Diurnal modulation of semidiurnal internal tides in Luzon Strait

Ocean Modelling 59–60 (2012) 1–10

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Diurnal modulation of semidiurnal internal tides in Luzon Strait Dong-Ping Wang ⇑ School of Marine and Atmospheric Sciences, Stony Brook University, Stony Brook, NY, United States

a r t i c l e

i n f o

Article history: Received 29 June 2012 Received in revised form 21 August 2012 Accepted 6 September 2012 Available online 29 September 2012 Keywords: Internal tide Luzon Strait Nonlinear internal wave

a b s t r a c t A three-dimensional model is applied to the South China Sea to study the generation of internal tides in the Luzon Strait, a mixed diurnal dominant barotropic tidal environment. The computed internal tidal flux is in good agreement with limited observations and with estimates from the barotropic tidal inversion. The diurnal internal tides are simple harmonic oscillations. The semidiurnal internal tides, on the other hand, are highly nonlinear with striking daily amplitude variations. Comparison with an analytical mixed internal tide–lee wave model shows that the diurnal modulation is caused by interaction of barotropic diurnal tidal currents with the linear internal waves. The highly nonlinear semidiurnal internal tides likely is responsible for the observed diurnally modulated nonlinear internal waves in the South China Sea. The semidiurnal internal tides also appears to be the major source for the enormous vertical mixing in the Luzon Strait. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction In Luzon Strait the barotropic tidal currents are the mixed diurnal dominant. The K1, O1 and M2 have comparable amplitudes while S2 is relatively small. Strong semidiurnal and diurnal internal tides are generated by barotropic tidal currents over the two submarine ridges. Niwa and Hibiya (2004), in a three-dimensional model study (horizontal resolution 1/32°) of semidiurnal M2 internal tides in the East and South China Seas, estimated that about 14 GW of barotropic M2 tidal energy is converted into baroclinic tidal energy in the Luzon Strait. Using a similar model approach (horizontal resolution 1/12°), Jan et al. (2007) estimated that about 12 GW of barotropic K1 tidal energy is converted into baroclinic tidal energy in the Luzon Strait. Including all four major tidal constituents, Jan et al. (2008) arrived at a total conversion rate of about 30 GW. Alford et al. (2011) however reported a smaller total energy conversion rate of 24.1 GW, based on results from a high-resolution (2.5 km) internal tidal model of Simmons et al. (2011). Anyhow, the Luzon Strait is a major source for the internal tides. For comparison, the estimated total internal tidal energy conversion over the 2500-km long Hawaii Ridges is about 25 GW (Zaron and Egbert, 2006). Nonlinear internal waves (NLIWs) with amplitudes >100 m are a common feature in the northern South China Sea (e.g. Klymak et al., 2006). These waves propagate westward across the basin, shoal and dissipate at the shallow continental shelves (e.g. Liu et al., 1998; Alford et al., 2010; Fu et al., 2012). Two types of NLIWs have been identified: a-waves that arrive regularly at the same ⇑ Fax: +1 (631) 632 8820. E-mail address: [email protected] 1463-5003/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ocemod.2012.09.005

time each day, and b-waves that arrive an hour later each day (Ramp et al., 2004; Duda et al., 2004). The a-waves are larger and have a rank-ordered packet structure, whereas the b-waves are smaller and often consist of a single solitary wave. Both a- and b-waves are strongly modulated in a fortnightly cycle. Also, the a-waves appear to be generated at the peak ebb currents and b-waves at the weaker peak flood currents (Alford et al., 2010). It is suggested that the amplitudes of NLIWs depend on the strength of the peak barotropic tidal currents (Ramp et al., 2010). Evolution of nonlinear internal waves have been investigated both theoretically (e.g. Helfrich and Grimshaw, 2008) and numerically (e.g. Shaw et al., 2009; Buijsman et al., 2010a). These studies indicated that the initial evolution of internal solitary waves mainly depends on the ratio between the nonlinearity and rotation. The internal solitary wave is formed when the nonlinearity overcomes the Coriolis dispersion. (Eventually, the nonlinearity is balanced by nonhydrostatic dispersion.) Li and Farmer (2011) found that under realistic situation this criterion is satisfied only in the semidiurnal internal tides which have larger amplitude (nonlinearity) and steeper slope (Coriolis dispersion) compared to the diurnal internal tides. They also found that the amplitudes of the leading a-waves have no apparent relationship with the diurnal barotropic tides but are correlated with the semidiurnal barotropic tides. Their results provide strong evidence that the nonlinear internal waves are derived from the semidiurnal tides. In this study, we use a three-dimensional model to test the hypothesis that the semidiurnal internal tides are strongly modulated by the barotropic diurnal tidal currents. While asymmetric barotropic tidal forcing has been emphasized in relation to the generation of nonlinear internal waves, no previous model study has recognized the diurnal variability of semidiurnal internal tides.

