Hideaki Kunishi

Tidal Exchange through a Strait: A Numerical Experiment Using a Simple Model Basin

Abstract In order to investigate the mechanism of tidal exchange through a strait, we numerically track the Lagrangian movement of water particles over a full cycle of the M2 tide. As a result, it is found that the spatially rapid changes of the amplitude and the phase lag of the M2 current in the vicinity of the strait cause the exchange of an extremely large amount of water through the strait. The tidally-induced residual circulation in the vicinity of the strait also plays an important role in the water exchange. The calculated exchange ratio over one tidal cycle is ∼87%, i.e., the greater part of the outer water entering into the inner basin through the strait stays in the inner basin while an equal amount of basin water is drawn out after a cycle of the M2 tide. This fact also suggests that the major part of the water exchange through a strait is generated, not by turbulent diffusion, but by the dynamic process of the tidal current.

A numerical simulation of the distribution of water temperature and salinity in the Seto Inland Sea

Constant flows, as well as oscillatory tidal flow, play an important role in the long-term dispersion of water in the Seto Inland Sea. Two kinds of numerical model (1-line and 2-line models) of the Seto Inland Sea have been developed to determine the role of density-induced currents, one type of the constant flow, in water dispersion in the Inland Sea. The seasonal variations of temperature, salinity and density fields are simulated and the density-induced current field is predicted at the same time. It is found that the most appropriate value of the longitudinal eddy diffusion coefficient,Kx, is 5×106−7×106 cm2sec−1. The value of the overall mean dispersion coefficient is of the order of 107cm2sec−1 (Hayami and Unoki, 1970). Consequently, it is suggested that 50–70% of the total dispersion in the Seto Inland Sea can be attributed to currents other than density-induced currents,i.e., tidal currents, tide-induced currents and wind-driven currents.In winter, both density and velocity fields, calculated using the 1-line model, satisfy the conditions for the existence of a coastal front in Kii Channel and in the eastern Iyo-nada.

Tidal exchange through Naruto, Akashi and Kitan Straits

In order to re-evaluate the water volume exchange through Naruto Strait, we have performed a numerical experiment (nonlinear barotropic model including the actual depth of water and the details of shoreline) where trajectories of a number of labeled particles are calculated during a full cycle of the M2 tidal current. The ratio of water volume exchanged through Naruto Strait to that through Akashi Strait is found to be twice as large as the previously estimated value. The calculated water exchange rate is 104 % for Naruto Strait, 52 % for Akashi Strait and 28 % for Kitan Strait. In the case where the tide-induced residual current is excluded from the calculated velocity field (i.e. considering only the M2 current), the calculated exchange rate maintains the level of 68 % for Naruto Strait, 18 % for Akashi Strait and 11 % for Kitan Strait, respectively. The mechanism of tidal exchange through these three straits is discussed, and it is shown that a suitable exchange rate is obtained by starting the calculation of trajectories of labeled particles at the time of either a maximum ebb- or flood-current.

Breaking of wind waves and the sea surface wind stress

In the conventional treatment of the coefficient of sea surface wind stress by plotting it against 10-m wind speed, there are inevitable discrepancies among results of various investigators. The reason is considered to lie primarily in the fact that the state of the sea surface or of waves is disregarded, which may have great influence on the sea surface wind stress.Former concepts concerning the conditions which control the sea surface wind stress are discussed, and it is shown that a more universal expression may be obtained by plotting the coefficient against a kind of roughness Reynolds number:Re2*=u*H/ν, whereu* is the friction velocity of air, ν the kinematic viscosity of air, andH the characteristic wave height.H is used here to treat some data in wind-wave tunnels, as a tentative variable, one step towards a more rigorous approach to the problem.This variableRe2*, orRe4*=u*w/L/vw=2πgu*w/vwn1, where the subscript ω represents values for water,L andn1 the characteristic wave length and frequency, respectively, is also the condition describing the air entrainment or the breaking of wind waves. In this case, these Reynolds numbers are interpreted as the quantity describing the intensity of turbulence of the water surface itself. It is shown, using data from our wind-wave tunnel experiments, that the breaking commences asRe2* reaches 1×103, or asRe4* reaches 3×103. Simultaneously, the stress-coefficient begins to increase sharply at this value ofRe2*. This phenomenon is understood as an increased momentum transfer from the air to the water through “boundary penetration of turbulence” caused by the breaking of wind waves. Further, it is suggested that there is a possibility that this excess momentum transfer does not increase wave momentum, but reinforces drift current.

