P8.3 Dynamics of the Surface Layer Over the Ocean As Revealed

P8.3 DYNAMICS OF THE SURFACE LAYER OVER THE OCEAN AS REVEALED

FROM FIELD MEASUREMENTS OF THE ATMOSPHERIC PRESSURE

£ ¾ ¿ Tihomir Hristov½ , Kenneth Anderson , James Edson and Carl Friehe

½ Dept. of Earth and Planetary Sciences, Johns Hopkins University, Baltimore, Maryland

¾ SPAWARSYSCEN-SD, 2858, San Diego, California

¿ Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

Dept. of Mechanical and Aerospace Engineering, Univ. of California, Irvine, California

1. INTRODUCTION on the turbulent pressure fluctuations to provide a random (Langevin) force acting onto the water surface that The spatio-temporal structure of the pressure field in the ensures wind-to-wave energy transfer. Fields in the wind atmospheric boundary layer is a significant element in the correlated with the water surface motion are considered layer’s dynamics. Pressure affects the atmospheric re- negligible and are ignored. Within this model the wave fractive conditions, thus pressure statistics is required in amplitude increases linearly in time, thus evolving in a studies of electromagnetic propagation. Although indi- way similar to a Brownian particle driven by the random rect experimental estimates and models have suggested forcing from its surrounding particles. The most efficient important role for the pressure field both over land and wind-to-wave energy transfer is found to occur from a over the ocean, infrequent measurements have so far pressure component advected with the wave and hav- provided only scarce data (Wyngaard (1998)). ing the same wavenumber. Several theories for wind-

In turbulent boundary layers, the kinetic energy (TKE) wave interaction (Miles (1957); Davis (1969); Belcher and

ÔÛ is balanced through the pressure transport Þ .A Hunt (1993)), although differing in their details, are built belief commonly shared before the Kansas experiment on the concept of fluid-mechanical instability. In that con- has been that the pressure transport term is small. How- cept the waves perturb the mean air flow and the turbu- ever, indirect estimates as well as direct pressure mea- lence in the wind, so the perturbation provides the feed- surements have indicated that over land the pressure back from wind to waves ensuring wave growth. Nu- transport can be a significant TKE gain if the surface layer merical models (Gent and Taylor (1976); Chalikov and is unstable (Wyngaard (1992, 1998)). Over the ocean Makin (1991); Meirink and Makin (2000)) have also em- the atmospheric surface layer is modified by the pres- ployed such a concept, while direct numerical simula- ence of ocean waves, as shown by wind velocity anal- tions have arrived to it (Sullivan et al. (2000); Sullivan and ysis of Hristov et al. (1998, 2003). The distribution of McWilliams (2002)). air pressure on the water surface determines the rate of Observations have so far provided limited information kinetic energy transfer from wind to the surface waves. on the physics of wind-wave interaction. Particularly, the Consequently, past experiments have conducted direct random force mechanism has never been addressed ex- measurements of the interfacial pressure (Shemdin and perimentally. Here the basic distinction between the way Hsu (1967); Dobson (1971); Elliott (1972); Snyder (1974); the random force and the instability mechanisms oper- Snyder et al. (1981)) or have considered extrapolated ate, is used to formulate an experimental criterion for estimates Papadimitrakis et al. (1986); Hasselmann and their relative efficiency. The random force mechanism Bosenberg¨ (1991); Hare et al. (1997); Donelan (1999). requires that the pressure field in the wind is predomi- However, the interfacial pressure distribution does not nantly turbulent, while, in contrast, the instability mecha- specify the spatio-temporal structure of the wave-induced nism relies on the organized wave-induced fields of ve- air flow, which uniquely selects the physical mechanism locity and pressure in the air. Therefore, to compare the of wind-wave coupling. Experimental determination of efficiency of the random force versus the instability mech- that structure requires measurements in the surface layer anism, we need to compare the intensities of the turbulent over the ocean. and wave-induced pressure in the air flow. On the other Broadly, the wind-wave interaction models and theo- hand, recognizing a particular instability mechanism in ries proposed so far have considered two categories of measured data requires finding an agreement between physical mechanisms. The Phillips (1957) theory relies the theoretically predicted and the observed structure of the wave-induced flow (Hristov et al. (2003)). £ Corresponding author address: Tihomir Hristov, Depart- ment of Earth and Planetary Sciences, Johns Hopkins Uni- In this work, from experimental perspective, we ad- versity, 3400 North Charles Str., Baltimore, MD 21218, dress the relative intensity of random force and instability e-mail: [email protected] mechanisms. We also discuss the issue of TKE balance

