A numerical coupled model for studying air–sea–wave interaction

A numerical coupled model for studying air–sea–wave interactionLe Ngoc Ly Citation: Physics of Fluids (1994-present) 7, 2396 (1995); doi: 10.1063/1.868751 View online: http://dx.doi.org/10.1063/1.868751 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/7/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The generation of edge waves by a wavemaker Phys. Fluids 8, 2060 (1996); 10.1063/1.869008 Homoclinic connections and period doublings of a ship advancing in quartering waves Chaos 6, 209 (1996); 10.1063/1.166166 A coupled bispectral, temporal and spatial coherence function of the pressure field, scattered from amoving sea surface J. Acoust. Soc. Am. 98, 2896 (1995); 10.1121/1.414267 Quantification of nonlinear phenomena in laboratorygenerated random sea waves J. Acoust. Soc. Am. 98, 2950 (1995); 10.1121/1.414091 Signatures of deterministic chaos in radar sea clutter and ocean surface winds Chaos 5, 613 (1995); 10.1063/1.166131

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A numerical coupled model for studying air-sea-wave interaction Le Ngoc Ly Department of Oceanography, Naval Postgraduate School, Monterey, California 93943-5100

(Received 11 January 1995; accepted 6 June 1995)

A numerical coupled model of air-sea-wave interaction is developed to study the influence of ocean wind waves on dynamical, turbulent structures of the air-sea system and their impact on coupled modeling. The model equations for both atmospheric and oceanic boundary layers include equations for: (1) momentum, (2) a k-e turbulence scheme, and (3) stratification in the atmospheric and oceanic boundary layers. The model equations are written in the same form for both the atmosphere and ocean. In this model, wind waves are considered as another source of turbulent energy in the upper layer of the ocean besides turbulent energy from shear production. The dissipation E at the ocean surface is written as a linear combination of terms representing dissipation from mean flow and breaking waves. The E from breaking waves is estimated by using similarity theory and observed data. It is written in terms of wave parameters such as wave phase speed, height, and length, which are then expressed in terms of friction velocity. Numerical experiments are designed for various geostrophic winds, wave heights, and wave ages, to study the influence of waves on the air-sea system. The numerical simulations show that the vertical profiles of E in the atmospheric and oceanic boundary layers (AOBL) are similar. The magnitudes of E in the oceanic surface zone are much larger than those in the atmospheric surface zone and in the interior of the oceanic boundary layer (OBL). The model predicts E distributions with a surface zone of large dissipation which was not expected from similarity scaling based on observed wind stress and surface buoyancy. The simulations also show that waves have a strong influence on eddy viscosity coefficients (EVC) and momentum fluxes, and have a dominated effect on the component of fluxes in the direction of the wind. The depth of large changes in flux magnitudes and EVC in the ocean can reach to lo-20 m. The simulations of surface drift currents confirm that the currents are overestimated if the surface waves are not considered. 0 1995 American Institute qf Physics.

I. INTRODUCTION

The influence of ocean surface wind waves on surface layers of the atmosphere and ocean has been studied experi- mentally by many investigators, including Roll,’ Benilov,zT3 Kitaigorodskii,4 Dubov,5 and modeled numerically by Kitaigorodskii,” Benilov,23 and LY.~*~ These processes are very complicated as the relationship between wind energy, wind-induced waves, drift currents, and turbulence is not completely known These processes need more theoretical, numerical! and experimental studies.

In contrasts to atmospheric boundary layers over land, where mean velocity shear is the main source of turbulence energy, the turbulence in the upper layer of ocean is gov- erned not only by mean velocity shear, but also by surface waves. Surface waves play an important role in the transfer of momentum and energy from one region to another and as a supplier of the energy for drift currents and turbulence. The character of the transport and supply is regulated by the turbulence of the atmospheric and oceanic boundary layers. The influence of waves on surface layers of the atmosphere and ocean has been well noted.

In practice, wind waves contain a considerable amount of momentum and energy, redistribute them over great dis- tances, and supply energy for drift currents and turbulence due to breaking. A part of momentum and energy is transferred directly from wind to drift currents, while another part goes into surface waves: The dissipation of the wave breaking intensifies turbulence in the ocean mixed layer.

The ocean wave effects on the air-sea system, and on momentum and energy fluxes, current and turbulence, are not taken into account in most existing air-sea coupled models. Some previous investigators as Kitaigorodskii,4 Benilovzv3 considered the waves to be the basic mechanism generating turbulence, while studying the turbulence properties of the upper oceanic layer. Recently Drennan et al. ,8 based on observations very near the surface, proposed a structure in which the uppermost layer is directly mixed by wavebreak- ing,,is relatively shear-free, and has a constant (with depth) dissipation rate. They also proposed an intermediate layer into which kinetic energy is diffused from above. The dissipation rate in this layer is decayed very rapidly with depth (approximately z 14). Below this layer is a layer where the dissipation rate decays more slowly with depth (approximately z- ’ ). It is likely that oceanic turbulence properties relate both to mean shear velocity and surface waves (Ly6).

This problem is very important not only in studies of the mechanism of energy transfer in an air-sea-wave system, but also in parameterization of the atmospheric and oceanic boundary-layer processes taking into account surface waves in mesoscale and large scale models (e.g. ocean circulation, atmospheric, large scale air-sea interaction, and climate models).