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We note that the diurnal modulation is different from the diurnal inequality; the former is a nonlinear process involving the internal tides whereas the latter is simply a superposition of (linear) semidiurnal and diurnal barotropic tides. Section 2 describes the model setup and the energy equation. Section 3 provides an analysis of the energy transfer and energy flux. Section 4 compares the three-dimensional model results with the analytical solution of Hibiya (1986) after Li and Farmer (2011). The implication of the model results is discussed in Section 5.

is implemented near open boundaries to dampen the baroclinic fluctuations. No surface atmospheric forcing is included. However, the initial nearsurface T/S gradients would be rapidly destroyed by strong mixing associated with the large internal tides. This suggests that an external heating is required to maintain the surface mixed layer. To compensate for lack of surface heating, the model T/S in the upper 50 m are relaxed to the climatology with a 5-day time scale. 2.2. Energy equations

2. Numerical model 2.1. Model setup The model used in this study is the Princeton Ocean Model (POM), adapted for the parallel processing (http://www.imedea.uib-csic.es/users/toni/sbpom/) (Jordi and Wang, 2012). POM is a hydrostatic, three-dimensional, finite-difference, sigma-coordinate model with the Mellor–Yamada level-2.5 turbulence closure. It has been extensively used in studies of internal tides with realistic topography, e.g. Cummins and Oey (1997), Holloway and Merrifield (1999), Niwa and Hibiya (2004), Jan et al. (2007) and Carter et al. (2008). The model domain covers the northern South China Sea, between 105.5°E  126°E and 16°N  23°N (Fig. 1). The horizontal resolution is 0.03°. There are 51 sigma levels, concentrated in the top (20 levels) and bottom (14 levels) 10% of the water column. The bathymetry is based on the 500-m resolution depth archive (117°E  123°E, 18°N  23°N) provided by the National Science Council of Taiwan and the 1-min resolution global surface (ETOPO1) provided by the National Geographical Data Center, NOAA. The initial temperature and salinity (T/S) profiles are based on the summer mean (May–October, 2008) derived from the Simple Ocean Data Assimilation (SODA) (http://www.atmos.umd.edu/ ~ocean/). The climatology mean profile shows a gradual shoaling of the thermocline from the West Philippine Sea to the South China Sea. We note that in reality the Kuroshio migrates in and out of Luzon Strait with a sharp temperature front. The climatology mean temperature gradient is too smooth to support a strong vertical shear. This, together with the condition of zero transport at open boundaries, eliminates the presence of the Kuroshio Current. The model is forced at lateral open boundaries with the tidal elevation and barotropic velocity derived from the TPXO7.2 (http://volkov.oce.orst.edu/tides/TPXO7.2.html). The four major constituents (M2, S2, K1 and O1) are included. The Flather condition is used to radiate surface gravity waves out of the model domain, and the Orlanski radiation condition is used to radiate the internal disturbances (e.g. Cummins and Oey, 1997). A narrow (3 horizontal grids) sponge layer of enhanced lateral diffusion

The derivation of the barotropic and baroclinic energy equations in sigma coordinates follows Floor et al. (2011). The horizontal velocity is decomposed into the barotropic and baroclinic components,

v a ¼ v a þ v 0a

ð2:1Þ

where the subscript a = x, y. The barotropic component, denoted by an overbar, is the vertical average.

