3.1. Barotropic Tide
 Since the internal tides in HYCOM are generated by the interaction of the barotropic tide with the bottom topography, an accurate barotropic tide is needed in order to produce an accurate internal tide. To assess the accuracy of the simulated barotropic tide in HYCOM, each of its eight tidal constituents (computed from total SSH, which is dominated by the barotropic tide) are compared to those from an altimetry constrained barotropic tide model (TPXO7.2; an update to that described by Egbert et al. ). A recent assessment of altimeter-constrained models [Ray et al., 2011] suggests M2 RMS errors of about 1.5 cm or less in the deep ocean and anywhere from two to ten times larger errors in shallow water, depending on location. The TPXO7.2 model has comparable statistics, while nonassimilative global tide models have much larger RMS errors.
 Results from this comparison for M2 and K1 (the largest amplitude semidiurnal and diurnal constituents) are shown in Figures 2 and 3, respectively. Qualitatively the tidal amplitudes and phases in HYCOM are similar to the results from TPXO, but there are differences. One difference is that HYCOM includes internal waves, resulting in small amplitude, small horizontal scale perturbations to both the amplitudes and phases in Figures 2b and 3b. Another difference is that our HYCOM tide simulation is a forward (nonassimilative) calculation and our barotropic tides therefore are not as accurate as those in barotropic data-assimilative global tidal models such as TPXO, or in regional models forced by data-assimilative barotropic models at their boundaries [e.g.,Cummins et al., 2001; Merrifield et al., 2001].
Figure 2. Amplitude (cm) of M2 surface tidal elevation in (a) TPXO7.2 (an update to that described by Egbert et al. ), a barotropic tide model constrained by satellite altimetry, and (b) HYCOM simulations in which the tide is unconstrained by satellite altimetry. Lines of constant phase plotted every 45° in Figures 2a and 2b are overlaid in white.
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 To quantify the differences between HYCOM and TPXO surface tidal elevations, we calculate the mean square error (MSE),
where A and ϕ are tidal amplitude and phase, respectively. The MSE for the eight constituents forced in the model are given in Table 1 with total MSE of 9.52 cm and MSE for the leading semidiurnal constituent (M2) of 7.48 cm and leading diurnal constituent (K1) of 2.25 cm. The geographical distribution of the MSE for M2 and K1 are shown in Figures 4a and 5a. The semidiurnal errors are largest around the continental margins and Southern Ocean where differences in the bathymetry of TPXO and our model are the largest. Two large regions of error are found in the central North and South Pacific. For the diurnal tides the errors are largest around the continental margins and over much of the Atlantic, Indian and Southern Oceans. The errors in the tidal elevations can arise from a combination of errors in the amplitude A and the phase ϕ. The MSE in (1) can be rewritten as
where MSE consists of contributions resulting from differences from the in phase (cosine) and quadrature (sine) terms for the constituent of interest. For all constituents, the global MSE is approximately equally divided between the in phase and quadrature terms. For the semidiurnal tide, the large central North Pacific error is predominately in phase and the central South Pacific is predominately in quadrature (maps not shown). However, significant quadrature errors in the North Pacific and in phase errors in the South Pacific are found in the same regions.
Table 1. Global-Averaged Amplitude, Phase, and Total RMS Errors of HYCOM Surface Tidal Elevations Measured Against TPXO7.2, Following the Error Derivation in(3)a
| ||Amplitude Error||Phase Error||Total Error|
Figure 4. (a) M2surface tidal elevation error for the HYCOM surface tidal elevation measured against TPXO. The contributions to the surface tidal elevation error resulting from errors in (b) tidal amplitude only and (c) amplitude-weighted phase following the derivation in(3). Units are in centimeters.
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 We are not just interested in the generation of barotropic and internal tides, but how errors in the barotropic tide translate into errors in the internal tides. As an alternative to (2), we can partition the MSE into contributions from differences in the amplitude only and from the cosine of the differences in the phases weighted by the geometric mean of the amplitudes (amplitude-weighted phase errors),
with the first term on the right hand side denoting a contribution to the surface tidal elevation error resulting from errors in tidal amplitude only (MSEamplitude) and the second term (MSEphase) from errors in the amplitude-weighted phase. As can be seen from(3), if either model has small amplitude, then the amplitude-weighted phase error will be small regardless the difference in phase. To illustrate the value of this partitioning, consider the case of two sine waves which differ only in phase. From(2), the relative contributions to the in phase and quadrature errors will vary depending upon the phase difference. However, from (3), the amplitude error, MSEamplitude, will be zero regardless of the phase difference and the error will be only in the amplitude-weighted phase MSEphase. Maps of the total (MSE), amplitude (MSEamplitude) and amplitude-weighted phase (MSEphase) errors for M2 and K1 are shown in Figures 4 and 5. The globally averaged statistics for all eight constituents in HYCOM are listed in Table 1 with the errors approximately evenly split between amplitude and phase. However, unlike the in phase and quadrature errors from (2), for the M2tide, the large error regions in the Pacific are predominately amplitude-weighted phase errors. The barotropic tide generates the baroclinic tides through topographic interactions. These generation regions are not uniformly distributed around the globe. In particular, large barotropic phase errors in the Pacific generation regions will lead to large baroclinic phase errors associated with the timing of the generation of the internal tide. Thus, the MSE for the internal tides will be large due to the timing errors, while the amplitudes will compare well.
