RBULENCE 



though some of our new knowledge 

 implies clear-cut limitations in this 

 respect. 



The Origin of WIT — Classical 

 theory concerns the rapid growth of 

 perturbations on an internal front 

 (inversion) in a fluid, called Kelvin- 

 Helmholtz instability, which leads to 

 large-amplitude Kelvin-Helmholtz 

 (K-H) waves. The rolling-up of these 

 waves under the action of wind shear, 

 and their subsequent breaking, like 

 ocean waves breaking on the shore, 

 produces turbulence. 



The process may be described sim- 

 ply, as follows: Suppose that we have 

 two fluids of different density and 

 that we arrange them in a stable 

 stratification with the lighter one on 

 top. Then we set the fluids in motion, 

 with one of the two moving faster 

 than the other, or in the direction 

 opposite to the other. If the density 

 change across the interface is strong 

 enough and the shear is not too great, 

 smaller perturbations will be damped 

 out and the interface will come back 

 to rest. But when the shear is strong 

 relative to the density gradient, the 

 situation is unstable and the pertur- 

 bations will grow rapidly with time; 

 vortices are created, as though a tum- 

 bleweed were being rolled between 

 two streams of air. 



The condition leading to unstable 

 K-H waves and turbulence is that the 

 ratio of buoyancy forces (working to 

 damp vertical perturbations) to shear- 

 ing forces (working to enhance them) 

 should be less than 1. One-fourth of 

 this ratio is the gradient Richardson 

 number, Ri, which is defined as 



Ri 



'ft 



m 



a) 



where g is the acceleration of gravity, 

 is potential temperature, dO/dz is 

 the vertical gradient of 6 (positive 

 whenever the atmosphere is more 

 stable than in the neutrally buoyant 

 or adiabatic case), V is the horizontal 

 wind velocity, and 3V/3z is the wind 



shear. A result obtained in 1931 that 

 the critical Ri leading to K-H insta- 

 bility is 1/4 has been confirmed. 

 More precisely, Ri > 1/4 is sufficient 

 for stability, and Ri ^ 1/4 is neces- 

 sary, but not sufficient, for instability. 



The entire process has been dem- 

 onstrated by Thorpe in laboratory 

 fluid experiments and by Woods in 

 thin, hydrostatically stable sheets in 

 the summer thermocline of the Medi- 

 terranean Sea. Both of these experi- 

 ments show the development of beau- 

 tifully formed billows, or K-H waves 

 which roll up into vortices and finally 

 break. And both demonstrate the gen- 

 eral validity of the critical Ri sC 1/4. 



Evidence from tlie AtmospJiere — 

 Ludlam has observed the existence of 

 the K-H instability mechanism in the 

 atmosphere by the presence of billow 

 clouds, but only rarely are the com- 

 bination of cloud and stability con- 

 ditions just right to produce the 

 lovely roll vortices in the clouds that 

 are seen in the laboratory and the 

 sea. The observation of their com- 

 mon presence in the atmosphere has 

 awaited the use of ultrasensitive ra- 

 dars capable of detecting the weak 

 perturbations in refractive index (due 

 to temperature or humidity perturba- 

 tions) which mark sharp inversions. 

 Using three powerful radars at Wal- 

 lops Island, Virginia, Atlas and his 

 colleagues first reported the radar de- 

 tection of clear air turbulence at the 

 tropopause; Hicks, Angell, Hardy, 

 and others have reported K-H waves 

 and turbulence in clear air layers 

 marked by static stability, large wind 

 shear, and small Richardson number. 



Undoubtedly the most striking evi- 

 dence of the K-H process as a cause 

 of WIT, and of its common occur- 

 rence at interval fronts, are the ob- 

 servations made possible by the use of 

 a unique new ultrasensitive FM-CW 

 (Frequency Modulated Continuous 

 Wave) microwave radar at the Naval 

 Electronics Laboratory Center, San 

 Diego. This radar is capable of one- 



meter vertical resolution, roiij; 

 hundredfold increase over that pre- 

 viously available with radars of com- 

 parable sensitivity. With this new 

 tool, it has been reported that K-H 

 waves are a virtually ubiquitous fea- 

 ture of the marine inversion over San 

 Diego at altitudes up to about one 

 kilometer. Indeed, the atmospheric 

 K-H waves observed in this manner 

 are commonly as beautiful in form as 

 those produced in the laboratory and 

 observed in the sea. (See Figure IV- 

 10) It is worth noting that the unex- 

 pectedly classical form of the waves, 

 and their great frequency of occur- 

 rence within the marine inversion, 

 recommends the southwest coast of 

 the United States as an atmospheric 

 laboratory for studies of WIT. 



What the Data Show — The fact 

 that the observed K-H waves are fre- 

 quently restricted to exceedingly thin 

 layers, sometimes only a few meters 

 in depth, and rarely with amplitudes 

 as large as 100 meters, explains why 

 the previously available high-sensi- 

 tivity radars of poor resolution could 

 not identify them. In other words, 

 the K-H wave structure was simply 

 too small to be seen and the echoes 

 appeared merely as thin, smooth lay- 

 ers marking the base of the inversion. 



The new data also indicate that, 

 though K-H wave activity may be in 

 progress, the associated turbulence 

 will not be intense unless the waves 

 grow to large amplitude prior to 

 breaking. This has been demon- 

 strated by the erratic perturbations of 

 the height of the radar-detected layer, 

 indicative of moderate turbulence, 

 which resulted from the breaking of 

 K-H waves of 75-meter amplitude. 

 In general, waves of significantly 

 smaller amplitude appear not to pro- 

 duce appreciable turbulence. 



Work now in progress shows that 

 the turbulent kinetic energy following 

 the breaking of the roll vortex of a 

 K-H wave is directly proportional to 

 the kinetic energy of the vortex im- 



109 



