1. Introduction
This paper examines the resonance of diurnal tides in the open and coastal oceans and the coupling between the open-ocean and coastal diurnal tides. We are motivated by the inherent interest in understanding the global tidal system and by recent studies which strongly suggest that tides of the ice age, during which lower sea levels implied a much reduced area of continental shelves, were much larger than those of today (e.g., Thomas and Sündermann 1999; Egbert et al. 2004; Arbic et al. 2004b, 2008; Uehara et al. 2006; Griffiths and Peltier 2008, 2009; Green 2010; Hill et al. 2011).
The topic of tidal resonance has a long and rich history. The open-ocean tides have been argued to be near resonant in both admittance studies of observations (Wunsch 1972; Garrett and Greenberg 1977; Heath 1981) and in studies noting the similarities between free oscillations computed in both idealized and realistic geometries (Rao 1966; Longuet-Higgins and Pond 1970; Platzman et al. 1981; Platzman 1991; Zahel and Müller 2005; Müller 2007) and the actual ocean tides. A number of studies have argued that selected coastal regions around the globe are resonant to semidiurnal or diurnal tidal forcing (e.g., Garrett 1972; Clarke 1991; Sutherland et al. 2005; Arbic et al. 2007; Cummins et al. 2010, among others).
Here, as in two previous papers, we add to the discussion of tidal resonance through experimentation with global forward tide models. Arbic et al. (2009, hereafter AKG) and Arbic and Garrett (2010, hereafter AG) explored the resonant attributes of the global semidiurnal tides. Following up on our work in AKG and AG, here we conduct a systematic exploration of resonance in a forward near-global model of the diurnal tides. The forward ocean tide model simulations are interpreted primarily with a model of free oscillations computed for realistic global ocean geometries and ocean physics and including frictional terms and the full self-attraction and loading effect (Müller 2007). Note that the set of free oscillations are often called “normal modes”, and we will at times refer to the global free oscillations model as a “modal synthesis” model. Note also that both the forward ocean tide model and the set of free oscillations are “numerical.” To avoid confusion, in this paper the term “numerical” will not generally be used in the description of either model.
Ocean tides are impacted by several factors. For instance, they are influenced by solid-earth body tides and the self-attraction and loading (SAL) term (Hendershott 1972), as well as by damping (quadratic bottom boundary layer drag and topographic internal wave drag). These factors will be described later in the paper. Below we describe the impact of oceanic length and depth scales.
The structure of the tides is also dependent upon the spatial structure in the astronomical tidal forcing. The equilibrium tidal forcing of the largest semidiurnal tides in the ocean (M2, S2, N2, and K2) is proportional to the spherical harmonic 
The factors that control diurnal tides will be examined in detail here. In the forward ocean tide model we vary the frequency ω, zonal spatial structure, and meridional spatial structure of the astronomical diurnal tidal forcing. The set of global free oscillations offers predictions of tidal sensitivity to forcing frequency. The spatial structures associated with the global free oscillations also permit an examination of the sensitivities of diurnal tides to both zonal and meridional spatial structure in the astronomical tidal forcing.
Because tides strongly depend upon water depths and basin length scales, they respond sensitively to changes in coastal ocean geometry. An additional factor in the sensitivity of open-ocean tides to changes in coastal regions is the substantial fraction of globally integrated tidal dissipation taking place in such regions (Egbert and Ray 2003). As previously noted, forward tide models suggest that tides of the ice ages were much larger than those of today. During ice ages, lower sea levels led to the removal of significant areas of present-day shelves. Motivated by this reduction in shelf area, and by the inherent interest in understanding the coupling of open-ocean and shelf tides, AKG and AG examined the response of open-ocean semidiurnal tides to the blocking out of regions of significant coastal semidiurnal tides. Here we perform experiments in which regions of significant coastal diurnal tides are blocked out.
It can be difficult to ascertain whether changes in geometry or in dissipation lie behind the impacts of the blocking experiments. The blocking experiments for semidiurnal tides in AKG and AG were interpreted with two highly simplified analytical models of open ocean–coastal tidal coupling. The coupled oscillator model of AG considers the shelf and the open ocean to each be a damped single spring system; the two are then coupled together. The one-dimensional nonrotating shallow water model of AKG considers the open ocean and shelf to both be boxes. Their dynamics are coupled because the smaller box is open to the larger box on one side. In both of these analytical models, if the natural periods of the shelf and open ocean are both set to be near the forcing period, removal of the shelf leads to substantial perturbations—generally, increases—in the open-ocean tides. This suggests that even a very small oscillator can have a profound effect on a nearby much larger oscillator it is coupled to, especially if both are near resonance. The AG and AKG analytical models allow for both the effects of coastal resonance and coastal tidal dissipation on the open-ocean tides; see for instance Eqs. (12)–(18) of AG.