The aim of the present investigation is to develop a numerical model of the interacting atmospheric and oceanic turbulent flows, taking ocean surface waves into account to study wind-induced wave influence on the air-sea system. In this study, the effect of dissipation from ocean surface waves

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Atmospheric Boundary Layer

Oceanic Boundary Layer

t . . . . . . Y

ij~.~ -......._....__.._...............~..........~........~.~....~.....~.....~~ - z,

FIG. 1. The model domain showing the atmospheric and oceanic boundary layers with the wave layer. Here H, and H, are height (depth) of boundary layers; U,, and II,, are the large scale wind (current) at the top (bottom) of boundary layers; U,, is the surface current; other symbols can be found in the text.

is modeled by using boundary conditions for the turbulent energy dissipation at the air-sea interface in a k-r turbulence scheme of a coupled model. In this model we will consider the effect of fully developed wind waves. The effect of higher-fi-equency waves are treated as in traditional physical oceanography by means of the roughness length, zo.

II. PROBLEM FORMULATION

A. A coupled model with k-e turbulence scheme.

The domain of the model is shown in Fig. I which includes the atmospheric and oceanic boundary layers and the wave layer. The model equations for both atmospheric and oceanic boundary layers include equations for: (1) momentum, (2) turbulent kinetic energy (TKE), (3) dissipation rate, (4) eddy viscosity or turbulent exchange coefficient (EVC) Cthe Kolmogorov equations), and (5) stratification in the atmosphere and ocean. Both the atmospheric and oceanic equations are written in the same form (Ly’,“) with the subscript i=a for atmospheric variables and i=s for oceanic ones. In a general form, the momentum equations for atmospheric and oceanic Ekman boundary layers (EBL), which are assumed horizontally homogeneous with zero mean vertical motion, can be written:

3Ui 1’ dPj 1 d - yg -fVi= - p z _ p z ? (l)

i I i i i piU~W~-pi~

L 1

f3Vi 1 dPi 1 d - at+fu,V--- pi ayi - j-z i

dVi Pi”[wl-bQ,zi

1 7 !a

where U and V are components of fluid mean velocity in the atmosphere and ocean; f=2Cl sin 4 is the Coriolis param-

eter with fi the earth’s angular velocity and qb the latitude; Xi, yi, zi are coordinates such that the xi-axes are in the direction of surface stress vector, z+xes are in the direction, up for atmosphere and down for ocean (see Fig. 1) ; pi are the densities of air and seawater; Pi are pressures; rui are kinematic viscosity. The prime denotes a turbulent quantity. The terms LJ~W~ and uiwi are known components of the Reynolds stresses representing momentum fluxes in the atmospheric and oceanic boundary layers.

The quantities ki and cl are obtained from the k-e turbulence scheme. Ly9 has deveIoped and applied a k--E scheme to the atmospheric and oceanic boundary layers, and compared results with the length scale approach and avail- able observed data. The k--E turbulence scheme includes TKE, e, and Kolmogorov equations. The TKE for the atmosphere and ocean is determined from the budget equation which has a form

dki -dUi --Vi “3i Fl - tEali -++J;- -v;w;-f--wWfs;

dZi dZi lyli sOi 1 - “4i g k,! WI - a2iEiT

where Sai are reference values of potential temperature (i = a j, 0, or of seawater density (i = sj, ps . The constants a (see Ly9) in the TKE equation have the same values for the atmosphere and ocean ((Y, i= Ly?i= 1; @2i=O.O46; a4i =0.73).

Energy dissipation, &i, for both the atmosphere and the ocean is determined from the following equation

d- ef -P4i-&~W~-fl2i- *

dZi ki (4)

The second order moments in (l)-(4) are parameterized by the K-closure (Panofsky and Dutton”) as

I- w;S; = Khi&Si ; -klWl=Kkidiiki; (5)

-&f Wf =KsidziCi.

The eddy viscosity coefficients (EVC) for momentum are expressed in terms of TKE and energy dissipation using Kol- mogorov equation

Kmi- a2ikflEi, (6)

where ~a~=O.O46 as in (3) determined from experimental data in the surface layer for both the atmospheric and oceanic equations. The constants a and /? link the EVC for heat and dissipation with the EVC for momentum as following,

Khi= a~iliT,i ; Kei= P3iKmi 7

Kki= a4iKmi ; K,i= ,84iKmi e ~(71

The constants pi have the following values (Ly9): PIa= l.43;p2,= l.87;~3,= l.0;p4a=0.60: for the atmO- sphere, /?ts= l.43;p2,= 1.97;8,,= 1.45;p4,-0.73: for the

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ocean. These values are close to those used by Stubley and Rooney” for the atmospheric boundary layer, and by Kundut2 for the ocean mixed layer. More general discussion of the constants for equation (5) can be found in Therry et al. l3

To simplify the modeling process and to focus on air- sea-wave interaction, vertical distributions of air temperature (Si=So= 19,) and seawater density (Si=S,=p,) are ap- proximated by typical observed distributions in the atmospheric and oceanic boundary layers (see Ly6). The vertical gradient of air temperature is determined by the formula

dqBn=-QoaI(po,cplu..~,)+(yd-y,), (8)

where pea is the reference value of air density, cP is the specific heat for air at constant pressure, u*, is the friction velocity of the atmosphere, yd is the dry adiabatic lapse rate, r, is the actual lapse rate at the upper part of the atmospheric boundary layer, and Qoa is the heat flux at the interface. The vertical distribution of seawater density is determined by the following formulae (Ly6)

dZ,Ps=~Pss(z,--z,)+(r1-rz)o(z,-z,)+-r2, (9)

where S(z,--z,) is the Dirac delta function and (T(z~-- z,) is the unit function. Here i is a characteristic magnitude of the jump in seawater density, zC is the depth of the density-jump layer (Ly6), and I?, and r2 are the vertical gradients of seawater density in the water layer lying below and above the density-jump layer. Values for R”, I?, , rZ and z, used in simulations are typical values taken from observed data.

