A Derivation of the Scaling Law, the Power Law and the Scaling Relation for Fetch-Limited Wind Waves Using the Renormalization Group Theory

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1 Journal of Oceanograhy, Vol. 54,. 641 to A Derivation of the Scaling Law, the Power Law and the Scaling Relation for Fetch-Limited Wind Waves Using the Renormalization Grou Theory KOJI UENO Meteorological College, Asahi-cyo, Kashiwa, Chiba , Jaan (Received 23 December 1997; in revised form 22 May 1998; acceted 27 June 1998) A state of wind waves at a fetch is assumed to be transformed into another state of wind waves at a different fetch by the renormalization grou transformation. The scaling laws for the covariance of water surface dislacement and for the one-dimensional and twodimensional sectrum and the ower law for the growth relation are derived from the fact that the renormalization grou transformation constitutes a semigrou. The scaling relation or the relation among the exonents of the ower law is also derived, using the two assumtions that the renormalization grou transformation is alicable to fetchlimited wind waves and that the saturated range exists, which imlies that the directional distribution function of energy in the wave number region much larger than the eak wave number does not deend on wave number. Keywords: Wind waves, renormalization grou, ower law, scaling relation, directional sectrum. 1. Introduction The growth relation and the sectral density for wind waves obey the scaling laws. The dimensionless wave energy and the dimensionless wave eriod are functions of the dimensionless fetch. Furthermore observations (Mitsuyasu, 1968; Hasselmann et al., 1973) indicate that the dimensionless wave energy is roortional to the dimensionless fetch and the dimensionless wave eriod is roortional to the cube root of the dimensionless fetch. The growth relation obeys the ower law. This ower law has not been derived from the scaling law of the growth relation but has been obtained emirically. The sectrum of wind waves has a similar form and is saturated with 4-ower frequency deendence at high frequencies (Toba, 1973). These ower laws suggest that renormalization grou theory is alicable to the henomena of wind waves. The renormalization grou theory was alied for the first time to the second order hase transition of a ferromagnetic body, which occurs at the Curie temerature T C. The magnetization m below T C is a decreasing function of T and vanishes at T C. For T very close to T C, the ower law behavior m (T C T) β (1) is a common feature. As T aroaches T C, the magnetic suscetibility χ diverges. The divergence is characterized by the exonents γ and γ : χ (T T C ) γ T > T C, (2) χ (T C T) γ T < T C. (3) The secific heat C has a singularity at T C. This singularity is characterized by the exonents α and α : C (T T C ) α T > T C, (4) C (T C T) α T < T C. (5) In general, these abnormal henomena are referred to as critical henomena. The exonents α, β and γ are called critical exonents. Critical exonents are not indeendent: there is a scaling relation between these critical exonents; as for the above examle, α + 2β + γ = 2. The renormalization grou (abbreviated as RG) method roosed by Wilson has been alied to critical henomena (Ma, 1976). The scaling laws and the scaling relations are obtained using the RG theory. The RG theory enables us to calculate critical exonents either by exansion or numerically. The RG theory has been alied to a variety of fields of research such as olymers, chaos and ercolation. We aly the RG theory to the similarity between two states of wind waves at different fetches. The scaling laws, the ower laws and scaling relations for wind waves are derived. In Section 2 the method of the RG is exlained briefly. The scaling law that is associated with the auto-covariance of water surface dislacement at a oint and the frequency sectral density is derived by alying the RG theory to the auto-covariance of water surface dislacement at a oint. In Section 3 the scaling law associated with two-dimensional Coyright The Oceanograhic Society of Jaan. 641

