*3.3. Energy Transfer*

In order to analyze energy transfer and internal wave generation, we introduce the depth-integrated barotropic and baroclinic energy equations [24,41,42].

$$
\frac{
\partial
}{
\partial t
}(\overline{E\_{k0}} + \overline{E\_{p0}}) + \nabla \cdot \overline{F\_0} = -\overline{\mathcal{C}} - \overline{\mathcal{E}\_0} \tag{2}
$$

$$\frac{\partial}{\partial t}(\overline{E\_k'} + \overline{E\_p'}) + \nabla \cdot \overline{F'} = \overline{\mathbb{C}} - \overline{\mathbb{c}'} \tag{3}$$

where subscript <sup>0</sup> and superscript <sup>0</sup> indicate barotropic and baroclinic, respectively. *E<sup>k</sup>* is the kinetic energy, *E<sup>p</sup>* is the available potential energy, *F* is the energy flux, *C* is the barotropic to baroclinic energy conversion rate that connects the two equations, and *e* is the dissipation term including the conversion and radiation processes and bottom drag. As this paper mainly focuses on the baroclinic responses of barotropic forcing, we diagnose only the relative kinetic and potential energy terms and the conversion rate between the two.

$$
\overline{\mathbb{C}} = p\_b' \mathcal{W} \tag{4}
$$

$$\overline{E\_{k0}} = \frac{1}{2}\rho\_0(\mathcal{U}^2 + V^2)H \tag{5}$$

$$\overline{E\_{p0}} = \frac{1}{2} \rho\_0 g \eta^2 \tag{6}$$

$$\overline{E\_k'} = \frac{1}{2}\rho\_0 \int\_{-d}^{\eta} (u'^2 + v'^2 + w^2) dz \tag{7}$$

$$\overline{E'\_p} = \frac{g^2}{2\rho\_0} \int\_{-d}^{\eta} \frac{\rho'^2}{N\_b^2} dz \tag{8}$$

where *p* 0 *b* is the perturbation pressure at the bottom; *<sup>W</sup>* <sup>=</sup> <sup>−</sup>*U*<sup>~</sup> · 5*<sup>H</sup>* is the vertical velocity at the bottom due to barotropic flow over variable topography; *ρ*<sup>0</sup> is the reference density; *u* <sup>0</sup> = *u* − *U* and *v* <sup>0</sup> = *v* − *V* are the zonal and meridional baroclinic velocity, respectively; *w* is the vertical velocity; and *ρ* <sup>0</sup> = *ρ* − *ρ<sup>b</sup>* is the perturbation density due to wave motions, where *ρ<sup>b</sup>* is the background density during the selected spring-neap cycle.

Overall, *C* represents the conversion rate from barotropic to baroclinic mode. Generally, *C* should be the sink term for Equation (2) and the source term for Equation (3), which also means that baroclinic terms gain energy from barotropic terms, on average. However, *C* can be a sink for baroclinic components, for example, *C* becomes negative when *W* and *p* 0 *b* are out of phase [24,43]. In our case, *C* is positive most of the time (Figure 12a), which represents energy transfer from barotropic tide to baroclinic tide. Negative *C* can reach nearly half of the maximum positive value during spring tide, which suggests a strong local dissipation of the baroclinic tide for this period.

For kinetic energy, the barotropic part *Ek*<sup>0</sup> and the baroclinic part *E* 0 *<sup>k</sup>* mainly change according to barotropic and baroclinic velocity, respectively. *Ep*<sup>0</sup> mainly changes with sealevel height and represents the potential energy due to surface waves. Equation (8) is an exact expression for the baroclinic potential energy if the fluid is linearly stratified [44]. In our case, *N<sup>b</sup>* at L1 is slowly varying and almost constant, which suggests that this expression is suitable for evaluating the local available potential energy. *E*0 *<sup>p</sup>* directly measures the strength of isopycnal perturbations.

Figure 12b–e shows the changes for those energy components. The time series of *Ek*<sup>0</sup> and *Ep*<sup>0</sup> both show a significant spring-neap cycle and peaks in one day, which matches with the local barotropic tidal signal. The baroclinic energy components exhibit different features. *E* 0 *k* reaches the highest peak when the local barotropic current is directed eastward. However, this behavior changes slowly and does not show a second narrow peak when the local barotropic current is directed westward. Especially for *E*0 *p* , it shows a weaker correlation to the boundary forcing, and thus more nonlinear characteristics comparing to barotropic energy components.