1 1 11 (a) (b) (c) 0.8 10 ) η 0 9 10 0.6 uu

8 S −1 0.4 10 7

Coherence (u, 0.2 6 −2

Along−wind velocity [m/s] 10 0 −2 −1 0 −1 0 0 20 40 60 80 100 10 10 10 10 10

1 (d) (e) (f) 5 0.8 −2 ) 10 η 0 0.6 pp S −3 −5 10 0.4 Pressure [Pa] Coherence (p, 0.2

−10 −4 10 0 −2 −1 0 −1 0 0 20 40 60 80 100 10 10 10 10 10

0 (g) 10 (h) (i) 0.4 2.5 −1 10 0.2 2 −2 10 ηη 1.5ηη 0 S S −3 10 1 Wave Height [m] −0.2 −4 10 0.5

0 −2 −1 0 −1 0 0 20 40 60 80 100 10 10 10 10 10 Time [s] Frequency [Hz] Frequency [Hz]

Figure 1: A flow regime frequently observed during CBLAST. (a) Timeseries of along-wind air velocities from the four ultrasonic anemometers over 100s; (b) Power spectra of along-wind air velocities calculated from 1 hour of data; (c) coherence between the along-wind air velocities and the surface elevation; (d) timeseries of the atmospheric pressure fluctuations at 2 levels; (e) pressure fluctuations power spectra calculated from 1 hour of data; (f) coherence between the pressure fluctuations and the surface elevation; (g) timeseries of the water surface elevation; (h) and (i) power spectra of the water surface elevation. over the ocean in the context of our experimental data. 3m. Instances of flow distortion from the tower are rel- atively rare, although very clear in the data. Rain occa- 2. EXPERIMENTAL SETTING sionally disrupted the sonic anemometers work.

The data analyzed here were collected at the WHOI- 3. DATA INTERPRETATION operated Air-Sea Interaction Tower (ASIT) off the coast of Massachusetts. Four sonic anemometers sampled the Figure 1 summarizes the features of a flow regime fre- wind velocity at 20Hz at levels of 7.2m, 9m, 11m, 13.4m quently observed during the experiment. The data show above the ocean surface. Two pressure sensors were that the along-wind velocity is mostly turbulent, as indi- positioned at heights of 9m and 11m. A TSK instrument cated by the timeseries and the power spectra, with some measured remotely the water surface elevation beneath suggestion of small wave influence in the coherence func- the sonic anemometers and pressure sensors. Data tion. In contrast, the pressure appears almost entirely or- were collected continuously from late June through early ganized and modulated by the waves. Comparing the November of 2003. For most of that period winds ranged pressure and the wave height signals, it is clearly no- between 1 and 12 m/s and significant wave heights were ticeable that the wave crests approximately coincide in below 1.5m, except for short storm periods when winds time with the minima and the troughs with the maxima reached 20m/s and significant wave heights exceeded of the pressure. A distinct peak at the wave frequency