2 sectral density is derived by alying the RG theory to the cross-covariance of the water surface dislacements at two different laces. In Section 4 an exonent of a linear transformation of the RG grou is determined and the fetch deendence of the sectral level in the frequency range higher than the eak frequency is discussed. The directional energy distribution derived from the RG theory is also comared with observational results. Section 5 is devoted to conclusions. 2. An Alication of the Renormalization Grou Theory to the Auto-Covariance Function of the Water Surface Dislacement We first survey this section briefly. We begin by defining a transformation as follows. At the first ste we take the average of each s successive data of time series of the water surface dislacements of wind waves at a oint at fetch F and divide the averages by a constant s a. In the second ste we shrink the time interval by a factor s. Then we get the new time series of water surface dislacement at the same time interval as that of the original time series. The transformation R s defined by these successive rocedures is called the RG transformation. This new time series of water surface dislacements is assumed to be the same as the time series of water surface dislacement of wind waves at fetch F/s y. This assumtion makes this RG transformation R s constitute a semigrou, which has no inverse transformation. This is because the roduct of the two transformations can be defined and these transformations satisfy the associative law. The transformation R s transforms wind waves with the wave height H and the wave eriod T at fetch F into wind waves with the wave height H/s a and the wave eriod T/s at fetch F/s y. The roduct R ss of R s and R s transforms wind waves with the wave height H at fetch F into wind waves with the wave height H/(ss ) a and the wave eriod T/(ss ) at fetch F/(ss ) y. The forms of s a and s y are derived from the fact that the transformation R s constitutes a semigrou. The scaling law and the ower law associated with wind waves are derived from the factors 1/s a and 1/s y in the transformation R s. 2.1 Renormalization grou theory We consider the time series of water surface dislacements η(t 0 ), η(t 1 ),..., and η(t ). These are random variables. We define the time series of the mean values of water surface dislacements η (t i ) according to λ s η ( t i )= 1 s s 1 η t si+n, ( 6) n=0 where λ s is a constant, t i is t si. The time interval of t is s times that of t. Let the time interval of t be equal to that of t. It is assumed that the robability distribution P(η, µ) and the arameter µ are transformed into P(η, µ ) and µ = R s µ resectively. This transformation can be exlicitly written as P( η,µ ) s 1 s 1 = P( η,µ ) sλ s δ sλ s η ( t i ) η( t si+n ) dη( t si+n ), ( 7) i n=0 n=0 where δ is the Dirac delta function. The form of Eq. (7) is derived in a similar way as follows. Suose that X 1, X 2 and P(X 1, X 2 ) are random variables and a robability density function. We can obtain the new robability density function P (X) of ax X 1 + X 2 by P ( x)= ap( x 1, ax x 1 )dx 1 = Px ( 1, x 2 )aδ ( ax ( x 1 + x 2 ))dx 1 dx 2. 8 The above rocedure is called the renormalization grou transformation. This transformation does not change the functional form of the robability density. The arameter µ is transformed to the arameter µ. In the case of fetchlimited wind waves, the arameter µ consists of fetch F and friction velocity u of wind. This is because a stationary state of fetch-limited wind waves is determined by the fetch and the friction velocity of wind. A state of fetch-limited wind waves at fetch F and under a friction velocity of u is assumed to be transformed into another state of fetchlimited wind waves at fetch F and a friction velocity u by the RG transformation. We assume that u = u and 1/F = f(s, 1/F). This means that wind waves at fetch F are similar to those at fetch F under the same wind velocity. Critical exonents for critical henomena are determined by the behavior of the RG in the vicinity of a fixed oint of the RG. A fixed oint µ* of the RG is