v a ¼

1 D

Z

g

v a dz

ð2:2Þ

H

and D = g + H is the total water depth. The vertical velocity can be split into the barotropic and baroclinic components,

v z ¼ v z þ v 0z

ð2:3Þ

where

zþH D   @g @g @H þ ðr  1Þv a v z ¼ r þ v a @xa @t @xa

r¼

v 0z ¼ rv 0a

@g @H þ ðr  1Þv 0a þx @xa @xa

ð2:4Þ

ð2:5Þ

ð2:6Þ

Here x is the vertical velocity in sigma coordinates, x ¼ Ddr=dt. The barotropic vertical velocity is induced by barotropic tidal currents moving up/down the sloping bottom. The hydrostatic kinetic energy density is defined as Ek ¼ 1=2qo v a v a . The vertically integrated barotropic kinetic energy equation can be written as

Tendency þ Advection þ Barotropic pressure flux divergence ¼ Barotropic conversion þ Nonlinear interaction þ Dissipation

Tendency ¼

k @E @g þ g qo g @t @t

Fig. 1. Topographic map of the northern South China Sea (depth contour interval = 1000 m). The Luzon Strait is marked by the red box. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

D.-P. Wang / Ocean Modelling 59–60 (2012) 1–10

Advection of kinetic energy ¼

k @Dv a E @xa

Barotropic pressure flux divergence Z 1 @Dv a g @Dv a P þ dr ¼ g qo r @xa @xa 0 Barotropic kinetic energy conversion ¼ D

Z

1

g q0 v z dr

ð2:7Þ

0

k is barotropic where q0 ¼ q  qo ; qo , is a reference density, and E  is a conkinetic energy. The depth-averaged hydrostatic pressure P tribution from the baroclinic tides to free surface fluctuations. The p ¼ q gz, and the vertically intebarotropic potential energy is E o grated barotropic potential energy equation is,

p @E @g ¼ g qo g @t @t

ð2:8Þ

The change of barotropic potential energy equation is purely associated with the free surface fluctuations. The baroclinic kinetic energy equation has a similar form as the barotropic kinetic energy equation.

Tendency þ Advection þ Baroclinic pressure flux divergence ¼ Baroclinic conversion þ Nonlinear interaction þ Dissipation

Advection of kinetic energy ¼

Z

@Dv a E0k dr @xa

1

0

Baroclinic pressure flux divergence ¼

Z 0

1

Z 0

3.1. Energy balance

1

g q0 v 0z dr

ð2:9Þ

We note that advection of baroclinic kinetic energy is associated with the total horizontal velocity. For the baroclinic potential energy density, E0p ¼ q0 gz, the vertically integrated baroclinic energy equation is

Tendency þ Advection ¼ Total buoyancy flux þ Diffusion Advection of potential energy ¼

Z

1

0

Total buoyancy flux ¼ D

Z

@Dv a E0p dr @xa

1

g q0 v z dr

 b  rH where the subscript b indicates the properrewritten as p0b u ties evaluated at the bottom. The form drag is defined as p0b rH, the force acting on the topography. There will be a drag if the pressure on the face of the slope upstream is different from that on the face downstream. The barotropic kinetic energy conversion is the work done by form drag on the fluid, that is, energy lost by the barotropic tides. The baroclinic kinetic energy conversion Eq. (2.9) is the buoyancy flux due to baroclinic tides, ghq0 w0 i. Energy is transferred from baroclinic potential energy to baroclinic kinetic energy if heavy fluids sink or light fluids rise. Likewise, the total buoyancy flux Eq. (2.10), the sum of barotropic and baroclinic  þ w0 Þi, represents energy exchange energy conversion, ghq0 ðw between the total kinetic and potential energy. In Floor et al. (2011) a linear equation of state is used, and potential density is equivalent to potential temperature. This is an idealized case. In general, potential density is a function of potential temperature, salinity, and the reference pressure (Mellor, 1991). A deeper reference pressure leads to a higher potential density, and consequently, a larger bottom pressure. Because form drag is directly proportional to bottom pressure, depending on the reference pressure there would be a significant difference in estimates of barotropic kinetic energy conversion (see the next section). This issue apparently has not been addressed in the internal tide literature. We use 2000 m for the reference level, similar to that used in the global baroclinic tide model of Simmons et al. (2004).

