測 地 学 会 誌,第47巻,第1号 Journal of the Geodetic Society of Japan (2001),551-557頁 Vol.47, No.1, (2001), pp.551-557 Non-equilibrium Characteristics of the Long-Period Ocean Tides from Dynamics and Energetics Takashi Takanezawa*, Koji Matsumoto, Masatsugu Ooe and Isao Naito National Astronomical Observatory, 2-12 Hoshigaoka-cho, Mizusawa-shi, Iwate-ken 023-0861, Japan (Received October 3, 2000; Revised February 7, 2001; Accepted February 15, 2001) Abstract Dynamics and energetics of the long-period ocean tides are investigated using a global barotropic ocean model. Estimated oceanic Qs of the long-period tides are about 3, which show that the ocean is heavily damped system. Some far-reaching tidal energy fluxes, however, exist. Equatorial eastward flux from the coast of South Africa to the Pacific especially stands out. LFrom the point of view of wave dynam ics, contrary to the tidal energy flux in the tropics, wave's energy propagates in the westward direction. With the relation between the energy flux of wave and group ve locity, the westward direction shows that the energy flux corresponds to long Rossby wave. Moreover, this relation clearly shows existence of short Rossby waves in the mid latitudes. LFrom the point of view of planetary scale dynamics, zonally averages of dy namic tide and tidal currents show existence of global non-equilibrium response, which is similar to a particular solution of constant depth ocean. The global response may corresponds to the deviation from the equilibrium tide in zonally averaged admittance, which is found in the analysis of altimeter data. 1. Introduction Though the ocean is heavily damped system for the long-period tides, non-equilibrium features can be found. Short Rossby wave (Wunsch,1967; Carton, 1983), long Rossby wave (Kantha et al., 1998) and Kelvin-like gravity wave (Miller et al., 1993) are proposed to be candidates for causes of the non-equilibrium features. However, where these waves exist is not clearly shown. This paper uses dynamics and energetics to determine the type and location of these waves that contribute to the non-equilibrium features of the long-period ocean tides. We also show existence of global response that is similar to O'Conner's (1992) particular solution. 2. Equations for Energetics and Dynamics We used solution of our constructed barotropic ocean model, of which grid size is half degree and computational method is finite difference method. Further details of the model can be referred from another paper in this proceedings (Takanezawa et al., 2001). For the followingarguments, we will show some equations for energetics and dynamics. *now at Faculty of Science, Kagoshima University, 1-21-35, Korimoto, Kagoshima-shi, Kagoshima-ken 890-0065, Tel/Fax: +81-99-285-8960, E-mail: [email protected] 552 Takashi Takanezawa, Koji Matsumoto, Masatsugu Ooe and Isao Naito An equation of energy balance for a barotropic ocean model can be expressed as follows. •¬(PE+KE)+‡™h•EP=RT+RD+RC (1) KE=phu2/2: kinetic energy, PE = pgƒÄ2/2: potential energy P=pg• (hƒÄu): tidal energy flux RT=pghu• ‡™h(ƒÅ+ƒÄ)=pgƒÅ(aƒÄ/at)+pgƒÄ(aƒÄ/at): rate of tidal work RC=phu• (f•~u): rate of work by Coriolis force (theoretically zero but some value remains because of finite grid size of model) RD = phu• F: rate of dissipation p: density of sea water, h: ocean depth u: tidal current, g: gravity accelerationƒÄ : tidal height, ƒÅ: primary tide generating potential esecondary tide generating potential (loading and self-attraction of sea water) F: friction (bottom friction and eddy viscosity in our model) f : Coriolis force, ‡™h: horizontal differential Integrating over the whole ocean and/or averaging over one tidal period give following equations. •¬(<KE>)+<PE>)=<RT>+<RD>+<RC>(3) < RT>+<RD>+<RC>=0 (4) where <> and -mean integrations over the whole ocean and averaging over one tidal period, respectively. Q-value of the ocean to the tidal force is conventionally defined as follows. Q = 2ƒÎE/‡™E (5) E=<[ph(u20+v20)+pgƒÄ20]/4>: mean energy of system ‡™ E=<RT>• T: energy loss per one tidal period u0, v0: amplitudes of eastward and northward tidal current velocitiesƒÄ0 : amplitude of tidal height, T: tidal period In the limit of linearity, the ocean tides can be represented by two kinds of expressions (Wunsch et al., 1997). ƒÄ =(equilibrium)+‡”nan(normalmode)n(6) ƒÄ =(particularsolution)+(homogenoussolution)(7) Wunsch et al. argued that superposing normal modes is mathematically valid but is not preferable to obtain the long-period ocean tides since the ocean is heavily damped system for the long-period tides and many spatially complicated modes would exist. Then they derived the Laplace's