invariant under R s, i.e., R s µ* = µ*. (9) At a fixed oint the system has erfect self-similarity. This is because the original system and the transformed system are described by the robability distribution with the same arameter µ*. We guess the fixed oint from an analogy between henomena of wind waves and ferromagnetism at a critical oint. In the case of critical henomena in ferromagnetic matter a correlation length of the sin density is infinite at a critical oint and the critical oint is a fixed oint. Magnetization, secific heat, suscetibility and so on are functions of the correlation length. The state of wind waves is a function of wavelength or wave eriod. If the emirical relation T F 1/3 is valid in the limit of F, wave eriod T and wavelength λ then tend to infinity. The oint F = K. Ueno

3 is a fixed oint of the RG in the arameter sace µ. The eriod of wind waves at fetch F is exressed by T(1/F). Period T and fetch F is transformed into T/s and 1/f(s, 1/F) by the RG R, i.e., T(1/F)/s = T( f(s, 1/F)). (10) The critical oint of wind waves where T tends to infinity is given by F = F C. In the limit of F F C both T(1/F) and T( f(s, 1/F)) tend to infinity. If there is one critical oint, 1/F C = f(s, 1/F C ). (11) This indicates that 1/F C is the fixed oint under the RG. For F = F C, T is equal to. Therefore the oint where T is the fixed oint under the RG. This also shows that the oint where F = is the fixed oint under the RG. We derive the roerties of the RG in the vicinity of a fixed oint of the RG. A set of transformations {R s, s 1} has the roerty that R s R s = R ss, (12) λ s λ s = λ ss. (13) This is because the roduct of two RG transformations is an RG transformation. The factor has the s deendence λ s = s a, where a is indeendent of s. The recirocal fetch f(s, 1/F) is exanded around the fixed oint 1/F c = 0. We have + 1/ F = f s,0 1/ F f s,1/ F 1/ F 1/ F = O ( 1/ F) 2 We ignore terms of O((1/F) 2 ) in comuting 1/F. Using 0 = f(s, 0), we obtain where 1/F = R s (1/F), (15) 1. Total energy 2. Wave eriod E F α (α = 1). (17) T F β (β = 1/3). (18) 3. Power sectral density of water surface dislacement in the saturated range of frequency much higher than the eak frequency P ω (ω) ω γ (γ = 4). (19) 4. Mean wave height of individual waves with the eriod of T H T δ (δ = 3/2). (20) We derive these ower laws and the relations among exonents α, β, γ and δ using the RG theory. This henomenological alication of the RG theory cannot give the values of the exonents. The friction velocity u is assumed to be invariant under the RG transformation. For simlicity, we exlicitly write only 1/F as the arameter of wind waves. The covariance of water surface dislacement η(t) is defined by G(t, F) <η(0)η(t)> P(η(t), 1/F), (21) where < > P denotes an average value with P. Average values calculated with P(η(t/s), s y /F) are simly related to those calculated with P(η(t),1/F). We obtain Gt, ( F)= η( 0)η t () P η(),1/ t F = λ 2 s η( 0)η t / s P ( η( t / s),s y / F) = s 2a Gt/ ( s, F / s y ). 22 ( 1/ F) f s,1/ F R s 1/ F =0. We get G(t, F) = s 2a G(t/s, F/s y ). (23) Now we have a linear transformation R s. In a similar way to the derivation of the s deendence of λ s, we have R s = s y. (16) 2.2 Scaling laws and ower laws for wind waves The following relations for wind waves are known (Mitsuyasu, 1968; Hasselmann et al., 1973; Toba, 1973; Tokuda and Toba, 1981): Setting s = F 1/y, we have the scaling law G(t, F) = F 2a/y G(t/F 1/y, 1). (24) Setting t = 0, we have G(0, F) = F 2a/y G(0, 1). (25) The covariance G(0, F) is the energy E of wind waves at A Derivation of the Scaling Low and the Power Law for Wind Waves 643