3. Energetics

@Dv 0a P0 dr @xa

Baroclinic kinetic energy conversion ¼ D

3

ð2:10Þ

0

The energy budget can be summed up in two different ways. The exchange between total barotropic energy, Eqs. (2.7) and (2.8), and total baroclinic energy, Eqs. (2.9) and (2.10), is through barotropic energy conversion. This is commonly used in the energy diagnostics in the internal tide model (Niwa and Hibiya, 2004). Alternatively, the exchange between total kinetic energy, Eqs. (2.7) and (2.9), and total potential energy, Eqs. (2.8) and (2.10), is through the total buoyancy flux. The tendency terms are negligible in a quasi-equilibrium state, and the nonlinear and advection terms generally are small. All energy equation terms are calculated from the hourly model velocity and T/S output except dissipation/ diffusion (including any numerical error) which is treated as imbalance in the energy budget. The energy budget can be better understood in the more familiar Cartesian coordinates (Pedlosky, 2003). The barotropic kinetic  where the bracket indicates energy conversion Eq. (2.7) is ghq0 wi integration over the water column. Using the hydrostatic equation and integrating by parts, the barotropic energy conversion can be

A standard model run is for 40 days starting from the rest. The hourly model data from day 15 to day 40 are used in the analysis. We focus on the Luzon Strait between 120°E  122.5°E and 18°N  22.5°N (Fig. 1). To calculate mean energy balance, the temporal averaging is over a fortnightly cycle from day 15 to day 30. Also, to distinguish between semidiurnal and diurnal tides, model output are filtered (5-th order Butterworth filter) to produce the lowpass (>33 h), bandpass (17–33 h), and highpass (<17 h) time series. The band- and high-pass data correspond to the diurnal and semidiurnal tides respectively, and the lowpass data is the slowly varying mean state. In previous model studies, energy flux typically is calculated from harmonic fits to model time series. We use the original time series in order to explore the nonlinear nature of the internal tides. Fig. 2 shows mean baroclinic energy flux for the diurnal and semidiurnal tides. For diurnal tides, the eastward energy flux is relatively uniform whereas the westward energy flux is concentrated to the south. For semidiurnal tides, the westward flux is uniform but the eastward flux is concentrated to the north. In both cases, the fluxes are radiated out of both the eastern and western ridges. The model baroclinic energy fluxes are similar to those shown in Alford et al. (2011) from a three-dimensional 2.5-km isopycnal-coordinate model after Simmons et al. (2011). We have added several observations listed in Alford et al. (2011) for comparison. The internal tides are affected by the local topography, winds and the Kuroshio, and the observations which were taken during different times, showed large fluctuations. We included only observations that were either long-term from the profiling moorings or were taken during the ‘average’ tidal conditions from the shipboard lowered ADCP-CTD. The agreement between model and observation is very good. The mean absolute amplitude differences are 4.4 and 2.8 KW/m for the semidiurnal and diurnal internal tides respectively. Conversion of barotropic to baroclinic tides mostly occurs over the eastern and western ridges (not shown) (Niwa and Hibiya,