In this paper we will show that the Sea of Okhotsk is resonant to diurnal tidal forcing and that removal of the Sea of Okhotsk leads to larger perturbations to the open-ocean diurnal tides than does removal of other coastal regions of similar area. This result is consistent with the simple interpretive models of AKG and AG, which continue to predict a large “back effect” of resonant coastal tides upon the open ocean when their governing parameters are changed from those suitable for semidiurnal tides to those suitable for diurnal tides. However, we will not utilize the simple interpretive models in this paper nearly as extensively as in AKG and AG, in part for the sake of brevity, in part owing to a reviewer who objected to the lack of rotation effects in either simple model, and in part due to the fact that even removal of regions of small coastal diurnal tidal elevations and dissipation can significantly impact the open-ocean tides (a prediction not consistent with the simple models). Note that recomputing the normal modes of Müller (2009) under various blocking scenarios or other altered ocean geometries would be far too time consuming to be practical for the present paper.
This paper is organized as follows. In section 2, we discuss the modal synthesis (global free oscillation) model. Details of the forward ocean tide model, the tuning required to optimize its topographic internal wave drag for diurnal tides, and its accuracy with respect to satellite-altimetry-constrained tide models, are presented in section 3. Section 4 presents the sensitivity tests to the frequencies and spatial structures in the tidal forcing, conducted with the forward ocean tide model and the set of global free oscillations. Section 4 also displays the results of the blocking experiments. Finally, section 4 demonstrates that the Sea of Okhotsk is resonant to diurnal tidal forcing and that blocking out the Sea of Okhotsk perturbs the open-ocean tides more than blocking out other regions of similar area. In section 5 we summarize our results.
2. The modal synthesis model
Ocean tides have been described (e.g., Platzman 1991) as a superposition of global free oscillations. Here we use a set of barotropic global free oscillations and their adjoint counterparts computed on a 1° horizontal grid including the North Pole, linearized friction terms, and full self-attraction and loading effects (Müller 2007).
As described in Eq. (8), the sensitivity of the ocean's response to the tidal forcing is determined by the distance between the forcing and free oscillation frequency |ω − ωk,2|, the free oscillation decay time 
3. Forward ocean tide model
This section discusses the forward ocean tide model. As in AKG and AG, the forward model is the tide model of Arbic et al. (2004a, hereafter AGHS). A few details of the model are given here; AGHS can be consulted for full descriptions of model details including the preparation of topography. In this paper we use a latitude–longitude grid running from 86°S to 82°N, as in AGHS, AKG, and AG. In contrast to most of the simulations in AGHS, AKG, and AG, the simulations here use ⅛° horizontal resolution instead of ½° resolution. Consistent with Egbert et al. (2004), in Arbic et al. (2008) and Müller et al. (2011) we found that this reduced the error of the forward model measured with respect to altimetry-constrained models. To allow for spinup, all of the simulations in this paper are run for at least 60 days, and analysis is undertaken on the latter part of the record (last 1–4 days). Note that, since all of the experiments in this paper are forced by just one frequency, there are no difficulties in separating nearby frequencies as there are in the actual ocean.
Our forward ocean tide model incorporates a parameterized topographic internal wave drag scheme (Garner 2005), which is motivated by inferences from both in situ microstructure data (Polzin et al. 1997) and satellite-altimetry-constrained tide models (Egbert and Ray 2000, 2001, 2003) of enhanced dissipation in regions of rough topography. Parameterized topographic internal wave drag improves the accuracy of forward tide models (Jayne and St. Laurent 2001; Carrere and Lyard 2003; Egbert et al. 2004; AGHS; Lyard et al. 2006; Uehara et al. 2006; Griffiths and Peltier 2008, 2009; Green 2010) because tidal amplitudes are controlled by the strength of the topographic drag (Egbert et al. 2004; AGHS). In AGHS the wave drag was tuned to minimize the discrepancy in open-ocean M2 elevations between the forward model and the satellite-altimetry-constrained GOT99 model of Ray (1999). As discussed below, in the present paper we retain the same topographic internal wave drag scheme, but optimized for K1 tides. Simulations for the present paper are conducted with a cutoff depth for activating topographic internal wave drag of 100 m, rather than 1000 m as in AGHS. As noted in Arbic et al. (2008) this improves the accuracy of the forward model and allows for a smaller value of the multiplicative factor discussed in AGHS. For all of the diurnal simulations presented in this paper, the multiplicative factor is set to 3. It should be noted that, despite the different tuning factors in the AGHS scheme and say, for instance, the wave drag scheme in Jayne and St. Laurent (2001), the spatially averaged drag strengths of the two schemes are very similar, as pointed out by AGHS. Thus, the suggestion is that a certain average strength of drag is needed to dissipate the proper amount of energy, regardless of the exact details of the wave drag scheme used.
The effects of deformation of the solid earth by the loading of ocean tides and the perturbations in the gravitational potential due to the self-gravitation of both ocean tides and the load-deformed solid earth are collectively known as the self-attraction and loading (SAL) term (Hendershott 1972). As in AGHS (see also Egbert et al. 2004), the SAL term is computed here with an iterative method. As in AGHS and AKG, we display results from the third iteration of the SAL term.