4 fetch F. From these equations, we see that the ower law E F 2a/y, (26) T F 1/y. (27) Using mean eriod T m F 1/y, we have the scaling law Setting t = 0, we get G(t, F) = T m 2a G(t/T m, 1). (28) G(0, F) = T m 2a G(0, 1) H 2 T m 2a. (29) When a is 3/2, this relation between the wave height and the wave eriod is the 3/2-ower law roosed by Toba (1972). By the definition of α and β, we get We define S(ω, F) as From (23) we have Setting s = F 1/y, we have Setting s = ω 1, we have α = 2a/y, (30) β = 1/y. (31) S( ω, F) dtg( t, F)e iωt. 32 S(ω, F) = s 2a+1 S(sω, F/s y ). (33) S(ω, F) = F (2a+1)/y S(F 1/y ω, 1). (34) S(ω, F) = ω (2a+1) S(1, Fω y ). (35) From (34) we see that the eak angular frequency ω is roortional to F 1/y. Using ω F 1/y, we have and S(ω, F) = ω (2a+1) S(ω/ω, 1), (36) S(ω, F) = ω (2a+1) S(1, (ω/ω ) y ). (37) In the saturated range of frequency much higher than the eak frequency the sectral density does not deend on fetch F, we get S(ω, F) ω (2a+1). (38) The exonent γ of the ω deendence of frequency sectral density in the saturated range of frequency much higher than the eak frequency is 2a + 1. If the saturated range does not exist and the sectral level in the frequency range much higher than the eak frequency deends on the fetch, we need an exonent ε defined by When ω is large enough, we assume Then we get from (35) P ω (ω) F ε ω γ. (39) S(1, Fω y ) (Fω y ) b. (40) P ω (ω) F b ω (2a+1)+yb. (41) Comaring (39) with (41), we have γ = 2a + 1 yb and ε = b. The mean height of individual waves with eriod T is a function of eriod T and fetch F, i.e., H = h(t, F). (42) Wind waves at fetch F are transformed into those at fetch F/ s y by the RG transformation: Setting s = T, we get h(t, F) = s a h(t/s, F/s y ). (43) H = T a h(1, F/s y ). (44) The mean wave height of waves with the eriod of T smaller than T m does not deend on the fetch. Therefore we obtain H T a. (45) Comaring (20) with (45), we obtain δ = a. We summarize the exonents as follows: α = 2a/y, β = 1/y, γ = 2a + 1, and δ = a. Since α = 1 and β = 1/3, a = 3/2 and y = 3. Eliminating a and y, we obtain the scaling relations: α = β (γ 1), (46) γ = 2δ + 1. (47) These equations hold exactly when the emirical values α = 1, β = 1/3, γ = 4 and δ = 3/2 are substituted. If the saturated range does not exist, γ = 2a + 1 yb and ε = b. Eliminating a, b and y, we obtain the other scaling relation ε = α + β(1 γ). (48) 644 K. Ueno

5 3. An Alication of the Renormalization Grou Theory to the Cross-Covariance Function of the Water Surface Dislacement The time series of the water surface dislacement η is defined on a discrete lattice with sacings X and Y; η is a function of a lattice oint (X i, Y j ) and a discrete time t k. The new variable η is defined as s a η X i s, Y j s, t k s z s 1 η X si + m X,Y sj + n Y,t k. 49 m=0,n=0 s 2 The new variable η of (49) is the average of s 2 water surface dislacements. This is the same as the RG transformation in Section 2 excet for the additional feature t t/s z, with the new exonent z, which lays a role similar to that of the exonent a. New η (X i, Y j, t k ) is described by P(η, µ ), where µ = R s µ and µ = (u, 1/F). It is assumed that the RG transformation R s is u = u and 1/F = s y /F. These rocedures mean that the shrinkage of the horizontal scale by a factor s and of time scale by a factor s z and of vertical scale by a factor s a transform wind waves with wave height H, wavelength λ and wave eriod T at fetch F into those with wave height H/s a, wavelength λ/s and wave eriod T/s z at fetch F/s y. The cross covariance of water surface dislacement is defined as G(X, Y, t, F) = <η(0, 0, 0)η(X, Y, t)> P, (50) where P is a robability density function. A transformation is defined as X X/s, (51) Y Y/s, (52) t t/s z, (53) η(x, Y, t) s a η(x/s, Y/s, t/s z ), (54) F F/s y. (55) After the transformation of G, we have G(X, Y, t, F) = s 2a G(X/s, Y/s, t/s z, F/s y ). (56) Setting s = F 1/y, we have G(X, Y, t, F) = F 2a/y G(X/F 1/y, Y/F 1/y, t/f z/y, 1). (57) Setting X = Y = t = 0, we obtain the wave energy G(0, 0, 0, F) = F 2a/y G(0, 0, 0, 1). (58) Setting X = Y = 0, we obtain the auto covariance G(0, 0, t, F) = F 2a/y G(0, 0, t/f z/y, 1). (59) From Eqs. (58) and (59), we obtain the energy and the wave eriod: E F 2a/y, (60) T F z/y. (61) We obtain a = 3z/2 and y = 3z from Eqs. (17) and (18). Setting t = 0, we have G(X, Y, 0, F) = F 2a/y G(X/F 1/y, Y/F 1/y, 0, 1). (62) This indicates the wavelength λ F 1/y. The wavelength is roortional to the square of the wave eriod. We have y = 3/2, z = 1/2 and a = 3/4. Defining the exonent β as λ F