With these assumptions, the equations of motion 

 are reduced to: 



pu u 



9u „ 9 



— =fv- — 

 9t ax 



— = - tu pu V ■ 



at ax 



ay 

 a 



p\i V 



pv V 



a 



9z 



a 

 az 



pu w 



(1.1) 



pv w 



(1.2) 



Equations 1.1 and 1.2 are then integrated from 

 the sea surface to the mixed layer depth D, in 

 accordance with the highly turbulent nature of 

 the mixed layer. 



The question of how much of a density gradient 

 is enough to put a boundary on the mixed layer is 

 a valid one. Because of the two dimensional 

 nature of this model, and the inclusion of the 

 changing thermocline structure, I hold that any 

 gradient is enough to satisfy the definition. A 

 large gradient will put a strong bound on the 

 layer because of stratification. A small gradient 

 will allow more of the wind stress to act on the 

 fluid below the layer definition, and this action 

 will be seen as a deepening of the thermocline. 

 For the case of initially small stratification, 

 deepening will occur until an equilibrium condi- 

 tion is met, as we shall see in the later sections of 

 this report. 



The mixed layer is conceptualized as sliding 

 over the stable fluid below, since the density gra- 

 dient at its lower interface provides for little tur- 

 bulence at the base of the mixed layer. Thus the 

 mixed layer acts as a slab of water moving in 

 body. (Pollard and Millard, 1970; Denman and 

 Miyake, 1973.) 



The reason for this integration, and the subse- 

 quent abandonment of the Ekman (1905) devel- 

 opment is for several reasons. First, I differ with 

 Ekman's assumption of a vertically homogeneous 

 fluid (constant A^), since the mixed layer is 

 bounded below by the thermocline, which places 

 a limit on the downward penetration of wind 

 energy. 



Ekman dynamics could be considered for use 

 in a development such as this if certain conditions 

 were met, as a laminar flow situation, and little 

 turbulent mixing (low A^). Laminar flow is an 

 implicit assumption in Ekman's development, 

 since the only communication between "layers" 

 in his equations is their boundary stresses. 

 Laminar flow is a necessary condition for low 

 turbulent mixing. I hold that the only way for 

 Ekman dynamics to be used in this problem 



would be if the depth of frictional resistance was 

 to be less than the mixed layer depth in a low 

 turbulence situation, since the thermocline pre- 

 vents further downward penetration of wind 

 energy (Pollard, 1970). At 45°N, the necessary A 

 for a depth of frictional resistance of 45 meters, a 

 reasonable estimate of most layer thicknesses, is 

 1.1 X 10^ gm cm"^sec"\ which is one order of 

 magnitude less than the usually accepted range 

 for A^of lOHo lOl 



All of these preceding conditions listed as 

 necessary for Ekman dynamics are violated by 

 the turbulence in the mixed layer. The flow is not 

 laminar, as the wind energy increases turbulence 

 in the mixed layer. This wind induced turbulence 

 increases the A^ which further deepens the theo- 

 retical mixed layer required for Ekman dynam- 

 ics to be used. For a wind increase from 4 m sec' 

 to 18 m sec', the corresponding rise in A is from 

 58 gm cm' sec"' to 2520 gm cm' sec"' (Neumann 

 and Pierson, 1966). Therefore, at a wind speed of 

 18 m sec"', the resulting depth of frictional resist- 

 ance, and the required depth of the mixed layer 

 for Ekman dynamics to exist, would be 219.66 

 meters. This is much deeper than the vast major- 

 ity of observed mixed layer thicknesses. 



The Reynolds stresses in the vertical are set 

 equal to the overall vertical stress. 



After integrating from the sea surface to D, 

 assuming p=l, and dividing by D after the 

 integration, equations 1.1 and 1.2 become 



an 





Tox 



Tdx 



a 





= fv- 



. 





. 



at 





D 



D 



ax 



av 





7oy 



7"Dy 



a 







= -fu - 







. 







at 





D 



D 



ax 



ay 



ay 



U V 

 (1.3) 



v'v' 



(1.4) 



where rox is the stress at the sea surface in the x 

 direction, toy the surface stress in the y direction, 

 and Tdx, tdy is the stress on the bottom of the 

 mixed layer over the stable fluid below. By the 

 elimination of the internal vertical Reynolds 

 stresses I hold that the only vertical stresses that 

 can occur in the mixed layer are those caused by 

 boundary processes. 



Since predictions of the horizontal Reynolds 

 stresses in the mixed layer cannot be made, and 

 prediction of the amount of momentum transfer 

 through the thermocline is extremely difficult, 

 the equations will have to be altered if they are to 

 be used in any but a "frictionless" form. 



