MODELING OF THE REYNOLDS STRESS IN THE BURSTING LAYER AFFECTED BY TYPHOON

When large and intensive water surface displacements are caused by developed wind waves due to a typhoon, it is impossible in the Eulerian coordinate system to measure water particle velocities continuously in a domain between the wave trough level and the mean water level. Consequently, the domain between the wave trough level and the mean water level becomes a void zone where the Reynolds stress cannot be described. By treating the sea surface boundary layer including the void zone as a bursting layer, we modeled the Reynolds stress in the bursting layer. Validity of that modeling was verified by performing comparisons with experimentally obtained results.


INTRODUCTION
Correct descriptions of the Reynolds stress in a sea surface boundary layer generated by strong the winds of a typhoon are extremely important for highly accurate predictions of typhoon intensity and transportation phenomena of surface waters affected by the typhoon because latent heat fluxes characterizing the typhoon intensity and velocity fields determining the amount of sea surface transportation are dominated by the Reynolds stress in the layer (Skachko et al. 2009;Ogasawara and Yasuda 2004).
The Reynolds stress in the sea surface boundary layer is described by application of the Reynolds average rule based on the assumption of the averaged sea surface to velocity fields in the layer. However, waves generated by strong winds of the typhoon produce large vertical displacements of the water surface that might exceed 20 m. In the Eulerian coordinate system, it is impossible to measure water particle velocities continuously in a domain between the wave trough level (WTL) and the mean water level (MWL) because of the large water-surface displacement. Consequently, the Reynolds average rule is inapplicable to velocity fields above the WTL. The domain between the wave trough level and the mean water level therefore becomes a void zone where the Reynolds stress cannot be described.
Although the sea surface boundary layer including the void zone is extremely important for determination of the typhoon intensity and transportation phenomena in the sea surface affected by the typhoon, conventional models (Craig and Banner 1994;Mellor and Blumberg 2004) using the Reynolds equation treat the sea surface boundary layer including the void zone as an unresolved zone with thickness defined as roughness length z 0 (Soloviev and Lukas 2006). Furthermore, the sea surface affected by strong winds becomes covered with developed windwave breakers. The turbulent flow fields in the sea surface boundary layer comprise a mixture of the mean flow, the cascade turbulence from the mean flow, the wave motion, and the turbulence generated by wave-breaking and airflow separation. Therefore, the Boussinesq eddy viscosity assumption, which is used in conventional models described above, becomes insufficient to describe all the turbulent components in the sea surface boundary layer.
In this study, an experiment using a double-bottomed wind-wave tank (Ogasawara and Yasuda 2004) is performed to obtain information related to the total mass flux, the vertical distribution of the mean flow velocity and the turbulent flow components affected by strong winds, which are required to describe the Reynolds stress. The sea surface boundary layer including the void zone is treated as the bursting layer, which is defined as a boundary layer affected by various actions such as wave breaking, airflow separation, and air bubbles, using a macroscopic approach to the assumption of the averaged sea surface (Murakami and Yasuda 2008). Then, the Reynolds average rule is applied to the velocity fields in the bursting layer. Furthermore, the Reynolds stress in the bursting layer is separated into a low-frequency Reynolds stress that originates from the mean shear flow and a high-frequency Reynolds stress that originates from the overall breaker action based on the experimental results. Finally, the bursting-layer model is developed to enable description of the low-frequency Reynolds stress and the high-frequency Reynolds stress in the bursting layer (i.e. the domain of roughness length z 0 ). The model validity is verified through comparison with experimental results.

MODELING CONCEPT
The void zone described above originates from the procedure by which the actual space is converted into an averaged space (bursting layer) based on the Reynolds average rule. It is necessary that the modeling to determine the vertical distribution up to the MWL of the mean velocity u in the averaged space. Ogasawara and Yasuda (2004) showed that the value of the total mass flux A q can be determined using a double-bottomed wind-wave tank. If the value of A q is known, then the value up to the MWL of the horizontal velocity u can be determined to satisfy the following equation.
In that equation, h is defined as a distance from the mean water level to the upper bottom in the windwave tank with a double bottom. The solution of the Reynolds equation in the bursting layer must agree with the value of u . Based on this consideration, the Reynolds stress is formulated by giving the vertical distribution up to MWL of u as the solution of the Reynolds equation in the bursting layer, and by then solving that equation as an inverse problem. Therefore, the modeling in this study is distinguished as a macroscopic approach based on information related to the vertical distribution of the horizontal mean velocity.

EXPERIMENTS
Laboratory experiments were performed using a 15.4-m-long, 0.4-m-wide, and 1.0-m-deep double-bottomed wind-wave tank, as illustrated in Fig. 1. The tank partially fulfills the continuity of the mass flux and reduces the influence of the return flow on wind-driven currents. The space between the upper and lower bottoms of the tank functions as a duct to circulate the currents as a return flow, allowing measurement of the return-flow velocity, which is only disturbed by the reverse pressure gradient within the tank. Therefore, the true wind-driven current, resulting solely from wind stress, is measurable according to the velocity of this measured return flow (Ogasawara and Yasuda 2004).
The still water depth was fixed as 60 cm. Reference wind speed U r at the wind-tunnel entrance is set as 12.0 m/s, yielding fully breaking waves. Table 1 presents values of wave parameters at measuring point W03, where H s is the significant wave height, T s is the significant wave period, and f p denotes the peak frequency of the wave spectrum. The friction wind velocity u *a was calculated using the empirical equation proposed by Wu (1980). Water particle velocity fields were visualized using 50µm-diameter nylon particles and a relative density of 1.02 as tracers. Nylon particles were irradiated from the tank floor using a 5-mm-thick 5 W power green light laser sheet. Images showing their positions were captured using two synchronized cameras of 480 × 480 pixel resolution at a 125 fps rate. The resultant images facilitated measurement of high-speed velocity fields immediately below the WTL. Velocity fields were examined using a high-resolution PIV system (Ogasawara et al. 2002), which can analyze high-speed and intensively fluctuating water particle velocities that are affected by breaking waves.