B. Boundary conditions

1. At the air-sea interface

Stresses and velocities are assumed to be continuous across the interface as is traditional in air-sea interaction modeling

ba4wXoa= -bs~:w~)lzos. (10)

Boundary conditions for velocities are

U&a) Lo, = U,(zsj IZ”*’ (11)

VakaNzon= vs(41zos’ (12)

where zna and zos are the roughness lengths of air and water at the air-sea interface, respectively.

The surface boundary conditions for the turbulent kinetic energy can be written in the traditional form for the atmospheric boundary layer over land (Monin and Yaglomr4), and for the air-sea boundary layer problem iLy6) as follows

ki(z&i= a;1’2U2*jr (13)

where u*i is the friction velocity of the air and water flows, and aZi is a nondimensional constant identical to the one in equation (3).

By dropping a buoyancy term in the energy dissipation budget equation (4) for the air-sea interface which were shown to have a small effect in the turbulent kinetic energy

budget analysis, Ly” solved the e-equation (4) analytically at the air-sea interface and obtained boundary conditions for R at the surface in the form:

%(Gz)lLOn= -J& iLq~,+(l -9dexp(-dl~~z0n 04)

for the atmosphere (i=a) where

qhz=&alP2a; q2a= (~$33~~ iL2p4, j lJ2 05) and

&.A4 Los =~3,,~~1s+~~-4~s~~~~~-cdzs~l~~~os 06)

for the ocean (i=s) where

Bls=PlsfP?s; q2s=(Ly:~p2El~2p4s)1'2. (17)

Here von Karman’s constant assumed to be i = 0.40.

2. Away from the interface

The top (bottom) boundary conditions of the atmospheric (oceanic) boundary layers are such that the velocities tend toward constant values while the momentum flux, turbulence (TKE), and dissipation (E) tend toward zero.

C. Ocean waves and similarity theory

1. Ocean waves and turbulent energy dissipation

At the ocean upper layer, ocean waves are an important source of turbulent energy besides the shear driven turbulent energy. Wave breaking is considered to play an important role in enhancing turbulence in the ocean upper layer in re- leasing the turbulent kinetic energy. In the real world, even with only small-scale wave breaking, the transformation of wave energy into turbulence (due to wave breaking) can dra- matically change the character of turbulence just below the wavy surface (Kitaigorodskii’5). The rate of turbulent kinetic energy dissipation (E) in the upper oceanic layers has great significance for surface layer mixing, oceanic turbulent structure, and energy transport across the interface. There has been great effort in recent years to acquire observational estimates of E in near surface layers (Kitaigorodskii et al. ;I6 Gargett;t7 Drennan et aL8). In addition, surface ocean waves ’ play an important role in the velocity field of the ocean surface .layer. It is known that there is a connection between energy dissipation and wave parameters. Dobroklonsky’s studied the influence of waves on turbulence by obtaining an equation for the eddy viscosity in a trochoidal wave. Benilov” used similarity theory to obtain the relationships between turbulent kinetic energy (TKEC), turbulent-energy dissipation, length scale, sea state, and depth for the case in which the breaking waves are the principal source of turbulence, as follows:

,-6WlX , (18)

where up, h, and A are the wave phase velocity, height, and length, respectively, y is a nondimensional constant, and z is the depth. According to equation (18), the energy dissipation rate must decrease rapidly with depth, which is in fairly good agreement with the estimates from observed data by Stewart

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and Grant.tg More details about various estimates of E, are given by Monin and Ozmidov.“’ Hereafter, subscript “0” shows the variables at the air-sea interface. The ocean turbulent energy dissipation of shear production at the air-sea interface is given in equations (14) and (16). In the case where waves are taken into account, the turbulent energy dissipation E at the air-sea interface has the form:

where qls and qzs are the same as in equation (17), the first term on the right side is the turbulent energy dissipation from the mean flow field, and the second term on the right side (.eO,,,) is the turbulent energy from the surface waves. The energy dissipation at the surface, .eow, from equation (18) has the form:

The equation (18) was derived for gravitational surface waves of small amplitude h/h< 1. Benilov3 indicated that since the wave spectrum is fairly narrow, equation (18) can be applied to evaluate sow, using mean values of wave height and length. Equation (18) neglects the presence of the surface wave spectrum. This apparently accounts for the large scatter obtained in estimates of y. Equation (19) shows that the energy dissipation at the air-sea interface results from shear production and from breaking waves. It is noted that in the case of no surface waves, equation (19) reduces to equation (16) and we have a situation described by Ly.” The parameter y is difficult to choose. It can be determined from measurements of the dissipation profile in the ocean.

2. Dim&sional relationships

The ocean wind waves which are discussed subsequently are assumed to be fully developed waves. Then the wave motion at the ocean surface can be characterized by wave length, X, wave height, h, and phase speed at the spectral peak, up. From the dispersion relation of deep water surface waves we have

0”=gi, w=2dT, .i=2?r/h, C21)

where w is the angular frequency, g is the acceleration of gravity, T is the wave period, and kx is the wave number.

