing, and compare these results to estimates based on island sea level records. Because usually tidal sea level variability is dominated by large-scale (relative to the island) structure, the 20–100 cm a1—see Eq. (1c)—part of the tide (which does not drive currents at an island or seamount) will dominate and often mask the pressure response to tidal currents. However, when near-resonant conditions happen to prevail, then it is possible that the perturbation to island sea level tides is a few centimeters (see section 3a), which might conceivably be detectable. A stronger pressure signal is consistently found near the outer reef edge than at the coast (e.g., Fig. 4), but then azimuthal dependence may be important and such records may not be plentiful. Thus, a thorough catalog of island tides versus altimeter tides might find a few cases where the island tidal heights are at substantial variance with the ambient. Perhaps, the most observable, albeit indirect and qualitative, hint of tidal near-resonance would be the detection of anomalously productive reef ecosystems.
Acknowledgments
Conversations with Ruth Musgrave, Steve Lentz, and Carl Wunsch were very helpful. Discussions with Eric Hochberg helped stimulate this work. Three helpful reviewers led to real improvements in the presentation.
Data availability statement
No data were used in this research.
APPENDIX A
The Outer Boundary Condition in a Superinertial Free Wave Problem
Dale and Sherwin (1996) carefully examine the problem of superinertial nearly trapped waves propagating along a straight coast in an ocean with continuous stratification and a sloping bottom. They express their offshore boundary condition in terms similar to (5) and (6) here, but they encounter difficult choices because it is not always possible to have solutions that both radiate energy offshore and that decrease in amplitude offshore. The following is an attempt to rationalize this situation.
Assume for now that l is real and that ω is complex, as would be the case in an unforced problem initialized with sinusoidal alongshore variations. Then the imaginary part of
At first glance, the result Im(γm) > 0, giving offshore amplitude increase, seems implausible. However, it actually makes sense. Consider the initial value problem. The Kelvin-like wave starts out with a certain amplitude and then decays with time due to offshore radiation. As time goes by at a fixed offshore distance, the amplitude of the internal waves radiated offshore also decay with time because the fundamental wave is decaying. But, farther offshore, the internal waves are “older” and have been generated nearshore earlier in time when the basic Kelvin-like wave had a larger amplitude. Thus, waves radiating away from the coast can have a larger amplitude farther offshore. There is then nothing wrong with having Im(γm) > 0, and no choice need be made between offshore decay and offshore radiation.
The situation is murkier if one happens to assume (as did Dale and Sherwin) that ω is real and that l is complex. A physical setup that would correspond to real ω would be a wave maker at some y location with a constant frequency, and where the complex l corresponds to alongshore propagation and decay. In this case, with Re(γm) > 0 one can again get solutions with Im(γm) either positive or negative. But, in this case, there is now offshore growth/decay at the same rate for all y, including right at the wave maker. One might expect that near the wave maker, little wave energy has been lost offshore, but that older waves (that might increase offshore) would be found farther away (in y) from the wave maker. However, the offshore growth/decay rate is the same for all y when ω is real. This does not have such an obvious physical interpretation.
These sorts of questions do not arise in the present, forced, context because ω and the azimuthal wavenumber (±1) are both real and determined by the tidal forcing.
APPENDIX B
Descriptions of Model Runs
The descriptions of the different model runs, being tightly confined by space, are undoubtedly too terse in Table 1. In the following, better descriptions are provided for each island run. The corresponding seamount (“s”) runs (Table 2) are identical to the same-numbered island run except for deleting the island and making the depth inside the reef 56 m everywhere.
This basic case is described in the text (section 3a) and in figures. The reef is 10 m deep, f = 0.2 × 10−4 s−1, δ = 0, and R = 0.05 cm s−1 everywhere (except near the outer boundary). The maximum bottom slope is 0.237.
For this case, there is no shelf. A wall is placed at r = 24.9-km radius, the outer edge of the flat part of the reef. Depth at this new coastal wall is 10 m, the same as the coastal depth in case 1; N2 and R are as in case 1. This case is meant to test the extent to which the topography over the reef and inshore makes a significant difference in resonant frequency. The
variance is computed just offshore of the coastal wall. In this example, there is no reef. The depth at the coastal wall is 10 m (as in case 1), and then depth increases linearly with radius out to r = 26.9 km, where the depth is 200 m. Farther offshore, the topography is identical to case 1; N2 and R are as in case 1. This case is meant to test the importance of a reef for resonant frequency.
In this case, the insular slope slopes much more gently than in case 1, but the topography of the reef and inshore is identical to case 1. Specifically, for r = 19.9–75 km, the bottom slope is uniformly 0.076. The depth in the flat bottom region offshore of 75 km is 4400 m, the same as in case 1; N2 and R are as in case 1.
Case 5 is the same as case 4 except that, like case 2, a 10-m coastal wall is placed at the outer edge of the reef, r = 19.9 km; N2 and R are as in case 1. The
variance is computed just offshore of the coastal wall. This case is the same as case 1 except that the stratification in the main thermocline is substantially stronger. Specifically, the maximum N2 is roughly doubled, but the N2 profile is identical to case 1 below 400 m. The first mode internal wave speed is 287 cm s−1, compared to 277 cm s−1 in case 1; R is as in case 1.
Case 7 is identical to case 1 except that N2 is doubled everywhere. Thus, c1 = 406 cm s−1; R is as in case 1.
In this case, the water depth only extends down to 3000 m. Inshore of this isobath, topography is the same as case 1; N2 and R are as in case 1.
This case is identical to case 1 except that f = 0.3 × 10−4 s−1.
This case is identical to case 1 except that 10 km are added to the radius of the coast and of all isobaths. Thus, the bottom slope remains as in case 1 but the island is bigger.
This example is identical to case 1 except that δ = 1 in (1b), so that the ambient velocity field traces out a frequency-dependent ellipse rather than a constant (with frequency) rectilinear motion.
This case is meant to emulate the island of Bermuda in a relatively realistic manner. The topography and stratification are the same as in case 1, but δ = 1 and f = 0.779 × 10−4 s−1.
This case is meant to consider the result of island-trapped wave dispersion curves not reaching up to ω = f; i.e., N2 and f are chosen so that the maximum value (evaluated at the bottom) of Nα/f = 0.9 < 1, where α is the bottom slope. It is anticipated that there will be no leaky mode resonances in this case. Specifically, f = 0.779 × 10−4 s−1 and the case 1 N2 is divided by 200 everywhere.
This case is identical to case 1 except that the strength of the bottom friction is doubled everywhere. Specifically, R = 0.10 cm s−1 everywhere (except near the outer boundary).
For this case, the reef is 5 m deep everywhere, as opposed to case 1 where the reef is 10 m deep. Also, the bottom frictional coefficient is spatially variable, and strongest over the reef. Specifically R = 1 cm s−1 on the reef, and tapers to 0.05 cm s−1 everywhere away from the reef top (except near the outer boundary where it goes to zero).
This case is identical to case 1 except that the reef is 5 m deep instead of 10 m. The bottom frictional coefficient is constant at R = 0.05 cm s−1 everywhere except near the outer boundary.
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