2 2 10 is present over the turbulent background in the pressure. 360 25 19 The wave-pressure coherence, which here compares the 0 12 ] 10 270 5 intensities of the wave-induced and the turbulent pres- −1 [s −2 [deg] 180 u sure, approaches 1 and is signiﬁcant over a broad fre- 10 25 u A 19 Φ 90 quency range even for low-energy wave modes. Clearly, −4 12 10 5 the data demonstrate that in the wind the organized wave- 0 −4 −2 0 2 −4 −2 0 2 10 10 10 10 10 10 10 10 induced pressure dominates and therefore in this ﬂow z/z z/z regime the instability mechanism of wind-wave interac- c c tion is favored to the mechanism of random (Langevin) 25

] 19 0 force. Two circumstances discussed below offer a likely 10 12 −1 270 5 m

explanation. Also, predominantly wave-modulated pres- ⋅ [deg]

−2 p sure suggests an analytic estimate for the pressure trans- 10 25 Φ [Pa

p 19

port term in the TKE equation. A 12 180 5 −4 10 −4 −2 0 2 −4 −2 0 2 10 10 10 10 10 10 10 10 4. DISCUSSION z/z z/z c c If a flow allows to ignore the fluid’s compressibility, the Vertical profiles of the wave-induced along- static pressure in that flow satisfies the Poisson equation: Figure 2: wind and pressure fluctuations obtained from numerical

¾ solutions of the Rayleigh equation, according to the crit-

Ö Ô Ö ¡ ´Ú ¡ÖµÚ

ical layer theory of Miles (1957). Different curves corre- Ù

From here, extending Kolmogorov’s arguments regard- spond to speciﬁc values of the wave age parameter £ ,

ing the spectral scaling of the turbulent velocity to the as indicated in the symbol list. The amplitude of the

¨ Ù

turbulent pressure leads to the conclusion that turbu- along-wind velocity ﬂuctuations (upper left), the phase

´ µ

lent pressure spectrum decays with scale as Ô of the along-wind velocity ﬂuctuations (upper right), the

´ µ

, i.e. faster than velocity’s spectral decay of Ù amplitude of the pressure ﬂuctuations (lower left), the

¨ Þ

Ô . Let us introduce the wavenumber correspond- pressure ﬂuctuations phase (lower right). is the crit-

ing to the integral scale in the atmospheric boundary ical height.

layer, Ò , the variance of a turbulent quantity retained

Î ´ µ ´Õ µÕ

by wavenumbers and larger,

pressure is projected vertically beyond the height .

and let Û denote a representative wavenumber

ÔÛ for the ocean surface waves. For typical conditions The pressure transport term Þ in the TKE

¾ equation has often been considered small and due to the

½¼

Ú ÒØ Û Ú Û . Thus for wavenumbers and lack of pressure measurements that term has been es-

larger the turbulent velocity ﬁeld Ù retains a variance frac-

¾¿

timated as a residual of all the other terms. The obser-

Î ´ µ Î ´ µ ´ µ

Û Ú Ù ÒØ Û Ú ÒØ tion Ù , while the tur-

bulent pressure retains a considerably smaller variance vation that the pressure ﬁeld over the waves is predomi-

nantly organized suggests a corollary regarding the pres-

Î ´ µ Î ´ µ´ µ

Û Ú Ô ÒØ Û Ú ÒØ fraction, Ô .

A previous ﬁeld experiment (Hristov et al. (2003)) has sure transport there. Let us present the pressure Ô and

found a good agreement between the observed struc- the vertical velocity Û ﬂuctuations in the wind as consist-

¼ ¼

Ô Û

ture of the wave-induced ﬂow and the one predicted by ing of turbulent and wave-coherent components

¼ ¼

Ô Û Ô Ô · Ô Û Û · Û

the critical layer theory of Miles (1957). This gives us , i.e. and . Since most of the

¼ ¾ ¾

´Ô µ ´ Ôµ

some conﬁdence in the theory’s description of the wave- pressure signal is wave-coherent, i.e.

induced ﬁelds. In this theory the turbulence in the wind and therefore Ô , that wave-coherent pressure will cor- Û

is responsible for creating a constant-stress layer over relate only with the wave-coherent vertical velocity , i.e.