β, we obtain β = 1/y. The Fourier transform of G reads Sk ( x, k y,ω, F)= s 2a+2+ z Ssk ( x, sk y, s z ω, F / s y ). ( 63) Two-dimensional wave number sectral density S 2 (k x, k y, F) is defined by S 2 ( k x, k y, F) Sk ( x, k y,ω, F)dω. 64 Integrating (63) with resect to ω, we have S 2 ( k x, k y, F)= s 2a+2 S 2 ( sk x, sk y, F / s y ), ( 65) S 2 ( k,θ, F)= s 2a+1 S 2 ( sk,θ, F / s y ). ( 66) One-dimensional wave number sectral density S 1 (k, F) is defined by S 1 ( k, F)= S 2 ( k x, k y, F)kdθ. 67 Integrating (66) with resect to θ, we obtain Setting s = 1/k, we obtain S 1 (k, F) = s 2a+1 S 1 (sk, F/s y ). (68) A Derivation of the Scaling Low and the Power Law for Wind Waves 645

6 S 1 (k, F) = k (2a+1) S 1 (1, Fk y ). (69) Assuming that the wave number sectral density S 1 (1, Fk y ) in the saturated wave number region much higher than the eak wave number k does not deend on fetch F, we obtain The exonent γ is defined by We have S 1 (k, F) k (2a+1), k >> k. (70) S 1 (k, F) k γ. (71) γ = 2a + 1. (72) We determine the general form for the two-dimensional sectral density of wind waves. The directional distribution function defined by Dk,θ, ( F) S 2 k,θ, F S 1 k, F Setting s = 1/k, we obtain = S 2 sk,θ, F / s y S 1 sk, F / s y = Dsk,θ, ( F / s y ). 73 D(k, θ, F) = D(1, θ, Fk y ). (74) If the two-dimensional sectral density S 2 and the one-dimensional density S 1 in the saturated range of wave number much larger than the eak wave number does not deend on fetch F, the directional distribution function D(k, θ, F) at a fixed fetch does not deend on wave number when wave number k is enough large. This is because D(k, θ, F) is a function of Fk y and θ. If the saturated range does not exist, that is, the directional distribution function at a given wave number deends on the fetch, the directional distribution function at a given fetch deends on the wave number. Using k F 1/y, we have D(k, θ, F) = D(1, θ, (k/k ) y ). (75) This indicates that the directional sreading is a function of the normalized wave number k/k. We obtain two-dimensional sectral density S 2 (k, θ, F) = S 1 (k, F)D(k, θ, F) = s 2a+1 S 1 (sk, F/s y )D(sk, θ, F/s y ). (76) Setting s = 1/k, we get the scaling law for the two-dimensional sectrum S 2 (k, θ, F) = S 1 (k, F)D(k, θ, F) = k (2a+1) S 1 (1, Fk y )D(1, θ, Fk y ). (77) Substitution of a = 3/4 and y = 3/2 gives us S 2 (k, θ, F) = k 5/2 S 1 (Fk 3/2 )D(1, θ, Fk 3/2 ). (78) Using S 2 (k x, k y, F)k S 2 (k, θ, F), we have S 2 (k x, k y, θ, F) = k 7/2 S 1 (Fk 3/2 )D(1, θ, Fk 3/2 ). (79) In the saturated range Table 1. Relations among exonents α, β, γ, β, γ, ε and arameters a, b, y, z. Definition One-dimension Two-dimension a 3/2 3/4 y 3 3/2 z 1/2 α E F α 2a/y 2a/y β T F β 1/y z/y β λ F β 1/y γ P(ω) ω γ 2a + 1 (2a + z)/z γ S 1 (k) k γ 2a + 1 ε P(ω) F ε ω γ b γ P(ω) F ε ω γ 2a + 1 yb scaling relation α = β(γ 1) α = β(γ 1) α = β (γ 1) ε = α + β(1 γ) 646 K. Ueno

7 S 2 (k x, k y, F) k 7/2 D sat (θ), (80) where D sat (θ) D(θ, Fk 3/2 ) (F ). This exression is the same as roosed by Phillis (1985). Integrating (63) with resect to k x and k y, we get the frequency sectral density P ω (ω, F) = s 2a+z P ω (s z, F/s y )), (81) where P ω (ω, F) dk x dk y S(k x, k y,ω, F). Setting s = z 1/ω, we obtain P ω (ω, F) = ω (2a+z)/z P ω (1, Fω y/z )). (82) The frequency sectral density P ω (ω, F) in the saturated frequency range much higher than the eak frequency does not deend on fetch F. Therefore we have P ω (ω, F) ω (2a+z)/z, ω >> ω, (83) which is the same as was derived in Subsection 2.2. Assuming that a = 3/4 and z = 1/2, we have P ω (ω, Fω y/z ) ω 4. (84) The relations among the exonents α, β, γ, and the arameters a, y, and z for one-dimension and two-dimension are summarized in Table Discussion We determine arameter a of the one-dimensional case. According to Masuda and Kusaba (1987), the nonlinear energy transfer (N.L.) and the energy inut (E.I.) should have the magnitudes and N. L.~ ω E Eω 4 E. I.