The satellite altimetry SSH data used in this paper are collected by multiple altimetry missions from 1993 to 2021. In the order of launch time, they are TOPEX/Poseidon, ERS-1, ERS-2, Geosat Follow-On, Jason-1, Envisat, Jason-2, CryoSat-2, SARAL/AltiKa, Haiyang-2A, Jason-3, Sentinel-3A, Sentinel-3B, Haiyang-2B, and Jason-CS/Sentinel-6 (Fig. ). The combined data set from 1993 to 2019 is about 100 satellite years long. We use the satellite along-track SSH data downloaded from the Copernicus Marine Service (10.48670/moi-00146). The SSH measurements have been processed by standard corrections for atmospheric effects, surface wave bias, and geophysical effects . The ocean barotropic tide, polar tide, solid Earth tide, and loading tide are corrected using theoretical or empirical models . Mesoscale correction is made using the gridded SSH fields downloaded from the Copernicus Marine Service (10.48670/moi-00148). The satellite along-track SSH data in 2020 and 2021 are used to evaluate the new N2 internal tide model as independent data (Sect. ). Extracted from the 27-year-long data, our N2 internal tide model contains only the 27-year coherent component. Their temporal variation (or incoherent component) is not addressed in this paper.

The observation of internal tides by satellite altimetry may be affected by an issue called tidal aliasing because the satellite repeat cycles are much longer than the semidiurnal and diurnal tidal periods. Here we examine possible tidal aliasing issues with N2 and M2 internal tides. In one 160 km by 160 km fitting window (Sect. ), there are typically about 7.84 × 104 SSH data from multiyear multi-satellite measurements. Using their observation times, we can calculate their phase lags with respect to the N2 tidal cycle (12.6583 h). Figure a gives the histogram of their phase lags over one N2 tidal cycle. For comparison, Fig. b shows the histogram with respect to the M2 tidal cycle (12.4206 h). The results show that the SSH data overall evenly distribute over one N2 or M2 tidal cycle, without extreme biases. In particular, their distribution on M2 is smooth, suggesting that there is no tidal aliasing issues for M2. Their distribution over N2 is a little bumpy, suggesting that the resulting N2 internal tides may have larger errors. The uneven distribution stems from the orbital configurations of the satellite missions. Fortunately, as shown in this study, the new mode-1 N2 internal tides can overcome background noise in some regions.

The core technique of our mapping procedure is plane wave analysis. By this method, internal tides are determined by fitting horizontal plane waves in one given fitting window (160 km by 160 km in this study), in contrast to harmonic analysis at one single site. This method has been described in detail in our previous studies . For each tidal constituent, there may be multiple internal tides of arbitrary propagation directions at each site, due to their multiple source regions and long-range propagation. Plane wave analysis can resolve these internal tides by propagation direction. We will fit five mode-1 N2 internal tidal waves at each site. Our five-wave representor follows 1Σm=15Amcos⁡kxcos⁡θm+kysin⁡θm-ωt-ϕm, where x and y are the east and north Cartesian coordinates, t is time, and ω and k are the frequency and horizontal wavenumber of the target internal tides, respectively. Three parameters need to be determined for each internal tidal wave: amplitude A, phase ϕ, and direction θ. To determine one wave, the amplitude and phase of a single plane wave are determined by the least-squares fit in each compass direction (with 1∘ increment). When the resulting amplitudes are plotted as a function of direction in polar coordinates, an internal tidal wave appears to be a lobe. The direction of the first wave is thus determined from the biggest lobe. Thus, the amplitude A, phase ϕ, and propagation direction θ of one internal tidal wave are determined. Afterward, its signal is predicted and subtracted from the original data, which removes the wave itself and its side lobes. This procedure can be repeated to extract an arbitrary number of waves one by one. The resulting internal tidal waves are sorted with descending amplitudes.

