A perfect focus of the internal tide from the Mariana Arc

Authors


Abstract

[1] The Mariana Arc of ridges and islands forms an ∼1300-km-long arc of a circle, ∼630 km in radius centered at 17°N, 139.6°E. The hypothesis that the westward-propagating internal tides originating from the arc converge in a focal region is tested by examining the dominant M2 internal tides observed with air-launched expendable bathythermographs (AXBTs) and altimetric data from multiple satellites. The altimetric and AXBT observations agree well, though they measure different aspects of the internal tidal motion. M2 internal tides radiate both westward and eastward from the Mariana Arc, with isophase lines parallel to the arc and sharing the same center. The westward-propagating M2 internal tides converge in a focal region, and diverge beyond the focus. The focusing leads to energetic M2 internal tides in the focal region. The spatially smoothed energy flux is about 6.5 kW/m, about four times the mean value at the arc; the spatially un-smoothed energy flux may reach up to 17 kW/m. The size of the focus is close to the Rayleigh estimate; it is thus a perfect focus.

1. Introduction

[2] Submarine ridges in the open ocean are efficient generation sites of internal tides, because barotropic tidal currents usually flow across isobaths [Ray and Mitchum, 1996, 1997; Egbert and Ray, 2000; Simmons et al., 2004]. Ocean mixing around submarine ridges is enhanced [Kunze et al., 2006; Carter et al., 2008; MacKinnon et al., 2008], and a significant fraction of the tidal energy propagates away up to 3000 km as low-mode internal tides and dissipate elsewhere [Ray and Mitchum, 1996; Zhao and Alford, 2009]. Due to their significant effects on turbulent mixing, ocean acoustics, submarine navigation, and ecosystems, internal tides have long been a research interest in physical oceanography [e.g., Wunsch, 1975; Munk and Wunsch, 1998; Alford, 2003; Rudnick et al., 2003; Garrett and Kunze, 2007].

[3] The Izu–Bonin–Mariana (IBM) arc system in the West Pacific is such a submarine ridge. It is over 2800 km long, extending from Japan to Guam, lying along the eastern margin of the Philippine Sea. At the IBM arc system, the barotropic tide loses energy at a rate of about 50 ± 10 GW [Egbert and Ray, 2000], compared to about 20 GW at the 2500-km-long Hawaiian Ridge [Zaron and Egbert, 2006]. This study focuses on the Mariana Arc and the West Mariana Ridge, the southern part of the IBM arc system (Figure 1). Both of them are about 1300 km long and oriented in south–north direction. They are generally parallel, with a separation distance of 200–300 km. Assuming the 50 GW barotropic tidal energy loss distributes evenly along the IBM arc system, the loss rate at the Mariana Arc and the West Mariana Ridge would be about 20 GW. The detailed partition between the two ridges is unknown to us; however our analysis in this study suggests that the shallower Mariana Arc plays a dominant role. About 800 km to the west exists the Kyushu–Palau Ridge, another potential generation site.

Figure 1.

Topographic map of the study area, superimposed with the ground tracks of multiple satellite missions: ERS (black), GFO (red), T/Po (yellow), and T/Pt (green). The Mariana Arc is described by an ideal arc (black curve) of 630 km in radius (arrow), centered at 17°N, 139.6°E (cross). The M2 and S2 barotropic current ellipses, predicted using TPXO6.2, are in red and cyan, respectively, and their Greenwich phases are indicated by lines. Both the M2 and S2 constituents rotate clockwise. The AXBT deployment sites are indicated by black dots.

[4] Our research interest was triggered by the Mariana Arc's special geometric shape. It can be simply described by an ideal arc (Figure 1, black curve) of a circle, ∼630 km in radius (arrow) centered at 17°N, 139.6°E (cross). Based on the TPXO6.2 prediction [Egbert and Erofeeva, 2002], both the M2 and S2 barotropic tidal currents are rectilinear, generally normal to the arc (ellipses). We hypothesize that the westward-propagating internal tides originating from the arc make a focus at its center, so that energetic internal tides appear 630 km away from the arc.

[5] This hypothesis is tested by examining the M2 internal tides observed by air-launched expendable bathythermographs (AXBTs) and by altimetric data from multiple satellites. During the Impact of Typhoons on the Ocean in the Pacific (ITOP) project in 2010, upper ocean temperature profiles were sampled by 71 AXBTs (Figure 1, black dots), and the internal-tide induced isothermal displacements were derived. In addition, the internal-tide induced centimeter-scale sea surface displacements can be detected by satellite altimetry [Ray and Mitchum, 1996]. This technique has been significantly improved by using the combined data from multiple satellite altimeters.