~ ω E g 2 ω u g 2 2 ( = Eω 3 ) 3 ( g 4 85) = Eω 3 u 2 g 2 86 around the eak frequency resectively. This energy inut is based on the emirical exression roosed by e.g. Plant (1982). The ratio of (N.L.) to (E.I.) is E 2 ω 6 /(u 2 g 2 ). We transformed these quantities at fetch F into those at fetch F/ s y by substituting sω and E/s 2a into ω and E. We obtain ( N. L. ) sω ~ s 33 2a N. L. ω, 87 and ( E. I. ) sω ~ s 3 2a E. I. ω, 88 ( N. L./E. I. ) sω ~ s 23 2a N. L./E. I. ω. 89 If the ratio of (N.L.) to (E.I.) does not change with fetch, we have a = 3/2. Conversely seaking, since we have already had a = 3/2 in the revious section, a = 3/2 indicates the ratio of (N.L.) to (E.I.) does not change with fetch. If we estimate the energy inut by the emirical formula obtained by Snyder et al. (1981), we have the energy inut ω u E. I.~ ω E After the same transformation we have g = Eω 2 u g. 90 ( N. L./E. I. ) sω ~ s 7 4a N. L./E. I. ω. 91 The condition that this ratio does not deend on fetch gives us a = 7/4. If a = 7/4, the sectral tail in higher frequency range has 4.5-ower deendence on frequency. We have the 7/4-ower law for wave height and wave eriod instead of the 3/2-ower law. Kahma and Calkoen (1992) roosed fetch-limited growth relations by reanalyzing the WAM database including data of JONSWAP and Lake Ontario. A set of their exonents is α = 0.9 and β = These exonents and γ = 4 do not satisfy the scaling relation α = β(γ 1). The JONSWAP sectrum has a tail whose level slightly deends on fetch. The JONSWAP sectrum has ε of 0.22 and γ of 5 defined by (39). The JONSWAP fetch relation ω F 0.33 gives the relation P ω (ω) ω 0.22/0.33 ω 5. Using β = 0.27 (ω F 0.27 ) roosed by Kahma and Calkoen (1992), we get P ω (ω) F ω 5 and have ε of To estimate ε from α, β and γ, we use the scaling relation ε = α + β (1 γ). Substituting 0.9, 0.27 and 5 into α, β and γ resectively, we get ε = The value of 0.18 agrees well with of the JONSWAP sectrum. Donelan et al. (1985) roosed a directional sectrum. Their sectral level in the angular frequency range higher than 1.6ω is roortional to ω Using the fetch relation roosed by Kahma and Calkoen (1992), we get ε = The one-dimensional sectrum of Donelan et al. has a tail with γ = 4. Substituting 0.9, 0.27 and 4 for α, β and γ resectively, we get ε = The Donelan et al. (1985) result (0.104) agrees well with 0.09 of ε obtained from the scaling. These agreements indicate that in both the cases the deendence of the level of the sectral tail on fetch is consistent with the exonent of ω. Mitsuyasu et A Derivation of the Scaling Low and the Power Law for Wind Waves 647

8 al. (1980) roosed F 2/7 and F 1/21 deendence of the sectral level of the tail with a ω 5 deendence and with a ω 4 deendence resectively. Their fetch relation has α = 1 and β = 1/3. A set of α = 1, β = 1/3 and γ = 5 gives ε a value of The other set of α = 1, β = 1/3 and γ = 4 gives ε a value of 0. These ε are aroximately equal to 2/7 = and 1/21 = The scaling relation ε = α + β (1 γ) is useful to determine the exonent ε of the fetch deendence of the sectral level of the tail in the high frequency range. Our derivation shows that the directional distribution function in the saturated region much larger than the eak wave number does not deend on wave number from the renormalization grou transformation of fetch-limited wind waves and from the existence of the saturated range. Mitsuyasu et al. (1975) measured the directional sectra of wind waves with a cloverleaf buoy in an oen sea. They found that the width of the directional distribution increases with frequency in the frequency region higher than the eak frequency. The one-dimensional sectra ranged from the sectra of laboratory wind waves with a shar eak to a Pierson-Moskowitz sectrum with a rather broad eak. These sectra did not have tails with a 4-ower deendence on frequency. Therefore, the directional distribution of the sectra with a 4-ower deendence on frequency in the saturated range is not clear. Hasselmann et al. (1980) measured the directional sectra using a itch/roll buoy. Their result on the frequency deendence of the directional sreading is similar to that of Mitsuyasu et al. (1975). Donelan et al. (1985) observed the directional sectra of wind waves using an array of wave staffs