Tuning the forward model for K1 tides
This subsection discusses the tuning of the forward model's parameterized topographic internal wave drag scheme for K1 tides. Our goal is to minimize discrepancies between the forward model and satellite-altimetry-constrained tide models. Here we examine kinetic energies as well as elevations. The K1 kinetic energies in forward global tide models with drag schemes optimally tuned for M2 elevations are weaker than K1 kinetic energies found in either current meter observations or altimetry-constrained models (P. Timko 2011, personal communication; Timko et al. 2012). We conjecture that the low diurnal kinetic energies result from the fact that models employing a wave drag tuned for M2 will damp K1 too strongly. Free internal waves are not generated by diurnal tides at topography poleward of about 30° latitude because of the requirement that the Coriolis parameter f be less than the forcing frequency (Gill 1982). As noted by, for instance, Jayne and St. Laurent (2001), topographic internal wave drag should be corrected by a frequency-dependent factor 
First column: List of diagnostics used to measure performance of forward model simulations of the K1 tide. Second column: diagnostics computed from a forward K1 simulation with “original drag,” that is, a drag scheme optimally tuned for M2 elevations. Third column: same as second column only with the M2 drag scheme corrected by the factor 
A summary of these diagnostics computed from a K1 simulation with the original wave drag (tuned for M2, as in AGHS), K1 simulations with K1 correction methods 1 and 2, and output from the altimetry-constrained models TPXO (Egbert et al. 1994) and GOT99 (Ray 1999) is provided in Table 1. Some of the TPXO and GOT99 values are taken from Table 1 in Egbert and Ray (2003). The quantities which can be computed strictly from elevations–ηrms, percent variance captured, potential energy, and “Power input”–compare well to TPXO and GOT99 for all three forward simulations. “Dissipation” matches “Power input” well, as it must in energy balance. In the simulation utilizing wave drag tuned for M2 (“Original drag”), the K1 kinetic energy is, indeed, much weaker than in TPXO, as anticipated. The percent of kinetic energy in shallow waters is also much smaller than the value in TPXO. In addition, the percent of dissipation in deep waters is larger than in TPXO or GOT99. The two solutions with drag corrected for K1 match the TPXO and GOT99 values more closely overall, although the “Percent variance captured,” “Power input,” and “Dissipation” diagnostics actually compare somewhat less well than in the solution with M2 drag. The “K1 correction method 1” matches the percentage of deep dissipation better than “K1 correction method 2”; “K1 correction method 1” is therefore taken as the baseline, or nominal, K1 solution in the remainder of the paper.
The K1 elevation amplitudes from GOT99 (Ray 1999), the optimally tuned forward ocean tide model (“K1 correction method 1”), and the modal synthesis model are displayed in Fig. 1. The three models match each other fairly well, although differences are clearly visible. Note, for instance, the overly strong tides in the southern Sea of Okhotsk in the forward ocean tide model. The ηrms and percent variance captured diagnostics for the modal synthesis model are given in Table 2.
Amplitude of K1 elevation (m) in (a) GOT99 (Ray 1999), a highly accurate satellite-altimetry-constrained tide model, (b) optimally tuned forward ocean tide simulation (K1 correction method 1), referred to as the “nominal” simulation in the text, and (c) the modal synthesis (global free oscillation) model.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
Second column: diagnostics computed from the modal synthesis model's simulation of the K1 tide. Third column: diagnostics computed from the satellite-altimetry-constrained tide model GOT99 (Ray 1999) after interpolation to a 1° grid.
The dissipation map from our optimally tuned forward model of K1 is shown in Fig. 2. It looks qualitatively similar to the K1 dissipation map in Egbert and Ray (2003), but there are notable differences in some locations, for example, Patagonia. Dissipation from quadratic bottom boundary layer drag is substantial in the Sea of Okhotsk and other shallow sea regions. The topographic internal wave drag dissipates substantial energy in the open Pacific and Indian Oceans in latitudes equatorward of 30°.
Dissipation map (milliwatts per squared meter) of K1 from our “nominal” simulation.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
4. Results of sensitivity experiments
In this section, we discuss the sensitivity of the forward ocean tide model and the global free oscillation model to changes in the frequency, zonal structure, and meridional structure in the astronomical diurnal tidal forcing. Our initial emphasis is on the open ocean, but later in the section we present an analysis of the frequency sweep in the Sea of Okhotsk, which is shown to be resonant to diurnal tidal forcing. We conclude this section by examining the sensitivity of open ocean diurnal tides to the removal of regions of strong coastal diurnal tides such as the Sea of Okhotsk.