We can write the phase speed uP taking equation (21) into account:

s ‘g 0

‘I2 gT +=-= = =-. 0 k 2?T t22j

In the case of developed surface waves there is a connection between phase speed, up and friction velocity u *a by similarity analysis

lLp=C,“U*(l, (23)

where c, is wave age which has a broad scope of values for developed waves (e.g., c,=22 by Longuet-Higgins21 and Kagan et a1.,“2 c,~-~O by Kitaigorodskii,4 and c,--33 by

Benilov et aZ.23). The other elements of the developed waves, as wave period T and wavelength A, can be expressed in terms of the friction velocity of air flow (Tarnop- olski et aLz4)

2-r T=cwy ~~a,

Inserting equations (21)-(25) into equation (2oj, we have co,,, in terms of wave height and the friction velocity of air flow, u*, , as:

h3g” EOW= Y (w*nm #.

W-3

Many observed wave phenomena can be described by linear theories. From the linearized theory we can write a simple condition for breaking as h w2=g12 and the maxi- mum wave slope as (hk”)max=(hti2/g)min= l/2 (for more details see Longuet-Higgins;‘l Lamb;“” Monin et al.““). Then we have

h hi 1 -=-=-. A 2,rr 4n- (27)

Equation (26), by taking equations (23)-(27) into account, can be written as a function of the friction velocity as

The continuity boundary condition for momentum flux at the air-sea interface (5) can be written in the following form:

(29)

Then we have 112

u*a* (30)

Studies by Kitaigorodskii,” BenilovZ3 and Benilov et aLz3 show that Charnock’s” relationship for the roughness length (zeJ does not contain a dependence on the interaction characteristics such as wave age, up lu*, . They also showed that,Chamock’s zon is two orders of magnitude smaller than those of measurements and theoretical estimates. The conse- quence from the above is-that Charnock’s constant varies over a large range. Here we use Benilov’s roughness length, zoa (at the upper bound of the wave layer), which is obtained from the analysis of observed data (Benilov;’ Benilov et a1.23)

zc,=O.ll 0.52-$ - 14.0-2.76

x 10-3 (31)

where v = 1 5X 10B3m2s-’ is the kinematic viscosity of the a * air. Benilov” indicated that the roughness length presented in


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equation (31) covers a broad scope of the wave age parameter: 56u,lu,, G90. They also indicated that for the regime that generates wind waves, the wave age parameter is in the interval between 5-33. The sea surface in this situation corresponds to the hard rough wall. The wave age parameter equal to 33-40 corresponds to fully developed ocean wind waves. The sea surface in this case appears to be close to the smooth wall. In the regime of diminishing swells the wave age parameter, ~~230-40, is in an aerodynamic relation where the sea surface appears to be smoother than the smooth hard wall.

According to Ly6 the roughness length for the lower boundary of the wave layer, zOsr can be written as Charnock’s” analog as:

( i

2 Pa U*S

zos=a - -, Ps g

(32)

where the function a(p,lp,) may be taken as an empirical constant equal to 0.01-0.05, which hereafter we will call Charnock’s constant c,. By inserting equation (29) into equation (32), the roughness length zos can be written in the form

ZOS =cc(Pa&JkPs. (33)

Substituting equations (28), (30), and (31) into (19), we have the turbulent energy dissipation at the air-sea interface expressed in terms of the friction velocity of air flow, u+.:, :

pa 1’2 1 ~s!G)IZOs=P*a j- I 1 s

ffchfU-41”) c

3

Xexpl- cdl + cfcTJj4 . w 1 (34)

D. Nondimensionalization

For convenience, the problem is solved in nondimensional form- To focus on coupling the physics of air-sea- wave interaction, we will study the coupling of Ekman planetary boundary layer flows which are generally accepted as stationary and horizontally homogeneous. Some of the nondimensional quantities are adopted as follow:

Z*i=fZil(LU*i)=ZilLi;

kni= (Y:~2kilU~i;

sin=i2EJlyU2,i);

4ni=(- l)‘+‘+i/Uii;

~ni'(-l)'+l~ilU~i; (35)

K,i=Kil(~U~iLi>;

where Li are typical height (depth) scales for Ekman’s layer in the atmosphere and ocean; r is equal to 1 for the atmospheric and 2 for oceanic characteristics.

As mentioned before, in order to focus on air-sea-wave interaction, we consider stationary, horizontally homogeneous flow of Ekman boundary layers in the atmosphere and ocean. Again, the subscripts n indicate any quantity in nondimensional form and i = a represents the physical characteristics of airflow and i=s represents those of seawater flow. By denoting the vertical lhtxes of horizontal momentum as

$i=-UiW(; 'pi= -q!w[ 06)

and substituting nondimensional variables in equation (35) into the system of equations (l)-(9), we have the following set of nondimensional equations for stationary flows: momentum:

d~ni~,i+ni+ cPnifKni=Ot dinizniYni- 4nilKni=O; (37)

turbulent kinetic energy:

+ ~4idLni(KnidZniknij = 0;

dissipation:

(38)

+ p4i(KniIknijd~,i(K,id:~i&,i) = 0; (39)

turbulent exchange coefficient:

K,i= &~ik~ilEni, (40)

where a set of constants C&p) is derived from the set of ai and pi as before. The nondimensional buoyancy terms in the TKE and E equations are