2. The Altimetric Observations

2.1. Data and Method

[6] Altimetric along-track gridded sea surface height anomaly (SSHA) data from multiple satellite missions can be used to extract internal tides. TOPEX/Poseidon (T/P), Geosat Follow-On (GFO), and European Remote Sensing (ERS) missions are described in Table 1; their ground tracks in the study area are shown in Figure 1. All these data products have been processed for sea surface condition, atmospheric, and geophysical corrections [Chelton et al., 2001]. There are two sets of T/P SSHA data. One is taken along its original tracks (T/Po) from 1992 to 2002, and the other is along its tandem tracks (T/Pt) from 2002 to 2005. The ERS data are merged from ERS-2 and Envisat, 15.5 years long and 162 cycles in total. For T/P, ERS, and GFO, M2 tidal signals alias to 62.11, 94.49, and 317.11 days, respectively. According to the Rayleigh criterion, all four data sets are long enough to separate M2 from the other major tidal constituents, the annual cycle, and the semi-annual cycle [Le Provost, 2001].

Table 1. Altimetric SSHA Data Sets From Multiple Satellite Missions
 SatelliteM2 AliasDataNumber of
DatasetAltimeter(Days)PeriodCycles
T/PoT/P original62.111992–2002364
T/PtT/P tandem62.112002–2005113
GFOGFO317.112000–2008176
ERSERS-294.491995–200385
 Envisat94.492003–201077

[7] A plane-wave fitting technique has been developed to extract internal tides from satellite altimetric SSHA data [Ray and Cartwright, 2001; Zhao and Alford, 2009]. In a given fitting region, amplitude a, phase ϕ, and propagation direction θ of an M2 internal tide η(a, ϕ, θ) are estimated by fitting plane waves to a series of altimetric SSHA data ssha(x, y, t), where x and y are the east and north coordinates, and t is time. The plane-wave fit is performed in a given fitting region. The denser ground tracks of multi-satellite altimetry allow a smaller fitting region of 160 km × 160 km in this study (Figure 2a, black boxes), compared to previous single-satellite estimates [Ray and Cartwright, 2001; Zhao and Alford, 2009].

Figure 2.

The altimetric observations of M2 internal tides. (a) Examples of the plane-wave fitting technique. The gray lines show the ground tracks of all satellite missions. The black boxes show the fitting regions (160 km × 160 km). The red curves show the plane-wave fitted amplitudes versus compass direction. The blue arrows indicate the selected internal tidal waves. (b) M2 internal tides. The yellow-red colors indicate the magnitudes of energy fluxes of M2 internal tides. The blue arrows indicate the energy flux vectors. The isophase lines of 90° and 270° are shown in gray and magenta, respectively. In both Figures 2a and 2b, the ideal Mariana Arc and its center are shown by a black curve and a black cross, respectively; the AXBT deployment sites are shown by black dots; the bathymetric contours at 1000, 2000, and 3000 m are shown in green. (c) Amplitudes of M2 internal tides plotted versus latitude in the focal region (Figure 2a, brown box): point-wise harmonic analysis (brown dots), 0.06-degree binned average of the point-wise harmonic analysis (green), and plane-wave fit (blue). The black lines show three Rayleigh focus sizes with radii of 100, 120, and 140 km, and amplitudes normalized to 60 mm.

[8] The mode-1 M2 internal tide fit is

equation image

where ω0 is the M2 frequency, and k0 is the wavenumber. Based on hydrographic profiles in the World Ocean Atlas, k0 is estimated to be 0.0413 rad/km, i.e., a wavelength of about 152 km. Because ω0 and k0 are known, for each compass direction θ, the amplitude a and phase ϕ are determined by least-squares fit using altimetric SSHA data. When a is plotted with respect to θ in polar coordinates, an M2 internal tide appears as a lobe (Figure 2a, red curves). For each lobe, the maximal amplitude is selected, and thus one M2 internal tide η(a, ϕ, θ) can be determined. For each plane-wave fitted M2 internal tide η(a, ϕ, θ), its energy flux F can be calculated following F = 1/2Fna2, where Fn is 0.95 (kW/m)/cm2 for mode-1 M2 internal tides in this study area [Chiswell, 2006; Zhao and Alford, 2009].

2.2. Results

[9] Amplitudes of the plane-wave fitted M2 internal tides in sixteen fitting regions are presented in Figure 2a (red curves). In all cases, there is more than one lobe, suggesting multiple M2 internal tides and side lobes exist in each fitting region. In the three fitting regions between the West Mariana Ridge and the Mariana Arc, the westward internal tides have greater amplitudes. It implies that both ridges generate internal tides and that the shallower Mariana Arc is a stronger generation site. Our research interest is the M2 internal tides radiated from the Mariana Arc. Thus in each fitting region, only one dominant internal tide is selected (blue arrows): One eastward wave is selected to the east of the arc; to the west one westward wave is selected. As expected, the propagation directions of these internal tides are normal to the Mariana Arc.