a. Sensitivity to forcing frequency
This subsection focuses on simulations of the forward and modal synthesis models with the same basin geometry and topography as the nominal solution, but in which the forcing frequency ω in the astronomical diurnal tidal forcing Eq. (9) is allowed to vary. For the sake of brevity, global amplitude maps of these simulations are not shown. Instead, in Fig. 3a, we plot global ηrms values from the forward ocean tide model and modal synthesis model (averaged over latitudes equatorward of 66° and in water depths exceeding 1000 m), as a function of the forcing period. Both the forward and modal synthesis models display clear peaks near 22 and 33 h, and a less prominent peak near 26 h. To assess the realism of the frequency sweeps conducted with the forward model, Fig. 3a includes results on the diurnal constituents Q1, O1, P1, and K1 from the satellite-altimetry-constrained tide models TPXO7.2 (Egbert et al. 1994) and GOT99 (Ray 1999). The TPXO7.2 and GOT99 results have been rescaled by the ratio of A(1 + k2 − h2) evaluated for K1 to A(1 + k2 − h2) evaluated for the constituent in question (see Table 3), the underlying assumption being that tides are linear to first order. Discrepancies between the forward ocean tide model and the altimetry-constrained models (and between the two altimetry-constrained models themselves) are clearly seen. The agreement between the forward and modal synthesis models and the altimetry-constrained models is close for K1 and (scaled) P1, but is not as close for (scaled) O1 and (scaled) Q1.
(a) Globally and temporally averaged ηrms values (cm) computed over the deep ocean (water depths 1000 m and greater) equatorward of 66° in simulations of the forward and modal synthesis models with K1-like astronomical forcing and realistic ocean geometry but with a varying forcing period (equivalently, varying frequency). Additionally, results from the satellite-altimetry-constrained models TPXO7.2 and GOT99, in which Q1, O1, and P1 results are rescaled as if their astronomical forcing amplitude were equal to that of K1, are shown. (b) Shape factor 
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
Second, third, and fourth columns: Periods, astronomical forcing amplitudes, and Love number combination 1 + k2 − h2, respectively, for the four largest diurnal constituents listed in the first column. Fifth column: scale factors [the ratio of the value of A(1 + k2 − h2) for K1 to that of the constituent in question] used to plot the TPXO7.2 and GOT99 results on diurnal constituents in Figs. 3, 7, and 8.
In Figs. 3b and 3c we show the shape factors and decay times of all free oscillations in the diurnal spectrum, as determined by Müller (2008). Obviously not every free oscillation yields a peak in the frequency sweep. Only if friction is low (i.e., the decay time is large) and the shape factor is large (which corresponds to a high spatial coherence of the corresponding adjoint free oscillation with the forcing field) is a peak likely to be generated. Decay times are apparently the most important factor, since the 26.20-h mode has a fairly low shape factor, but still yields a small peak in the frequency sweep. The 32.64-h mode has the largest decay time and shape factor, explaining its dominance in the frequency sweep.
The amplitudes and phases of the 21.97-, 26.20-, and 32.64-h normal modes are displayed in Fig. 4. The 21.97- and 26.20-h modes display a strong signature in the North Pacific, while the 32.64-h mode displays a strong signature of an Antarctic Kelvin wave.
Amplitude and Greenwich phase (°), respectively, of tidal elevations for the (a),(b) 21.97-h mode, for the (c),(d) 26.20-h mode, and for (e),(f) the 32.64-h mode. Amplitude and phase are normalized as in Müller (2008).
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
The largest peak in Fig. 3a corresponds to the free oscillation with the 32.64-h period, which represents the first-order Antarctic Kelvin wave and Kelvin wave propagating along the North Pacific coasts (Figs. 4e and 4f). This free oscillation is one of the major contributors to the main diurnal tides (Müller 2008), although the 32.64-h mode lies farther from the diurnal band than many other modes shown in Figs. 3b and 3c. The importance of this free oscillation stems from its large damping time of about 54 h and its large shape factor. The two modes at 21.97 and 26.20 h, seen as peaks in the frequency sweep, are also important in the modal synthesis of diurnal tides. All three modes shown in Fig. 4 have a signature of the Antarctic Kelvin wave, but only the 26.20- and 32.64-h modes significantly contribute to the tidally forced one. The 32.64-h mode has 49% of its total energy in the Southern Ocean, while the 21.97- and 26.20-h modes have 17% and 24% of their energy in the Atlantic Ocean, respectively, and 58% of their total energy in the Pacific Ocean. A detailed discussion of the features of all modes used for the modal synthesis model and their contribution to the diurnal tides can be found in Müller (2008, 2009).
A conclusive result is that the globally averaged ηrms values displayed in Fig. 3a are quite consistent between the forward ocean tide model and the modal synthesis model. The peaks near 22 and 33 h, for instance, nearly coincide between the two models.
b. Sensitivity to zonal structure of forcing
The ηrms values, in the open ocean and in the shelf, are plotted versus N in Fig. 5a. The vertical line in Fig. 5a marks the value of N corresponding to the realistic case (N = 1). As can be seen in Fig. 5a, predictions from the modal synthesis model match general trends seen in the forward ocean tide model results, although there are clear discrepancies. Figure 5b displays the dependence of the shape factor on the zonal wavenumber for modes with periods between 10 and 80 h. This figure clearly shows clusters of large shape factors between N = 1 and 4, which aligns with the strong oceanic response seen in Fig. 5a.