tidal equations directly with spectral element ocean model. However, the expression by normal modes is useful for interpreting the dynamics of non-equilibrium features in the case where the normal modes are not overlapped spatially or only the one of modes predominate. In such a case, since nonlinearity of energy with tidal height can be Non-equilibrium Characteristics of the Long-Period Ocean Tides 553 neglected, we can use wave's energy to determine the type of wave . Hence, we will define potential and kinetic energies of wave and energy flux of wave as follows. PEw=pg(ƒÄ-ƒÅ-ƒÄ)2/2 (8) KEw=phu2/2(9) Pw=pgh(ƒÄ-ƒÅ-ƒÄ)u(10) The definition of potential energy of wave seems to be doubtful at a glance , but it is valid in consideration that tidal currents of long-period ocean tides are close to the geostrophic flow. Here, we also define (ƒÄ-ƒÅ-ƒÄ) as dynamic tide following Miller et al . (1993). The energy flux of wave is related with group velocity by a well-known relation (e .g., Gill, 1982; Pedlosky, 1987), Pw=EwCg(11) Ew=PEw+KEw: mean energy density Cg: group velocity of wave Since the direction of group velocity depends on the type of wave, that of energy flux of wave can be used to determine the type of wave. On the second expression (eq.7), O'Connor (1992, 1995) obtained the particular solution for constant depth ocean and homogeneous solutions for meridional bounded ocean. His particular solution can express tidal currents simpler than Hough's function can. In this paper, we will use only his particular solution for discussion on global non-equilibrium response. Table 1. Oceanic Q for long-period tide. 3. Results and Discussion We show the Q-values of ocean to the long-period tides in Table 1. These values are about 3 except for Ssa, which show that the ocean is heavily damped system for the long- period tides. The Q-value of Ssa is very larger than those of other constituents, but it doesn't mean larger difference from the equilibrium or existence of problem on model. This value is due to small dissipation, and in the case of complete equilibrium no tidal current exists and hence the Q-value becomes infinity. Figure 1 shows divergence of Mf tidal energy flux and flux itself. Complicated distri bution of the divergence is due to strong effect of that of the rate of tidal work. Positive area means that the moon or sun work on the ocean, and negative area means opposite situation. The distribution of divergence suggests that the tidal energy hardly propagate far away. However, in Figure lb some large scale tidal energy fluxes, especially far-reaching tidal energy flux from off the Africa to the East Pacific Rise can be found. To know what kinds of waves correspond to these tidal energy fluxes, we use the energy flux of wave defined , 554 Takashi Takanezawa, Koji Matsumoto, Masatsugu Ooe and Isao Naito Fig.1. Mf tidal energy flux and its divergence. Fig.2. Energy flux of wave and group velocity of Mf tide. Fig.3. Energy flux of waves of Mtm and Mm tides. Non-equilibrium Characteristics of the Long-Period Ocean Tides 555 Fig.4. Contour of f/H (f:Coriolis parameter; H:ocean depth) . Contour interval= 10-8m-1s-1. Fig.5. Energy flux of wave of Mf tide in the case of excluding the Arctic Ocean . Fig.6. Zonal average of Mf dynamic tide and tidal velocities. The solid lines mean the value of our model and the dotted lines mean O'Connor's particular solution. 556 Takashi Takanezawa, Koji Matsumoto, Masatsugu Ooe and Isao Naito by eq.(10) (Figure 2a). In the tropics, contrary to the tidal energy flux, which stretches over the Indian Sea and Pacific Ocean and propagates eastward, the energy flux of wave is limited in the Pacific Ocean and propagates westward. This westward direction of the propagation means that the corresponding wave is long Rossby wave. Figure 2b shows that the group velocity obtained by the relation eq.(11) has reasonable value for Rossby wave. A similar westward energy flux of wave can be found in Mm constituent but cannot be found in Mtm (Figure 3). Also eastward fluxes in the mid-latitudes of North Pacific Ocean are found in Mf and Mm constituents. They are considered to correspond to short Rossby waves since they propagate eastward and attenuates as the tidal period becomes longer. This tendency of the attenuation is predicted by Carton (1983). Also fluxes in the Southern Pacific Ocean should be short Rossby waves since they propagate along the contours of f/H (Figure 4). In the Atlantic Ocean, Kelvin-likewave propagations can be found. These waves are estimated to be excited in the Arctic Ocean by comparison with the flux of a model that is run with the Arctic Ocean excluded (Figure 5).