[10] Over the whole study area, plane-wave fits are conducted on overlapping 0.1° × 0.1° grids. The westward and eastward M2 internal tides are then determined by selecting the dominant waves. Then the energy fluxes are calculated. The resultant outgoing propagation of M2 internal tides is shown in Animation S1 in the auxiliary material. In Figure 2b, the yellow–red colors indicate the energy flux magnitudes, and the blue arrows indicate the energy flux vectors. The Mariana Arc is spatially inhomogeneous in the generation of M2 internal tides, similar to the Hawaiian Ridge [Zaron and Egbert, 2006]. A strong generation site is near Saipan (15°11′N, 145°45′E), where the eastward and westward energy fluxes reach 5.4 and 3.8 kW/m, respectively. Averaged along the arc, the mean westward and eastward energy fluxes both are about 1.7 kW/m. Note that the altimetric estimates represent the lower bounds, because only the temporally and spatially coherent internal tides are extracted [Ray and Cartwright, 2001].

[11] The isophase contours of 90° and 270° appear as a series of concentric arcs, parallel to the Mariana Arc (Figure 2b). A wavelength of 152 km can be derived from these concentric arcs, consistent with the theoretical value. The eastward internal tides diverge. The westward M2 internal tides converge in the focal region, and produce strong internal tides. The SSHA amplitude reaches 37 mm (Figure 2c, blue lines), equivalent to an energy flux of 6.5 kW/m. This flux is about four times greater than the mean westward M2 energy flux at the arc. Beyond the focus, the westward-propagating internal tides diverge, and their isophase lines change from concave to convex shapes. Portions of the isophase lines are distorted, likely due to interference with the M2 internal tides generated at the Kyushu-Palau ridge.

[12] The plane-wave fitted results are spatially smoothed in 160 km × 160 km fitting windows. In contrast, the point-wise harmonic fitted M2 internal tides are not spatially smoothed. Figure 2c shows comparisons of the smoothed and un-smoothed amplitudes in the focal region (Figure 2a, brown box). The brown dots indicate all the point-wise harmonic fitted M2 signals falling inside the focal region. It reveals that the un-smoothed amplitudes (Figure 2c, brown dots) usually are greater than the smoothed amplitudes (blue curves). At the center of the focal region, the un-smoothed amplitudes may be greater than 60 mm, which is equivalent to an energy flux of 17 kW/m, comparable to the near-field values at the Hawaiian Ridge [Rudnick et al., 2003; Carter et al., 2008].

[13] As shown in Figure 2c, the focal region is not a singular point. Theoretically, the radius r of the focal region follows r = 1.22/D, where f is focal length, λ is wavelength, and D is the diameter of the aperture [Rayleigh, 1880]. Here f = 630 ± 50 km, λ = 152 km, and D = 1000 ± 100 km (the meridional extent of the arc), yielding r ≈ 120 ± 20 km. Rayleigh focal regions with radii of 100, 120, and 140 km, and amplitudes of 60 mm are superimposed in Figure 2c (black curves). The green curve in Figure 2c indicates the 0.06-degree binned average of the point-wise harmonic amplitudes. This curve has a radius of about 100 km in the meridional direction. It confirms that our altimetric observations (brown dots and green curve) agree well with the theoretical estimates (black curves), strongly suggesting a perfect focus.

3. The AXBT Observations

[14] On 8 September 2010, 71 AXBTs were deployed from a C130 operated by the 53rd Air Force Reserve squadron (Figure 3a, black dots). The AXBT data were received on the aircraft, decoded, and quality controlled. The experiment consisted of 17 stations, centered at 16.7°N, 142.4°E, about 200 km long on each side. The station pattern was repeated four times during the 8-hour flight to sample more than half an M2 period. According to the TPXO6.2 prediction [Egbert and Erofeeva, 2002], the deployments were during the spring tide. The temperature profiles from surface to 800 m deep were sampled at a vertical resolution of about 2 meters. The temperature profiles cluster (Figure 3b), in good agreement with the World Ocean Atlas (WOA) monthly temperature profile in September. However, there are strong variations over the 8-hour period. From the isothermal contours (Figure 3c), the isothermal displacements can be calculated (Figure 3d).

Figure 3.