 When the model barotropic tide is compared to the 102 pelagic gauges described by Shum et al. , the RMS errors increase slightly to 7.80 cm for M2 and 10.22 cm for all eight constituents, but are still lower than in the work by Arbic et al. [2004, 2010]. Data assimilation for TPXO reduces the errors in the shallow water tidal models relative to the 102 pelagic gauges to ∼1.6 cm for M2 and ∼3 cm for the eight constituents [Shum et al. 1997]. When compared to a data-assimilative model (TPXO7.2), our model tides are comparable to other nonassimilative shallow water tide models, ∼7 cm [Jayne and St. Laurent, 2001; Arbic et al., 2004] and ∼5 cm for M2 [Egbert et al., 2004]. Note that Jayne and St. Laurent , Arbic et al. , and Egbert et al.  all utilized a rigorous (i.e., nonscalar) SAL correction, in contrast to the HYCOM results shown here. Egbert et al.  find that 5–10% random errors in the bathymetry can lead to ∼8 cm RMS differences in the M2 amplitude.
3.2. Internal Tide
 The barotropic tides interact with topography to generate internal tides. Thus, errors in the barotropic tides or bathymetry will lead to errors in the internal tides. Globally, the barotropic tide errors are split almost evenly between amplitude and phase errors. Since phase errors in the barotropic tide will cause phase errors in the baroclinic tides, the traditional RMS error statistic for the baroclinic tidal heights may not be a good measure of the model performance. For example, consider the M2 internal tides in the northeastern Pacific, shown in Figure 6, where the amplitude of the M2 internal tides from the model (red) and altimeter (black) are plotted. Qualitatively, the amplitudes of the observed altimetric tide and model tide agree well with RMS amplitudes of 0.361 cm and 0.346 cm for the altimeter and model, respectively. However, RMS error of the complex amplitudes, including the phase of the tides, is 0.232 cm. If we partition this difference into amplitude and phase errors following (3), then the amplitude error is 0.128 cm while the phase error is 0.193 cm. Thus, most of the differences between the model and altimeter internal tides arise from phase errors. Given the errors in the phase of the barotropic tide and model bathymetry errors, it is not very surprising that phase errors may dominate the internal tides. Making sure we convert the proper amount of energy from the barotropic tide into the baroclinic tide is an important first step in the evaluation of the model tides. We will therefore use the area-averaged absolute value of the amplitude as the statistic for our comparisons. Using the absolute value of the amplitude and areal averaging reduces the sensitivity of the statistics to phase errors.
Figure 6. M2internal tide amplitude along ascending tracks from the HYCOM (red) and altimeter-based analysis (black). For each track, the line showing the coordinates of the track represents a zero amplitude for the tides on that track. The short-scale smoothness is due in part to the application of the band-pass filter and is not due to the response method used in the altimetric-based analysis.
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 The global M2along-track altimetric tidal analysis (Figure 7a) exhibits several internal tide generation regions (“hot spots”) near Madagascar, Hawaii, east of the Philippines and the tropical south and southwest Pacific. Internal tides radiating over long distances are also evident, for example between the Aleutian Islands and the Hawaii hot spot [e.g., Cummins et al., 2001]. Amplitudes fall sharply and are relatively low outside these hot spot regions, although close analysis can reveal internal tide signals even in “quiet” regions such as the southeast Pacific. HYCOM exhibits similar features to those noted in the altimetric tidal analysis (Figure 7b).
Figure 7. The M2internal tide amplitude from the (a) altimetric-based and (b) HYCOM tidal analyses. The five subregions denoted by black boxes in Figure 7b are used to compute the area-averaged amplitudes inTable 2.
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 To quantitatively assess how well the internal tide results from HYCOM compare with the altimetric-based analysis, area-averaged amplitude is computed over five subregions centered on internal tide generation regions (black boxes inFigure 7b). In addition, area-averaged statistics are also computed over the world ocean outside of these five hot spot regions. All four semidiurnal constituents largely share these hot spot regions, and summary statistics for these constituents are shown inTable 2.
