a. Global distribution of M₂ internal tide energy fluxes

In most parts of the ocean, M₂ tides are the dominant components of tides. They can be extracted accurately from the satellite altimeter data in which they show up as strong signals (Kantha and Tierney 1997; Ray and Cartwright 2001; Tian et al. 2003). The method to calculate the energy flux of M₂ internal tides was described in detail by Tian et al. (2003). Here we will describe the method only briefly along with an illustrative example.

Ten years of TOPEX/Poseidon altimeter tidal data (Benada 1997) from October 1992 to June 2002 constitute the baseline of this study. Tracks forming a diamond-shaped domain are chosen to obtain a relatively accurate 2D structure of the internal tide field.

First, to determine the main tidal components, we compute the frequency spectra of TOPEX/Poseidon time series data. This is followed by harmonic analysis from which the tidal height of all main tidal components, especially that of M₂ tide, is calculated. Second, the baroclinic components of the M₂ tide are separated from the barotropic ones according to their wavelength difference. Simple high-pass filtering (cutoff at approximately 400 km) of tidal heights along the tracks can adequately remove the barotropic tides in the open ocean (Ray and Mitchum 1997). Then, the baroclinic tidal waves are treated as a superposition of several plane waves, which propagate in different directions. Although the magnitude of the horizontal wavenumber of M₂ internal tide is uniquely determined by the linear dispersion relationship, the wavenumbers in different directions, however, have different projections on a specific altimeter track. Hence the corresponding peaks in the wavenumber spectrum along the altimeter track are distinguishable. Thus, the number of the internal waves in the superposition can be determined according to the number of peaks in the wavenumber spectrum, that is,

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where ζ is the baroclinic tidal height, A_i and B_i are the amplitudes, ω_i is frequency, k_i and l_i are x and y components of the wavenumbers respectively, and i = 1, 2, . . . , n. Usually the peaks are obvious enough to be picked out by eye, and hence a threshold spectral amplitude is not assigned for the identification of individual peaks. Because every plane wave in Eq. (1) is horizontally two-dimensional, the baroclinic tidal heights along all of the four altimeter tracks, which take the form of a diamond-shaped domain, are used to determine k_i and l_i by the least squares method. Third, the velocity and energy flux of the internal tides are calculated. That the Brunt–Väisälä frequency profiles have the form of a Dirac function (Conkright et al. 1999) helps in the derivation of the analytical solutions to the dynamic equations of internal tides (Baines 1982). Here the x coordinate is assigned to be the wave propagation direction, letting K_i = k²_i + l²_i, i = 1, 2, . . . , n, where the subscript i represents the index of the superposition component. Assuming that the velocity potential of the first baroclinic mode of internal tides has the form Ψ_i = φ_i exp(iK_ix), where φ_i satisfies

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and where

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N₀ is the Brunt–Väisälä frequency, d is the thermocline depth, and h is the water depth, the velocities u_i and w_i of the first baroclinic mode of internal tides are calculated by u_i = −∂Ψ_i/∂z and w_i = −∂Ψ_i/∂x. The total velocity is then u = Σⁿ_i=1(u_in_i + w_iz), where n_i is the horizontal unit vector of the ith wave component, and z is the vertical unit vector. Last, the energy flux of the internal tides is calculated by P(z) = p(z) · u(z), in which p(z) is the fluctuation pressure. Note that we only obtain one energy flux of the internal tide for each diamond-shaped domain.

As an illustrative example given below, the energy flux of M₂ internal tide near the Ryukyu Trench has been calculated using the above equations. A diamond-shaped calculation area is shown in Fig. 1. Ten years of TOPEX/Poseidon tidal data at point A (27.8°N, 132.8°W), marked in red circle in Fig. 1, are shown in Fig. 2a. The spectrum at point A is shown in Fig. 3. Note that according to the folding frequency, the corresponding spectrum peak of M₂ tide should be at 62.1 days (Ray and Mitchum 1997). The M₂ tidal heights after the harmonic analysis are shown in Fig. 2b. Simple high-wavenumber-pass filtering along the tracks (cutoff is 400 km) is used to obtain baroclinic M₂ tidal heights. The baroclinic tidal heights of the time series at point A are shown in Fig. 2c. The high portion of the wavenumber spectrum along track B marked in red in Fig. 1 is shown in Fig. 4, which shows clearly the presence of three peaks. This is true also for the high portion of the wavenumber spectrum along other tracks in this calculation area. Thus, the baroclinic tides in this area are treated as a superposition of three plain waves. Applying a least squares method, A_i, B_i, k_i, and l_i can be accurately determined by Eq. (1). Then, applying Eq. (2), the velocities of M₂ internal tides can be calculated. Last, the vertically integrated energy flux of M₂ internal tides in this area is calculated to be 3.9 kW m⁻¹, as indicated by the black arrow in Fig. 1.

