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Numerical Simulation of the Internal Tidal Wave Field in the Global Ocean !"# $%, &'()*+,-., &/01/234 7-3-1, niwa@eps.s.u-tokyo.ac.jp

567 89, &'()*+,-., &/01/234 7-3-1, hibiya@eps.s.u-tokyo.ac.jp

Yoshihiro Niwa, Grad. Schl. of Science, The Univ. of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan

Toshiyuki Hibiya, Grad. Schl. of Science, The Univ. of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan

Internal tidal waves are ubiquitous phenomena in a stratified ocean where they are generated in response to tidal

flows incident upon bottom topography. In this study, we investigate the global distribution of the semidiurnal

and diurnal internal tidal wave fields using a hydrostatic sigma-coordinate numerical model. The model results

show that energetic internal tidal waves is generated over the limited prominent topographic features such as

those in the Indonesian Archipelago, the continental shelf slope in the East China Sea, and the mid-ocean ridges

such as Hawaiian Ridge. By comparing the model results with current observations, it is found that the generated

internal tidal waves can propagate a very long distance of more than several thousand kilometers across the open

ocean. The conversion rate from the surface to internal tides integrated over the global ocean amounts to about

800GW, and nearly 25% of which is dissipated in the deep ocean.

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Fig.1: The whole model domain together with the bottom topography. Block-shaped areas enclosed by thick yellow line

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M2 internal tidal wave.

Fig. 3: The model predicted distribution of vertical isopycnal

displacement of the M2 internal tidal waves at a depth of

1000m in the mid-latitude North-Western Pacific.

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Fig. 4: As in Figure 2 but for the K1 internal tidal wave.

Fig. 5: As in Figure 3 but for the K1 internal tidal wave.

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20days, 30days from the start of the calculation. The results of calculation with a 30-day damping are shown.

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Fig.7: Locations of the long-term mooring stations at which observed

current data and the model results are compared.

Fig.8: Scatter plot comparing the first vertical-mode semidiurnal internal

tidal wave kinetic energy predicted by the model (abscissa-axis) and

that observed at locations shown in figure 5 (ordinate-axis). The

model results with 5-day damping (upper-left), 15day-damping

(upper-right), 30-day damping (lower-left), 60-day damping

(lower-right) are compared.

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Fig. 8: As in Figure 7 but for K1 internal tide.

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Fig9: Conversion rates of the M2, S2, K1 and O1 internal tides

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wave energy integrated over the full water column within

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Fig.11: Depth distribution of the dissipation rate of the M2, S2, K1,

and O1 internal tidal energy.

Fig.12: Cumulative depth distribution of the dissipation rate of

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Z[\]

(1) Munk, W. H., and Wunsch, C., “Abyssal recipes II:

energetics of tidal and wind mixing”, Deep-Sea Res.,

45(12)(1998), pp.1977-2010

(2) Niwa, Y., and Hibiya, T., “Numerical study of the spatial

distribution of the M2 internal tide in the Pacific Ocean”, J.

Geophys. Res., 105(C6)(2001), pp.13933-13943

(3) Alford, M.H., “Redistribution of energy available for ocean

mixing by long-range propagation of internal waves”,

Nature, 423(6936)(2003), pp.159-162

(4) Alford, M.H. and Zhao, ZX., ” Global patterns of low-mode

internal-wave propagation. Part I: Energy and energy

flux”, J. Phys. Oceanogr., 37(7) (2007), pp.1829-1848

(5) Simmons, H.L., Hallberg, R.W., and Arbic B.K.,” Internal

wave generation in a global baroclinic tide model”,

Deep-Sea Res., 51(25-26) (2008), pp.3043-3068

(6) Simmons, H.L.,” Spectral modification and geographic

redistribution of the semi-diurnal internal tide”, Ocean

Modelling, 21(3-4) (2008), pp.126-138

(7) Smagorinsky, J.S., “General circulation experiments with

the primitive equations, I, The basic experiment”, Mon.

Weather Rev., 91(3)(1963), pp.99-164

(8) Pacanowski R.C. and Philander S.G.H., “Parameterization

of vertical mixing in numerical models of tropical oceans”,

J. Phys. Oceanogr., 11(11) (1981), pp.1829-1848

(9) Matsumoto, K., Takanezawa T., and Ooe M., “Ocean tide

models developed by assimilating Topex/ Poseidon

altimeter data into hydrodynamical model: a global model

and a regional model around Japan”, J. Oceanogr., 56

(5)(2000), pp.567-581

(10) Niwa Y., and Hibiya, T., “Three-dimensional numerical

simulation of M2 internal tides in the East China Sea”, J.

Geophys. Res., 109(C4)(2004), doi:10.1029/2003

JC001923

(11) Munk, W. H., “Internal waves and small-scale processes”,

in Evolution of Physical Oceanography, edited by B. S.

Warren and C. Wunsch, MIT Press, Cambridge, Mass.,

(1981), pp. 264-291