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Fig. 2. Baroclinic energy flux (kW/m) for (left) semidiurnal and (right) diurnal internal tides. Data from Alford et al. (2011) are marked in red. Depth contour interval = 1000 m. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2004; Alford et al., 2011). Fig. 3 shows snapshots of the temperature field over a daily period at the center section, 20.5°N. For clarity, only the depth range between 300 and 1500 m is shown. The western ridge, located at 121°E, is about 2000 m below the sea surface. In the Luzon Strait, the barotropic tides are quite asymmetric with strong ebb currents that occur once a day and otherwise weak currents that reverse about every 6 h (the top panel in Fig. 4). The internal tidal activity is most dramatic over the eastern ridge. Maximum ebb current occurs at day 33.75. During the accelerating stage (from day 33.5–33.75), cold water surges up the inner flank, overflows the top, and slides down the outer flank. The rapid drawdown extends well above the ridge, bringing subsurface warm water with it. As the tides relax (day 33.75–34), the water surges back, and the isotherms are displaced markedly down slope along the inner flank. As the tides reverse again (day 34–34.25), the sequence is reversed. On the deep western ridge, the isotherm oscillations are very large (hundreds of meter) along its inner flank (not shown). However, because the stratification is weak, the deep fluctuations make relatively minor contribution to energy conversion. The impact of the western ridge is mainly felt through a narrow beam emanated along the inner flank that creates large temperature fluctuations in the mid-water column. Table 1 lists the total energy flux and energy conversion in Luzon Strait. The nonlinear and advection terms are small, 0.3– 0.5 GW, and are not included. For semidiurnal tides (‘D2’), barotropic flux into Luzon Strait is 64.9 GW, and barotropic flux out of Luzon Strait is 45.5 GW. This leaves a flux divergence of 19.4 GW. Conversion from barotropic kinetic energy to baroclinic potential energy is 20.1 GW, indicating negligible dissipation in the barotropic tides. (Obviously, barotropic conversion cannot be greater than flux divergence, the small ‘error’ likely is due to the uncertainties in the reference pressure and filtering; see later discussion.) Out of 20.1, 11.5 GW is converted into baroclinic kinetic energy, and the remaining stays in the potential energy pool. The

total baroclinic flux out of Luzon Strait is 8.6 GW, of which 4.8 GW is into the Pacific and 3.8 GW into the South China Sea. The difference between baroclinic energy conversion and baroclinic energy flux, 3 GW, is attributed to dissipation of the baroclinic tides. In the potential energy pool, 1 GW can be attributed to advective flux. The remaining 8 GW is available for diapycnal mixing. For diurnal tides (‘D1’), barotropic flux into Luzon Strait is 61.8 GW and outflux is 45.4 GW, leaving a flux divergence of 16.4 GW. Conversion from barotropic kinetic energy to baroclinic potential energy is 14.1 GW, and the remaining 2 GW is dissipated by the barotropic tides. Out of 14.1 GW, 11.1 GW goes into baroclinic kinetic energy. The total baroclinic flux out of Luzon Strait is 7.6 GW. The difference between baroclinic conversion and baroclinic energy flux, 3.5 GW, is dissipated by the baroclinic tides. In the potential energy pool of 3, 2.2 GW is advected. Thus, unlike semidiurnal tides, vertical mixing due to baroclinic diurnal tides is relatively small, 1 GW. Also, as noted earlier, calculation of barotropic energy conversion and baroclinic energy flux is influenced by the choice of a reference pressure. If potential density is referenced to the surface (instead of 2000 m), barotropic kinetic conversion is reduced by about 3 GW and baroclinic flux by about 2 GW for both the semidiurnal and diurnal tides. In calculating energy balance, we assume there is no interaction between the semidiurnal and diurnal tidal bands. This in general is not valid as the energy spectrum is continuous. Table 1 includes the energy flux terms calculated for the total tidal band (‘D1 + D2’). The differences are small for baroclinic flux between ‘D1’ + ‘D2’ and ‘D1 + D2’. On the other hand, barotropic fluxes in and out of Luzon Strait are increased by about 6 GW in ‘D1 + D2’ compared to ‘D1’ + ‘D2’. In other words, a fraction of barotropic energy flux is lost by treating D1 and D2 separately. Table 1 also includes the results from a separate model run with the semidiurnal tides only (‘M2 + S2’). There is little difference (<1 GW) in semidiurnal tidal

D.-P. Wang / Ocean Modelling 59–60 (2012) 1–10

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Fig. 3. Snapshots of the temperature field at the center section (20.5°N). Time in day, and temperature contour = 0.5 °C. Depth range is between 300 and 1500 m. The western ridge is about 2000 m below the surface, located at 121°E.

