1. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 1 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 1  INTEGRATED MODEL FOR A WAVE BOUNDARY LAYER Vladislav POLNIKOV Obukhov Institute for Physics of Atmosphere of RAS Moscow, Russia polnikov@mail.ru

2. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 2 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 2  Abstract A new semi phenomenological model is constructed, to estimate friction velocity via 2D wave-spectrum S and mean wind W at the standard horizon. The model is based on the momentum-flux balance equation averaged over a wave-zone covering the space from troughs to crests of waves. In the model there are only two constituents of the full momentum flux: the wave part,, corresponding to the energy transfer to waves, and the tangential part,, do not related to the energy transfer. The wave part is sheared into to contributions: one of them, supported by energy containing waves, is expressed via wind W, whiles the second one, supported by short waves, is expressed via. The tangential part is parameterized by means of similarity methods under assumption that it corresponds to an effective viscosity momentum-flux realized in the wave-zone between through and crests. The model is verified on the basis of simultaneous measurements of 2D-spectrum S, wind W, and friction velocity, described in Babanin&Makin (JGR,2008). Results of verification show that a relative error for the magnitude of ( ) 2 is of the order of 20%, what is fairly good, taking into account the experimental errors.

3. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 3 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 3  Introduction A wave boundary layer (WBL) model is aimed to relate the friction velocity,, as a main parameter of an atm. boundary layer, with the local wind W at the standard horizon z ( usually z=10m ) and 2D wave-spectrum, or its analog, as the feature of the wave field. There are a lot of such WBL-models: pure analytical, empirical, numerical, and phenomenological. The so called semi-phenomenological models( Polnikov, 2009 ) are the most prospective for a practical use (Janssen, 1991; Chalikov, 1995; Zaslavskii, 1995; Kudryavtsev&Makin,1999,2001; and so on). Here, we restrict ourselves by this type of models. As we do not consider a wind profile W(z), the models under investigation are classified as the integrated ones.

4. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 4 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 4  Introduction (continued) The main equation of WBL-model has the form of balance eq.(BE) (1) written for the momentum flux from wind to the air-sea interface. Usually, the modelling flux is treated in the form of the sum (2) where,, is the wave, viscosity, and turbulent part (respect.) of the total (measured) flux (all fluxes are normalized by the air density). The first term is usually represented in the kind, (3) where (4) is the rate of energy input from wind to waves with empirical. The problem is in treatment of and, depend. on averaging (1).

5. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 5 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 5  Introduction (continued) Trad-ly, BE (1) is averaged along the wavy interface surface (Fig.1) Fig.1. under assumption that vertical thestructure of WBL is horizontally invariant. It allows to treat, as in the case of “hard wall” turbulence. But from Fig. 1, it is evident that the air flow structure depends on the phase of wave: troughs, crests, rear or forward side of surf ! This leads to principal uncertainties in specification of and and in solving BE (1) ( i.e. in the WBL-model construction ).

6. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 6 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 6  New treatment of BE Instead of treat BE at wave surf, we do it in a whole wave-zone (Fig. 2), covering the space between troughs and crests of waves. That is equivalent to the ensemble averaging of Eq. (1)!!! Fig. 2. Due to the mixing all summands in (2), we may rewrite Eq.(1) in the form (5) with only 2 terms in the l.h.s. First is tradit-l. Second is to be found.

7. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 7 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 7  New treatment (continued) Ensemble averaging of BE ensures a proper scales enlargement, corresponding to using BE in the wave spectral representation. Besides, the averaging leads to the horizontal homogeneity and time stationarity of the WBL-system. Wave term in (5) has the traditional treatment. It corresponds to the energy transfer from wind to waves and has the form (3) (wave term of BE is called the “form drag”). The second term is not related to the energy transfer. It ensures a tangential stress in the wave-zone (tangential term of BE, or “skin drag”). Expression for is not known. It is to be found on the basis of statistical hydromechanics (similarity method using non-dimensional parameters of the system). If is found, equation (5) is to be solved numerically as what close the task of the integrated WBL-model construction.

8. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 8 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 8  Parameterization of Usually,, is given by Eqs. (3,4) as the function of spectrum S (3) (4) where the input function IN(S) includes some empirical growth increment function  (W,...) (which is assumed to be known). Though, the integral in (3) is bad converging at high freqencies, thus it needs a wide frequency-space for integration, where spectrum S is not well known.

9. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 9 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 9  Parameterization of (continued) Due to this, spectrum S is to be shared in 2 parts: (6) where ( following to Elfouhaily T.B. et al, JGR,1997 ), S L is the low-frequency part (k<1r/m) known by num. calcul-ns or measur-nts, S H is the high-frequency part (1<k<1000)r/m known via model presentation. In such case, one can represent in the form of 2 summands (7) where the arguments of functions in (7) are the integrated parameters of the wind-wave system (see below). and are the proper parts of normalized by.

10. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 10 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 10  Parameterization of (continued) Using the model for S H from Kudryavtsev et al (JGR,2003), we did calculate S H and tabulate function (Fig. 3). Fig. 3. The upper lines correspond to k<1r/m, lower lines are the series for 1<k<4r/m, step 0.2r/m. As far as can be easily calculated, we state that the total wave stress function is fully parameterized (i.e. known).

11. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 11 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 11  Parameterization of To parameterize, we use the well known formula (Monin,Yaglom,1967). (9) The turbulent viscosity coeff. K and wind profile deriv. should be expressed via u * and non-dim parameters for wave field. To do this, we take into account the fact that that W(z) is linear with z in the wave-zone (Fig.4) (Polnikov, 2010). Herewith (10) where c p is the pha.spd. and H is the m.w.h. Fig.4.

12. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 12 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 12  Parameterization of (continued) Due to linearity of the wind profile in the wave-zone, the latter could be treated as the analog of the viscous layer with a constant value of turbulent viscosity coefficient K (indep of z). Therefore, parameterization for K takes the form (11) where is the non-dim function of the wave field parameters, which is to be found in the course of verification. In the simplest way, we put (12) as far as the wave steepness is the most important param. among others. (This point would be sophisticated in a later study). Thus, the tangential part of momentum flux is parameterized as and the integrated WBL-model is ready.

13. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 13 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 13  Specification of the WBL-model The certain specification of our WBL-model (BE) has the kind (13) where the integration domain for estimation of is given by ratio, d is the local depth, is tabulated ( Fig.3), and c t =0.8 is the fitting coeff. for, found by verification. For increment (at present stage) we use the form (Yan,1987) (14) with the fixed ratio (15) to involve external wind W explicitly into BE written for u * (W,S) Solution of eq. (13) with respect of u * ensures the result of WBL-model.

14. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 14 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 14  Verification of the WBL-model To verify model (13), the joint measurements of u * and S( ,  ), described in Babanin,Makin (JGR,2008), were used. Results are shown in tab. 1. It is seen that the mean error for a drag coefficient, given by the ratio, (16) is about 15-20%, what is fairly good, taking into account the experimental errors. This result is very encouraging for future using the model in practice. Further extended verificaion is planned. No of run Meas. Ser number d,md,m W 10, m/s u *, m/s 10ε = =10k p Н Cd Е, 10 -3 Cd М, 10 -3 Def, % 10100040.869.90.380.911.471.55+06 20100550.8711.80.510.841.871.45-22 3 0102480.9314.80.671.002.051.811.81-12 41110511.1412.90.571.331.952.24+15 51111561.1412.60.571.281.952.19+12 61112241.1411.90.501.271.771.84+04 71114021.1413.00.561.261.861.99+07+07 81115381.1411.60.481.311.712.00+17 91412151.0910.10.391.171.491.491.4900 10101412501.0911.00.451.251.671.75+05 11413051.0014.10.640.882.061.41-31-31 12121512491.109.90.360.781.321.40+06 13131513181.109.400.350.641.390.890.89-36 14141513421.109.100.330.871.321.83+39 15151514101.19.70.330.881.161.78+54 16161614250.716.90.261.021.421.31-07 17172014460.896.70.261.361.51 -00-00 18182611480.845.90.281.112.251.01-54 19192711000.956.10.280.68(!)2.111.18-44 202815440.965.10.161.110.981.11+13 213116380.949.30.440.69(!)2.241.04-53-53 223118231.1219.80.981.042.452.24-08 233118451.0415.00.630.891.761.34-24 243119080.9312.90.530.911.691.56-07-07 253120211.0213.70.571.031.731.98+14 263121110.869.30.350.831.421.34-06

15. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 15 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 15  List of main references Babanin A. V., Makin V.K. Effect of wind trend and gustiness in the sea drag: Lake George study. J. Geophys. Res. 2008. V. 113. P. 1-18. С02015. Elfouhaily T.B., et al. A unified directional spectrum for long and short wind-driven waves. J. Geophys. Res. 1997. V. 107. P. 15781–15796. Makin V.K., Kudryavtsev V.N. Coupled sea surface-atmosphere model. Pt.1 Wind over waves coupling. J. Geophys. Res. 1999. V. 104, #C4. P.7613-7623. Kudryavtsev V.N., Makin V.K. The impact of air-flow separation on the drag of the sea surface. Boundary Layer Met. 2001, V.98. P.155-171. Polnikov V.G. Comparative analysis of WBL models. Izvestiya. Atmosph. Ocean. Phys. 2009. Т 45, №5. P. 583-597. Polnikov V.G. Features of air flow in the trough-crest zone of wind waves. ArXiv:1006.3621. Yan L. An improved wind input source term for third generation ocean wave modelling. Scientific report WR-87-8. KNML, The Netherlands. 1987. 27p.

16. Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 16 Образец заголовка Образец текста Второй уровень Третий уровень Четвертый уровень Пятый уровень 16  Acknowledgements The author is kindly grateful to Prof. Vladimir Kudryavtsev for numerous discussions of the topic and calculation codes for the high-frequency spectrum. Additionally, I express my high gratitude to Prof. Alex Babanin for his kind support with the experimental data. The work was done under the financial support of the Russian Foundation for Basic Research, project #09-05-00773_a.