The irrotational wave-drift current associated with the one-dimensional eq uilibrium wave spectrum is calculated, and by comparison with observation, it is shown that in many situations this current may make an important contribution to surface-drift velocities. One of the interesting features of the irrotational current profile is a remarkable affinity to the logarithmic profile of similarity theory, both formally and quantitatively

Bye, John A.T.

English

Yale University

      P.O. BOX 208118 | NEW HAVEN CT 06520-8118 USA | PEABODY.YALE. EDU   JOURNAL OF MARINE RESEARCH The Journal of Marine Research, one of the oldest journals in American marine science, published important peer-reviewed original research on a broad array of topics in physical, biological, and chemical oceanography vital to the academic oceanographic community in the long and rich tradition of the Sears Foundation for Marine Research at Yale University.  An archive of all issues from 1937 to 2021 (Volume 1–79) are available through EliScholar,  a digital platform for scholarly publishing provided by Yale University Library at  https://elischolar.library.yale.edu/.  Requests for permission to clear rights for use of this content should be directed to the authors, their estates, or other representatives. The Journal of Marine Research has no contact information beyond the affiliations listed in the published articles. We ask that you provide attribution to the Journal of Marine Research.  Yale University provides access to these materials for educational and research purposes only. Copyright or other proprietary rights to content contained in this document may be held by individuals or entities other than, or in addition to, Yale University. You are solely responsible for determining the ownership of the copyright, and for obtaining permission for your intended use. Yale University makes no warranty that your distribution, reproduction, or other use of these materials will not infringe the rights of third parties.  This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. https://creativecommons.org/licenses/by-nc-sa/4.0/   The Wave-drift Current' John A. T. Bye Scripps Institution of Oceanography University of California, San Diego La J olla, California ABSTRACT The irrotational wave-drift current associated with the one-dimensional equilibrium wave spectrum is calculated, and by comparison with observation, it is shown that in many situations this current may make an important contribution to surface-drift velocities. One of the interesting features of the irrotational current profile is a remarkable affinity to the logarithmic profile of similarity theory, both formally and quantitatively. I. Introduction. Consider in a general manner the conditions near the surface of a wind-driven sea. First, consider a spectrum of nearly irrotational surface waves and an associated irrotational wave-drift current that may be defined as the Stokes mass transport of each component: S = a2 akr2 kz, where a is the wave amplitude, a is the wave frequency, k is the wave number, and z is the depth (measured downward from the surface) integrated over the spectrum. Second, consider a spectrum of turbulent eddies and planetary motions varying from large-scale current systems and tidal motions to scales of the order of the smallest wave motions. At a given place, the total drift current is formally the sum of the above two components. It is the purpose of this paper to evaluate the wave-drift current, and, by comparison with observations, to show that in many situations it may make an important contribution to the total drift experi enced on or near the sea surface. Because of the complexity of the problem, only a completely irrotational theory is given, but in the dis-cussion an attempt is made to consider qualitatively the possible effects on the drift profile of the diffusion of vorticity through the surface boundary. Basically, of course, the principles of the analysis of Longuet-Higgins (1953) apply, but 1. Accepted for publication and submitted to press I 5 October 1966. 2. T o terms o(a1). 95 Journal of Marine Research [25, I the difficulties of a complete treatment for a wind-driven sea are formidable because of the nature of the boundary conditions and the role of the convective as well as the conductive terms in the diffusion of vorticity. F (a-) 2. The Equilibrium Wave-drift Pro.file. The wave-drift current is calculated for the one-dimensional energy spectrum F(a) = /3g2 a-5, where g is the acceleration of gravity; f3 = 1.5x ro- 2, which is the estimate given by Longuet-Higgins (1962). This Figure r. The equilibrium energy spectrum. spectrum, in the range of frequencies be-tween the upper and lower frequency bounds, au and a L (Fig. I), approximates the conditions along the direction of the wind in a fully developed sea. The upper bound (au) is estimated to be the frequency at which surface tension and gravity combine to give a mnimum wave speed (i.e., au = 60 seC1 ) and the lower bound is the frequency of the energy maximum of the spectrum. First, by definition the effective amplitude (a) of a small range (La) of wave components in the spectrum about a mean frequency (a) is given by a2 = 2F(a)La. Therefore, by substituting eq. (2) into eq. (3), a2 = 2/3g2 a-5 La, (3) and, substituting this equation into eq. ( r) for the mass transport, using the wave equation gk = a2, S is obtained as a function of a: S = 2/3ga-2 e<- 2 u'z)/g La. The total wave-drift current is simply or, in the limit o, <J u U = 2/3ga- 2 e<-w'z)/g da. <IL (4) Bye: The Wave-drift Current Finally, integrating eq. (4), and assuming au )) a L, U = 2/3 g (~ eHa~z)/g - (2 n z/ g)'l2 P'° ) a L 2 aL(z/g)11' , where P is the normal probability function. 