The frequency (ω) and horizontal wavenumber (k) of the target internal tides are needed in plane wave analysis. The M2 and N2 tidal periods (equivalent to frequencies) are from the Moon's orbital motion around the Earth . They are 12.4206 and 12.6583 h, respectively. The local phase speed (equivalent to wavenumber) of the target internal tides is theoretically determined from the World Ocean Atlas 2018 (WOA18) . The WOA18 provides climatological hydrographic profiles on a spatial grid of 0.25∘ latitude by 0.25∘ longitude. Ocean depth is based on the 1 arcmin topography database constructed using in situ and satellite measurements . For a given ocean depth and stratification profile, the vertical structures and eigenvalue speeds of internal tides are obtained by solving the Sturm–Liouville equation : 2d2Φ(z)dz2+N2(z)c2Φ(z)=0, subject to free-surface and rigid-bottom boundary conditions, where N(z) is the buoyancy frequency profile and c is the eigenvalue speed. The phase speed cp can be calculated from the eigenvalue speed following 3cp=ωω2-f2c, where ω and f are the tidal and inertial frequencies, respectively. Note that the phase speed is a function of longitude and latitude .

Our three-step mapping procedure consists of two rounds of plane wave analysis and a spatial two-dimensional (2D) bandpass filter in between . In this paper, the mapping process is illustrated by showing intermediate results in Fig. . An interested reader is referred to the above papers for more details.

In step 1, mode-1 N2 internal tides are mapped by plane wave analysis as described above. The N2 internal tides are mapped from along-track SSH data onto a spatially regular grid. In this paper, our fitting window is chosen to be 160 km by 160 km, consistent with wavelengths of mode-1 N2 internal tides. The resulting N2 internal tides are at a 0.2∘ longitude by 0.2∘ latitude grid. At each grid point, five mode-1 N2 internal tidal waves of arbitrary propagation directions are determined. The vector sum of these five waves gives the internal tide solution. Figure a shows the mode-1 N2 internal tide field obtained in this step. It gives obvious internal tide signals (e.g., around the Hawaiian Ridge) but the non-tidal noise is high. In step 2, the spatially regular N2 internal tide field is cleaned by a 2D bandpass filter in overlapping 850 km by 850 km windows. The N2 internal tide field is first converted to the 2D wavenumber spectrum by Fourier transform. The spectrum is truncated to [0.8, 1.25] times the local wavenumber. The truncated spectrum is converted back to the internal tide field by inverse Fourier transform. Figure b shows the cleaned N2 internal tide field. Now the N2 internal tide signals are much cleaner. However, Fig. b cannot resolve multiple internal tidal waves yet. In step 3, plane wave analysis is called again to decompose the filtered internal tide field into five internal waves of arbitrary propagation directions. The second-round plane wave analysis is the same as the first-round plane wave analysis, except that the input is the filtered internal tide field in step 2. In the end, the resulting five waves are saved with their respective amplitudes, phases, and directions. Figure c shows the five-wave superimposed internal tide field. It is very similar to Fig. b because this step only decompose the internal tide field. The five-wave decomposition allows us to separate internal tides of different propagation directions. They will be used to extract long-range internal tidal beams in the ocean (Sect. ).

We map both the mode-1 N2 and M2 internal tides following the same three-step procedure. They are constructed from the same satellite altimetry data but using their respective wave parameters (frequency and wavenumber). Figure  shows the resulting N2 and M2 internal tide fields. Internal tides in shallow waters (< 1000 m) are discarded. The new M2 internal tides are almost identical to those obtained in previous studies using slightly different satellite data . Here we find that the N2 and M2 internal tides have similar spatial patterns and that the N2 amplitudes are about 20 % of the M2 amplitudes. The largest N2 amplitudes are about 5 mm, compared to 20–30 mm for M2 internal tides. To account for this factor, their color map ranges are different by a factor of 5. Figure  gives SWOT ground tracks in its 1 d fast-repeating phase (green lines). It shows that strong mode-1 N2 internal tides occur under some SWOT swaths, for example, those off the California coast, in the New Caledonia region, in the western North Pacific, and on the Amazon continental shelf. In these regions, the N2 internal tides cannot be neglected in the study of sub-mesoscale dynamics. Conversely, the upcoming SWOT data also offer a great opportunity to explore N2 internal tides.