(a) Globally and temporally averaged ηrms values (cm), computed over latitudes equatorward of 66° in water depths exceeding 1000 m (open ocean) and in depths shallower than 1000 m (shelf), in simulations with astronomical forcing like that of K1 but with a varying zonal wavenumber N in Eq. (15). The extra vertical line represents the realistic case (N = 1). (b) Shape factors 
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
c. Sensitivity to meridional structure of forcing
As shown in Fig. 6, the forward ocean tide model and modal synthesis model both display a peak in response for degree l = 3. To help us understand which modes are responsible for this strong response to third-degree forcing, in Figs. 6b and 6c we plot the shape factors and weighting coefficients [see Eq. (6) and Eq. (8)] of free oscillations in the diurnal spectrum for degrees l = 1, 2, 3, 4, and 5, respectively. These figures show that the peak seen in Fig. 6a for l = 3 is primarily produced by the 21.97- and 26.20-h modes. The adjoint counterparts of these modes fit best to third degree–shaped forcing, and thus, combined with their long decay times (see Fig. 3c) and near resonance, they dominate with large weighting coefficients. However, when the tidal forcing is second degree, the shape factors of the 21.97- and 26.20-h modes are reduced by a factor of ~2. This implies that the second-degree tidal forcing is not as effective as the third-degree tidal forcing, for tides of diurnal frequency.
(a) Globally and temporally averaged ηrms values (cm), computed over latitudes equatorward of 66° and in water depths exceeding 1000 m (open ocean), in simulations with astronomical forcing like that of K1 but with a varying degree l of the spherical harmonic in Eq. (19). (b) Shape factor and (c) weighting coefficients ak plotted as a function of mode number, that is, the period of the free oscillation in hours multiplied by 100. Colored lines in (b),(c) represent the solutions for different degree of the spherical harmonic forcing function.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
The results presented here on the sensitivity of diurnal tides to meridional forcing structure are consistent with results of previous studies of the third-degree diurnal tides in the North Atlantic (Cartwright 1975; Ray 2001). In these previous studies the relatively large M1 tide observed in the North Atlantic was explained by the correlation (shape factors) of free oscillations with the periods of 23.7 and 25.7 h, which were computed by Platzman et al. (1981). The free oscillations with these periods also correlated better with third-degree forcing than with second-degree forcing. Furthermore, the spatial structure of the 26.20-h mode of the present study is similar to that of the 25.7-h mode of Platzman et al. (1981), with two amphidromes located in the North and South Atlantic (see Fig. 4).
To summarize the results of sections 4a, 4b, and 4c, the modal synthesis model predicts the sensitivity of the forward ocean diurnal tide model to the frequency, zonal structure, and meridional structure of the astronomical tidal forcing with a fairly high degree of skill. This adds to evidence, previously accumulated by other means, that open-ocean tides can be reasonably thought of as a superposition of damped oscillatory normal modes. We next turn our attention to the resonance of coastal diurnal tides and to the impact on open-ocean diurnal tides of blocking out regions of prominent coastal diurnal tides.
d. Coastal tidal sensitivity to forcing frequency
We now use the frequency sweep simulations of the forward tide model to better understand the coastal diurnal tides. In Figs. 7 and 8 we show the tidal amplitudes and phases at six locations of large coastal diurnal tides, plotted against the forcing period 2π/ω in the frequency sweep. The amplitudes and phases in Figs. 7 and 8 are taken from specific model gridpoints, given by the latitudes and longitudes listed in the titles of the various subplots. As in the open-ocean results shown in Fig. 3a, the coastal results in Figs. 7 and 8 include rescaled TPXO7.2 and GOT99 results. As with the open-ocean results, discrepancies between the forward ocean tide model and the altimetry-constrained models (and between the two altimetry-constrained models themselves) are clearly seen. However, as the forcing period varies, the forward ocean tide model usually exhibits the same trends as the satellite-constrained models.
Amplitude (m) and Greenwich phase (°) of tidal elevations in the Sea of Okhotsk, Bristol Bay, and Gulf of Tonkin in forward tide model experiments with astronomical forcing like that of K1 but with a varying forcing period (equivalently, varying frequency). Results from the satellite-altimetry-constrained models TPXO7.2 and GOT99 in which Q1, O1, and P1 results are rescaled as if their equilibrium forcing amplitude were equal to that of K1, are also shown. Results are taken from specific model grid points, identified by the latitudes and longitudes in the titles of each subplot.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
As in Fig. 7, but for the Sahul Shelf, northwest Australia, and Ross Sea.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
Having shown that the forward model has some degree of skill in reproducing the frequency sensitivity in coastal zones, we now examine coastal resonance. The coastal locations in Figs. 7 and 8 all display peaks in their amplitude values somewhere within the period band of 22–33 h. For example, in the Sea of Okhotsk (Fig. 7a) a peak is seen near 22 h, while in Bristol Bay (Fig. 7c) broad peaks appear near both 22 and 33 h. The Gulf of Tonkin (Fig. 7e) and Ross Sea (Fig. 8e) peak near 33 h. The peaks near periods of 22 and 33 h are driven by the global free oscillations with periods of 21.97 and 32.64 h. Both are major components of the diurnal tides.