97 (5) Eq. (5) has several interesting features that will be discussed below and then compared with observations. (i) THE WAVE-SURFACE CURRENT. By re-expressing eq. (4) rn terms of F(a) at z = o, the surface current, au Us = 2a3;(a) da, aL is the third moment of the energy spectrum. Evaluating by means of eq. (5), 2{3g 30 Us = -- = - ems/sec . ClL ClL (6) Hence, the magnitude of Us is inversely proportional to the frequency of the maximum in the energy spectrum. (ii) THE WAVE-CURRENT PROFILE. In Fig. 2, U is plotted against z for a L = Io see1 and I see1 . The notable feature is that the central sections of both profiles have an approximate logarithmic slope as indicated by the arrows in Fig. 2. If eq. (5) is expressed in terms of the nondimensional depth, t = (ai z)/g, and is then divided by a£/2{3g, UaL _ e-•~ (2 =)1/2 1:rJ2p"' - - - - (," 2~112 • 2{3g (7) Eq. (7) is a universal profile for the nondimensional velocity, Ua£/2{3g, which is valid for all CJ£. Wave-drift current profiles for specific CJL are therefore similar and may be obtained on a semilogarithmic graph by a translation along the depth axis, In z = In t + In (g/ai), and by a change in scale along the velocity axis. This is readily observable in Fig. 2. There is indeed a close analogy between eq. (7) and the universal logarithmic profile over solid surfaces, Journal of Marine Research [25,1 (8) where k is VonKarman's constant and Zo is a roughness length; U* = (r/e)* is the frictional velocity of the fluid of density, e, and r is the boundary-layer stress. Eq. (8) represents a family of curves of the same general nature: (i) the velocity 2 fJ g/a L corresponds to the velocity U* /k, and both expressions contain a universal constant; (ii) the nondimensional parameter, ~, cor-responds to z/zo, or in other words, the length, gfaL, is analogous to the roughness length, Zo. In § 3, when the magnitude of the logarithmic slopes of these profiles is considered, this analogy is shown to have even greater significance. /0-2 ~-~~---~--~---~--~--~ a-L = I sec-1· 104 ~---'-----'---....I.....-_ _J_ ___ ...J._ _ __J 0 5 10 15 20 25 30 U ems/sec Figure 2. Calculated wave-drift profiles. Bye: The Wave-drift Current 99 3. Comparison with Observations. Observations of phenomena near the sur-face of lakes have been discussed (Bye 1965). It is assumed below that these observati ons are applicable in a full y developed sea (in the absence of non-local currents, e.g., tidal currents or large-scale circulati ons) and may be compared with the analysis. (i) THE SURFACE CURRENT. Under a wide vari ety of conditions, the surface current is given by the relati ons (Van Dorn 1953 and Bye 1965, respectively) Us ~ .033 Ua ~ 22 u., (9) where Ua is the wind speed measured at I m; U* = (-r: /e)* is the frictional velocity of water in which e is the density of water and -,; is the surface stress. Now, substituting for Ua in eq. (9) from the empirical relationship ( I )I /2 u - l amax = 3 2 g a , where amax is the frequency of the maximum in the energy spectrum, which is derived from the N eumann wave spectrum for a fully developed sea (Kins-man 1965: 395), and equating aL with amax, u _ 40 s - ( 1 oa) aL and u =~ * (1 ob) aL Eq. (10a) agrees in form with the present analysis, and the irrotational surface current, though somewhat smaller ( o. 7 5), is in approximate agreement with the observati onal estimate. (ii) THE CuRRENT PROFILE. The only observations of the drift-current profile near the surface appear to have been made in a steady wind of 7-8 m/sec that had been blowing for several hours with a fetch of 14.5 km (Bye 1965). The profile was observed to have the logarithmic equation Now, substituting for u. from the empirical eq. (10b), U = - 22 - - In - ; 1.8( I Z) aL k Zo 100 Journal of Marine Research hence the velocity difference (6.U) between two levels, Zr and z2, is given by 4.4 Zr 6. U = U2 - Ur = - In - . CJL Z2 (11) Compare 6.U in eq. (11) with the velocity difference (6.Um) measured directly from the approximately logarithmic central regions of the wave-drift profiles in Fig. 2; either may be used since the profiles are similar; "U - 5.5 I zr u. m-- n . (12) CJL Z2 Again there is order of magnitude agreement between observation and the irrotational theory, although the irrotational logarithmic slope is larger than the observed, in contrast to the estimates of the surface current. In view of the approximations involved in the comparison, however, this difference is probably not significant. Therefore, provided the above observations are accepted as applicable to our analysis, it appears that the total drift associated with a fully developed sea may be explained essentially as an irrotational wave-drift current, the profile of which shows a remarkable affinity to the logarithmic profile of similarity theory, both