b. Estimation of mixing rate caused by M₂ internal tides

From the viewpoint of energy balance, we can estimate the energy dissipation of M₂ internal tides in the upper ocean. Assuming that the ocean is at a steady state, the energy balance of M₂ tides is in the form of

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where ε, S, and P are the energy dissipation, energy source, and energy flux of M₂ internal tides, respectively. The M₂ internal tides derived from the TOPEX/Poseidon data are the first mode, which contains most of the energy (Cummins et al. 2001; Ray and Cartwright 2001). Thus, we constrain our estimations to the first baroclinic mode. In the upper layer of the inner ocean, which is far from the coasts and with rough bottom topography, the source term S can be omitted; that is, energy dissipation balances net internal tide flux in the inner ocean. Hence the energy balance Eq. (3) can be simplified as

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Integrating (4) in a suitable control volume Ω, the average energy dissipation of M₂ internal tides in the upper ocean is

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where the control volume comprises several adjacent diamond-shaped areas, including horizontally the TOPEX/Poseidon orbits and vertically the upper layer from the ocean surface to the main thermocline bottom. Then the eddy diffusivity caused by the M₂ internal tides is calculated based on an assumed relationship with average energy dissipation and stratification (Osborn 1980),

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where the Brunt–Väisälä frequency N can be calculated from ocean profile data (Conkright et al. 1999).

A flowchart showing the logistics of the method is given in Fig. 5.

a. Global energy flux and latitudinal distribution of mixing rate

Figure 6 shows the global distribution of the M₂ internal tide energy fluxes calculated from the TOPEX/Poseidon data. One can see that the energy flux is high near the coasts and areas with abruptly changing bottom topography, as in the Ryukyu Trench, midocean ridges, and fracture zones, and propagates into the inner ocean. Theoretically we can calculate a flux in each TOPEX/Poseidon diamond. In real practice, however, it is a time-consuming task, and we do not have enough computer time to accomplish the task. Therefore, the energy fluxes shown in Fig. 6 are sparser than those in the TOPEX/Poseidon diamonds.

Spatial distribution of the energy dissipation caused by M₂ internal tides is shown in Fig. 7. Because the control volume is composed of several adjacent TOPEX/Poseidon diamonds, one energy dissipation estimate requires several energy fluxes. In addition, one can see from the procedures of the method in Fig. 5 that there are not enough energy fluxes to estimate the energy dissipation in a few areas and Eq. (4) is not applicable in a few other areas. Thus, the number of dots in Fig. 7 is much less than that of the arrows shown in Fig. 6. To illustrate by a real case how the energy dissipation diagram as shown in Fig. 7 is constructed, a part of the area in Fig. 6 south of the Aleutian Trench is extracted and displayed in Fig. 8. One can see that there are six energy fluxes (marked I–VI) in or near a control area composed of four TOPEX/Poseidon diamonds (encompassed by red lines in Fig. 8) south of the Aleutian Trench. In this domain, the average depth of the main thermocline bottom is 300 m and the average Brunt–Väisälä frequency above the main thermocline bottom is 6 cph (Conkright et al. 1999), as shown by the embedded figure in Fig. 8. The fluxes I–IV flow into the control volume, while fluxes V and VI flow beyond the edge of the domain. The energy dissipation of M₂ internal tides in this domain is assumed to be the summation of fluxes I–IV. The calculated energy dissipation in this domain is 1.29 × 10⁻⁸ W kg⁻¹, and the estimated mixing rate caused by M₂ internal tides is 1.73 × 10⁻⁵ m² s⁻¹.