We have examined the possible cross talk between the N2 and M2 internal tides in our mapping procedure. We map N2 internal tides using two different data sets. The first is the original satellite altimetry SSH data set (Sect. ). The second is the M2-corrected data set. In other words, the M2 internal tides are predicted using our empirical model and subtracted from the original data. We find that the resulting N2 internal tides from the two data sets are almost the same. The variance of their differences is < 1 % that of the N2 internal tides. Likewise, we map M2 internal tides using both the original and N2-corrected data sets and find that the impact of N2 on M2 is negligible. Our analysis reveals that the N2 and M2 internal tides do not cross talk in our mapping method. This is because the 27-year-long satellite data from 1993 to 2019 are sufficient long to unambiguously separate the N2 and M2 tidal constituents (about 14 min apart).

Model errors in our N2 and M2 internal tide models are estimated using background internal tides. In contrast to N2 and M2 internal tides, which are mapped using tidal periods of 12.6583 and 12.4206 h, respectively, background internal tides are mapped using the same satellite altimetry data but for tidal periods between N2 and M2. Specifically, we map 13 sets of background internal tides using 13 different tidal periods that are linearly interpolated between N2 and M2 (Fig. ). The other mapping parameter, wavenumber (equivalently phase speed), can be obtained using Eq. (). The same strategy has previously been employed to estimate barotropic tide errors. For example, study the fortnightly tidal cycles (MSf and Mf) of tidal mixing using satellite sea surface temperature data. study the fortnightly variability in Chl a using satellite sea surface color data. In both studies, tidal errors are estimated using signals at fake or false tidal frequencies near the real tidal constituents.

We thus obtain 13 background internal tides in the central Pacific (Fig. c, box). Their regional mean SSH amplitudes in this region are 0.8 ± 0.1 mm, compared to 1.66 and 7.75 mm for N2 and M2 (Fig. ). Note that the SSH amplitudes of the 13 background internal tides are almost the same, showing no significant tidal cusps around the N2 or M2 internal tides. In addition, we have calculated the correlation coefficients among these 15 sets of internal tides (including N2 and M2). All correlation coefficients are < 0.05, suggesting that these background internal tides are independent of each other and of the N2 and M2 internal tides. In other words, background internal tides are signals we obtain where there are no tidal constituents. We suggest that the model errors in N2 and M2 can be represented by background internal tides. In this study, we pick one tidal period (12.6074 h) for a global run to obtain background internal tides (model errors), considering that a global run is time-consuming. Figure c gives the resulting background internal tides (model errors). It reveals that model errors are large in regions of strong mesoscale motions because model errors are mainly leaked mesoscale signals. Figure  shows that the N2 internal tides are noisier than M2 because the small-amplitude N2 internal tides are easily affected by model errors. On a global average, the error variance is about 25 % of the N2 variance, and only 1 % of the M2 variance.

Our N2 and M2 internal tide models are evaluated using independent satellite SSH data collected in 2020 and 2021. For each SSH measurement of known time and location, the internal tide signal is predicted using the model under evaluation and subtracted from the SSH measurement. Variance reduction is the variance difference before and after the internal tide correction. The variance reductions for all SSH measurements are binned into 2∘ by 2∘ boxes. The global map of N2 variance reduction is shown in Fig. a. The M2 internal tide model is evaluated in the same way and shown in Fig. b. Note that the color map ranges for N2 and M2 differ by a factor of 25, that is, the square of the factor of their amplitudes.