Fig. 4. Section averaged baroclinic flux at the western edge of the analysis box (120°E): (top) total barotropic transport (blue) and semidiurnal component (red); (middle) semidiurnal flux; (bottom) diurnal flux. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

energy flux between ‘D2’ and ‘M2 + S2’. We also have carried out a model run with 40 sigma levels eliminating the high resolution (10 levels) near the bottom. Since strong mixing/dissipation mostly occurs near the steep slope, the test is to find if the vertical resolution

would impact the energy budget. For both barotropic and baroclinic tides, the difference is small, <0.1 GW. It is useful to compare our results with some previous model estimates. In TPXO, a two-dimensional global barotropic tidal

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D.-P. Wang / Ocean Modelling 59–60 (2012) 1–10

Table 1 Energy flux (GW). Flux is positive leaving the Luzon Strait. Barotropic

D2 D1 D1 + D2 M2 + S2

Baroclinic

Flux east

Flux west

Conversion

Flux east

Flux west

Conversion

64.9 61.8 134.4 65.4

45.5 45.4 96.2 45.2

20.1 14.1 36.3 21.3

4.8 3.8 9.1 5.6

3.8 3.8 8.0 4.6

11.5 11.1 24.0 13.0

model with data inversion, conversion of barotropic tides into baroclinic tides is taken into account by adjusting the barotropic model solution to fit tidal harmonics derived from along-track TOPEX/ POSEIDON and JASON sea surface height observations (Egbert and Erofeeva, 2002). As such, flux divergence should incorporate energy loss to the baroclinic tides. The fluxes in(out) of Luzon Strait from TOPIX are 55.0 (35.2), 8.9 (4.6), 39.3 (29.2), and 29.2 (21.7) GW for M2, S2, K1, and O1 respectively. Grouping into D1 and D2, flux divergence are 17.7 and 24.1 GW respectively from TOPIX, which compare favorably with 16.4 and 19.4 GW from the model. We also can compare with the previous three-dimensional models of internal tides. Niwa and Hibiya (2004) considered only the M2 tide, and their barotropic (baroclinic) flux divergence are 15 (7.4) GW (S2 would add another 10%), which agree well with 19.4 (8.6) GW for D2. Including all four tidal constituents, Jan et al. (2008) obtained 30.2 (12.2) GW, which is consistent with 35.8 (16.2) GW for ‘D1’ + ‘D2’. Also, Alford et al. (2011) reported 24.1 (14.6) GW. Their baroclinic flux divergence agrees well with our value, but the barotropic flux divergence appears to be far too small.

3.2. Temporal variability The mean semidiurnal and diurnal baroclinic fluxes averaged over a fortnightly cycle are comparable. Their temporal variations, on the other hand, are quite different. Fig. 4 shows the baroclinic flux into the South China Sea and the corresponding barotropic tidal transport in the Luzon Strait. For diurnal flux, the short-term variability is dominated by the second harmonic (1/2-day) fluctuations. This is expected for simple harmonic oscillations as the flux is proportional to the velocity squared. The tidally averaged (lowpass filtered, cutoff = 26 h) diurnal flux is smooth, and its fortnightly variation follows the fortnightly cycle of the diurnal barotropic transport. The tidally averaged (lowpass filtered, cutoff = 17 h) semidiurnal baroclinic flux, in contrast, shows large diurnal modulation, indicating strong nonlinearity. The fortnightly envelopes of the semidiurnal and diurnal baroclinic energy flux are proportional to their respective fortnightly envelopes of the barotropic transport squared. There is no phase difference for the semidiurnal internal tides, but the diurnal energy flux leads the diurnal transport by about 1 day. Fig. 5 shows the baroclinic flux into the Pacific. For diurnal flux, there is little difference between the east- and westward flux. For semidiurnal flux, the second harmonic (1/4-day) fluctuations are much more pronounced in the eastward flux compared to that in the westward flux. The daily modulations only become apparent during the second half of the fortnightly cycle when the diurnal barotropic transport is larger. Also, the eastward flux is ahead of westward flux for both the semidiurnal and diurnal internal tides. Fig. 6 shows the model run with the semidiurnal tides only (‘M2 + S2’). The baroclinic flux is stronger eastward than westward, 20% higher (Table 1). Most importantly, the diurnal modulation is completely absent without the barotropic diurnal tides. The depth-integrated kinetic energy (KE) and available potential energy (APE) are computed for the semidiurnal and diurnal internal tides. The APE is calculated after Vallis (2006)