Formation of Water Masses and Fronts due to Density-induced Current System*

Several numerical experiments were carried out on the formation of water masses and their fronts such as observed in the Kii Channel in winter. Such water masses and fronts may caused by density-induced current system. The phenomenon is assumed to take place in the vertical two-dimensional plane not involving the effect of the earths rotation. The linear momentum equation and the diffusion-advection equations of salinity and temperature are integrated with respect to time under a vertically hydrostatic condition. The result is that two rolls which correspond to the onshore water mass and the offshore water mass are formed with an accompanying front between them. The apparent diffusion coefficient reaches a relatively great amount inside the water masses and drops down to the eddy diffusivity level at the front. The dependence of the synoptic distributions of the temperature and salinity on several parameters is also examined. Finally another experiment is carried out involving the effect of the earths rotation, which results in a rather different distribution pattern from that of the non-rotating model.

Characteristics of a front formed by cooling of the sea surface and inflow of the fresh water

Two numerical studies (Endoh, 1977;Harashimaet al., 1978) have been proposed on a front formed by a coupling effect of cooling of the sea surface and inflow of the fresh water in a vertical two-dimensional plane without the rotation of the earth. It is, however, not easy to interpret their numerical results. A simple interpretation will be proposed by an analytical study in this paper.It is found that local convection due to the density inversion, which is expressed by the convective adjustment of the vertical diffusion coefficient in the actual numerical calculations, plays an important role on the front formation.The characteristics of the front is also clarified in the case of steady state. Namely, simple functional dependences are obtained of the position and the width of the front, the horizontal and the vertical velocities and the distribution of the buoyancy and the salinity in the neighborhood of the front on the horizontal coordinate, the cooling rate, the eddy coefficients of diffusion and viscosity, the water depth and the vertically averaged horizontal fluxes of buoyancy and salinity.

Water exchange between adjacent vortices under an additional oscillatory flow

It is shown that the coupling effect of the steady vortices and the Eulerian oscillatory flow yields the 8-shaped Lagrangean motion through which adjacent vortices intercommunicates, inducing water exchange between them. The water exchange coefficient is fairly large. This coupling effect is considered to play an important role in the water exchange across the narrow strait which is accompanied with a strong tidal current and a pair of tidal residual circulations.

On the coefficient of shear diffusion

The approximate relation of the coefficient of shear diffusion against time is shown by means of a two-level model.

A dispersion equation and dispersion coefficient for material transport in Inland Seas

To analyse material transport in inland seas, a horizontal two-dimensional dispersion equation is derived, and the dispersion coefficient due to the combined effect of vertical turbulent mixing and vertical shear of both a steady current and a tidal current is studied. In the present study, the assumption that velocity is uniform in horizontal planes is not necessary, and velocity has a free vertical profile; thus the dispersion coefficient formulated is general, and is represented by a tensor of the second order. The properties of the dispersion coefficient in the horizontal two-dimensional dispersion model are also investigated, and it is shown that the time-averaged dispersion coefficient due to the tidal current over a tidal period is approximately half that due to the steady current, if the velocity amplitude and the vertical profile of the tidal current are the same as those of the steady current (a similar result was presented byBowden (1965) for horizontal one-dimensional models). Finally, the dispersion coefficient in Hiuchi-Nada (Hiuchi Sound) in the central part of the Seto Inland Sea is evaluated by using the model. The values of the dispersion coefficient in that region range from 103 cm2 s−1 to 105 cm2 s−1 when vertical turbulent diffusivity is taken to be 50 cm2 s−1.

Bispectra of wind-waves and wave-wave interaction

Bispectra of wind-waves in wind tunnels were calculated in order to understand the characteristics of the nonlinear wave-wave interaction in actual wind-wave field. It is shown that the nonlinearity in wind-waves increases in magnitude with the development of wind-waves and that the characteristics of nonlinearity in wind-waves in the early stage of development differ from those in the late stage. It is shown that the bispectra are classified into five types (I∼V), and that the bispectral type changes from the type I to the type V as the wind-waves develop from the stage of the “initial-wavelets” to that of the “sea-waves”. The relations between frequencies of the component waves interacting each other are discussed in each bispectral type.