ÔÛ Ô Û the ocean with averaged ﬂow velocity obeying the law . Relaying again on the shear ﬂow insta-

bility model (Lin (1955); Miles (1957)), the wave-induced

Í ´Þ µ ´Ù µ ÐÓ ´Þ Þ µ ¼

of the wall, i.e. £ . The pertur-

Ô Ù Û

pressure and velocities are related by

´Þ µ

bation of the equilibrium ﬂow Í by the waves is gov-

½ ¾

erned by the Rayleigh equation with controlling parame-

Ô ´ Ù µ Ù · ª Û

¼ (1)

´ Ù µ

ter £ , known as wave age. Important dynamic sig- Þ

niﬁcance is assigned to the critical height , where the

Ù £ where is the air density, is the turbulent friction ve-

wind speed matches the phase speed of the considered ª

locity, is the von Karman’s constant, is the Charnok’s

Í ´Þ µ ´Ù µ ÐÓ ´Þ Þ µ¼

wave mode, i.e. where £ .

ÐÓ ´Þ Þ µ ÐÓ ´Þ Þ µ Ù Þ

¼ ¼ £

parameter, , , is ¾

Figure 2 presents numerical solutions of the Rayleigh ¾

Þ ªÙ ´ µ ¼

the coordinate in vertical direction, and £ Ù

equation for the wave-induced along-wind velocity and is the sea surface roughness. From here, the real part of Ô

the static pressure . The solutions predict a distinctly dif- the “pressure ﬂux” is expressed as

Ù´Þ µ Ô ´Þ µ

ferent behavior of the ﬁelds and . While the ve-

¾Ù Ù

£ £

Ù´Þ µ Þ

´ ÙÛ · Ù Ûµ ´Þ Ù µ ´ ÔÛµ locity shows considerable decay with , for a broad

£ (2)

´ Ù µ

range of wave ages £ the pressure remains virtu-

´Þ Ù µ ally unchanged both in amplitude and phase. Therefore, £ being the wave-coherent momentum ﬂux asso-

almost unattenuated surface value of the wave-induced ciated with the wave mode of phase speed . Integrating

´Þ Ù µ

£ over the wave spectrum of all phase speeds , Dobson, F.: 1971, Measurements of atmospheric pressure on

we obtain for the pressure transport term: wind-generated sea waves. J. Fluid Mech., 48, 91–127.

