Chinese Journal of Oceanology and Limnology Vol. 27 No. 1, P. 129-134, 2009 DOI: 10.1007/s00343-009-0129-5
Estimates of global M2 internal tide energy fluxes using TOPEX/POSEIDON altimeter data* ZHANG Yanwei (张艳伟)**, LIANG Xinfeng (梁鑫峰), TIAN Jiwei (田纪伟), YANG Lifen (杨丽芬) Physical Oceanography Laboratory, Ocean University of China, Qingdao 266100, China
Received Aug. 8, 2007; revision accepted Sept. 5, 2007 Abstract TOPEX/POSEIDON altimeter data from October 1992 to June 2002 are used to calculate the global barotropic M2 tidal currents using long-term tidal harmonic analysis. The tides calculated agree well with ADCP data obtained from the South China Sea (SCS). The maximum tide velocities along the semi-major axis and semi-minor axis can be computed from the tidal ellipse. The global distribution of M2 internal tide vertical energy flux from the sea bottom is calculated based on a linear internal wave generation model. The global vertical energy flux of M2 internal tide is 0.96 TW, with 0.36 TW in the Pacific, 0.31 TW in the Atlantic and 0.29 TW in the Indian Ocean, obtained in this study. The total horizontal energy flux of M2 internal tide radiating into the open ocean from the lateral boundaries is 0.13 TW, with 0.06 TW in the Pacific, 0.04TW in the Atlantic, and 0.03 TW in the Indian Ocean. The result shows that the principal lunar semi-diurnal tide M2 provides enough energy to maintain the large-scale thermohaline circulation of the ocean. Keyword: TOPEX/POSEIDON altimeter; M2 internal tide; vertical energy flux; horizontal energy flux
1 INTRODUCTION The meridional overturning circulation (MOC, which is also called the large-scale thermohaline circulation) in the ocean carries approximately one third of heat flux from low latitudes to high latitudes as a major component in the climate system. Internal tides, which are generated by the interaction between barotropic tides and the bottom topography, have been implicated as a major source of mechanical energy for MOC (Munk and Wunch, 1998). Thus, various methods have been applied to calculate the energy conversion from the barotropic tides to internal tides. Egbert and Ray (2000) empirically calculated the tidal energy dissipation in the deep ocean using 6-year satellite altimeter data and the value could reach up to 1 TW, which is nearly the half of the necessary energy to maintain the MOC. St. Laurent and Garrett (2002) improved the linear internal wave generation theory (Bell, 1975) and estimated the M2 internal tide vertical energy flux in the East Pacific Rise and the Mid-Atlantic Ridge. They showed that only 30 ± 10% of the baroclinic tide energy dissipated near the generation site, while the rest propagated into the inner ocean. Nycander (2005) applied the linear theory for small bottom
slope to global bottom topography, finding a plausible flux of 1.2 TW into internal tides at the M2 frequency. Many researches find that a large fraction of the energy loss from the barotropic tide in the open ocean is associated with steep isolated features (Egbert and Ray, 2000; Niwa and Hibiya 2001; Simmons et al., 2004). Although indirect evidence of internal tides has been found over gentle topography, direct in situ measurements of internal tide generation have thus far concentrated on abrupt topography. The measurements by Gregg (1987) and released tracers experiments by Ledwell et al. (1998) show that the mixing rates are roughly 1.0×10-5 m2 s-1 in the upper ocean interior; meanwhile, some deep-ocean microstructure observations in the Brazil Basin over rough bathymetry displayed enhanced mixing rates as large as 1.0×10-3 m2 s-1 (Polzin et al., 1997). Many forecasts are confirmed from theories and numerical simulations in field programs at
* Supported by the National Basic Research Program of China (973 Program, No. 2005CB422303), the International Cooperation Program (No. 2004DFB02700), and the National Natural Science Foundation of China (No. 40552002). The TOPEX/POSEIDON data are provided by Physical Oceanography Distributed Active Archive Center (PO DACC) ** Corresponding author:
[email protected]
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Mendocino Escarpment (Althaus et al., 2003) and the Hawaiian Ridge (Lee et al., 2006; Martin et al., 2006; Nash et al., 2006; Rudnick et al., 2003). However, all these estimates are made in very limited regions, and the global energy conversion is still unknown. In this paper, a simplified method is developed to extract the M2 barotropic tidal current from TOPEX/POSEIDON altimeter data, with which the M2 internal tide vertical energy flux in the interior ocean and horizontal energy flux from the boundary are calculated and discussed. At last, a global distribution of M2 internal tide energy flux is presented.