APE ¼

Z

E 1 D g ðzðqÞ  zÞ2 dq; 2

ð3:1Þ

where z is the height of an isopycnal surface, and the overbar indicates the mean (time averaged) interface. Eq. (3.1), expressed in density coordinate, is a full nonlinear description of the available potential energy (Fu et al., 2012). Fig. 7 shows section-averaged KE and APE at the western edge of the analysis box (120°E). The KE and APE are highly coherent. They have comparable amplitudes in the semidiurnal tides, but KE is much greater in the diurnal tides. The APE/KE ratios averaged over a fortnightly cycle are 0.87 and 0.38 respectively for the semidiurnal and diurnal tides. For linear internal waves, the APE/KE ratio is ðx2  f 2 Þ=ðx2 þ f 2 Þ, where x is the wave frequency and f is the Coriolis parameter, corresponding approximately to 0.78 and 0.36 respectively for the semidiurnal and diurnal tides. The agreement is quite good especially for the diurnal tides which are essentially linear. 4. Linear mixed internal tide–lee wave theory The nature of tidally generated internal waves depends on the ratio of the tidal excursion length to the topography horizontal scale. If the tidal excursion is small compared to the topography length, the classical linear internal tide theory is applicable (Baines, 1982). On the other hand, when the tidal excursion is comparable to or larger than the topography length, the barotropic tidal currents will interact with the internal waves. The classical lee wave theory is in the limiting case of a steady mean flow. Hibiya (1986) formulated an analytic solution for the mixed internal tide–lee waves of a linearized two-layered ocean. We apply their solution to the South China Sea to test if the theory could explain the diurnal modulation clearly revealed in the fully nonlinear three-dimensional (3D) model. Our approach follows Li and Farmer (2011) who have adopted the analytical solution to study the generation of nonlinear internal waves in the South China Sea. The interface displacement of a westward propagating internal tide according to Hibiya (1986) is

g

Z 0

t

  Z t @ 0 Udt þ cðt  sÞ ds h x @x s

ð4:1Þ

where c is phase velocity (=3 m/s) and U is amplitude of the barotropic tidal currents. A double ridge system is assumed

" #  2 x ðx þ 100Þ2 hðxÞ ¼ exp  2 þ 0:5 exp  2 b b

ð4:2Þ

Here b is the ridge’s half width (=25 km). The eastern ridge is located at x = 0. The western ridge is half the eastern ridge’s height, and is located at x = 100 km. Because the model is linear Eq. (4.1), the actual ridge height is irrelevant. The amplitude and phase of the barotropic tidal currents are set to 5.5, 2.2, 5.2, 3.8 cm/s and 14°, 110°, 165°, 81° respectively for M2, S2, K1, and O1. The relative amplitude and phase among the four tidal constituents are according to the model’s barotropic tidal transport. The absolute magnitude is somewhat arbitrary, representing the barotropic tidal currents over the ridge. (In a linear model, the bottom is flat and the

D.-P. Wang / Ocean Modelling 59–60 (2012) 1–10

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Fig. 5. Section averaged baroclinic flux at the eastern edge of the analysis box (122.5°E): (top) total barotropic transport (blue) and semidiurnal component (red); (middle) semidiurnal flux; (bottom) diurnal flux. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Section averaged baroclinic flux of the semidiurnal forcing only: (top) semidiurnal transport (middle) westward flux; (bottom) eastward flux. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

tidal currents are incorporated as a body force.) A larger tidal forcing would increase the nonlinearity, but does not affect the basic results. Fig. 8 shows the solution of Eq. (4.1) for a one-week period at x = 200 km, or at approximately 120°E. The results are similar

to Li and Farmer (2011). The interface is distorted showing two daily lows of different amplitudes. Dividing the interface oscillation into diurnal and semidiurnal bands, it is clear that the diurnal tide is a harmonic oscillation but the semidiurnal tide is diurnally modulated. These results are consistent with our analysis of the

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D.-P. Wang / Ocean Modelling 59–60 (2012) 1–10

Fig. 7. Section averaged energy density at the western edge of the analysis box (120°E): (top) total barotropic transport (blue) and semidiurnal component (red); (middle) semidiurnal energy density: kinetic energy (blue) and available potential energy (red); (bottom) diurnal energy density: kinetic energy (blue) and available potential energy (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Interface from two-layered model: (a) total barotropic velocity (blue) and semidiurnal component (red); (b) total interface; (c) semidiurnal (red) and diurnal (blue) interface. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

internal tidal energetics. Indeed, the amplitude modulation of the available potential energy which is proportional to the interface squared, is quantitatively similar to the 3D model results. Fig. 9 shows temperature profiles during the same one-week period from the 3D model results at a comparable location

