SECT. 5] RIPPLES 729 



Another interaction between air and water — Kelvin and Helmholtz' in- 

 stability — may be of importance at very high wind speeds. If air were inviscid 

 so that wind could flow with speed independent of height, then it can be shown 

 (Lamb, 1954, Art. 268) that waves of length A would become unstable when 

 the wind speed equals (1 +5)s-'2c(A), where s is the ratio of air to water density 

 and c{X) is the phase velocity of waves in the absence of wind forces. Since c 

 has a minimum of 23 cm sec-i for A= 1.7 cm, these waves become unstable for 

 a wind speed of 6.6 m sec~i. 



The instability is caused by the reduction of pressure over crests (and increase 

 over troughs) due to the Bernoulli effect. When the reduction of pressure is 

 sufficiently great, the normal restoring forces of a wave due to gravity and 

 surface tension are annulled and the wave crest is blown "into spindrift" 

 (Kelvin). Several authors have looked in nature for such an effect, some finding 

 and some not finding it (e.g. Munk, 1947; Lawford and Veley, 1956; Mandel- 

 baum, 1956) at 6.6 m sec~i wind speed (measured at typical anemometer 

 heights). A theoretical study by Miles (1959a) shows, however, that allowance 

 for a reasonable velocity profile in the air alters the critical velocity radically — 

 an approximate value of 20 m sec~i (when measured well above the sea) or 

 more appears to be necessary. It is barely possible that the sharp increase of 

 mean square slope found in laboratory channels at high wind speeds (Cox, 

 1958; Schooley, 1958; cf. also Fig. 1 at 23 m sec-i) is the expression of this 

 Kelvin-Helmholtz-Miles instability. 



References 



Burling, R. W., 1959. The spectrum of waves at short fetches. Dent. Hydrog. Z., 12, 19-117. 

 Cox, C. S., 1958. Measurements of slopes of high-frequency wind waves. J. Mar. Res., 



16, 199-230. 

 Cox, C. S. and W. H. Munk, 1954. Statistics of the sea surface derived from sun glitter. 



J. Mar. Res., 13, 198-227. 

 Cox, C. S. and W. H. Munk, 1955. Some problems in optical oceanography. J. Mar. Res., 



14, 63-78. 

 Crapper, G. D., 1957. An exact solution for progressive capillary waves of arbitrary 



amplitude. J. Fluid Mech., 2, 532-540. 

 Katzin, M., 1957. On the mechanisms of radar sea clutter. Proc. Inst. Rad. Eng., 45, 44-54. 

 Keulegan, G. H., 1951. Wind tides in small closed channels. J. Res. Nat. Bur. Standards, 



46, 358-381. 

 Lamb, H., 1945. Hydrodynamics (6th ed.). Dover Publications, New York, 738 pp. 

 Lawford, A. L. and V. F. C. Veley, 1956. Change in the relationship between wind and 



surface water movement at higher wind speeds. Trans. Amer. Geophys. Un., 37, 



691-693. 

 Lock, R. C, 1954. Hydrodynamic stability of the flow in the laminar boundary layer 



between parallel streams. Proc. Ca^nbridge Phil. Soc, 50, 108-124. 

 Mandelbaum, H., 1956. Evidence for a critical wind velocity for air-sea boundary processes. 



Trans. Amer. Oeophys. Un., 37, 685-690. 

 Miles, J. W., 1957. On the generation of surface waves by shear flows. J. Fluid Mech., 3, 



185-204. 

 Miles, J. W., 1959. On the generation of surface waves by shear flows. Pt. 2. J . Fluid 



Mech., 6, 568-582. 