formally and quantitatively. The thickness (ow) of this wave-boundary layer, defined as the depth at which the wave-drift current is reduced to e- r Us, is given by 0 - 100 . w- - 2-, CJL (cf. Fig. 2) Ow is about 0.25 of the average height (Hr/ro) of o. 1 of the surface waves, which, from the Sverdrup-Munk empirical relation (Kinsman 1965: 390), may be written = .31 u;, by eq. (9) 500 by eq. (10) Hence, the wave drift occurs mainly above the trough lin e.3 3. Drift in the context of this paper is essentially Lagrangian, such as might be observed with small neutrally buoyant particles to which surface floats approximate. From an Eulerian point of view, the entire Stokes mass transport occurs rigorously in motions above the trough line. Bye: The Wave-drift Current IOI (iii) THE ROUGHNESS LENGTH (zo) OVER THE SEA. N ote also that the present theory predicts that the governing length in the air-water interface has the dimensions g/a{. This may be the reason for some of the experimental scatter in determinati ons of the roughness length, zo, over the sea. It is more appropriate to relate Zo to the frequency of the energy maximum in the wave spectrum than to the wind speed or fri cti onal velocity. Only in a full y developed sea, in which eq. (10b) is valid, is it expected that zo would have the dimensional form proposed by Elli son ( I 9 56 ), where oc is a constant. u2 Zo = oc --.! , g Eq. ( I 3) may well represent the limitin g curve fo r z under full y developed conditions. However, small er values of zo obtain in fetch-limited situations or where the durati on of the wind has not been suffici ent to obtain the equili-brium wave spectrum. 4. Discussion. This analysis will be discussed two ways. First, consider the energeti cs of the wave-boundary layer. It may appear that such a discussion is not pertinent, since an ir rotational theory has been proposed. However, if it is assumed that a turbulent stress (-r) in the boundary layer is provided by some mechanism outside the present theory (for example, wave breaking) there must be an energy transfer from the wave-drift current. In particular, if -. is ap-proximately constant with depth, the rate of work on the surface by the tan-gential stress is exactl y balanced by the energy Hux from the wave-drift cur-rent to turbulence; thus, 0 W = -rU8 = -r ~(~~)dz. C0 In other words, energy from the tangential stress is " filt ered" by the wave spectrum to become an input to turbulence at all levels in the boundary layer. The tangential energy Hux effecti vely passes through rather than into the wave spectrum. Of course, this energy Hux to turbulence is in addition to that deri ved from normal pressure flu ctuati ons through wave-breaking mechanisms. Second, consider qualitati vely the effect on the drift profile of the vorti city that undoubtedly penetrates the sea surface through processes such as those described above. Consider now the approach to a full y developed sea. As the durati on of the wind increases, longer and longer wavelengths attain their equilib rium-energy state until each component in the spectrum is finall y in equilibrium. N ow, equilibrium simply means that a component is, on an average, just maintained by the surface forces; Lamb (1932: 623-624) has shown that it may be completely irrotati onal, even when viscous effects are 1 0 2 'Journal of Marine Research [25,1 included. Hence, as the sea develops and as longer and longer wavelengths attain equili brium, it is expected that an essentiall y ir rotational boundary layer develops downward from the surface. In the limitin g case of the full y devel-oped sea, this layer is approximated by the wave-boundary layer discussed in the present paper. However, at depths greater than the thickness of the layer [ cf. § 3 (ii )], the ir rotational component of the drift profil e is almost certainly of minor importance, and the fl ow is dominated by rotati onal factors beyond the scope of the present analysis. It is concluded, therefore, that (i) in a full y developed sea, the ir rotational drift current may be the dominant component of the total drift current, and (i i) under developing or fetch-limited conditions, the importance of the ir-rotational drift is probably intimately related to the conditions themselves and to their rate of change. Simultaneous observations on the growth of the wave spectrum and the drift profi le under a variety of conditions are required to obtain the true picture. REFERENCES BYE,]. A. T. 1965. Wind-driven circulation in unstratifi ed lakes. L imnol. Oceanogr. I O: 451- 458. ELLISON, T. H . 1956. Atmospheric turbulence, 111 surveys in mechanics; pp. 400-430. Cambridge Uni-versit y Press, London. 47 5 pp. KINSMAN, BLAIR 1965. Wind waves. Prentice-Hall, New Jersey. 676 pp. L AM B, H ORACE 1932. Hydrodynamics. Dover, New York . 748 pp. LONGUET-HIGGINS, M. S. 1953. Mass transport in water waves. Philos. Trans., (A) 245: 535- 581. 1962. The directi onal spectrum of ocean waves, and processes of wave generation. Proc. roy. Soc. Lond., 265 A : 280-315. V AN D ORN, w. G. 1953. Wind stress on an artificial pond. J. Mar. Res., r 2: 249- 276. 

The wave-drift current

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The wave-drift current

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