The spectral peaks in the coastal regions, and their consistency with the frequencies of the major free oscillations for diurnal tides, reflect that the tides are driven in these regions by global free oscillations. The global free oscillations represent coupled shelf–open ocean resonances. We next attempt to separate, to the extent that we can, the coastal resonances from the global free oscillations. To do this we extend the admittance analysis of Garrett (1972) to selected coastal regions of interest, using the frequency sweep simulations from the forward ocean tide model.
To identify the presence of coastal resonance, we investigate the amplitude–phase relationships between the “inside” coastal system and the corresponding “outside” open-ocean system. Garrett (1972) developed an approach using three semidiurnal tidal constituents inside and outside of the Gulf of Maine–Bay of Fundy system and derived the coastal resonance period of 13.3 h, for this system. A particular feature of his method is that he acknowledged the much stronger forced M2 compared to the other two considered tidal constituents, that is, S2 and N2. Thus, nonlinearities in the tidal response induced by nonlinear bottom friction were explicitly included. In the present study, we will use the frequency sweep simulations. Since each simulation within the frequency sweep consists of one constituent only, we can neglect these considerations of nonlinear response, meaning that the value of the parameter “c” in Garrett (1972) is one.
The Garrett (1972) method applied to the frequency sweep experiments and Sea of Okhotsk. The inside point is located at (61°N, 163.5°E), and the outside grid point at (53.5°N, 160.5°E). (top) The amplitude at the inside Sea of Okhotsk point vs forcing period. (middle) The amplitude of the response function [see Eq. (25)] of the coastal system. (bottom) The phase of the response function. Blue crosses depict all frequency sweep experiments, red crosses show experiments used for the resonance fit, and the circle denotes the frequency chosen as the reference frequency.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
This result clearly reflects that the Sea of Okhotsk itself is resonant with a resonance period of ~22 h. For all regions shown in Figs. 7 and 8, the aforementioned analysis has been applied, and we find that the Sahul Shelf and Gulf of Tonkin harbor coastal resonances with respective natural oscillation periods of about 27 and 29 h (not shown here for the sake of brevity), as well, whereas the other coastal regions in Figs. 7 and 8 cannot be shown to be resonant by this technique. Application of the admittance analysis to the most important region for semidiurnal tides identified in AKG—the Hudson Strait—demonstrates that it is also resonant (not shown here for the sake of brevity).
e. Back effect of shelf upon open ocean in blocking experiments
We now explore the back effect of coastal diurnal tides on open-ocean diurnal tides in forward ocean tide model experiments with basin geometries that are realistic except for the fact that coastal regions of interest are blocked off one at a time. Figures 10b,e–15b,e display elevation amplitudes in K1 experiments in which portions of the Sea of Okhotsk, Bristol Bay, the Bering Sea, the Sahul Shelf, the northwest Australian Shelf, the Ross Sea, the Gulf of Tonkin, the west coast of South America, the Bay of Bengal, Bass Strait, and the Gulf of Mexico, have been blocked out one at a time. The Blocked Large Sea of Okhotsk region (Fig. 10b) blocks off the majority of the sea in which depths are shallower than 1000 m. The Blocked Small Sea of Okhotsk region (Fig. 10e) is defined by the region where diurnal tidal elevations within the sea exceed 2.0 m in the nominal simulation. The ocean gridpoints omitted in the Blocked Sahul Shelf simulation include the gridpoints omitted in the Blocked Australian Shelf simulation. For comparison, Figs. 10a,d–15a,d show the amplitudes in these regions for the nominal experiment (also referred to as “unblocked”). In each of the blocked experiments a large portion of the coastal region in question is removed so that features of interest such as previously large coastal tides are now absent. Several of the regions were chosen for having large amplitude coastal diurnal tides, but some regions were chosen for other reasons. The Bering Sea has large diurnal tidal dissipation (Fig. 2), but relatively small diurnal tidal amplitudes (Fig. 11d). The west coast of South America, Bay of Bengal, Bass Strait, and Gulf of Mexico were chosen as “control” regions, in which neither the coastal diurnal tidal elevations or dissipation are particularly strong.1 The maximum depth of the grid points blocked out never exceeds 1000 m, except in the Blocked Ross Sea experiment. In that experiment, we enforce the criterion that depths of removed grid points not exceed 2000 m; in the nominal simulation, the Ross Sea possesses large tidal elevations in the depth range between 1000 and 2000 m.
(a),(b) The K1 elevation amplitudes (m) in Sea of Okhotsk in (a) nominal (Unblocked) and (b) Blocked Large Sea of Okhotsk simulations. (c) Global map of K1 elevation amplitude differences (m) between Blocked Large Sea of Okhtosk simulation and Unblocked simulation. (d)–(f) As in (a)–(c), but for Blocked Small Sea of Okhotsk simulation. Note change in scale between (c) and (a),(b) and between (f) and (d),(e).