2 EXTRACTING M2 BAROTROPIC TIDAL CURRENT In this section, the M2 barotropic tidal current is extracted using ten-year TOPEX/POEISDON sea level height data along satellite tracks. The orbital period of TOPEX/POEISDON is 10 days. The distance between two neighboring resolution cells along satellite ground track is 5.75 km and the equatorial cross-track separation is 315 km. The M2 tidal amplitudes ζ0 at each point can be calculated with harmonic analysis. First, the frequency spectra of the time series sea level heights are estimated at each point to determine the main tidal components. For example, the altimeter sea level height data of an arbitrarily chosen point (27.8°N, 132.8°W) is shown in Fig.1a; and Fig.1b is the corresponding frequency spectrum. The corresponding period of the maximal spectral peak is obtained for 62.7 days, which is the alias M2 tidal period. Considering that the ascending and descending tracks of TOPEX/POEISDON satellite have the same angle to the equator, the eastward and northward gradient of the M2 tidal amplitude can be calculated as follows:
∂ζ
∂ζ ∂ζ 0 = ( 0 + 0 ) / 2cos θ ; ∂x ∂l ∂m ∂ζ ∂ζ ∂ζ 0 = ( 0 − 0 ) / 2sin θ (1) ∂y ∂l ∂m where ζ 0(x, y) is the M2 tidal amplitudes, l and
m are the directions of the ascending and descending tracks, θ is the angle between the satellite tracks and the equator.
Momentum equation of barotropic tide is considered as: ∂u (2) + f × u = − g ∇ (ζ − ζ E ) h ∂t
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where u is the barotropic tidal current, f is the Coriolis parameter, ζ is the tidal elevation, ζE is the equilibrium tidal height, and ∇ h stands for horizontal gradient. Note that there are no friction and energy dissipation in Eq.2 and the calculation with St. Laurent and Garret model (2002) of these effects is introduced in the next section. In a small area, the change of ζE is so small that the horizontal gradient can be considered as zero. Therefore, Eq.2 can be simplified to be ∂u (3) + f × u = − g∇ ζ h ∂t Because the frequency of M2 tide ω is a constant, the tidal elevations can be expressed in the form of ζ = ζ0(x, y)exp(iωt) and the tidal currents can be expressed in the same way, u=U0exp(iωt), v=V0 exp(iωt) (Smith and Young, 2002; Laurent and Garrett, 2002), where θ0 is the relative temporal phase between u and v. Substitute these two solutions into Eq.3, and then the analytical expressions of U0, V0 are obtained as g (iω U = 0
∂ζ
∂ζ 0 + f 0) ∂x ∂y − b ± b 2 − 4 ac 2a ω2 − f 2
(4) g (iω V = 0
∂ζ
∂ζ 0 − f 0) 1 ∂y ∂x ⋅ 2 2 cos ϑ0 ω − f
(5)
Therefore, the M2 tidal ellipses including semi-major and semi-minor axes are obtained with U0, V0, and further the transportation ellipse using (U, V)=H(u, v). The distribution of the M2 barotropic tidal transportation ellipse of global ocean (Fig.2) shows a single point to the solutions to Eqs.4 and 5 when ω=f. This point is termed the critical latitude. Since the critical latitude for M2 tide is 74.5°, while the range of TOPEX/POSEIDON data is between 66°N and 66°S, our results do not suffer from the critical latitude problem.