Rni=!Rna ,Rns),

where

(41)

R,,= uo+~lzna (42)

with v. =gi4( yd- y)/ f?,,f’ which is a parameter for stratification at the upper part of the atmospheric boundary layer (LaykhtmaP)

R,,=(k”4g/fpos){[(Apsmr12/~~( 1 +a2)]tan-‘a

+ r&9,

where a=m(z,-z,,)

(43)

b= - ,d’Qoa I( Qoa~acpf&J = La bn, (44)

and ,ii is a nondimensional surface heat flux that is used to specify the surface stability (Tennekes29); L, as in (35); L,,= - u~,l[~(gle,,)(Q,,lc,p,>l is Monin-Obukhov length scale; Ap, is the difference in seawater density between the upper and lower boundaries of the density-jump layer in the ocean; I? 12= rl - rz[ see (9)]; m is a parameter controlling the shape of the seawater density both in the shape of the jump layer and in the matching between the two regions of constant gradient; rr is a constant and v. can be taken as a constant (rr=3.1416 and va=O.54); Qoa is the

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surface heat flux; do, and pas are reference air temperatures and seawater densities in the atmosphere and ocean, respectively. A value of I;;= - 100 corresponds to a surface heat flux of about 500 Wm.-’ (Ly and Takle3’).

By use of equations (lo), (13), (35) and (37), the boundary conditions for nondimensional momentum fluxes and turbulent kinetic energy can be written in simpler form as

VjtriiZdl~O,rl= 1; (P,*i(Zni)lza,i=o; kni(Zni)lzoni= l. (I43

At the boundaries away from the interface, the nondimensional momentum fluxes and turbulent kinetic energy have the following form

(Pnitzd lz,,,t = (Pni(Zni) Iz,,li40;

Icni(Zni> lzcsnf-iO; (46)

Eni(Zni)lz,,i-+o*

Next, we will find nondimensional roughness lengths and nondimensional dissipation rate at the interface. From equation i35) we have

L zo,~a=..fzoa l&x, ; ZOns=fzOsliU*s. (47)

Applying equation (29) to (47), we have

z,tos= if, /PO) 1’2znOa * (48)

The nondimensional roughness length, zOan, can also be written based on equation (33) as - LOan =zo,lL, . Then nondimensional roughness lengths can always be found from equations (31), (35) and (47).

Applying equation (35) to (14) and using Charnock’s roughness zoa= cc&/g, which is a good estimate of roughness length for a shear driven flow, we have the nondimensional form of the turbulent energy dissipation, E,, , on the atmospheric side of the air-sea interface, as follows:

%akna)lz,Oa = ~C4l,,+(l-4l,)exp(-42.)1. (49) c

On the oceanic side of the interface we have equation (34) in the nondimensional form as

~d~,~~)l~,,~~= i

j+-Ih+(l -hJexphbN

+( ;j “,,;;~)4] > (50)

where constants qln,42a,~ls,q2s are defined by (15) and (17); and E,,,, is a nondimensional form of equation (28). Then equations (49) and (50) are surface boundary conditions on dissipation for the dissipation equation i39).

E. Air-sea coupling

The atmospheric and oceanic boundary layers are coupled by the boundary conditions at the air-sea interface (LYNCH) which result in the following equations:

~Uga(cosa,.SmocosLy,)Iu*,

= - id,,,cz+ Cp4,,dlz,,,~ 61)

= idz,,hca + +&,,4ns&,,~ 62)

where U~i and Vgi are the geostrophic velocity components, and CY~ are the angles between the surface wind and the geostrophic wind (i =a> and geostrophic current (i = s). The nondimensional vertical flux components of horizontal momentum in the x and y directions for both flows are &i and ~ni (see Ly6). Using equations (51) and (52), it is easy to obtain expressions for the geostrophic aerodynamical drag coefficient (C@) and angle (a,) between the vectors of the surface wind and the geostrophic wind in the following forms:

(iC,d)2=(l -2mOcosP+m~)[(dz,,~~~+Cpdr,,~ns)2

+ (d,,,~na+ Cp4,,nJ21- ‘3 (53)

tancr,=[mo(Nosit@+cosp)- I]

XINo-mo(cosp-N&r@)]-‘,

where

ilo-U,,lU,,;

(54)

/3-360°=as-~a; (55)

c, = (p, lp,) 112= gj;

No = ( d,,,&a + Cpdz,,&) (d,,,qna + CpdZ,apns) - ’ (56)

and p is the angle between the geostrophic wind and geostrophic current.

Iii. COMPUTING SCHEME

The systems of nonlinear equations for both atmospheric and oceanic boundary layers are solved by using the matrix and simple pivotal condensation methods (see Ly31). The model equations can be solved separately for the atmosphere and the ocean by using the boundary conditions for TKE (equation 45) and for E (equations 49 and 50). The grid used for the simulations consisted of 200 irregularly spaced points for each boundary layer. The domain of integration ranges over OSz,+2 for each boundary layer.