Table 2. Area-Averaged Amplitudes of Semidiurnal Internal Tides From the Altimetric-Based (Upper Value) and HYCOM (Lower Value) Tidal Analyses Computed Over the Five Subregions Depicted inFigure 7ba
|Hawaii|| || || || |
|East of Philippines|| || || || |
|Tropical South Pacific|| || || || |
|Tropical SW Pacific|| || || || |
|Madagascar|| || || || |
|Rest of world oceanb|| || || || |
 The area-averaged amplitude is found to agree well across the five hot spot subregions for the four semidiurnal constituents. The average percent discrepancy ((|hycom − altim|/|altim|) × 100) across the five hot spot subregions for all four constituents is ∼15%, with M2 having the lowest average percent discrepancy (∼9%) and N2having the highest (∼26%). Across the four semidiurnal constituents the largest discrepancy is noted for the world ocean outside of the five hot spot regions, where the average percent discrepancy is ∼91% with the model underestimating the internal tide energy compared to the altimeter. Inaccuracies in the simulated barotropic tide, which generates the internal tide, account for part of the discrepancy. For example, the model internal tides are too weak in the North Atlantic, where the model barotropic tide is weaker than the data-assimilative barotropic tide (Figure 2). Another source of the discrepancies, mesoscale leakage, will be discussed later in this section.
 The global K1 internal tide amplitudes from the altimetric analysis and HYCOM are shown in Figure 8. The altimetric and HYCOM tidal analyses exhibit three main hot spot regions: near the Philippines, the central Indian Ocean and the central tropical Pacific. The average percent discrepancy across the three hot spot subregions for all four diurnal constituents (Table 3) is ∼23%, with K1 having the lowest average percent discrepancy (∼3%) and Q1 having the highest (∼57%). Across the four diurnal constituents the average percent discrepancy for the world ocean outside of the three hot spot regions and equatorward of 30° is ∼37%.
Figure 8. The K1internal tide amplitude from the (a) altimetric-based and (b) HYCOM tidal analyses. Areas where mesoscale variability contaminates the altimetric-based tidal analysis are identified by the red circles in Figure 8a. The three subregions denoted by black boxes in Figure 8b are used to compute the area-averaged amplitudes inTable 3.
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Table 3. As in Table 2 but for the Three Subregions Depicted in Figure 8b, Which Are Hot Spots for Diurnal Internal Tidesa
|Central Indian Ocean|| || || || |
|Philippines|| || || || |
|Central Tropical Pacific|| || || || |
|Rest of world ocean equatorward of 30°|| || || || |
 Poleward of 30° latitude the altimetric-based tidal analysis exhibits significantly higher amplitudes than the HYCOM analysis (Figure 8). These high-amplitude areas (circled regions inFigure 8a) coincide with areas of high-mesoscale activity, including the Kuroshio, Gulf Stream, Antarctic Circumpolar Current (ACC) and Brazil-Malvinas confluence. This pattern is consistent across all four of the diurnal constituents.
 As mentioned in the introduction, propagating diurnal internal tides do not exist poleward of approximately 30° [Gill, 1982]. HYCOM diurnal tidal amplitudes obtained from hourly samples satisfy this theoretical constraint (Figure 8b). However, as discussed in the literature [Tierney et al., 1998; Carrère et al., 2004; Ray and Byrne, 2010], the altimetric analysis (Figure 8a) shows features that result from the leakage of mesoscale activity into tidal frequency estimates. This leakage is visually evident across all the diurnal constituents, where internal tides do not propagate, and it can be seen in the semidiurnal constituents as well. For example, mesoscale leakage can be clearly seen in S2 altimetric internal tidal amplitudes (Figure 9a), with large amplitudes in the Kuroshio, Gulf Stream and ACC regions not present in HYCOM (Figure 9b). It is worth emphasizing, however, the extremely small amplitudes in both Figures 8 and 9. In each case the color bar spans only 5 mm. It is thus understandable that detection and mapping of such small signals is extremely challenging for satellite altimetry, even after almost two decades of data.
Figure 9. The S2internal tide amplitude from the (a) altimetric-based and (b) HYCOM tidal analyses. Areas where mesoscale variability contaminates the altimetric-based tidal analysis are identified by the red circles in Figure 9a.
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 Quantitative evidence of mesoscale leakage in the semidiurnal constituents is also evident in Table 2, where the average percent discrepancy over the world ocean outside of the hot spot regions is 80%. This discrepancy is significantly higher than the diurnal case (∼37%) because the latter statistic was computed over the 30°S–30°N latitude range, effectively filtering out large areas of mesoscale leakage (e.g., Gulf Stream, Kuroshio, ACC).