In the evaluation, the true N2 internal tides (variance σN22) in the model will remove the N2 internal tides in the independent data, leading to positive variance reduction, while the model errors (variance σϵ2) will increase the variance of the independent data, leading to negative variance reduction. Together, we obtain positive variance reduction where σN22>σϵ2, and negative variance reduction where σN22<σϵ2. Figure a shows positive variance reduction in the global ocean, suggesting that the true N2 internal tides are greater than model errors. In particular, in regions of strong N2 internal tides such as the Hawaiian Ridge and the Amazon continental shelf, patches of positive variance reduction are observed because the strong N2 internal tides can overcome model errors, while negative variance reduction usually occurs in regions of weak N2 internal tides such as the eastern equatorial Pacific and the Southern Ocean. The regions of strong mesoscale motions are dominated by negative variance reduction, where weak N2 internal tides are overwhelmed by large model errors (Fig. c). For comparison, Fig. b shows that the M2 internal tide model causes positive variance reduction throughout the global ocean, except for regions of strong mesoscale motions or strong temporal variation . This is because the strong M2 internal tides are almost always greater than the model errors σM22>σϵ2. In summary, our N2 internal tide model can reduce variance in some regions, although the N2 SSH amplitudes are just a few millimeters.

We further examine the relation between the N2 variance reduction shown in Fig. a and the variance difference between the N2 model and the model error. Figure c and d give the N2 model variance and the error variance that are computed from Fig. 4a and c, respectively. Note that the N2 model variance σ2 contains both true N2 internal tides and errors σ2=σN22+σϵ2. Under the condition that the N2 variance is greater σN22>σϵ2, we should have σ2>2σϵ2. We thus calculate the variance difference σ2-2σϵ2 and show the global map in Fig. e. To test this relation, we calculate the variance difference σ2-m⋅σϵ2 for m ranging from 0.5 to 3.5 with a step of 0.1. For each resulting variance difference map (e.g., Fig. e), we calculate its correlation coefficient with the N2 variance reduction (Fig. a). We get the best spatial correlation when the factor m is 2, consistent with our theoretical analysis. Note that all the above analyses are based on the 2∘ by 2∘ binned values. Figure f shows the mask region where the N2 model variance is greater than twice the error variance, indicating regions where the N2 model can be used to make internal tide correction. The mask covers regions of strong N2 and M2 internal tides such as the Hawaiian Ridge, the area off the California coast, the Amazon continental shelf, the western North Pacific, and New Caledonia.

We next examine the performance of the N2 and M2 internal tide models in making internal tide correction for SWOT. In Fig. , the green lines denote the SWOT ground tracks in its daily fast-repeating phase. We interpolate the N2 and M2 variance reductions onto the SWOT ground tracks (neglecting its swath) and calculate the along-track mean variance reductions. For N2, the mean variance reductions in and outside the mask region are 0.73 and -0.25 mm2, respectively. The negative variance reduction suggests that the N2 model does not work well outside the mask region. Fortunately, the N2 model can make internal tide correction in the mask region where the N2 internal tides can overcome model errors. The variance reductions caused by the N2 model seem small, but keep in mind that (1) internal tides and sub-mesoscale motions both have millimeter-scale SSH amplitudes and (2) internal tides are much stronger in their source regions. For M2, the along-track mean variance reductions in and outside the mask region are 25.6 and 2.5 mm2, respectively. They suggest that the M2 model performs well both in and outside the mask region because the M2 internal tides dominate errors throughout the global ocean.

Our mode-1 N2 model reveals that N2 internal tides are widespread in the global ocean (Fig. a). In the Indian Ocean, they are observed in the Arabian Sea, the Bay of Bengal, and the Madagascar–Mascarene region. In the Pacific Ocean, N2 internal tides occur in regions such as the French Polynesian Ridge, the Hawaiian Ridge, the Indonesian seas, the western South Pacific, and the western North Pacific. In the Atlantic Ocean, N2 internal tides appear in regions including the Azores region, the Amazon shelf, the Bay of Biscay, and the Vitória–Trindade Ridge. Our M2 model reveals that mode-1 M2 internal tides are observed in the same regions (Fig. b). The N2 and M2 internal tides have similar spatial patterns, but the N2 amplitudes are about 20 % of the M2 amplitudes. To further quantify their relation, we give in Fig. a the scatterplot of the N2 and M2 SSH amplitudes. It shows that the N2 and M2 amplitudes largely follow the diagonal line with a ratio of 5. Their correlation coefficient is 0.69 (R in MATLAB function corrcoef).