(20.42°N, 120°E). The 3D model isotherms are highly distorted showing steep daily temperature depressions throughout the water column. Indeed, the isotherm amplitudes increase with depth as the stratification becomes less (not shown). The sharp temperature drops are primarily due to the highly nonlinear

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Fig. 9. Temperature profiles in the upper 600 m (local water depth = 3500 m) at the western edge (20.42°N, 120°E): (left) total temperature; (middle) diurnal component; (right) semidiurnal component. Temperature contour intervals are 1 °C between 8 and 10 °C and 2 °C above.

semidiurnal tides, reinforced by the nearly synchronized diurnal tidal oscillations. The 3D model results are much more striking compared to the analytical solution; nevertheless, the basic features are very similar. 5. Discussion and conclusion The three-dimensional model shows striking diurnal modulation of the semidiurnal internal tides in the South China Sea. Comparison with the analytical solution of Hibiya (1986) confirms that the diurnal modulation is caused by interaction between the barotropic tidal currents and tidally generated internal waves. The model temperature variations, marked by steep daily isotherm fluctuations throughout the water column, are consistent with the observed temperature profiles in Zhang et al. (2011) after Ramp et al. (2010). The prediction that the semidiurnal internal tidal energy flux is correlated with the semidiurnal barotropic transport, not with the total barotropic transport, also is consistent with Li and Farmer (2011). The diurnal modulation of semidiurnal internal tides could have important implication for prediction of nonlinear internal waves in the South China Sea. The strong contrast in the steepness of the two daily interface plunges likely is the main cause for the presence of two different types of the nonlinear internal waves, the a- and b-waves. The model’s barotropic flux divergence agrees with that estimated from the barotropic tidal inversion, and the model’s baroclinic energy fluxes are consistent with the observed energy flux. Moreover, by calculating the buoyancy flux, it is possible to estimate the contribution of internal tides to diapycnal mixing. The diurnal internal tides have little impact (1 GW), but the semidiurnal internal tides contribute an enormous buoyancy flux surplus of 8 GW. Based on chemical hydrographic data, Gong et al. (1992)

found that subsurface waters (100–600 m) in the Luzon Strait are considerably colder and more enriched in nutrients and depleted in oxygen compared to that of the West Philippine Sea. Vertical mixing in the Luzon Strait must be vigorous to bring up deep (below 600 m) water to compensate for rapid flushing from the Kuroshio Current. In this study, a smoothed temperature front over the Luzon Strait is imposed. Buijsman et al. (2010b) have compared the generation of nonlinear internal waves with and without a temperature front in an idealized two-dimensional nonhydrostatic model. They found only minor differences between the two cases. Their temperature front setting is similar to that used in our model. We have run a model simulation with a horizontally uniform shallow thermocline of the northern South China Sea. The total baroclinic flux decreases by about 18% for semidiurnal internal tides and 9% for diurnal internal tides. Also, the Kuroshio Current is not included in our study. Theoretical considerations indicate that generation and propagation of internal tides are strongly modulated by ambient (mean) vertical and horizontal stratification (Chuang and Wang, 1981; Baines, 1982). Lee et al. (2012) based on a long-term ADCP mooring on the continental slope, found that most of the observed internal tidal fluctuations were (75%) incoherent (not phase locked with astronomical forcing), contrary to the common belief. It is likely that the Kuroshio and mesoscale eddies add significant random perturbations into the regular barotropic tidal forcing. Acknowledgments The work was carried out while the author was on a sabbatical leave at the National Sun Yet-Sen University in Kaohsiung, Taiwan, sponsored by the National Science Council of Taiwan. This research

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utilized resources at the New York Center for Computational Sciences at Stony Brook University/Brookhaven National Laboratory which was supported by the U.S. Department of Energy under Contract No. DE-AC02-98CH10886 and by the State of New York.

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