in Lake Ontario. They obtained the sectra with tails which have a 4-ower deendence on frequency in the frequency range above 1.6ω. The directional width of two-dimensional sectra in this frequency range is constant. Their observational results suort our theoretical ones. However, Banner (1990) suggested that the directional sreading increases beyond 1.6ω u to much higher frequency and tends to be a constant. This indicates that the lower boundary of the saturated frequency range for directional sreading is much higher than that for onedimensional sectrum. The directional sreading of Mitsuyasu et al. (1975) and Hasselmann et al. (1980) is a function of the wave age. By contrast, the Donelan et al. (1985) result showed no deendence on the wave age. Young et al. (1996) found that the directional sreading of fetch-limited waves in water of finite deth is a function of ω/ω and does not deend on the wave age. The results of Donelan et al. (1985) and Young et al. (1996) suort our result that the directional sreading is a function of the normalized wave number. However, our results can be alied to the directional sreading of fetch limited waves under a constant wind velocity. According to our defintion, the RG transformation does not change the wind velocity. We cannot derive anything about the relation between two sea states under different wind velocities. 5. Conclusions A state of wind waves at a fetch is assumed to be transformed into another state of wind waves at a different fetch by the renormalization grou transformation. The scaling laws for the covariance of water surface dislacement and for the one-dimensional and two-dimensional sectrum and the ower law for the growth relation are derived from the fact that the renormalization grou transformation constitutes a semigrou. The scaling relation or the relation among exonents of the ower law is also derived. The exonent of the fetch deendence of the level of the sectral tail at higher frequencies is estimated from the exonents of the growth relation and of the frequency deendence of the tail of sectrum using the scaling relation. These values agree well with the observational results. It is derived from the two assumtions: that the renormalization grou transformation is alicable to fetch-limited wind waves and that the saturated range exists that the directional distribution function of energy in the wave number region much larger than the eak wave number does not deend on wave number, although the lower boundary of the saturated frequency range for directional sreading may be much higher than that for a one-dimensional sectrum. References Banner, M. L. (1990): Equilibrium sectra of wind waves. J. Phys. Oceanogr., 20, Donelan, M. A., J. Hamilton and W. H. Hui (1985): Directional sectra of wind-generated waves. Phil. Trans. R. Soc. Lond., A315, Hasselmann, K., T. P. Barnett, E. Bouws, H. Carlson, D. E. Cartwright, K. Enke, J. A. Ewing, H. Giena, D. E. Hasselmann, P. Kruseman, A. Meerburg, P. Müller, D. J. Olbers, K. Richter, W. Sell and H. Walden (1973): Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dtsch. Hydrogr. Z. Sul., A8(12), 95. Hasselmann, D. E., M. Dunkel and J. A. Ewing (1980): Directional wave sectra observed during JONSWAP J. Phys. Oceanogr., 10, Kahma, K. K. and C. J. Calkoen (1992): Reconciling discreancies in the observed growth of wind-generated waves. J. Phys. Oceanogr., 22, Ma, S. (1976): Modern Theory of Critical Phenomena. W. A. Benjamin, New York, 561. Masuda, A. and T. Kusaba (1987): On the local equilibrium of winds and wind-waves in relation to surface drag. J. Oceanogr. Soc. Jaan, 43, Mitsuyasu, H. (1968): On the growth of the sectrum of windgenerated waves I. Re. Res. Inst. Al. Mech., Kyushu Univ., 16, Mitsuyasu, H., F. Tasai, T. Suhara, S. Mizuno, M. Ohkusu, T. Honda and K. Rikiishi (1975): Observations of the directional sectrum of ocean waves using cloverleaf buoy. J. Phys. Oceanogr., 5, Mitsuyasu, H., F. Tasai, T. Suhara, S. Mizuno, M. Ohkusu, T. Honda and K. Rikiishi (1980): Observation of the Power 648 K. Ueno

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