3 M2 INTERNAL TIDE ENERGY FLUX RADIATED INTO THE INTERIOR OCEAN The total internal tide energy flux radiating into the interior ocean is obtained as the sum of two independent parts, the vertical one from the ocean bottom and the horizontal one from the lateral,
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Fig.1 (a) Altimeter sea level height data of an arbitrarily chosen point (27.8°N, 132.8°W), (b) Corresponding frequency spectrum of the above sea level height data
Fig. 2 Global distribution of M2 barotropic tidal ellipses inferred from the TOPEX/POSEIDON altimeter data
boundaries. These two parts are calculated separately, using two different methods. The internal tides are generated when barotropic tides meet the bottom topography and then radiate into the interior ocean. For the vertical energy flux of internal tides generated in deep ocean, the linear internal wave generation model of Laurent and Garrett (2002) is chosen, and then the internal tide energy is expressed as two parts: one is the vertical energy flux Ef0 under the assumption of infinite depth, and the other is the energy modification Efc considering the influence of finite depth. In this model, the generated energy flux is: Ef = Ef0 - Efc, where (( N b2 − ω 2 )(ω 2 − f 2 ))1 / 2 1 ρ0 2 ω (6) ∞ ∞ 2 2 2 2 2 ( u e cos ϑ + ve sin ϑ ) K φ ( k , l ) dkdl −∞ −∞
Ef0 =
(( N b2 − ω 2 )(ω 2 − f 2 ))1 / 2 1 ρ0 ω 2 k0 k0 2 2 2 2 2 (u e cos ϑ + ve sin ϑ ) K φ ( k , l ) dkdl − k0 − k0
Efc =
(7)
where ρ0 is the sea water density; Nb is the Brunt-Väisälä frequency near the bottom, which is calculated with Levitus data obtained in 1998; f, the Coriolis parameter, k0 =
1
ω2 − f 2 2 ) ; 2H N 2 − ω 2 π
(
ue and ve are the maximum velocities along the semi-major axis and the semi-minor axis, respectively; l k
ϑ = tan −1 − α , α is the angle between the major-axe of the tidal
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ellipse and eastward direction; K2=k2+l2, φ(k, l) is the 2D wavenumber spectrum of the bottom topography; and k, l are eastward and northward wavenumber, respectively. The global ocean is divided into 5°×5° grids, and the corresponding φ(k, l) in each grid is calculated using the ETOPO5 bottom topography data with the resolution of 5°×5°. Fig.3a shows an arbitrarily chosen bottom topography based on ETOPO5 data and Fig.3b is the corresponding wavenumber spectrum φ(k, l). The spectra have been
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normalized so the integrated spectrum gives the __ mean square height h of topography in all 5°×5° grids as: ∞ ∞ 2 −∞ −∞ φ ( k , l ) dkdl = h
(8)
The global vertical energy fluxes from bottom of M2 internal tides are calculated by using the M2 barotropic tidal currents obtained in Section 2 and Levitus data obtained in 1998 within every 5°×5°
Fig.3 (a) Arbitrarily chosen bottom topography based on ETOPO5 data; (b) Corresponding wavenumber spectrum of the chosen topography
Fig.4 Estimates of M2 internal tidal vertical energy flux (color) from bottom and M2 internal tidal horizontal energy flux (vectors) from boundaries
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ZHANG et al.: Estimates of global M2 internal tide energy fluxes using TOPEX/POSEIDON altimeter data
cell (color codes in Fig.4). In the Pacific, the mid-ocean ridge lies in the eastern part of the ocean basin, but the slope of ridge is quite smooth, so that the energy conversion above the mid-ocean ridge in the Pacific is small, while that in the western Pacific is much larger for having many deep ocean trenches and archipelagoes including Hawaii Islands, Micronesia, and Melanesia. The above distribution is similar to corresponding to the areas of large energy dissipation obtained by Egbert and Ray (2000). In the Atlantic Ocean, the mid-ocean ridge lies in the middle and parallels to the coastlines. The slope of the mid-ocean ridge is steep. In the Indian Ocean, the mid-ocean ridge is almost in the middle of the ocean basin with three branches. Therefore, the vertical energy fluxes in these two oceans are large and uniform. Integrating