LOW AND HIGH-FREQUENCY REYNOLDS STRESSES
Velocity components of currents driven by strong winds comprise a mixture of the mean flow, the cascade turbulence from the mean flow (low-frequency turbulence), the wave motion, and the turbulence generated by breaking waves (high-frequency turbulence), as apparent from the shape of the velocity spectrum presented in Fig. 2. To model the breaker stresses that cause the bursting layer, it is necessary to extract the breaker-generated turbulence from the mixture of these turbulent flow components. Thereby, three cutoff frequencies are provided from the shape of the velocity spectrum presented in Fig. 2: f cc between the spectral band of the mean flow and the low-frequency turbulence; f lc between the low-frequency turbulence and the wave motion; and f hc between the wave motion and high-frequency turbulence. Based on them, the spectral band is separable into four domains. Velocities u of currents driven by strong winds are treated as the sum of the mean flow velocity c u , the cascade turbulence velocity l u , the wave motion velocity w u , and the velocity of the breaker-generated turbulence h u : The mutual correlations among components must be known when the Reynolds average rule is applied to these components. The reverse-Fourier transform is then performed to investigate the values of the correlation coefficients between each velocity component. Table 2 presents experimental values of the correlation tensor among horizontal and vertical velocity components at depths of z = -3, -6, and -10 cm for respective frequency bands. Results show that the value of the cross-correlation between the velocity components in different frequency bands is of negligible order at each water depth. Even for breaker-affected velocity fields comprising a mixture of various turbulent flow components with different origins, only components in the same frequency band of the correlation tensor have considerable values. Thereby, the following Reynolds average rule becomes applicable.
Furthermore, the averaging rule below can apply to the cross-correlation term uw  , as Table 2 shows.
In that equation, l l u w  is the low-frequency Reynolds stress, which can be described by the following Boussinesq eddy viscosity assumption because l l u w  originates from the mean shear flow.
Here, l  is a low-frequency eddy viscosity coefficient, which is generated only by the mean shear flow.
Because the orthogonal relation holds between u w and w w of the irrotational wave, their inner product w w u w  becomes 0. However, the actual value of the inner product is not negligible; rather it is significant because w w u w  comprises the turbulence generated by wave-breaking, the cascading turbulence from mean shear flow and rotational wave motion. Their primary energy source is presumed to be waves and breakers generated by strong winds in addition to that of h h u w  . Therefore, The total Reynolds stress uw  in the bursting layer is balanced with the momentum flux by the strong winds. The following relation holds for all depths in the bursting layer under the assumption of the constant flux layer.
Consequently, the high-frequency Reynolds stress t t u w  is represented as Equations (10) and (13)

VERTICAL DISTRIBUTION OF THE HORIZONTAL MEAN VELOCITY
The vertical distribution up to the MWL of the mean velocity u is determined to make its vertically integrated value agree with the total mass flux of the wind-driven currents obtained from our experiments. The value and the vertical distribution of the modeled formulation must correspond to those of the following logarithmic law at the interface between the bursting layer and the logarithmiclaw layer ( In that equation,  represents von Karman's constant; b u is the value of mean velocity at Consequently, we infer that the assumption of the mixing length theory is applicable to the calculation of the low-frequency eddy viscosity coefficient l  in the bursting layer.     Figure 5 presents a comparison of the vertical distribution of the horizontal mean velocity u between the computational result obtained using the bursting-layer model and the experimental one. Moreover, the vertical distribution of u , calculated using a model proposed by Craig and Banner (1994), is also presented in Fig. 5. The Craig-Banner model is incapable of describing the steep vertical distribution in the bursting layer. In contrast, the bursting-layer model accurately calculated the steep vertical distribution of the current velocity in the bursting layer. This result demonstrates that both the modelings of the low-frequency Reynolds stress l l u w  and the high-frequency Reynolds

CONCLUSIONS
This study was aimed at modeling of the Reynolds stress in the sea surface boundary layer including the void zone for cases in which the Reynolds stress cannot be described in the conventional models. The sea surface boundary layer was treated here as the bursting layer affected by various actions, such as wave breaking, airflow separation and air bubbles, and was modeled by describing their effects from a macroscopic viewpoint. Our experimental results demonstrated that the Reynolds stress in the bursting layer is separable into low-frequency Reynolds stress that originates from the mean shear flow and high-frequency Reynolds stress that originates from the overall breaker action.
Based on the concept of the bursting layer and the experimental results, we modeled the lowfrequency Reynolds stress and the high-frequency Reynolds stress in the bursting layer. Validity of the modeling was verified through comparisons with experimental results. This model enables us to describe the vertical distributions of the Reynolds stress and the horizontal mean velocity in the sea surface boundary layer, regarded to have a thickness of roughness length z 0 , which is treated in conventional models as the unresolved zone.