Figure 9 shows the respective latitudinal distributions of the mixing rate in the Pacific, the Atlantic, and the Indian Oceans. One can see that the latitudinal distribution of the mixing rate exhibits a near-symmetrical structure with respect to the equator. At 30°N and 35°S, the mixing rate reaches a maximum value of 3.8 × 10⁻⁵ m² s⁻¹ in the North Pacific and 2.5 × 10⁻⁵ m² s⁻¹ in the South Pacific. At 30.5°N and 27.5°S, it reaches a maximum value of 5 × 10⁻⁵ m² s⁻¹ in the North Atlantic and 3.6 × 10⁻⁵ m² s⁻¹ in the South Atlantic. At 29.5°S, it reaches a maximum value of 3.7 × 10⁻⁵ m² s⁻¹ in the south Indian Ocean. The mixing rate decreases both at the equator and toward the higher latitudes. At the equator it is only 10% of that at 28.9° in all of the three oceans, at the same order as a molecular mixing coefficient. This indicates that midlatitudes are the key regions for mixing in the upper ocean. These results may be related to parametric subharmonic instability (McComas and Müller 1981; Nagasawa et al. 2000; Hibiya et al. 2002), which can enhance the mixing rate caused by M₂ internal tides at 28.9°. Hence, we may conclude from the discussion that the mixing rate should not be set as a constant, as practiced in many climate models, but rather as a parameter varying with latitudes.

b. Error analysis

The quality of the calculation of the M₂ internal tides was tested using the ADCP data observed in the South China Sea from 20 August 2000 to 4 November 2000. The ADCP was located at 20°34.8510′N, 118°24.4610′E. Using the above method, we processed the TOPEX/Poseidon (T/P) data concurrent with the ADCP observations. The details of such comparisons have been previously discussed by Tian et al. (2003).

The control volume is composed of several T/P diamonds. The horizontal size of the control volume is determined by the size of the diamond. Because linear approximations were applied to estimate the energy dissipations, small control volumes are required. Enlargement of the control volume would make the size of the control volume several times larger. This would challenge the validity of applying Eq. (4) in a large area, and could lead to larger errors. The above consideration would therefore favor the choice of the smallest possible control volume. On the other hand, if the control volume is too small, there would be insufficient energy flux to estimate the energy dissipation. The lower boundary of the control volume is the bottom of the thermocline. The depth of the thermocline is from −300 to −900 m, as was measured by Conkright et al. (1999). We estimated that

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where ρ₀ is the seawater density, l is the length of a leg of the control volume, and d is the depth of the thermocline. One could see that, even if the error of d is as large as 50 m, the relative error of the dissipation rate ε would still be no larger than 20%.

The calculations are based on 10 years of TOPEX/Poseidon altimeter tidal data. The nominal bias of tidal data is 1.5 cm (Benada 1997). The wavenumber of baroclinic M₂ internal tides is of the order of 0.01 km⁻¹, the average depth of the thermocline is assumed to be 500 m, and the average water depth is about 4000 m. The estimated bias of the energy flux in Fig. 6 is about 1000 W m⁻¹. We also estimate the error caused by the nominal bias of tidal data, using the real parameters including the wavenumber, depth of the thermocline, and the water depth in 50 random T/P diamonds. The estimated mean error of the M₂ internal tidal energy flux is 846 W m⁻¹, the maximum error is 1142 W m⁻¹, and the minimum is 634 W m⁻¹. If the average Brunt–Väisälä frequency is taken to be 3 cph in the inner ocean, the estimated error of mixing rate is roughly 1 × 10⁻⁵ m² s⁻¹.

There are some volumes in which the fluxes are insufficient to calculate the summation. There are also a few volumes near large topography, for example, near the east of Australia, in which the sum of fluxes is negative. This means that Eq. (4) does not apply in such areas and, as a consequence, not all energy fluxes in Fig. 6 are used to calculate the energy dissipation shown in Fig. 7.

The error is as large as one-quarter of the mixing rate caused by M₂ internal tides near 28.9°N/S and is even larger than that near the equator. However, our main purpose in this study is to show the quality of the latitudinal structure of mixing rate from the viewpoint of energy balance. The error as discussed above should therefore have no effects on the latitudinal pattern that we concluded. Besides, the pattern of mixing rate variation with latitude in the North Pacific (Fig. 9a) is consistent with those shown in Fig. 2 of Gregg et al. (2003).