the global oceans, the M2 internal tidal vertical energy flux is 0.96 TW in total, of which 0.36 TW is in the Pacific, 0.31 TW in the Atlantic and 0.29 TW in the Indian Ocean. However, since the area of the Pacific is almost the sum of the Atlantic and the Indian Ocean, the vertical energy flux density in the Indian Ocean is the largest with the value of 3.9×10-3 W m-2, and the second is 3.8×10-3 W m-2 in the Atlantic, in the Pacific it is 2.2×10-3 W m-2. The method described in detail by Tian et al. (2003) is used to calculate the horizontal energy flux radiated into the ocean from the lateral boundaries. Vectors in Fig.3 show the net horizontal energy flux of M2 internal tide generated in the margins of the oceans, which is also extracted from TOPEX/ POSEIDON altimeter data. By integrating them along the ocean margins, the total horizontal energy flux into the interior ocean is calculated at 0.13 TW, with 0.06 TW in the Pacific, 0.04 TW in the Atlantic and 0.03 TW in the Indian Ocean. The ocean margins are the main regions generating internal tide, because the depth is smaller and tidal range is larger. Furthermore, the complex bottom topography such as islands trenches and archipelagoes distributes mainly in the ocean margins. Some parts of the internal tide energy disperse in the generating areas, and the other parts radiate into the interior ocean. Even now, the marginal area is so small comparing with the wide ocean basins; the horizontal energy flux from internal tide into the interior ocean is merely 1/10 of the vertical one from the bottom. In summary, total M2 internal tide energy flux radiating into the interior ocean is about 1.09 TW, including 0.42 TW, 0.35 TW and 0.32 TW in the Pacific, Atlantic and Indian Ocean, respectively.
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4 DISCUSSION AND CONCLUSION There are some previous works on the extraction of internal tides from altimeter data. Ray (2001) calculated the global tidal ellipse of M2 tides and K1 tides. The main features of our results and Ray’s are similar. However, his model requires a big load of calculation, whereas only the amplitudes of M2 tides are calculated in our model. For example, the M2 tides are large in the areas to the east of Madagascar, and to the south of Alaska and the Northern Atlantic. But the M2 tides are small to the east of Japan. Therefore, the field measurements of current velocities in the South China Sea (SCS) are used to test the method. The ADCP was moored at 118°24.461′E/20°34.851′N from August 20, 2000 to November 4, 2000. Meanwhile, long-term tidal harmonic-analysis technique is used to separate M2 tides from ADCP current data. Table 1 shows clearly that results from our method, which agree well with that from the ADCP observation. This agreement makes us confident to estimate the energy flux of internal tide with this model. Table 1 Comparison of barotropic tidal currents derived from the ADCP and from the TOPEX/POSEIDON data ADCP Data
TOPEX/POSEIDON Data
Amplitude/ms-1
Phrase/rad
Amplitude/ms-1
Phrase/rad
U
0.082
1.919
0.073
2.021
V
0.061
0.224
0.059
0.206
Munk and Wunsch (1998) indicated that mechanical energy of 2 TW is required to maintain MOC, of which M2 tides provide 0.6 TW. According to the estimation by Egbert and Ray (2000), the global tides provide 1 TW energy to MOC. Results in this paper show that all the vertical energy flux caused by M2 internal tide is 0.96 TW, with 0.36 TW in the Pacific, 0.31 TW in the Atlantic and 0.29 TW in the Indian Ocean. Thereby, adding the net horizontal M2 internal tide energy flux radiated from the ocean margins, i.e., 0.06 TW for the Pacific, 0.04 TW for the Atlantic and 0.03 TW for the Indian Ocean, the total M2 internal tide energy flux radiated into the interior ocean is obtained for about 1.09 TW. This result is a little greater than that by Egbert and Ray (2000) and Simmons et al. (2004) and close to that of Nycander (2005). In our case, the M2 internal tide can provide solely the enough energy to maintain MOC.
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