The AXBT observations of M2 internal tides. (a) Sites of 71 AXBT deployments (black dots), and three examples (color curves) of plane-wave fits using the AXBT-derived isothermal displacements at 200, 400, and 600 m, respectively. The green arrows indicate the selected dominant westward internal tides. (b) Temperature profiles of the AXBT measurements (red) and the World Ocean Atlas (WOA, black). Note that the ocean is about 4800 m deep. (c) Temperature, superimposed with isothermal contours of 6–28°C with 2°C interval. (d) Isothermal displacement. (e) Vertical structure of the displacement of mode-1 internal tide, normalized to 1 at 1150 m (black), and the fitted displacements at different depths, scaled by 1:18 (green). (f) Greenwich phase and (g) propagation direction of the altimetric and AXBT observed M2 internal tides. The AXBT results in Figure 3f are shifted by 180°, because for mode-1 internal tides the isothermal displacement and the sea surface displacement are in antiphase.

[15] The isothermal displacements induced by M2 internal tides are extracted with the plane-wave fitting technique. Using all the AXBT data gives a fitting region of 200 km × 200 km. The isothermal displacements are used here, in contrast to the SSHA data for satellite altimetry. Altimetry has only one layer of SSHA data at the sea surface; however the AXBT data represents multiple layers of isothermal displacements. Thus an M2 internal tide can be estimated at each layer by the plane-wave fitting technique. The plane-wave fits at depths of 200, 400, and 600 m (Figure 3a, color curves) reveal multiple internal tides and side lobes, with the westward internal tides dominant (green arrows). The fitted amplitudes at all depths are plotted (Figure 3e, green) overlapping onto the mode-1 vertical structure (black) normalized to 1 at 1150 m, i.e., the depth of the mode-1 node. Plotting the AXBT amplitudes at a scale of 1:18 suggests the M2 internal tide has an 18-m amplitude in this region.

[16] To compare the phase (Figure 3f) and propagation direction (Figure 3g) of the altimetric and AXBT M2 internal tides at 16.7°N, 142.4°E, the AXBT-derived phases are shifted by 180°, because for mode-1 internal tides the isothermal displacement and the sea surface displacement are in antiphase. Figures 3f and 3g reveal that the altimetric and AXBT observations generally agree, though they measure different aspects of the internal tidal motion. Modal decomposition shows that a 10-mm sea surface displacement equals to a 3.5-m isothermal displacement. Therefore, the 18-m isothermal amplitude has a sea surface amplitude of 51 mm, much larger than the 26-mm amplitude derived from satellite altimetry. The discrepancy is understandable considering that the altimetric data are from 1992–2010, while the AXBT measurements are from an 8-hour period at the spring tide. Note that the AXBT experiment is about 300 km to the east of the focal region, thus does not sample the most energetic internal tides.

4. Summary

[17] The M2 internal tides originating from the Mariana Arc have been examined using multi-satellite altimetric SSHA data and ITOP AXBT temperature measurements to confirm a focal region around the center of the Mariana Arc. Our major results include:

[18] 1. The altimetric and AXBT measurements of M2 internal tides agree in propagation direction and phase, though they sample different aspects of the internal tidal motion. The AXBT measurements give much larger internal wave amplitudes at the sea surface than the altimetric measurements, likely because of different time coverages: 18 years for multi-satellite altimetry, and 8 hours at the spring tide for AXBT.

[19] 2. Both the Mariana Arc and the West Mariana Ridge are generation sites of M2 internal tides. From the Mariana Arc, M2 internal tides radiate both westward and eastward, with isophase lines parallel to the arc, and sharing the same center.

[20] 3. The westward-propagating M2 internal tides from the Mariana Arc converge in a focal region centered 17°N, 139.6°E, with strong M2 internal tides converging in a focal region of a size comparable to that of a perfect focus. In the focal region, the spatially smoothed energy flux is about 6.5 kW/m, about four times the along-arc mean value; while the spatially un-smoothed energy flux may reach up to 17 kW/m.

[21] These results are intriguing, but are limited by low spatial and/or temporal resolution in the data sets. The altimetric results contain only the temporally coherent components, and thus underestimate the real strength. The AXBT observations are low in spatial resolution at only 17 stations. Its temporal resolution is high, but its record length is only 8 hours. These limitations prevent quantitative analysis; dedicated numerical and field experiments are needed to further understand these processes.

Acknowledgments

[22] This work was supported by the Office of Naval Research through the ITOP project (N00014-08-1-05777). We are grateful to the people and aircraft of the 53rd Air Force Reserve squadron, “Hurricane Hunters”. The altimeter products were produced by Ssalto/Duacs and distributed by Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO), with support from Cnes (http://www.aviso.oceanobs.com/duacs/).

[23] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.

Ancillary