We extract the N2 and M2 barotropic tides from TPXO.8 and show them in Fig. . We find that the N2 and M2 barotropic tides have similar spatial patterns and that the N2 amplitudes are about 20 % of the M2 amplitudes. We examine the relation between the N2 and M2 barotropic tides as well. Figure b shows the scatterplot of the N2 and M2 barotropic amplitudes. It shows that N2 and M2 have a very tight relation, with a correlation coefficient of 0.96. show that the M2 and N2 barotropic-to-baroclinic energy conversion maps have similar spatial patterns and that their amplitudes differ by a factor of 25 (see their Fig. 1). The N2 and M2 relation (spatial pattern and amplitude ratio) is the same for both barotropic and baroclinic tides. Because N2 and M2 have close tidal periods (12.6583 and 12.4206 h), their generations over the same topographic features should be the same (distinguishing their slight differences may improve our understanding of internal tide dynamics in the future). In addition, it is reasonable that the N2 and M2 internal tides have a relatively weak relation (Fig. a) because the long-range propagation of internal tides is affected by an inhomogeneous ocean environment.

In this section, we study the long-range mode-1 N2 internal tidal beams. We have fitted five mode-1 N2 internal tidal waves at each grid point by plane wave analysis. Taking advantage of the five-wave fits, we can decompose the N2 internal tide field into the northward (0–180∘ counterclockwise from due east) and southward (180–360∘) components by propagation direction. Each component contains internal tidal waves with propagation directions falling in the given range (Fig. ). The decomposed components clearly show well-defined long-range N2 internal tidal beams, which are characterized by larger amplitudes and cross-beam co-phase lines (not shown here for clarity; see Figs.  and ). There are numerous long-range N2 internal tidal beams, which radiate from the strong generation sites mentioned above. For example, northward N2 beams are observed to originate from the French Polynesian Ridge, the Macquarie Ridge, and the Amazon shelf. Southward N2 beams are observed to originate from the Andaman Islands, the Lombok Strait, the Hawaiian Ridge, the French Polynesian Ridge, the Mendocino Ridge, and the Azores, among others. Note that the M2 long-range internal tidal beams have been well studied in previous studies Fig. 5 therein. To avoid repetition, the M2 internal tidal beams are not shown here. Together, we observe that the N2 and M2 internal tides have similar long-range beams. In this study, we examine two long-range internal tidal beams as examples.

First, we examine the southward internal tides from the Amukta Pass, Alaska. The M2 long-range beam from Amukta Pass has been studied recently . Figure a shows the southward N2 internal tides in the central North Pacific (Fig. b, blue box). For comparison, the southward M2 internal tides are shown in Fig. b. Both tidal constituents can travel from the Aleutian island chain to the Hawaiian Ridge over 3000 km away. Their propagation directions are about -78∘ from due east. The black lines in Fig.  show the 0 and 180∘ co-phase charts. Figure c shows their phase difference, which increases with propagation because N2 internal tides travel faster than M2 internal tides according to Eq. (). In the propagation, their phase difference increases with propagation. Along the dashed line from source (52.6∘ N, 189∘ E) to far field (26∘ N, 195∘ E), their phase difference increases from 65 to 305∘. The overall phase change is 240∘. It takes about 18 tidal cycles for the N2 and M2 internal tides to travel along the path.

Figure  shows southward internals tides in the region off the California coast (Fig. b, cyan box). This region is chosen for a detailed investigation because it contains one site for the SWOT calibration/validation field campaign. The green lines in this figure indicate the SWOT swaths in its fast-repeating phase . The crossover region of the ascending and descending swaths is the SWOT calibration/validation site. This region is dominated by the southward internal tides from the Mendocino Ridge. Note that this region is also affected by internal tides in other propagation directions . Additionally, there are southwestward internal tides from the Monterey Bay. The two internal tidal beams intersect around the SWOT campaign site. As explained earlier, our N2 model can make internal tide correction for SWOT. Figure  shows that N2 and M2 internal tides are very similar, although the N2 fluxes are much weaker. Both N2 and M2 beams can be tracked from 40 to 20∘ N for > 2000 km. They both bifurcate around 32∘ N near Fieberling seamounts (32.5∘ N, 232.3∘ E) for unknown reasons. The dashed line delineates the beam from 40.3 to 22∘ N along 128∘ W. This line is about 2000 km long. Along this line, the N2 and M2 phase difference increases from 40 to 160∘ over about 14 M2 or N2 tidal cycles.

