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ABSTRACT 



Experiments by Faller (1963) revealed the existence and nature of a shear- 

 ing instability in laminar Ekman boundary flov. Barcilon (196^) formulated 

 the appropriate linear stability equations and obtained partial and approximate 

 solutions, based on the large Reynolds number methods appropriate to other 

 related problems. Numerical solutions of the perturbation differential equa- 

 tions have been obtained for a large number of variations of the three variable 

 parameters, a , the dimensionless wave number, e , the angle of the instability 

 bands relative to the geostrophic flow, and R, the Reynolds number. Solutions 

 are presented for the eigenvalues, the real and imaginary velocity components, 

 and for the eigenvectors, the actual modes of the normal, parallel, and vertical 

 velocity components . 



Two essentially different unstable modes are found, only one of which cor- 

 responds to Faller 's measured results. This solution, which we call the normal 

 mode, derives its energy from the mean flow component normal to the bands. It 

 is characterized by a band orientation lying between the geostrophic and surface 

 flow directions, i.e. e is positive and less than ^5°' The critical Reynolds 

 number is about 110, compared to Faller' s measured value of 125, and the critical 

 values of a , e and the small real wave velocity are in good agreement with 

 the experiments. This instability is adequately explained (Faller, I963, also 

 see Stuart, 1955, for a related problem) by inviscid theory, and is associated 

 with the principal point of inflection in the normal mean velocity profile . 



The other instability mode obtains its energy from the mean flow shear com- 

 ponent parallel to the instability bands, and we therefore denote it as the 

 parallel mode. At the critical point the bands have an angle e < , that 

 is a direction outward from the geostrophic flow, and a wave length about 

 double that of the normal mode bands. The critical Reynolds number is about 

 60, thus in the experimental realizations the parallel" mode should be present 

 before and together with the normal mode . Faller apparently observed this 

 instability qualitatively but erratically. His dye visualization method was 

 probably most responsive to nearly stationary bands, while the parallel mode 

 bands have a large inward real wave velocity. 



Further analysis of the dynamics of the parallel mode reveal similarities 

 and differences with other types of hydrodynamic instability. Rotation is of 

 critical importance, as can be determined by comparison of numerical solutions 

 or the perturbation equations with and without the Goriolis terms. Analytic 

 solutions can be obtained from a simplified version of the perturbation equa- 

 tions, which compare fairly closely with the complete numerical solutions. 

 A similarity can be seen to exist with baroclinic instability for cases of 

 short wave length and neutral static stability. 



Comparisons between the instabilities present in laminar boundary flow 

 with the finite amplitude disturbances existing in a real turbulent boundary 

 layer are at best qualitative . It may be postulated, however, that either or 

 both of the modes of energy transfer described here could form the principal 

 energy- containing eddies of the atmospheric planetary boundary layer. 