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
As in Fig. 10, but for Blocked Bristol Bay and Blocked Bering Sea. For the difference maps the scale is reduced from that in Fig. 10.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
As in Fig. 10, but for Blocked Sahul Shelf and Blocked northwest Australian Shelf. For the difference maps the scale is reduced from that in Fig. 10.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
As in Fig. 10, but for Blocked Ross Sea and Blocked Gulf of Tonkin. For the difference maps the scale is reduced from that in Fig. 10.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
As in Fig. 10, but for Blocked West Coast of South America and Blocked Bay of Bengal. For the difference maps the scale is reduced from that of Fig. 10.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
As in Fig. 10, but for Blocked Bass Strait and Blocked Gulf of Mexico. For the difference maps the scale is reduced from that in Fig. 10.
Citation: Journal of Physical Oceanography 43, 7; 10.1175/JPO-D-12-054.1
We emphasize the impact of blocking by also showing the difference in elevation amplitudes between the blocked and unblocked simulations (Figs. 10c,f–15c,f). In all of the blocked simulations, amplitude differences of 5–10 cm or more are easily seen, and the blocking affects the tides on basinwide and even global scales. The globally averaged rms signals ηrms for the blocking simulations are shown in Table 4. In most of the blocked simulations, ηrms increases from its value in the nominal simulation, as in the semidiurnal simulations of AKG.
Globally averaged diagnostics (see text) for various blocking experiments described in section 4e and shown in Figs. 10–15.
The globally integrated dissipation rates calculated via Eq. (14) for the nominal and blocked simulations are also provided in Table 4. The global dissipation rate upon removal of specific coastal areas is sometimes larger but usually smaller than the global dissipation rate in the nominal simulation. This contrasts with the results of semidiurnal blocking simulations in AKG, which typically saw a rise in global dissipation rates with blocking.
In Table 4 we also show the area blocked out in each blocking simulation, and the perturbation to open-ocean diurnal tides per area blocked out. Thus we see, for instance, that the large perturbations in the Blocked Gulf of Mexico, Blocked Sahul Shelf, and Blocked Large Sea of Okhotsk simulations are in part due to the large areas blocked out in these simulations. The largest perturbation per area blocked out is seen in the Blocked Small Sea of Okhotsk simulation. In section 4d the Sea of Okhotsk was shown to be resonant to diurnal tidal forcing. Thus, the large response to its blocking is consistent with the arguments of AKG and AG that the blocking of regions of resonant coastal tides strongly affects open-ocean tides. Also consistent with this notion is the fact that blocking the Hudson Strait, found here and in earlier papers (Arbic et al. 2007; Cummins et al. 2010) to be resonant to semidiurnal tidal forcing, has the largest effect on open-ocean semidiurnal tides of all regions examined in the blocking experiments of AKG. In further defense of the importance of coastal resonance in blocking experiments, we note that the blocking of the Sahul Shelf and Gulf of Tonkin, also found here to be resonant to diurnal forcing, both elicit large perturbations to open-ocean diurnal tides. Consistent as well with the simple models in AG and AKG—see for instance Eqs. (12)–(18) in AG—is the fact that blocking the Bering Sea, site of large diurnal tidal dissipation but relatively small diurnal elevations, elicits a large perturbation in open-ocean diurnal tides. Complicating these arguments, however, is the fact that blocking regions with neither large diurnal tides nor significant diurnal tidal dissipation—such as the west coast of South America, the Bay of Bengal, the Bass Strait, and the Gulf of Mexico—also elicits large perturbations to open-ocean diurnal tides. It appears that strong resonance, strong dissipation, and simple coastal geometrical considerations irrespective of coastal resonances, are all capable of shaping the response of the open ocean to coastal tides.
We end this section with two notes. First, since clear deviations appear between our forward model and altimetry-constrained models in various coastal regions (i.e., the Sea of Okhotsk, Fig. 1), the results of the blocked experiments should be taken with some care. Second, in addition to performing the K1 blocking experiments described in this section, we also reran the M2 Blocked Hudson Strait experiment in AKG at ⅛° resolution and confirmed that it yielded similar results as the AKG Blocked Hudson Strait experiment, which was performed at ½° resolution. The latter point demonstrates that the main AKG results are not artifacts of the relatively coarse horizontal resolution employed in that study.
5. Summary and discussion
We have systematically investigated the resonance of diurnal tides using a forward near-global ocean tide model and a modal synthesis (global free oscillation) model. The forward ocean tide simulations presented here build upon similar simulations conducted for semidiurnal tides by AKG and AG. The set of global free oscillations (normal modes) computed for realistic ocean geometries and ocean physics are taken from Müller (2007, 2008, 2009).