Donelan, M. A.: 1999, Wind Induced Growth and Attenuation of

¼ ¼

Ô Û Ô Û § Þ

Þ (3) Laboratory Waves, Clarendon Press, Oxford. 183–194. Þ Elliott, J.: 1972, Microscale pressure fluctuations near waves be- where § stands for the total wave-induced momentum ing generated by wind. J. Fluid Mech., 54, 427–448. flux, i.e. the momentum flux associated with the whole wave spectrum. Since the critical layer theory is a linear Gent, P. R. and P. A. Taylor: 1976, A numerical model of the air one, the amount and the spectral distribution of energy flow above water waves. J. of Fluid Mech., 77, 105–128. in the wave field determines the magnitude of the wave- Hare, J., T. Hara, J. Edson, and J. Wilczak: 1997, A similarity induced momentum flux. In conditions with low intensity analysis of the structure of airflow over surface waves. Journal of the wind turbulence air motion is substantially wave- of Physical Oceanography, 27, 1018–1037. driven and the wave-coherent momentum flux can ex- Hasselmann, D. and J. Bosenberg:¨ 1991, Field measurements ceed 50% of the total momentum flux. In conditions with of wave-induced pressure over wind sea and swell. J. Fluid high turbulent intensity, and especially when the wave Mech., 230, 391–428. field is not yet developed, at the heights of our instru- Hristov, T., C. Friehe, and S. Miller: 1998, Wave-coherent fields ments the wave-induced momentum flux is a smaller frac- in air flow over ocean waves: Identification of cooperative tion (1% to 10%) of the total momentum flux. behavior buried in turbulence. Phys. Rev. Letters, 81, 5245– 5248. 5. SUMMARY Hristov, T., S. Miller, and C. Friehe: 2003, Dynamical coupling of In theoretical and numerical studies two types of phys- wind and ocean waves through wave-induced air flow. Nature, ical mechanisms have been employed in describing 422, 55–58, doi:10.1038/nature01382. wind-wave interaction: the mechanism of random force Lin, C.: 1955, The theory of hydrodynamic stability. Cambridge (Langevin force) and the mechanism of fluid-mechanical University Press, Cambridge. instability. Measurements of wind velocity and atmo- Meirink, J. F. and V. Makin: 2000, Modelling low-Reynolds- spheric pressure presented here illustrate a frequently number effects in the turbulent air flow over water waves. J. observed air flow regime where the velocity is dominated Fluid Mech., 415, 155–174. by turbulence, while the pressure is mostly organized and Miles, J. W.: 1957, On the generation of surface waves by shear correlated with waves. Such flow structure clearly favors flows. J. Fluid Mech., 3, 185–204. the instability mechanism of wind-wave interaction. We consider two circumstances that lead to and may explain Papadimitrakis, Y. A., E. Y. Hsu, and R. L. Street: 1986, The such organized-pressure flow regime. First, the turbulent role of wave-induced pressure fluctuations in the transfer pro- cesses accross an air-water interface. J. Fluid Mech., 170, cascade delivers little pressure variance (much less than 113–137. velocity variance) at scales of the surface waves’ wave- lengths. Second, theory predicts substantial decay for Phillips, O. M.: 1957, On the generation of waves by turbulent the organized velocity with height, while the organized wind. J. Fluid Mech., 2, 417–445. pressure remains unattenuated up to the critical height Shemdin, O. and E. Y. Hsu: 1967, Direct measurement of aero-

Þ dynamic pressure above a simple progressive gravity wave. . Considering the strong organized component in the pressure signal, we proposed an analytical estimate of J. Fluid Mech., 30, 403–416. the pressure transport term through the wave-coherent Snyder, R., F. Dobson, J.A.Elliott, and R. Long: 1981, Array mea- momentum flux. surements of atmospheric pressure fluctuations above surface gravity waves. J. Fluid Mech., 102, 1–59. Acknowledgments. Discussions with O. Phillips, M. Snyder, R. L.: 1974, A field study of wave-induced pressure fluc- Donelan, M. Banner, W. Drennan, P. Sullivan, J. tuations above surface gravity waves. J. Marine Research, 32, McWilliams, M. Parlange, C. Higgins, and C. Meneveau 497–531. contributed to this work. We gratefully acknowledge the Sullivan, P. and J. McWilliams: 2002, Turbulent flow over water support by the Office of Naval Research, ONR N00014- waves in the presence of stratification. Physics of Fluids, 14, 02-1-0872. 1182–1195. Sullivan, P., J. McWilliams, and C.-H. Moeng: 2000, Simulation of turbulent flow over idealized water waves. J. Fluid Mech., References 404, 47–85. Belcher, S. and J. C. R. Hunt: 1993, Turbulent shear flow over Wyngaard, J.: 1992, Atmospheric turbulence. Annual Review of slowly moving waves. J. Fluid Mech., 251, 109–148. Fluid Mechanics, 24, 205–233.

Chalikov, D. and V. Makin: 1991, Models of the wave boundary Wyngaard, J. C.: 1998, Convection viewed from a turbu- layer. Boundary-Layer Meteorology, 56, 83–99. lence perspective. Buoyant Convection in Geophysicsl Flows, E. Plate, E. E. Fedorovich, D. X. Viegas, and J. C. Wyngaard, Davis, R. E.: 1969, On high reynolds number ﬂow over a wavy eds., Kluwer, NATO Advanced Science Institute Series, 23– boundary. J. Fluid Mech., 36, 337–346. 39.