The computing scheme for the coupled air-sea boundary layer problem has the following order: the simulation process for the atmospheric boundary layer is done by solving equation (37) for the momentum fluxes with boundary conditions (45) and (46), and the Kj(zj) = Zj approximation for EVC. The TKE equation for the atmosphere is then solved with “new” fluxes +j and ‘pj, and boundary conditions in (45) and (46). The E equation (39) for the atmosphere is solved with the surface boundary condition in (49) and the “diminished” boundary condition at the top of the atmospheric boundary layer. .Then Kj (EVC) is calculated from equation (40). The iterative process for the atmosphere is terminated when Kj satisfies the following convergence con-


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dition for some prescribed E”: IIKj”‘(zj) - Ky+‘)(zjjII ~EI[[Ky’(z~)ll. This condition implies that the numerical SO- lution approaches the true solution as P+M and &z,~+O. This is the concept of convergence. The same procedure is then followed for the ocean. After obtaining the distributions of nondimensional functions, one can easily calculate dimensional functions by using equation (35).

Distribution of Dissipation in the Atmosphere y&y-- b I t ! ! -7-j

IV. NUMERICAL SIMULATIONS

The numerical experiments are designed for various geostrophic winds, LJRa, wave heights, and wave ages, C,, to study the influence of ocean wind waves on turbulent fluxes, turbulent and dynamical structures, and the resulting impacts on coupled modeling. Our study is of the wave effects on distributions of momentum fluxes, turbulent structure, and air-sea interaction characteristics.

The rate of turbulent kinetic energy dissipation (E) in the upper oceanic layer, particularly, the distribution of E in the oceanic surface layer, has great significance in problems relating to mass transfer across the air-sea interface, the mixing of surface layer water, dispersal of buoyant pollut- ants, testing of similarity theory on turbulent structure, and the modeling of oceanic mixed layers. In recent years a great effort has been expended to estimate the vertical distributions of E near the surface (see Gargett;17 Drennan et al.;s Agrawal et al 32) . .

Shay and Gregg33 directly measured the turbulent dissi-

FIG. 2. Distribution of dissipation, Diss (Diss in cm2/s3 and height Za in m), for the surface layer of the atmospheric boundary layer at various surface wave states (with wave parameter y) for wind U,,= 11 ~lts’-‘: (a) y=O (no surface wind wave condition); (b) y= 1; (c) y= 10’; (d) y= 104.

pation rate in an oceanic convective mixed,layer during a cold-air outbreak. They presented the vertical distribution of measured E [rate of dissipation of turbulent kinetic energy (TKE) per unit mass] in the oceanic boundary layers (OBL) normalized by the surface buoyancy flux Ji (see also Gargetti7). The height (depth) z is nQrmalized by boundary layer height (depth) D. The observations in the ocean were taken in a situation where large surface heat loss (coupled with moderate winds) was the dominant driving force for turbulence in most of the surface layer (see also Gargett17). Dimensionless dissipation profiles from the atmospheric mixed layer were replotted by Shay and Greggj3 based on the atmospheric experiment data to compare with dissipation distribution in the ocean. It is noted that both.datasets were collected under convective condition of the atmosphere and ocean.

and oceanic surface layers for various surface wave states for geostrophic wind lJgn= 11 ms-’ are shown in Figs. 2 and 3, respectively. In these numerical experiments, we vary y=O, 1.0, lo2 and 10” to represent various surface wave states. The thin solid curve shows the E distribution under the “no waves” condition. The thick solid curve shows the largest wave condition ( y= 10’). It is noted that, as in a paper by Shay.and Gregg,33 vertical profiles of E in the ABL and OBL are similar. It is also noted that E in a surface zone is much larger than in the interior of the OBL. These values are much larger than in the ABL. It can be seen from these figures that surface waves strongly influence .a distributions in the surface layers of both the ABL and OBL. Here, waves add more dissipation through breaking to the surface dissipation. Our model predicts E distributions with a surface

They showed that profiles of dissipation E in the atmospheric and oceanic ‘boundary layers are similar. It is also clear from their work that E in a surface zone is much larger than in the interior of the oceanic boundary layer (OBL). Shays and Gregg33 also showed that this zone is much thicker in the oceanic boundary layer than in the atmospheric boundary layer (ABL). The E in the oceanic surface layer is much larger than in the atmosphere. It is interesting that these E values are much larger than expected from similarity scalings based on observed wind stress and surface buoyancy flux (see Kitaigorodskii et al.; I6 Gregg34). It was concluded by Gregg”4 that these observations must be taken as evidence of the inadequacy of atmospheric prediction of the constant- stress boundary layer theory in the near-surface OBL.

Distribution of Dissipation in the Ocean

q7Fy-Jl’H

+5-. ..,.... . . . . . . . . . . . / .

:i

.,.‘.. , _ : /. I -4- ,.. .< : I ,- , -3.5 -

i4 3 9-

* I/. II: ..__... . I___ . . . . . . . . . . . . . . . .

-2.5- j .z .

I 7..

/ .._...... .,..

:

-1.5 - .,.. -- . 1 . . . . ..I.... i

..f..

-,I /; ; ; , i 1 - -1 0 1 2 3 4 5

Log Diss

Vertical distributions of dissipation in the atmospheric FIG. 3. Same as Fig. 2 but for the oceanic surface layer.



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Distribution of Momentum Fluxes in lhe Atmosphere

I

FIG. 4. Vertical profiles of x and y components of momentum fluxes (qY q in ,n2/s2, and Za in m) in the atmospheric boundary layer for various geostrophic winds U,, and wave ages C,=40: (a) 4 for Usd= 11 ms-‘; (b) cp for U,, = 11 ins-‘; (C) 4 for U,, = 17 ins-‘; (d) cp for U,, = 17 tnsL’.

zone of large dissipation values which were not expected from similarity scaling based on observed wind stress and surface buoyancy.

