SEAWAY 



13 



5.75 {t/pY'", and the intercept at u = gives log Zt,." 

 Under conditions of a fluid flow along a rough station- 

 ary surface, and in particular in the case of wind over 

 land area analyzed by G. I. Taylor (lOKl), the surface 

 does not absorb the energj'. The loss of momentuni in 

 the air is accompanied by dissipation of the kinetic 

 energy in eddies, turbulence, and finally in the form of 

 heat. The organized kinetic energy of the potential uir 

 flow is in part disorganized, lost in the form of heat, and 

 thus is no longer available. An entirely different situa- 

 tion exists in very mobile sea wa\'es in which the energy 

 is transmitted from air to water. To a large extent the 

 air does work on the moving water siu'face by normal 

 pressures, .so that kinetic energy given up by the air I'e- 

 appears as the kinetic energy of the potential wave mo- 

 tion. Only a part of the kinetic energy of wind is dis- 

 sipated in friction in tiie form of air and water turbu- 

 lence. The theory of a Ixunulary layer at a mol)ile or 

 oscillating surface has not yet been developed, and so in 

 practice it becomes necessary to assume that the turbu- 

 lent boundary-layer relationships developed for fixed 

 surfaces remain valid for the mobile w;iter .surface. The 

 empirically derived coefficients, however, may not be the 

 same in the two ca.ses in view of the fundamental distinc- 

 tion of the two phenomena. Clearly realizing this distinc- 

 tion Neumann (1948, 1949a) speaks of the "effective 

 tangential force coefficient," which is compo.sed of con- 

 tributions of both the dynamic, i.e., pressure, drag and 

 the frictional drag, for which alone the turbulent-bound- 

 ary-layer expressions are truly valid. This represents 

 an assumption that the pressure drag of a moving, wavy 

 surface affects the air-velocity disti-ibution u(z) in the 

 same functional form as the frictional drag. Practical 

 application of the method appears to confirm this as- 

 sumption, but apparently no deeper in\-estigation of this 

 ([uestion and no crucial experiments were made. In 

 connection with the foregoing Xcumann (1948, 1949a) 

 emphasized that the roughness parameter Zo in eiiuatioii 

 (25) is a purely nominal quantity, characteristic of the 

 sea surface but bearing no direct relationship to the ap- 

 parent roughness of the sea. In fact, as will be .shown 

 later, the roughness parameter Zu is often shown to de- 

 crease with increasing wind and apparent sea roughness. 

 In the computation of the drag of the earth's surface 

 by G. I. Taylor (1916) the dimensions of the roughness 

 (ground undulations, trees and so on) were small as com- 

 pared to the height z used, and z was therefore obtained 

 by measurements over the ground without ambiguity. 

 In oceanography the usual measurements of the wind 

 velocity are made at a relatively small height o\'er large 

 waves. A more specific definition of the height z is 

 therefore needed. Roll (1948) expressed 2 as ^' + 

 H/2, where z' is the height above wave crests, ami // 

 the wave height. This definition is close to but not 

 identical with measuring z over the undisturbed water 

 surface. Howe-\-er, there was a kink in his plotted ciu've 

 of log z versus u at a low value of z. Neumann (1949a) 

 suggested that z = z' + H be used; i.e., the lowest level 

 of wave troughs be used as the reference level. In the 



Fig. 1 3 Variation of mean air velocity versus height 

 in vicinity of ground 



plots made by him on this basis the kinks disappeared 

 and the desired straight-line ])lot resulted. At a wind of 

 4 mps (about i;^ fps) and wave height // = 50 cm (about 

 1.6 ft) shown by Roll's (1948) measurements, Neumann 

 estimates Zw = 2 cm (0.79 in.).' 



Francis (1951) applied the method described in the 

 ([uotation from his work given before to his measurements 

 in a wind flume. At low z in proximity to the waves, 

 however, the plot of u versus log z exhibited violent 

 kinks, and only for a short range of the larger z heights 

 measured was the plot linear. Although the measure- 

 ments showed wide scatter, Francis obtained a mean 

 \'alue of 5.0 for the coefficient in equation (25), which is 

 clo.se to the theoretically expected 5.75. It was ap- 

 parently impossible to evaluate zo (and therefore r) from 

 the.se plots. 



3 Energy Balance in Waves and Energy Dissipation 



The energy of a wave system grows with distance l)y the 

 amount of the energy recei\-ed from the wind less the 

 amount dissipated l)y internal friction. An elementary 

 analysis of this process will ho given. 



Consider a stretch of sea of unit width, traversed by 

 imaginary control planes located at fetches F^ and Fo. 

 The mean rate of energy gain E per square foot o\-er the 

 distance F^ — Fi is expressed as 



clE/d.i- = 



E. - El 



F, - Fi 



(26) 



' For a recent di.scus.sion on ]jroiX'rtie.s of the boundary layer at 

 the sea surface the reader is referred to Ellison ( 1956 )( see p. 105). 