We calculate the depth-integrated energy flux of mode-1 N2 internal tide from their SSH amplitudes and a transfer function (Fn). The transfer function is calculated using the WOA18 climatological hydrography and the Sturm–Liouville equation . The same calculation method has also been derived by . In this study, we follow our method (previously for mode-1 M2 internal tides) to obtain the transfer function for mode-1 N2 internal tides. It is a function of ocean depth, tidal frequency, mode number, latitude, and stratification. The transfer functions for N2 and M2 are very close because their tidal periods are close. At each grid point, we thus obtain five energy fluxes for the five internal tidal waves following F=12FnA2, where A is the SSH amplitude. The vector sum of the five energy fluxes gives the final energy flux at this site. In this study, we compare the N2 and M2 internal tide energy fluxes in two regions. An interested reader can examine other ocean regions. We show that their energy fluxes have similar spatial patterns. The results show that the mode-1 N2 internal tides can be observed by satellite altimetry, although they are much weaker than the M2 internal tides. Following the same procedure, we have computed the depth-integrated internal tide energies from SSH amplitudes. The globally integrated area-weighted energies for the N2 and M2 internal tides are 1.8 and 30.9 PJ, respectively. The N2-to-M2 ratio is about 5.8 %, larger than the theoretical value of 4 % because N2 contains larger error variance. As explained earlier, the error variance is about 25 % of the N2 variance but only 1 % of the M2 variance.

Figure  shows the N2 and M2 energy fluxes in the western South Pacific. In this study, it is trimmed to 30∘ S–0∘, 145∘ E–125∘ W. Colors show flux magnitudes, and black arrows show flux vectors. This region is chosen because (1) it features various topographic obstacles such as mid-ocean ridges and island chains and (2) the New Caledonia region is one site for SWOT calibration/validation field experiments . There are numerous N2 and M2 internal tidal beams in this region. They are dominantly generated over topographic features. For example, N2 and M2 internal tidal beams radiate from many straits surrounding the Coral Sea . The internal tidal beams can be in any horizontal propagation direction. From the French Polynesian Ridge, internal tides mainly propagate southward and northward. From the Kermadec Arc and New Caledonia, the outgoing internal tidal beams usually travel eastward or westward. The energy fluxes of N2 and M2 internal tides have similar spatial patterns. Figure  shows seven SWOT swaths in this region (green lines). Among them, the two swaths in the New Caledonia region (black box) overlap with strong N2 internal tides whose contribution cannot be neglected. In addition, the two swaths cross the French Polynesian Ridge, where one should pay an attention to N2 and M2 internal tides in the study of mesoscale and sub-mesoscale processes.

Figure  shows the N2 and M2 energy fluxes in the North Atlantic Ocean (2∘S–53∘ N, 58–3∘ W). Figure  is in the same format as Fig. . Internal tides in this region have attracted much attention in recent years . In particular, internal tides on the Amazon continental shelf have been intensively studied recently, partly because of the co-existence of internal tides and internal solitary waves . Our satellite observation reveals that strong N2 and M2 internal tides occur around notable topographic features including the Mid-Atlantic Ridge, the Amazon continental shelf, the Azores region, the Bay of Biscay, the Canary Islands, and the Cabo Verde islands. The longest internal tidal beams for both N2 and M2 are the southward internal tidal beams from the Azores . The two beams can be tracked over 2000 km. In this region, there are four SWOT swaths in its fast-repeating phase, which overlap remarkable N2 and M2 internal tidal beams.