The vertical distributions of the x and y components of stress in the atmospheric surface layer for various geostrophic winds, lJan= 11 and 17 ms- ‘, and wave ages, C,,= 20 and 40, are’presented in Fig. 4 (also, see Fig. 6). Fig. 5 is for various lJga, but the same wave age C,=40. Our estimates of wind wave heights are 1.5 and 2.3 m for Us, equal to 11 and 17 MS- ‘, respectively. As expected, winds strongly affect the stress distributions. From these figures we can see that wave ages of 20 and 40 have stronger influence on stress than winds of I1 and 17 ms - ’ up to lo-20 r?z height.

0.25

Distrfbution of Momentum Fluxes in the Atmosphere

‘---~-- T

_ r---

-. ‘\

..--I

.o.iJ- r -3 2 1 . ..--.-.-I -1 ..-..............._.. 0 L -_A 1 ---A 2 LOe=

FIG. 5. Same as Fig. 4, but all cases for wind U,, = 17 tns-‘: (a) and (b) for C,,=20; and (c) and (dj for C,;-40.

Distribution of Momentum Fluxes in the Ocean I 1 --7

FIG. 6. Same as Fig. 4, but for the oceanic boundary layer.

We can see very sirn@r pictures in Figs. 6 and 7 for the oceanic surface layers. From Figs. 6 and 7 we can see a large change in stress magnitude between lo-20 m. These changes in stress magnitude in the oceanic surface layer must be related to wind and wave conditions at the air-sea interface. .This suggests that waves may affect stress distributions in the oceanic upper layer to lo-20 m depth. From these figures of stress distribution in the surface ABL and OBL we see that waves have a dominant effect on the x components in comparison with y components. This happens because our model uses the traditional assumption of zero angle between wind and wind stress at the an-sea interface (see LYLE). This traditional assumption corresponds to the assumption that the y component of friction velocity, u f W ; , is negligible with respect to the x component, ~fwl (Ly35).

Vertical profiles of eddy viscosity coefficient (EVC) for ABL and OBL conditions of Uga = 11 and 17 m, and

X104 Distribution of Momentum Fluxes in the Ocean -.~?--~--~ T~~.~--, ~. -”

.___ - ---- -------- --.____ -.

3- - 4 .- \

\ \ a

\ \ \

2-

-21-l -4 -3 ~2

L&s 0 1 2

FIG. 7. Same as Fig. 6. but (a) and (b) for C,=20, and (c) and (d) for C,” = 40.


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Distribution of EVC in the Atmosphere

: : : ‘\ :;bj .c,

5~o~~~~,~~~~~~-:;::.::!--:-;:.::: -.&‘; 1 1 2 3 4 5 0 7 8 * ,.

Ka

FIG. 8. Vertical profiles of eddy viscosity coefficient, Ku (Ka in m2/s and Za in m) for the ABL at various winds CU,,), and wave ages (C,): (a) CJT.= 17 ms-’ and C,= 20; (b) lJ,,= 11 ms ” and C,= 40; (c) U,,= 17 and C,= 40.

C,= 20 and 40, are shown in Figs. 8 and 9. The model produced “classical” profiles of EVC for both ABL and OBL (see Ly6). The EVC “classical” profiles increase their magnitudes away from the air-sea interface, obtaining maxima near 100 m height for ABL and 10 m depth for OBL. The EVC decrease their magnitudes approaching the top/bottom of ABUOBL. It is interesting to note that wind-generated waves (wave height and age) have very similar effects on EVC in the ABL/OBL as effects of atmospheric and oceanic stratification (see Ly’). It is expected that the wave effect on EVC will be greater in unstable and less in stable stratifica- tions in ABL/OBL, because the vertical mixing will be stronger in the former and weaker in the latter (see also Ly6). Here, we see the wave age (20 and 40) stronger affect on EVC profiles than wind speeds of 11 and 17 nzs - ’ which correspond to wave heights of 1.5 and 2.3 m from our esti- mation. From these figures again we can see that the wave

Distribution of EVC in tie Ocean

FJG. 9. Same as Fig. 8, but for OBL (ks in cm2/s).

FIG. 10. Dependence of surtace drift current on 10 m height wind speeds for various wave ages (C,=20, 30 and 40).

effect on EVC distribution in the oceanic upper layer can also reach to roughly lo-20 ti depth. These wave-affected depths for the ocean surface layers agree with recent measurements by Kitaigorodskii et al. I6 which showed a wave- affected region of about 10 times the wave amplitude. Thorpe36 suggested a wave-affected layer of depth approximately 0.2 of the surface wavelength. Osborn et aZ.37 used upward-looking acoustic instruments and shear probes mounted on a submarine in the Pacific Ocean and showed that their result is qualitatively consistent with the results of Kitaigorodskii et ul. I6 and Thorpe.36

Dependence of surface drift current on 10 m height winds at various wave ages of 20, 30, and 40 is shown in Fig. 10. Surface currents are larger by about 28 and 30% for wave ages of 40 in comparison with those of 20 at winds Vlo= 10 and 6 ms -I, respectively. This shows that even for small winds the surface currents are strongly affected by developed wind waves with a wave age equal to 40 (in comparison with 20). The older waves more strongly affect surface drift current than the younger waves in this case. Numerical simulations confirm that surface drift currents in the presence of wind waves of age 40 can reduce their magnitudes up to 50% in comparison with the ‘<no wave” case (not shown). Physically, this can be interpreted that, in the presence of wind waves, a part of the energy transferred to the surface water by winds goes into the generation and maintenance of wind wave, i.e., not all wind energy is transferred to drift currents. These results agree with Ly’s6 numerical study of the wave and surface drift current relationship. This relationship is still not well know and deserves to be checked against observations.