Figure 12f shows the time-smoothed results of each of the above components with the same color. In order to analyze the interaction between the components, we applied a suitable amplification factor. The largest lag difference among energy components is between *E*0 *<sup>p</sup>* and the other components. Except for *E*<sup>0</sup> *p* , the other three energy components show the same fortnightly variability that matches well with the local barotropic forcing. The lag of *E*0 *p* indicates that the maximum baroclinic disturbance is not generated during the maximum barotropic forcing. At L1, the lag is approximately two days, which indicates that the maximum baroclinic disturbance occurs two days after the maximum spring

tide. As we analyzed before, the baroclinic potential energy and barotropic components exhibit different features. The baroclinic potential energy depends on the perturbation density, which depends on local stratification and dynamic field. The stratification shows a fortnightly variability resulting from the advection of buoyancy by rectified baroclinic flows. In general, the fortnightly stratification variability, i.e., the phase of baroclinic potential energy and the phase of barotropic forcing do not match at most locations. Thus, our result shows a lag or lead relation between the maximum baroclinic potential energy and the maximum barotropic forcing on fortnight time scale. Additionally, the time-smoothed result of *C* shows a comparable lag, which suggests that the maximum barotropic to baroclinic energy conversion is affected by this fortnightly stratification variability.

**Figure 12.** (**a**) Time series of depth-integrated barotropic to baroclinic conversion rate *C*, (**b**) barotropic kinetic energy *Ek*<sup>0</sup> , (**c**) baroclinic kinetic energy *E* 0 *k* , (**d**) barotropic potential energy *Ep*0, and (**e**) baroclinic potential energy *E* 0 *<sup>p</sup>* at L1. (**f**) Time-smoothed results of the above-mentioned components with the same color.

Here, we introduce a harmonic fit *f*(*t*) = *A* cos (*ωt* + *P*) + *M* of the time-smoothed results, where *ω* is the selected spring-neap cycle frequency corresponding to 14.78 days, *t* is the time, *A* is the amplitude, *P* is the phase, *M* is the time average of a spring-neap cycle, and *f* is the harmonic fitted result. In order to evaluate the goodness of this fit, the coefficient of determination *R* = 1 − ∑ *n i*=1 (*yi*−*f i* ) 2 (*yi*−*yav*) 2 is calculated here, where *f<sup>i</sup>* is the predicted value from the fit, *y<sup>i</sup>* represents the observed data, and *yav* is the mean of observed data. *R* is generally a value between 0 and 1, and a value closer to 1 indicates a better fit.

Figure 13 shows the harmonic fit results of *Ek*<sup>0</sup> and *C* for model days 24 to 38. The average *R* of *Ek*<sup>0</sup> and *C* in the Luzon Strait is 0.9664 and 0.9097, respectively, which suggests that most of the spring-neap variation of barotropic and baroclinic tides in the Luzon Strait can be explain by this fit. Strong *Ek*<sup>0</sup> is mainly distributed in channels between islands and seamounts (Figure 10c), and is accompanied by intense spring-neap variation (Figure 13a). The main generation sites and dissipation sites of baroclinic tides, suggested by time-averaged *M* (Figure 13f), are mainly distributed along the two ridges [7,23,31] and are accompanied by intense spring-neap variation as well (Figure 13d). *P* suggests the arrival time of spring tide. The arrival time of the maximum *Ek*<sup>0</sup> (Figure 13b) and *C* (Figure 13e) during the selected period can be calculated from the phase. For *Ek*<sup>0</sup> , the arrival time shows differences along the isobath in general. For *C*, only the sites where *R* is greater than 0.9 and where the average *<sup>C</sup>* is greater than 2 *<sup>W</sup>* · *<sup>m</sup>*−<sup>2</sup> are presented because we mainly focus on the generation process, the arrival time is also quite similar in the main generation sites.

**Figure 13.** (**a**) The amplitude *A*, (**b**) arrival time calculated from phase *P* and (**c**) time average *M* of selected spring-neap cycle of depth-integrated barotropic kinetic energy *Ek*<sup>0</sup> , and (**d**–**f**) conversion rate *C*.

Figure 14 shows the estimated lag between *C* and *Ek*<sup>0</sup> calculated from the phase difference. Only the generation sites are presented as well as Figure 13e. Obviously, on the fortnightly time scale, the *C* is not phase-locked to *Ek*<sup>0</sup> on potential generation sites of the Luzon Strait. Similar to L1, which we analyzed above, there are many other sites where *C* lags behind *Ek*<sup>0</sup> . Meanwhile, during this spring-neap period, *Ek*<sup>0</sup> may also lag behind *C* in some locations. The coexistence of both positive and negative lags is an expected result as well as we can see the stratification in the Luzon Strait is fortnightly redistributed (Figure 11). The difference in the arrival time between the maximum barotropic tidal forcing and the maximum barotropic to baroclinic conversion rate are not only a local relation at L1 but are also a more general pattern, which suggests a lead-lag relation between barotropic tidal forcing and maximum barolinic response within the fortnightly tidal cycle.

**Figure 14.** The estimated lag between conversion rate *C* and depth-integrated barotropic kinetic energy *Ek*<sup>0</sup> at main generation sites of baroclinic tides in the Luzon Strait. Positive means *C* lags behind *Ek*<sup>0</sup> .