Prior to conducting simulations with the forward ocean tide model, the topographic internal wave drag is tuned specifically for diurnal tides. In global tide models which are optimally tuned for M2, kinetic energies for the K1 constituent are lower than those seen in current meter observations (P. Timko 2011, personal communication; Timko et al. 2012). Here, we adjust the wave drag by a well-known frequency-dependent factor and find that the discrepancies between the forward ocean tide model diurnal kinetic energies and those in satellite-altimetry-constrained tide models are substantially reduced as a result.
Simulations conducted with the forward ocean diurnal tide model and an array of forcing frequencies exhibit peaks in the globally averaged amplitude near periods of 22 and 33 h. Predictions of sensitivity to forcing frequency made by the modal synthesis model closely mimic the results from the forward ocean tide simulations. Specifically, the free oscillations with periods of 21.96 and 32.64 h yield peaks quantitatively consistent with the peaks near 22 and 33 h in the forward model results.
Sensitivity to zonal and meridional structure in the astronomical diurnal tidal forcing is also examined here. Forward ocean diurnal tide model simulations performed with variations in the zonal forcing structure show that globally averaged tidal amplitudes in the shelf regions peak at a zonal wavenumber equal to one. A broad peak near this wavenumber is also seen in the globally averaged open-ocean tidal amplitudes. The predictions made by the modal synthesis model once again resemble the forward ocean tide model results reasonably well. Forward ocean tide model simulations in which the meridional forcing structure is varied show that tidal elevations peak when the astronomical diurnal tidal forcing is of third degree. This result is consistent with predictions made from the modal synthesis model and with the relatively strong third-degree diurnal tides observed in the North Atlantic Ocean (Cartwright 1975; Ray 2001).
Taken together, the results presented here and in our previous studies of semidiurnal tides (AKG and AG) add to previously accumulated evidence (e.g., Platzman 1991; Müller 2008) that it is appropriate to view the semidiurnal and diurnal ocean tides as a system of forced-damped normal modes (free oscillations).
This paper also presents a discussion of diurnal tides in specific coastal locations and the impact of those locations on the open-ocean diurnal tides. Admittance analysis of the frequency sweep results demonstrates that the Sea of Okhotsk, Sahul Shelf, and Gulf of Tonkin are resonant to diurnal tidal forcing. Similarly, admittance analysis of the frequency sweep in AKG demonstrates that Hudson Strait is resonant to semidiurnal tidal forcing. A set of simulations in which specific coastal locations are blocked out demonstrates that blocking yields considerable alterations in tidal elevation amplitudes and phases, across basinwide and even global scales. Consistent with the predictions of simple analytical models in AG and AKG on the importance of coastal resonance, the Blocked Sea of Okhotsk and Blocked Hudson Strait experiments yield the largest perturbations to open-ocean diurnal and semidiurnal tides, respectively, of our blocked experiments. Regions of high diurnal tidal dissipation, such as the Bering Sea, yield large perturbations to the open-ocean diurnal tides as well, also in line with predictions from the simple analytical models in AG and AKG. Complicating the analysis, however, is the fact that blocking out regions in which neither diurnal tidal elevations nor dissipation are high, also often leads to significant perturbations to open-ocean diurnal tides. Evidently, the tides are quite sensitive to nearly any perturbations to geometry or to dissipation, particularly so in regions where the coastal tides are resonant. The blocking experiments are relevant to understanding tides of the ice age, during which lower sea levels entail a reduced area of continental shelves. Simulations of the ice age tides (e.g., Thomas and Sündermann 1999; Egbert et al. 2004; Arbic et al. 2004b, 2008; Uehara et al. 2006; Griffiths and Peltier 2008, 2009; Green 2010; Hill et al. 2011) strongly suggest that ice age tides were larger than those of the present day. The work presented in AKG, AG, and this paper provides some context with which to understand the results of ice-age tide simulations.
Acknowledgments
The authors thank two anonymous reviewers whose careful and extensive comments led to several important changes to this manuscript. BKA gratefully acknowledges support from National Science Foundation (NSF) Grants OCE-0924481 and OCE-09607820. AWS and LZ were supported by a Research Experience for Undergraduates (REU) supplement to OCE-09607820, while WJG was supported by an REU supplement to OCE-0924481. M.M. acknowledges support from DFG (Deutsche Forschungsgemeinschaft) Grant MU3009/1-1. We gratefully acknowledge Mike Messina for help in setting up the forward ocean tide model simulations for this project, which were carried out on the supercomputer provided by the University of Michigan Office of Research Cyberinfrastructure (ORCI). Important preliminary computations were done on the high-performance supercomputing cluster (HPC) at Florida State University. For the Florida State calculations BKA and WJG gratefully acknowledge assistance from Brian Rivera, Jordan Yao, Dan Voss, and Paul van der Mark.
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We thank an anonymous reviewer for suggesting additional blocking experiments, in the Bering Sea and in some “control” regions, beyond what was presented in the original manuscript. It should be noted that the diurnal tides in the Gulf of Mexico are considerably stronger than the very weak diurnal tides in the North Atlantic. Thus, compared to other nearby locations, the Gulf of Mexico diurnal tides are not small.