Dependence of the geostrophic aerodynamic’ drag coefficient (Cgd= ugalUgn) on geostrophic wind is shown in Fig. 11 (after Ly9). The shaded area shows the relationship by Grant and Whiteford3* using data collected by aircraft over the ocean. The solid and open circles are JASIN and KONTUR data, respectively, collected during the 1,978 JASIN experiment in the North Atlantic and the 1981

2404 Phys- Fluids, Vol. 7, No. 10, October 1995 Le Ngoc Ly


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b - 1

8. 0 Y h 0,. . I .Q I”” ” B vi2J?mmB~~pl l-/lllli r, ////!I

! tqsnd

a . JASIN *ata g ylgta IJJ -+.-, ~ f--~--~ ____ ~7~- 0 10 20 30 40

Geostraphic Wind, I& (m/s)

PIG. 11. The geostrophic aerodynamic drag coefficient, CBd , as a function of geostrophic winds (after Ly9). The shaded area shows the range of values of C*d . Solid and open circles are JASIN and KONTUR data, respectively. The model C,, is shown for the wave age C,U = 37.5 (solid line).

KONTIJR experiment in the German Bight area of the North Sea (Grant and Whiteford”). Our model output is the solid line for C,=37.5, which shows that Cgd is nearly indepen- dent of geostrophic wind speed. The model values of CRd for the wave age of 37.5 is in very good agreement with the data of JASIN and KONTUR experiments, and modeling result of Cgd ‘-v LY.’

V. SUMMARY AND REMARKS

A numerical model of air-sea-wave interaction is developed to study the influence of ocean wind waves on dynamical and turbulent structures of the atmospheric and oceanic boundary layers and their impact on coupled modeling. The model equations for both atmospheric and oceanic boundary layers include equations for: (1) momentum, (2) k--E turbulence scheme (Lya), and (3) stratification in the atmosphere and ocean. These equations are written in the same form for both the atmosphere and ocean. In this model, ocean wind waves are considered another source of turbulent energy in the upper layer of the ocean besides turbulent energy from shear production. The dissipation E at the surface of the oceanic boundary layer is written in the form of a linear combination of terms represented by dissipation from mean flow and breaking waves which is estimated by similarity theory and observed data. The dissipation from breaking waves is written in terms of wave parameters such as wave phase speed, height, and length, which are then expressed in terms of friction velocity by using similarity theory.

The numerical simulations show that vertical profiles of dissipation E in the ABL and OBL are similar. Our simulation E profiles using a coupled model for various surface wind wave states and geostrophic wind equal to 11 ms - ’ are similar to observed by Shay and Gregg.s3 Observations of E in the oceanic surface zone are much larger than those in the atmospheric surface zone and in the interior of the OBL. This kind of E distribution in the near-surface layer of the OBL has been taken as evidence for the inadequacy of the constant stress boundary layer theory.

The wave ages have stronger influences on stress (factor of 2.75) in lo-20 m oceanic surface layers than geostrophic winds of 11 and 17 ms-’ and its corresponding wave heights

(factor of 2.4). The depth of large changes in stress magnitudes in surface layers can reach to IO-20 m. This suggests that waves may affect stress distribution in the oceanic upper layer to lo-20 m depth. Surface waves have a dominant effect on the x component of stress because our model uses a traditional assumption of zero angle between wind and wind stress at the air-sea interface (Lys5).

The model produced “classical” profiles of the eddy viscosity coefficient (EVC) for both the atmospheric boundary layer (ABL) and oceanic boundary layer (OBL). Wind- generated waves (wave height and age) have strong and similar effects on EVC profiles in the ABL/OBL as effects of atmospheric and oceanic stratification. From the EVC profile in the OBL it can be seen that wave-affected depths can reach to roughly lo-20 m depth which agrees with observa- z;;;3Py Kitaigorodskii et aZ.,16 Thorpe,36 and Osborn

Surface drift currents are larger by about 28 and 30% for wave ages C, of 40 in comparison with those for 20 at 10-m height winds of 10 and 6 ms-‘, respectively. Numerical simulations confirm that surface drift currents in the presence of wind waves of age 40 can reduce their magnitudes up to 50% in comparison with those in the “no wave” case. Our simulations also show that older waves more strongly affect surface drift currents than the younger waves. The relationship between surface drift current and wind-generated waves is still not well known and needs more study. The model geostrophic aerodynamic drag coefficients are in good agreement with measurements.

ACKNOWLEDGMENTS

The support of the Waves in the Ocean Program of the Office of Naval Research (ONR) under Dr. Edwin P. Rood, and partly, by the ONR Marine Boundary Layer (MBL) Pro- gram is gratefully acknowledged. The author expresses special thanks to Dr. A. Yu. Benilov of the Stevens Institute of Technology for his discussions, and interest in an early ver- sion of the model. Special thanks go to Mr. Thach Le (Ecole Centrale, Paris, France) for his help in computer graphics. The comments and suggestions of anonymous reviewers were helpful.

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A numerical coupled model for studying air–sea–wave interaction